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JOURNAL NMR OF MAGNETIC 33, 627-641(1979) RESONANCE Shifts in Paramagnetic Systems: A Nonmultipole Method R.M. GOLDING DepartmentofPhysical Chemistry, Received UniversityofNew February AND L.C. South Australia 24, 1978; Wales, revised Expansion STUBBS Box 1, P.O., June Kensington, 2033, N.S. W., 19, 1978 NMR shifts arising from the electron orbital angular momentum and the electron spin dipolar-nuclear spin angular momentum interactions using a nonmultipole expansion method are examined for a d electron in a crystal field of octahedral symmetry and with a tetragonal component. Bonding with the paramagnetic center is included. The NMR shifts are illustrated as isoshielding contour maps, and the results are compared with the multipole expansion method. It is shown that even when experimental data are successfully fitted by considering the NMR shifts as arising from the Fermi contact and the dipolar interactions, such an interpretation may lead to incorrect conclusions about the interactions within the paramagnetic system. INTRODUCTION Over the last few years the interpretation of the NMR shifts in d” and f” paramagnetic systems has been based upon the electron-nuclear interactions arising from the Fermi contact interaction, the electron orbital angular momentum, and the electron spin dipolar-nuclear spin angular momentum interactions. The latter two terms have been treated as a multipole expansion in R, where R is the distance between the NMR nucleus and the electron-bearing atom. In particular, the dipolar term in the expansion which is related to the magnetic susceptibility anisotropy (I) has been used almost exclusively, although more recently the higher multipole terms (2-5) have been evaluated for some specific examples. In a recent paper (6) we have shown that the use of such an expression may lead to serious errors in the interpretation of the NMR shift through this mechanism, especially if only the dipolar term is considered. In some cases the Fermi contact interaction has been used to interpret the NMR shifts; see, for example, the NMR shifts in a series of transition metal ion dithiocarbamate complexes (7) and the 170 NMR shifts of the aqueous trivalent lanthanide ions (8,9), where the NMR shift, Al?, is given by AB = a(Sz)lgnrCLt+ r11 Here a is the hyperfine interaction constant. In other cases the dipolar term in the multipole expansion has been used; see, for example, the proton NMR shifts insome 627 OOZZ-2364/79/03062~21$02.00/0 Copyright 0 1979 by Academic Press. Inc. All rights of reproduction in any form reserved. Printed in Great Britain GOLDING 628 lanthanide complexes components (1) xoo/ (IO), where AND STUBBS in terms of the magnetic ABIB = -(/..d477N,~,, -&xx + xyyIH3 ccd 0 - 1) + %xXx-xy,,) sin’ 0 cos 2@]/3R3. susceptibility 1.21 The polar coordinates of the NMR nucleus in relation to the paramagnetic center are (R, 0, @), where the z-axis is the principal axis of the complex. However, both the Fermi contact contribution given by [l] and the dipolar contribution given by [2] may be important (II). Recently (12) a method has been developed whereby, from a set of experimental results, an estimate may be made of the various contributions to the NMR shifts. However, few publications have been devoted to the determination of the region where the dipolar contribution given by [2] may yield a good approximation. This is of special importance when there is delocalization of the unpaired electrons from the paramagnetic center. For example, it has been shown/6) that the 14N NMR shift in the Fe(CN)63- ion may be interpreted as arising dominantly from the interaction with the nitrogen nucleus of the unpaired electrons in the nitrogen p orbital of the 2T2 molecular orbitals. In this paper we examine the NMR shifts arising from the electron orbital angular momentum and the electron spin dipolar-nuclear spin angular momentum interactions using the nonmultipole expansion method (13, 14) (i.e., applicable for any R) for a number of specific cases. We include the effect of bonding with the paramagnetic center. The results are compared with the multipole expansion method. Finally, we show that even when the multipole expansion method gives errors, experimental shifts may still be fitted successfully using a combination of [l] and [2] but will lead to incorrect conclusions about the interactions within the paramagnetic system. THEORY The principal values a,,, cr,+,, and cZZ of the NMR screening tensor u are determined by considering the magnetic field interaction as parallel to the X, y, and z directions and averaged assuming a Boltzmann distribution. The contribution to the NMR shift, AB, is given by AB=~B(u,,+a,,+cr,,), [31 where (X) refers to the Boltzmann average of the hyperfine interaction represented by the Hamiltonian 21N ’ 1 3(& * s)rN ’ 1 x = E gNpSpN ---T-+gs [41 rh rN NMR SHIFTS IN PARAMAGNETIC 629 SYSTEMS . electron f ZrV r / tl electron - bearing nucleus FIG. 1. The coordinate system. In this paper we take gS to be equal to 2 exactly. Here rN is the radius vector of the electron about the nucleus with nuclear spin angular momentum I, as shown in Fig. 1. The system we consider is a d’ system in a strong crystal field of octahedral symmetry, where bonding effects have been incorporated using a molecular orbital approach. The paramagnetic site is surrounded by six ligands lying at a distance *Z?L along each of the three axes. It was assumed that the ligands could be treated as single atoms with three p orbitals available for bonding. The p orbitals parallel to the X, y, and z axes for the jth ligand are written as Ixi), Iyj), and IZj), respectively. The numbering of the ligands is as shown in Fig. 2. The appropriate molecular orbitals are (1.5) &>+ W2)[lY5)- Iyd+ Id- aId +@/ml) - lzd+ l&l, 1%) - Ml, ~15~+~~/~~~1~,~-I~~>+lY*>-lY3~1, NMR FIG. 2. The numbering of the ligands [a 630 GOLDING AND STUBBS where the d orbitals are taken to be Slater-type 3d orbitals and are defined as l-5= (Wh)“2~z exp(-kbrh 14 = (2~:/3dI’= exp(-O2rL lbl It>=CW%~)“2x~ exp(-ML and similarly the p orbitals are taken to be Slater-type 2p orbitals defined as Ix) = @V~)1’2x exd-khr), IY>= (P:l.~)“~yexd-kbr), II71 12) = (/~:/T)~‘~z exp(-&r). The coefficients a and b in the molecular orbitals are defined from k’=a2+$b2+3ab(djp), PI 1 =a2+b2+4ab(djp), where k’ is the orbital reduction factor and (dip) gives the amount of overlap. The ‘T2 ground state of the d’ system is split by spin-orbit coupling into two levels with respective eigenvalues of -11;’ and f’, where 5’ = (2k’where fh and &, are the spin-orbit respectively. The NMR shift is given by AB -=--B 341~ kT l)& + (1 -k’)&, coupling constants for the d and p orbitals, E + F exp(3[‘/2kT) + G{l -exp(35’/2kT)}kT/[’ 1 + 2 exp(3[‘/2kT) [91 where E = $(2k’+ l)(D -2L), F = $(l - k’)(D + 12L), G = &(k’+ 2)(0 +4L), where D=u~Do(R)+: ,g Di(Ri), I L=a’Lo(R)+: 1 .i Li(Ri), I 1 where Ri is the vector pointing from the NMR nucleus to the ith ligand. As an example, Rs is shown in Fig. 2. The functions Do(R) and Lo(R) arising from the d NMR SHIFTS IN PARAMAGNETIC 631 SYSTEMS orbitals are D,(R)= - 1 +y Y&O, 0) S(t,) +fg 7r1’*[Yoo(o, [loal @)]iv(to) and Lo(R) = “(~)1’2[~(~)1’2Y~~(B, P;R~ 21 40,+;(;)1’2Y40(8, +; (i) 1’2Y4-4(@, @)]K(to)+~ where to = 2&R ?r”*[Y00(0, @) @)]M(to), [lob1 and s(t)=[l-e-‘n~o$]’ ~(t,=[l-e-‘j~~+~~,~)], N(t)=[e-(-~+~+;+t+l)], . . K(r)=[l-emtn~o$]. M(t)=[e-f;+;+t+l)]. . . The functions Di(Ri) and Li(Ri) (i = 1, . . . , 6) arising from the p orbitals on the ith ligand are given by Dl(Rd= 1 5 II2 Y42(01, @l) j 2 [ 0 1 5 l/2 Y4-2(@1, @l) H(t1) +T 02 I 72?* -m +)‘“[;(?)l’* 1 3 112 y2-2(@*, +z 0 2 Y*2(01, 01) 1 wl)+~ + f @l) Y40(@1, -$ Y20(@1, 4p: ~ Qjl) w "2 [YOO(@l,@dlJ(td, LIlaI 63’ GOLDING Ll(Rl)=~(~)1f2~~(~)“2 Y22t0,r +i 3 STUBBS (PI)-; Y2&91, @I) lf2 ( ) 2 2 yz-Z(@l, @l,i T(fl) [Yd@l, D2(R2)= AND 72~“~ 1 5 1’2 PTR; j [ 0 rllbl @l)la~l)~ 2 [llcl L2(R2)=-~(~)1’2[~(~)1’2Y22(~2,~2)+~Y2,(B2,~2) +I- 2 0 32 4 -3 Rs If2 Y2-2(02, 1 @2) T(t2)+5g 7P2[YOO(@2? @2)1C(f2L 7Tlf2 0-5 LdR5) =$(;) [lldl [llel 1’2[Y20(@5, G'dlT(fs)+~ ~'~"P'-oo~& @5)1C(t5), 5 and D3(R3) = D1CR3); L3(R3) = L(R31, DA&) = D2W; LdW = DdR6) = D&b); L2NA LtdRd = LdRd, [llfl NMR SHIFTS IN PARAMAGNETIC SYSTEMS 633 where ri = 2PlRi and Hw=[l-e-‘n~o$], z(c)=[l-e-r(~~5+~~,~)], J(t)+‘( -;+,+I)], T(t)=[l-e-‘.~o$]. C(t) = [e-‘(t+ l)]. When bonding effects are neglected (that is, taking a2 = 1 and b’= O), the results reduce to those given by Golding and Stubbs (14). In arriving at this expression for the NMR shift many molecular hyperfine integrals had to be evaluated. The full set of integrals for the 2p orbitals is given in the Appendix. The results given by Golding and Stubbs (14,16) for the components of the hyperfine interaction tensors for the *T2E” and *TziJ’ levels may be extended to incorporate bonding effects by replacing, in the appropriate formulas for the tensor components, (5lLsk), (5lTalakL b#‘ahL GtlTasld, (dTapk>, ~~I%&), (77I&k&415% b$cdih), where Tap = (3rNnrNP - rkLp)lrk by respectively. (hJ&lC>, 634 GOLDING AND STUBBS DISCUSSION Initially we examine the case for a d’ transition metal ion in a strong crystal field of octahedral symmetry where the transition metal ion is surrounded by six ligands all 0.2 nm from the central atom along the three axes. We choose o1 = 2.275/au and p2 = 2.2/ao, the temperature as T = 300 K, the bonding coefficients as a = 0.8154 and b = -0.6818, and the spin-orbit coupling constants as & = 400 cm-’ and &, = 110 cm-‘. The overlap integral (dip) = 0.0584. The contours of equal chemical shift AB/B in parts per million in the xy plane as determined from the expression given in the previous section are shown in Fig. 3. The contour map is very complex and is markedly different for the case when no bonding is considered; see Fig. 4. In FIG. 3. An isoshielding diagram for the case of the NMR nucleus in the xy plane for a d’ transition metal ion in a strong crystal field of octahedral symmetry, where bonding effects have been considered. Ligands are located at kO.2 nm from the central metal ion along the three axes. The contours of equal chemical shift AB/B are in parts per million. (A bar over a number indicates a negative value.) NMR FIG. 4. An isoshielding metal ion in a strong crystal contours of equal chemical SHIFTS IN PARAMAGNETIC SYSTEMS 635 diagram for the case of the NMR nucleus in the xy plane for a d’ transition field of octahedral symmetry, for the case when bonding is not considered. The shift AB/B are in parts per million. Fig. 3 near the transition metal ion and the six ligands, AB/B is nearly isotropic. There are a number of regions where AB/B is a maximum or a minimum and regions where AB/B changes rapidly with distance. The pattern in Fig. 3 gives an insight into the very complex manner in which the NMR shift varies throughout the molecular system. Next we examine the reliability of considering the NMR shift in specific regions in space for the system given by Fig. 2 using the multipole expansion approach. A comparison of the exact solution with all the multipolar terms and only the dipolar term is given in Table 1 at different values of R along the (loo), (1 lo), and (111) axes. In this particular case the dipolar term arises only from the ligand part of the electronic wavefunction. From Table 1 we observe that consideration of the dipolar A TABLE 1 - 244.203 - 177.257 --820.166 145.109 - 22.274 -2.350 -6.011 - 1.124 - 0.606 -0.355 -0.221 -0.144 - 0.097 0.2 0.25 0.3 0.4 0.35 0.45 0.5 0.55 0.6 0.65 0.7 -1294.244 (a) Exact 0.1 0.15 0.05 R (nm) AB/B a (100) - 0.606 -0.355 -0.221 -0.144 - 0.097 --____.--..- 2.350 - 6.023 -1.124 -23.381 -313.601 - - 545.230 -350.816 -57,714.116 (b) Multipolar along ic) -0.566 -0.335 -0.210 -0.138 - 0.094 -..... -_-- -2.105 -5.133 - 1.032 - 17.523 - 139.897- -22.454 - 141.984 Dipolar ..____.~ - 13.543 axis (ppm) 0.699 1.408 0.367 2.955 11.232 6.090 28.633 23.388 -803.132 0.205 0.120 0.074 0.047 0.032 -... - .-~ ..___ -- ._ _~ (a) Exact AB/B a (110) 0.205 0.120 0.074 0.047 0.032 I,__- . 0.699 1.408 0.367 2.958 12.076 6.145 708.003 43.189 89.840.682 (b) Multipolar along 0.155 0.095 0.060 0.039 0.027 ,__ .-_ 0.475 0.874 0.266 1.589 1.802 2.477 - 10.759 -3.842 - 12.953 (4 Dipolar axis (ppm) 0.188 0.130 0.091 0.064 0.046 _--.-... 0.384 0.495 0.272 0.439 --4.703 0.429 - 134.595 -25.103 -935.934 (4 Exact AB/B 0.188 0.130 0.091 0.064 0.046 0.384 0.495 0.272 0.438 -5.289 -0.451 -712.930 -41.367 -95,644.784 a (111) - 0.233 0.153 0.104 0.072 0.05! _. . .” _. - 0.571 0.904 0.361 1.364 0.743 1.691 - 9.734 - 3.492 - 12.803 Cc) Dipolar axis (ppmj USING (a) THE EXACT (b) Multipolar along COMPARISON, FOR DIFFERENT VALUES OF R ALONG (100). (llO), AND (111) AXES, OF THE RESULTS FOR AB/E SOLUTION; (b) THE MULTIPOLAR EXPANSION; AND (C)THE POINT-DIPOLE APPROXIMATION 3 VI 3 z 8 8 Et NMR SHIFTS IN PARAMAGNETIC 637 SYSTEMS term only gives rise to substantial errors at distances less than 0.7 nm from the paramagnetic center. However, by considering all the multipolar terms the agreement with the exact solution occurs at distances greater than 0.4 nm. In many cases experimental paramagnetic NMR shifts are analyzed in terms of the Fermi contact and the dipolar interactions given by Eqs. [l] and [2]. To explore the possible problems of interpretation associated by such a procedure we examine the AB/B results obtained from the previous section over a temperature range and reconsider the data as a set of experimental results. We attempt to analyze the data when a ligand atom is the NMR nucleus as arising from a sum of the Fermi contact and the dipolar interactions given by Eqs. [l] and [2]. In this paper we choose (S,) in Eq. [l] as given in Ref. (17) for a d’ ion in a crystal field of octahedral symmetry. In the evaluation of ,YIIand ,Y~we include a small tetragonal distortion from octahedral symmetry. The quantities ~11and xL are determined from expressions in Ref. (15), where S is a measure of the distortion from octahedral symmetry. The S value is adjusted to give the best fit of the theoretical values of AI? to the expression AB = a(Sz)+b(x~~-xJ. In this case the best fit is obtained over the temperature range 200 to 400 K by choosing a very small distortion, and the results are summarized in Table 2. From Table 2 it follows that the analysis of the data using Eqs. [l] and [2] would incorrectly imply that the NMR shift arises dominantly from the Fermi contact interaction with a small contribution from the dipolar term in the multipole expansion. Although the case above was chosen when the ligand was the NMR nucleus, similar analyses may be made when the NMR nucleus is at any other position within the molecular system. TABLE 2 SUMMARY OF THE RESULTS OF FI-ITING A SET OF THEORETICAL DATA FOR AB/B AS ARISING FROMA SUM OFTHEFERMI CONTACTANDDIPOLAR INTERACTIONS AS GIVEN BY Eos. [l] AND [2], FOR THE CASE OF A d’ TRANSITION METAL ION IN A STRONG CRYSTAL FIELD OF OCTAHEDRAL SYMMETRY,~HERE BONDING EFFECTS HAVE BEEN CONSIDERED A AB/B fitted T W) ABIB (ppm) Exact Contribution from [l] 200 220 240 260 280 300 320 340 360 380 400 -644.4 -706.8 -752.1 -784.2 -806.2 -820.2 -828.0 -831.0 -830.3 -826.8 -820.9 -938.8 -932.1 -923.9 -914.3 -903.5 -891.6 -878.9 -865.5 -851.7 -837.5 -823.2 to Eqs. [l] Contribution from [2] 293.0 225.9 173.2 131.6 98.6 72.2 51.0 34.1 20.4 9.5 0.7 and [2] Total -645.8 -706.2 -750.7 -782.7 -804.9 -819.4 -827.9 -831.4 -831.3 -828.0 -822.5 63X GOLDING AND STUBBS Hence, we have demonstrated that it does not necessarily follow that a good fit ot a combination of Eqs. [I] and [2] to a set of experimental data results in a correct understanding of the origin of the NMR shift. Finally, employing results given in (14), we examine the AB/B contour map for a d’ octahedral metal ion system where the crystal field environment is of tetragonal symmetry with the tetragonal distortion along the z axis. We neglect bonding in this example. The tetragonal distortion component of the crystal field interaction was chosen in the form 6(/z -2), where S, the distortion parameter, was taken as 1000 cm- I. With & = 400 cm-‘, /?2 = 2.2/ ao, and T = 300 K, the position of the NMR nucleus relative to the d-electron-bearing atom to give rise to a specific NMR FIG. 5. An isoshielding diagram for the case of the NMR nucleus in the xy plane for a d’ transition metal ion in a strong crystal field of tetragonal symmetry for S = 1000 cm-‘, & = 400 cm-‘, p2 = 2.2/ao, and T = 300 K. The contours of equal chemical shift AB/B are in parts per million. (A bar over a number indicates a negative value.) NMR SHIFTS IN PARAMAGNETIC SYSTEMS FIG. 6. An isoshielding diagram for the case of the NMR nucleus in the zx plane for a d’ transition metal ion in a strong crystal field of tetragonal symmetry for 6 = 1000 cm-‘, & = 400 cm-‘, p2 = 2.2/a,, and T = 300 K. The contours of equal chemical shift AB/B are in parts per million. (A bar over a number indicates a negative value.) shift, ABIB, was determined as a set of isoshielding contours. The AB/B contours in the xy and IX planes are shown in Figs. 5 and 6, respectively. In the xy plane (see Fig. 5), the isoshielding lines form a complex pattern for R values less than 0.2 nm, whereas at larger values of R the contours, as expected, are independent of @. Similarly, in the zx plane (see Fig. 6), the isoshielding lines form a complex pattern for R values less than 0.3 nm. At very large values of R the pattern approaches the contours proportional to (3 co? 0 - 1)/R’. These diagrams illustrate clearly the complex patterns formed by the isoshielding lines for a wide range of R, namely, R = 0 to 0.5 nm. To investigate further the method of using [l] and/or [2] to interpret paramagnetic NMR shifts, we choose a set of AB/B values over a temperature range for this case 641) GOLDING AND ‘TABLE, A SUMMARY OF ‘I-HE RESULTS FOR AB/B AS AR~SINCJ FROM A INTERACTIONS AS GIVEN BY TRANSITION METAL ION IN A STUBBS 3 OF FI’ITING A SET OF THEORETIC‘AI 11~1, SUM OF THE FERMI CONTACT ANU DIPOI AR Eos. [l) AND [2], FOR THF. CASE OF A ‘1” STRONG CRYSTAL FIELD OF TETRAGONAI SYMMETRY AB/B fitted to Eqs. [l] and [2] ABIB (ppm) T W Exact Contribution from [l] Contribution from [2] Total 200 220 240 260 280 300 320 340 360 380 400 -35.56 -30.19 -25.37 -21.12 -17.41 -14.19 -11.41 -9.03 -6.98 -5.24 -3.74 -2.69 -2.67 -2.65 -2.62 -2.59 -2.55 -2.52 -2.48 -2.44 -2.40 -2.36 -32.91 -21.51 -22.71 -18.49 -14.82 -11.65 -8.92 -6.S7 -4.55 -2.83 -1.35 -35.60 --30.18 --25.35 -21.11 -17.41 -14.20 - 1 1.43 -9.05 -6.99 m-5.23 -3.71 when the NMR nucleus is 0.2 nm along the x axis. The analysis of fitting the temperature dependence of AB/B for the exact solution for a tetragonal distortion component given by 6 = 1000 cm-’ to a combination of [l] and [2] is summarized in Table 3. Such an analysis yields a distortion parameter and a distance R from [2] of 650 cm-’ and 0.220 nm, respectively, with substantial contributions from both the Fermi contact and dipolar interactions. The result further supports our previous finding that care needs to be exercised in interpreting NMR results of paramagnetic systems. APPENDIX: MOLECULAR HYPERFINE INTEGRALS FOR 2p ORBITALS (a) The Radial Series The method of evaluating the molecular hyperfine Golding and Stubbs (14). Define the radial integral m R:=‘(t) = 4@:(-Rf 1 r;-Lb,(R, 0 and Cl” (t) = Rc2’(t) ” 3 V,,(r) = R;‘(t), W,,(t) = R?‘(t), integrals is that given by rN) drN NMR SHIFTS IN PARAMAGNETIC 641 SYSTEMS where t = 2/3,R and b,(R, TN)= (rJr41’2Zn+1,2 (2P1r4Kn+3,2 (Wlr,) - (r.Jra)1’2Zn--1,2 (2P1r4Kn+1/2 G&r>), where r< is the smaller of R and rN and r, is the larger of R and rN and Z, and K, are the modified Bessel functions. The following radial integrals for 2p orbitals are required: U*(t)=&[;-(f+t+2+~) e-j, vlit)=-P:[;-(t+2+;) e-j, VAtI= -p:[(f-f)+(it+Y+Y+F+F) e-11, Wdt)=p:[~-(l+;) e-j, w*it)=#-$)+(2+y+$+$-q e-j, w4(t)=P’[(f-fy+y)-(3+T+~+~+~+~)e 8 32 200 760 1680 1680 (6) The Integrals (Pi) Tao / Pii> Employing the notation of Griffith (IS), define -G”(@, a) = YLO(@, @), ZE4(0, @) = (1/21’2)[YL-M(0, Z%(O, @) = (i/2”‘)[ @)+ YL.f(@, YL-M(O, @)- Y&(0, CD)], @)I. For the radial parts of the integrals define f,(a)= U2+aV1+(2-a)V3+ W, The integrals ( PiI Tea) Pi) are (x~T~,~x)=Z(~)“~~~(O)~~~~~, oi-~(~)“;,(o)zk;‘(e, @) - r]. 642 GOLDING AND STUBBS NMR SHIFTS IN PARAMAGNETIC SYSTEMS 643 644 GOLDING AND STUBBS 645 646 GOLDING (c) The Integrals ( ?Pijldrf,) The integrals (‘Pi/I&&I iblhdr~ Iv) = -(E) i(xIINylritly)= “‘(VI = --i(3”‘(VI+ i(ylldr;tilz~ = -(fi)1’2(VI+ = - (fi) + W2)Z:“: (0, @I, W2)Z:“:(@,@), i<xLlr~ly> iW.dr~l~) Pi} Pi) are given by -(&)“i(Vl+ +Y AND STUBBS W2)Z2,(0, O)+q(V,+ O)+~(~)1’2(VI+ W2)Z:“:(@, 1’2( VI + WZE -(fj)“2(VI+ @) (@, @), @I, W2)Z:“:(O,@), v1+ +T cv1+ i(zllNz/rilx>= W2)Z2,(0, (V1+ Wo)Zoo(@, @), ~(yL/&lz>= -(fi)1’21V~+W2)Z:“:(@, i<zI&.J&lx>= Wo)Zoo(O, @), -(E)1’2(VI+ @)+f(;y2(v,+ w,)z:‘:(@, Wo)Zoo(@, W2)Z:“:(0, W2)Z*o(O, @) @I, @). REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. R.J.KURLANDAND B.R.McGARvEY, J. Magn.Reson.2,286 (1970). A. D.BUCKINGHAM AND P.J. STILES, Mol.Phys.24,99 (1972). P.J. STILES, Mol. Phys.27,501 (1974). 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