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Anisotropy of the indirect nuclear spin-spin coupling constant
CHEMICAL PHYSICS LETTERS
Volume 4, number 8
ANISOTROPP NUCLEAR
OF
SPIN-SPIN
THE
INDIRECT
COUPLING
MJIICH~L
1 January 1970
CONSTANT
BARFIELD
Received 29 November 19G9
Anisotropics in the indirect nuclear spin-spin coupling constants arise from cross-terms between the and &polar terms in the electron-nucienr Hamiltonian. Calculated anisotropic H-H and l%-H coupling constnnts. which nrc based on valence-bond wavefunctions. arc non-negligible in comparison with the isotropic coupiing constants, but nre much smnller than the direct dipole-dipole coupling conrontnct
stants.
Experimental studies of high resolution nuclear magnetic resonance (NMR; spectra in SOIids fl, 2] and in partially oriented molecules [3-S] have raised questions of the importance of the anisotropies of the indirect nuclear spin-spin coupling constants, The anisotropic spin coupling term in the spin Hamiltonian, which is used to analyze NMR spectra in nematic liquid crystais, includes contributions from direct and indirect terms. Recent experimenial results [6,7] suggest substantial indirect coupling anisotropies, especially between directly-bonded nuclei. Theoretical values are of particular interest as most analyses of Iih%Rspectra of oriented molecuks assume that the indirect anisotropic coupling constants are negligible and neglect anharmonieity effects on the direct dipole coupling. Contributions to the indirect nucfear spin-spin coupling constants from the contact term, $$3) = (16&fi/3f
c y Sfr ts -r k,N N IZN k N’
(I)
in the electron-nuclear Hamiltonian are rigorously isotropic [El,9 J, In eq. (1) B is the Bohr magneton, YN is the magnetogyrir: ratio of nucleus N, 6(rk$ is the Dirac delta function, and Sk and f~ are the electron and nuclear spin operators, respectively. Both isotropic and anisotropic contributions to nuclear spin-spin coupling arise from the electron-nuclear dipok- Hamiftonian, ‘I
where r&J is the vector which connects electron R and nucleus N. Dipolar contributions to indirect Z-H coupling arise in second-order perturbation theory, but these are small in comparison with the total isotropic coupling constants [8]. Therefore, it appears that the most important contributions to the anisotropic, indirect eou-
pling constants in organic molecules are those
which arise from cross-terms between the contact and dipolar Hamiltonian operators. From second-order perturbation theory [8j and the spin dependence of the one particfe density matrix f9& the contact-dipolar nuclear spin coupling tensor between nucIei N and N’ is given bY J(23) orfiNN’=
Volume 4, number 6
CHEAIICALPHYSICSLETTERS
where (Y,/3 =x,y, and z and Q1( 0~~ /l;l) is the transition spin density connecting singlet and triplet states having energies I.,!?0 and 3E,, respectively, and the summation is over all of the triplet levels. Transition spin densities evaluated at nucleus N are denoted by Q~(OK~ ilN;lN) and TN~~ is a dipolar tensor operator for nucleus N,
Valence-bond
calculations
of h-let
and
trip-
let-state energies, transition spin densities and contact nuclear spin-spin coupling constants have been reported [lo-121 for a large number of hydrocarbons. Dipolar integrals for use in eq. (3) were obtained by the formulas and methods of McConneil and Strathdee [13]. In the valencebond (VB) approximation, transition spin densities are given by [lo] eq. (5)
where Y denotes an atomic orbital, and in the superposition diagram of singlet and triplet cannonical structures, ijz is the number of islands, and f Tl is +1 if atomic orbital Y is part of an island &hich contains a broken bond and is in the even subset; fyi is -1 if ?-is a part of an island which contain H a broken bon$ but is in the odd subset. In all other cases fjl = 0. Since it is reasonable to assume that maximum anisotropic indirect coupling will occur for directly bonded atoms, a calculation was performed for the hydrogenOmolecule with 3E1 - lE0 = 9.0 eV, r(H-H) = 0.74A and effective nuclear charges set equal to unity. Where the internuclear axis is parallel to the z axis, the contactdipolar coupling tensor is diagonal with principal components, J_EAH, = J($LHr = +33.7 Hz, and J(23) = -67.4 Hz. The corresponding isotropic zz HH’ coupling constant based on the contact term with parameters is 464 Hz, which is to be compared with a value of 278 Hz, inferred from the experimental value [14] for the HD molecule. The contact-dipolar coupling constant components in H2 arise from the simultaneous interaction of each electronic spin with the two nuclear spins via the two mechanisms. For example, if the spin of proton H is oriented along the internuclear or z axis, then the antiparallel arrangement of the electron spin centered CII H’ is favored by the dipolar interaction. Since this elec-
these
tron interacts with H’ via the contact term, an antiparallel orientation is favored here, also. Thus, the parallel arrangemcpt of the two protons is of lower energy, negative
and J?~~~. should be
in sign.
More interesting, perhaps, are theoretical estitnates of directly bonded 13C--H anisotropic coupling constants, since a value for (Je -J,)C of +1890 Hz has been suggested [S] from an anaaysis af the NMR spectrum of methyl fluoride in a nemakic
liquid
crystal_
As
more
electrons
2~
contribute to the anisotropy of the 13C-11 coupling constant, the situation is much mure diEEicuit than that for H A simple calculation based on a two electron
-43’C-H
frqmwt
consisting
of
the 1s atomic orbital on hydrogen and a tecrahedral hybrid on carbon with s*(C-H) = I.lOi gives a diagonal anisotropy tensor with (
$123’ - iyCH =-17.1
Hz. The corresponding
calculated isotropic coupling constant is J (3) _ CH = 139 Hz. The anisotropic l3C-H coupling tensor based on an eight electron fragment for the methane molecule and the VB exchange integral param-
eters of Karplus and Bersohn [15] is J(23) &3C-H
=
i 19.47
-1.64
-1.64
22.78
I-2.94
-3.54
-2.94 I -3.54 j Hz, 1 -42.25 1
where the z axis is parallel to one of the C-H bonds and the xz plane coincides with one of the CH2 planes. Not only is there an enormous disparity in the magnitudes of these results and those suggested by Krugh and Bernheim [S], but but they are of opposite sign. Anisotropic vinical H-H coupling tensors were calculated for a four electron $thanic fragment @-c,c’-IL’) with s-(C-H) = l.lOA, r(C-C) = 1.54A, L HCC = 109’28’, and effective nuclear charges of 1.0 and 3.18 for hydrogen and carbon, respectively. Where the z axis is parallel to the C-C bond, the respective ~WZS and garde anisotropic coupling tensors are J(23) @HH -1.56 =
0.00
0.00
0.00 3.90
1.89
0.00 1.85 -2.34
l-0.89 and
0.53 -0.18 i 1 0.53 0.65 0.81 Hz.
! -0.18
0.81
O-24 I
The calculated isotropic coupling constants, based on the same exchange integral parameters, 519
CHRMCAL PHYSICSLETTERS
Volume 4, number 8
are 21.1 and 3.2 Hz, respectively [IlJ. These values of the anisotropic indirect coupling constants are negligible in comparison with the direct terms IlS]. Calculations of anisotropic geminal H-H coupling constants were based on the eigtt electron ethylenic fragment which is depicted in fig, 1. In these calculations it was assumed that r(C- H) = l.Osii, r(C=C) = X,337& L HCH = 120°, and exchange integral parameters are those given by Karplus and Fraenkel [l?]. The calculated geminal H-H coupling tensor for the sys;em of axes depicted In fig. 1 is 2,,(23)
I-5.51
22.85
= 22.85
16.41
0.00
0.00
cu@HH’
0.00 1
0.00 Hz. -1c.90
The corresponding calculated isotropic H-H coupling constant is 13.48 Hz. Since this value is considerably larger than typical geminai ethylenic coupling constants, the most uncertain of the exchange integrals, a&‘, c), was reduced from 0.106 eV to 0.036 eV. This change gives a calculated isotropic geminal H-H coupling of &.I6 Hz and the corresponding geminal H-H auisotropic ccupling tensor is I -4.98
= 23.54 2J(23) o!PHH’ 0.00
23.54
0.00 I
16.37
0.00
0.00
-11.39
Hz.
Y
1.
520
The author would like to thank Dr. C. S. Yannoni for indicating the need for theoretical work in this area and for providing helpful comments on the manuscript. This work was supported in part by a grant from the National Science Foundation. Facilities of the University of Arizona Computer Center are gratefully acknowledged. REFERENCES [f] U. Haeberfen and J. S. Waugh, Phys. Rev. 175 (1968)
453. fZl E. Andrew and L. F. Farnell. Mol. Phys. 15 (l968) 157. [3] A. Saupe and G. Englert, Phys. Rev. Letters 11 (1963) 462. [4] A. D. Buckingham and K. A. McLauchlan, Progress
in nuclear masnetic resonance soectrasconv. Vol. &ess, New York.*J.967) p. 65.
Xt
Eight eiectron fragment of the ethylene moieplane with the y ads parailel to the C-C bond.
cule. The atoms lie in the xy
tropic coupling constants, is in agreement with data obtained by other techniques. However, the authors flB] emphasize the difficulty of applying corrections for vibrational motions, and the greatest source of uncertainty may be due to anharmonicity effects. The neglect of coupling between non-bonded protons in the analyses of the NMR spectra of oriented molecules appears to be justified. It is interesting to note, however, that the anisotropic components can be significantly larger than isotropic couplixq constants. the corresponifing
I1 (Pergamon
Therefore, in contrast to the isotropic contact coupling constant [la] the anisotropic coupling tensor is relatively inserzzitive to the bond order between the coupled atoms. The NMR spectrum of ethylene in a nematic crystal has been analyzed [19] and the geometry, which is based on neglect of the indirect, aniso-
Fig.
1 January 1970
161 L. C.Snvder, 3. Chem. Phvs. 43 11965‘14041. iGj T. R.K&gh and R.A. Berr;heim, j. A&. Chem.Soc. 91 0.969) 2385. 171 C. S. Yanncni, J. Chem, Phys., to be published. fS] N. F. Ramsey, Phys. Rev. 91 (1953) 303. fSJ R. McWeeny and Y. Mizuno, Proc. Roy. Sot. (London) A259 (1961) 554. [lOI M.Barfieid, J. Chem. Phys. 48 (1868) 4458. [ll] hi. Bar-field, J. Chem. Phys. 46 f1967) 811; 48 (1968) 4463; 49 (l968) 2145. [12] M. Barfield and B. Chakrabarti, J. Am. Chem.Soc. 91 (1969) 4346. [X3] H. M. McConneIl and J. Strathdee, Mol. Phys. 2 (1353) 129. Note the corr~tian given hr RM.
Pitzer. C. W. Kern and W.N. Lipscomb. J. Chem.
Phys. 37 0962) 267. [14] T. F. Wimett. Phrs. Rev. 91 (1963) 476. [IS] S$rpIus and R. Bersohu, J. Chem. Phys. 51 (1969) [Is] D. X~Sifverman and B. P. Dailey. J. Chem. Phys. 51 (1969) 655. [I?] M. Karplus and G. K. Fraenkel. J. Chem. Phys. 35 (1961) 1312. {IS] M. Barfield and M.Karplus, J. Am. Chem.Soc. 91 (1969) 1. 1191 W. Bovee, C. W. Hilbers and C. MacLean, Mol. Phys. 17 (1969) 75.
Volume 4, number 8
ANISOTROPP NUCLEAR
OF
SPIN-SPIN
THE
INDIRECT
COUPLING
MJIICH~L
1 January 1970
CONSTANT
BARFIELD
Received 29 November 19G9
Anisotropics in the indirect nuclear spin-spin coupling constants arise from cross-terms between the and &polar terms in the electron-nucienr Hamiltonian. Calculated anisotropic H-H and l%-H coupling constnnts. which nrc based on valence-bond wavefunctions. arc non-negligible in comparison with the isotropic coupiing constants, but nre much smnller than the direct dipole-dipole coupling conrontnct
stants.
Experimental studies of high resolution nuclear magnetic resonance (NMR; spectra in SOIids fl, 2] and in partially oriented molecules [3-S] have raised questions of the importance of the anisotropies of the indirect nuclear spin-spin coupling constants, The anisotropic spin coupling term in the spin Hamiltonian, which is used to analyze NMR spectra in nematic liquid crystais, includes contributions from direct and indirect terms. Recent experimenial results [6,7] suggest substantial indirect coupling anisotropies, especially between directly-bonded nuclei. Theoretical values are of particular interest as most analyses of Iih%Rspectra of oriented molecuks assume that the indirect anisotropic coupling constants are negligible and neglect anharmonieity effects on the direct dipole coupling. Contributions to the indirect nucfear spin-spin coupling constants from the contact term, $$3) = (16&fi/3f
c y Sfr ts -r k,N N IZN k N’
(I)
in the electron-nuclear Hamiltonian are rigorously isotropic [El,9 J, In eq. (1) B is the Bohr magneton, YN is the magnetogyrir: ratio of nucleus N, 6(rk$ is the Dirac delta function, and Sk and f~ are the electron and nuclear spin operators, respectively. Both isotropic and anisotropic contributions to nuclear spin-spin coupling arise from the electron-nuclear dipok- Hamiftonian, ‘I
where r&J is the vector which connects electron R and nucleus N. Dipolar contributions to indirect Z-H coupling arise in second-order perturbation theory, but these are small in comparison with the total isotropic coupling constants [8]. Therefore, it appears that the most important contributions to the anisotropic, indirect eou-
pling constants in organic molecules are those
which arise from cross-terms between the contact and dipolar Hamiltonian operators. From second-order perturbation theory [8j and the spin dependence of the one particfe density matrix f9& the contact-dipolar nuclear spin coupling tensor between nucIei N and N’ is given bY J(23) orfiNN’=
Volume 4, number 6
CHEAIICALPHYSICSLETTERS
where (Y,/3 =x,y, and z and Q1( 0~~ /l;l) is the transition spin density connecting singlet and triplet states having energies I.,!?0 and 3E,, respectively, and the summation is over all of the triplet levels. Transition spin densities evaluated at nucleus N are denoted by Q~(OK~ ilN;lN) and TN~~ is a dipolar tensor operator for nucleus N,
Valence-bond
calculations
of h-let
and
trip-
let-state energies, transition spin densities and contact nuclear spin-spin coupling constants have been reported [lo-121 for a large number of hydrocarbons. Dipolar integrals for use in eq. (3) were obtained by the formulas and methods of McConneil and Strathdee [13]. In the valencebond (VB) approximation, transition spin densities are given by [lo] eq. (5)
where Y denotes an atomic orbital, and in the superposition diagram of singlet and triplet cannonical structures, ijz is the number of islands, and f Tl is +1 if atomic orbital Y is part of an island &hich contains a broken bond and is in the even subset; fyi is -1 if ?-is a part of an island which contain H a broken bon$ but is in the odd subset. In all other cases fjl = 0. Since it is reasonable to assume that maximum anisotropic indirect coupling will occur for directly bonded atoms, a calculation was performed for the hydrogenOmolecule with 3E1 - lE0 = 9.0 eV, r(H-H) = 0.74A and effective nuclear charges set equal to unity. Where the internuclear axis is parallel to the z axis, the contactdipolar coupling tensor is diagonal with principal components, J_EAH, = J($LHr = +33.7 Hz, and J(23) = -67.4 Hz. The corresponding isotropic zz HH’ coupling constant based on the contact term with parameters is 464 Hz, which is to be compared with a value of 278 Hz, inferred from the experimental value [14] for the HD molecule. The contact-dipolar coupling constant components in H2 arise from the simultaneous interaction of each electronic spin with the two nuclear spins via the two mechanisms. For example, if the spin of proton H is oriented along the internuclear or z axis, then the antiparallel arrangement of the electron spin centered CII H’ is favored by the dipolar interaction. Since this elec-
these
tron interacts with H’ via the contact term, an antiparallel orientation is favored here, also. Thus, the parallel arrangemcpt of the two protons is of lower energy, negative
and J?~~~. should be
in sign.
More interesting, perhaps, are theoretical estitnates of directly bonded 13C--H anisotropic coupling constants, since a value for (Je -J,)C of +1890 Hz has been suggested [S] from an anaaysis af the NMR spectrum of methyl fluoride in a nemakic
liquid
crystal_
As
more
electrons
2~
contribute to the anisotropy of the 13C-11 coupling constant, the situation is much mure diEEicuit than that for H A simple calculation based on a two electron
-43’C-H
frqmwt
consisting
of
the 1s atomic orbital on hydrogen and a tecrahedral hybrid on carbon with s*(C-H) = I.lOi gives a diagonal anisotropy tensor with (
$123’ - iyCH =-17.1
Hz. The corresponding
calculated isotropic coupling constant is J (3) _ CH = 139 Hz. The anisotropic l3C-H coupling tensor based on an eight electron fragment for the methane molecule and the VB exchange integral param-
eters of Karplus and Bersohn [15] is J(23) &3C-H
=
i 19.47
-1.64
-1.64
22.78
I-2.94
-3.54
-2.94 I -3.54 j Hz, 1 -42.25 1
where the z axis is parallel to one of the C-H bonds and the xz plane coincides with one of the CH2 planes. Not only is there an enormous disparity in the magnitudes of these results and those suggested by Krugh and Bernheim [S], but but they are of opposite sign. Anisotropic vinical H-H coupling tensors were calculated for a four electron $thanic fragment @-c,c’-IL’) with s-(C-H) = l.lOA, r(C-C) = 1.54A, L HCC = 109’28’, and effective nuclear charges of 1.0 and 3.18 for hydrogen and carbon, respectively. Where the z axis is parallel to the C-C bond, the respective ~WZS and garde anisotropic coupling tensors are J(23) @HH -1.56 =
0.00
0.00
0.00 3.90
1.89
0.00 1.85 -2.34
l-0.89 and
0.53 -0.18 i 1 0.53 0.65 0.81 Hz.
! -0.18
0.81
O-24 I
The calculated isotropic coupling constants, based on the same exchange integral parameters, 519
CHRMCAL PHYSICSLETTERS
Volume 4, number 8
are 21.1 and 3.2 Hz, respectively [IlJ. These values of the anisotropic indirect coupling constants are negligible in comparison with the direct terms IlS]. Calculations of anisotropic geminal H-H coupling constants were based on the eigtt electron ethylenic fragment which is depicted in fig, 1. In these calculations it was assumed that r(C- H) = l.Osii, r(C=C) = X,337& L HCH = 120°, and exchange integral parameters are those given by Karplus and Fraenkel [l?]. The calculated geminal H-H coupling tensor for the sys;em of axes depicted In fig. 1 is 2,,(23)
I-5.51
22.85
= 22.85
16.41
0.00
0.00
cu@HH’
0.00 1
0.00 Hz. -1c.90
The corresponding calculated isotropic H-H coupling constant is 13.48 Hz. Since this value is considerably larger than typical geminai ethylenic coupling constants, the most uncertain of the exchange integrals, a&‘, c), was reduced from 0.106 eV to 0.036 eV. This change gives a calculated isotropic geminal H-H coupling of &.I6 Hz and the corresponding geminal H-H auisotropic ccupling tensor is I -4.98
= 23.54 2J(23) o!PHH’ 0.00
23.54
0.00 I
16.37
0.00
0.00
-11.39
Hz.
Y
1.
520
The author would like to thank Dr. C. S. Yannoni for indicating the need for theoretical work in this area and for providing helpful comments on the manuscript. This work was supported in part by a grant from the National Science Foundation. Facilities of the University of Arizona Computer Center are gratefully acknowledged. REFERENCES [f] U. Haeberfen and J. S. Waugh, Phys. Rev. 175 (1968)
453. fZl E. Andrew and L. F. Farnell. Mol. Phys. 15 (l968) 157. [3] A. Saupe and G. Englert, Phys. Rev. Letters 11 (1963) 462. [4] A. D. Buckingham and K. A. McLauchlan, Progress
in nuclear masnetic resonance soectrasconv. Vol. &ess, New York.*J.967) p. 65.
Xt
Eight eiectron fragment of the ethylene moieplane with the y ads parailel to the C-C bond.
cule. The atoms lie in the xy
tropic coupling constants, is in agreement with data obtained by other techniques. However, the authors flB] emphasize the difficulty of applying corrections for vibrational motions, and the greatest source of uncertainty may be due to anharmonicity effects. The neglect of coupling between non-bonded protons in the analyses of the NMR spectra of oriented molecules appears to be justified. It is interesting to note, however, that the anisotropic components can be significantly larger than isotropic couplixq constants. the corresponifing
I1 (Pergamon
Therefore, in contrast to the isotropic contact coupling constant [la] the anisotropic coupling tensor is relatively inserzzitive to the bond order between the coupled atoms. The NMR spectrum of ethylene in a nematic crystal has been analyzed [19] and the geometry, which is based on neglect of the indirect, aniso-
Fig.
1 January 1970
161 L. C.Snvder, 3. Chem. Phvs. 43 11965‘14041. iGj T. R.K&gh and R.A. Berr;heim, j. A&. Chem.Soc. 91 0.969) 2385. 171 C. S. Yanncni, J. Chem, Phys., to be published. fS] N. F. Ramsey, Phys. Rev. 91 (1953) 303. fSJ R. McWeeny and Y. Mizuno, Proc. Roy. Sot. (London) A259 (1961) 554. [lOI M.Barfieid, J. Chem. Phys. 48 (1868) 4458. [ll] hi. Bar-field, J. Chem. Phys. 46 f1967) 811; 48 (1968) 4463; 49 (l968) 2145. [12] M. Barfield and B. Chakrabarti, J. Am. Chem.Soc. 91 (1969) 4346. [X3] H. M. McConneIl and J. Strathdee, Mol. Phys. 2 (1353) 129. Note the corr~tian given hr RM.
Pitzer. C. W. Kern and W.N. Lipscomb. J. Chem.
Phys. 37 0962) 267. [14] T. F. Wimett. Phrs. Rev. 91 (1963) 476. [IS] S$rpIus and R. Bersohu, J. Chem. Phys. 51 (1969) [Is] D. X~Sifverman and B. P. Dailey. J. Chem. Phys. 51 (1969) 655. [I?] M. Karplus and G. K. Fraenkel. J. Chem. Phys. 35 (1961) 1312. {IS] M. Barfield and M.Karplus, J. Am. Chem.Soc. 91 (1969) 1. 1191 W. Bovee, C. W. Hilbers and C. MacLean, Mol. Phys. 17 (1969) 75.