- Email: [email protected]
Nonempirical quantum mechanical calculations of three contributions to the variation of nuclear magnetic shielding constants with intermolecular interactions. I. Method and applications to water and methane
JOURNAL
OF
MAGNEnc
RESONANCE
55.64-77
(1983)
Nonempirical Quantum Mechanical Calculations of Three Contributions to the Variation of Nuclear Magnetic Shielding Constants with Intermolecular Interactions. I. Method and Applications to Water and Methane C. GIESSNRR-FQETTRE
AND S. FERCHIOU
Luboratoire de Biochimie Thiorique Associk au CNRS, Institut de Biologie Physico-Chimique, 13, rue Pierre et Marie Curie, 75005 Paris, France Received March 30. 1983 Nonempirical quantum mechanical procedures are proposed for the calculation of (a) the “geometric” magnetic shielding of a nucleus N of a molecule A by the electrons of a molecule B, (b) the variation of the shielding constant aN because of the polarization of molecule A by molecule B, and (c) the sum of the contributions of the exchange and charge transfer mechanisms between mok.cuks A and B to the variation of ON. The method proposed utilizes gauge invariant atomic orbitals and gives for the different contributions results which are gauge invariant. Results concerning H20. . * H20, cI&. - * CH,, HrO. * . H+, and CH4. . . H+ are reported and discussed.
To calculate the chemical shift variations due to intermolecular interactions by an ab initio method, the procedure utilized is to calculate the magnetic shielding constants of the nuclei of interest in the isolated molecule and in the complex treated as a supermolecule (1). The desired quantity is then given by the difference between the values obtained from the two computations. For a nucleus N of a molecule A the chemical shift variation from the interaction of A with a molecule B is given by ANN
=
UNAB-
aNA
t11
where UNABand CNA are the magnetic shielding constants of nucleus N in the AB supermolecule and the isolated molecule A, respectively (Z-9). Such computations have in most cases given results that are in good agreement with experimental data and useful for their interpretation in terms of intra- and intermolecular contributions to the measured variations of the magnetic shielding constants studied (2, 9). However, it is not possible from these studies to determine the relative importance of the different type of contributions like the “geometric” (general name proposed by Lau and Vaughan (10) for the ring current plus local magnetic susceptibility anisotropy effects (I 1)) or the polarization (or electric field) (12) contributions. In the present study we propose within the framework of nonempirical computations a decomposition scheme of the chemical shift variations caused by intermolecular 0022-2364f83
$3.00
Copyright 8 1963 by AcaUemic Press. Inc‘. All rights of reproduction in any form reserved.
64
INTERMOLECULAR
SHIELDING
EFFECTS
65
interactions. Interest in the calculation of the different contributions to the variations of shielding constants which are obtained from supermolecule computations is twofold. On the one hand such calculations will indicate which fraction of A6 each contribution is responsible for, and how its importance varies according to the type of molecular interaction considered and to the intermolecular arrangement. They will also provide, on the other hand, a test for the semiempirical methods of calculations of intermolecular chemical shift variations (11-18). Such tests appear particularly important for the polarization contribution. If the role of this term is clearly established for protons participating in a hydrogen bond (12, 14, 19) this is not the case for the other protons of the molecules nor for the nonhydrogen nuclei of these molecules. Accurate studies have shown that an external electric field does produce a variation of the magnetic shielding constants of the nuclei of a molecule (20) but the results which have been reported cannot be utilized straightforwardly to evaluate the chemical shift variations due to the electric field created by the electronic and nuclear charge distribution of a molecule although experimental and theoretical results tend to support that they are far from negligible in many cases (4, 21). METHOD
Since intermolecular problems deal with variable distances, it is particularly important for such studies to obtain results which are strictly independent of the choice of the system of coordinates used for the computations. To satisfy this criterion, we utilize gauge invariant atomic orbitals and carry out the calculations within the fiamework of the self-consistent perturbation treatment proposed by Ditchfield (22). The calculation of the different contributions to the variation of the magnetic shielding constant of the nucleus N of a molecule A with intermolecular interactions is achieved by carrying out three computational steps. (a) Total variation. The c$? component of the magnetic shielding tensor of a nucleus N of a molecule A of a system AB made of the two molecules A and B is given by
(22): & = (G%I~Hm~NBIe3)+ (K&MpNBIai)
PI
where $!a is the ground state wavefunction of the AB system #$$ the perturbation of I& due to the external magnetic field H, XHnrrNu is the second order perturbation operator which introduces the magnetic field H and the nuclear moment (c(~) of nucleus N and ZrN8 is the first order perturbation operator depending only on j&N. (b) Geometric (or direct) contribution. If we suppose that there is no interaction between molecules A and B we can write ‘Z*~=‘X*+~~ EAB = EA + EB and (hMB>
= 0
131
where $A and I/~ are the molecular wavefUnctions of molecule A and B, respectively.
66
GIESSNER-PRETTRE
AND FERCHIOU
If these relations [3] are taken into account, the expression giving ST8 [41 takes the form
or more compactly with obvious notation. Taking into account the particular forms of & expression giving (rN can be written
and +% given in [3] and [6] the
where we see that the hrst two terms depend only on the wavefunction of molecule A while the two following terms depend on the wavefunction of molecule B only. Therefore if we consider, for example, a particular nucleus N of the AB system belonging to molecule A and if we put and we can write UN = UNA + A&
[lOI
where uNA is the shielding constant of this nucleus in the isolated molecule A while Aa”,, is the “geometric” contribution (according to Lau and Vaughan’s definition (10)) of molecule B to the shielding of nucleus N of molecule A. The inspection of formula [7] shows that u ~~ and A6m can be calculated independently since UNA does not depend upon 1c/Band A& does not depend on #A. Hence it is possible to calculate the shielding of nucleus N due to the electrons of molecule B without introducing the wavefimction of molecule A in the computation. Aside from one-electron integrals which depend upon the location of N with respect to the atoms of molecule B the only quantities required for the computation of A$$ are $$ and $p. The unperturbed wavefunction $J: is obtained from an ab initio SCF computation using Gaussian basis functions and@ is calculated by the self-consistent perturbation method using GIAO (22). Then for each location of the nucleus N we calculate the one-electron integrals and (PIfl’“%) 1111 q are the gauge invariant basis functions of molecule B used in the of *p. To obtain meaningful and gauge invariant values of A&B the of these integrals which introduce the nuclear magnetic moment have out using the system of coordinates used throughout the calculation of This requirement is due to the gauge dependence of $#“ even when (24). (PI% Ha“NBjq)
where p and computation computations to be carried #! and +k. using GIAO
INTERMOLECULAR
SHIELDING
EFFECTS
67
(c) Polarization contribution. The calculation of this contribution utilizes polarized wave functions for A and B defined in the fashion of Morokuma (25); thus, the polarized wave functions @‘, 1c/$’of the two molecules A and B are calculated by a SCF procedure using a Hamiltonian which takes into account the presence of the other molecule but neglects completely intermolecular overlap. We have +l(p, = &V$ and (v@‘l@) = 0 = hf ore but the two Hamiltonians &“A and Xa contain explicitly the intermolecular nuclear attraction and electronic repulsion. We then use the polarized wavefunction $2 and @’ as zeroth order wavefunctions for the self-consistent perturbation calculation of the magnetic shielding constant. For the nucleus N of molecule A @LA is given by u$l = (~~q~Ha”Nql+bp)
+ (~~pJP”q@-).
[I21
The nuclear attraction and electron repulsion between molecules A and B is taken into account in the computation of #g“‘r’, therefore the pq element of the perturbation Hamiltonian takes the form
+ PC&
(
(pqlrs) - Jj (pslrq)4&~
>I}
1131
where 6, is equal to one if the two atomic orbitals concerned are located on the same molecule and zero otherwise. Once & is obtained, the polarization contribution to AS is defined by Asi; = (TEA- (TNA El41 (d) Exchange plus charge transfer contribution. The sum of these two contributions can be obtained by subtracting from the total A6 the geometric and polarization contributions; therefore we have A,g+’ = A& - A& - A&.
[I51
This difference can be identified as the exchange plus charge transfer contributions since it takes into account the overlap of the two wavefunctions (exchange) and introduces in the perturbation treatment the intermolecular excitations (occupied orbitals of A to vacant orbitals of B and vice versa) which are responsible for the charge transfer interactions. Concerning the charge transfer contribution we want to point out that the meaning of the term charge transfer is not identical in the case of calculation of magnetic shielding constants and in the case of intermolecular interaction energy. For the nucleus N of molecule A the vacant orbitals of molecule B contribute to ON not only through the perturbation of the ground state wave function of molecule A (as for calculations of AEd) but also through the perturbation treatment giving $2. Basis sets. The computations have been carried out for each of the cases considered with two different basis sets that we use currently for theoretical NMR calculations (4-Q, because it is of interest to determine how the different contributions are sensitive qualitatively and quantitatively to the choice of the basis functions. One of the basis set is minimal and the second is a split valence set defined in (23).
68
GIEXBER-PRETTRE
AND FERCHIOU
H2 31
\
‘(
CL,,
.H2 ,cf<,
/ H2
H1
(b)
“\ 0_.._... ,,+ /
H’
H2
FIG. I. Atom numbering and geometric arrangement in the (a) water-water, (b) methane-methane, water-proton, and (d) methane-proton systems.
RESULTS
(c)
AND DISCUSSION
In Fig. 2 and 3 are reported the values of the geometric, polarization, and exchange plus charge transfer contributions to A.6 (also reported) for the five different nuclei of the water dimer for various intermolecular distances: The results clearly show that for each nucleus at any intermolecular distance the geometric contribution is an order of magnitude smaller than the total chemical shift variation. In addition the values obtained with the two difkrent basis sets for this effect are almost identical numerically. For intermolecular distances larger than the equilibrium distance (2.95 A with the minimal basis set and 2.80 A with the extended one) the polarization contribution appears numerically important with respect to A6 even for the nuclei which are not engaged in the hydrogen bond. On the other hand the magnitude of this effect appears much larger for the oxygen nuclei than for the protons. The comparison of the value of A6p obtained with the two different basis sets shows that the extension of the basis produces an increase of the absolute value of this contribution, a result which parallels the increase of the polarization component of intermolecular interaction energies with the extension of the basis (26). The curves of Figs. 2 and 3 representing the variation of the exchange plus charge transfer contributions show that this term is negligible for large intermolecular distances
‘\ \\
; I
1H
-2-
i 02
6 d(Q-Q) %
*
FIG. 2. Variation c&ulated with the minimal basis set of three contributions to A6 and of their sum for the different nuclei Of the water dimer as a function of the 0. . .O d&an& (fj = 15(Y). The curves blended into the abscissa’s axis are not reported. * * *, Geometric contribution; - - -, polarization contribution; _.-.- , charge transfer plus exchange contribution; -, total.
2 t-
Ad iw *.
i
\ \
‘. \ - . . -.._ ....!_.____. >:&.->...:.T..--- --I--0’2 , 4
2-
A \ 4- ‘1, , \
s
70
GIFSSNER-PRETTRE
AND FERCHIOU
INTERMOLECULAR
SHIELDING
EFFECTS
71
but increases very rapidly for small ones. This is in keeping with the fact that the exchange and charge transfer mechanisms are short-range effects (25, 27-29). It is important to notice that for the equilibrium distance this contribution can be, for the proton, for example, as large, in absolute value, as the polarization elXxt. The comparison of Figs. 2 and 3 shows that the value calculated for this charge transfer plus exchange contribution is extremely sensitive to the basis set used; it is particularly pronounced in the case of O2 since at the equilibrium distance AP” is equal to +0.6 ppm from minimal basis set computations and -10.4 ppm if we use the extended basis. In Figs. 4 and 5 are reported the variation of A6 and of the different contributions for the equilibrium distance, as a function of the angle B defined in Fig. 1. The value of 180” corresponds to the colinearity of the 0,-H, bond with the bisector of the HOH angle of molecule labeled 2. We see from these figures that the variation of AP is negligible for each nucleus and that APc is approximately constant for the protons. On the contrary the variation of this last contribution is downfield for O2 and upfield for 0, when B decreases from 180 to 90”. This result indicates that the charge transfer plus exchange effect is, for the nonhydrogen atoms, very sensitive to the intermolecular orientation. This angular dependence of Ascf+” is to be compared with the corresponding variation obtained for the charge transfer component of the intermolecular interaction energy of the water dimer (30). A6 p.p.m. L
pgn. I
0
6-
9,o -.
-2-
y
.....-
yo
qo 180 L '.- . ..___.._........ 8' _ _-. -______
-,.
--..
4-
-.-.
-.-._
___---,,-I' 2-
IH
0
/I
'
9p -. .
01
l?O_
'$0 _. __.180 _.
0'
A6 PPm. 1 90___......10 -........ 150 I . . . . . .._._ --.-.-.__-__.
0
e
-1
I+===-H2
FIG. 4. Variation calculated with the minimal basis set of three contributions to A6 and of their sum for the different nuclei of the water dimer as a function of 0 (Ro.. .. = 2.95 A). (For notations see Fig. 2.)
72
GESSNER-PRETTRE
AND FERCHIOU
pgm A 8-
pp.m.
i’LL
6-
90
\ ‘\
120
150 180 ----.- -.-_._____..._ _..._.-8' '. __.--.-Lc.-.-.---.----___
‘\ ‘,.’ /
4-
I’
/*---‘\
‘.
-1.___
I/ ,’ 2-
I
H
A6
A6 mm. 1
0 -1
;--
90
.-.-.-.-.-.-_ __-----
_--.-
-..-....ice . i80 *. 120 “1’
FIG. 5. Variation calculated with the extended basis set of three contributions to Ad and of their sum for the different nuclei of the water dimer as a function of 0 (Ro.. .o = 2.80 A). (For notations see Fig. 2.)
Methane-Methane The results are displayed on Figs. 6 and 7. As expected from the magnetic isotropy of the molecule, it appears that the geometric contribution is negligible. Also, because of the nonpolar character of methane, the polarization contribution is negligible for the protons and has some numerical importance only for the carbon nucleus and only for an intermolecular distance smaller than the equilibrium value calculated with dispersion (3.540 A (32)). The results obtained clearly show that the global downfield shift calculated for all the atoms is principally due to the charge transfer plus exchange contribution; in addition this shift is larger by an order of magnitude for the carbon nucleus than for the protons. H*O..
.H+andCH4..*H+
In Figs. 8 and 9 are reported the calculated values of the different contributions to the chemical shift variations of the nuclei of a molecule of water interacting with a proton (see Fii. lc for the geometrical arrangement) as a function of the 0 - - - H+ distance. In this case there are only two contributions to AS, namely the polarization
INTERMOLECULAR
SHIELDING
EFFECTS
73
FIG. 6. Variation calculated with the minimal basis set of three contributions to A6 and of their sum for the different nuclei of the methane dimer as a function of the C . - -C distance. (For notations see Fig. 2.)
and the charge transfer ones. The geometric effect is equal to zero since there are no electrons on the isolated proton; for the same reason there is no exchange term between the two interacting entities. The curves reported show clearly that the charge transfer contribution is larger in absolute value than the pohuization effect at short intermolecular distances and negligible at larger distances, while the polarization contribution decreases only slowly at large intermolecular distances. Therefore A6 varies as A6” when the proton is at short distances and as A&’ when it is at larger distances. In the case of the methane-proton system, Figs. 10 and 11 show that the charge transfer contributions becomes large only at very short distances and that it is downfield hb
PP.m
A6
hb
p.p.m. I -602 -1 t-
4 dlC-Cl d H2
-8 -
I I i I I I I
FIG. 7. Variation calculated with the extended basis set of three contributions to Ab and of their sum for the dillkent nuclei of the methane dimer as a function of the Cm - -C distance. (For notations see Fig. 2.)
74
GIESSNER-PRETTRE
AND FERCHIOLJ
-6
/
/
FIG. 8. Variation calculated with the minimal basis set of two contributions to A8 and of their sum for the nuclei of water interacting with a proton as a function of 0. - -H+ distance. (For notations see Fig. 2.)
for the three types of nuclei as in the case of water. For this molecule when the approaching proton is at large distance the polarization contribution remains important for all the protons but not for the carbon nucleus although this atom is closer to the perturbing charge than the proton HZ. This result is due to a compensation of the polarization of the CHr bonds on one hand and of the CH2 ones on the other, since they are in opposite directions. From Figs. 8 to 11 we can notice that the sign of A#’ is in every case the one expected from the electron migration due to the polarization of the molecule by the positive charge of the proton (32). The comparison of Figs. 8 and 9 on one hand and 10 and 11 on the other shows that the charge transfer contribution to the chemical shift variation of the nonhydrogen nuclei increases considerably when the basis set is extended. The general negative sign, corresponding to a downfield shift, obtained for A?P of every nucleus of the water and methane molecules interacting with a proton indicates that the charge transfer mechanism produces a magnetic deshielding of the nuclei of the molecules when they are, without ambiguity, electron donating. CONCLUSION
The results that we have obtained for the different contributions to intermolecular chemical shift variations, as calculated by the ab initio procedures proposed here, show that for molecules like water and methane, short-range effects, namely the
INTERMOLECULAR
SHIELDING
EFFECTS
75
A6 P.P.1
0
‘i
-2
0
!
FIG. 9. Variation calculated with the extended basis set of two contributions to A6 and of their sum for the nuclei of water interacting with a proton as a function of the 0. * - H+ distance. (For notations see Fig. 2.)
-b-
FIG. 10. Variation calculated with the minimal basis set of two contributions to A6 and of their sum for the nuclei of methane interacting with a proton as a function of the C - - - H+ distance. (For notations see Fig. 2.)
AND FERCHIOU
GIESSNER-PRE’ITRE
, dlC- W
?d(C-Ii’)
7
-15 -
0. ; -l-
3 .,.-4---
5
,
;----Ya-r~ 7 a
6
-2
A
/i : I
i
1; i
H2
FIG. 11. Variation calculated with the extended basis set of two contributions to A6 and of their sum for the nuclei of methane interacting with a proton as a function of the C - - - H+ distance. (For notations see Fig. 2.)
exchange (or repulsion) and charge transfer effects, can be, for some nuclei, principally for the nonhydrogen atoms, the largest ones in absolute value. In the case of methanemethane interactions the global downfield shift obtained for ‘%Z and ‘H is due to these contributions. This finding shows that the decreased shielding which is measured for both type of nuclei when the gas pressure increases (33) is not entirely attributable to van der WaaIs forces (which are not included in the present treatment) in agreement with the previous deductions of Jackowski et al. (3). It appears from the present study that the short-range contributions which have been completely omitted in semiempirical treatments of A6 (12-14) can definitely not be neglected, particularly for the nuclei situated in the intermolecular interaction site. In case of polar molecules like water or of interaction with a charged species . H+), the polarization contribution to A6 remains important (Hz00 . . H+orCH4.. for large intermolecular distances for all the nuclei as can be seen from the calculations. ACKNOWLEDGMENT We thank Dr. A. Pullman for many helpful discussions as well asfor pertinent advice about the pnzzentation of the results. REFERENCES 1. A.
PULLMAN, “Structure and Conformations of Nucleic Acids and Protein Nucleic Acids Interactions,” (M. Sundaralingam and S. T. Rao, Eds.), p. 457, University Park Press, Baltimore, 1975. 2. R. DITCHFIEI~), J. Chem. Phys. 65,3123 (1976); R. DITCHFIELD ANO R. E. MCKINNIZY, JR., Chem.
Phys.
13, 187 (1975).
INTERMOLECULAR
SHIELDING
EFFECTS
77
3. K. JACKOWSKI, W. T. RAYNES, AND A. J. SADLEJ, Chem. Phys. Lett. 54, 128, (1978); I. H. RILLEY AND W. T. RAYNES, Mol. phys. 38, 353 (1979). 4. C. GIESSNER-PREITRE AND A. PULLMAN, Chem. Phys. Len. 77,444 (198 I). 5. F. RIBAS PRADO, C. GIESSNER-PREITRE, A. PUUMAN, J. F. HINTON, D. HA-L, AND K. R. METZ, Theor. Chim. Acta 59,55 (1981); F. RIBAS--, C. GIESSNER-F’RXIRE, AND B. PULLMAN, Org. Magn. Reson. 16, 103 (1981). 6. F. RIBA.+-, C. GIESSNER PRFITRE, J-P. DAUDEY, A. PIJLL~UN, J. F. HINTON, G. YOUNG, AND D. HARPOOL. J. Magn. Reson. 37,43 1 ( 1980). 7. R. HOLLER AND H. LIXHKA, Chem. Phys. Lett. 84,94 (198 1). 8. M. JASZUNSKI AND A. J. SADLW, Theor. Chim. Acta 30,257 (1973); A. J. SADLEJ, Chem. Phys. Lett. 30, 432 (1975); A. J. SADLEJ, M. ZAUCER, AND A. AZMAN, Mol. Phys. 35, 1397 (1978). 9. J. S. TSE, Chem. Phys. Lett. 92, 144 ( 1982). 10. K. F. LAU AND R. W. VAUGHAN, Chem. Phys. Lett. 33, 550 (1975). II. J. A. POPJX, J. Chem. Phys. 24, 1111 (1956); Proc. R. Sot. A 239, 54 1 ( 1957); H. MCCONNELL,, J. Chem. Phys. 27, 226 (1957). 12. W. G. SCHNEIDER, H. J. BERNSTEIN, AND J. A. POPLE, J. Chem. Phys. 28,603, (1958); A. D. BUCKINGHAM, T. ~CHAEFFER, AND W. G. SCHNEIDER, J. Chem. Phys. 32, 1227 (1960); H. F. HAMEKA, Nuovo Cimento 11, 382 (1959). 13. P. J. Snm, Chem Phys. Lett. 30,259 (1975); J. S. SCHRAM AND S. CAWLEY, Collect. Czech. Chem. Commun. 36,2986 (1971). 14. C. GIESSNER-PRETIXE, B. PIJLL.MAN, AND J. CAILLET, Nucleic Acids. Res. 499 (1977); A. PULLMAN, H. BERTHOD, C. GIESSNER-PRETIRE, J. F. HINTON, AND D. HARPOOL, J. Am. Chem. Sot. 100, 3991 (1978). 15. B. D. ARTER AND P. G. SCHMIDT, Nucleic Acids Res. 3, 1437 (1976); D. R. LIGHFOOT, K. L. WONG, D. R. KEARNS, B. R. REID, AND R. G. SHIJLMAN, J. Mol. Biol. 78, 71 (1973). 16. C. K. MITRA, M. H. SARMA, AND R. H. SARMA, J. Am. Chem. Sot. 103,6727 (1981). 17. L. S. KAN AND P. 0. P. Ts’o, Nucleic Acids Res. 4, 1633 (1977); D. M. CHENG, L. S. KAN, P. 0. P. Ts’o, C. GIESSNER-PRETTRE, AND B. PULLMAN, J. Am. Chem. Sot. 102, 525; L. S. KAN, P. N. BORER, D. M. CHENG, AND P. 0. P. Ts’o, Biopolymers 19, 1641 (1980). 18. C. GIISNER-AND B. PULLMAN, Biochern. Biophys. Res. Commun. 70,578 (1976); C. GIESSNERPRETTRE, B. PULLMAN, P. N. BORER, L. S. KAN, AND P. 0. P. Ts’o, Biopolymers l&2277 (1976); G. GO~IL AND R. V. Hosu~, “Conformation in Biological Molecules Nuclear Magnetic Resonance Basic Principals and Progress,” (P. Diehl, E. Fluck, and R. Kosfeld, Eds.), 20,56, Springer-Verla8, New York, 1982. 19. J. W. AKITT, J. Chem. Sot. Faraday Trans. I 73, 1622 (1977). 20. T. W. MARSHALL AND J. A. POPLE, Mol. Phys. 1, 199 ( 1958); B. DAY AND A. D. BUCXNGIUM, Mol. Phys. 32, 343 (1976); A. J. SADLEJ AND W. T. RAYNES, MO/. Phys. 35, 101 (1978). 21. K. SEIDMAN AND G. E. MACIEL, J. Am. Chem. Sot. El,3254 (1977); L. W. LAQUES, J. B. MACASIULL, AND W. WELTNER, JR., J. Phys. Chem. 83, 14 12 (1979); H. J. SCHNEIDER, W. GREITAG, W. GXHWENDTNER, AND G. MALDENER, J. Magn. Reson. 36,273 (1979); M. E. VAN DONNELAN, J. W. DE HAAN, AND H. M. BUCK, Org. Magn. Reson. 13,447 (1980); C. PICCIM-LEOPARDI AND J. REISS, J. Magn, Reson. 46, 60 ( 198 1). 22. R. DIT~HFIELD, Mol. Phys. 27, 789 (1974). 23. F. RIBAS PRADO AND C. GIESSNER-PRETTRE, J. Magn. Reson. 47, 103 (1982). 24. C. GIESSNER-PRETTRE AND B. PULLMAN, .I. Am. Chem. Sot. 104, 70 (1982). 25. K. MOROKUMA, J. Chem. Phys. 55, 1236 (1971). 26. H. BERTHOD AND A. PULLMAN, Zsr. J. Chem. 19,299 (1980). 27. M. DREYFUS AND A. PULLMAN, Theor. Chim. Acta 19.20 (1970). 28. K. SUZUKI AND K. IGUCHI, J. Chem. Phys. 77,4594 (1982). 29. B. JEZIORSKI AND M. VAN HEMERT, Mol. Phys. 31, 713 (1976). 30. N. GRESH, P. CLAVERIE, AND A. PULLMAN, Znt. J. Quantum Chern. 22, 199 (1982). 31. W. KOLOS, G. RANGHINO, E. CLEMENTI, AND 0. NOVARO, Znt. J. Quantum Chem. 17,429 (1980). 32. A. PLJLLhfAN, Chem. Phys. Lett. Xl,29 (1973). 33. W. T. RAYNES, Mol. Phys. 17, 169 (1969); K. JACKOWSKI AND W. T. RA~VES, Mol. Phys. 34,465 ( 1977).
OF
MAGNEnc
RESONANCE
55.64-77
(1983)
Nonempirical Quantum Mechanical Calculations of Three Contributions to the Variation of Nuclear Magnetic Shielding Constants with Intermolecular Interactions. I. Method and Applications to Water and Methane C. GIESSNRR-FQETTRE
AND S. FERCHIOU
Luboratoire de Biochimie Thiorique Associk au CNRS, Institut de Biologie Physico-Chimique, 13, rue Pierre et Marie Curie, 75005 Paris, France Received March 30. 1983 Nonempirical quantum mechanical procedures are proposed for the calculation of (a) the “geometric” magnetic shielding of a nucleus N of a molecule A by the electrons of a molecule B, (b) the variation of the shielding constant aN because of the polarization of molecule A by molecule B, and (c) the sum of the contributions of the exchange and charge transfer mechanisms between mok.cuks A and B to the variation of ON. The method proposed utilizes gauge invariant atomic orbitals and gives for the different contributions results which are gauge invariant. Results concerning H20. . * H20, cI&. - * CH,, HrO. * . H+, and CH4. . . H+ are reported and discussed.
To calculate the chemical shift variations due to intermolecular interactions by an ab initio method, the procedure utilized is to calculate the magnetic shielding constants of the nuclei of interest in the isolated molecule and in the complex treated as a supermolecule (1). The desired quantity is then given by the difference between the values obtained from the two computations. For a nucleus N of a molecule A the chemical shift variation from the interaction of A with a molecule B is given by ANN
=
UNAB-
aNA
t11
where UNABand CNA are the magnetic shielding constants of nucleus N in the AB supermolecule and the isolated molecule A, respectively (Z-9). Such computations have in most cases given results that are in good agreement with experimental data and useful for their interpretation in terms of intra- and intermolecular contributions to the measured variations of the magnetic shielding constants studied (2, 9). However, it is not possible from these studies to determine the relative importance of the different type of contributions like the “geometric” (general name proposed by Lau and Vaughan (10) for the ring current plus local magnetic susceptibility anisotropy effects (I 1)) or the polarization (or electric field) (12) contributions. In the present study we propose within the framework of nonempirical computations a decomposition scheme of the chemical shift variations caused by intermolecular 0022-2364f83
$3.00
Copyright 8 1963 by AcaUemic Press. Inc‘. All rights of reproduction in any form reserved.
64
INTERMOLECULAR
SHIELDING
EFFECTS
65
interactions. Interest in the calculation of the different contributions to the variations of shielding constants which are obtained from supermolecule computations is twofold. On the one hand such calculations will indicate which fraction of A6 each contribution is responsible for, and how its importance varies according to the type of molecular interaction considered and to the intermolecular arrangement. They will also provide, on the other hand, a test for the semiempirical methods of calculations of intermolecular chemical shift variations (11-18). Such tests appear particularly important for the polarization contribution. If the role of this term is clearly established for protons participating in a hydrogen bond (12, 14, 19) this is not the case for the other protons of the molecules nor for the nonhydrogen nuclei of these molecules. Accurate studies have shown that an external electric field does produce a variation of the magnetic shielding constants of the nuclei of a molecule (20) but the results which have been reported cannot be utilized straightforwardly to evaluate the chemical shift variations due to the electric field created by the electronic and nuclear charge distribution of a molecule although experimental and theoretical results tend to support that they are far from negligible in many cases (4, 21). METHOD
Since intermolecular problems deal with variable distances, it is particularly important for such studies to obtain results which are strictly independent of the choice of the system of coordinates used for the computations. To satisfy this criterion, we utilize gauge invariant atomic orbitals and carry out the calculations within the fiamework of the self-consistent perturbation treatment proposed by Ditchfield (22). The calculation of the different contributions to the variation of the magnetic shielding constant of the nucleus N of a molecule A with intermolecular interactions is achieved by carrying out three computational steps. (a) Total variation. The c$? component of the magnetic shielding tensor of a nucleus N of a molecule A of a system AB made of the two molecules A and B is given by
(22): & = (G%I~Hm~NBIe3)+ (K&MpNBIai)
PI
where $!a is the ground state wavefunction of the AB system #$$ the perturbation of I& due to the external magnetic field H, XHnrrNu is the second order perturbation operator which introduces the magnetic field H and the nuclear moment (c(~) of nucleus N and ZrN8 is the first order perturbation operator depending only on j&N. (b) Geometric (or direct) contribution. If we suppose that there is no interaction between molecules A and B we can write ‘Z*~=‘X*+~~ EAB = EA + EB and (hMB>
= 0
131
where $A and I/~ are the molecular wavefUnctions of molecule A and B, respectively.
66
GIESSNER-PRETTRE
AND FERCHIOU
If these relations [3] are taken into account, the expression giving ST8 [41 takes the form
or more compactly with obvious notation. Taking into account the particular forms of & expression giving (rN can be written
and +% given in [3] and [6] the
where we see that the hrst two terms depend only on the wavefunction of molecule A while the two following terms depend on the wavefunction of molecule B only. Therefore if we consider, for example, a particular nucleus N of the AB system belonging to molecule A and if we put and we can write UN = UNA + A&
[lOI
where uNA is the shielding constant of this nucleus in the isolated molecule A while Aa”,, is the “geometric” contribution (according to Lau and Vaughan’s definition (10)) of molecule B to the shielding of nucleus N of molecule A. The inspection of formula [7] shows that u ~~ and A6m can be calculated independently since UNA does not depend upon 1c/Band A& does not depend on #A. Hence it is possible to calculate the shielding of nucleus N due to the electrons of molecule B without introducing the wavefimction of molecule A in the computation. Aside from one-electron integrals which depend upon the location of N with respect to the atoms of molecule B the only quantities required for the computation of A$$ are $$ and $p. The unperturbed wavefunction $J: is obtained from an ab initio SCF computation using Gaussian basis functions and@ is calculated by the self-consistent perturbation method using GIAO (22). Then for each location of the nucleus N we calculate the one-electron integrals and (PIfl’“%) 1111 q are the gauge invariant basis functions of molecule B used in the of *p. To obtain meaningful and gauge invariant values of A&B the of these integrals which introduce the nuclear magnetic moment have out using the system of coordinates used throughout the calculation of This requirement is due to the gauge dependence of $#“ even when (24). (PI% Ha“NBjq)
where p and computation computations to be carried #! and +k. using GIAO
INTERMOLECULAR
SHIELDING
EFFECTS
67
(c) Polarization contribution. The calculation of this contribution utilizes polarized wave functions for A and B defined in the fashion of Morokuma (25); thus, the polarized wave functions @‘, 1c/$’of the two molecules A and B are calculated by a SCF procedure using a Hamiltonian which takes into account the presence of the other molecule but neglects completely intermolecular overlap. We have +l(p, = &V$ and (v@‘l@) = 0 = hf ore but the two Hamiltonians &“A and Xa contain explicitly the intermolecular nuclear attraction and electronic repulsion. We then use the polarized wavefunction $2 and @’ as zeroth order wavefunctions for the self-consistent perturbation calculation of the magnetic shielding constant. For the nucleus N of molecule A @LA is given by u$l = (~~q~Ha”Nql+bp)
+ (~~pJP”q@-).
[I21
The nuclear attraction and electron repulsion between molecules A and B is taken into account in the computation of #g“‘r’, therefore the pq element of the perturbation Hamiltonian takes the form
+ PC&
(
(pqlrs) - Jj (pslrq)4&~
>I}
1131
where 6, is equal to one if the two atomic orbitals concerned are located on the same molecule and zero otherwise. Once & is obtained, the polarization contribution to AS is defined by Asi; = (TEA- (TNA El41 (d) Exchange plus charge transfer contribution. The sum of these two contributions can be obtained by subtracting from the total A6 the geometric and polarization contributions; therefore we have A,g+’ = A& - A& - A&.
[I51
This difference can be identified as the exchange plus charge transfer contributions since it takes into account the overlap of the two wavefunctions (exchange) and introduces in the perturbation treatment the intermolecular excitations (occupied orbitals of A to vacant orbitals of B and vice versa) which are responsible for the charge transfer interactions. Concerning the charge transfer contribution we want to point out that the meaning of the term charge transfer is not identical in the case of calculation of magnetic shielding constants and in the case of intermolecular interaction energy. For the nucleus N of molecule A the vacant orbitals of molecule B contribute to ON not only through the perturbation of the ground state wave function of molecule A (as for calculations of AEd) but also through the perturbation treatment giving $2. Basis sets. The computations have been carried out for each of the cases considered with two different basis sets that we use currently for theoretical NMR calculations (4-Q, because it is of interest to determine how the different contributions are sensitive qualitatively and quantitatively to the choice of the basis functions. One of the basis set is minimal and the second is a split valence set defined in (23).
68
GIEXBER-PRETTRE
AND FERCHIOU
H2 31
\
‘(
CL,,
.H2 ,cf<,
/ H2
H1
(b)
“\ 0_.._... ,,+ /
H’
H2
FIG. I. Atom numbering and geometric arrangement in the (a) water-water, (b) methane-methane, water-proton, and (d) methane-proton systems.
RESULTS
(c)
AND DISCUSSION
In Fig. 2 and 3 are reported the values of the geometric, polarization, and exchange plus charge transfer contributions to A.6 (also reported) for the five different nuclei of the water dimer for various intermolecular distances: The results clearly show that for each nucleus at any intermolecular distance the geometric contribution is an order of magnitude smaller than the total chemical shift variation. In addition the values obtained with the two difkrent basis sets for this effect are almost identical numerically. For intermolecular distances larger than the equilibrium distance (2.95 A with the minimal basis set and 2.80 A with the extended one) the polarization contribution appears numerically important with respect to A6 even for the nuclei which are not engaged in the hydrogen bond. On the other hand the magnitude of this effect appears much larger for the oxygen nuclei than for the protons. The comparison of the value of A6p obtained with the two different basis sets shows that the extension of the basis produces an increase of the absolute value of this contribution, a result which parallels the increase of the polarization component of intermolecular interaction energies with the extension of the basis (26). The curves of Figs. 2 and 3 representing the variation of the exchange plus charge transfer contributions show that this term is negligible for large intermolecular distances
‘\ \\
; I
1H
-2-
i 02
6 d(Q-Q) %
*
FIG. 2. Variation c&ulated with the minimal basis set of three contributions to A6 and of their sum for the different nuclei Of the water dimer as a function of the 0. . .O d&an& (fj = 15(Y). The curves blended into the abscissa’s axis are not reported. * * *, Geometric contribution; - - -, polarization contribution; _.-.- , charge transfer plus exchange contribution; -, total.
2 t-
Ad iw *.
i
\ \
‘. \ - . . -.._ ....!_.____. >:&.->...:.T..--- --I--0’2 , 4
2-
A \ 4- ‘1, , \
s
70
GIFSSNER-PRETTRE
AND FERCHIOU
INTERMOLECULAR
SHIELDING
EFFECTS
71
but increases very rapidly for small ones. This is in keeping with the fact that the exchange and charge transfer mechanisms are short-range effects (25, 27-29). It is important to notice that for the equilibrium distance this contribution can be, for the proton, for example, as large, in absolute value, as the polarization elXxt. The comparison of Figs. 2 and 3 shows that the value calculated for this charge transfer plus exchange contribution is extremely sensitive to the basis set used; it is particularly pronounced in the case of O2 since at the equilibrium distance AP” is equal to +0.6 ppm from minimal basis set computations and -10.4 ppm if we use the extended basis. In Figs. 4 and 5 are reported the variation of A6 and of the different contributions for the equilibrium distance, as a function of the angle B defined in Fig. 1. The value of 180” corresponds to the colinearity of the 0,-H, bond with the bisector of the HOH angle of molecule labeled 2. We see from these figures that the variation of AP is negligible for each nucleus and that APc is approximately constant for the protons. On the contrary the variation of this last contribution is downfield for O2 and upfield for 0, when B decreases from 180 to 90”. This result indicates that the charge transfer plus exchange effect is, for the nonhydrogen atoms, very sensitive to the intermolecular orientation. This angular dependence of Ascf+” is to be compared with the corresponding variation obtained for the charge transfer component of the intermolecular interaction energy of the water dimer (30). A6 p.p.m. L
pgn. I
0
6-
9,o -.
-2-
y
.....-
yo
qo 180 L '.- . ..___.._........ 8' _ _-. -______
-,.
--..
4-
-.-.
-.-._
___---,,-I' 2-
IH
0
/I
'
9p -. .
01
l?O_
'$0 _. __.180 _.
0'
A6 PPm. 1 90___......10 -........ 150 I . . . . . .._._ --.-.-.__-__.
0
e
-1
I+===-H2
FIG. 4. Variation calculated with the minimal basis set of three contributions to A6 and of their sum for the different nuclei of the water dimer as a function of 0 (Ro.. .. = 2.95 A). (For notations see Fig. 2.)
72
GESSNER-PRETTRE
AND FERCHIOU
pgm A 8-
pp.m.
i’LL
6-
90
\ ‘\
120
150 180 ----.- -.-_._____..._ _..._.-8' '. __.--.-Lc.-.-.---.----___
‘\ ‘,.’ /
4-
I’
/*---‘\
‘.
-1.___
I/ ,’ 2-
I
H
A6
A6 mm. 1
0 -1
;--
90
.-.-.-.-.-.-_ __-----
_--.-
-..-....ice . i80 *. 120 “1’
FIG. 5. Variation calculated with the extended basis set of three contributions to Ad and of their sum for the different nuclei of the water dimer as a function of 0 (Ro.. .o = 2.80 A). (For notations see Fig. 2.)
Methane-Methane The results are displayed on Figs. 6 and 7. As expected from the magnetic isotropy of the molecule, it appears that the geometric contribution is negligible. Also, because of the nonpolar character of methane, the polarization contribution is negligible for the protons and has some numerical importance only for the carbon nucleus and only for an intermolecular distance smaller than the equilibrium value calculated with dispersion (3.540 A (32)). The results obtained clearly show that the global downfield shift calculated for all the atoms is principally due to the charge transfer plus exchange contribution; in addition this shift is larger by an order of magnitude for the carbon nucleus than for the protons. H*O..
.H+andCH4..*H+
In Figs. 8 and 9 are reported the calculated values of the different contributions to the chemical shift variations of the nuclei of a molecule of water interacting with a proton (see Fii. lc for the geometrical arrangement) as a function of the 0 - - - H+ distance. In this case there are only two contributions to AS, namely the polarization
INTERMOLECULAR
SHIELDING
EFFECTS
73
FIG. 6. Variation calculated with the minimal basis set of three contributions to A6 and of their sum for the different nuclei of the methane dimer as a function of the C . - -C distance. (For notations see Fig. 2.)
and the charge transfer ones. The geometric effect is equal to zero since there are no electrons on the isolated proton; for the same reason there is no exchange term between the two interacting entities. The curves reported show clearly that the charge transfer contribution is larger in absolute value than the pohuization effect at short intermolecular distances and negligible at larger distances, while the polarization contribution decreases only slowly at large intermolecular distances. Therefore A6 varies as A6” when the proton is at short distances and as A&’ when it is at larger distances. In the case of the methane-proton system, Figs. 10 and 11 show that the charge transfer contributions becomes large only at very short distances and that it is downfield hb
PP.m
A6
hb
p.p.m. I -602 -1 t-
4 dlC-Cl d H2
-8 -
I I i I I I I
FIG. 7. Variation calculated with the extended basis set of three contributions to Ab and of their sum for the dillkent nuclei of the methane dimer as a function of the Cm - -C distance. (For notations see Fig. 2.)
74
GIESSNER-PRETTRE
AND FERCHIOLJ
-6
/
/
FIG. 8. Variation calculated with the minimal basis set of two contributions to A8 and of their sum for the nuclei of water interacting with a proton as a function of 0. - -H+ distance. (For notations see Fig. 2.)
for the three types of nuclei as in the case of water. For this molecule when the approaching proton is at large distance the polarization contribution remains important for all the protons but not for the carbon nucleus although this atom is closer to the perturbing charge than the proton HZ. This result is due to a compensation of the polarization of the CHr bonds on one hand and of the CH2 ones on the other, since they are in opposite directions. From Figs. 8 to 11 we can notice that the sign of A#’ is in every case the one expected from the electron migration due to the polarization of the molecule by the positive charge of the proton (32). The comparison of Figs. 8 and 9 on one hand and 10 and 11 on the other shows that the charge transfer contribution to the chemical shift variation of the nonhydrogen nuclei increases considerably when the basis set is extended. The general negative sign, corresponding to a downfield shift, obtained for A?P of every nucleus of the water and methane molecules interacting with a proton indicates that the charge transfer mechanism produces a magnetic deshielding of the nuclei of the molecules when they are, without ambiguity, electron donating. CONCLUSION
The results that we have obtained for the different contributions to intermolecular chemical shift variations, as calculated by the ab initio procedures proposed here, show that for molecules like water and methane, short-range effects, namely the
INTERMOLECULAR
SHIELDING
EFFECTS
75
A6 P.P.1
0
‘i
-2
0
!
FIG. 9. Variation calculated with the extended basis set of two contributions to A6 and of their sum for the nuclei of water interacting with a proton as a function of the 0. * - H+ distance. (For notations see Fig. 2.)
-b-
FIG. 10. Variation calculated with the minimal basis set of two contributions to A6 and of their sum for the nuclei of methane interacting with a proton as a function of the C - - - H+ distance. (For notations see Fig. 2.)
AND FERCHIOU
GIESSNER-PRE’ITRE
, dlC- W
?d(C-Ii’)
7
-15 -
0. ; -l-
3 .,.-4---
5
,
;----Ya-r~ 7 a
6
-2
A
/i : I
i
1; i
H2
FIG. 11. Variation calculated with the extended basis set of two contributions to A6 and of their sum for the nuclei of methane interacting with a proton as a function of the C - - - H+ distance. (For notations see Fig. 2.)
exchange (or repulsion) and charge transfer effects, can be, for some nuclei, principally for the nonhydrogen atoms, the largest ones in absolute value. In the case of methanemethane interactions the global downfield shift obtained for ‘%Z and ‘H is due to these contributions. This finding shows that the decreased shielding which is measured for both type of nuclei when the gas pressure increases (33) is not entirely attributable to van der WaaIs forces (which are not included in the present treatment) in agreement with the previous deductions of Jackowski et al. (3). It appears from the present study that the short-range contributions which have been completely omitted in semiempirical treatments of A6 (12-14) can definitely not be neglected, particularly for the nuclei situated in the intermolecular interaction site. In case of polar molecules like water or of interaction with a charged species . H+), the polarization contribution to A6 remains important (Hz00 . . H+orCH4.. for large intermolecular distances for all the nuclei as can be seen from the calculations. ACKNOWLEDGMENT We thank Dr. A. Pullman for many helpful discussions as well asfor pertinent advice about the pnzzentation of the results. REFERENCES 1. A.
PULLMAN, “Structure and Conformations of Nucleic Acids and Protein Nucleic Acids Interactions,” (M. Sundaralingam and S. T. Rao, Eds.), p. 457, University Park Press, Baltimore, 1975. 2. R. DITCHFIEI~), J. Chem. Phys. 65,3123 (1976); R. DITCHFIELD ANO R. E. MCKINNIZY, JR., Chem.
Phys.
13, 187 (1975).
INTERMOLECULAR
SHIELDING
EFFECTS
77
3. K. JACKOWSKI, W. T. RAYNES, AND A. J. SADLEJ, Chem. Phys. Lett. 54, 128, (1978); I. H. RILLEY AND W. T. RAYNES, Mol. phys. 38, 353 (1979). 4. C. GIESSNER-PREITRE AND A. PULLMAN, Chem. Phys. Len. 77,444 (198 I). 5. F. RIBAS PRADO, C. GIESSNER-PREITRE, A. PUUMAN, J. F. HINTON, D. HA-L, AND K. R. METZ, Theor. Chim. Acta 59,55 (1981); F. RIBAS--, C. GIESSNER-F’RXIRE, AND B. PULLMAN, Org. Magn. Reson. 16, 103 (1981). 6. F. RIBA.+-, C. GIESSNER PRFITRE, J-P. DAUDEY, A. PIJLL~UN, J. F. HINTON, G. YOUNG, AND D. HARPOOL. J. Magn. Reson. 37,43 1 ( 1980). 7. R. HOLLER AND H. LIXHKA, Chem. Phys. Lett. 84,94 (198 1). 8. M. JASZUNSKI AND A. J. SADLW, Theor. Chim. Acta 30,257 (1973); A. J. SADLEJ, Chem. Phys. Lett. 30, 432 (1975); A. J. SADLEJ, M. ZAUCER, AND A. AZMAN, Mol. Phys. 35, 1397 (1978). 9. J. S. TSE, Chem. Phys. Lett. 92, 144 ( 1982). 10. K. F. LAU AND R. W. VAUGHAN, Chem. Phys. Lett. 33, 550 (1975). II. J. A. POPJX, J. Chem. Phys. 24, 1111 (1956); Proc. R. Sot. A 239, 54 1 ( 1957); H. MCCONNELL,, J. Chem. Phys. 27, 226 (1957). 12. W. G. SCHNEIDER, H. J. BERNSTEIN, AND J. A. POPLE, J. Chem. Phys. 28,603, (1958); A. D. BUCKINGHAM, T. ~CHAEFFER, AND W. G. SCHNEIDER, J. Chem. Phys. 32, 1227 (1960); H. F. HAMEKA, Nuovo Cimento 11, 382 (1959). 13. P. J. Snm, Chem Phys. Lett. 30,259 (1975); J. S. SCHRAM AND S. CAWLEY, Collect. Czech. Chem. Commun. 36,2986 (1971). 14. C. GIESSNER-PRETIXE, B. PIJLL.MAN, AND J. CAILLET, Nucleic Acids. Res. 499 (1977); A. PULLMAN, H. BERTHOD, C. GIESSNER-PRETIRE, J. F. HINTON, AND D. HARPOOL, J. Am. Chem. Sot. 100, 3991 (1978). 15. B. D. ARTER AND P. G. SCHMIDT, Nucleic Acids Res. 3, 1437 (1976); D. R. LIGHFOOT, K. L. WONG, D. R. KEARNS, B. R. REID, AND R. G. SHIJLMAN, J. Mol. Biol. 78, 71 (1973). 16. C. K. MITRA, M. H. SARMA, AND R. H. SARMA, J. Am. Chem. Sot. 103,6727 (1981). 17. L. S. KAN AND P. 0. P. Ts’o, Nucleic Acids Res. 4, 1633 (1977); D. M. CHENG, L. S. KAN, P. 0. P. Ts’o, C. GIESSNER-PRETTRE, AND B. PULLMAN, J. Am. Chem. Sot. 102, 525; L. S. KAN, P. N. BORER, D. M. CHENG, AND P. 0. P. Ts’o, Biopolymers 19, 1641 (1980). 18. C. GIISNER-AND B. PULLMAN, Biochern. Biophys. Res. Commun. 70,578 (1976); C. GIESSNERPRETTRE, B. PULLMAN, P. N. BORER, L. S. KAN, AND P. 0. P. Ts’o, Biopolymers l&2277 (1976); G. GO~IL AND R. V. Hosu~, “Conformation in Biological Molecules Nuclear Magnetic Resonance Basic Principals and Progress,” (P. Diehl, E. Fluck, and R. Kosfeld, Eds.), 20,56, Springer-Verla8, New York, 1982. 19. J. W. AKITT, J. Chem. Sot. Faraday Trans. I 73, 1622 (1977). 20. T. W. MARSHALL AND J. A. POPLE, Mol. Phys. 1, 199 ( 1958); B. DAY AND A. D. BUCXNGIUM, Mol. Phys. 32, 343 (1976); A. J. SADLEJ AND W. T. RAYNES, MO/. Phys. 35, 101 (1978). 21. K. SEIDMAN AND G. E. MACIEL, J. Am. Chem. Sot. El,3254 (1977); L. W. LAQUES, J. B. MACASIULL, AND W. WELTNER, JR., J. Phys. Chem. 83, 14 12 (1979); H. J. SCHNEIDER, W. GREITAG, W. GXHWENDTNER, AND G. MALDENER, J. Magn. Reson. 36,273 (1979); M. E. VAN DONNELAN, J. W. DE HAAN, AND H. M. BUCK, Org. Magn. Reson. 13,447 (1980); C. PICCIM-LEOPARDI AND J. REISS, J. Magn, Reson. 46, 60 ( 198 1). 22. R. DIT~HFIELD, Mol. Phys. 27, 789 (1974). 23. F. RIBAS PRADO AND C. GIESSNER-PRETTRE, J. Magn. Reson. 47, 103 (1982). 24. C. GIESSNER-PRETTRE AND B. PULLMAN, .I. Am. Chem. Sot. 104, 70 (1982). 25. K. MOROKUMA, J. Chem. Phys. 55, 1236 (1971). 26. H. BERTHOD AND A. PULLMAN, Zsr. J. Chem. 19,299 (1980). 27. M. DREYFUS AND A. PULLMAN, Theor. Chim. Acta 19.20 (1970). 28. K. SUZUKI AND K. IGUCHI, J. Chem. Phys. 77,4594 (1982). 29. B. JEZIORSKI AND M. VAN HEMERT, Mol. Phys. 31, 713 (1976). 30. N. GRESH, P. CLAVERIE, AND A. PULLMAN, Znt. J. Quantum Chern. 22, 199 (1982). 31. W. KOLOS, G. RANGHINO, E. CLEMENTI, AND 0. NOVARO, Znt. J. Quantum Chem. 17,429 (1980). 32. A. PLJLLhfAN, Chem. Phys. Lett. Xl,29 (1973). 33. W. T. RAYNES, Mol. Phys. 17, 169 (1969); K. JACKOWSKI AND W. T. RA~VES, Mol. Phys. 34,465 ( 1977).