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Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 www.elsevier.com/locate/pnmrs Spin –spin coupling tensors as determined by experiment and computational chemistry Juha Vaaraa,1, Jukka Jokisaarib,*, Roderick E. Wasylishenc,2, David L. Brycec,3 a b Department of Chemistry, P.O. Box 55 (A. I. Virtasen aukio 1), FIN-00014 University of Helsinki, Helsinki, Finland NMR Research Group, Department of Physical Sciences, P.O. Box 3000, FIN-90014 University of Oulu, Oulu, Finland c Department of Chemistry, University of Alberta, Edmonton, Alberta, Canada T6G 2G2 Accepted 2 September 2002 Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Scope of the review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. NMR spin Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Symmetry aspects and tensorial properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Nonrelativistic theory of the spin– spin coupling tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. High field approximation in NMR spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. NMR in isotropic media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Liquid crystal NMR spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Liquid crystal solvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3. J tensor contribution to Dexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4. Vibration and deformation effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5. Limitations in the quantitative determination of J tensors. . . . . . . . . . . . . . . . . . . . . . . . 2.3.6. Qualitative determination of J aniso . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.7. Results derived from LCNMR experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Solid-State NMR spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Solid-State NMR determination of J tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3. Results from single crystal studies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4. Results from studies of stationary powders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5. Results from spinning powder samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 234 234 237 237 240 240 241 242 242 242 244 246 247 248 250 254 254 255 256 258 260 * Corresponding author. Tel.: þ 358-8-553-1308; fax: þ358-8-553-1287. E-mail addresses: [email protected] (J. Jokisaari), [email protected] (J. Vaara), [email protected] (R.E. Wasylishen), [email protected] (D.L. Bryce). 1 Tel.: þ358-9-191-50181; fax: þ358-9-191-50169. 2 Tel.: þ1-780-492-4336; fax: þ 1-780-492-8231. 3 Also at: Department of Chemistry, Dalhousie University, Halifax, Nova Scotia, Canada, B3H 4J3. 0079-6565/02/$ - see front matter q 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 7 9 - 6 5 6 5 ( 0 2 ) 0 0 0 5 0 - X 234 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 2.5. High-resolution molecular beam spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. NMR relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Quantum chemical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Correlated ab initio methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Density-functional theory methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Basis set requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Effects of nuclear motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Relativistic effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. Solvation and intermolecular forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Couplings for large systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8. Quantum chemical results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1. Symmetric components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2. Antisymmetric components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction 1.1. Scope of the review Our aim is to examine the recent experimental and theoretical research involving the nuclear spin –spin coupling tensor, the indirect coupling mediated by the electronic structure, JMN ; between general magnetic nuclei M and N in closed-shell molecules. The principal experimental techniques in the field are NMR spectroscopy of molecules dissolved in liquid crystalline media (liquid crystal NMR, LCNMR) or solid samples either as powders or single crystals. Interpretation of hyperfine data taken from molecular beam experiments is also discussed in this context. Quantum chemical electronic structure calculations provide a theoretical means to study this property. We focus on the developments since the previous review on LCNMR and computational methods, which was written in 1982 [1]. The solid state NMR literature prior to 1990 has, in turn, been reviewed in Ref. [2]. We have omitted many references to classic papers as they were given in Refs. [1,2]. Of the new material, we include only references reporting properties of the spin – spin coupling tensor as opposed to solely the isotropic coupling constants, i.e. 13 of the trace of J. Concerning quantum chemical data, only results of non-empirical work carried out either by ab initio or density-functional theory will be included. The list of relevant, yet omitted semiempirical papers 267 271 272 272 274 276 277 279 281 282 283 283 295 296 298 includes Refs. [3 – 9]. Despite the fact that corrections for relativistic effects, rovibrational motion, and environmental (solvent) effects have not been extensively applied to the tensorial properties of J, we devote some space to these issues as they are likely to be subjects of increased interest in the near future. We have tried to be comprehensive but it is inevitable that some important papers have been overlooked. We apologize for these oversights. Our review covers literature published prior to autumn 2001. 1.2. NMR spin Hamiltonian The NMR spin Hamiltonian for spin- 12 nuclei is written in its general form (in frequency units) as 1 X g I ·ð1 2 sM Þ ·B0 HNMR ¼ 2 2p M M M X þ IM ·ðD0MN þ JMN Þ·IN : ð1Þ M,N HNMR is a phenomenological, effective energy expression designed to reproduce the transition energies between the Zeeman states of nuclear magnetic dipole moments mM ¼ gM "IM ð2Þ placed in the external magnetic field B0 : Here "IM is the spin angular momentum of nucleus M and J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 IPPP Nomenclature B3LYP CAS(SCF) CC CCSD CCSDPPA CCSDT CCSD(T) CDFT CI CISD CLOPPA CP CPMAS DFT DHF DNA DSO DZP EFG EOM EOM-CC FC FCI FOPPA FPT FWHH GGA GIAO HFA three-parameter Becke – Lee – Yang– Parr complete active space (self-consistent field) coupled cluster coupled cluster singles and doubles coupled cluster singles and doubles polarization propagator approximation coupled cluster singles, doubles and triples coupled cluster singles, doubles and non-iterative triples current density-functional theory configuration interaction configuration interaction singles and doubles contributions from localized orbitals within the polarization propagator approximation cross polarization cross polarization magic-angle spinning density-functional theory Dirac – Hartree – Fock deoxyribonucleic acid diamagnetic nuclear spin-electron orbit double-zeta plus polarization electric field gradient equations-of-motion equation-of-motion coupled cluster Fermi contact full configuration interaction first-order polarization propagator approximation finite perturbation theory full width at half height generalized gradient approximation gauge-including atomic orbital high-field approximation LC LCNMR LDA LR MAS MBER MBMR MCLR MCSCF MO MP2 MQMAS NMR NOE NQR PAS PES PPA PSO QCISD(T) RAS(SCF) REX RHF RPA SCF SD SOPPA SOPPA(CCSD) 235 inner projections of the polarization propagator liquid crystal liquid crystal nuclear magnetic resonance local density approximation linear response magic-angle spinning molecular beam electric resonance molecular beam magnetic resonance multiconfiguration self-consistent field linear response multiconfiguration self-consistent field molecular orbital second-order Møller – Plesset perturbation theory multiple quantum magic-angle spinning nuclear magnetic resonance nuclear Overhauser enhancement nuclear quadrupole resonance principal axis system potential energy surface polarization propagator approximation paramagnetic nuclear spin-electron orbit quadratic configuration interaction singles, doubles and noniterative triples restricted active space (self-consistent field) relativistic extended Hückel restricted Hartree –Fock random phase approximation self-consistent field spin dipole second-order polarization propagator approximation second-order polarization propagator approximation with coupled cluster singles and doubles amplitudes 236 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 SOS SOS-CI SOS-DFPT SS TEPS sum-over-states sum-over-states configuration interaction sum-over-states density-functional perturbation theory solid state tetraethyldiphosphine disulfide the magnetogyric ratio, gM ; is a nuclear property. In addition to the interaction with B0 ; the Hamiltonian of Eq. (1) contains coupling terms describing the interaction of mM with the fields arising from the dipole moments of other magnetic nuclei N of the system. Conceptually HNMR is obtained by averaging the full molecular Hamiltonian H over all its degrees of freedom apart from B0 and the set of nuclear spins {IM } as HNMR ðB0 ; {IM }Þ ¼ kHðB0 ; E ; {ri }; {si }; {RM }; {IM }ÞlE;{ri };{si };{RM } ; ð3Þ where the effects of external electric fields E , the positions of the nuclei {RM } and electrons {ri }; as well as the spins of the latter {si } are absorbed in the parameters sM ; JMN ; and D0MN of HNMR : The functional form of HNMR can be seen from an expansion of the energy appropriate to H in terms of the small perturbations caused by B0 and the mM ; around B0 ¼ mM ¼ 0; EðB0 ; {mM }Þ ¼ E0 þ EB0 ·B0 þ X EmM ·mM þ M þ X M mM ·EmM ;B0 ·B0 þ X 1 B ·E ·B 2 0 B0 ;B0 0 TLC TMPS TZP VAS ZORA thermotropic liquid crystal tetramethyldiphosphine disulfide triple-zeta plus polarization variable angle spinning zeroth-order regular approximation the hyperfine coupling tensor of the nucleus M. Both properties vanish for a closed-shell system. EB0 ;B0 is related to magnetizability (susceptibility). Higher order dependencies on B0 ; appearing as a magnetic field dependence in the parameters of HNMR have been speculated upon [10,11] and even found [12 – 14]. The influence of terms in HNMR higher than quadratic in I has not been experimentally observed, although the forms in which the terms would appear have been investigated [15]. The diagonal ðM ¼ NÞ occurrences in the coupling term correspond to either the (true) nuclear quadrupole coupling between the nuclear electric quadrupole moment and the electric field gradient tensor at the nuclear site [16,17], or pseudoquadrupole coupling where magnetic hyperfine operators produce, to second order, energy terms bilinear in IM [18,19]. We will not consider these properties here. Comparing Eqs. (1) and (4), the parameters sM and JMN are obtained by searching for energy terms with particular functional dependencies on B0 and IM : The terms bilinear in the two correspond to the nuclear shielding tensor, the Cartesian et component of which is ›2 EðmM ; B0 Þ sM;et ¼ det þ : ð5Þ ›mM;e ›B0;t m ¼0;B ¼0 M mM ·EmM ;mN ·mN þ · · ·; M,N sM corresponds to the modification, caused by the presence of the electron cloud, of the Zeeman interaction of bare nuclei with B0 ; ð4Þ where the nomenclature Ea ¼ ›E=›ala¼0 ; etc. is used. There are thus, in principle, terms linear, quadratic, cubic, etc. in mM (IM ; by Eq. (2)) in the expansion. The properties EB0 and EmM are related to the permanent magnetic moment of the molecule and 0 HZ ¼ 2 1X 1 X mM ·B0 ¼ 2 g I ·B ; h M 2p M M M 0 ð6Þ expressed in frequency units. The det in Eq. (5) takes HZ into account and makes the definition of shielding consistent with HNMR of Eq. (1). J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 237 The terms bilinear in IM and IN correspond to the spin – spin coupling tensor, The general symmetric and antisymmetric Cartesian components of J are JMN;et ¼ 2D0MN;et S JMN; et ¼ þ " ›2 EðmM ; mN Þ gM gN : 2p ›mM;e ›mN;t mM ¼0;mN ¼0 ð7Þ D0MN ¼ 2 " m0 3R R 2 1R2MN gM gN MN MN : 2p 4p R5MN ð8Þ The direct dipolar coupling tensor D0MN contains information about the internuclear vectors RMN ¼ RM 2 RN ; which makes it an important tool in investigating molecular structure by NMR spectra obtained in anisotropic media, as well as molecular beam experiments, as will be discussed below. D0MN is traceless and symmetric, in particular axially symmetric with respect to the direction of RMN in the absence of motion (for asymmetry induced by motion, see Refs. [20,21]). The electronic, indirect coupling tensor is often discussed using the related reduced coupling tensor KMN;et ¼ 2p 1 J " gM gN MN;et ð9Þ to remove the parametric dependence on the magnetogyric ratios. This enables studies of trends in indirect spin – spin coupling between various elements and/or isotopes without the need to take into account the nuclear factors. 1.3. Symmetry aspects and tensorial properties ðJMN;et þ JMN;te Þ 2 JMN det ð12Þ 1 2 ðJMN;et 2 JMN;te Þ; ð13Þ and A JMN; et ¼ In analogy with the nuclear shielding vs. the nuclear Zeeman interaction, J constitutes a (usually but not always) small electronic perturbation to the direct through-space magnetic P dipole – dipole interaction of bare nuclei, HDD ¼ M,N IM ·D0MN ·IN ; where 1 2 corresponding to the rank-2 and -1 contributions, respectively. Whereas the nuclear site symmetry in a given molecular system determines which components of sM are non-vanishing, the local symmetry about the internuclear vector determines the situation for JMN : The number of independent components in J for a number of point group symmetries was reported in Ref. [22]. Ref. [3] revisited the problem concerning coupled nuclei that are exchanged through a local symmetry operation. The paper contains an explicit listing of independent components of both JS and JA in most important point group symmetries (see also [23]). In the general case, JMN and JNM differ only in their antisymmetric components [3]: S S JMN; et ¼ JNM;et ; A A JMN; et ¼ 2JNM;et : ð14Þ In particular, JA has a non-vanishing component only if it generates the totally symmetric representation of the local point group. To first order JA does not affect NMR spectra; however in strongly coupled systems perturbations have been predicted [23 –25], but not observed experimentally so far. In principle, JA contributes to the relaxation rates T1 and T2 ; as discussed in Section 2.6. Other mechanisms are typically much more efficient, however. Examples where JS and/or JA influence T1 or T2 have not been reported [26]. For a recent ab initio calculation of JA ; see Ref. [27]. In general, J is described by a 3 £ 3 matrix, expressable as a sum of zeroth-, first-, and secondrank tensors, 1.4. Nonrelativistic theory of the spin – spin coupling tensor JMN ¼ JMN 1 þ JAMN þ JSMN : We limit ourselves to the case of molecules with a closed-shell singlet electronic ground state. Ramsey’s paper on the non-relativistic theory of J appeared in 1953 [28]. It, among his other classic works on molecular magnetic properties, was recently treated in a perspective article [29]. Here ð10Þ The rank-0 contribution, J1, corresponds to the isotropic spin –spin coupling constant, JMN ¼ 1 3 TrJMN ¼ 1 3 ðJMN;xx þ JMN;yy þ JMN;zz Þ: ð11Þ 238 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 we choose to restate the theory in modern response theory [30] notation that is both compact and lends itself naturally for discussions of many of the practical methods of approximate electronic structure calculation of J. The latter are thoroughly reviewed in Ref. [31]. The standard basis for the non-relativistic treatment of molecular electromagnetic properties is provided by the Breit –Pauli Hamiltonian, HBP [32,33], correct to order a2 in the fine structure constant a ¼ e2 =ð4pe 0 "cÞ: The assumptions underlying HBP break down for systems with heavy nuclei, in which case a genuinely relativistic theory must be applied. Comments on recent research in relativistic calculations of J will be included in Section 3.5. Ramsey’s theory is obtained from HBP by looking for the energy terms of the required hIM ·JMN ·IN form and including all the contributions up to the order a4 : This involves three Oða2 Þ quantum mechanical operators that contribute through second-order expressions, and one Oða4 Þ operator that gives a first-order (expectation value) term. We list the operators below, using the atomic unit system for simplicity.4 It is useful to divide the operators into singlet and triplet operators depending on whether or not, respectively, they include a dependence on the electron spin si : The singlet operators are the diamagnetic and paramagnetic nuclear spin-electron orbit operators, X 1ðriM ·riN Þ 2 riN riM 1 ð2Þ HDSOðMNÞ ¼ a4 gM gN IM · ·IN 3 3 2 riM riN i X liM ; 3 i riM IM £ riM ; 3 riM ð17Þ corresponding to the magnetic field from the nuclear magnetic dipole moment of nucleus M, in the point dipole approximation. The triplet operators relevant in the present context arise from the electronic spin Zeeman interaction with the magnetic field from the point dipole nucleus [34]. They are the Fermi contact and spin – dipole interactions, X 4p 2 ð1Þ HFCðMÞ ¼ a ge gM dðriM Þsi ·IM ð18Þ 3 i and ð1Þ HSDðMÞ ¼ 2 X 3riM riM 2 1riM 1 2 a ge gM si · ·IM ; 5 2 riM i ð19Þ respectively. Here, ge is the free electron gvalue for which the latest standard value is 2.0023193043737(82) [35] and dðriM Þ is the Dirac delta function at nucleus M. From Eq. (7), limiting ourselves to the electronic terms only, we obtain five contributions to the indirect coupling tensor 1 ›2 EðIM;e ; IN;t Þ JMN;et ¼ 2p ›IM;e ›IN;t I ¼I ¼0 M;e N;t ð15Þ ð16Þ The diamagnetic coupling is obtained from the bilinear operator of Eq. (15) as a ground state expectation value respectively. Here, riM ¼ ri 2 RM is the position vector of electron i with respect to the position of nucleus M, and liM ¼ 2iriM £ 7i is the (field-free) angular momentum with respect to the same reference point. Eqs. (15) and (16) are obtained from the gauge-invariant expression for the electronic kinetic 4 AM ðri Þ ¼ a2 gM SD SD=FC þ JMN; et þ JMN;et : ð20Þ and ð1Þ ¼ a2 gM IM · HPSOðMÞ energy, including the contributions to the momentum from the vector potential In the a.u. system, the numerical values of the following constants are equal to unity: "; e, me ; and 4pe 0 : Then, the speed of light in vacuum, c ¼ 1=a and the permeability of vacuum m0 ¼ 4pa2 ; in a.u. ¼ DSO JMN; et ¼ DSO JMN; et þ PSO JMN; et þ FC JMN; et 1 4 a gM gN k0l 4p X det ðriM ·riN Þ 2 riN;e riM;t £ l0l: 3 3 riM riN i ð21Þ The DSO terms generally contribute to the trace as well as symmetric and antisymmetric parts of J. Typically, J DSO is either numerically small in comparison with the other contributions to the coupling constant, or occasionally (in couplings involving hydrogen, particularly JHH ) largely cancelled by the PSO term to be J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 discussed below. In the past, the efficient calculation of the molecular integrals of the DSO operator was a practical bottleneck; currently, however, the method in Ref. [36] is often used. The remaining coupling tensor contributions involve second-order perturbation theory expressions. ð1Þ For these, two first-order perturbations HM ¼ hM;e IM;e ð1Þ 0 0 and HN ¼ hN;t IN;t give rise to a general term JMN;et contributes to all parts of the J tensor listed in Eq. (10). J PSO is often the second-most important contribution to the coupling constant after J FC : For example, in systems where the valence s-electrons of a certain atom contribute little to bonding, the J PSO contribution to coupling involving this atom may exceed J FC : This happens particularly in couplings to the 19F nucleus as well as other halogen nuclei. X k0lðhM;e IM;e þ h0N;t IN;t ÞlnlknlðhM;e IM;e þ h0N;t IN;t Þl0l 1 ›2 ¼ 2p ›IM;e ›IN;t n–0 E0 2 En I ¼ 1 X k0lhM;e lnlknlh0N;t l0l þ k0lh0N;t lnlknlhM;e l0l 1 Rh ; h0 S : ; 2p n–0 2p M;e N;t 0 E 0 2 En In the final identity, the spin – spin coupling tensor is expressed as a linear response function [30]. The subscript zero indicates that the static limit, corresponding to time-independent perturbations, is taken. By definition, RA; BS0 is symmetric with respect to the order in which the operators A and B occur, and it includes contributions of both the AB and BA successions. No double-counting nor associated numerical prefactors occur in the case of J due to the fact that the operators hM;e and h0N;t refer to different nuclei. Depending on the spin rank of the perturbations, singlet or triplet, the singlet closed-shell ground state l0l is coupled to singlet or triplet excited electronic states, lnl ¼ lnS l or lnT l; respectively, in the sum-over-states expression of Eq. (22). Fig. 1 illustrates the different couplings allowed by the electronic spin symmetry. The singlet operators referring to the two nuclei, PSO(M ) and PSO(N ), couple to each other, whereas the triplet operators FC(M ), FC(N ), SD(M ), and SD(N ) couple among themselves. Operators from the two different spin ranks do not mix, unless electronic spin-orbit coupling is allowed for in third-order perturbation theory [28, 37,38]. These relativistic contributions are Oða6 Þ; however. Like the DSO term, the PSO term ** ++ X liM;e X liN;t 1 4 PSO JMN;et ¼ a gM gN ; ; ð23Þ 3 3 2p riM i i riN 0 239 M;e ¼IN;t ¼0 ð22Þ Computationally, the calculation of the PSO term involves solving for the first-order wave functions, e.g. through solving linear response equations [30], with respect to the three imaginary ðt ¼ x; y; zÞ; corresponding to the operators hPSOðNÞ t Cartesian components of the liM vector operator in Eq. (16). The situation thus resembles that encountered in the calculation of s, where firstorder wave functions with respect to the components of the orbital Zeeman operator are usually solved for. In contrast to s, however, the ‘natural’ gauge origin for the nuclear magnetic dipole field is at the nucleus in question. Hence, there is no need to apply special techniques such as the gaugeincluding atomic orbital (GIAO) ansatz in the calculation of the spin –spin coupling tensors [31]. Fig. 1. Schematic illustration of the second-order processes contributing to the nuclear spin–spin coupling interaction. 240 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 The Fermi contact term !2 1 4p FC a4 g2e gM gN det JMN;et ¼ 2p 3 ++ ** X X £ dðriM Þsi;e ; dðriN Þsi;e i i ð24Þ 0 is isotropic, JFC ¼ 1J FC ; and thus only contributes to the isotropic coupling constant. It is often the dominant term. It can be shown that the Cartesian components of the two electronic spin vectors occurring in Eq. (24), as well as in other second-order expressions involving two triplet operators, must be the same [31]. Due to the isotropic spatial structure of the FC operator, one then has to solve only one linear response equation, e.g. with respect to dðrN Þsz ; to obtain J FC : This term poses otherwise heavy computational requirements, however, through the need (a) to describe electron correlation (the N-electron problem) very accurately and (b) to use a good one-electron basis set flexible enough in the atomic core region. These matters will be discussed in Section 3. The contribution of the spin-dipolar term SD JMN; et 1 a4 2 gg g ¼ 2p 4 e M N 2 X X 3riM;n riM;e 2 den riM £ si;n ; 5 riM n¼x;y;z i S 0 DSO PSO SD FC JMN ¼ JMN þ JMN þ JMN þ JMN S DSO;S PSO;S SD;S SD=FC JMN; et ¼ JMN;et þ JMN;et þ JMN;et þ JMN;et R 2 X 3riN;n riN;t 2 dtn riN si;n 5 riN i of the FC and SD operators contributes to JS only. Often the SD/FC mechanism dominates numerically the anisotropic properties of J. It is a sum of two response functions, where the FC and SD interactions refer to both nuclei in turn. The two responses may be physically different in a coupling tensor between non-equivalent SD=FC nuclei. Interestingly, the separation of JMN into the SD(M )/FC(N ) and SD(N )/FC(M ) contributions has only been investigated in a few papers [39 –42] to the authors’ knowledge. The information already gathered when calculating the other contributions to J is sufficient to evaluate the SD/FC terms as well. Often the wave function responses necessary for JSD are not calculated at the highest possible level. Then, the FC and SD/FC terms that give the often dominant contributions to J may be obtained from solving the first-order wave functions with respect to the two FC perturbations involved, at the best available theoretical level. Summarising the contributions from the terms discovered by Ramsey to the different-rank tensorial properties of JMN ; ð27Þ A DSO;A PSO;A SD;A JMN; et ¼ JMN;et þ JMN;et þ JMN;et : ð25Þ is often small although a priori non-negligible in the general case. JSD can be broken into contributions with tensorial ranks 0, 1, and 2. It is computationally the most demanding mechanism, generally requiring solutions to six response equations corresponding to the six independent Cartesian components of the riN riN operator that appears in the nominator of hSDðNÞ : Finally, the traceless and symmetric [22] cross-term 1 4p a4 2 SD=FC g g g JMN; et ¼ 2p"*3* 2 e M N ++ 2 X X 3riN;e riN;t 2 det riN dðriM Þsi;e ; si;e 5 riN i i 0 ** ++ # 2 X X 3riM;t riM;e 2 det riM si;t ; dðriN Þsi;t þ 5 riM i i 0 ð26Þ As mentioned above, the related direct dipolar coupling tensor also consists only of the symmetric contribution, i.e. D0 ¼ D0 S : 2. Experimental methods 2.1. High field approximation in NMR spectroscopy The magnetic field of the NMR spectrometer is generally taken to coincide with the z0 axis of the laboratory coordinate system ðx0 ; y0 ; z0 Þ : B0 ¼ B0 z^ 0 : When the Zeeman interaction of the bare nuclei, Eq. (6), is large compared with the other interactions, it is sufficient to treat the energy of the nuclear spin system by first order perturbation theory, X ENMR ¼ EZ þ kmM lHs lmM l þ X M,N M kmM mN lHD0 þ HJ lmM mN l; ð28Þ J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 where the lmM l and lmM mN l denote the unperturbed one- and two-spin Zeeman states. This high field approximation (HFA) is broken by strong quadrupolar interactions [43] (experimental examples are given later for this situation). Collecting the terms of Eq. (1) that contribute to ENMR under the HFA, gives HFA HNMR ¼2 B0 X g I 0 þ Hs þ HD0 þ HJ ; 2p M M M;z ð29Þ and for the time average, kTz0 z0 l ; T þ T aniso ¼ set ¼ 12 ð3 cos ue z0 cosutz0 2 det Þ: The assumption B X Hs ¼ 0 g s 0 0I 0; 2p M M M;z z M;z kset Tet l ¼ Set kTet l and for the coupling interactions X 0 1 HD0 ¼ DMN;z0 z0 IM;z0 IN;z0 2 ðIMþ IN2 þ IM2 INþ Þ 4 M,N ð31Þ X 1 HJ ¼ JMN;z0 z0 IM;z0 IN;z0 2 ðIMþ IN2 þ IM2 INþ Þ 4 M,N X 3 þ J ðI I þ I I Þ; ð32Þ 4 M,N MN Mþ N2 M2 Nþ where the tracelessness of D0 has been used. Eqs. (31) and (32) involve the ladder operators for nuclear spins, IM2 ¼ IM;x0 2 iIM;y0 : IMþ ¼ IM;x0 þ iIM;y0 et where ue a is the angle between the e and a axes. It then follows that X Tz0 z0 ¼ cos ue z0 cos utz0 Tet et ¼ 1X 3 e Tee þ 2X1 3 et 2 ð3 cos ue z0 cos utz0 2 det ÞTet ð35Þ ð36Þ ð37Þ ð38Þ corresponds to neglecting correlation between rotation and internal (vibrational) motion of the system. It defines the traceless and symmetric orientation tensor, S [44]: Set ¼ kset l ¼ 12 k3 cos ue z0 cos utz0 2 det l: ð39Þ S carries information on the probability distribution of molecular orientation with respect to B0 : As discussed below, the assumption expressed by Eq. (38) has been abandoned in the modern LCNMR determination of D0 [45 – 47]. Eq. (36) defines the isotropic and anisotropic parts of the NMR tensors, T¼ ð33Þ The spectral observables in the HFA correspond to the time average of the components of the NMR tensors, T ¼ s; D0 , and J, along the direction of the external magnetic field, kTz0 z0 l: Transformation of T between any two sets of Cartesian axes, ðe ; t; nÞ and ða; b;cÞ; may be accomplished using X Tab ¼ cos ue a cos utb Tet ; ð34Þ 1X 2X kTee l þ ks T l; 3 e 3 et et et where e and t denote any of the molecule-fixed coordinates ðx; y;zÞ; and where the operator for the shielding interaction is ð30Þ 241 1 3 TrT ¼ 1 3 ðkTxx l þ kTyy l þ kTzz lÞ ð40Þ and T aniso ¼ 2 2X S : kTl ¼ S kT l; 3 3 et et et ð41Þ respectively, using Eq. (38) for the latter equality. In these equations, the time averaging has been explicitly indicated using the angular brackets k l. This notation will be dropped in most of the following. It should be remembered, however, that the NMR parameters to be discussed are timeaveraged quantities. 2.2. NMR in isotropic media In the gas phase or in ordinary liquids, the molecules have no orientational order to a first approximation. Consequently, Set ¼ 0 and the static 242 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 spin Hamiltonian takes the form X B0 X gM ð1 2 sM ÞIM;z0 þ JMN IM ·IN : 2p M M,N iso HNMR ¼2 ð42Þ The spectral observables are reduced to their isotropic parts, Eq. (40), the shielding constant sM ¼ 1 3 TrsM ; ð43Þ and the spin – spin coupling constant J defined in Eq. (11) (time-averaged). While sM determines the chemical shift of M, the JMN are responsible for the fine structure of the spectra [48 – 50]. 2.3. Liquid crystal NMR spectroscopy 2.3.1. Introduction In 1963, Saupe and Englert [51] proposed the use of LCs as a medium to create an anisotropic orientation distribution for solute molecules. In a LC environment, solute molecules undergo translational and rotational motion sufficiently fast that intermolecular dipole – dipole interactions vanish. On the other hand, intramolecular dipolar interactions, the anisotropic contributions of the s and J tensors, as well as the quadrupole coupling tensors, average to non-zero values. Consequently, LCNMR can be used for the determination of molecular structures, components of s, J and quadrupole coupling tensors (for nuclei with spin $ 1). For spin systems consisting only of spin- 12 nuclei, the Hamiltonian of Eq. (29) becomes [52] B0 X g ð1 2 sM 2 saniso M ÞIM;z0 2p M M X X 1 aniso þ JMN IM ·IN þ DMN þ JMN 2 M,N M,N LC HNMR ¼2 ð3IM;z0 IN;z0 2 IM ·IN Þ; ð44Þ where DMN ¼ 1 2 0 aniso DMN ð45Þ is commonly denoted the direct dipolar coupling. One should, however, note that sometimes another definition, D ¼ D0 aniso ; is also used. Furthermore, in the solid state context the related quantity * + m0 "gM gN 1 RDD ¼ 8p2 R3MN ð46Þ is used. A noteworthy feature of LCNMR is the fact that peak widths (FWHH) of a few Hz and even better than 1 Hz in 1H NMR spectra are possible in favorable circumstances. Because of the high NMR receptivity of 1H nuclei, a good signal-to-noise ratio may be obtained with short accumulation times. As a result, peak positions and spectral parameters may be determined with a high degree of accuracy. Spectral analysis is very similar to that of isotropic systems except that spectra of molecules in LCs are very rarely first order, since dipole – dipole couplings are typically on the order of kHz [53]. The 1H NMR spectrum of benzene and 13C6-benzene in an isotropic solution and in a LC, shown in Fig. 2, illustrates the superb resolution available for solute molecules in LCs. The spectrum of 13C6-benzene also reveals one of the limitations of the method; when the number of interacting nuclei increases, the spectrum becomes very complicated, and consequently its analysis may be difficult or impossible. In practice, systems consisting of up to 10 – 12 spin- 12 nuclei may be analyzed, depending upon the symmetry of the system. The LCNMR method as a means to derive J tensors is in principle quite straightforward. However, in order to obtain reliable, solvent-independent results, molecular vibrations and the correlation between vibrational and reorientational motion must be properly taken into account. A comprehensive review article on the anisotropies of s and J as determined using LCNMR appeared in 1982 [1]. Since then, however, remarkable progress has taken place [54], particularly in the characterization of J tensors. One should regard old data, particularly those that report small anisotropies, with caution. 2.3.2. Liquid crystal solvents The most important LC solvents in studies of the structure of low molar mass molecules as well as the characterization of s, J, and quadrupole coupling tensors are those known as thermotropic (TLC) that J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 243 cases uniaxial, meaning that there exists cylindrical symmetry around n, and that the directions n and 2 n are equivalent, i.e. the phase is apolar. For a more detailed description of the physical properties of LCs, see Ref. [55]. Fig. 3 gives a schematic illustration of a nematic phase. When a LC sample is placed in an external magnetic field, B0, the sample becomes magnetized. The magnetization M is given by [56] 1 X Me ¼ x B; ð47Þ m0 t et t where m0 is the permeability in vacuo and the xet are the components of the diamagnetic (volume) susceptibility (magnetizability) tensor, xd ; which is diagonal in the uniaxial phase.The energy density, rB ; due to the magnetization can be represented as ð B0 rB ¼ 2 M·dB 0 ¼2 Fig. 2. Top: 1H NMR spectrum of benzene in an isotropic solution. Middle: 400 MHz 1H NMR spectrum of benzene oriented in a liquid-crystalline solution. Bottom: 400 MHz 1H NMR spectrum of 13 C6-benzene oriented in a liquid-crystalline solution. feature liquid-crystalline mesophases within certain temperature ranges. In a few cases, lyotropic LCs have also been used. The disadvantage in applying lyotropics is the generally small orientational order of solute molecules, which leads to correspondingly small anisotropic contributions to the Dexp and chemical shifts. Investigations of J tensors have generally been performed on solute molecules dissolved in nematic phases of TLCs. The nematic phase exists at temperatures immediately below the isotropic phase. In nematic phases, LC molecules possess only a short range positional order. However, they tend to align with their long axes5 parallel to a common axis which defines the director, n, of the LC phase. Nematic phases are in almost all known 5 In most applications described in this context, the LCs consist of elongated molecules. Such LCs are called calamitic. B20 2 xd þ Dxd P2 ðcos uBn Þ ; 3 2m0 ð48Þ where xd ¼ 13 Trxd is the isotropic diamagnetic susceptibility, Dxd is the anisotropy of the susceptibility tensor, and P2 ðcos uBn Þ ¼ 12 ð3 cos2 uBn 21Þ is the second-order Legendre polynomial (uBn being the angle between B0 and n). Eq. (48) determines the orientation of the director with respect to the external magnetic field (note Fig. 3. Molecular orientational order in a nematic phase. Reprinted with permission from Ref. [55]. Copyright (1998) Wiley–VCH. 244 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 that xd , 0 always): 1. when Dxd . 0; rB reaches a minimum with P2 ðcos uBn Þ ¼ 1, i.e. uBn ¼ 08 and thus n places itself parallel with B0, and 2. when Dxd , 0; rB is a minimum with P2 ðcos uBn Þ ¼ 2 12 ; i.e. uBn ¼ 908 and thus n places itself perpendicular to B0. TLCs with both positive and negative Dxd values exist and have been used in studies of J tensors within solute molecules. Furthermore, appropriate mixtures of these two kinds of LCs have been utilized. 2.3.3. J tensor contribution to Dexp As seen from Eq. (44), the experimental anisotropic coupling, Dexp MN ¼ DMN þ 1 aniso 2 JMN ; ð49Þ includes a contribution from the J tensor. From Eq. (41), this contribution can be represented as aniso JMN ¼ X 2 P2 ðcos uBn Þ SD et JMN;et ; 3 et Fig. 4. The coordinate systems used in the determination of the anisotropic properties of NMR tensors in uniaxial liquid crystals. In the illustrated example the director of the liquid crystal phase, n, is perpendicular to the external magnetic field B0 : The laboratoryfixed axis system is ðx0 ; y0 ; z0 Þ where B0 ¼ ð0; 0; B0 Þ: The moleculefixed axis system is ðx; y; zÞ: In the most general case, Eq. (50) can be written as nh i aniso JMN ¼ 23 JMN;zz 2 12 ðJMN;xx þ JMN;yy Þ SD zz D þ 12 ðJMN;xx 2 JMN;yy ÞðSD xx 2 Syy Þ ð50Þ where JMN;et is the component of J in the molecule-fixed coordinate system. SD et in turn is the component of the Saupe orientational order tensor (with respect to n) [44] as defined in Eq. (39). The factor P2 ðcos uBn Þ changes the reference direction from B0 to n. Fig. 4 illustrates the different coordinate systems involved. The number of independent components of S is determined by molecular symmetry (Table 1). The values of the components of S are obtained from the Dexp provided that at least one internuclear distance within the solute molecule is known or assumed. Particularly in the theoretical description of orientational order, Wigner matrices [57] are used because of their convenient transformation properties. In the early LCNMR literature, the so-called Snyder motional constants [58] were also used. Table 2 gives the relationships between the order parameters in the various representations. In this review, however, the Saupe orientational order tensor is used exclusively. S SD þ2JMN;xy xy þ S 2JMN;xz SD xz þ P2 ðcos uBn Þ: S 2JMN;yz SD yz ð51Þ The factor DJMN ¼ JMN;zz 2 1 2 ðJMN;xx þ JMN;yy Þ ð52Þ defines the anisotropy of J with respect to the molecular z axis. In practically all studies of J, the solute molecules possess high symmetry so that their orientation can be described with two or only one orientational order parameter (Table 1). For solute molecules with C2v ; D2 ; or D2h symmetry, Eq. (51) reduces to h i aniso D D 1 JMN ¼ 23 DJMN SD zz þ 2 ðJMN;xx 2 JMN;yy ÞðSxx 2 Syy Þ £ P2 ðcos uBn Þ ð53Þ and for molecules with at least a 3-fold symmetry axis to an even simpler form aniso ¼ JMN 2 3 DJMN SD zz P2 ðcos uBn Þ: ð54Þ J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 245 Table 1 Independent second-rank orientational order parameters for molecules of various symmetry in uniaxial liquid crystal phases (From Ref. [57]) Table 2 Relations between the components of the Saupe orientational order tensor [44], averages of Wigner rotation matrix elements [57], and Snyder motional constants [58] Molecular point group Saupe tensor component Wigner matrix element Szz kD20;0 l Sxx 2 Syy 3 1=2 Syz i 38 1=2 ðkD20;2 l 2 kD20;22 lÞ 3 1=2 Cxy 20 Sxz 2 38 1=2 ðkD20;1 l þ kD20;21 lÞ 3 1=2 Cxz 20 Syz i 38 1=2 ðkD20;1 l þ kD20;21 lÞ 3 1=2 Cyz 20 C1, Ci Cs, C2, C2h C2v, D2, D2h C3, S6 C4, C4h, S4 C4v, D2d, D4h, D4 C5, C5h, C5v D4d, D5, D5h, D5d C3h, C6, C6h, C6v D3h, D6, D6h, D6d C1, C1v, C1h, D1h Number of independent order parameters 5 3 2 1 Order parameters Szz, Sxx 2 Syy, Sxy, Sxz, Syz Szz, Sxx 2 Syy, Sxy Szz, Sxx 2 Syy Szz In Eqs. (53) and (54), the z-axis is chosen to be along the n-fold ðn $ 2Þ symmetry axis of the molecule. The molecule-fixed ðx; y; zÞ frame is not in the general case the principal axis system (PAS) of either the S tensor or J. If we assume ða; b; cÞ to be the PAS(J ) of the J tensor, Eqs. (53) and (54) transform to n aniso 1 2 JMN ¼ 3 Dc JMN SD zz ½ð3cos ucz 21Þ þhc ðcos2 uaz 2cos2 ubz Þ h D 2 2 1 þðSD 2S Þ xx yy cos ucx 2cos ucy þ 3 hc io ðcos2 uax 2cos2 uay 2cos2 ubx þcos2 uby Þ P2 ðcosuBn Þ ð55Þ and aniso 1 2 D D ¼ 3 Dc JMN ½SD JMN zz ð3cos ucz 21ÞþðSxx 2Syy Þ ðcos2 ucx 2cos2 ucy ÞP2 ðcosuBn Þ; ð56Þ respectively. In Eqs. (55) and (56), Dc JMN ¼JMN;cc 2 12 ðJMN;bb þJMN;aa Þ 2 ðkD20;2 l þ kD20;22 lÞ Snyder motional constant 1 1=2 5 3 1=2 5 C3z2 2r2 Cx2 2y2 is the J tensor anisotropy in its PAS, and 3 JMN;aa 2JMN;bb hc ¼ 2 Dc JMN ð58Þ is the asymmetry parameter. ucz ; for example, is the angle between the c axis of PAS(J ) and the z axis of the molecule-fixed frame. These equations show that in order to determine the J tensor components in its PAS by applying LCNMR, it is necessary to know the angles between the principal axes and the axes of the molecule-fixed frame. This information is typically not available. Another revealing way to look at the problem is to choose the coordinate system in which the S tensor is diagonal, i.e. in the principal axis system, PAS(S ), of S, (1,2,3). In this frame, the molecular orientational order is determined by two independent order parameters, S33 and S11 2 S22 : Consequently, if the rotation – vibration correlation effects (see Section 2.3.4) are neglected, any dipolar coupling and the anisotropic indirect contribution can simply be written as 2 DMN ¼ 2 12 FMN ½SD 33 ð3 cos uMN;3 2 1Þ D 2 2 þ ðSD 11 2 S22 Þðcos uMN;1 2 cos uMN;2 Þ ð57Þ P2 ðcos uBn Þ ð59Þ 246 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 and aniso ¼ JMN 1 D3 JMN SD 33 þ 2 ðJMN;11 2 JMN;22 Þ i D ðSD 11 2 S22 Þ P2 ðcos uBn Þ: 2 3 factor is omitted. Similarly, h In Eq. (59), the factor F is defined as * + m0 "gM gN 1 FMN ¼ ¼ RDD MN : 8p2 R3MN aniso S ¼2SD JMN 33 ½3D3 JMN þ h ðJMN;11 2JMN;22 ÞP2 ðcosuBn Þ: ð61Þ One should note that uMN;a ða ¼ 1; 2;3Þ is now the angle between the RMN vector and the a axis of PAS(S ), D3 JMN as well as JMN;11 2 JMN;22 are given in the PAS(S ), and, consequently, are not the same as those in the other coordinate frames. Eqs. (59) and (60) clearly illustrate the fact that although D and J aniso possess similar dependence on the orientational order parameters they generally do not vanish under the same conditions; the zero condition for the former is 3 cos2 uMN;3 2 1 S11 2 S22 ¼2 2 ; S33 cos uMN;1 2 cos2 uMN;2 ð62Þ i.e. determined by the molecular geometry alone, whereas J aniso vanishes when S11 2 S22 22D3 JMN ¼ : S33 JMN;11 2 JMN;22 ð65Þ ð60Þ ð63Þ with S33 – 0: Thus, the experimentally determined Dexp ¼ D þ ð1=2ÞJ aniso may have a (sizable) non-zero value even though the dipole–dipole coupling, D, is vanishingly small [59,60]. In papers dealing with NMR spectra of biomacromolecules partially oriented in dilute LC solutions, Eq. (59) is generally written in the form [61] h 2 DMN ¼ 2 12 FMN SD 33 ð3 cos a 2 1Þ i ð64Þ þ 32 hS3 sin2 a cos 2b P2 ðcosuBn Þ; D D where hS3 ¼ ðSD is the asymmetry and 11 2 S22 Þ=D3 S D D D D D3 S ¼ S33 2 1=2ðS11 þ S22 Þ ¼ ð3=2ÞSD 33 the anisotropy of the S tensor, and the angles a ¼ uMN;3 and b (angle between the 1-axis and the projection of RMN in the 12-plane) are polar angles defining the orientation of the internuclear vector RMN in PAS(S ). In order to take into account the internal motion of the RMN vector, a scaling factor is generally introduced [61]; however, in the present case this Thus, in the situation where DMN ¼0; " 2 3 cos2 a 2 1 aniso D JMN ¼ 2S33 3D3 JMN 2 3 sin2 acos2b # ðJMN;11 2 JMN;22 Þ P2 ðcosuBn Þ ð66Þ can be non-vanishing. 2.3.4. Vibration and deformation effects Since the Dexp values are averages over internal molecular vibrations, it was recognized in the late 1960s that they should be corrected for the vibrations [62]. However, more than ten years passed before Sýkora et al. [63] published a theory and a general computer program, VIBRA, became available to correct dipolar couplings for harmonic vibrations. Later, another program (AVIBR) was developed to compute the effects of anharmonic vibrations [64]. In 1966 Snyder and Meiboom [65] and a few years later Ader and Loewenstein [66] recorded NMR spectra of tetramethylsilane and methane in LC solutions, respectively, and detected a small dipolar splitting despite the fact that the molecules should not be oriented because of their high symmetry. This observation was ascribed to a slight distortion of the solute molecule by the anisotropic force exerted by the solvent. The anisotropic interactions visible in the spectra arise in this case from correlation between internal molecular vibrational and reorientational motion with respect to the anisotropic solvent frame. Consequently, the 23 separation ksMN R23 MN l ¼ SMN kRMN l; Eq. (38), is not strictly valid. Here, sMN ¼ P2 ðcosuMN;B Þ is the component of set ; Eq. (37), along RMN : Then, the dipole – dipole coupling must be represented as * + m0 "gM gN sMN P2 ðcosuBn Þ: ð67Þ DMN ¼ 2 8p 2 R3MN In 1984, a general theory was presented to take the deformation effect into account [45,47]. Five years later, the computer program MASTER [67] was published; this program computes the vibrational J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 and deformation contributions to the dipole – dipole couplings. For small-amplitude motion, the various contributions to the dipolar coupling can be separated as follows: ah h d DMN ¼ Deq MN þ DMN þ DMN þ DMN : ð68Þ Deq is the dipole-dipole coupling corresponding to the equilibrium structure of the molecule, Dah arises from the anharmonicity of the vibrational potential, Dh is the contribution from the harmonic vibrations, and Dd is the deformation contribution. The experimental couplings thus are eq ah h d Dexp MN ¼ DMN þ DMN þ DMN þ DMN þ 1 aniso 2 JMN : ð69Þ It has been experimentally realized that undistorted structures of solute molecules may be derived through the use of proper mixtures of LCs. Some LCs lead to positive structural deformations whereas others cause negative deformations. When mixed in a proper molar ratio, they produce an environment that does not distort solute molecular structure. The methane molecule is a suitable deformation reference [68]. The method can be expected to work for solutes that interact with the solvent LCs in a manner qualitatively similar to that of CH4. There exists some evidence that not only molecular structure deformation but also the apparent deformations of the s and J tensors are cancelled (or at least reduced) in such LC mixtures [69]. Table 3 lists the importance of each contribution to Dexp in benzene. 2.3.5. Limitations in the quantitative determination of J tensors In order to obtain a reliable value for J aniso ; which is often small and in many cases comparable in magnitude with the vibrational and deformation correction terms (see Table 3), the molecular structure should be determined as completely as possible with the aid of Dexp values. This necessarily means a full analysis of data taking into account the molecular vibrations and rotation-vibration correlation. In such an analysis, the number of unknown parameters may exceed the number of Dexp couplings obtainable from one experiment, i.e. the problem becomes underdetermined. Occasionally this can be overcome by carrying out experiments in several LC solvents 247 and performing a joint analysis of the couplings [69,71,72]. In order to keep the problem of a solute molecule in one LC solvent (such as those listed in Table 4) overdetermined, one must have a sufficient number of Dexp couplings in which the J aniso contribution can be considered negligible. If the spin system consists of N interacting nuclei, the total number of available Dexp couplings is NðN 2 1Þ=2: If the molecule has no symmetry, the number of orientational order parameters is five and the number of coordinates 3N: However, a basic property of dipolar couplings is that they do not define absolute but only relative internuclear distances, i.e. the shape of a molecule. For the determination of absolute order parameters and distances, one internuclear distance has to be assumed. This means that the number of adjustable coordinates is 3ðN 2 2Þ: Consequently, in this general case N has to fulfill the condition NðN 2 1Þ=2 2 5 2 3ðN 2 2Þ $ 0: ð70Þ This means that the derivation of atomic coordinates and orientational order parameters necessitates N $ 7; in other words, 21 Dexp couplings with negligible anisotropic contribution are necessary for this purpose. Thus, the determination of some of the J tensors becomes feasible only if the number of couplings exceeds 21. For a planar molecule with, e.g. C2v symmetry (two independent order parameters), N has to satisfy the inequality NðN 2 1Þ=2 2 2 2 2ðN 2 2Þ $ 0 ð71Þ from which N $ 5: The condition of insignificant ð1=2ÞJ aniso (as compared to D ) is usually valid for X1H (X ¼ 1H, 13C, 14N, 15N, 19F, etc.) couplings. Uncertainty in the determination of J aniso may appear in the case of the spin system consisting of different kinds of nuclei, for instance I and S spins. Namely, the NMR spectrum renders it possible to determine only the sum l2Dexp IS þ JIS l: If JIS cannot be determined in the same experimental conditions (the same solvent, temperature, concentration, etc.) as the sum, additional uncertainty may be introduced to Dexp ; and consequently to J aniso as well. In principle, JIS can be determined in LC phases by performing, e.g. variable angle spinning (VAS) experiments [73]. The anisotropic contribution, J aniso ; for molecules with high symmetry depends exclusively on the 248 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 Table 3 Various contributions (in Hz) to D exp in benzene Couplinga D eq Dh Dd D calc D exp 3 2701.16 2134.94 287.65 22108.96 2269.33 268.43 246.20 2248.79 247.88 231.10 9.78 0.98 0.40 158.55 5.08 0.48 0.18 1.83 0.03 20.05 21.99 20.27 20.14 214.23 20.58 20.08 20.01 20.44 0.05 0.08 2693.37 2134.23 287.39 21964.64 2264.83 268.03 246.02 2247.40 247.80 231.07 2693.368(7) 2134.220(9) 287.417(12) 21964.637(14) 2264.790(9) 268.131(9) 246.022(12) 2248.217(21) 247.569(20) 231.613(32) Coupling ratioc Experimental valueb Theoretical value DoHH =Dm HH DoHH =DpHH m DHH =DpHH DoCC =Dm CC DoCC =DpCC p Dm CC =DCC 5.166(5) 7.932(5) 1.535(6) 5.218(4) 7.852(11) 1.505(2) 5.1962 8.0000 1.5396 5.1962 8.0000 1.5396 DHH DHH 5 DHH 1 DCH 2 DCH 3 DCH 4 DCH 1 DCC 2 DCC 3 DCC 4 b aniso The difference between D calc and D exp for the CC couplings is due to ð1=2ÞJCC [70]. On the bottom are also shown the ratios of the experimental 1H1H and 13C13C couplings. a The number in the upper left corner indicates the number of bonds between the interacting nuclei (analogous to the notation for the indirect spin–spin coupling), although dipolar coupling is a through-space interaction. b The figure in parentheses gives the experimental error in units of the last digit(s). c The ortho, meta, and para couplings are denoted by o, m, and p, respectively. orientational order parameter of the symmetry axis, as shown in Eq. (54). Therefore, the determination of DJ; once Szz and the molecular structure are known, is relatively straightforward. In contrast, in less symmetric molecules DJ and other components of J, such as Jxx 2 Jyy in Eq. (51), can be derived only if the ratio of the orientational order parameters, such as ðSxx 2 Syy Þ=Szz ; can be changed by choosing another LC solvent (then, of course, one has to assume that the J tensor is independent of solvent). If the ratio remains constant or changes only by a small amount, only the combination of the order parameters and J tensor components can be determined. 2.3.6. Qualitative determination of J aniso Molecular symmetry can constrain the ratios of D couplings. This can be used to reveal whether Dexp includes a significant contribution from ð1=2ÞJ aniso exp or not. If the examination of the ratio Dexp MN =DOP reveals a deviation from the corresponding ratio of direct dipolar couplings (D ¼ Dexp 2 ð1=2ÞJ aniso ¼ Deq þ Dah þ Dh þ Dd ; see Eq. (69)), the couplings (or at least some of them) may be affected by ð1=2ÞJ aniso : Benzene is a good example of a molecule in which symmetry completely defines the ratios of the D couplings. The order parameter of each MN-direction in the ring plane is equal to 2ð1=2ÞSzz (Szz is the order parameter of the 6-fold symmetry axis), consequently the following equation is valid (in the first order approximation) Dexp gM gN kR23 MN l MN exp ¼ gO gP kR23 DOP OP l " !# 21 21 1 kR23 kR23 MN l OP l DJMN £ 12 2 DJOP 3 F 0MN F 0OP ð72Þ where F 0MN ¼ FMN kR23 MN l: The expression in the square brackets equals 1, i.e. the ratio of the D couplings J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 Table 4 Liquid crystals used in studies of J tensors by their code name and composition Code name Composition EBBA HAB Phase 4 N-( p-ethoxybenzylidine)-p-n-butylaniline p,p0 -di-n-heptylazoxybenzene Eutectic mixture of p-methoxy-p0 -nbutylazoxybenzenes Mixture of Phase 4 and p-ethoxy-p0 -nbutylazoxybenzenes Mixture of phenylcyclohexanes, biphenylcyclohexane and phenylcyclohexane esters Mixture of azoxy compounds and a biphenyl ester Mixture of trans-4-n-propyl-(4-cyanophenyl) -cyclohexane (24%), trans-4-n-pentyl -(4-cyanophenyl)-cyclohexane (36%), trans-4-n-heptyl-(4-cyanophenyl)-cyclohexane (25%), trans-4-n-pentyl-(40 -cyanobiphenyl-4)cyclohexane (15%) Mixture of 4-n-trans,trans-bicyclohexyl-40 -carbonitrile (36%), 4-n-propyl-trans,transbicyclohexyl-40 -carbonitrile (34%), 4-n-heptyl -trans,trans-bicyclohexyl-40 -carbonitrile (30%) Mixture of alkylphenylcyclohexanes, alkylcyclohexanebiphenyls, and bicyclohexanebiphenyls Mixture of alkylbicyclohexanes and alkyltricyclohexanes Phase 5 Phase 1221 ZLI 997 ZLI 1132 ZLI 1167 ZLI 1982 ZLI 2806 corresponds to the ratio of the rovibrational averages (derived from the dipolar couplings corrected for vibrations and deformation effects) of the inverse cube distances between interacting nuclei, if (1) DJMN and DJOP vanish simultaneously, or (2) DJMN = 23 DJOP ¼ ðgM gN =gO gP ÞkR23 MN l=kROP l: For the cases M;N;O;P all equal to either 1H or 13C, Eq. (72) reduces to the form Dexp kR23 MN l MN exp ¼ DOP kR23 OP l 1 23 23 1 2 0 ðDJMN kRMN l 2 DJOP kROP lÞ ; ð73Þ 3F where F 0 ¼ F 0MN ¼ F 0OP and the condition (2) above 23 becomes DJMN =DJOP ¼ kR23 OP l=kRMN l: It follows from pthe ffiffi hexagonal symmetry of benzene that Rm ¼ 3Ro and Rp ¼ 2Ro (Ri is the distance between protons or carbons in ortho (o ), meta (m ) or para ( p ) positions with respect to each other). Thus, for the interacting nuclei of the same 249 pffiffi isotopic species, Do : Dm : Dp ¼ 1 : 3=9 : 1=8 < 1 : 0:1925 : 0:1250: It has been found that the 1H1H couplings in benzene indeed fulfill these ratios aniso and thus the JHH contributions obviously are negligible. In contrast, significant deviations have been detected in the ratios of the 13C13C couplings (see Ref. [70] and Table 3). Other interesting and illustrating cases are provided by linear solute molecules. The ra structure, i.e. the structure determined from the D couplings corrected for harmonic vibrations, is internally consistent. Consequently, there is no shrinkage effect and the internuclear distances are additive. For example, in ethyne (C2H2), RCC ¼ 2R0CH 2 RHH ; where RHH is the distance between the average positions of the hydrogen atoms, and RCC and R0CH are the corresponding one- and two-bond distances between hydrogen and carbon positions. The use of this relation renders possible the derivation of the following equation [72]: 8 2 1 3 < g 3 exp DJCC ¼ pCC DCC 2 42 H Szz : gC 2 1 3 1 2 D 2 pCH 2 Dexp 2 J S CH zz CH 3 323 9 2 = 1 gH 3 exp 2 3 5 2 pHH DHH ; gC ð74Þ where the anisotropy of JHH is assumed to be negligible, and pMN ¼ 1 þ DhMN =DMN is the harmonic correction factor (which is independent of molecular orientation for molecules with at least a 3-fold symmetry axis). One should emphasize in this context that the vibrational corrections have to be calculated exclusively for the purely dipolar part of the experimental coupling, i.e. for D ¼ Dexp 2 ð1=2ÞJ aniso : Eq. (74) can be approximated by a linear equation DJCC ¼ AD2 JCH þ B ð75Þ where experiments gave average values of 6.107 and 2112:7 Hz for A and B, respectively [72]. The ab initio calculated point [72], (D2 JCH ¼ 28:2 Hz, DJCC ¼ 47:5 Hz), is relatively close to the abovementioned straight line. Consequently, this finding can be regarded as the first experimental evidence of the anisotropy of a 13C1H spin – spin coupling 250 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 tensor. In principle, a similar procedure can be applied to the one-bond JCH but in practice it is restricted by the uncertainty (a few percent) of the vibrational correction factor 1 pCH : Another interesting example of a linear molecule is carbon diselenide (CSe2). In this case the harmonic corrections to the D couplings are presumably small, and thus are omitted. Furthermore, omitting the deformation contributions we obtain for the ratio of the experimental couplings Dexp gC 1 2 13 DJSeC R3SeC =F 0SeC SeC exp ¼ 8 gSe 1 2 83 DJSeSe R3SeC =F 0SeSe DSeSe ð76Þ (a series expansion cannot be applied here because DJ may be relatively large) where the numerical values of F 0SeC and F 0SeSe are 5780:3 and 4398:6 Å3 Hz, respectively. If neither JSeC nor JSeSe is anisotropic, the ratio should equal 8gC =gSe < 10:513: The experimentally determined ratio is ^3:66 ^ 0:05 : the sign of the Dexp SeSe coupling cannot be determined from the experimental spectra without knowing DJSeSe : The coupling Dexp SeC can be deduced to be negative on the basis of good agreement between experimental and calculated nuclear shielding tensor anisotropies, DsC and DsSe ; Dexp SeC was used to solve for the orientational order parameter that has to be positive [8]. Consequently, either one or most likely both of the two tensors possess a sizable anisotropy. Relativistic extended Hückel (REXNMR) [7,74] calculations estimated for the ratio DJSeC =DJSeSe the value of , 0:2: Using this result one obtains the following anisotropies: DJSeSe < 2654 Hz and DJSeC < 2131 Hz or DJSeSe < 1212 Hz and DJSeC < 242 Hz with the positive and negative sign of the ratio of the experimental D couplings, respectively. Which one of the two possible solutions is closer to the truth, cannot be determined from the experimental data. REXNMR calculations favor the latter solution since they yield þ330 and þ1330 Hz for DJSeC and DJSeSe ; respectively [8]. It is evident that the aniso anisotropic contribution ð1=2ÞJSeSe ¼ ð1=3ÞDJSeSe Szz exp dominates in DSeSe and determines the sign of the coupling. The above procedure can be applied even more generally. Namely, if SMN for the nuclei M and N is the same as SOP for the nuclei O and P in any molecule, in other words the axes passing through the nuclear pairs MN and OP are parallel, the ratio DMN =DOP is independent of the orientation. If the corresponding ratio of the experimental couplings is found to deviate from the ratio of the purely dipolar couplings, it suggests a J aniso contribution to at least one of the couplings. 2.3.7. Results derived from LCNMR experiments The LCNMR results for J tensors derived for a number of ‘model systems’ since 1982 are shown in Table 5. The molecules investigated possess high symmetry (only one or two orientational order parameters are needed to describe their orientation) and in most cases they contain hydrogen atoms. The importance of hydrogen atoms follows from the fact, as stated above, that the indirect aniso contribution ð1=2ÞJXH to Dexp XH can generally be neglected, and consequently the determination of molecular structure and orientational order parameters can be based on these couplings. Much emphasis is given to the investigation of X13C (X ¼ 13C, 14N, 15N, 19F) coupling tensors. There are two reasons for this; first, the tensors can be theoretically computed with reasonable effort and good accuracy allowing for comparison between experimental and calculated results, and second, the Dexp XC couplings are used to determine orientational order parameters of LC molecules [82 –85] and biomacromolecules dissolved in dilute liquidcrystalline solvents [61]. In order for the couplings to be applicable in the latter cases it is necessary to know the size of the indirect contribution as compared to the respective Dexp XC or DXC : As pointed out above, D and ð1=2ÞJ aniso do not vanish simultaneously, and therefore in certain circumstances the anisotropic contribution may even dominate in Dexp : There are only a few LCNMR studies dealing with couplings between heavy nuclei. The only examples since 1982, as shown in Table 5, are the 77 Se13C and 77Se77Se coupling tensors in carbon diselenide [8] and the 199Hg13C coupling tensor in dimethyl mercury [81]. Earlier on, results for 111,113 Cd13C [86], 29Si13C and 119Sn13C [87] and 77 31 Se P [88,89] were published but they can be regarded as more or less qualitative as compared to what is achievable today with all the necessary corrections to Dexp : J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 251 Table 5 Indirect spin–spin coupling tensors as determined experimentally by LCNMR Coupling and molecule a Results and comments Reference A relation, based on the additivity of the ra distances, between DJCC and D2JCH was derived. The relation was found to be linear: DJCC ¼ 6.107 £ D2JCH 2 112.7 Hz. MCLR calculations predict the point (D2JCH ¼ 28.2 Hz, DJCC ¼ 47.5 Hz) which is fairly close to the straight line; substitution of 47.5 Hz for DJCC in the above relation leads to D2JCH of 26.2 Hz, in perfect agreement with the calculations. Thus this finding may be considered as the first experimental evidence of a non-negligible 13C1H spin– spin coupling tensor anisotropy. Values from 49.20 ^ 0.03 to 49.26 ^ 0.03 Hz were obtained for 2JCH in the isotropic state of the LCs (ZLI 1167 and in three mixtures of ZLI 1167 and Phase 4) used in the determination of the tensor anisotropy. For the details, see text. [72] 13 1 C H coupling Ethyne C2H2 13 13 C C coupling Acetonitrile CH3CN Benzene C6H6 Ethane C2H6 Ethene C2H4 DJCC values of 115 ^ 24 and 112 ^ 25 Hz were determined for acetonitrile dissolved in [75] ZLI 1132 and ZLI 1167 LCs, respectively. In both cases, experimental dipolar couplings were corrected only for harmonic vibrations, i.e. correlation between vibrational and reorientational motion was neglected. For JCC, values of 58.0 ^ 0.2 and 57.5 ^ 0.3 Hz were determined in acetone-d6 solution and in the isotropic state of the ZLI 1167 LC, respectively. [71] A new method was developed in order to take into account correlation between vibration and rotation. The method is based on considering only torques acting on the bonds between light atoms of a molecule. The NMR data obtained for acetonitrile in five LCs (ZLI 1132, ZLI 1167, Phase 4, EBBA, and a mixture of ZLI 1167 and Phase 4) were treated applying the above procedure and a joint analysis of the dipolar couplings. This leads to kDJCC l ¼ 30 ^ 33 Hz: For the JCC, see above. [70] The values of D exp were corrected for both vibrational and deformation effects. Experiments performed in three LCs gave the following anisotropies for the nJCC tensors: LC D1JCC D2JCC D3JCC ZLI 1167 21.2 25.2 8.7 Phase 4 17.5 22.5 10.7 MIX 13.8 23.9 9.1 The respective average values are: 17.5, 23.9 and 9.5 Hz. (MIX is a 58:42 wt% mixture of ZLI 1167 and Phase 4.) The coupling constants, determined in the isotropic state of the LCs, range as follows: 1 JCC: 55.811 ^ 0.004…55.98 ^ 0.01 Hz 2 JCC: 22.519 ^ 0.009… 2 2.434 ^ 0.007 Hz 3 JCC: 10.090 ^ 0.006…10.12 ^ 0.02 Hz. DJCC was determined separately in five LC solvents (ZLI 1167, ZLI 1982 and three mixtures [72] of ZLI 1167 and Phase 4). Vibrational and deformation contributions were taken into account and the internal rotation around the CC bond was treated quantum mechanically. The anisotropy values range from 49 to 61 Hz, the average being 56 Hz. JCC ranges from 34.498 ^ 0.015 to 34.558 ^ 0.006 Hz in the isotropic state of the LCs used. Due to the D2h point group symmetry of ethene, two order parameters, Szz and Sxx 2 Syy, are [72] aniso ; is given by needed to describe its orientation. Therefore, the anisotropic contribution, JCC Eq. (53). As the experiments in the six LCs (ZLI 1167, ZLI 1982, ZLI 2806, and three mixtures of ZLI 1167 and Phase 4) do not yield independent information to determine both DJCC and JCC,xx 2 JCC,yy, the asymmetry parameter (JCC,xx 2 JCC,yy)/JCC,zz was constrained to be the same in different solvents. Least-squares fit of the mean value of the anisotropy (DJCC was allowed to change from one LC to another) and the tensor asymmetry parameter led to (with dipolar couplings corrected both for vibrational and deformation effects): kDJCC l ¼ 11 Hz; and kJCC;xx 2 JCC;yy l ¼ 244 Hz: There is, however, quite a large variation in the individual anisotropy values, from 3 to 21 Hz, when determined in different LCs. JCC varies between 67.45 ^ 0.02 and 67.62 ^ 0.01 Hz in the isotropic state of the LCs used. (continued on next page) 252 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 Table 5 (continued) a Coupling and molecule Results and comments Reference Ethyne C2H2 Fixing D2JCH to the ab initio calculated value of 28.2 Hz and applying the experimentally determined relationship between DJCC and D2JCH gives DJCC ¼ 59.5 Hz. JCC ranges from 169.63 ^ 0.02 to 169.819 ^ 0.014 Hz in the LCs heated to the isotropic state. For the details, see text and 13C1H coupling/ethyne [72] A similar procedure as described above in determination of DJCC was applied to the determination of D2 J15 NC The analysis resulted in D2 J15 NC ¼ 218 ^ 7 Hz: A value of 2.9 ^ 0.2 Hz was measured for D2 J15 NC in CDCl3. Experiments were performed in five LC solvents (ZLI 1167, ZLI 1132, Phase 4, EBBA, and a 58:42 mixture of ZLI 1167 and Phase 4). Besides the correction of D couplings for vibrational and deformation effects, also the deformation contribution to the JNC coupling tensors was taken into account using an adjustable parameter. The DJ from a joint analysis of five sets of Dexp NC (altogether 10 couplings and four parameters) led to the following results: kDJC14 N l ¼ 8:7 ^ 1:7 Hz; and kDJ14 NC l ¼ 42:8 ^ 2:8 Hz: The first coupling is over the single bond whereas the latter is over the triple bond. Scaling of these results to correspond to the 15N13C couplings leads to 212.2 and 260.0 Hz, respectively. JC14 N ¼ 7:63 ^ 0:04 Hz and J14 NC ¼ 6:30 ^ 0:09 Hz when determined in a CDCl3 solution. [71] 14,15 N13C coupling Acetonitrile CH3CN Methylisocyanide CH3NC 19 13 F C coupling Difluoromethane CH2F2 Fluoromethane CH3F p-Difluorobenzene p-C6H4F2 Experiments were carried out at several temperatures in three LCs (ZLI 1132, ZLI 1167 and Phase 5). The experimental dipolar couplings were corrected for harmonic vibrations and deformation effects. Furthermore, the contribution of the anharmonicity of the vibrational potential was partially considered by estimating the diagonal cubic stretching force constants from the respective harmonic ones. For symmetry reasons, the orientation of CH2F2 is described by two independent orientional order parameters, Szz and Sxx 2 Syy, and aniso ; in principle allow the determination of consequently the anisotropic contributions, JFC both DJFC and JFC,xx 2 JFC,yy. In practice, however, experiments did not yield enough independent data, and therefore, in performing a joint analysis of the experimental data, the ratio DJFC/(JFC,xx 2 JFC,yy) was fixed to the corresponding ab initio value. Then DJFC ¼ 13.5 Hz and JFC,xx 2 JFC,yy ¼ 2360 Hz. JFC ¼ 2236:01 ^ 0:05… 2 236:186 ^ 0:006 Hz in the isotropic state of the LCs, whereas in the gas phase a value of 2233.91 ^ 0.11 Hz was measured. Experiments were carried out in eight LCs (ZLI 1167, EBBA, Phase 4, Phase 1221, and four mixtures of ZLI 1167 and EBBA). When correcting the dipolar couplings only for harmonic vibrations, DJFC ranges from 24955.3 ^ 260.2 to þ 689.8 ^ 62.5 Hz in the individual LCs. Performing a joint fit to 44 couplings corrected for both harmonic vibrations and aniso ; leads to deformation effects, and taking into account the deformation contribution to JFC kDJFC l ¼ 404 ^ 31 Hz: JFC ¼ 2161.62 ^ 0.26… 2 161.20 ^ 0.24 Hz in the isotropic state of the LCs used. In a recent paper, the kDJFC l was determined by using spectra recorded at eight temperatures in one LC (ZLI 1132) and applying a joint analysis of the set of dipolar couplings corrected for both vibrational and deformation effects. The resulting kDJFC l ¼ 350 Hz: JFC ¼ 2161.30 ^ 0.04 Hz in the isotropic state of the LC. Experiments were performed in five LCs (ZLI 1167, ZLI 1132, ZLI 1695, Phase 4, and a mixture of ZLI 1167 and Phase 4 in which Dexp CH of methane is vanishingly small). Only aniso harmonic vibrations were considered. Due to the C2v symmetry of the molecule, the n JFC n n n depend on both D JFC and JFC,xx 2 JFC,yy, see Eq. (53). In these particular LCs the ratio of the orientational order parameters, (Sxx 2 Syy)/Szz, varies so that the two nJFC tensor properties could be determined for n ¼ 3 and 4: Coupling tensor DJ Jxx 2 Jyy 3 JFC 8 ^ 9 Hz 25 ^ 10 Hz 4 JFC 111 ^ 17 Hz 2130 ^ 18 Hz The z axis of the molecule-fixed frame lies along the CF bond. [75] [76] [77] [78] [69] [78] [79] J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 253 Table 5 (continued) Coupling and molecule 29 Si13C coupling Methylsilane CH3SiH3 Se13C coupling Carbon diselenide CSe2 a Results and comments Reference Experiments were performed in five mixtures of various LCs (ZLI 1167 þ Phase 4, ZLI 2806 þ Phase 5, ZLI 1132 þ EBBA, ZLI 997 þ ZLI 1982, and ZLI 997 þ ZLI 1167). The D exp were corrected for both harmonic and anharmonic vibrations, apart from the deformation effects. In calculating the anharmonic corrections, a similar approximation was applied as described for difluoromethane. The experiments did not yield enough independent information to resolve the two tensorial properties (see above), and therefore the ratio DnJFC/(nJFC,xx 2 nJFC,yy) was fixed to the value given by the ab initio calculation. The results are shown below: Coupling tensor DJ Jxx 2 Jyy 1 JFC 400 ^ 90 13 ^ 3 2 JFC 239 ^ 2 220.5 ^ 1.1 3 JFC 17.6 ^ 0.2 13.7 ^ 0.1 4 JFC 220.0 ^ 0.9 235 ^ 2 The z axis of the molecule-fixed frame lies along the CF bond. The nJFC couplings were found to be independent of the LC solvent in the isotropic state (at 355 K): 1JFC ¼ 2242.61 Hz, 2JFC ¼ 24.29 Hz, 3JFC ¼ 8.18 Hz, and 4JFC ¼ 2.67 Hz. [60] The harmonic force field was calculated at the semiempirical level with two parametrizations (AM1 and PM3) and at the ab initio MP2 level. The anharmonic vibrations were treated as described for difluoromethane. A quantum mechanical approach was applied to average couplings over the internal rotation. The analysis of the set (obtained from experiments in the ZLI 1167 and ZLI 2806 LCs) of corrected ‘best’ experimental dipolar couplings led to an average value of 289 Hz for DJSiC. The use of harmonic force fields derived from calculations at various levels results in DJSiC values that range from 286 to 2108 Hz. JSiC is 251.59 ^ 0.03 and 251.55 ^ 0.02 Hz in the isotropic state of the LCs used. [80] exp The ratio of the experimental anisotropic couplings, lDexp SeC =DSeSe l ¼ 3:66; was found to deviate from 10.513, which is the value of the ratio for the case that DJSeC and DJSeSe are simultaneously vanishingly small. Utilization of the REXNMR calculations in the analysis of the experimental data leads to the value of either 2131 or þ242 Hz, depending upon whether the sign of the ratio of the experimental couplings is positive or negative, respectively. For details, see text. JSeC ¼ 2226.59 ^ 0.36 Hz in CDCl3 solution and 2226 ^ 6 Hz in the isotropic state of the ZLI 1132 LC. [8] 77 199 Hg13C coupling Dimethylmercury (CH3)2Hg The DJHgC ranges from 655 ^ 56 to 864 ^ 15 Hz when determined in four LCs (ZLI 1167, [81] Phase 4, and two mixtures of ZLI 1167 and Phase 4). Only a harmonic force field was taken into account when calculating corrections for D exp. JHgC varies from 690.3 to 693.8 Hz in the isotropic state of the LCs used. 19 19 F F coupling In this particular case, the deformation contribution to 1 Dexp [78] Trifluoromethane CHF3 CH appeared to be exceptionally large for the two LCs used (ZLI 1132 and ZLI 1167). Experiments were performed at several temperatures and the set of corrected D exp was analysed using D2JFF as a free parameter but keeping the DJ of the other coupling tensors fixed to their ab initio values. The resulting D2JFF is 2200 Hz. 2 JFF could not be determined experimentally because of the chemical equivalence of the 19F nuclei. For details, see 19F13C coupling/p-difluorobenzene p-Difluorobenzene p-C6H4F2 Coupling tensor DJ Jxx 2 Jyy 5 JFF 230 ^ 15 Hz 236 ^ 15 Hz [79] 236.5 ^ 0.5 Hz 238.4 ^ 0.5 Hz [60] 5 JFF ¼ 17.445 Hz in the isotropic phase of the ZLI 997 (32.1 wt%)/ZLI 1982 (67.9 wt%) LC mixture. (continued on next page) 254 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 Table 5 (continued) Coupling and molecule Se77Se coupling Carbon diselenide CSe2 a Results and comments Reference For the details, see 77Se13C coupling/carbon diselenide and text. Depending upon the sign of the ratio of the D exp, DJSeSe is either 2654 or þ 1212 Hz. Both aniso contributions that are much larger than of these are so large that they lead to the ð1=2ÞJSeSe . J the purely dipolar contribution DSeSe in Dexp SeSe was not determined experimentally SeSe because of the magnetic equivalence of the 77Se nuclei [8] 77 a For the code names and composition of LC solvents, see Table 4. 2.4. Solid-State NMR spectroscopy 2.4.1. Introduction In the case of crystalline solids, the constituent molecules are oriented in certain directions determined by the crystal structure. Large-amplitude reorientational motion is usually not possible and only small-amplitude lattice vibrations contribute to the motional effects on NMR properties6. All the interactions incorporated in HNMR can in principle contribute to solid-state spectra, thus the corresponding peaks are generally broad compared to those of liquid and gaseous samples. In a static sample, the angles in Eq. (35) between the z0 direction of observation and the molecule-fixed axes are constant. There is no need for rotational averaging of Tz0 z0 as in isotropic media and LCs. The Hamiltonian corresponding to spin-1=2 nuclei in the solid state thus takes the SS HFA ¼ HNMR : form specified in Eqs. (29) – (32), HNMR The solid-state NMR observables vary depending on the nature of the NMR sample. A single crystal sample is one coherent block of matter. Apart from inevitable defects and vibrational motion, the lattice vectors remain constant throughout the sample. It often is difficult to grow large enough single crystals for NMR experiments, and a powder sample must be used. The latter consists of randomly oriented crystallites that are small single crystals themselves. The single crystal samples are studied using a 6 This is not strictly true, e.g. for guest species in molecular sieves that have large enough cavities to allow large-amplitude rotation and/or translation. The situation from the point of view of the NMR observables of the guest then resembles that in LCs or isotropic systems, depending on how hindered the motion is. goniometer whose rotation axis, z00 in the goniometerfixed frame ðx00 ; y00 ; z00 Þ; is at the angle of u with respect to B0 : The components of the NMR tensors along the direction of the field can again be obtained from the transformation Eq. (35) as Tz0 z0 ¼ c0 þ c1 cosw þ s1 sin w þ c2 cos 2w þ s2 sin 2w; ð77Þ where w is the turn angle of the goniometer and c0 ¼ 1 2 ðTx00 x00 þ Ty00 y00 Þsin2 u þ Tz00 z00 cos2 u; c1 ¼ TxS00 z00 sin 2u; c2 ¼ 1 2 s1 ¼ TyS00 z00 sin 2u; ðTx00 x00 2 Ty00 y00 Þsin2 u; ð78Þ s2 ¼ TxS00 y00 sin2 u: The different coordinate systems used are illustrated in Fig. 5. The spectrum is a periodic function of w; thus the five constants in Eq. (77) are available by using one known angle u: The spectral observables are the six (five if T ¼ D0 ) independent components of T1 þ TS in the goniometer-fixed frame. By using either several u values or different mounting directions of the sample to the goniometer, the full T1 þ TS is available. For discussions of the effect of TA ; for which a second-order treatment has to be adopted, see Refs. [23,24,25]. So far, there is no evidence from solid-state NMR for TA : Finally, to obtain the NMR observables from solid powder samples, Eq. (35) may be written with e ; t ¼ a; b; or c, i.e., the transformation is between the laboratory-fixed frame and PAS(T ), T þ TzS0 z0 ¼ sin2 u cos2 wTaa þ sin2 u sin2 wTbb þ cos2 uTcc ; ð79Þ J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 255 on a stationary powder sample or a sample that is spun about some axis relative to B0 : In principle, samples may be spun at a rate that is relatively fast or slow with respect to all internal NMR interactions. Most often the angle between the spinning axis and B0 is the so-called magic angle, but spectra may be acquired for spinning at any angle b: Each of the methods for characterizing J has its own advantages and drawbacks; however, as with all methods for determining DJ; the accuracy of the final results depends strongly on knowledge of the direct dipolar coupling constant, RDD ; Eq. (61). In the solidstate literature it is customary to define the experimentally measured dipolar coupling constant as Fig. 5. The coordinate systems used in the determination of the anisotropic properties of NMR tensors in single crystalline solids. u is the angle between the rotation axis z00 of the goniometer-fixed frame ðx00 ; y00 ; z00 Þ and the external magnetic field B0 : The laboratoryand molecule-fixed frames are as in Fig. 4. where u and w are the spherical coordinates that now specify the orientation of B0 in the ða; b; cÞ frame of one crystallite. The distribution of u and w; due to the differently oriented crystallites, gives rise to a powder pattern from which the principal values of the tensors can be identified. The Tii (i ¼ a; b; c) are thus the NMR observables of powder samples. The tensor T1 þ TS is completely specified in PAS(T ) by the principal values Taa ; Tbb and Tcc : Alternatively, T, Dc T; and hc (Eqs. (40), (57), and (58)) can be used. Practical details of the analysis of single-crystal NMR spectra may be found in Refs. [90 – 93]. 2.4.2. Solid-State NMR determination of J tensors As discussed above, solid-state NMR spectroscopy offers the potential of providing a wealth of information on anisotropic NMR interaction tensors. Solid-state NMR techniques for the characterization of J may be divided into three categories, based on the nature of the sample and whether it is examined as a stationary sample or a spinning sample. Stationary samples may be either a single crystal or a powder sample. NMR measurements on single crystals are performed as a function of the orientation of the single crystal in the applied B0 : Under special circumstances it may also be beneficial to spin a single crystal. NMR measurements on powder samples can be performed Reff ¼ RDD 2 DJ 3 ð80Þ for coincident dipolar and J tensors. In cases where Reff is very similar in magnitude to RDD ; corrections for motional averaging become very important. It is relatively straightforward to correct RDD for rovibrational effects for diatomic molecules in the gas phase, given the availability of high-quality experimental data. Similarly, more complicated corrections may be made for small molecules in liquid crystal media (vide supra ). However, for molecules in the solid state, how to carry out such corrections is not obvious. In total, there are a very limited number of accurate and precise measurements of the complete J tensor available from solid-state NMR due primarily to the large number of parameters that are involved in the analysis. To obtain reliable experimental J tensors, the molecule, spin system, and type of experiment to be carried out must be very carefully chosen such that the number of assumptions that must be made is minimized. The following discussion of the available data delineates some of the assumptions that are commonly made in the analysis of NMR spectra for the extraction of DJ; and provides an evaluation of the reliability of several of the reported results. The results to be discussed will generally be restricted to the period 1990– 2001, as literature on the experimental measurement of J by solid-state NMR methods has been covered in the review of Power and Wasylishen [2] (see also Refs. [23,94]). It is important to emphasize that the values of DJ 256 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 that we report in the solid-state context are defined as Dc J ¼ Jk 2 J’ ; or in the more general case, Dc J ¼ Jcc 2 ðJaa þ Jbb Þ=2: Some literature uses a so-called reduced anisotropy dJ ¼ ð2=3ÞDc J: 2.4.3. Results from single crystal studies The limited data on J available from single-crystal NMR experiments are summarized in Table 6. While single-crystal NMR experiments offer the potential to provide some insight into the orientation and asymmetry of J, in all reports to date these properties have been dictated by symmetry and J has been found to be axially symmetric within experimental error. One of the most frequently cited references on the subject of anisotropic J is the 31P single-crystal NMR experiment on tetraethyldiphosphine disulfide (TEPS) of Tutunjian and Waugh [101]. This study, and their subsequent one on the structurally similar compound tetrabutyldiphosphine disulfide [102], reported particularly large values for DJPP ; e.g. 2.2 or 8.8 kHz for TEPS. A reinvestigation of the same coupling in TEPS via single-crystal NMR by Eichele et al., in 1995 [92] revealed that the inadvertent neglect of a factor of 3/2 in the analysis of Tutunjian and Waugh was the likely cause of the apparently substantial values of DJ: The value determined by Eichele et al., DJ ¼ 462 Hz, represents an upper limit and is more in line with known 1 JPP : A 31P NMR study of a single crystal of the related compound, tetramethyldiphosphine disulfide (TMPS) resulted in an upper limit of 450 Hz for DJPP [96]. In any case where Reff and RDD are of the same sign and Reff is less than RDD ; the resultant value of DJ must be an upper limit since the effects of motional averaging are not known accurately. The most convincing evidence for DJ comes from two single-crystal NMR investigations carried out by Lumsden et al., on 1:1 and 1:2 mercury phosphine complexes [99,100]. The large values of DJ199 Hg31 P ; on the order of 4 –5 kHz, provide conclusive evidence for non-Fermi contact coupling mechanisms. It is possible that the anisotropy arises solely from the anisotropic SD/FC term, and J is nevertheless dominated by the FC term. It seems very unlikely, however that while the SD/FC mechanism would be active, the SD and/or PSO mechanisms would not contribute in a substantial way to both J and DJ: Table 6 Indirect nuclear spin–spin coupling tensors determined from single-crystal NMR spectroscopy Coupling and molecule 207 Pb19F coupling PbF2 (cubic) Results and comments Reference DcJ ¼ 8130 ^ 300 Hz (preferred) or 210 ^ 300 Hz. J ¼ ^(2150 ^ 50) Hz (negative sign preferred). [95] DcJ ¼ 462 Hz (preferred) or 10362 Hz. J is axially symmetric within experimental error. DcJ # 450 Hz [92] DcJ ¼ 1220 ^ 75 or 2600 ^ 75 Hz J ¼ ^(225 ^ 10) Hz [97] J ¼ ^(170 ^ 40) Hz B(pseudodipolar) ¼ 2300 ^ 70 or 2990 ^ 70 Hz [98] DcJ ¼ 5404 ^ 150 Hz (site 1) and 5385 ^ 150 Hz (site 2). J ¼ 8199 ^ 25 Hz J is axially symmetric within experimental error, with the unique component coincident with the unique component of D0 DcJ ¼ 4000 ^ 500 Hz, J ¼ 5550 Hz J is axially symmetric within experimental error, with the unique component coincident with the unique component of D0 . [99] 31 31 P P coupling Tetraethyldiphosphine disulfide Tetramethyldiphosphine disulfide 113,115 [96] 31 In P coupling InP 115 In31P coupling InP 199 Hg31P coupling HgPCy3(NO3)2 (Cy ¼ cyclohexyl) Hg(PPh3)2(NO3)2 [100] J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 Sears et al., have analyzed 19F NMR spectra of a single crystal of cubic lead fluoride [95]. The crystal was first oriented such that [100] axis was along the direction of B0 ; for which both the 207Pb19F direct dipolar and anisotropic J coupling are zero. By examining spectra acquired for other crystal orientations, JPbF was characterized, with preferred values of J ¼ 22:15 ^ 0:05 kHz and DJ ¼ 8:13 ^ 0:3 kHz. The 31P NMR spectrum of a stationary single crystal of InP was analyzed by accounting for contributions to the second moment, M2 ; from direct 115 31 In P dipolar coupling, 31P31P homonuclear dipolar coupling, isotropic J coupling, and anisotropic J coupling [98]. Employing a similar strategy to what was used for PbF2, a single crystal of InP was first oriented such that [100] axis was along the direction of B0 and subsequent moment analysis of the 31P free induction decay as a function of crystal orientation provided values for the isotropic and anisotropic parts of J115 In31 P : Tomaselli et al., also carried out a study of J113;115 In31 P in undoped InP [97]. Triple-resonance NMR experiments were performed on both powder and single crystal samples, under both stationary and magic-angle spinning (MAS) conditions. One of the key experimental methods was to cross-polarize (CP) from 113In nuclei to 31P, and acquire the 31P spectra while decoupling 115 In nuclei. The value of J113;115 In31 P ; ^ð225 ^ 10Þ Hz, was determined using this triple-resonance technique on a powder sample, from the splitting induced by 31P coupling to the spin-9=2 113In nuclei. Insight into the magnitude of DJ113;115 In31 P was afforded by a 113In31P CP experiment on a single crystal spinning at the magic angle. As shown in Fig. 6, the signal build-up upon crosspolarization at the þ1 sideband matching condition was simulated to successfully yield the value of Reff ¼ ^ 230 ^ 25 Hz. In combination with the known In –P bond length, the two possible values of DJ113;115 In31 P were found to be þ1220 ^ 75 Hz and þ2600 ^ 75 Hz. Of the numerous studies of the JInP in InP [98,103,104], the study of Tomaselli et al., provides the most convincing results. Furthermore, it is one of the most reliable determinations of DJ for the case where lReff l is not larger than lRDD l: The problems that arise in the interpretation of spin – spin coupling tensors in situations where the D0 257 Fig. 6. (a) Total intensity of the 161.196 MHz 31P NMR signal under conditions of cross-polarization from 113In to 31P in an indium phosphide single crystal spinning at the magic angle at a rate of 10 kHz, as a function of contact time tCP : The circles represent data obtained under J CP conditions and the crosses represent data obtained for the þ 1 sideband matching condition. The fit to these data corresponds to lJ113 In31 P l ¼ 225 Hz: (b) Same as part (a), with an expansion in the region tCP ¼ 0 – 1 ms: The solid line fit to the data points represented by crosses corresponds to an effective dipolar coupling constant of 230 Hz, with error limits of ^25 Hz denoted by the dashed lines. Reprinted with permission Ref. [97]. Copyright (1998) by the American Physical Society. and J tensors are non-coincident have been discussed for MAS powdered samples (vide infra). For experiments involving either powdered or single crystal samples, an important point is that only an effective dipolar coupling tensor may be measured. If the D0 and J tensors are not forced to be in the same PAS by symmetry, there is generally no unambiguous way to analyze the dipolar couplings to gain information on 258 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 the orientation, anisotropy, or asymmetry of the D0 and J tensors individually. In practice, the only way to confidently access information on J is to carefully choose a spin system of appropriate symmetry, for which the assumption of coincident D0 and J tensors is well-founded. In such cases, the asymmetry of J will likely be close to zero as a result of the requisite symmetry. As discussed in Section 2.3.3 and below, evidence exists that the largest component of J does not always lie along the internuclear vector, where the largest component of D0 lies. 2.4.4. Results from studies of stationary powders The J tensors characterized by analysing NMR spectra of stationary powder samples are presented in Table 7. The major difficulty in determining precise values of DJ from stationary powder samples lies in the uncertainties associated with motional averaging of D0 : This is exemplified by the 1981 study of CH3F in an argon matrix by Zilm and Grant [116], where after correcting RDD 19 F13 C by about 3% for motional averaging, a value of DJ19 F13 C ¼ 1200 ^ 1200 Hz resulted, i.e. one cannot state with confidence that DJ is non-zero. It is clear that in order for reliable, precise, nonzero values of DJ to be determined, the values of Reff and RDD must differ significantly, beyond the point where the difference could be attributed to motional averaging. The absolute minimum difference in Reff and RDD in order for a credible value of DJ to be determined may be stated as approximately 10%, though convincing and careful arguments should be presented to convince the reader that such a difference is not due solely to motional averaging of Reff : One must also always bear in mind that the effect of DJ is reduced by a factor of three when it is manifested as part of Reff (see Eq. (80)). This fact further complicates the extraction of very accurate and precise values of DJ: Three papers have reported on unexpectedly large values of DJPP for 1,2-bis(2,4,6-tri-t-butylphenyl) diphosphene and tetraphenyldiphosphine and DJPC for 2-(2,4,6-tri-t-butylphenyl)-1,1-bis(trimethylsilyl) phosphaethene and 2-(2,4,6-tri-t-butylphenyl)phosphaethyne determined from analyses of stationary powder samples [105,107,109]. For all of these compounds except 1,2-bis(2,4,6-tri-t-butylphenyl) diphosphene, it is likely that the relatively small discrepancies in the measured Reff and the RDD calculated from known bond lengths are due to motional averaging rather than due to substantial values of DJ: For example, for the 31P and 13C nuclei involved in the double bond in 2-(2,4,6-tri-t-butylphenyl)-1,1-bis(trimethylsilyl)phosphaethene, an assumption of negligible anisotropy in J results in an NMR-derived bond length of 1.72 Å, which is only 3% longer than the X-ray value, 1.665 Å [105]. Motional averaging is known to account for differences of approximately 1 – 4% between NMR-derived bond lengths and those determined from X-ray crystallography [117 – 120]. The large values of DJPP and DJPC reported in Refs. [105,109] are clearly suspect. In the case of 1,2-bis(2,4,6-tri-t-butylphenyl) diphosphene [107], however, the value of Reff ; 2800 ^ 100 Hz, was found to be greater than RDD ; 2345 Hz. The difference of 455 Hz cannot be attributed to motional averaging since such averaging serves to reduce Reff to a value less than RDD : The preferred value of DJ; 2 1380 Hz, is certainly unexpectedly large for a phosphorus spin pair. It is important to note that this system was treated as an A2 spin system (i.e. where the 31P are magnetically equivalent) while subsequent studies indicated that it is in fact an AB spin system [108]. It is conceivable that the assumption of an A2 system could introduce considerable error into the value of Reff determined from the 2D spin-echo experiment, since J ¼ 580 Hz, would contribute to the observed splitting in the F1 dimension. The actual value of Reff could be as low as 2220 Hz, which is 5% less than RDD : Several reliable values of DJ199 Hg31 P are known from 31P NMR spectroscopy of stationary powder samples [113 –115]. For the series [HgPR3(NO3)2]2 (see Table 7), J199 Hg31 P ranges from 8008 to 10566 Hz, and DJ199 Hg31 P is on the order of 5 kHz, with errors of less than 10%. In these systems, DJ=3 (< 1670 Hz) makes a larger contribution to Reff than does RDD (< 645 Hz). Since the magnitude of Reff is larger than RDD for all of these mercury– phosphorus compounds, one can be confident that the source of the difference is due to DJ: Presented in Fig. 7 is an example of the rotation plots generated in the 31P NMR analysis of a single crystal of Hg(PPh3)2(NO3)2. J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 259 Table 7 Indirect nuclear spin–spin coupling tensors determined from NMR spectroscopy of stationary powder samples (results in Hz) Coupling and molecule 31 13 P C coupling 2,4,6-t-Bu3C6H2PyC(SiMe3)2 2,4,6-t-Bu3C6H2PyC(SiMe3)2 2,4,6-t-Bu3C6H2CP 199 Hg13C coupling K2Hg(CN)4 31 31 P P coupling 1,2-Bis(2,4,6-tri-t-butylphenyl)diphosphene Tetraphenyldiphosphine 115 31 In P coupling Br3InP(4-(CH3O)C6H4)3 195 31 Pt P coupling trans-Pt(PPh3)2Cl2 trans-Pt(PCy3)2Cl2 trans-Pt(PEt3)2Cl2 cis-Pt(PPh3)2Cl2 cis-Pt(Et3)2Cl2 Cl2Pt(PPh2CH2PPh2) Cl2Pt(PPh2CH2CH2PPh2) Cl2Pt(PPh2CH2CH2CH2PPh2) 199 Hg31P coupling (EtO)2P(O)Hg(OOCCH3) (EtO)2P(O)HgI (EtO)2P(O)Hg(SCN) [HgPR3(NO3)2]2 R ¼ phenyl m-tolyl p-tolyl mesityl p-MeOPh cyclohexyl [HgP(o-tolyl)3(NO3)2]2 Results and comments Reference DcJ ¼ 777 or 15 117, J ¼ 91. See text for discussion. DcJ ¼ 1008 or 10 638, J ¼ 90. Coupling to aryl 13C. See text for discussion. DcJ ¼ 1233 or 19 821, J ¼ 59. See text for discussion. [105] [105] DcJ ¼ 950 ^ 60, J ¼ 1540 ^ 2 [106] DcJ ¼ 21380 or 15 420 (J ¼ 580 ^ 20 [108]). See text for discussion of possible errors in the spectral analysis for this compound. DcJ ¼ 300, J ¼ 2200 ^ 100. See text for discussion. [107] DcJ ¼ 1178 ^ 150 (preferred) or 22558 ^ 150 J ¼ 1109 ^ 9 [110] DcJ ¼ 1865 ^ 250, J ¼ 2624 ^ 25 DcJ ¼ 1602 ^ 250, J ¼ 2420 ^ 25 DcJ ¼ 1536 ^ 250, J ¼ 2392 ^ 25 DcJ ¼ 2184 ^ 600 or 3282 ^ 600 (site 1) DcJ ¼ 1968 ^ 600 or 3498 ^ 600 (site 2) DcJ ¼ 2037 ^ 600 or 3429 ^ 600 (site 3), J ¼ 3727 ^ 25 (site 1), 3910 ^ 25 (site 2), 3596 ^ 25 (site 3) DcJ ¼ 1356 ^ 600 or 4104 ^ 600 J ¼ 3448 ^ 25 DcJ ¼ 2130, J ¼ 3064 DcJ ¼ 1660, J ¼ 3591 DcJ ¼ 840, J ¼ 3354 [111] [111] [111] [111] DcJ ¼ 2700 ^ 250, J ¼ 13 324 ^ 15 DcJ ¼ 1500 ^ 250, J ¼ 12 623 ^ 15 DcJ ¼ 1600 ^ 250, J ¼ 12 119 ^ 15 [113] [113] [113] [114] DcJ ¼ 4545 ^ 500, J ¼ 9572 ^ 15 DcJ ¼ 5235 ^ 200, J ¼ 9165 ^ 15 DcJ ¼ 5470 ^ 200, J ¼ 9144 ^ 15 DcJ ¼ 5560 ^ 500, J ¼ 10 468 ^ 15 (site 1) DcJ ¼ 5560 ^ 500, J ¼ 10 566 ^ 15 (site 2) DcJ ¼ 4765 ^ 250, J ¼ 9327 ^ 15 (site 1) DcJ ¼ 3740 ^ 375, J ¼ 9309 ^ 15 (site 2) DcJ ¼ 5525 ^ 200, J ¼ 8008 ^ 15 DcJ ¼ 5170 ^ 250, J ¼ 9660 In this case, the maximum possible splitting Dn 2 Jiso in the absence of an anisotropic J tensor is 1200 Hz. The experimental measurement of larger splittings provides unambiguous evidence for a significant DJ: A series of cis and trans platinum phosphines of the type Pt(PR3)2Cl2 has been investigated by 31P NMR [105] [109] [111] [112] [115] [111]. This study has provided several large values of DJ195 Pt31 P ; on the order of 1–2 kHz (Table 7). As for the HgP couplings, Reff differs substantially from RDD ; thus providing convincing evidence for the existence of large anisotropy in J. For example, in the case of transPt(PCy3)2Cl2, RDD is 822 Hz from the Pt–P bond length of 2.337 Å [111], while Reff is only 35% of this value. 260 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 Fig. 7. Variation in the 199Hg31P effective dipolar coupling obtained at 81.03 MHz for rotation of a single crystal of Hg(PPh3)2(NO3)2 about the crystal-holder X-axis. The signals due to two crystallographically distinct sites are represented by squares and circles. The horizontal line at 1200 Hz represents the maximum possible coupling due solely to direct dipolar coupling and in the absence of DJ. Since the experimental data points surpass this limit, DJ199 Hg31 P is non-zero for this compound. Reprinted with permission from Ref. [100]. Copyright (1995) American Chemical Society. Phosphorus-31 dipolar-chemical shift NMR experiments carried out on a series of metal compounds featuring cyclic phosphino ligands yielded three values of DJ195 Pt31 P [112]. For the series Cl2Pt(PPh2CH2PPh2), Cl2Pt(PPh2CH2CH2PPh2), Cl2Pt(PPh2CH2CH2CH2PPh2), DJ195 Pt31 P was found to decrease with increasing ring size, with a maximum value of 2130 Hz for Cl2Pt(PPh2CH2PPh2). The reported values are reliable in that the measured Reff 195 Pt31 P are significantly different from the predicted dipolar coupling constants, well beyond any reasonable differences due to motional averaging. For example, for Cl2Pt(PPh2CH2PPh2), Reff is approximately 25% of the value of RDD : Anisotropy in J199 Hg13 C in partially 13C-enriched K2Hg(CN)4 was determined from 199Hg spectra of stationary samples [106]. The symmetry of the tetracyanomercurate anion guarantees axial symmetry of J. The measured value of Reff was found to be 60% less than the value of RDD obtained from the Hg – C bond length of 2.152 Å [121]; such a large difference clearly cannot be accounted for by considering motional averaging effects alone. The value of DJ obtained, 950 ^ 60 Hz, is in good agreement with the value obtained for dimethylmercury in a LC solvent, 864 Hz [81,122]. Wasylishen and co-workers analysed the 31P NMR spectra of MAS and stationary samples of solid Br3InP(4-(CH3O)C6H4)3 (Fig. 8) and obtained values of J115 In31 P ¼ 1109 ^ 9 Hz and DJ ¼ 1178 ^ 150 Hz [110]. The presence of a 3-fold symmetry axis about the indium –phosphorus bond guarantees axial symmetry of D0 as well as J. Analysis of the spectrum of a stationary sample, shown in Fig. 8(b), provided a value for Reff of 230 ^ 50 Hz, which differs significantly from RDD ¼ þ623 Hz; determined from the bond length. The analysis also demonstrates the different effects of the direct and indirect spin – spin coupling interactions on each of the ten 31P subspectra arising because of the allowed indium spin states (Fig. 8(c)). 2.4.5. Results from spinning powder samples Values of DJ determined by analysing NMR spectra of spinning powdered samples are summarized in Table 8. In this section, we will discuss selected representative examples in detail. As with all methods for determining reliable values of DJ; experiments where powdered solid samples are spun at an angle with respect to B0 rely on a priori knowledge of RDD : Reliable estimates of RDD may be calculated from a relevant internuclear distance determined from a diffraction experiment. Many efforts to measure DJ have involved MAS; however, by spinning the sample about an axis off the magic angle, one can in principle access a value of Reff which is scaled by ð3 cos2 b 2 1Þ=2; where b is the angle between the rotation axis and B0 : Of course all anisotropic interactions will be scaled by rapid sample spinning. By obtaining high-quality NMR spectra at several angles b, the J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 261 Fig. 8. (a) 81.033 MHz 31P CPMAS NMR spectrum of solid Br3In-P(4-(CH3O)C6H4)3, depicting splittings due to the isotropic J115 In31 P coupling constant of 1109 ^ 9 Hz. (b) 31P CP NMR spectrum of a stationary powder sample of Br3In-P(4-(CH3O)C6H4)3, with splittings due to the effective dipolar coupling between 115In and 31P evident. (c) Simulation of the spectrum shown in part (b), from which a value of DJ115 In31 P ¼ 1178 ^ 150 Hz is extracted. Shown in part (d) are each of the ten 31P subspectra arising due to the allowed indium spin states. Reprinted with permission from Ref. [110]. Copyright (1994) American Chemical Society. spectrum of the stationary sample may be inferred by extrapolating the frequencies of shoulders and singularities to b ¼ 08: This could be advantageous if the NMR spectrum under consideration consists of several peaks which overlap as b ! 0: Regardless of the angle, it is desirable to spin very fast to reduce the number of spinning sidebands, and concentrate the signal intensity in the centreband. In cases where this is not possible, the intensity from the sidebands must be added to the centreband after acquisition, using spectral processing software. We may divide the methods for obtaining DJ from spinning samples into three general categories based on the types of nuclei involved: (i) a spin-1/2 nucleus coupled to a quadrupolar nucleus; (ii) a heteronuclear spin-1/2 pair; (iii) a homonuclear spin-1/2 pair. In general, the methods rely on the assumption that the spectrum may be successfully analysed as an isolated spin pair. In principle, solid-state NMR experiments on a spin system consisting of a pair of either homonuclear or heteronuclear quadrupolar nuclei will, in favourable cases, also yield information on DJ: For example, Wi and Frydman [140] have outlined the methodology for extracting DJ from multiple quantum MAS (MQMAS) [141,142] spectra by carrying out experiments involving 14N11B, B11B, and 55Mn55Mn spin pairs; however, non-zero values of DJ obtained using these methods have not been reported. To date, there have been some general assumptions regarding the orientation and symmetry properties of J that have been required to facilitate the determination of DJ in spinning samples. First, it is inevitably assumed that the asymmetry parameter of J tensor, h ¼ hc ; is zero. This is a valid assumption for geometrical arrangements of high symmetry, i.e. C3v or higher. For systems of lower symmetry, h may fortuitously be close to zero. However, one must be aware that in general there is no requirement for h to be zero. The second assumption, which is almost always made, is that J is coincident with D0 : For 1 J; this implies that the largest components of both J and D0 are along the bond axis. One of the major reasons for this assumption has been the lack of evidence to the contrary, although recent high-level ab initio and DFT calculations have provided this evidence for several systems, in particular many one-bond interhalogen couplings, where the largest component of J is perpendicular to the bond axis [27,143– 146]. In general, an incorrect assumption of coincident J and 11 262 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 Table 8 Indirect nuclear spin–spin coupling tensors determined from solid-state NMR of spinning samples Coupling and molecule 199 Results and comments Reference Dc 2 J199 Hg14 N ¼ 69 ^ 15 Hz; 2 J199 Hg14 N ¼ 20:6 ^ 2:0 Hz Dc 2 J199 Hg15 N ¼ 297 ^ 15 Hz; 2 J199 Hg15 N ¼ 29:0 ^ 2:0 Hz 199 Hg MAS NMR [106] Reff ¼ 24020 ^ 350 Hz, 1J ¼ ^1300 ^ 10 Hz Reff ¼ 23740 ^ 350 Hz, 1J ¼ ^1260 ^ 10 Hz Reff ¼ 23260 ^ 350 Hz, 1J ¼ ^1530 ^ 10 Hz 119 Sn MAS NMR. The precise extraction of DcJ from Reff is hampered by the lack of knowledge of the precise and accurate Sn– F bond lengths for these compounds D2c J ¼ 23150 and 22950 Hz (2 sites), J ¼ ^2275 Hz Reff ¼ 24000 ^ 400 Hz, DcJ ¼ 240 to 21320 Hz The large range of possible values for DcJ arises due to the lack of knowledge of the precise Sn–F bond lengths for this compound. [123] 14,15 Hg N coupling K2Hg(CN)4 119 Sn19F coupling Me3SnF (i-Bu)3SnF Ph3SnF Mes3SnF (n-Bu)3SnF Mn31P coupling Mn2(CO)9PPh3 [124] 55 PhCH2C(O)Mn(CO)4(PPh3) MesCH2C(O)Mn(CO)4(PPh3) PhCH2C(O)Mn(CO)4[P(C6H11)3] MesCH2C(O)Mn(CO)4[P(C6H11)3] PhCH2Mn(CO)4(PPh3) PhCH2Mn(CO)4[P(tolyl)3] PhCH2Mn(CO)4[P(PhF)3] 63 Cu31P coupling Several Triphenylphosphine-copper(I) complexes 63,65 Cu31P coupling [(PBz3)2Cu][CuBr2] [(PBz3)2Cu][PF6] 77 Se31P coupling (Me)3PSe (Ph)3PSe 113 Cd31P coupling Cd(NO3)2·2PMe2Ph 119 Sn35Cl coupling (Benzyl)3SnCl SnCl2(acac)2 Ph3SnCl DcJ ¼ 1027 or 5400 Hz, J ¼ ^297 Hz 31 P MAS NMR. The two possible values of DcJ arise because the sign of C55QMn is not known. hQ is assumed to be zero. DcJ ¼ 678 ^ 42 Hz, J ¼ 216 ^ 4 Hz DcJ ¼ 589 ^ 24 Hz, J ¼ 233 ^ 2 Hz DcJ ¼ 639 ^ 41 Hz, J ¼ 220 ^ 2 Hz DcJ ¼ 495 ^ 10 Hz, J ¼ 232 ^ 2 Hz DcJ ¼ 412 ^ 13 Hz, J ¼ 202 ^ 2 Hz DcJ ¼ 508 ^ 22 Hz, J ¼ 196 ^ 3 Hz DcJ ¼ 538 ^ 30 Hz, J ¼ 204 ^ 1 Hz 31 P MAS NMR. See text for discussion of these data, in particular the small reported errors. [125] DcJ 31 ¼ 600 Hz, J ¼ 900–2000 Hz P MAS NMR. In some cases, the sign of C63QCu and magnitude of hQ are estimated by the method of Vega [128]. [127] DcJ ¼ 750 ^ 50 Hz, J ¼ 1535 ^ 10 Hz DcJ ¼ 720 ^ 50 Hz, J ¼ 1550 ^ 10 Hz 31 P MAS NMR. Interaction tensor orientations are dictated by symmetry to be coincident. Similarly, the value of hQ for copper is zero by symmetry. [129] DcJ ¼ 640 ^ 260 Hz, J ¼ 2656 Hz from 31P MAS NMR, DcJ ¼ 550 ^ 140 Hz, J ¼ 2656 Hz from 77Se CPMAS NMR DcJ ¼ 590 ^ 150 Hz, J ¼ 2735 Hz from 31P CPMAS NMR [130] DcJ ¼ 22600 or 21101 Hz, J ¼ 2285 Hz From 113Cd-31P rotary resonance MAS spectra. Two possible values arise due to lack of knowledge concerning the absolute sign of Reff (Reff ¼ 0 ^ 250 Hz) [131] DcJ ¼ 2438 Hz with ‘substantial possible error’, J ¼ 227 Hz DcJ ¼ 2740 Hz, J ¼ ^276 Hz 119 Sn MAS NMR. C35QCl is assumed to be negative, and hQ is assumed to be zero. For SnCl2(acac)2, the magnitude of C35QCl is estimated. DcJ ¼ 2350 Hz. 119Sn MAS NMR. C35QCl is assumed to be negative, and hQ is assumed to be zero. [132] [126] [130] [133] J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 263 Table 8 (continued) Coupling and molecule 119 Sn55Mn coupling ( p-XC6H4)3SnMn(CO)5 X ¼ CH3 X ¼ H (a,b) X ¼ H (c) X ¼ H (d) X ¼ OCH3 X ¼ F (a) X ¼ F (b) X ¼ Cl X ¼ SCH3 X ¼ SO2CH3 Ph3SnMn(CO)5 (3 sites) [Mn(CO)5]2SnPh2 207 Pb55Mn coupling Ph3PbMn(CO)5 [Mn(CO)5]2PbPh2 119 Sn117Sn coupling (benzyl3Sn)2O 125 Te123Te coupling (Me4N)2Te2 Results and comments Reference [134] DcJ ¼ 622 ^ 42 Hz, J ¼ 132 ^ 2 Hz DcJ ¼ 354 ^ 4 Hz, J ¼ 135 ^ 2 Hz DcJ ¼ 515 ^ 33 Hz, J ¼ 141 ^ 3 Hz DcJ ¼ 352 ^ 4 Hz, J ¼ 141 ^ 2 Hz DcJ ¼ 398 ^ 13 Hz, J ¼ 149 ^ 1 Hz DcJ ¼ 401 ^ 13 Hz, J ¼ 165 ^ 2 Hz DcJ ¼ 566 ^ 16 Hz, J ¼ 151 ^ 2 Hz DcJ ¼ 305 ^ 12 Hz, J ¼ 160 ^ 2 Hz DcJ ¼ 501 ^ 21 Hz, J ¼ 170 ^ 1 Hz DcJ ¼ 584 ^ 15 Hz, J ¼ 250 ^ 3 Hz 119 Sn MAS NMR (X ¼ H: four molecules (a–d) in the unit cell; X ¼ F: two (a,b).) DcJ ¼ 353 ^ 8 Hz, J ¼ 135 ^ 1 Hz DcJ ¼ 345 ^ 8 Hz, J ¼ 142 ^ 2 Hz DcJ ¼ 507 ^ 55 Hz, J ¼ 141 ^ 1 Hz 119 Sn MAS NMR RDD ¼ 2560 ^ 4 Hz, Reff ¼ 223.6 ^ 0.6 Hz, J ¼ 139 ^ 1 Hz DcJ not reported. 119Sn MAS NMR Recrystallized from octane: Reff ¼ 47.4 ^ 12.4 Hz, J ¼ 250 ^ 4 Hz (site A), Reff ¼ 190 ^ 15 Hz, J ¼ 253 ^ 4 Hz (site B), Reff ¼ 267 ^ 22 Hz, J ¼ 275 ^ 8 Hz (site C), Reff ¼ 52.9 ^ 4.3 Hz, J ¼ 274 ^ 7 Hz (site D). DcJ not reported. 207Pb MAS NMR Recrystallized from benzene– octane: Reff ¼ 49.7 ^ 0.4 Hz, J ¼ 251 ^ 1 Hz (site A), Reff ¼ 65.8 ^ 2.3 Hz, J ¼ 247 ^ 1 Hz (site B), Reff ¼ 97.0 ^ 6.5 Hz, J ¼ 273 ^ 3 Hz (site C), Reff ¼ 52.7 ^ 2.2 Hz, J ¼ 274 ^ 1 Hz (site D) DcJ not reported. 207Pb MAS NMR RDD ¼ 293 ^ 3 Hz, Reff ¼ 5.6 ^ 0.4 Hz, J ¼ 228 ^ 1 Hz DcJ not reported. 207Pb MAS NMR [135] [136] [135] [136] D2c J ¼ 1263 ^ 525 Hz (preferred), or 429 ^ 525 Hz J ¼ ^950 Hz. 119Sn off-MAS NMR. The two tin atoms are crystallographically equivalent. Spinning off the magic angle (e.g. 568) reintroduces the effective dipolar coupling between 119Sn and 117Sn nuclei in the linear Sn –O –Sn fragment. The value of 1263 Hz is preferred if the supposition that 2J is positive holds. Note that a ‘reduced anisotropy’, dJ ¼ ð2=3ÞDc J; is reported in Ref. [137]. [137] DcJ ¼ 24270 ^ 800 Hz, J ¼ ^2960 ^ 5 Hz 123 Te MAS NMR. The two Te atoms are crystallographically equivalent. Information on the tellurium chemical shift tensor is extracted from the spinning sideband manifold of uncoupled 123Te nuclei, via the method of Herzfeld and Berger [139] [138] 2 All results rely on the assumption that J is axially symmetric and coincident with the direct dipolar tensor, unless otherwise stated. D0 tensors will have a large impact on the resulting value of DJ; however for many systems this is likely to be a valid assumption. Less general assumptions which are sometimes invoked to extract DJ will be discussed as appropriate in the sections below. Coupled spin-1/2 and quadrupolar nuclei. Most MAS studies of DJ have involved a spin-1/2 nucleus 264 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 coupled to a half-integer spin quadrupolar nucleus, S (Table 8). The spin angular momentum of the quadrupolar nucleus is not completely quantized by B0 ; rather, the largest component of the electric field gradient tensor competes with B0 to determine the final direction of quantization. As a result, MAS does not completely average Reff to zero for a spin pair involving a quadrupolar nucleus. This phenomenon has been extensively discussed in the literature; the perturbation approach of Olivieri is most useful [133, 147,148]. According to the perturbation approach, under conditions of rapid MAS, the spectrum of the spin-1/2 nucleus depends on d, the residual dipolar coupling constant, d¼ 3CQ SðS þ 1Þ 2 3m2 ½RDD ð3 cos2 bD 2 1 20nS Sð2S 2 1Þ þ hQ sin2 bD cos 2aD Þ 2 þ hQ sin2 bJ cos 2aJ Þ: 1 DJð3 cos2 bJ 2 1 3 ð81Þ This form of d accounts fully for the relative orientations of the electric field gradient (EFG), D0 ; and J tensors, and is valid in the regime where CQ , 4Sð2S 2 1ÞnS : Here, CQ is the quadrupole coupling constant of the quadrupolar nucleus S, nS is the Larmor frequency of spin S, hQ is the asymmetry parameter of the EFG tensor for S, bD and aD are the polar angles that describe the orientation of D0 in the PAS of the EFG tensor of the quadrupolar nucleus. Analogously, bJ and aJ are the corresponding angles which describe the orientation of J in PAS(EFG). When CQ is of the same order of magnitude as 4Sð2S 2 1ÞnS ; the perturbation treatment which yields Eq. (81) is no longer valid and complete diagonalization is required [149]. Clearly, there are several parameters which must be determined accurately in order to obtain convincing evidence for anisotropy in J. CQ and hQ may be measured independently via an NMR experiment on the quadrupolar nucleus. In many cases, however, the value of CQ is prohibitively large for NMR experiments at moderate field strengths and in these situations, nuclear quadrupole resonance (NQR) experiments may provide CQ : It is important to note, however, that the sign of CQ is not provided by direct observation of the quadrupolar nucleus. Knowledge of the relative signs of CQ and Reff are nevertheless extremely important when attempting to determine DJ from a MAS spectrum of the spin-1/2 nucleus. Furthermore, a relatively straightforward analysis via Eq. (81) is only feasible if some knowledge of the relative orientations of the three interaction tensors is available. In particular, the relative orientations of the EFG, D0 ; and J tensors may confidently be assigned only in cases where high symmetry dictates the orientations of these tensors. In situations of lower symmetry, e.g. where hQ is not zero, it is extremely difficult to make statements concerning the relative orientations of the three interaction tensors. For example, one cannot state with certainty that J is coincident with D0 : This requires that all parameters in Eq. (81) be considered independently, and thereby renders the already formidable task of extracting DJ even more daunting. In general, therefore, the most reliable values of DJ which are determined by observing the MAS NMR spectrum of a spin-1/2 nucleus coupled to a quadrupolar nucleus are those for which hQ is zero. Most of the values of DJ which have been extracted from analysis of the MAS spectrum of a spin-1/2 nucleus coupled to a quadrupolar nucleus involve 119Sn or 31P. For example, analysis of the 31P CPMAS spectra of two linear bis(tribenzylphosphine) cuprate(I) salts, wherein 31P is coupled to 63Cu and 65 Cu, both spin-3/2 nuclei, yielded precise values of þ 720 ^ 50 Hz and þ 750 ^ 50 Hz for DJ [129]. Shown in Fig. 9 is a demonstration of the sensitivity of the simulated 31P MAS NMR spectra of [(PBz3)2Cu][CuBr2] to the magnitude as well as the sign of DJ: In these cases, molecular symmetry of the P– Cu –P fragment [150] guarantees that the EFG and J tensors are axially symmetric, and also provides strong indications that the EFG, J, and D0 tensors will Q be coincident. The values of CCu were obtained via 31 NQR experiments. P CPMAS measurements were made at three applied field strengths, and the analysis involved a complete diagonalization of the Hamiltonian, rather than the perturbation approach discussed above. Christendat et al. have provided several values of DJ for 55Mn31P and 119Sn55Mn spin pairs in a series of compounds [126,134,135]. For several complexes of relatively low symmetry involving J207 Pb55 Mn and J119 Sn55 Mn ; it was recognized that in cases where J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 Fig. 9. Simulated 31P CPMAS NMR spectra of [(PBz3)2Cu][CuBr2] at 4.7 T, demonstrating the influence of the sign and magnitude of DJ65=63 Cu31 P ¼ 750 Hz: Reprinted with permission from Ref. [129]. Copyright (1998) Academic Press. symmetry does not dictate the relative orientations of the EFG, D0 ; and J tensors, quantitative information concerning DJ may not be extracted with confidence [135,136]. For other systems of higher symmetry, analyses of the high-quality 31P MAS NMR spectra for a series of tertiary phosphine substituted alkyl- and acyltetracarbonylmanganese(I) complexes relied on assumptions concerning the sign of C55QMn ; the value of hQ (approximately zero), and the relative orientations of the EFG, D0 ; and J tensors. These assumptions, which are based on crystal symmetry, calculations on model systems, symmetry arguments, and prior data on similar compounds are generally well founded. However, the C55QMn are not known independently. Given the large number of approximations which must be made for these systems, the very small errors on DJ which are reported seem optimistic. For example, DJ55 Mn31 P for MesCH2C(O)Mn(CO)4[P(C6H11)3] is reported as 495 ^ 10 Hz. Such a small error would imply that the error in Reff is only 3.3 Hz; this is 265 implausible considering that X-ray structures are not available for many of the complexes. An error in the manganese– phosphorus bond length of just 0.002 Å will lead to an uncertainty in RDD of more than 3.3 Hz. Additionally, the reported values of Reff were not corrected for vibrational averaging, a procedure which would be required to claim such a small error in DJ: Errors as small as 4 Hz were reported for C6H4SnMn(CO)5 [134]. The quadrupolar nucleus involved in J does not have to be of half-integer spin; the possibility of anisotropic coupling to 14N ðI ¼ 1Þ has been discussed by Olivieri and Hatfield [151]. However, for silicon nitride and associated compounds involving SiN spin pairs, no conclusive evidence for DJ was found. Anisotropic coupling between the spin-1/2 nucleus 199Hg and 14N was found by analysing the 199Hg MAS NMR spectra of K2Hg(CN)4, where D2 J199 Hg14 N was found to be þ 69 ^ 15 Hz [106]. The tetrahedral geometry of the tetracyanomercurate anion once again provides the symmetry necessary for a confident analysis of the spectra. This is the first D2 J to be determined for a solid. The scarce previous reports of two-bond coupling anisotropies are from LCNMR results (vide supra). Coupled heteronuclear spin-1/2 pairs of nuclei. Several values of DJ have been determined for heteronuclear spin-1/2 pairs of nuclei (Table 8). Many studies have employed slow MAS followed by either a Herzfeld –Berger [152] analysis or analysis by the method of moments [153] for the individual subspectra arising from the two possible spin states of the coupled spin-1/2 nucleus, as described by Harris et al. [123,124,154]. These methods of analysis are typically only valid for an isolated spin-1/2 nucleus; however if each subspectrum is treated independently, then ‘effective’ tensor components may be extracted and interpreted to provide a value of Reff ; since the relative intensities of each sideband for the two subspectra are dependent on Reff : An obvious prerequisite to this analysis is that J is large enough for the subspectra to be resolved. This method has been employed, for example, by Grossmann et al. to extract DJ77 Se31 P ¼ þ590 ^ 150 Hz from the 31P CPMAS NMR spectrum of triphenylphosphine selenide [130]. The value determined in this manner for trimethylphosphine selenide, DJ77 Se31 P ¼ þ640 ^ 260 Hz [130], is in excellent agreement with the 266 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 value determined in a nematic LC phase, þ 680 ^ 60 Hz [89,94]. Although the molecules possess only C1 symmetry in the solid state, their pseudo-C3 symmetry lends support to the assumption that J is nearly axially symmetric, and coincident with D0 : It should be cautioned that this slow MAS method is critically dependent on good signal-to-noise ratios for as many spinning sidebands as possible. Reports by Harris and co-workers on SnF couplings have provided several precise values of Reff ; however, the determination of reliable values of DJ has been hampered by a lack of precise Sn –F bond lengths from which RDD may be calculated [123,124]. Marichal and Sebald have presented an interesting determination of a substantial two-bond J coupling anisotropy D2 J119 Sn117 Sn ¼ þ1263 ^ 525 Hz for solid (benzyl3Sn)2O [137]. This is in contrast to the negligible value which was reported for (cyclohexyl3Sn)2S [155]. The linear arrangement of the Sn – O – Sn moiety in (benzyl 3Sn)2O allows for the usual assumptions concerning the symmetry and orientation of J. In this case, off-magic-angle spinning (e.g. 568) was employed to determine DJ for the two crystallographically equivalent tin atoms (Fig. 10). By spinning off the magic angle, Reff is reintroduced with a scaling factor of 20:0619: Since the chemical shift parameters may be determined beforehand using standard MAS, the only parameter to be optimized in simulating the off-angle spectra is Reff ; with the usual assumptions regarding the relative tensor orientations. It is critical in these types of experiments to have independent accurate knowledge of the spinning angle b. In the case of tin, this is facilitated by simulating the spectrum arising solely due to an isolated, uncoupled 119Sn nucleus. There are several potential methods for the determination of DJ for heteronuclear spin-1/2 pairs which remain almost entirely unexploited. The area of dipolar recoupling under MAS conditions has been the focus of intense research in solid-state NMR for several years [156,157]. Such experiments are available for both heteronuclear and homonuclear spin pairs. Ideally, dipolar recoupling experiments selectively reintroduce a direct dipolar interaction of interest while suppressing chemical shift interactions and unwanted additional dipolar interactions. The direct dipolar coupling is then interpreted to provide distance information. What is often ignored in these Fig. 10. (a) Off magic-angle (568) spinning 119Sn NMR spectrum of (benzyl3Sn)2O acquired at 4.7 T. Simulation (part (b)) of the scaled powder patterns denoted by asterisks allows for the extraction of an effective 119Sn, 117Sn dipolar coupling constant, from which a value of DJ119 Sn117 Sn may be determined. Reprinted with permission from Ref. [137]. Copyright (1998) Elsevier. experiments is that the measured quantity is Reff rather than RDD : Thus, the opportunity exists in all dipolar recoupling experiments to measure DJ: Of course, the same limitations apply to all experimental measurements of DJ; e.g. the need for an accurate independent measurement of RDD : To our knowledge, the only DJ which has been measured via a dipolar recoupling technique is DJ113 Cd31 P in Cd(NO3)2·2PMe2Ph, which was found to be 21200 ^ 700 Hz via rotary resonance recoupling [158]. It is clear that the potential exists to apply heteronuclear dipolar recoupling experiments in order to determine DJ for a wider variety of spin-1/2 pairs. Coupled homonuclear spin-1/2 pairs of nuclei. As with heteronuclear spin-1/2 pairs, there are several dipolar recoupling experiments which may be applied to homonuclear spin-1/2 pairs [156,159,160]. Although the experiments themselves involve different pulse sequences depending on whether the spin J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 system is heteronuclear or homonuclear, the end result is the same: Reff between the spins is reintroduced under conditions of MAS. Although no precise experimental determinations of non-zero values of DJ exist from MAS experiments for homonuclear spin pairs, the potential certainly exists, especially for relatively heavy spin-1/2 nuclei such as 119Sn. Dusold et al. have investigated the potential of using an iterative fitting technique for the extraction of DJ from MAS NMR spectra of homonuclear spin1/2 pairs [131,161]. For example, the possibility of anisotropic J between the phosphorus nuclei in Cd(NO3)2·2PPh3 was addressed by carrying out a full iterative optimization of all parameters involved. The conclusion reached is that there is no significant anisotropy when coincident D0 and J tensors are assumed; however, if non-coincident tensors are considered, no definite conclusions may be made about the magnitude of the anisotropy in J. This work demonstrates the importance of considering the relative orientations of D0 and J. 2.5. High-resolution molecular beam spectroscopy A less well-recognized source of J is the hyperfine structure in molecular beam and high-resolution microwave spectra [143,162 – 167]. The case of diatomic molecules is particularly simple, and in favourable cases both the isotropic and anisotropic portions of J may be extracted with a high degree of precision [143,168]. The high-resolution spectra of diatomics also provide information on the quadrupolar, spin – rotation, and s tensors [163,165,169]. Since these experiments are performed on gaseous samples at very low pressures, intermolecular effects on the interaction tensors are negligible. This has the advantage of providing very accurate experimental J (or K) tensors which may be used to establish the reliability of first-principles calculations. In addition, due to the simplicity of the molecules which are studied, the hyperfine data allow for particularly meaningful interpretations of J and K in terms of the local electronic structure. Molecular beam spectroscopy allows for the investigation of rotational transitions (e.g. J ¼ 1 ˆ 0), and more importantly, the investigation of so-called hyperfine structure within a single rotational state (Fig. 11). 267 There are several versions of ‘molecular beam’ spectroscopy, including molecular beam electric resonance (MBER), molecular beam magnetic resonance (MBMR), molecular beam maser spectroscopy, molecular beam absorption spectroscopy, and molecular beam deflection measurements. Much of the reliable information on J has come from MBER and MBMR, and in recent years almost all of the highly precise data have come from MBER measurements in the laboratory of Cederberg [171]. There, a spectrometer built by Norman Ramsey in 1970 is still used to provide extremely high-quality data on diatomics. Molecular beam spectroscopy differs from most other forms of spectroscopy in that a beam of molecules is detected rather than electromagnetic radiation of some type. The MBER spectrometer is composed of five main parts: the beam source, the A state selector, the C transition region, the B state selector, and the detector. MBER relies on the second-order Stark effect to carry out rotational state selection in the A and B regions, by applying an inhomogeneous electric field to alter the trajectories of molecules with differing rotational angular momentum quantum numbers M. Only molecules with permanent dipole moments will experience the Stark effect and therefore only these molecules are suitable for MBER spectroscopy. In practice, a very weak electric field is applied such that the results may be extrapolated to zero field. Additionally, at least one of the nuclei must be quadrupolar in order to split the energy levels such that information on J is accessible. MBMR may be used to investigate the magnetic hyperfine structure in molecules which lack a permanent dipole moment, such as homonuclear diatomics, e.g. iodine [172]. One of the key features of molecular beam spectroscopy is the very high resolution and narrow lines which may be obtained; positions of the lines in the spectra may be measured with uncertainties of less than 1 Hz [173]. The hyperfine Hamiltonian in the absence of external fields for a diatomic molecule such as potassium monofluoride, 39K19F, may be written as h21 Hhf ¼ VK : QK þ c1 IK ·J þ c2 IF ·J þ c3 IK ·dT ·IF þ c4 IK ·IF ð82Þ If we are interested in the J ¼ 1 rotational level, higherorder terms such as the nuclear magnetic octupole and nuclear electric hexadecapole interaction are zero. The 268 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 Fig. 11. Energy level diagram for a molecular beam electric resonance experiment on a diatomic molecule composed of a spin- 12 and a spin- 32 nucleus such as 87Rb19F. Adapted from the diagram for 87Rb19F shown in Ref. [170]. Second-order quadrupolar effects were evaluated numerically and are shown for the F1 ¼ 3=2 and 5/2 levels. first term describes the interaction of the potassium nuclear electric quadrupole moment with the EFG tensor and the next two terms describe the K and F spin– rotation interactions. The last two terms describe the sum of the direct and indirect nuclear spin– spin coupling interactions between K and F, with dT denoting here the traceless part of the interaction tensor. Shown in Fig. 11 is an energy-level diagram for a 1 S diatomic molecule composed of a spin-3/2 nucleus and a spin-1/2 nucleus, such as 39K19F or 87Rb19F [170]. The levels all exist within a single rotational–vibrational state, in this case n ¼ 0; J ¼ 1: In this diagram, the pure J ¼ 1 state is first perturbed by the quadrupolar and spin–rotation interaction associated with the spin-3/2 nucleus. When the spin-1/2 nucleus is considered, its spin – rotation constant as well as the spin – spin coupling tensors cause further splittings of the energy levels. Measurement of the allowed transitions provides enough data to solve for quadrupole coupling, c1 ; c2 ; c3 ; and c4 : The quantum numbers F1 and F are defined as F1 ¼ I1 þ J and F ¼ F1 þ I2 ; where I1 is the angular momentum quantum number of the spin-3/2 nucleus and I2 is the angular momentum quantum number of the spin-1/2 nucleus. The selection rules for the electric dipole transitions are DF1 ¼ 0; ^1; ^2; DF ¼ 0; ^1; ^2; and DMF ¼ 0; ^1: If one notes the form of the Hamiltonian, the parameter c4 is readily identified with the isotropic J. The parameter c3 provides the tensor part (D0 and J) of the total spin–spin coupling tensor [143]: DJMN ; ð83Þ c3 ¼ RDD 2 3 where RDD is the direct dipolar coupling constant. c3 may thus be described as an effective dipolar coupling constant, Reff : The NMR interaction tensors must be axially symmetric for a 1 S diatomic molecule, and therefore the complete J is entirely described by J and DJ (or equivalently, by c4 and c3 ). The parameter c3 is frequently written as a sum of the direct and indirect contributions, i.e. c3 ¼ c3 (direct) þ c3 (indirect) or c03 þ c003 [165,174]. The relationship between c3 and J familiar to NMR spectroscopists was originally J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 described by Ramsey [165]. It should be noted that other symbols are also used to represent c3 and c4 : Reff is equal to c3 ; which is also sometimes denoted dT [175], or S [176–178], or simply d [179,180]. The c4 ¼ J; is also denoted d [165,179,180] (which is also used for chemical shifts in the NMR literature!) and dS [181]. The reader should be aware that in some of the older literature, c3 and c4 are defined differently, i.e. with extra factors.7 The relationship between c3 and the vaguely defined ‘tensor part of the electron-coupled spin–spin interaction’ has been alluded to in the literature [74,163, 182], but only very rarely has it been explicitly stated that c3 is equal to RDD 2 DJ=3: The relationship is not widely appreciated by NMR spectroscopists. Certainly some of the molecular beam literature extracts c3 (indirect) from the full value of c3 and the rovibrationally averaged value of c3 (direct); however interpretation of these data in the language of NMR J has been lacking. While it is certainly true in many NMR experiments and some molecular beam experiments that the contribution from the anisotropic J is swamped by the contribution of D0 ; there are just as certainly many cases where valuable information concerning J may be extracted by a careful analysis of the data. English and Zorn [174] provided a summary of the available values of c3 and c4 for alkali fluorides in 1967, and also extracted c3 (indirect). Had there been interest in converting c3 (indirect) to DJ; much of the 1967 data on c3 had such large relative errors that in many cases it would have been difficult to determine the sign of DJ; let alone the precise value. As with other experimental methods for determining accurate and precise values of DJ; D must be known with high precision. Bond lengths in diatomics are frequently determined to more than five significant figures, thus providing very precise direct dipolar coupling constants, RDD : Over the past few decades since the summary of English and Zorn [174], very precise values of c3 and c4 ; e.g. to five significant figures, have become available for a wide variety of diatomics (Table 9). Müller and Gerry separated the direct and indirect portions of c3 for five monofluorides; however the values of c3 (indirect) were not discussed in terms of DK [178]. Bryce and Wasylishen have extracted 7 See, for example, footnote b in Table III of Refs. [176,177]. 269 several reliable values of DK from the very highresolution hyperfine data which are now available [143] (Table 10). In combination with high-level ab initio calculations, these high-quality experimental data have provided some insight into periodic trends in K. For example, periodic trends are clearly evident for both the isotropic and anisotropic portions of K for the thallium halides. The reduced K coupling is negative and increases in magnitude as the atomic number of the halogen increases. The reduced anisotropic coupling is positive, and increases in magnitude as the atomic number of the halogen increases. The thallium halides are also interesting in that the ratio of DJ to D is very large; in TlI this ratio is nearly 1500! This clearly demonstrates the potential hazards of neglecting the 2DJ=3 term in the interpretation of measured Reff in NMR experiments. It is also interesting to note that in many cases shown in Table 10, the magnitude of DK is greater than K. Ref. [143] provides further investigations of the periodic trends in K in diatomic molecules. Two recent studies of cesium fluoride [193] and lithium iodide [186], for example, provide striking demonstrations of the sensitivity of the molecular beam method to the value of J. The values of c3 and c4 are clearly sensitive to the vibrational state of the molecule (Table 9). Thus, molecular beam experiments on diatomics provide a unique opportunity to learn about the rotational – vibrational dependence of J [200]. We also emphasize that the sign of c3 and c4 are determined in molecular beam and microwave experiments. To our knowledge, almost no information on J (c3 and c4 parameters) has been extracted for polyatomic molecules. Even for some 1 S diatomics it is difficult to precisely determine c3 and c4 ; e.g., GaF [201] and see Table 9, simply due to poor resolution, signal-to-noise, or relatively small values of these parameters. The Hamiltonian described by Dyke and Muenter for polyatomic molecules neglects the effects of J [163]. One polyatomic molecule for which c3 has been measured is methane [202,203]. However, the two-bond value, c3(H,H) ¼ 20.9 ^ 0.3 kHz, does not provide any useful information on anisotropy in J, since c3 may be accounted for fully in this 270 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 Table 9 Magnetic hyperfine spin–spin coupling tensors available from molecular beam and microwave experiments Molecule c3 (kHz) c4 ¼ J (kHz) Year Ref. to original hyperfine literature H19F 7 LiH 7 LiH 7 LiH 7 LiD 7 19 Li F 143.45(3) 11.346(7)(n ¼ 0, J ¼ 1) 11.03(8)(n ¼ 1, J ¼ 1) 11.329(12)(n ¼ 0, J ¼ 2) 1.7430(70) 11.4292(42) 2 0.2122(86)ðn þ ð1=2ÞÞ þ 0.0039(29)ðn þ ð1=2ÞÞ2 1.1789(78) 1.0710(61) 0.62834(68) 2 0.0050(11)ðn þ ð1=2ÞÞ 3.85(25) 3.7(2) 0.4269(15) 2 0.0042(2)ðn þ ð1=2ÞÞ 2 0.00021(9)[J(J þ 1)] 0.3922(16) 2 0.0029(5)ðn þ ð1=2ÞÞ þ 0.00014(11)[J(J þ 1)] 0.4749(27) 2 0.0065(10)ðn þ ð1=2ÞÞ 0.2606(15) 2 0.0035(5)ðn þ ð1=2ÞÞ 0.035(12) 2.45(37) 0.79681 ^ 0.00036 2 (0.00642 ^ 0.00027)ðn þ ð1=2ÞÞ 0.92(12)(n ¼ 0, J ¼ 1) 0.92713(53) 2 0.00917(93)ðn þ ð1=2ÞÞ þ 0.00097(29)ðn þ ð1=2ÞÞ2 0.028(2) 2 0.000(3)ðn þ ð1=2ÞÞ þ 0.0002(7)ðn þ ð1=2ÞÞ2 Not determined 0.3026(50) 2.859(9) 7.15a (n ¼ 0, J ¼ 1) 6.93(26)(n ¼ 1, J ¼ 1) 5.202(146)a 2.62(3)(n ¼ 0, J ¼ 1) 2.62(1)(n ¼ 0, J ¼ 2) 3.50(15) 20.09(10)(n ¼ 0, J ¼ 2) 20.13(10)(n ¼ 0, J ¼ 2) 20.56(50) 20.13(15) 21.55(8)(n ¼ 0, J ¼ 2)b 21.65(5)(n ¼ 0, J ¼ 2) 21.68(8)(n ¼ 0, J ¼ 2) 21.77(5)(n ¼ 0, J ¼ 2) 22.59(2)(n ¼ 0, J ¼ 3) 22.48(10)(n ¼ 0, J ¼ 3) 1.58(5)(n ¼ 0, J ¼ 13) 1.528(18)(n ¼ 0, J ¼ 13) 1.519(18)(n ¼ 0, J ¼ 15) 0.50(2) 0.135(10)(n ¼ 0, J ¼ 1) 0.17(4)(n ¼ 1, J ¼ 1) 0.160(5) (n ¼ 0, J ¼ 2) 0.005(10) 0.1744(21) 2 0.0042(21)ðn þ ð1=2ÞÞ 1987 1975 1975 1975 1975 1992 [176,177] [183] [183] [183] [183] [184] 0.0711(89) 0.0604(70) 0.06223(36) þ 0.00041(26)ðn þ ð1=2ÞÞ 0.150(250) 20.2(2) 0.0859(18) 1972 1972 1999 1964 1965 1987 [185] [185] [186] [187] [188] [189] 0.078(3) 1987 [189] 0.0578(13) 0.0317(7) 0.009(6) 0.86(40) 0.23766 ^ 0.00032 2 (0.00245 ^ 0.00022)ðn þ ð1=2ÞÞ 0.61(10)(n ¼ 0, J ¼ 1) 0.62745(30) 2 0.00903(22)ðn þ ð1=2ÞÞ 1988 1988 1984 1972 2002 [173] [173] [190] [191] [192] 1967 1999 [174] [193] 0.060(4) þ 0.002(5)ðn þ ð1=2ÞÞ þ 0.0006(12)ðn þ ð1=2ÞÞ2 0.306(30) 1.0667(65) 0.840(6) 4.86(28)(n ¼ 0, J ¼ 1) 6.47(84) (n ¼ 1, J ¼ 1) 5.73(105) 22.15(3)(n ¼ 0, J ¼ 1) 22.11(1)(n ¼ 0, J ¼ 2) 213.3(7) 21.52(10) 21.54(10) 21.11(50) 21.28(15) 26.39(8)(n ¼ 0, J ¼ 2) 26.35(5)(n ¼ 0, J ¼ 2) 26.91(8)(n ¼ 0, J ¼ 2) 26.84(5)(n ¼ 0, J ¼ 2) 26.57(1)(n ¼ 0, J ¼ 3) 26.67(5)(n ¼ 0, J ¼ 3) 3.66(3)(n ¼ 0, J ¼ 13) 3.708(22)(n ¼ 0, J ¼ 13) 3.701(23)(n ¼ 0, J ¼ 15) 1977 [194] 1976 1985 1977 1995 1995 1995 1972 1972 1964 1969 1969 1969 1969 1970 1970 1970 1970 1970 1970 1980 1999 1999 [181] [180] [195] [178] [178] [178] [196] [196] [182] [197] [197] [197] [197] [198] [198] [198] [198] [199] [199] [172] [179] [179] 7 Li81Br Li79Br 7 127 Li I 23 Na19F 23 Na19F 23 Na81Br 7 23 Na79Br 39 K19F K19F 39 35 K Cl 87 Rb19F 85 Rb19F 41 133 Cs19F Cs19F 133 133 Cs35Cl 23 Na39K Na2 35 19 Cl F 79 19 Br F 79 19 Br F 127 19 I F 115 19 In F 115 19 In F 205 19 Tl F 203 35 Tl Cl 205 35 Tl Cl 203 37 Tl Cl 205 37 Tl Cl 203 79 Tl Br 205 79 Tl Br 203 81 Tl Br 205 81 Tl Br 203 127 Tl I 205 127 Tl I 127 I2 127 I2 127 I2 23 a b Values for which no rovibrational dependence is given are for the n ¼ 0, J ¼ 1 state unless otherwise indicated. The value for c3 reported in Ref. [178] is of opposite sign due to use of a different convention. Extensive data are available for TlBr, e.g. for five vibrational states and two rotational states. J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 Table 10 Summary of precise indirect nuclear spin–spin coupling tensor data available from hyperfine structure in high-resolution rotational spectra of diatomic molecules [143] Molecule K/1020 NA22 m23 DK/1020 NA22 m23 Ref. to original hyperfine data LiH LiF LiBr LiI NaBr KF CsF CsCl Na2 ClF BrF IF InF TlF TlCl TlBr TlI I2a 2.89 3.92 5.15 6.65 9.76 10.9 41.8 39.4 127 75.7 171 252 286.4 2202 2224 2361 2474 763 21.22 3.94 18.1 18.4 43.9 23.8 46.5 67.9 25.71 281.8 2206 2257 89.9 173 262 448 664 2785 [183] [184] [185] [186] [189] [173] [193] [194] [180] [195] [178] [178] [196] [182] [197] [198] [199] [179] All results are for n ¼ 0, J ¼ 1 rotational–vibrational state except TlBr: n ¼ 0, J ¼ 2; TlI: n ¼ 0, J ¼ 3; and I2: n ¼ 0, J ¼ 13. a Data are from stimulated resonant Raman spectra. case by the DHH interaction, zero-point vibration, and centrifugal stretching effects [203]. Finally, we note that there has been a recent report of the c3 and c4 parameters for molecular iodine using stimulated resonant Raman spectroscopy. This is not a rotational spectroscopic technique; however, the results complement those discussed in this section. Wallerand et al. [179] have improved the precision in the values reported by Yokozeki and Muenter using MBMR [172] and also detected a slight rotational dependence of the parameters. The Raman data provide the following values for I2 : K ¼ ð763 ^ 5Þ £ 1020 NA22 m23 and DK ¼ ð2785 ^ 11Þ NA22 m23 for the n ¼ 0; J ¼ 13 state, and K ¼ ð761 ^ 5Þ £ 1020 NA22 m23 and DK ¼ ð2779 ^ 11Þ £ 1020 NA22 m23 for the n ¼ 0; J ¼ 15 state. Clearly, it would be of interest to further investigate the rotational dependence of this coupling tensor for a larger range of rotational states. Judging by the quality of much of the data discussed in this section, it is clear that molecular 271 beam experiments provide information on J tensors that is extremely valuable to NMR spectroscopists and theoreticians. The accuracy and precision to which the molecular beam data are determined, especially the recent results from Cederberg et al., provide unique opportunities to study and interpret the rotational –vibrational dependence of J. 2.6. NMR relaxation Nuclear spin – spin and spin –lattice relaxation may in principle occur via an anisotropic spin – spin coupling ðDJÞ mechanism [26,204]. Equations describing this phenomenon were outlined by Blicharski in 1972 [204]; although, to date relaxation by the DJ mechanism has not been identified experimentally. As with all experimental methods for the determination of DJ; complications arise due to the identical transformation properties of the direct dipolar and anisotropic J coupling Hamiltonians. In a system where relaxation of a particular nucleus may be predicted to arise solely from spin – spin coupling interactions (direct and indirect), the known geometry allows for the calculation of the spin relaxation rate based solely on the direct dipolar coupling relaxation mechanism. In such an ideal system, deviations from the predicted rate would be attributed to contributions from DJ; and as such, relaxation measurements on carefully chosen systems represent a means to characterize DJ experimentally. It is important to note that this is the sole method which offers the potential to measure DJ in an isotropic solution. In principle the method may be applied to oriented phases as well. Precise measurements of DJ by relaxation studies will pose several challenges, however, due to the difficulty in choosing an appropriate spin system and due to the assumptions which must be invoked concerning the orientation and asymmetry of J. Of course, in environments of high symmetry, e.g. linear molecules, the orientation and asymmetry of J are dictated by the local molecular symmetry. It is important to note that relaxation by DJ can either increase or decrease the rate of relaxation which would be predicted based solely upon the direct dipolar coupling mechanism, depending on the sign of DJ: This is exemplified in the following equation given by Blicharski for the spin –lattice relaxation rate 272 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 in the rotating frame: ! 1 1 1 2 ¼ 12 x : T1r T1r dip 3 3. Quantum chemical methods ð84Þ Here, the total rate of relaxation is equal to the relaxation due the pure dipolar interaction, ð1=T1r Þdip ; multiplied by a factor involving the ratio of DJ to RDD ; represented by x. Depending on the value of x, relaxation by DJ may either increase the total rate of relaxation (x , 0 or x . 6), decrease the total rate of relaxation ð0 , x , 6Þ; or completely interfere with the direct dipolar coupling mechanism such that the total rate is effectively zero ðx ¼ 3Þ: If x ¼ 6 fortuitously, no effect on the total rate of relaxation will be observed. Unambiguous experimental identification of contributions from DJ to the spin – spin or spin – lattice relaxation rate is an interesting and formidable challenge and will require a unique and carefully chosen spin system which is structurally wellcharacterized and for which a particular spin pair may be predicted to have a substantial DJ and a small but accurately-known direct dipolar coupling constant. Additionally, we note that in principle nuclear spin relaxation may also arise from the antisymmetric part of J. Identifying contributions to relaxation by such a mechanism would no doubt be at least as daunting as identifying contributions from DJ: Finally, one must realize that despite the practical difficulties associated with isolating and identifying contributions to nuclear spin relaxation arising due to anisotropic and antisymmetric J coupling, every spin – spin and spin – lattice relaxation time constant reported in the literature which has been measured based on the assumption of relaxation exclusively by the direct dipole –dipole mechanism contains contributions from the J (except in cases where J is forced to be perfectly isotropic by symmetry). This fact is inescapable, and relates back to the similar forms of the direct dipolar and indirect spin –spin coupling Hamiltonians. It is similarly true that all measured nuclear Overhauser enhancements (NOE) contain contributions from DJ which may not be related to internuclear distance in any straightforward manner; of course many NOEs of interest involve proton– proton couplings for which DJ=3 will be negligible compared to RDD : 3.1. Correlated ab initio methods A recent comprehensive review article discusses the technical aspects of quantum chemical calculation of spin –spin coupling and nuclear magnetic shielding tensors [31]. In general, calculations of J have rather different computational requirements compared to those of s: As already mentioned, there is no gauge origin problem; however, there are more mechanisms contributing to J than to s; as discussed earlier. This makes the number of necessary first-order wave functions much larger for J than in the s case. For couplings, ten responses are needed for each nucleus, whereas three suffice for all the sM regardless of the size of the system. Additional differences arise from the nature of the perturbation operators involved in the calculation of J. The fact that the FC and SD interactions couple the singlet ground state to triplet excited states, makes the restricted Hartree –Fock (RHF) reference state unsuitable as it may be unstable towards triplet perturbations [205]. This results in unphysically large magnitudes of the triplet terms in J (for a discussion, see, e.g. Refs. [206,207]). For example, the calculation of JCC in ethene (C2H4) is a wellknown failure case, where a Hartree – Fock linear response (SCF LR, equivalent to the random phase approximation, RPA) calculation based on the RHF reference state leads to values in the range of thousands to tens of thousands of Hz. The experimental result in solution is about 67.5 Hz [72]. Thus, in contrast to calculations of s; the simplest ab initio quantum chemical level, RHF, does not provide a meaningful starting point even for qualitative work. For the same reason, electron-correlated post-Hartree – Fock methods based on the RHF reference state may be suspect. In practice, multiconfiguration selfconsistent field (MCSCF) linear response (MCLR) [208] and coupled cluster (CC) [209 –212] methods (the latter without explicit orbital relaxation, i.e. only including relaxation implicitly through the CC amplitudes corresponding to the single excitations) have been found to be stable in this respect. Even systems that do not exhibit triplet instability at or close to their equilibrium geometries may suffer J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 from a near- or quasi-instability, which may lead to gross overestimation of the contributions from the triplet mechanisms. Another qualitative difference as compared to the shielding theory is that a better treatment of dynamic electron correlation is essential for reliable calculations of J tensors. The four hyperfine operators involved in the calculation of J, Eqs. (15),(16),(18),(19), sample different spatial regions of the electron cloud and couple to excited states of different spin symmetry. Due to the need to be able to accurately describe more physical features of the system than in the case of s; error cancellation has less room to operate in the calculation of J. Satisfactory results are in practice obtained at the coupled cluster singles and doubles (CCSD) excitation level as well as MCLR with large active molecular orbital (MO) space. A ruleof-thumb in the latter case is that about 95% or more of the total occupation of virtual MOs, based on the natural orbital occupation numbers obtained using, e.g. second-order Møller –Plesset (MP2) or configuration interaction singles and doubles (CISD) one-particle density matrices, should be included in the chosen active space. Despite the apparent challenge that J poses to computational methods, it is possible to reach quantitative agreement with experiment at least for small main-group systems. A prime example of this is a recent MCLR application [213] on the coupling constants of ethyne, Table 11. A brief list of the different implementations of J calculations introduced or relevant in the review period follows. † The sum-over-states (SOS) method [216] features an uncoupled property calculation using ab initio wave functions. As the response of the electron – electron interaction to the magnetic field perturbation is neglected, the method is physically not well-justified. The calculated results have to be scaled for comparison with experiment. † Finite perturbation theory (FPT) calculations of J FC have been carried out at various levels of ab initio theory (see, e.g. Refs. [217 – 219]). This is physically motivated, but the approach lacks the remaining spin – spin coupling terms 273 Table 11 Calculated (MCLR) spin – spin coupling constants in C2H2 compared to the experimental results extrapolated to the equilibrium molecular geometry (results in Hz) Method 3 MCLR [213] Experimental [214,215] 10.80 10.89 † † † † JHH 1 JCH 244.27 242.70 2 1 53.08 53.82 184.68 185.04 JCH JCC and is unable to provide the anisotropic properties. A major drawback of FPT is that numerical instabilities may arise when supplementing the basis set with large-exponent functions to better describe the FC perturbation (see Section 3.3). Ab initio implementation of contributions from localized orbitals within the polarization propagator-inner projections of the polarization propagator approach (CLOPPA-IPPP) has been presented by Contreras and co-workers [220]. The RPA level method operates with localized occupied and virtual orbitals and allows investigation of contributions to coupling from different localized MOs, as well as coupling pathways. The knowledge of MO contributions in principle makes it possible to reduce the dimension of the virtual space in calculations of second-order properties, without losing much quality in the results. Ref. [220] extends earlier semi-empirical [9,221,222] and ab initio [223] work. The equations-of-motion (EOM) method [224] is an intermediate ab initio approach between RPA and MP2. Analytic derivatives of the MP2 energy have been used by Fukui and co-workers [225], extending the earlier work by the same group based on FPT [226,227]. The computational cost of MP2 scales as N 5 where N is the number of basis functions. The second-order polarization propagator approach (SOPPA) of Oddershede and coworkers [228,229] is another analytic secondorder method roughly at or better than the MP2 level. While SOPPA (scaling as N 5 ) is still somewhat subject to the triplet instability problem, its accuracy is very useful for 274 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 qualitatively correct couplings in large systems that are not affected by the triplet instability. † The use of SOPPA with correlation amplitudes taken from a CCSD calculation forms the SOPPA(CCSD) method [230], a simpler version of which was earlier called CCSDPPA [231, 232]. SOPPA(CCSD) provides generally improved results as compared to SOPPA, although it is strictly consistent only to second-order. The CCSD amplitude calculation scales as N 6 ; while property calculations scale as N 5 : † Analytical implementation of the CCSD method has been performed by Bartlett and co-workers [210,211], replacing the earlier FPT version [209,233]. The different models include the socalled quadratic one, meaning unrelaxed analytic second derivatives of the CCSD energy, as well as the equation-of-motion coupled cluster (EOM-CC) approximation featuring unrelaxed, configuration interaction (CI)-like SOS formulation. The latter method is not size-extensive. The results are compared in Ref. [234]. More efficient versions of the CI-like method are discussed in Ref. [235]. The full CCSD linear response ðN 6 Þ is currently the most accurate black-box model for systems where static electron correlation is of little importance. Analytic CCSD(T) ðN 7 Þ featuring perturbative inclusion of triple excitations has been reported by Auer and Gauss [212], as well as the following methods using FPT [212]: full CCSDT with explicit triples ðN 8 Þ and the CC3 model of Ref. [236] with approximate triples ðN 7 Þ: † The MCLR method of Vahtras et al. [208] including both the complete active space (CASSCF) and restricted active space (RASSCF) models, has the possibility of extending the active space in principle all the way up to full configuration interaction (FCI). MCLR can be expected to be successful particularly for systems affected by static correlation. The convergence of the treatment of dynamical correlation is slow, however, exemplified by the JFH coupling constant in the HF molecule as a function of the size and treatment of the virtual active space in Table IV of Ref. [237]. The influence of correlating the semicore and core molecular orbitals has been investigated [238 – 241], as well as higher than singles and doubles excitations in the RASSCF model [239,241,242]. In contrast to the other approaches listed here, the need to choose the active molecular orbital space renders MCLR a non-black-box method, requiring insight in the electronic structure of the system under investigation. The scaling of the CASSCF model is factorial in the number of active molecular orbitals. To date, most applications are carried out using the SOPPA, SOPPA(CCSD), EOM-CCSD, and MCLR methods. Examples of their performance with respect to experimental J coupling constants can be found in Ref. [31]. The typical accuracy of state-of-the-art calculations for small molecules composed of light elements is 5 – 10%, with additional provisos for the presence of rovibrational and solvent effects. Hence, there is still room for improvement even in the treatment of small model systems. In particular, tractable CC models beyond CCSD are desirable, as they are both black-box methods and likely to be more easily extended for larger molecules than MCLR. Despite not having yet been applied to J, the linearscaling CC approaches [243] are promising in this respect. The CCSD(T) model has proven to be very successful for calculations of s [244]. However, this particular method for triples seems to reintroduce, in the J case, problems related to the triplet instability [212]. Instead, the performance of the numerical CC3 model was found promising by Auer and Gauss in Ref. [212], and an analytic derivative implementation of the method would be of substantial interest. 3.2. Density-functional theory methods Density-functional theory (DFT) [245] has become very popular in quantum chemistry due to the fact that it allows the inclusion of electron correlation effects roughly at the cost of the uncorrelated ab initio Hartree – Fock level methods, N 3 – 4 : The drawback of DFT is that there exists no systematic way of improving the exchange-correlation functional of electron density Exc ½r that lies at the heart of DFT, from one calculation to another. The J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 three ‘generations’ of Exc ½r functionals in use are the local density approximation (LDA), generalized gradient approximations (GGA), and various hybrid functionals, with results generally improving in this order. Whereas LDA is parametrized, in principle in an ab initio manner, based on the exchangecorrelation energy per particle in a uniform electron gas, the GGAs also parametrize density gradients semi-empirically. The hybrid functionals incorporate some specific fraction of the exact Hartree – Fock exchange. There is considerable research activity devoted to developing exchange-correlation functionals, with emerging hope for systematic progress [246,247]. In the presence of a magnetic field, Exc should not only be a functional of the electron density as in the field-free case, but it also should refer to the current density [248 – 250], coining the name current DFT (CDFT). A local model of CDFT has been tested in the context of calculating s [251]. The effect of including the current dependence was found to be very small in comparison with the remaining errors of DFT calculations. Most likely the same situation prevails for calculations of J, for which CDFT has not been applied so far. Since the RHF method is unsuitable for the computation of J, DFT holds a different status in calculations of this property as compared to s: DFT seemingly does not suffer from the triplet instability [252 – 254] in J calculations, making it by far the least computationally demanding method by which qualitatively correct J values may be calculated. Furthermore, its scaling with the system size currently makes it the only practical method for calculating J in large molecules. DFT calculation of s as well as JPSO using pure (i.e. non-hybrid) functionals (as well as omitting any current dependence of Exc ) can be carried out non-iteratively, in an uncoupled fashion, in contrast to the wave function methods. This is due to the imaginary character of the relevant perturbations, causing the corresponding first-order density change to vanish. Calculation of J by DFT necessitates additionally a coupled or response procedure due to the real FC and SD perturbations, unless FPT is used for these interactions. In the hybrid DFT framework, the presence of the exact Hartree – Fock exchange term 275 makes the coupled procedure necessary also for JPSO : A discussion of relevant DFT implementations for spin –spin coupling calculations follows: † Ref. [255] described an implementation of an uncoupled SOS procedure for all the secondorder terms in J, i.e. including also JFC and JSD for which SOS is not applicable. As hybrid functionals were used to obtain the Kohn – Sham orbitals [255], even the uncoupled calculation of the PSO term is inconsistent. The results are unsatisfactory and in poor agreement with experiment. † Malkin et al. [256,257] presented a combined FPT and sum-over-states density-functional perturbation theory (SOS-DFPT) method, with the possibility of using pure DFT LDA and GGA functionals. In the calculation of the FC and SD/FC terms, Eqs. (24) and (26), the FC operator on the chosen nucleus is applied as a finite perturbation in the spin-polarized unrestricted Kohn –Sham calculation. JSD is neglected because of the more complicated finite perturbations that would be necessary in this case. While not warranted a priori, the omission is in practice often a justified approximation on the basis of results obtained. JPSO is calculated using a sum-over states expression after a converged restricted Kohn – Sham calculation: PSO JMN; et / occ X virt 3 X kfk llM;e =rM lfa lkfa llN;t =rN3 lfk l k a xc 1k 2 1a 2 DEk!a ð85Þ where the orbital energy denominators 1k 2 1a have been subjected to the ‘Malkin correction’, xc DEk!a [258]. This has been viewed as an a posteriori attempt to model the current dependence of Exc or merely an ad hoc correction for the deficiencies in the orbital energy denominators. In any case, the modified PSO terms are in good agreement with correlated ab initio calculations [256,257]. Earlier, Dickson and Ziegler [259] reported a similar implementation without the Malkin corrections, with the PSO contributions apparently overestimated [144]. Slater-type basis functions were used in Ref. [259]. Ziegler and 276 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 co-workers [260] made an interesting connection xc of DEk!a to an approximate correction for the unphysical electronic self-interaction effect on the orbital energy differences. Excellent performance has been reported [256, 257] for this approach for JCC and JCH using the semi-empirical choice of the GGA functional, namely Perdew86 [261,262], known to give reliable isotropic electron spin resonance hyperfine couplings for non-p radicals [263]. However, the performance has been found to deteriorate for J FC towards group 17 elements, particularly for 19 F. Apparently the quantitative description of the FC-induced spin density becomes increasingly difficult for systems with lone pairs. However, the errors appear to be rather systematic, based on comparison with ab initio calculations. Despite the problems, this DFT method remains one of the most popular computational approaches to J in recent literature, with particularly many applications to large molecules. † Ref. [264] described a FPT implementation of both the SD and FC terms in the Gaussian suite of programs, enabling the use of hybrid functionals. † Full DFT implementations including also the JSD terms and using analytical derivative theory were reported in Refs. [252,253]. Both programs are capable of using also the hybrid functionals. Whereas the Hartree – Fock level of theory typically leads to overestimated spin density and, hence, FC contributions, the GGA functionals tend to underestimate the same quantities. Somewhat expectedly then, the quality of results improves significantly in the succession LDA ! GGA ! hybrid functionals, in the main-group systems investigated so far [252,253]. The problem with 19 F is not, however, solved by the hybrid functionals. Presently this method holds the greatest promise for solving chemical problems in large main-group systems. The performance of the popular B3LYP hybrid functional [265,266] for the anisotropic properties of J has recently been tested [42]. While not validated for J, in transition metal systems hybrid functionals do not appear to offer systematic improvement for other properties. † Autschbach and Ziegler have implemented the relativistic zeroth-order regular approximation (ZORA) for calculations of J tensors [144, 267]. This method includes both scalar relativistic [267] and electronic spin –orbit effects [144], with JSD calculated in connection with the latter. ZORA leads to modified hyperfine operators that can be interpreted in non-relativistic terms, however. Analytical derivative techniques were used. The method allows qualitatively accurate calculations for spin – spin couplings also involving heavy nuclei. The software used is limited to pure DFT functionals. The Xa approximation is used for the first-order exchange-correlation potential necessary in the coupled DFT calculation. In the pioneering study of Ref. [260], an approximate self-interaction correction at the LDA level was not found to lead to a systematic improvement of the total J, albeit J PSO as well as s were clearly improved. One reason for these mixed observations might be the fact that the magnetic field response of the potential term corresponding to the self-interaction correction was neglected. We note that this term is not present in the above-mentioned successful cases where the perturbation operators are purely imaginary, in contrast to J SD and J FC : In any case, further investigations along the direction of Ref. [260] would be very interesting. Concerning the problematic couplings to 19F, the application of methods providing localized orbital contributions to the calculated couplings in combination with DFT [220,221,268] might give increased insight [269]. 3.3. Basis set requirements The treatment of the many-body problem as well as the basis set requirements are demanding issues that must be addressed in calculations of J. The reason is 2-fold. First, the need for highly correlated wave functions places the corresponding demands on the basis set. JDSO has been found to be remarkably easy to calculate, with SCF wave functions and double-zeta plus polarization (DZP) basis sets giving accurate values [270 – 272]. The generally small magnitude of these terms contributes to the favourable situation as the error in the total J is dominated by the second-order J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 contributions. For them, correlated calculations are mandatory and the TZP basis set can be viewed as the entry level. The basis set convergence in the valence region may be expected to be faster with DFT as compared to the correlated ab initio methods. Second, the description of the hyperfine perturbations that sample the immediate vicinity of the atomic nuclei, necessitates more basis functions in the atomic core region than what is necessary for standard energetic properties. Particularly, the FC operator is difficult to represent using a small number of Gaussian functions. Hence, the standard basis sets used in quantum chemistry most often need to be supplemented with high exponent, tight, basis functions at least of s-type [273], if converged values of J are to be obtained. The tight functions typically increase the coupling constants by 5 –10% as compared to stateof-the-art basis sets for valence properties. A few systematic studies of the basis set requirements for J at ab initio level have been carried out. † The polarization propagator calculations for JHD were performed for the hydrogen molecule [274]. The need for tight s-functions was established, all the way up to exponents such as as ðHÞ ¼ 150; 000 whereas in ordinary basis sets max½as ðHÞ ¼ Oð100Þ: † The CASSCF LR results for JFH in HF [275] recommended systematically converging, although relatively expensive cc-pVXZ-sun basis sets that are based on the correlation consistent paradigm [276 – 279], decontracted in the s-function space, and supplemented with n tight primitives of this type. See also Ref. [280] for a related study. † System-dependent basis set prescription has been proposed, through using contraction coefficients from MO coefficients for the molecule under study [242,281,282]. This was slightly generalized, based on simple model hydrides containing the nuclei of interest, in Ref. [283]. † A pragmatic procedure more easily adopted in large systems has been followed in Refs. [284, 285]. There, use has been made of sets that build on the decontracted Huzinaga/Kutzelnigg (‘IGLO’) basis sets [286,287] commonly denoted BII –BIV or HII–HIV. These basis sets have been shown to perform very well for their size in Refs. [242,275]. 277 Uncontracting and supplementing them with n sets of tight s-type primitives (in some cases also p- and d-type) provides nice convergence behaviour of the properties that depend on the FC perturbation. These basis are designated as, e.g. HIVun. † In the locally dense basis set concept [288,289], a large basis with tight primitives is only used for the interesting part of the molecule, possibly only at the centres with the coupled nuclei, while the rest of the system is treated more approximately. This method was applied to 3 JHH in C2H5X (X ¼ H, F, Cl, Br, I) in Ref. [282] at the SOPPA level with encouraging results. Changes of the order of 0.3 Hz, or 3% of the total magnitude of the coupling, were observed due to the locally dense approximation. While the use of an all-electron basis set is normally necessary for the nuclei for which couplings are calculated, the reconstruction of the core response to hyperfine operators in a pseudo-potential framework [290,291] would be interesting in the context of J, as well. Nair and Chandra [292] have used energyoptimized bond-centred s- and p-primitive functions to significantly improve calculated coupling constants at the SCF level, with otherwise very modest basis sets. A systematic study at a correlated level would be in order. Interesting initial work has been carried out by Rassolov and Chipman [293,294] (see also the earlier paper by Geertsen [295]) where the delta function sampling of the wave function at the nucleus is replaced through integration by parts by a global operator covering an extended area (r < 0.1 a.u.) in space around the nucleus. This procedure eliminates, at least partially, the need for tight s-functions. Results are identical to those obtained with the d-function operator for the exact wave function. For approximate wave functions, errors are smaller than with the dfunction operator. Apart from the initial trials, the performance of the method has not been investigated in detail. 3.4. Effects of nuclear motion Zero-point and thermal motion of the nuclei, as well as the presence of a medium, affect 278 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 Fig. 12. Schematic diagram of the various factors affecting the comparison of experimentally and theoretically determined NMR parameters. Drawn after Ref. [296]. the parameters of HNMR : Ultimately, the comparison of accurate theoretical calculations with experimental results should take into account both. This has seldom been the case in the spin – spin coupling literature. Fig. 12 illustrates the different steps in the comparison. J depends on the geometrical parameters such as bond lengths (see, e.g. Ref. [297] for N2 and CO molecules) and angles [298]. Particularly the triplet coupling mechanisms exhibit large geometry dependence, the prototypical example being the increase of J FC in HD by orders of magnitude as the bond length is extended [299]. The origin of the effect is in the shared dissociation limit of the singlet ground state and the triplet excited state, and the consequently decreasing triplet excitation energy as the bond is extended (see also Ref. [300]). The dependence of 3 JHH on the dihedral angle, giving rise to the well-known Karplus plot, is exemplified for ethane (C2H6) in Ref. [301]. A comprehensive review on the rovibrational averaging of molecular properties was given in Ref. [302]. Computational modelling of rovibrational effects involves determining the J coupling hypersurface X ›J J ¼ Je þ Q ›Q k e k k ! 1X ›2 J þ Q Q þ ··· ð86Þ 2 kl ›Qk ›Ql e k l where J is now a component of the J tensor, Je its equilibrium geometry value, and the Qk are some nuclear displacement coordinates: either, for example local coordinates such as Dr ¼ r 2 re ; symmetry coordinates, or normal coordinates. The derivatives are the parameters of the property hypersurface. When determining the surfaces for tensorial quantities such as the components of JA and JS using molecular geometries displaced from the equilibrium, it is necessary to ensure that the Eckart conditions [303,304] are fulfilled by the coordinate representation used for the property tensors [305,306]. The property surface is averaged over the nuclear motion as X ›J kJlT ¼ Je þ kQ lT ›Qk e k k ! 1X ›2 J þ kQ Q lT þ · · ·; ð87Þ 2 kl ›Qk ›Ql e k l where the nomenclature kAlT specifies either the temperature average of A or its average in a particular rovibrational state, occupied with a certain temperature-dependent probability. The averages are, in turn, determined by the potential energy surface (PES) of the system. In a normal coordinate expansion, the second-order (harmonic) terms arise due to the quadratic potential surface, while the leading anharmonic contributions are due to the semi-diagonal components of the cubic force field. The effect of vibrational anharmonicity can be covered, to a good approximation, by carrying out a single-point calculation at the thermally averaged ra geometry, where ! 1 X ›2 J T kJl < Ja þ kQ2 lT : ð88Þ 2 k ›Q2k e k The expansion of Eq. (87) is usually truncated after the harmonic terms, causing typically only a small error. Once the property hypersurface is mapped out, there are different ways to perform the averaging. In the widely used perturbational method (see, e.g. Ref. [307]), the thermal vibrational averages kQk lT and kQk Ql lT ; as well as the rotational contribution to the former, are calculated based on the formulae given in Ref. [308]. For diatomics, properties averaged in individual rovibrational states are conveniently available by solving the rovibrational Schrödinger equation numerically [309]. Recently, an approach based on sampling geometries accessible to the rovibrational motion by semi-classical path integral simulation, has been advanced [310]. J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 The process of rovibrational averaging is computer and human resource-intensive, as many single-point calculations are needed to map the property and potential energy hypersurfaces. Refs. [311,312] describe an automated procedure for carrying out zero-point vibrational corrections by first finding the average geometry rz ¼ ra (0 K) and then performing the harmonic vibrational corrections at that point. The method features numerical derivatives of analytic single-point gradients and properties. Besides automation, its principal merit as compared to expansions at re is the smaller truncation error. Generalization to rovibrational averaging at finite temperatures is in progress [313]. Another automated implementation of the zero-point vibrational corrections to molecular properties, based on expansion about the re geometry, has been carried out [314]. Applications of rovibrational corrections to J include HD [208,274], HF [237], FHF2 [315], N2 [297,316], CO [297,316], OH2 [317], H2O [241,318], H3Oþ [317], CH4 [319,320], C2H2 [214,215], and SiH4 [321]. The MCLR, EOM-CCSD, SOPPA, and SOPPA(CCSD) methods have been used. Calculations at an a priori inadequate level where only the first-order Taylor expansion of the coupling constants is used, are reported for CH4 in Refs. [281,322] and for XH4 (X ¼ C, Si, Ge, Sn) in Ref. [323]. A FCI study of the FC contribution in H2 was carried out in Ref. [299]. The effect of thermal motion on the tensorial properties of J appears to have garnered almost no attention in recent literature, apart from the CASSCF 279 study of diatomic molecules by Bryce and Wasylishen [143]. Table 12 compares their calculated results for equilibrium geometry and in specific rovibrational states. The calculated (ro)vibrational corrections both for J and DJ are quite small in these systems. The accuracy of the calculations is not yet sufficient to assess their significance in comparison with the experiment. Ref. [78] reports an estimate of the rovibrational effect on D1 JFC in CH3F with essentially the same result. Ref. [324] reported J and DJ in HCN and HNC as a function of the length of the triple bond, but did not carry out averaging over nuclear motion. The magnitudes of all of the DJ increase with increasing bond length. The changes are, in most cases, smaller than those of the corresponding coupling constants, implying smaller rovibrationally induced changes for the anisotropic observables than for J. Galasso [325] reported a large dependence of D1 J (defined with respect to the direction of the internuclear axis between the heavy atoms) on the dihedral angle for N2H4, P2H4, and PH2NH2. The case of P2H4 had been studied earlier by Pyykkö and Wiesenfeld in Ref. [74]. Further studies on nuclear motion and rovibrational averaging effects on the tensorial properties of J would be of interest. 3.5. Relativistic effects Classic reviews on the effects of special relativity in chemistry have been given by Pyykkö [326,327]. Relativistic effects on atomic and molecular Table 12 Comparison of calculated 1J for diatomic molecules at equilibrium geometries and in specific rovibrational states. Results from Ref. [143] (results in Hz) Molecule Coupling LiH 7 LiF 19 7 KF 39 Na2 23 ClF 35 a Li1H F Li K19F Na23Na Cl19F J DJ J DJ J DJ J DJ J DJ re value Rovib. state Rovib. average Exp.a 152.47 212.39 193.10 177.43 76.59 109.22 1243.6 229.88 832.24 2805.68 n ¼ 0, J ¼ 1 151 213 199.0 176.9 78.2 109.5 1245 230 829 2800 135(10) 257(21) 172.3(32) 173.2(28) 57.8(13) 125.7(51) 1067(7) 248(15) 840(6) 2907(27) n ¼ 0, J ¼ 0 n ¼ 0, J ¼ 0 n¼0 n ¼ 0, J ¼ 1 Experimental microwave spectroscopic results for the specified rovibrational states. For references, see Ref. [143]. 280 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 electronic structure can be categorized into scalar relativistic and spin – orbit effects. The rough consequence of the former is a contraction of the atomic sand p-shells as well as expansion of the d- and f-shells, whereas the latter causes spin polarization even in closed-shell systems by mixing triplet excited states with the ground state. Generally, relativistic effects are larger for systems with heavy nuclei. The NMR and hyperfine properties, in general, are susceptible to relativity, even for light elements, as the quantum mechanical operators involved probe the region of the electron cloud close to the nuclei, where the electron velocities are large. In the context of spinHamiltonian parameters such as J, relativistic effects enter first through modification of the wave function due to (in the Pauli language) the mass –velocity, Darwin, and spin – orbit interactions. Second, completely new terms or combinations of non-relativistically uncoupled mechanisms may appear, such as the FC/PSO cross-terms [37,38] or second-order spin – orbit terms [38]. Third, the relativistic hyperfine operators themselves are different from their nonrelativistic limits [328 – 330]. For J, the leading relativistic correction terms are Oða6 Þ; two powers of a higher than the basic non-relativistic theory. A brief list of the currently available methods that include relativity in the calculation of J is as follows. † A posteriori multiplicative correction factors are obtained as the ratio of the Dirac – Fock and Hartree – Fock hyperfine integrals [328]. This is a semi-empirical correction, the applicability of which depends on the dominance of the FC contribution. Ref. [331] applies the idea in the DFT framework by borrowing the electron density at one of the coupled nuclei from a scalar relativistic atomic calculation. † Relativistic extended Hückel (REXNMR) [7,74] is a semi-empirical method based on the relativistic parametrization (obtained by Dirac –Fock atomic calculations) of the extended Hückel method. While the results are at best qualitatively correct, this is the relativistic method by which the largest number of studies of J have been carried out so far [3 –5,7,8,74]. Among the obtained results, the increase of the relative anisotropy DK=K due to relativistic effects [74,332] seems to be a general feature. † The CLOPPA RPA method with relativistic semi-empirical parametrization [9,221,222] in a formally non-relativistic framework features localized MO contributions. † Breit – Pauli corrections for the spin – orbit effect have been carried out through third-order perturbation theory [37,38,333]. A second-order correction was added in Ref. [38]. The method requires scalar relativistic effects for comparison with experiment. † The Pauli Hamiltonian in a scalar relativistic frozen core DFT framework has been used by Khandogin and Ziegler [331]. This is theoretically somewhat incomplete as relativistic modification of the wave function by the mass – velocity and Darwin interactions is used with non-relativistic hyperfine operators. The approach features FPT for J FC and neglects J SD : Results seem to be worse than in the simple modification of the FC contribution, discussed earlier. † The four-component Dirac – Hartree –Fock (DHF) LR model has been implemented and applied [334 –337]. Both scalar relativistic and spin –orbit effects are included in a fully relativistic framework. The method has in principle a simple structure due to only one relativistic hyperfine operator, with the diamagnetic term in particular arising from rotations between occupied electronic and virtual positronic states [336]. DHF LR needs to be extended beyond the RPA level for direct comparison with experiment, however. † The ZORA DFT method, already mentioned in Section 3.2, includes both scalar relativistic and spin – orbit effects. This is, for practical problems, the most applicable of the presently available methods, with potential for large systems as well. Applications already include the tensorial properties of J [144 – 146]. Autschbach and Ziegler [267] found the spatial origin of the relativistic increase of J FC in the distance range up to 1022 a.u. from the heavy nucleus. A comparison with frozen core calculations points out that the core tails of valence electrons are mainly responsible. The DHF method is a useful benchmark for more approximate methods of including relativity. More practical applications are to be expected from the transformed Hamiltonian methods, notably ZORA, but J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 most likely the Douglas – Kroll – Hess approach [338,339] and direct perturbation theory [340,341] will be implemented and applied in the context of J in the near future. There, one has to account for the picture change effects on the hyperfine operators [342, 343]. The relativistic pseudo-potential route can also be followed for couplings between light nuclei in systems where heavy elements are represented by scalar relativistic [344 –346] and spin – orbit pseudopotentials. For implementations and applications of this method to s and the electron spin resonance g-tensor, see Refs. [347,348], respectively. Examples of calculated relativistic effects on specific coupling tensors are deferred until Section 3.8.1. 3.6. Solvation and intermolecular forces Medium and solvation effects on J and other molecular properties have their origins in intermolecular interactions, repulsion, dispersion, electrostatic, and induction forces, as well as hydrogen bonding effects [349]. There are both indirect effects manifested through changes of the molecular geometry as well as direct electronic structure modifications that already have an effect at the gas-phase structure. Two approaches to account for medium effects are used in the recent J coupling literature (for older references see Ref. [350]). † The molecule can be placed in a cavity within a homogeneous, linear dielectric medium characterized by its dielectric constant, and subjected to the reaction field caused by the response of the environment to the charge distribution of the molecule. This was first reported for J in Ref. [351] within the spherical cavity model. This is an analytic derivative method covering the long-range electrostatic forces. In particular, hydrogen bonding is not within reach in a pure reaction field model. † The supermolecule method has been applied, where parts of the immediate molecular surroundings of the system under study are explicitly included in the finite field spirit. In principle one would want to include parts of the environment at least up to the first solvation shell, but in practice that is a difficult require- 281 ment in the J coupling context. So far applications have been limited to including a few molecules of the environment. The supermolecule method accounts for the shortrange intermolecular interactions, and can be combined with the reaction field model for longrange electrostatics by placing the supermolecule in a cavity. Proper treatment of the dispersion interaction is difficult in a supermolecule calculation. An additional complication of the method is caused by the basis set superposition error, for which the counterpoise method [352] is a pragmatic solution. Applications for J include C2H2 [353,354], C2H3F [238], H2O [355], CH3OH [356], CH3NH2 [356], HCN [357], H2S [357], and H2Se [351] using the reaction field method, and the first two molecules of the list using the supermolecule or combined method [354,355]. The effects on J caused by the solvation by one water molecule were investigated in Ref. [358] for CH2O, C2H2, and CH3OH. The hydrogen-bonded complexes formamide dimer and formamidine –formamide dimer were investigated in Ref. [359]. The effect of dimerization on the couplings in HCOOH was also studied. An earlier study of formamide solvated by four water molecules was reported in Ref. [360]. Transition metal compounds were studied at the supermolecule level by Autschbach and Ziegler [361, 362]. Ref. [363] reports a Hartree –Fock study of the FC effects on 1 JNLi in LiNH2, LiN(CH3)2, and (LiNH2)2 due to explicit solvation by one to four water molecules. Ref. [364] used IPPP-CLOPPA to investigate the effect of the electric field due to a solvent water molecule on 1 JCH in CH4 and HCN. The 1 JCH in the CH4 – FH and H2O – HCN supermolecular systems was investigated in Ref. [365]. Ref. [354] used the concept of intermolecular coupling constant surfaces, which is likely to be of large qualitative value in the interpretation of medium effects on J. Many recent applications on J mediated through hydrogen bonds [355,358,359,366 – 376] can be considered to fall into the supermolecule category, although the goal there is in calculating actual intermolecular couplings. Ref. [377] is a pioneering report on J mediated by the van der Waals interaction in He2. Ref. [378] gives an estimate of 282 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 the corresponding FC contribution in Xe2. Ref. [376] considers JNN over the hydrogen bond in a methyleneimine dimer and reports both the coupling constant and anisotropy as functions of the hydrogen bond geometry. The only study of intermolecular J featuring configurational sampling that we are aware of, is that reported in Ref. [379]. In that work, Monte Carlo simulations were used to estimate the coupling constant between the nucleus of the F2 ion and the nuclei in a water solvent. The intermolecular coupling surface was initially parametrized using quantum chemistry and the simulation was analysed assuming pairwise additivity of the interactions. The presentday computational resources should facilitate further research in this direction. Little is known about the medium effects on the tensorial properties of J. Generally, the effects on J are of sufficient magnitude to warrant investigation of the first and second-rank contributions as well. 3.7. Couplings for large systems To date, computational studies of J have mainly concentrated on small molecules of prototypical value for chemical purposes. Furthermore, the isotropic coupling constants have almost exclusively been the focus. The limited number of studies for large systems reflects the unfavourable computational scaling of the current correlated ab initio methods with the size of the system, discussed in Section 3.1. Despite this, a few computational tricks are already available to facilitate studies of medium- to large-sized molecules. † Chemically motivated model molecules. Cluster models of the environment. † Locally dense basis sets. Pseudo-potentials for inter- and intra-ligand couplings. † Tailor-made contraction of the basis set according to the molecular orbital coefficients of the system under study. † Bond-centred basis functions [292]. † Calculations only of the most demanding contributions (FC and SD/FC) at the highest level; lower-level methods for DSO, PSO, SD. These tools are naturally also available for DFT calculations of J for which several application papers have already appeared. These include one-bond metal-ligand couplings at the quasi-relativistic [331,380] and ZORA [144,267] levels, as well as solvation effects in the couplings in a coordinatively unsaturated transition metal compound [361,362]. In these works, explicit solvation was found to be absolutely necessary, along with the contribution of scalar relativistic effects, to produce a qualitatively correct description of the J coupling patterns. Through-space FF couplings were studied in different polycyclic organic fragments [264,268, 381]. It is noteworthy that through-space FF couplings can be calculated with DFT much more accurately than what would be expected on the basis of throughbond couplings in small molecules. Bryce and Wasylishen [382] investigated the XF (X ¼ H, C, F) couplings using the MCLR method and HF –CH4 and HF –CH3F complexes as model systems. Also coupling anisotropies were reported; in particular the FF coupling anisotropy was found to be large at small inter-fragment distances. JCH and JCC in some prototypical hydrocarbons, e.g. pyridine, were calculated in Ref. [256]. The dependence of 1 JCa Cb and 1 JCa Ha on the backbone conformation of a model dipeptide [383,384], the dependence of 3 JCC on the conformation of an open chain natural product fragment [385], the HH and CH couplings in Me a1 D -xylopyranoside [386], JCH in cyclohexane-related 1 systems [387], JFH in (HF)n clusters [388], internucleotide JNH and JNN between DNA base pairs [368], and 1 JHD in Os(II)-dihydrogen complexes [346] have also been investigated. In most of these applications, J SD has been neglected. An earlier investigation of JHD in Os(II)-dihydrogen complexes was carried out at the SCF and MP2 levels in Ref. [344]. Refs. [389 – 391] report a hybrid DFT (B3LYP) level study of n JCC and n JCH in 2-deoxy-b-D ribofuranose and related systems using a FPT calculation of J FC : In particular, the work at the DFT level can be carried out without having to resort to the awkward procedure of scaling low-level ab initio results based on benchmark calculations for smaller systems at a higher level [392 –394]. Ref. [255] reports an uncoupled DFT study of n JHH in terpenes. The results are hampered by serious methodological deficiencies as discussed before. In Ref. [216], the uncoupled SCF method was used for FC FC FC JHH ; JCH ; and JCC in a number of light main-group J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 molecules. In this context, FPT calculations for J FC at levels ranging from SCF to quadratic configuration interaction [QCISD(T)], carried out in Refs. [219, 395], are more consistent and superior in quality. EOM-CCSD was applied to the CC, CH, and HH coupling constants in the 2-norbornyl carbocation with excellent results [396]. The EOM method was applied to the HH, CH, and CC coupling constants in the cyclodecyl cation and related systems in Ref. [397]. The 1 JCC in bicyclo butane, tricyclopentane, and tricyclohexane, as well as octabisvalene were studied by this method in Ref. [398]. For related, earlier work, the reader is referred to Refs. [399 – 401]. FC A Hartree – Fock level study of 1 JNLi in model systems exhibiting the LiN-bond was carried out in FC Ref. [363]. 1 JCH in five- and six-membered heterocyclic compounds were studied in Ref. [402]. JHH ; JCH ; and JCC in bicyclo[1.1.1]pentane have been investigated at the SCF LR level [403]. 3.8. Quantum chemical results 3.8.1. Symmetric components Tables 13 – 24 display the results for components of J1 þ JS : In cases where the principal values of the tensors are specified, we do not report the directions of the corresponding principal axes, for reasons of space. The reader is asked to refer to the original publications where this information can in most cases be found. When converting between reduced coupling units, 1019 T2J21 and Hz, or between different isotopes of the same nucleus, use has been made of magnetogyric ratios tabulated in Ref. [408]. We divide the discussion into parts according to the methodology used. The early work is characterized by inadequate electron correlation treatment and/or, by today’s standards, modest basis sets. Following that, the more recent papers with up-todate methods are commented upon. Early work using Hartree – Fock-level methods. Lazzeretti et al. carried out a coupled Hartree – Fock þ 2 study of PH2 2 , PH3, and PH4 [40] as well as NH2 , þ 2 NH4 , and BH4 [41] using reasonable basis sets but neglecting the DSO contribution. The authors should be commended for reporting full information on their coupling tensors, including JA ; in contrast to most of 283 the work in the field. The calculated J were in qualitative agreement with the experiment, although error cancellation between both the tensor components and the different mechanisms may make the individual numbers not very trustworthy. In these papers, the individual SD(M )/FC(N ) and SD(N )/FC(M ) contributions to the SD/FC cross-term have been presented separately. For 1 J; roughly equally large contributions appear with opposite signs and the total SD/FC value of each tensor component is smaller in absolute terms than either the SD(X)/FC(H) or SD(H)/FC(X) values. The lighter the element X is (i.e., from P to N to B), the more the term with the FC interaction at H dominates the total SD/FC contribution. Lazzeretti et al. performed first-order polarization propagator (FOPPA) studies, equivalent to SCF or RPA level, on AlH2 4 and SiH4, with decent basis sets [406]. The full tensors were reported. The lack of electron correlation limits the reliability of the tensorial results, as can be seen from a comparison with the later SOPPA(CCSD) calculation for SiH4 [321]. The isotropic J values are reasonable, but are not in quantitative agreement with the available experimental data. The relatively large importance of the DSO contribution to DJ as compared to coupling constants is evident from this and other early work. Galasso [325] used the SOS-CI method of Nakatsuji [409] for nine dihydrides containing B, N, or P as heteroatoms, in a study reporting 1 J and D1 J with respect to the direction of the vector joining the heavy nuclei. The SOS-CI method comprises a noniterative calculation with all singly and some doubly excited configurations, and the results are roughly of Hartree – Fock quality. A combination of the modest 6-31G and 4-31G basis sets was employed. The available experimental J values are reproduced qualitatively. These systems have not been subjected to a modern study. For example, the SD contribution to JPP and DJPP P2H4 is not negligible. Pioneering work using correlated wave functions. Geertsen and Oddershede compared SOPPA calculations with lower-order methods for water in Ref. [228]. As the basis sets used were reasonably good, electron correlation was included, all the physical contributions were calculated, and results for the full tensor reported, the work remains as one of the most complete early papers on J. For 2 JHH ; the later 284 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 Table 13 Quantum chemically calculated components of the symmetric part J1 þ J S of the 1H1H spin–spin coupling tensors System Bonds Theorya Values Reference BH2 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2h 2i 2 2 2 2 2 2 2 2 2 2 3 3 3 3k 3l 3m 3k 3l 3n SCF/(s ) SOPPA(CCSD)/(s ) RAS/HIV RAS/HIII SCF/(s ) SCF/(s) RAS/HIII RAS/HIII SOPPA(CCSD)/(s ) CAS/(s ) CAS þ SOe/HIVu4 CAS/HIVu4 SOPPA/(s ) SOPPA/(s ) MP2/(s ) RAS/HIII RAS/HIII RAS/HIII FOPPA/(s ) SOPPA(CCSD)/aug-cc-pVTZ(m ) FOPPA/(s ) RAS/HIII RAS/HIII SCF/(s ) SCF/(s ) SCF/(s ) CAS þ SOe/HIVu4 CAS/HIVu4 MP2/6-31G(**) CAS þ SOe/HIVu4 CAS/HIVu4 CAS þ SOe/HIVu3 CAS/HIVu3 SOPPA(CCSD)/(s ) RAS/HIV ZORA DFT(GGA)/Slater RAS/HIV RAS/HIV RAS/HIII RAS/HIII RAS/HIII RAS/HIII Jaa ¼ 212.10, Jbb ¼ 219.96, Jcc ¼ 2 20.67 Jaa ¼ 28.25, Jbb ¼ 29.48, Jcc ¼ 225.63 DJ ¼ 5.3, Jxx 2 Jyy ¼ 15.3b, Jaa ¼ 4.5, Jbb ¼ 6.8, Jcc ¼ 2 8.5 DJ ¼ 28.3, Jaa ¼ 28.0, Jbb ¼ 28.8, Jcc ¼ 225.5 Jaa ¼ 8.96, Jbb ¼ 215.21, Jcc ¼ 234.89 Jaa ¼ 25.13, Jbb ¼ 221.83, Jcc ¼ 235.75 DcJ ¼ 218.97, J ¼ 222.91c,d DcJ ¼ 217.65, J ¼ 219.05c,d Jaa ¼ 1.21, Jbb ¼ 6.38, Jcc ¼ 233.33 Jaa ¼ 0.56, Jbb ¼ 5.48, Jcc ¼ 234.84 Jaa ¼ 0.75, Jbb ¼ 5.25, Jcc ¼ 234.54 Jaa ¼ 0.73, Jbb ¼ 5.19, Jcc ¼ 234.60 Jaa ¼ 1.16, Jbb ¼ 5.66, Jcc ¼ 234.22 Jaa ¼ 21.14, Jbb ¼ 1.69, Jcc ¼ 235.93 Jaa ¼ 20.93, Jbb ¼ 25.12, Jcc ¼ 248.93 Jaa ¼ 11.2, Jbb ¼ 211.8, Jcc ¼ 11.9 DJ ¼ 210.53, Jaa ¼ 25.37, Jbb ¼ 26.58, Jcc ¼ 222.61f DJ ¼ 6.06, Jxx 2 Jyy ¼ 16.65g, Jaa ¼ 3.35, Jbb ¼ 5.61, Jcc ¼ 211.04 Jaa ¼ 25.55, Jbb ¼ 27.19, Jcc ¼ 27.96 Jaa ¼ 0.01, Jbb ¼ 2.43, Jcc ¼ 5.34 Jaa ¼ 0.42, Jbb ¼ 21.71, Jcc ¼ 25.48 DJ ¼ 27.33, Jaa ¼ 29.50, Jbb ¼ 2 9.73, Jcc ¼ 2 26.49 DJ ¼ 21.96, Jaa ¼ 0.15, Jbb ¼ 3.69, Jcc ¼ 3.72 Jaa ¼ 5.92, Jbb ¼ 28.38, Jcc ¼ 29.42j DJ ¼ 4.26, Jaa ¼ 27.79, Jbb ¼ 223.52, Jcc ¼ 224.56 Jaa ¼ 5.04, Jbb ¼ 26.09, Jcc ¼ 26.93 Jaa ¼ 26.52, Jbb ¼ 215.73, Jcc ¼ 224.67 Jaa ¼ 26.70, Jbb ¼ 215.80, Jcc ¼ 224.85 Jaa ¼ 214.92, Jbb ¼ 221.07, Jcc ¼ 2 26.18 Jaa ¼ 213.38, Jbb ¼ 214.14, Jcc ¼ 2 24.81 Jaa ¼ 213.74, Jbb ¼ 214.91, Jcc ¼ 2 25.67 Jaa ¼ 213.26, Jbb ¼ 217.02, Jcc ¼ 2 23.03 Jaa ¼ 214.05, Jbb ¼ 218.59, Jcc ¼ 2 24.70 DJ ¼ 3.20, J ¼ 11.31 DJ ¼ 3.4, J ¼ 10.8 DJ ¼ 12, J ¼ 10 DJ ¼ 4.0, Jxx 2 Jyy ¼ 21.2b, Jaa ¼ 8.4, Jbb ¼ 9.7, Jcc ¼ 13.1 DJ ¼ 5.0, Jxx 2 Jyy ¼ 20.8b, Jaa ¼ 14.4, Jbb ¼ 15.8, Jcc ¼ 20.9 DJ ¼ 2.2, J ¼ 7.2 Jaa ¼ 0.7, Jbb ¼ 2 0.8, Jcc ¼ 2.4 Jaa ¼ 10.5, Jbb ¼ 10.6, Jcc ¼ 14.2 DJ ¼ 1.23, J ¼ 3.80 [41] [321] [72] [72] [41] [41] [324] [324] [318] [318] [38] [38] [280,318] [228] [227] [404,405] [78] [78] [406] [321] [406] [80] [80] [40] [40] [40] [38] [38] [227] [38] [38] [38] [38] [214,215] [72] [144] [72] [72] [72] [404,405] [404,405] [80] CH4 C2H4 C2H6 NH2 2 NHþ 4 CH3CN CH3NC H2O H2O H2O H2O H2O H2O H2O HCONH2 CH3F CH2F2 AlH2 4 SiH4 SiH4 CH3SiH3 CH3SiH3 PH2 2 PH3 PHþ 4 H2S H2S H2S H2Se H2Se H2Te H2Te C2H2 C2H2 C2H2 C2H4 C2H4 C2H6 HCONH2 HCONH2 CH3SiH3 Either the principal values of the symmetrized tensor (Jii, i ¼ a; b; c with lJaa l # lJbb l # lJcc l), specified Cartesian components (xx, xy, etc.), or the anisotropy with respect to a certain axis (e.g. Dz J ¼ Jzz 2 1=2ðJxx þ Jyy Þ), is indicated. Unless otherwise noted, in systems possessing a unique n-fold rotation axis (n $ 2), the anisotropy is defined with respect to that axis. Results in Hz. a (u ) designates the uncontracted form of the indicated standard basis, (m ) a more substantial modification of the indicated standard basis, and (s ) designates custom basis. Please refer to the original papers for complete details of the basis sets used. b Anisotropy with respect to the z direction of the CC bond, with the molecule in the xz plane. c Anisotropy defined in the principal axis frame of the tensor. d Insufficient information given for obtaining principal values. e Including corrections for the relativistic spin–orbit interaction. f Error in the original paper [78]. Jaa along the internuclear axis and Jcc makes an angle of 48 with the normal of the local HXH plane, towards the F atom. g Anisotropy with respect to z direction bisecting the FCF angle, with the F atoms in the xz plane. h Coupling between the CH3 group protons. i Coupling between the SiH3 group protons. j At the optimized geometry [40]. k cis-Coupling. l trans-Coupling. m Parameters averaged over trans and gauche positions. n Average coupling between the methyl and silyl groups. J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 Table 14 Quantum chemically calculated components of the symmetric part J1 þ J S of the a 285 13 1 C H spin–spin coupling tensors System Bonds Theory Values Reference CH4 CH4 C2H2 C2H2 C2H2 C2H4 1 1 1 1 1 1 SOPPA(CCSD)/(s ) MP2/(s ) SOPPA(CCSD)/(s ) RAS/HIV ZORA DFT(GGA)/Slater RAS/HIV [321] [227] [214,215] [72] [144] [72] C2H6 C6H6 HCN CH3CN CH3NC HCONH2 CH3F CH2F2 1 1 1 1 1 1 1 1 RAS/HIII CAS/(s ) RAS/HIV RAS/HIII RAS/HIII RAS/HIII RAS/HIII RAS/HIII CHF3 CH3SiH3 C2H2 C2H2 C2H2 C2H4 C2H6 C6H6 HNC CH3CN HCONH2 HCONH2 CH3SiH3 C6H6 CH3NC C6H6 1 1 2 2 2 2 2 2 2 2 2f 2g 2 3 3 4 RAS/HIII RAS/HIII SOPPA(CCSD)/(s ) RAS/HIV ZORA DFT(GGA)/Slater RAS/HIV RAS/HIII CAS/(s ) RAS/HIV RAS/HIII RAS/HIII RAS/HIII RAS/HIII CAS/(s ) RAS/HIII CAS/(s ) DJ ¼ 225.6, J ¼ 123.8b DJ ¼ 64.44, J ¼ 130.63b DJ ¼ 263.41, J ¼ 254.95 DJ ¼ 262.4, J ¼ 232.1 DJ ¼ 238.7, J ¼ 262 DJ ¼ 2.6, Jxx 2 Jyy ¼ 228.8c, Jaa ¼ 123.9, Jbb ¼ 158.0, Jcc ¼ 161.2 DJ ¼ 6.0, Jaa ¼ 102.7, Jbb ¼ 128.3, Jcc ¼ 128.5 DJ ¼ 28.0, Jaa ¼ 144.0, Jbb ¼ 190.8, Jcc ¼ 195.4 DJ ¼ 263.34, J ¼ 249.27 DcJ ¼ 228.82, J ¼ 142.43d DcJ ¼ 224.70, J ¼ 143.50d Jaa ¼ 161.5, Jbb ¼ 192.9, Jcc ¼ 195.2 DJ ¼ 6.10, Jaa ¼ 122.00, Jbb ¼ 149.84, Jcc ¼ 152.62 DJ ¼ 27.02, Jxx 2 Jyy ¼ 212.26e, Jcc ¼ 186.00, Jbb ¼ 184.15, Jaa ¼ 156.88 DJ ¼ 231.19, J ¼ 236.79 DJ ¼ 7.10, Jaa ¼ 96.73, Jbb ¼ 124.85, Jcc ¼ 125.63 DJ ¼ 31.12, J ¼ 51.73 DJ ¼ 28.2, J ¼ 50.1 DJ ¼ 39.3, J ¼ 52.3 DJ ¼ 5.2, Jxx 2 Jyy ¼ 6.0c, Jaa ¼ 21.1, Jbb ¼ 23.0, Jcc ¼ 28.1 DJ ¼ 21.8, Jaa ¼ 22.5, Jbb ¼ 26.6, Jcc ¼ 27.0 DJ ¼ 29.2, Jaa ¼ 21.7, Jbb ¼ 27.1, Jcc ¼ 213.5 DJ ¼ 33.34, J ¼ 16.44 DcJ ¼ 5.12, J ¼ 215.46d Jaa ¼ 0.4, Jbb ¼ 2.0, Jcc ¼ 6.0 Jaa ¼ 22.2, Jbb ¼ 24.1, Jcc ¼ 26.3 DJ ¼ 0.41, Jaa ¼ 2.79, Jbb ¼ 3.58, Jcc ¼ 3.84 DJ ¼ 3.3, Jaa ¼ 8.6, Jbb ¼ 12.5, Jcc ¼ 13.9 DcJ ¼ 5.21, J ¼ 2.63d DJ ¼ 26.9, Jaa ¼ 0.6, Jbb ¼ 25.3, Jcc ¼ 29.2 [72] [70] [324] [324] [324] [404,405] [78] [78] [78] [80] [214,215] [72] [144] [72] [72] [70] [324] [324] [404,405] [404,405] [80] [70] [324] [70] Either the principal values of the symmetrized tensor (Jii, i ¼ a; b; c with lJaa l # lJbb l # lJcc l), specified Cartesian components (xx, xy, etc.), or the anisotropy with respect to a certain axis (e.g. Dz J ¼ Jzz 2 1=2ðJxx þ Jyy Þ), is indicated. Unless otherwise noted, in systems possessing a unique n-fold rotation axis (n $ 2), the anisotropy is defined with respect to that axis. Results in Hz. a See footnote a in Table 13. b Anisotropy along the bond in question. c See footnote b in Table 13. d See footnotes c and d in Table 13. e See footnote g in Table 13. f Coupling to trans-hydrogen with respect to the oxygen atom. g Coupling to cis-hydrogen with respect to the oxygen atom. SOPPA calculation of Ref. [318] produced a different ordering for the two smallest principal values of the tensor, the reason for the difference being probably the basis sets used. The data for 1 JOH [228] has stood the test of time remarkably well. Galasso and Fronzoni applied the EOM method using the small 6-31G** basis set on a variety of simple organic molecules [407]. Information on the anisotropic properties of a number of 1 J was given, but limited to DJ with respect to the direction of the bond only. The results for J are in qualitative agreement with the available experimental data, although, for triple bonds the agreement is worse than for double bonds, and the couplings over 286 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 Table 15 Quantum chemically calculated components of the symmetric part J1 þ J S of the 15 N1H spin–spin coupling tensors System Bonds Theorya Valuesb Reference NH2 2 NH3 NHþ 4 HNC N2H4 BH2NH2 BH3NH3 HCONH2 HCONH2 PH2NH2 HCN CH3NC HCONH2 CH3CN 1 1 1 1 1 1 1 1f 1g 1 2 2 2 3 SCF/(s ) MP2/(s ) SCF/(s ) RAS/HIV SOS-CI/(s ) SOS-CI/(s ) SOS-CI/(s ) RAS/HIII RAS/HIII SOS-CI/(s ) RAS/HIV RAS/HIII RAS/HIII RAS/HIII Jaa ¼ 22.26, Jbb ¼ 46.39, Jcc ¼ 68.41 DJ ¼ 15.70, J ¼ 260.07 DJ ¼ 220.12, J ¼ 68.96c DJ ¼ 36.39, J ¼ 2112.61 DJ ¼ 21.82, J ¼ 285.36d,e DJ ¼ 20.28, J ¼ 298.44d DJ ¼ 21.52, J ¼ 288.48d Jaa ¼ 274.8, Jbb ¼ 2100.4, Jcc ¼ 2103.5 Jaa ¼ 274.6, Jbb ¼ 299.6, Jcc ¼ 2102.7 DJ ¼ 25.65, J ¼ 2103.82d,h DJ ¼ 219.51, J ¼ 26.44 DcJ ¼ 21.91, J ¼ 4.46i Jaa ¼ 215.6, Jbb ¼ 216.0, Jcc ¼ 217.4 DcJ ¼ 24.30, J ¼ 22.03i [41] [227] [41] [324] [325] [325] [325] [404,405] [404,405] [325] [324] [324] [404,405] [324] Either the principal values of the symmetrized tensor (Jii, i ¼ a; b; c with lJaa l # lJbb l # lJcc l), specified Cartesian components (xx, xy, etc.), or the anisotropy with respect to a certain axis (e.g. Dz J ¼ Jzz 2 1=2ðJxx þ Jyy Þ), is indicated. Unless otherwise noted, in systems possessing a unique n-fold rotation axis (n $ 2), the anisotropy is defined with respect to that axis. Results in Hz. a See footnote a in Table 13. b Results converted for 15N when necessary. c See footnote b in Table 14. d Anisotropy with respect to the axis joining the heavy nuclei of the system. See also footnote d in Table 13. e Values at the dihedral angle value corresponding to the equilibrium geometry. f See footnote f in Table 14. g See footnote g in Table 14. h Planar configuration, dihedral angle 908 [325]. i See footnotes c and d in Table 13. Table 16 Quantum chemically calculated components of the symmetric parts J1 þ J S of the 19 1 F H spin–spin coupling tensors System Bonds Theorya Values Reference HF HF HF HF HF HF CH3F CH2F2 1 1 1 1 1 1 2 2 DHF/cc-pVTZ(u ) SCF/cc-pVTZ(u ) CAS/cc-pV5Z CAS þ SOb/HIVu3 CAS/HIVu3 MP2/(s ) RAS/HIII RAS/HIII [335] [335] [143] [38] [38] [227] [78] [78] CHF3 p-C6H4F2 p-C6H4F2 2 3 4 RAS/HIII RAS/HII(m ) RAS/HII(m ) DJ ¼ 160.19, J ¼ 610.42 DJ ¼ 158.68, J ¼ 612.23 DJ ¼ 115.98, J ¼ 476.09 DJ ¼ 127.3, J ¼ 534.7 DJ ¼ 126.7, J ¼ 534.8 DJ ¼ 2715.92, J ¼ 570.01 DJ ¼ 256.73, Jaa ¼ 8.59, Jbb ¼ 37.00, Jcc ¼ 100.76 DJ ¼ 23.51, Jxx 2 Jyy ¼ 44.71c, Jaa ¼ 22.87, Jbb ¼ 45.38, Jcc ¼ 87.38 DJ ¼ 40.57, Jaa ¼ 60.17, Jbb ¼ 70.67, Jcc ¼ 107.11 DJ ¼ 16.0, Jxx 2 Jyy ¼ 15.1d, Jaa ¼ 2.6, Jbb ¼ 211.8, Jcc ¼ 12.4 DJ ¼ 21.4, Jxx 2 Jyy ¼ 24.6d, Jaa ¼ 2.7, Jbb ¼ 8.2, Jcc ¼ 9.6 [78] [60] [60] Either the principal values of the symmetrized tensor (Jii, i ¼ a; b; c with lJaa l # lJbb l # lJcc l), specified Cartesian components (xx, xy, etc.), or the anisotropy with respect to a certain axis (e.g. Dz J ¼ Jzz 2 1=2ðJxx þ Jyy Þ), is indicated. Unless otherwise noted, in systems possessing a unique n-fold rotation axis (n $ 2), the anisotropy is defined with respect to that axis. Results in Hz. a See footnote a in Table 13. b See footnote e in Table 13. c See footnote g in Table 13. d Anisotropy with respect to the z direction of the FF internuclear axis, with the molecule in the xz plane. J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 287 Table 17 Quantum chemically calculated components of the symmetric parts K1 þ K S of the X1H spin–spin coupling tensors other than those listed in Tables 13 –16 a System X Bonds Theory Values LiH BH2 4 B2H4 B2H6 B2H6 BH2NH2 BH3NH3 BH2PH2 BH3PH3 H2O H2O H2O H2O H2O H2O H2O HCONH2 HCONH2 HCONH2 AlH2 4 SiH4 SiH4 SiH4 CH3SiH3 CH3SiH3 PH2 2 PH3 PH3 PHþ 4 P2H4 BH2PH2 BH3PH3 PH2NH2 H2S H2S H2S HCl HCl HCl HCl HCl HCl H2Se H2Se HBr HBr HBr HBr H2Te H2Te 7 Li B 11 B 11 f B 11 g B 11 B 11 B 11 B 11 B 17 O 17 O 17 O 17 O 17 O 17 O 17 O 17 O 17 O 17 O 27 Al 29 Si 29 Si 29 Si 29 Si 29 Si 31 P 31 P 31 P 31 P 31 P 31 P 31 P 31 P 33 S 33 S 33 S 35 Cl 35 Cl 35 Cl 35 Cl 35 Cl 35 Cl 77 Se 77 Se 79 Br 79 Br 79 Br 79 Br 125 Te 125 Te 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3j 3k 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 CAS/cc-pV5Z SCF/(s ) SOS-CI/(s ) SOS-CI/(s ) SOS-CI/(s ) SOS-CI/(s ) SOS-CI/(s ) SOS-CI/(s ) SOS-CI/(s ) SOPPA(CCSD)/(s ) CAS/(s ) CAS þ SOh/HIVu4 CAS/HIVu4 SOPPA/(s ) SOPPA/(s ) MP2/(s ) RAS/HIII RAS/HIII RAS/HIII FOPPA/(s ) SOPPA(CCSD)/aug-cc-pVTZ(m ) MP2/6-31G(**) FOPPA/(s ) RAS/HIII RAS/HIII SCF/(s ) MP2/6-31G(**) SCF/(s ) SCF/(s ) SOS-CI/(s ) SOS-CI/(s ) SOS-CI/(s ) SOS-CI/(s ) CAS þ SOh/HIVu4 CAS/HIVu4 MP2/6-31G(**) DHF/cc-pVTZ(u) SCF/cc-pVTZ(u) CAS/aug-cc-pVQZ CAS þ SOh/HIVu3 CAS/HIVu3 MP2/6-31G(**) CAS þ SOh/HIVu4 CAS/HIVu4 DHF/(s ) SCF/(s ) CAS þ SOh/HIVu3 CAS/HIVu3 CAS þ SOh/HIVu3 CAS/HIVu3 DK ¼ 22.8, K ¼ 32.3b [143] DK ¼ 210.35, K ¼ 24.28c [41] DK ¼ 0.40, K ¼ 40.08d,e [325] DK ¼ 20.59, K ¼ 47.12d [325] DK ¼ 0.20, K ¼ 18.06d [325] DK ¼ 0.09, K ¼ 40.83d [325] DK ¼ 1.00, K ¼ 38.52d [325] DK ¼ 20.49, K ¼ 47.72d [325] DK ¼ 1.27, K ¼ 41.21d [325] Kaa ¼ 45.50, Kbb ¼ 48.37, Kcc ¼ 56.33 [318] Kaa ¼ 47.38, Kbb ¼ 49.94, Kcc ¼ 57.24 [318] Kaa ¼ 46.47, Kbb ¼ 50.14, Kcc ¼ 57.29 [38] Kaa ¼ 46.45, Kbb ¼ 50.27, Kcc ¼ 57.34 [38] Kaa ¼ 45.19, Kbb ¼ 49.49, Kcc ¼ 57.09 [280,318] Kaa ¼ 39.58, Kbb ¼ 47.03, Kcc ¼ 54.33 [228] DK ¼ 10.15, Kxx 2 Kyy ¼ 51.29, K ¼ 45.87i [227] Kaa ¼ 1.6, Kbb ¼ 2.6, Kcc ¼ 7.2 [404,405] Kaa ¼ 20.4, Kbb ¼ 20.8, Kcc ¼ 3.0 [404,405] Kaa ¼ 20.4, Kbb ¼ 21.0, Kcc ¼ 21.6 [404,405] DK ¼ 2.00, K ¼ 46.82c [406] DK ¼ 9.53, K ¼ 80.42c [321] DK ¼ 53.61, K ¼ 78.47c [227] DK ¼ 11.72, K ¼ 98.95c [406] DK ¼ 22.43, Kaa ¼ 73.90, Kbb ¼ 74.10, Kcc ¼ 81.72 [80] DK ¼ 21.07, Kaa ¼ 22.33, Kbb ¼ 24.78, Kcc ¼ 25.05 [80] Kaa ¼ 12.93, Kbb ¼ 14.95, Kcc ¼ 27.65l [40] DK ¼ 221.34, K ¼ 41.41 [227] DK ¼ 29.33, Kaa ¼ 33.49, Kbb ¼ 38.22, Kcc ¼ 60.50 [40] DK ¼ 9.84, K ¼ 135.96c [40] DK ¼ 216.33, K ¼ 32.01d,e [325] DK ¼ 24.99, K ¼ 64.24d [325] DK ¼ 28.36, K ¼ 82.96d [325] DK ¼ 29.84, K ¼ 39.80d,m [325] Kaa ¼ 22.80, Kbb ¼ 30.39, Kcc ¼ 66.24 [38] Kaa ¼ 23.74, Kbb ¼ 30.57, Kcc ¼ 66.19 [38] DK ¼ 19.62, Kxx 2 Kyy ¼ 285.75, K ¼ 40.98i [227] DK ¼ 71.40, K ¼ 26.27 [335] DK ¼ 70.84, K ¼ 27.46 [335] DK ¼ 51.9, K ¼ 50.0 [143] DK ¼ 54.68, K ¼ 37.62 [38] DK ¼ 54.51, K ¼ 37.66 [38] DK ¼ 256.63, K ¼ 20.75 [227] Kaa ¼ 24.46, Kbb ¼ 17.05, Kcc ¼ 121.69 [38] Kaa ¼ 6.21, Kbb ¼ 19.28, Kcc ¼ 121.30 [38] DK ¼ 216.29, K ¼ 215.82 [335] DK ¼ 206.10, K ¼ 4.81 [335] DK ¼ 140.05, K ¼ 34.07 [38] DK ¼ 138.43, K ¼ 34.69 [38] Kaa ¼ 228.98, Kbb ¼ 18.56, Kcc ¼ 191.72 [38] Kaa ¼ 8.44, Kbb ¼ 23.22, Kcc ¼ 190.97 [38] (continued on next page) 11 Reference 288 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 Table 17 (continued) a System X Bonds Theory Values Reference HI HI HI HI PbH4 127 1 1 1 1 1 DHF/(s ) SCF/(s ) CAS þ SOh/HIVu3 CAS/HIVu3 ZORA DFT(LDA)/Slater DK ¼ 369.82, K ¼ 2113.20 DK ¼ 340.17, K ¼ 212.97 DK ¼ 216.57, K ¼ 40.22 DK ¼ 213.78, K ¼ 41.01 DK ¼ 672, K ¼ 1121c [335] [335] [38] [38] [144] I I 127 I 127 I 207 Pb 127 Either the principal values of the symmetrized tensor (Kii, i ¼ a; b; c with lKaa l # lKbb l # lKcc l), specified Cartesian components (xx, xy, etc.), or the anisotropy with respect to a certain axis (e.g. Dz K ¼ Kzz 2 1=2ðKxx þ Kyy Þ), is indicated. Unless otherwise noted, in systems possessing a unique n-fold rotation axis (n $ 2), the anisotropy is defined with respect to that axis. Reduced couplings are given in units of 1019 T2 J21. a See footnote a in Table 13. b Corrections carried out for the n ¼ 0, J ¼ 1 rovibrational state. c See footnote b in Table 14. d See footnote d in Table 15. e See footnote e in Table 15. f Coupling to terminal hydrogen. g Coupling to bridging hydrogen. h See footnote e in Table 13. i Anisotropy with respect to the z direction along the C2 molecular symmetry axis. The y direction is perpendicular to the plane of the molecule. See also footnote d in Table 13. j See footnote f in Table 14. k See footnote g in Table 14. l See footnote j in Table 13. m See footnote h in Table 15. single-bonds are rather good. The method was limited to the one-particle/one hole excitation level, hence besides basis set limitations, further correlation contributions are to be expected. The calculated DJ may be compared with later theoretical work for ethene and ethyne (see below) that seems to have settled at qualitatively different total D1 JCC values, mainly due to the quite different magnitudes of the SD/FC contribution as compared to the EOM work. As in the case of isotropic J, the small basis sets and modest correlation treatment used in this study [407] of DJ seem to be the main reasons for the differences. The fact that the DSO coupling anisotropies are very different from what the current calculations are able to provide [72], is harder to understand as this contribution is not affected very much by correlation or basis set effects. Fukui et al. investigated the simple first- and second-row hydrides CH4, SiH4, NH3, PH3, H2O, H2S, HF, and HCl using FPT MP2 calculations and modest Pople-type basis sets [227]. While the agreement with the experimental isotropic JXH is reasonable in the first-row hydrides, the results for all JHH as well as JXH in the second row hydrides are disappointing. In addition to the fact that both the basis sets used as well as the MP2 correlation treatment leave lots of room for improvement, there seems to be something wrong in the calculated anisotropic properties. Later calculations (cited below) systematically disagree with the results of Fukui et al. [227] in the order of magnitude and even the sign of the individual Cartesian components of the tensors. MCSCF studies. Barszczewicz et al. carried out one of the first theoretical investigations of the tensorial properties of J that can be considered modern in terms of adequate treatment of the electron correlation problem and large one-electron basis sets [324]. The HCN, HNC, CH3CN and CH3NC systems were investigated at the RASSCF LR level using moderately large active spaces and the HIII and HIV basis sets. The J values were in semi-quantitative agreement with the experimental results, giving confidence also to the calculated anisotropic properties for which the experimental data set is much more sparse. The remaining errors in J for these systems may be caused by solvent effects to a large extent. It should be noted in this context that the calculated DJ for non-axial couplings in CH3CN and CH3NC were J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 Table 18 Quantum chemically calculated components of the symmetric part J1 þ J S of the a 289 13 13 C C spin–spin coupling tensors System Bonds Theory Values Reference C2H2 C2H2 C2H2 C2H2 C2H4 C2H4 C2H4 C2H6 C2H6 H2CyCyCH2 HCxC–CxCH HCxC–CxCH C6H6 CH3CN H2CyCyNH CH2CO OCyCyCO C6H6 CH3NC C6H6 1 1 1 1 1 1 1 1 1 1 1d 1e 1 1 1 1 1 2 2 3 SOPPA(CCSD)/(s ) RAS/HIV EOM/6-31G** ZORA DFT(GGA)/Slater RAS/HIV EOM/6-31G** ZORA DFT(GGA)/Slater RAS/HIII ZORA DFT(GGA)/Slater EOM/6-31G** EOM/6-31G** EOM/6-31G** RAS/HII RAS/HIII EOM/6-31G** EOM/6-31G** EOM/6-31G** RAS/HII RAS/HIII RAS/HII DJ ¼ 49.55, J ¼ 190.00 DJ ¼ 47.5, J ¼ 181.2 DJ ¼ 24.32, J ¼ 216.99 DJ ¼ 72.1, J ¼ 186.6 DJ ¼ 26.5, Jxx 2 Jyy ¼ 244.3b, Jaa ¼ 39.2, Jbb ¼ 83.6, Jcc ¼ 87.9 DJ ¼ 1.29, J ¼ 82.37c DJ ¼ 38.8, J ¼ 59.2c DJ ¼ 32.1, J ¼ 38.8 DJ ¼ 34.0, J ¼ 23.8 DJ ¼ 25.39, J ¼ 109.64c DJ ¼ 24.62, J ¼ 225.87 DJ ¼ 3.61, J ¼ 157.92 DJ ¼ 11.0, Jaa ¼ 44.9, Jbb ¼ 78.2, Jcc ¼ 89.5 DJ ¼ 36.57, J ¼ 71.97c DJ ¼ 11.19, J ¼ 111.21c DJ ¼ 212.74, J ¼ 112.43c DJ ¼ 6.36, J ¼ 221.07 DJ ¼ 212.7, Jaa ¼ 20.6, Jbb ¼ 20.8, Jcc ¼ 213.5 DJ ¼ 11.64, J ¼ 25.23c DJ ¼ 12.8, Jaa ¼ 13.3, Jbb ¼ 16.4, Jcc ¼ 27.6 [214,215] [72] [407] [144] [72] [407] [144] [72] [144] [407] [407] [407] [70] [324] [407] [407] [407] [70] [324] [70] Either the principal values of the symmetrized tensor (Jii, i ¼ a; b; c with lJaa l # lJbb l # lJcc l), specified Cartesian components (xx, xy, etc.), or the anisotropy with respect to a certain axis (e.g. Dz J ¼ Jzz 2 1=2ðJxx þ Jyy Þ), is indicated. Unless otherwise noted, in systems possessing a unique n-fold rotation axis (n $ 2), the anisotropy is defined with respect to that axis. Results in Hz. a See footnote a in Table 13. b See footnote b in Table 13. c See footnote d in Table 13. d Over the triple bond. e Over the single bond. Table 19 Quantum chemically calculated components of the symmetric part J1 þ J S of the 15 N13C spin– spin coupling tensors System Bonds Theorya Valuesb Reference HCN HNC CH3CN CH3NC CH3NC CH3CN H2CyCyNH CH2N2 HCNO HCONH2 1 1 1 1c 1d 2 1 1 1 1 RAS/HIV RAS/HIV RAS/HIII RAS/HIII RAS/HIII RAS/HIII EOM/6-31G** EOM/6-31G** EOM/6-31G** RAS/HIII DJ ¼ 254.64, J ¼ 219.83 DJ ¼ 250.49, J ¼ 210.47 DJ ¼ 250.70, J ¼ 221.55 DJ ¼ 247.06, J ¼ 212.57 DJ ¼ 217.04, J ¼ 219.26 DJ ¼ 27.66, J ¼ 2.82 DJ ¼ 25.89, J ¼ 231.87e DJ ¼ 26.92, J ¼ 228.54e DJ ¼ 29.16, J ¼ 259.16 Jaa ¼ 26.6, Jbb ¼ 214.9, Jcc ¼ 232.4 [324] [324] [324] [324] [324] [324] [407] [407] [407] [404,405] Either the principal values of the symmetrized tensor (Jii, i ¼ a; b; c with lJaa l # lJbb l # lJcc l), specified Cartesian components (xx, xy, etc.), or the anisotropy with respect to a certain axis (e.g. Dz J ¼ Jzz 2 1=2ðJxx þ Jyy Þ), is indicated. Unless otherwise noted, in systems possessing a unique n-fold rotation axis (n $ 2), the anisotropy is defined with respect to that axis. Results in Hz. a See footnote a in Table 13. b Results converted for 15N when necessary. c See footnote d in Table 18. d See footnote e in Table 18. e See footnote d in Table 13. 290 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 Table 20 Quantum chemically calculated components of the symmetric part J1 þ J S of the a 19 13 F C spin–spin coupling tensors System Bonds Theory Values Reference CH3F CH2F2 1 1 RAS/HIII RAS/HIII [78] [78] CHF3 p-C6H4F2 1 1 RAS/HIII RAS/HII(m ) p-C6H4F2 2 RAS/HII(m ) p-C6H4F2 p-C6H4F2 3 4 RAS/HII(m ) RAS/HII(m ) DJ ¼ 207.84, J ¼ 2156.56 DJ ¼ 10.39, Jxx 2 Jyy ¼ 2280.33b, Jaa ¼ 236.62, Jbb ¼ 2261.19, Jcc ¼ 2364.34 DJ ¼ 2173.34, Jaa ¼ 25.47, Jbb ¼ 2333.49, Jcc ¼ 2387.25 DJ ¼ 368.8, Jxx 2 Jyy ¼ 11.5c, Jaa ¼ 61.1, Jbb ¼ 2301.9, Jcc ¼ 2313.4 DJ ¼ 236.9, Jxx 2 Jyy ¼ 219.4c, Jaa ¼ 215.8, Jbb ¼ 64.5, Jcc ¼ 78.8 DJ ¼ 37.5, Jxx 2 Jyy ¼ 29.1c, Jaa ¼ 2.2, Jbb ¼ 223.5, Jcc ¼ 31.9 DJ ¼ 219.2, Jxx 2 Jyy ¼ 234.0c, Jaa ¼ 23.2, Jbb ¼ 25.4, Jcc ¼ 30.8 [78] [60] [60] [60] [60] Either the principal values of the symmetrized tensor (Jii, i ¼ a; b; c with lJaa l # lJbb l # lJcc l), specified Cartesian components (xx, xy, etc.), or the anisotropy with respect to a certain axis (e.g. Dz J ¼ Jzz 2 1=2ðJxx þ Jyy Þ), is indicated. Unless otherwise noted, in systems possessing a unique n-fold rotation axis (n $ 2), the anisotropy is defined with respect to that axis. Results in Hz. a See footnote a in Table 13. b See footnote g in Table 13. c See footnote d in Table 16. reported [324] in their PAS(J ). Thus, they should not be directly compared to experimental data obtained by LCNMR, as the latter refer to the frame used to represent the orientation tensor. The calculations of Ref. [324] nicely imply transferability of the properties of certain type of couplings between different molecules containing similar structural units. Kaski et al. [70] investigated the n JCH ðn ¼ 1; 2; 3; 4Þ and n JCC ðn ¼ 1; 2; 3Þ coupling tensors in benzene using CASSCF and RASSCF LR calculations. While the size of the molecule prohibited reaching definite convergence of results as a function of the size of the basis set (the standard HII and modified triple-zeta sets were used) and particularly the length of the determinantal expansion, convergence of the calculated results towards the experimental data could be established. In particular, the FC and SD/FC terms in the tensor were seen to dramatically decrease upon improving the correlation treatment. The PSO term was finally left as the dominant contributor to D1 JCC : For the n JCC that constituted the main objective, the experimental sign patterns of both J and DJ were reproduced. The magnitudes of most of the calculated parameters are somewhat overestimated. Together with the experimental findings, the results indicate that Table 21 Quantum chemically calculated components of the symmetric parts K1 þ K S of the X13C spin–spin coupling tensors other than those listed in Tables 14 and 18–20 System X Bonds Theorya Values Reference HCONH2 CH2CO OCyCyCO CH3SiH3 17 1 1 1 1 RAS/HIII EOM/6-31G** EOM/6-31G** RAS/HIII Kaa ¼ 237.3, Kbb ¼ 42.2, Kcc ¼ 2157 DK ¼ 101.1, K ¼ 276.88b DK ¼ 124.1, K ¼ 291.28 DK ¼ 98.76, K ¼ 100.7 [404,405] [407] [407] [80] O O 17 O 19 Si 17 Either the principal values of the symmetrized tensor (Kii, i ¼ a; b; c with lKaa l # lKbb l # lKcc l), specified Cartesian components (xx, xy, etc.), or the anisotropy with respect to a certain axis (e.g. Dz K ¼ Kzz 2 1=2ðKxx þ Kyy Þ), is indicated. Unless otherwise noted, in systems possessing a unique n-fold rotation axis (n $ 2), the anisotropy is defined with respect to that axis. Reduced couplings are given in units of 1019 T2 J21. a See footnote a in Table 13. b See footnote d in Table 13. J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 291 Table 22 Quantum chemically calculated components of the symmetric parts K1 þ K S of the NX spin–spin coupling tensors other than those listed in Tables 15 and 19 System X Bonds Theorya Values Reference BH2NH2 BH3NH3 CH2N2 N2H4 HCNO HCONH2 PH2NH2 11 1 1 1 1 1 2 1 SOS-CI/(s ) SOS-CI(s ) EOM/6-31G** SOS-CI(s ) EOM/6-31G** RAS/HIII SOS-CI(s ) DK ¼ 30.98, K ¼ 101.4b,c DK ¼ 20.7, K ¼ 18.2b,c DK ¼ 104.8, K ¼ 39.1b DK ¼ 91.33, K ¼ 4.4b,c,d DK ¼ 124.4, K ¼ 2171.9 Kaa ¼ 30.6, Kbb ¼ 239.1, Kcc ¼ 41.6 DK ¼ 179.3, K ¼ 230.42b,c,e [325] [325] [407] [325] [407] [404,405] [325] B B 15 N 15 N 17 O 17 O 31 P 11 Either the principal values of the symmetrized tensor (Kii, i ¼ a; b; c with lKaa l # lKbb l # lKcc l), specified Cartesian components (xx, xy, etc.), or the anisotropy with respect to a certain axis (e.g. Dz K ¼ Kzz 2 1=2ðKxx þ Kyy Þ), is indicated. Unless otherwise noted, in systems possessing a unique n-fold rotation axis (n $ 2), the anisotropy is defined with respect to that axis. Reduced couplings are given in units of 1019 T2 J21. a See footnote a in Table 13. b See footnote d in Table 13. c See footnote d in Table 15. d See footnote e in Table 15. e See footnote h in Table 15. the ð1=2ÞJ aniso contribution to Dexp CC in aromatic systems is in the range of 2% or less. The JCH tensors were only calculated using a modest CASSCF wave function, with the calculated 2 JCH having the wrong sign as compared to experiment and the other parameters overestimated. RASSCF LR calculations were reported for all the coupling tensors in formamide (HCONH 2 ) in Refs. [404,405]. This is a biologically relevant model molecule and displays a rich variety of NMR observables. The active spaces were large; however, the largest one was given only a single-reference wave function treatment (due to computational limitations at the time) with single and double excitations into the virtual orbitals. Consequently, there may still be room for improvement in the correlation treatment. The HIII basis set was used, hence additional error limits of a few % must be allowed due to the lack of tight functions. The results for J compare well with experiment apart from couplings to 17O for which experimental results are not available. Experiments [404,405] for the anisotropic observables could only verify the qualitative features of the calculated data due to the low experimental order parameters and hence large uncertainty. It is likely that most of the calculated anisotropic couplings are reliable, judging also from the convergence of the results in the sequence of improved wave functions. A possible exception is formed by the 17O couplings. Using the experimental S tensor obtained in the work, the calculated J aniso gives a negligible contribution to Dexp : The prototypical hydrocarbon series ethane, ethene, and ethyne was studied at the RASSCF LR level using large active spaces and the HIV (HIII for ethane) basis set [72]. The goal was to investigate the properties of 1 JCC as a function of the hybridization of the coupled carbons. Judging by the generally well calculated isotropic J for all the couplings, the anisotropic properties should also be of high quality. Indeed, a qualitative agreement of the theoretical D1 JCC (and JCC;xx 2 JCC;yy for C2H4) with the results of LCNMR experiments [72] was found. Both the theoretical and experimental results point out, together with the previous study on benzene [70], that the tensorial properties of JCC may be neglected in comparison with the direct coupling regardless of the hybridization. The anisotropy along the CC bond displays a minimum for the sp2-hybridized ethene, despite the monotonically decreasing JCC from ethyne to ethene and ethane. A SOPPA(CCSD) or a full CCSD calculation could be used to verify this. The different contributions to D1 JCC evolve from the SD/ FC dominance in the sp3 carbons to the large PSO term of the sp1 case. 292 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 Table 23 Quantum chemically calculated components of the symmetric parts K1 þ K S of the 19FX spin– spin coupling tensors other than those listed in Tables 16 and 20 System X Bonds Theorya Values Reference LiF BF OF2 ClF3 CH2F2 7 Li B 17 O 19 F 19 F 1 1 1 2c 2 CAS/cc-pV5Z CAS/cc-pV5Z RAS/cc-pCVQZ RAS/cc-pVQZ RAS/HIII [143] [143] [27] [27] [78] CHF3 p-C6H4F2 19 2 5 RAS/HIII RAS/HII(m ) NaF AlF ClF ClF ClF3 ClF3 KF BrF IF TlF 23 1 1 1 1 1g 1h 1 1 1 1 CAS(s ) CAS/aug-cc-pVQZ CAS/aug-cc-pVQZ ZORA DFT(GGA)/Slater RAS/cc-pVQZ RAS/cc-pVQZ CAS/(s ) ZORA DFT(GGA)/Slater ZORA DFT(GGA)/Slater ZORA DFT(GGA)/Slater DK ¼ 240.25, K ¼ 45.28b DK ¼ 129.4, K ¼ 261.3 Kaa ¼ 84.1, Kbb ¼ 286.7, Kcc ¼ 607 Kaa ¼ 6.4, Kbb ¼ 24.4, Kcc ¼ 83.1 DK ¼ 224.64, Kxx 2 Kyy ¼ 213.11d, Kaa ¼ 16.11, Kbb ¼ 34.19, Kcc ¼ 47.29 DK ¼ 221.81, Kaa ¼ 214.07, Kbb ¼ 17.82, Kcc ¼ 39.21 DK ¼ 23.40, Kxx 2 Kyy ¼ 23.58e, Kaa ¼ 20.23, Kbb ¼ 1.37, Kcc ¼ 4.95 DK ¼ 165.1, K ¼ 64.8 DK ¼ 188.4, K ¼ 2213 DK ¼ 2721, K ¼ 747f DK ¼ 2982, K ¼ 872 Kaa ¼ 211.7, Kbb ¼ 298.3, Kcc ¼ 638 Kaa ¼ 83, Kbb ¼ 2167, Kcc ¼ 528 DK ¼ 207.3, K ¼ 148.0b DK ¼ 22123, K ¼ 1886 DK ¼ 22955, K ¼ 2241 DK ¼ 2324, K ¼ 22034 11 F F 19 Na Al 35 Cl 35 Cl 35 Cl 35 Cl 39 K 79 Br 127 I 205 Tl 27 [78] [60] [143] [143] [143] [144] [27] [27] [143] [144] [144] [144] Either the principal values of the symmetrized tensor (Kii, i ¼ a; b; c with lKaa l # lKbb l # lKcc l), specified Cartesian components (xx, xy, etc.), or the anisotropy with respect to a certain axis (e.g. Dz K ¼ Kzz 2 1=2ðKxx þ Kyy Þ), is indicated. Unless otherwise noted, in systems possessing a unique n-fold rotation axis (n $ 2), the anisotropy is defined with respect to that axis. Reduced couplings are given in units of 1019 T2 J21. a See footnote a in Table 13. b Corrections carried out for the n ¼ 0, J ¼ 0 rovibrational state. c Coupling between equatorial and axial fluorine atoms. d See footnote g in Table 13. e See footnote d in Table 16. f Corrections carried out for the n ¼ 0, J ¼ 1 rovibrational state. g Coupling to the equatorial fluorine atom. h Coupling to the axial fluorine atom. Kaski et al. studied CH3SiH3 using RASSCF LR with a large active space [80]. The basis was the standard HIII set, thus a priori restricting the accuracy somewhat. The agreement with experiment is semiquantitative, with particularly 1 JSiC and D1 JSiC overand underestimated, respectively. This may be partially due to neglecting correlation of the Si semicore aniso would orbitals [240,257]. The neglect of 1 JSiC correspond to only a 1% error in the CSi bond length. Ref. [60] investigated the couplings to 19F in pC6H4F2 using the RASSCF LR method. For systems of this size, compromises in the basis set—a HII set supplemented with tight s-primitives—and the correlation treatment had to be made. The results generally show a qualitative agreement, of signs and orders of magnitude, as well as evolution of results when improving the wave function, with the observed J. The same applies for the anisotropic properties from the LCNMR experiment, although the analysis of the experimental data was not completely independent of the calculation. The experimental 1 JFC coupling tensor would likely be particularly difficult to reproduce theoretically. A calculation featuring a more efficient electron correlation treatment, as well as estimates of intermolecular and rovibrational effects would be interesting. The contribution of J aniso to the experimentally observable long-range 3;4 exp DFC and 5 Dexp FF couplings was estimated to exceed J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 293 Table 24 Quantum chemically calculated components of the symmetric parts K1 þ K S of the XY spin–spin coupling tensors other than those listed in Tables 13 –23 System XY Bonds Theorya Values Reference B2H4 B2H6 BH2PH2 BH3PH3 Na2 KNa P2H4 (CH3)3PSe TlCl TlBr TlI 11 11 1 1 1 1 1 1 1 1 1 1 1 SOS-CI(s ) SOS-CI(s ) SOS-CI(s ) SOS-CI(s ) CAS/Partridge CAS/Partridge SOS-CI(s ) DFT(GGA)(s ) ZORA DFT(GGA)/Slater ZORA DFT(GGA)/Slater ZORA DFT(GGA)/Slater DK ¼ 19.92, K ¼ 72.19b DK ¼ 2.47, K ¼ 22.95b DK ¼ 79.88, K ¼ 77.36b DK ¼ 50.22, K ¼ 34.81b DK ¼ 235.5, K ¼ 1480c DK ¼ 273.1, K ¼ 3230 DK ¼ 111.92, K ¼ 263.33b,d Kaa ¼ 2375, Kbb ¼ 21127, Kcc ¼ 21137 DK ¼ 2971, K ¼ 22185 DK ¼ 5926, K ¼ 23153 DK ¼ 8911, K ¼ 23818 [325] [325] [325] [325] [143] [143] [325] [130] [144] [144] [144] B B B B 11 31 B P 11 31 B P 23 Na23Na 23 Na39K 31 31 P P 31 77 P Se 35 205 Cl Tl 79 205 Br Tl 127 205 I Tl 11 11 Either the principal values of the symmetrized tensor (Kii, i ¼ a; b; c with lKaa l # lKbb l # lKcc l), specified Cartesian components (xx, xy, etc.), or the anisotropy with respect to a certain axis (e.g. Dz K ¼ Kzz 2 1=2ðKxx þ Kyy Þ), is indicated. Unless otherwise noted, in systems possessing a unique n-fold rotation axis (n $ 2), the anisotropy is defined with respect to that axis. Reduced couplings are given in units of 1019 T2 J21. a See footnote a in Table 13. b See footnote d in Table 13 and footnote d in Table 15. c Corrections carried out for the n ¼ 0 vibrational state. d See footnote e in Table 15. 3%. This situation results from (1) the aromatic system being able to convey the components of longrange J and (2) the small value, due to the R23 dependence on the internuclear distance, of the corresponding D. Lantto et al. studied fluorine-substituted methanes CH42nFn ðn ¼ 1; 2; 3Þ at the RASSCF level, using large active spaces and HIII basis sets [78]. While there are both correlation and basis set deficiency errors remaining (the basis particularly lacking tight functions and hence expected to be converged up to ca. 5%), the quality of the tensorial properties of JHH ; JCH ; JFC ; and JFF is expected to be at least semi-quantitative. The corresponding J values are very satisfactory, contrary to prior calculations at the DFT level [257,259]. The work, together with Ref. [60] for p-C6H4F2, provides the first reliable computational estimates for the couplings to 19F in the literature. In these systems, the need to calculate all of the contributions to the tensors is particularly clear. Especially JSD FF should not be neglected. The question of the value, even the order of magnitude, of D1 JFC in CH3F has attracted a lot of attention in the past (see Ref. [78] for some of the references). It seems that the current theoretical value of 208 Hz [78], has settled the issue. The contribution of J to Dexp was found to be in the 1– 1.5% range for 1 JFC and 2 JFF in the systems studied. Bryce and Wasylishen compared CASSCF calculations using medium-size active spaces and mostly correlation-consistent basis sets, to molecular beam spectroscopic data for light diatomic molecules containing elements ranging from the alkali metals to halogens [143]. This comparison is particularly fruitful as the experimental J and DJ are practically free from environmental effects. A qualitative agreement with experiment was reached, and further improvement may be sought both from larger active spaces and basis sets that contain tight functions. Contributions from the different coupling mechanisms were reported for both K and DK; giving rise to interesting preliminary trends for the two quantities across the Periodic Table. Briefly, the magnitude of the total K and DK; as well as the PSO and SD contributions to DK; increase from left to right along a given period in the Table. The magnitudes of K FC and DK SD=FC follow the opposite trend. While the FC contribution dominates in most (but not all, notably in 294 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 ClF) coupling constants, the dominance of DK SD=FC is far less clear. The results underline the general necessity to calculate all five coupling terms unless substantial experience has been gathered for the type of coupling and system under study. Further interesting trends were depicted in Ref. [143] concerning the change of K and DK down the columns (groups) of the Periodic Table. Based on calculations for the lighter systems and experimental data for the heavier ones, the magnitudes of K and DK were found to increase along the series XF (X ¼ B, Al, In, Tl), TlX (X ¼ F, Cl, Br, I), and XF (X ¼ Cl, Br, I). This is accompanied by a dramatic increase in both DK=K and the indirect contribution to Dexp : Bryce and Wasylishen carried out RASSCF calculations using medium-size active spaces on ClF3 and OF2 [27]. Comparison of the calculated JFF ðClF3 Þ and JFO ðOF2 Þ with their respective experimental counterparts was very successful, however a discrepancy was observed for the weighted-average JClF ðClF3 Þ: Besides the usual possible explanations (neglect of relativistic effects, rovibrational corrections and solvent modelling), the flexibility of the ccpVQZ basis set used for ClF3 may be inadequate in the core region. Other methods. In Ref. [130], 1 JSeP in (CH3)3PSe was investigated using the DFT/GGA method with the FPT/SOS-DFPT ansatz, thus omitting the JSD term. A fair agreement with the experimental solid-state coupling constant was obtained, 2 656 Hz (exp.) vs. 2 820 Hz (calc.). The calculated principal values were similarly overestimated as compared to the experimental ranges given in the paper. While separation of the possible model construction and methodological errors (particularly the lack of relativity) is difficult in this case, this level of agreement is certainly useful already. This work appears to be the first application of DFT to the anisotropic properties of J. The water molecule has been studied using the SOPPA(CCSD) method and a large basis set [318]. The results of a small CASSCF calculation using the same basis were quoted for comparison, as well as SOPPA results originating from Ref. [280]. Although the latter were at a slightly different molecular geometry, it appears that SOPPA is in this case a better approximation to the apparently very accurate SOPPA(CCSD) numbers than the basic CASSCF wave function used [318]. Full tensors were reported. Sauer et al. reported high-accuracy SOPPA(CCSD) calculations using good, augmented basis sets for the prototypical CH4 and SiH4 molecules [321]. While both the 1 JXH and 2 JHH are dominated by the FC mechanism, the anisotropic properties of the couplings obtain relevant contributions from SD/FC, PSO, and DSO mechanisms. It is noteworthy that the D1 K parameters have opposite signs in the two systems. The K values are, after the rovibrational treatment, in very good agreement with experiment. Hence, the anisotropic properties are also most likely reliable. Wigglesworth et al. carried out SOPPA(CCSD) calculations using large basis sets for all of the coupling tensors in C2H2 [214,215]. While J FC dominates the isotropic couplings, DJ SD=FC is the largest contribution only in JCH : The PSO mechanism is very important for both D2 JCC and D2 JCH ; and DSO in D3 JHH : The calculated J values are in good agreement with experimental estimates for the equilibrium geometry (see Table 11). The values of DJ agree well with the earlier calculations of Kaski et al. [72]. Relativistic effects. Visscher et al. compared nonrelativistic and fully relativistic (four-component) SCF results for the HX (X ¼ F, Cl, Br, I) series of molecules [335]. Relativistically optimized basis sets for Br and I were used, as well as uncontracted cc-pVTZ sets for the other elements. While the uncorrelated method is as such inadequate for J couplings, the results are indicative of the importance of relativistic effects on the couplings to a heavy atom. The conclusion is that relativity affects J significantly, particularly for HBr and HI, whereas the effects on DJ are smaller. There are opposite changes in the absolute magnitude of DJ and J, increase and decrease, respectively. The relative anisotropy DJ=J increases for the lighter members of the series, but HBr and HI feature a sign change in J. The effect of switching from a point-like nuclear model to a Gaussian distribution is ca. þ 1% for JIH : Vaara et al. investigated the H2X (X ¼ O, S, Se, Te) and HX (X ¼ F, Cl, Br, I) systems at both nonrelativistic and spin – orbit corrected CASSCF levels, using basis sets close to convergence [38]. The 1 KXH were modified by the SO-corrections towards J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 the experimental values (particularly for the H2X series), but the correction is not large enough to reproduce the experimental trend for the heavier systems. Comparison with the findings of Refs. [335, 336] underlines the importance of the scalar relativistic effects for couplings involving heavy atoms. For 2 JHH ; the SO correction is probably adequate, and quantitatively accurate values should be sought by improving the electron correlation treatment beyond the CASSCF level. The magnitude of D1 KXH increases due to the SO interaction, while that of 1 KXH decreases. Autschbach and Ziegler applied their two-component relativistic ZORA DFT method in Ref. [144]. The agreement of 1 KPbH with experiment at the LDA level is excellent for PbH4, implying also a good calculated value for the corresponding DK: In the other systems studied in this work, GGA was found to be superior to LDA. Qualitative agreement was obtained with experiment for both the 1 K and D1 K in the XF (X ¼ Cl, Br, I) series, with K PSO and the total value overestimated for ClF as compared to the MCSCF results of Ref. [143]. In the case of IF, the authors demonstrated the effects of relativity on the individual contributions. While scalar relativistic effects increase both K PSO and lDK PSO l; the spin – orbit interaction seems to partially cancel this effect. There is, similarly, a substantial effect of the scalar relativity on the (small) K FC contribution, and a very small opposing spin –orbit effect. These conclusions cannot be generalized to other systems, however, as exemplified by the TlX (X ¼ F, Cl, Br, I) series. There, the inclusion of scalar relativity in the model worsens the agreement of the calculated result with the experimental data, and the large spin – orbit contributions restore the qualitative agreement. For these systems, the choice of either LDA or GGA is irrelevant in comparison with the effect of relativity. The total K and DK under- and overestimate, respectively, their experimental counterparts. While the numbers calculated by the ZORA DFT method are not in fully satisfactory agreement with the experiment, the method has reached a useful level of accuracy for systems that have previously been beyond the reach of meaningful modelling. In the prototypical hydrocarbon series C2H2, C2H4, and C2H6, the values of 1 KCC are qualitatively correct in 295 Ref. [267] but further removed from the experiment than the non-relativistic RASSCF data of Ref. [72]. The calculated DFT DKCC as well as KCC decrease monotonically in the series, in contrast to DKCC at the MCSCF level [72], which has a minimum for ethene. 3.8.2. Antisymmetric components Table 25 displays the results for components of JA : In general, only very few reports of JA exist, although the antisymmetric components are available from practically all of the programs in current use. For consistency, the procedure of diagonalizing the J1 þ JS part and expressing the components of JA in the PAS(J ) frame should be adopted. Few papers report the full tensors from which the antisymmetric components can be extracted. Refs. [40,41,406] reported SCF level calculations for 1 J and 2 J in simple first- and second-row hydrides. As these numbers do not contain any electron correlation contribution, they should be used with caution. More reliable SOPPA(CCSD) calculations were carried out in Ref. [321] for the couplings in CH4 and SiH4. SOPPA, SOPPA(CCSD), and CASSCF LR were compared for 2 JHH using good basis sets for water in Ref. [318]. In 1984, a SOPPA calculation [228] was performed for H2O, but the antisymmetric component is overestimated for 2 JHH : For 2 JHH ; the antisymmetric components are noted to be very small. The value of 1 JAOH is even less than the value of 2 A JHH based on SOPPA, CASSCF LR, and SOPPA(CCSD) calculations [228,318]. The corresponding terms for BH [41], NH [41], and PH couplings [40], albeit from uncorrelated calculations, are only slightly larger. Whereas the antisymmetric components of the couplings involving proton seem to be negligibly small, the ab initio RASSCF work reported in Ref. [27] on couplings possessing Cs local symmetry in ClF3 and OF2 demonstrated similar order of magnitude of the antisymmetric components to the corresponding J. The same comments as before, concerning the data on J1 þ JS in these systems, apply here as well. The antisymmetry seems to increase rapidly as heavier elements are involved. The semi-empirical REXNMR results of Ref. [3] (not tabulated) for H2Te2 point to the same conclusion. 296 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 Table 25 Quantum chemically calculated absolute values of the components of the antisymmetric part J A of spin–spin coupling tensors System Coupling Bonds Theorya lValuel Reference BH2 4 CH4 NH2 2 NHþ 4 H2O H2O H2O H2O AlH2 4 SiH4 SiH4 PH2 2 PH3 PHþ 4 NH2 2 H2O H2O H2O H2O PH2 2 PH3 OF2 ClF3 ClF3 H1H H1H 1 1 HH 1 1 HH 1 1 HH 1 1 HH 1 1 HH 1 1 HH 1 1 HH 1 1 HH 1 1 HH 1 1 HH 1 1 HH 1 1 HH 15 1 NH 17 1 OH 17 1 OH 17 1 OH 17 1 OH 31 1 PH 31 1 PH 17 19 O F 19 19 F F 35 19 Cl F 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2e 1f SCF/(s ) SOPPA(CCSD)/(s ) SCF/(s ) SCF/(s ) SOPPA(CCSD)/(s ) CAS/(s ) SOPPA/(s ) SOPPA/(s ) FOPPA/(s ) SOPPA(CCSD)/aug-cc-pVTZ FOPPA/(s ) SCF/(s ) SCF/(s ) SCF/(s ) SCF/(s ) SOPPA(CCSD)/(s ) CAS/(s ) SOPPA/(s ) SOPPA/(s ) SCF/(s ) SCF/(s ) RAS/cc-pCVQZ RAS/cc-pCVQZ RAS/cc-pVQZ lJxzl ¼ 1.66b lJxzl ¼ 1.31b lJxyl ¼ 4.77c lJxzl ¼ 5.40b lJxyl ¼ 1.62c lJxyl ¼ 1.74c lJxyl ¼ 1.71c lJxyl ¼ 19.48c lJxzl ¼ 1.59b lJxzl ¼ 3.66b lJxzl ¼ 1.75b lJxyl ¼ 2.33c lJxzl ¼ 0.33b, lJyzl ¼ 2.52d lJxzl ¼ 2.17b lJxyl ¼ 0.62b lJxyl ¼ 0.21b lJxyl ¼ 0.09b lJxyl ¼ 0.19b lJxyl ¼ 0.44b lJxyl ¼ 2.35b lJxyl ¼ 1.04b lJyzl ¼ 109b lJxyl ¼ 292b lJxyl ¼ 150b [41] [321] [41] [41] [318] [318] [280,318] [228] [406] [321] [406] [40] [40] [40] [41] [318] [318] [280,318] [228] [40] [40] [27] [27] [27] a b c d e f 1 1 Results in Hz. See footnote a in Table 13. Component in the local symmetry plane. With the molecule in the xy plane and the C2 axis along the y direction. Symmetry plane is xy. See footnote c in Table 23. See footnote h in Table 23. 4. Conclusions An effort has been made to summarize recent progress in understanding indirect spin – spin coupling, J; tensors. From an experimental point of view, considerable progress has been realized in the three main methods used in their characterization. The importance of NMR of solute molecules in liquid crystal solvents (LCNMR) emerges in cases where the contribution from the J tensor to the experimental anisotropic coupling Dexp is relatively small. Key issues associated with this method are the contributions of the vibrational motions and of the correlation of the vibrational and reorientational motions (the deformation effects) to the dipolar couplings. After extensive theoretical studies in the 1980s, these contributions can be treated for systems with small amplitude motions using existing computer programs and available force fields. Proper treatment of these contributions is important for two reasons: first, the determination of accurate, solvent-independent molecular structures and orientational order parameters, and second, the separation of the often minute indirect anisotropic contribution, ð1=2ÞJ aniso ; from the corresponding Dexp : The method is restricted by the fact that the determination of the complete structure and orientation tensor requires a large number of Dexp couplings in which ð1=2ÞJ aniso is J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 safely negligible. Usually this means that couplings involving at least one proton have to be used. On the other hand, an increasing number of couplings result in more difficult analyses of the spectra. Therefore, LCNMR experiments performed in highly ordered LC environments are restricted to spin systems with high symmetry and at least a few hydrogen atoms. As shown earlier, reliable data have been derived by LCNMR, e.g. for the CC, NC, FC, SiC, and FF spin – spin coupling tensors in hydrocarbons, fluoromethanes, methyl cyanides and methylsilane, as well as fluorobenzene. In particular, the methodological progress is exemplified in the classic question of the value of D1 JFC in monofluoromethane, where modern LCNMR techniques have narrowed the range of experimental results to 350 – 400 Hz from the unrealistic early results that are scattered over literally thousands of Hz. JCC ; JNC ; and JFC are not only valuable from the electronic structure calculation point of view, but also because the corresponding Dexp are increasingly used in studies of protein structure and orientation in dilute liquid-crystalline solutions, as well as in studies of the orientational behaviour of liquid crystal molecules. In certain cases, ð1=2ÞJ aniso may even dominate Dexp and, hence, introduce large uncertainty in the structural and orientational order parameters. The amount of data obtained from NMR measurements on solid samples has increased enormously; however, there are still problems associated with correcting measured effective dipolar coupling tensors for motional averaging. Although molecular motion in solids is highly restricted, vibrations and librations will lead to some averaging of the dipolar interaction (typically 1– 5%). Quantitative corrections of the measured effective dipolar coupling constants, Reff ; for such motion are difficult if not impossible. Often researchers have failed to consider how molecular motion might influence the anisotropic spin – spin coupling constant data they report. There is clearly a need for further single-crystal NMR data on systems where the Reff are significantly different in magnitude than the direct dipolar coupling constants, RDD : Finally, one advantage of NMR investigations of solids is that spin-pairs that involve quadrupolar nuclei can be examined because quadrupolar nuclei often have relatively long nuclear relaxation times in the solid state compared to solution. In fact, 297 the presence of a quadrupolar nucleus can be critical in characterizing J-tensors. In such systems, it is very important to carry out measurements at more than one applied magnetic field strength. Also, it is important to recognize that the most reliable data will generally result from systems where symmetry demands that the electric field gradient tensor at the quadrupolar nucleus is axially symmetric. The availability of high-resolution molecular beam data is very important as it provides highly accurate and precise spin – spin coupling data on isolated diatomics which serve as most suitable experimental benchmarks for testing computational methods. Particularly significant is the recent work of Cederberg and co-workers where the vibrational dependence of spin – spin coupling constants is measured to a precision of better than 1 Hz. For example, in the case of CsF, J133 Cs19 F ¼ 0:62745ð30Þ 2 0:00903ð22Þ £ ðn þ ð1=2ÞÞ kHz; with one standard deviation of uncertainty estimates in the last two digits shown in parentheses. The development of quantum chemical methods, their efficient implementation, and the rapid increase of computer resources have revolutionized theoretical calculations of J. For small molecules consisting of light elements, the present ab initio methods are approaching quantitative agreement with experiment. Comparison of the experimental coupling constants with the most accurate calculations still leaves room for improvement in the latter. Regarding the rank-2 part of J, it is more difficult to assess the accuracy of the theoretical calculations because the errors associated with the experimental values are larger than for the isotropic part. Continued efforts in ab initio calculations of J are very well motivated. Systematically improving methods provide reliable benchmarks for more approximate approaches. Coupled cluster methods beyond CCSD are likely to constitute one of the main directions where progress can be expected. Parallelization and linear scaling techniques would increase the range of systems accessible to ab initio quantum chemical methods. For medium-size systems, the recent analytical DFT implementations for J calculations are promising; however, further benchmarking studies are still necessary. The available DFT exchange-correlation functionals have not been parametrized for 298 J. Vaara et al. / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 233–304 hyperfine properties or transition metal systems. Problems in the DFT performance are apparent already in couplings to fluorine, as discussed in this review. The long-term goal is to develop more systematic functionals with less or no need for empirical parametrization, as well as a better understanding of the quantitative role of the current dependence of the exchange-correlation functionals. The anisotropic properties of J increase in significance in systems containing heavier elements. There, one has to resort to comparison with experimental data when judging the accuracy of the practical (ZORA) DFT method that both includes relativity and is available for J. Correlated relativistic ab initio methods for J at four- and two-component levels would indeed be very desirable. The reliability of DFT is nevertheless at the present time sufficient to make qualitative conclusions of chemical trends and to be of substantial assistance in steering the direction of experimental work. The roles of rovibrational averaging, intermolecular and solvation effects, as well as configurational sampling in more complex systems, remain relatively unexplored in the context of J tensors. Acknowledgements JV and JJ would like to thank Jaakko Kaski, Perttu Lantto, Juhani Lounila, Kenneth Ruud, and Olav Vahtras for research cooperation, and Henrik Konschin for discussions (JV). REW and DLB thank Prof. James Cederberg for rubidium fluoride molecular beam data in advance of publication, and the members of the solid-state NMR group of the University of Alberta for valuable comments: Kirk Feindel, Guy Bernard, Michelle Forgeron, Kristopher Ooms, Kristopher Harris, Myrlene Gee, Renée Siegel, Takahiro Ueda, and Se-Woung Oh. 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