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Quantum chemical modelling of molecular properties - NMR shielding constant, spin-spin coupling constant, rotational g factor, spin-rotation constant, nuclear quadrupole coupling constant Effective NMR Hamiltonian High resolution NMR spectra are interpreted by means of effective NMR Hamiltonian. Its eigenfunctions correspond to the spin states of nuclei, and its eigenvalues to the energy levels of nuclear spins in external magnetic field: H NMR =− X K 1 X T mK (DKL + KKL ) mL B (1 − σ K ) mK + 2 T K6=L σ K is the shielding constant tensor, KKL is the tensor of (reduced) spin-spin coupling constant (indirect, through-electron coupling). DKL is the tensor of direct (through-space) spin-spin coupling constant. All effects connected with the electronic structure are contained in σ K and KKL . (In some cases the contribution containing the nuclear quadrupole coupling constant is added, but in isotropic medium it influences only the width of the sygnal.) The shielding constant can be treated as modification of external magnetic field by the field resulting from the presence of moving electrons. Bloc K = (1 − σ K ) B Effective NMR Hamiltonian in isotropic medium When molecules can rotate freely (gas, liquid), the effective NMR Hamiltonian has the form: NMR Hiso =− X B (1 − σK ) mK,Z K 1 X + KKL mT K · mL 2 K6=L σK (isotropic shielding constants) i KKL (isotropic reduced coupling constant) are traces of the appropriate tensors: σK = KKL = 1 Tr σ K 3 1 Tr KKL 3 Tensor DKL is traceless and averages out to zero. σK is connected with the chemical shield δK δK = σK (reference) − σK (sample) ∼ = σK (reference) − σK (sample) 1 − σK (reference) Shielding constant as energy derivative Nuclear shielding constant can be expressed as a second derivative of the energy with respect to external magnetic field and nuclear magnetic moment 2 ∂ E(B, mK ) ∂B ∂mK = σK − 1 B=0,mK =0 For closed shell systems (in nonrelativistic theory) shielding constant (similarly as magnetic susceptibility) can be expressed as a sum of diamagnetic part (calculated as an expectation value) and the paramagnetic part (calculated as linear response function): K(PSO) B,K σK,αβ = h0|Ĥαβ |0i + hhm̂α ; Ĥβ where para dia ii = σK,αβ + σK,αβ B,K Ĥαβ d2 Ĥ = dBα dBβ and K(PSO) Ĥβ = dĤ dBβ Shielding constant as energy derivative, continued σK , like any other second derivative of energy with respect to time-independent perturbations, can also be calculated from sum-over-states expression: σK E D B,K = 0 Ĥ 0 + 2 ED E D B,orb K(PSO) n n Ĥ 0 X 0 Ĥ n6=0 En − E0 Electronic Hamiltonian in magnetic field Standard Schrödinger Hamiltonian does not contain any terms containing magnetic moments of the nuclei (or external magnetic field). They appear in the Breit-Pauli Hamiltonian, which can be 2 order derived when relativistic effects are taken into account to αfs (αfs = c−1 is a fine structure constant). In the presence of magnetic perturbations, the non-relativistic molecular electronic Hamiltonian may in atomic units be written in the form X ZK 1X 1 1X 2 X tot mi · B (ri ) − π − + H (B, M) = 2 i i r 2 rij iK i iK + i6=j X 1 X ZK ZL X tot − MK · B (RK ) + MTK DKL ML 2 RKL K6=L K K>L We have here introduced the operators for the mechanical momentum π i = −i∇i + Atot (ri ) (1) where Atot (ri ) is the vector potential at the position of electron i, constructed such that its curl reproduces the magnetic induction Btot (ri ) arising from the external field and the NMR active nuclei: Btot (ri ) = ∇i × Atot (ri ) The vector potential and the magnetic induction may each be decomposed into one contribution from the external field and one contribution from each nucleus X tot AK (ri ) A (ri ) = AO (ri ) + K B tot (ri ) = B+ X BK (ri ) K The vector potential associated with the nucleus may for example be written as 2 MK × riK (2) AK (ri ) = αfs 3 riK andthe corresponding magnetic induction (the curl of AK ) as BK (ri ) = 2 2 riK 1 −αfs 2 8παfs − 3riK rTiK δ (riK ) MK MK + 5 riK 3 Ĥ B,K i Ĥ K(PSO) operators contributing to the shielding constant Consequently, Ĥ B,K i Ĥ K(PSO) operators (derivatives of the Hamiltonian) have the following form: Ĥ B,K = 2 αfs Ĥ e2 X (riO · riK ) 1 − riO rT iK 3 2me i riK K(PSO) = 2 2µB αfs X liK 3 ~ i riK Dependence of the calculated shielding constant on the gauge origin Both dia- and paramagnetic part of the shielding constant are gauge-dependent (they change their values when the gauge origin for the vector potential A is shifted) even for the exact wave function and therefore separation of the shielding into dia- and paramagnetic parts is somewhat arbitrary. As a consequence, their separate discussion makes sense only when the gauge origin is defined. The usual way is to put it on the nucleus of interest, and in para this case σK is connected with the nuclear spin-rotation constant (but the relation holds only in nonrelativistic approximation). The total shielding constant depends on the gauge origin for approximate wave function (except for wariational wave functions in the limit of complete basis set). This dependence can be eliminated by the use of London atomic orbitals (GIAOs): i ωµ (B, RM ) = exp − eAM · r χµ (RM ) ~ However, the division into dia- and paramagnetic parts is still gauge-dependent. Computational requirements of the shielding constants • Perturbation-dependent basis sets (London orbitals) are absolutely necessary. • A basis set should have a good description of outer core-inner valance shell, and also polarization functions. • The role od electron correlation varies with the system and nucleus. The electron correlation effects tend to be large in systems with multiple bonds. They are small in the case of 1 H shielding, but large for example for 17 O shielding. This phemonenon is connected with the relative magnitude of diaand paramagnetic parts: 1 H shielding is dominated by the diamagnetic part, which is less sensitive than the paramagnetic part to electron correlation. • Shielding constants are strongly affected by rovibrational effects and environment (including solvent). The role of relativity is also significant (starting from 31 P shielding). Spin-spin coupling constant as energy derivative The spin-spin coupling constant between nuclei K and L can be expressed as the second derivative of energy with respect to nuclear magnetic moments MK and ML . 2 KKL + DKL d E (M1 , M2 , · · · ) = dMK dML , MK =ML =···=0 The reduced coupling constants KKL is connected with the coupling constant JKL (measured in Hz) in the following manner: JKL = h γK γL KKL , 2π 2π In the isotropic phases the trace of the tensor is measured: JKL = 31 Tr KKL . DKL is a traceless tensor. DKL µ0 3RKL RKL − 1R2KL = 5 4π RKL Spin-spin coupling constant as energy derivative, continued KKL , like any other second derivative of energy with respect to time-independent perturbations, can be calculated from sum-over-states expression: KKL = 0 2 ∂ H 0 +2 ∂MK ∂ML ED E D ∂H ∂H X 0 ∂MK n n ∂ML 0 n6=0 En − E0 After evaluation of the derivatives (from the Breit-Pauli Hamiltonian with no relativistic terms) Ramsey formula is obtained KKL DSO = 0 hKL 0 + + PSO PSO T 0 hK nS nS (hL ) 0 2 EnS − E0 nS 6=0 FC T SD SD T X 0 hFC nT (hL ) + (hL ) 0 K + hK nT . 2 EnT − E0 n X T The first summation is over excited singlet |nS i states with energies EnS , the second is over excited triplet |nT i states with energies EnT . The spin-spin coupling constant is calculated in practice as a sum of an expectation value h0|Ĥ K,L(DSO) |0i (diamagnetic spin-orbit term, usually small) and terms calculated as linear response functions: hh Ĥ K(PSO) ; Ĥ L(PSO) ii (paramagnetic spin-orbit term) i hh Ĥ K(FC) + Ĥ K(SD) ; Ĥ L(FC) + Ĥ L(SD) ii (Fermi contact term and spin-dipole term). hDSO KL = hPSO K = hFC K = hSD K = e2 X (riK · riL ) 1 − riK rT iL , 3 3 2me i riK riL X liK 2 2µB αfs , 3 ~ i riK X 2 8π δ (riK ) 1, −αfs 3 i 4 αfs 2 −αfs X 3 riK rT − r 2 1 iK iK , 5 riK i α ≈ 1/137 is a fine structure constant, riK is a position of the electron i with respect to the nucleus K, I3 is a 3:3 unit matrix, P δ(riK ) is a Dirac delta, and si is spin of electron i. i denotes summation over all electrons, and T transpose of a vector. If only the isotropic part of the spin-spin coupling constant is of interest (which is usually the case), the FC-SD term does not need to be calculated, since it is purely anisotropic and does not contribute to isotropic JKL . JKL is then a sum of four terms: DSO PSO SD FC JKL = JKL + JKL + JKL + JKL Computational requirements of the spin-spin coupling. • Calculations of spin-spin coupling constants are relatively time-consuming: each magnetic nucleus introduces a new perturbation(3 components). • Electron correlation is crucial: Hartree-Fock produces inaccurate or nonsensical results. • DFT/B3LYP is suitable for most of the calculations — except for fluorine coupling constants. • Very accurate calculations for small molecules can be carried out by means of MCSCF i CC. (In such calculations also rovibrational effects should be accounted for.) • Standard basis sets of atomic orbitals are not suitable for calculations of spin-spin coupling constants: the FC term requires a large number of s funcitons with high exponents (inner core). The PSO and SD contributions, on the other hand, need outer core, valance and polarization functions (PSO is similar in this respect to the shielding constant). Different contributions have thus different requirements. What do we mean by ’inaccurate’ and ’nonsensical’ results? C2 H6 C2 H4 SCF Jeq exp Jtot 1 JCC 60.6 34.5 1 JCH 149.1 125.2 1 JCC 1270.2 67.6 1 JCH 754.5 156.3 T. Helgaker i M. Pecul Spin–spin coupling constants with HF and DFT methods, in: Quantum Chemical Calculation of Magnetic Resonance Properties. Theory and Applications. Problems with DFT in calculations of 19 F coupling constants HF Jeq B3LYP Jeq MCSCF Jeq CCSD Jeq exp Jtot HF 1 JHF 668.9 416.6 544.2 521.6 500 FHF− 2 JFF 656.8 24.9 358.1 438.6 ≈ 247 1 JHF 181.0 38.4 126.5 81.6 107.0 T. Helgaker i M. Pecul Spin–spin coupling constants with HF and DFT methods, in: Quantum Chemical Calculation of Magnetic Resonance Properties. Theory and Applications, ed by: M. Kaupp and M. Bühl and V. G. Malkin, Wiley-VCH (2004), pages 101—121. Examples of calculations. Spin-spin coupling constants in C2 H2 1 a J(CC) 1 J(CH) 2 J(CH) 3 J(HH) MCSCF-3/cc-pCVQZsu2 184.68 244.27 53.08 10.80 Eksperymenta 184.52 242.40 53.76 10.11 with vibrational corrections M. Jaszuński i K. Ruud Spin-spin coupling constants in C2 H2 , Chem. Phys. Lett. 336 (2001) 473. All spin-spin coupling constants (7587)in valiomycine C54 H90 N6 O18 (LDA/6-31G calculations). M. A. Watson, P. Sałek, P. Macak, M. Jaszuński, T. Helgaker, Chem. Eur. J. (2004) 10: 4627–4639. 62 Hz 77 Hz Coupling constants in C60 fullerene. M. Jaszuński, T. Helgaker, and K. Ruud Mol. Phys. (2003) 101: 1997—2002. Vibrational effects on spin-spin coupling constants [Hz] B3LYP Jvib exp Jtot HF 1 JHF −38.0 500 H2 O 1 JOH 5.4 −80.6 CH4 1 JCH 5.3 125.3 2 JHH −0.7 −12.8 1 JCC −10.0 174.8 1 JCH 4.6 247.6 2 JCH −3.0 50.1 3 JHH −0.1 9.6 C2 H2 T. Helgaker and M. Pecul Spin–spin coupling constants with HF and DFT methods, in: Quantum Chemical Calculation of Magnetic Resonance Properties. Theory and Applications, ed by: M. Kaupp and M. Bühl and V. G. Malkin, Wiley-VCH (2004), pages 101—121. Nuclear spin-rotation constant Interaction of magnetic moment of a nucleus with magnetic field caused by molecular rotation results in splitting of the rotational spectrum. ∆EK = −IT K MK J where IK is nuclear spin, J is a total molecular momentum, and nucl MK = Mel K + MK is a spin-rotation tensor. It is composed of two terms, nuclear and electronic, which can be expressed as: Mnucl K = 2 αfs µ N gK X L6=K i 2 RLK 1 − RLK RT LK −1 I ZL 3 RLK Mel K µN gK ∂ 2 E el =− 2π ∂mK ∂J mK ,J=0 −1 Mel hhĤ K(PSO) ; ˆlK ii K can be obtained as a linear response I Connection with the shielding constant Mel K is connected with the paramagneti part of the shielding constant, defined as a difference between the total shielding constant and its diamagnetic part with the gauge for the vector potential A on nucleus K. nucl Mel K = MK − MK i ~ µ N gK h −1 = σ K − σ dia K (RK ) I µB 2π Molecular magnetogiric ratio g molecular g factor A rotating molecule produces a magnetic field because of the motion of charged particles: electrons and nuclei. The associated magnetic moment can be expressed as: mJa µN X J gab Jb = ~ a,b gdzie µN is nuclear magneton, Jb is the rotation quantum number, and the molecular factor g J is a sum of contributions from nuclei and electrons. J g =g J,nucl +g J,el = −1 mp I X ZK 2 RK,CM 1 K − 4mp me −1 para I ξ (RCM ) 2 e − RK,CM RT K,CM I is the momentum of inertia, and ξ para is a paramagnetic part of magnetic susceptibility (discussed in one of the previous lectures). g influences high-resolution rotational specra even in the absence of external magnetic field. It is important for precise determination of molecular geometry. Nuclear quadrupole coupling constant Relaxation time in NMR (influencing the signal width) is strongly influenced, for nuclei with spin 1 or higher, by the interaction of nuclear electric quadrupole moment with electric field gradient: ∆ν1/2 ∝ (− eQ K V z z ~ K eQ Vz z K ~ 2 is the nuclear quadrupole coupling constant, NQCC, Vz z electric field gradient on nucleus K, eQ nuclear electric quadrupole moment. For nuclei with 1/2 spin (for which eQ = 0) electric field gradient does not affect the signal width. Electric field gradient on the nucleus influences also the fine structure of microwave spectrum (which allows for determination of nuclear quadrupole moments). Nuclear quadrupole coupling constant, continued Electric field gradient on the nucleus K, Vα β K , and therefore the nuclear quadrupole coupling constant, can be calculated as the expectation value: K K Vαβ = h0 | V̂αβ | 0i where the operator V̂αβ has the form: K Vˆαβ = 2 1 X 3r̂iK,α r̂iK,β − r̂iK δαβ e 5 4πǫ0 i riK − 2 X 3RLK,α RLK,β − RLK δαβ 1 e ZL 5 4πǫ0 RLK L6=K Computational requirements of nuclear quadrupole coupling constant Operator in the expression for nuclear quadrupole coupling constant is of r −3 type (like those in the expressions for the shielding constant and PSO contribution to the coupling constant), therefore the basis set requirements (extended outer core-inner valance region) and the electron correlation effects are similar. Programs for calculations of NMR parameters • Shielding constants in Gaussian 09 (HF, DFT, MP2), Dalton (HF, DFT, MCSCF, SOPPA), CFOUR (CC), ReSpect (DFT), ADF (DFT), NWChem (HF, DFT). • Molecular g factors in Dalton (HF, DFT, MCSCF, SOPPA). • Spin-spin coupling constants in Gaussian 09 (HF, DFT), Dalton (HF, DFT, MCSCF, SOPPA), CFOUR (CC), ReSpect (DFT), ADF (DFT), NWChem (HF, DFT - in developement). • Nuclear quadrupole coupling constants in Gaussian 09 (HF, DFT), Dalton (HF, DFT, MCSCF, SOPPA), ADF (DFT), CFOUR (CC)...