1292
nfAURIZIO CIGNITTI A N I
THOMAS L. ALLEN
Nonbonded Interactions and the Internal Rotation Barrier
by Maurizio Cignitti’ and Thomas L. Allen Department of Chemistry, Cnioersity of California, Dazis, California (Received August 6 , 1963)
The role of nonbonded interactions in internal rotation is examined from several different points of view. Semi-empirical formulas indicate fairly large contributions (ca. SOY0) of nonbonded H . ‘ H repulsions to the ethane barrier. T’alence-bond calculations consistent with the valence-bond method aiid potential functions derived from data on intermolecular interactions show relatively small contributions (ca. l5Y0 or less) from nonboiided interactions. It is proposed that hybridization by 2s and 2p hydrogen orbitals enhances the nonbonded repulsions, and that this may be an important factor in internal rotation.
Introduction A number of theories have been proposed to explain the energy barrier to internal rotation in ethane and similar molecules, but none has gained very wide acceptance.2 It is known that in the usual simple quantum mechanical treatments of ethane (either valencebond or molecular orbital, with s and p orbitals), only hydrogen-hydrogen interactions contribute to the b a ~ i e r . ~A. ~calculation of these interactions in the ethane molecule by Eyring in 1932 led to a predicted barrier of 0.36 kcal./mole, with the staggered configuration having the lower potential en erg^.^ It is now known that the equilibrium configuration was correctly predicted, but the magnitude of the barrier in ethane is 2.7-3.0 kcal./mole.2 (Because of uncertainty as to the shape of the potential function, the barrier might be as low as 1.55 kcal./mole.6) Perhaps partly as a consequence of the failure of nonbonded interactions to account for the size of the barrier, attention has shifted to more complex wave functions utilizing d and f h y b r i d i ~ a t i o n ,excited ~ , ~ valence-bond structure~,‘Q-~~ and electronic ~ o r r e l a t i o i i . ~However, ~ recent calculations of the contributions of nonbonded interactions to internal rotation barriers have given the values 1.77 kcal./m0le~~,~6 and 2.63 kcal./mole” for ethane, and correspondingly good results for a number of other molecules. With the introduction of a suitable angular dependency of the nonbonded interaction, the former result increases to the experimental value.16 Recent applications of the virial theorem to molecules with only hydrogens bonded to the axial atoms have shown that internal rotation The Journal of Physical Chemistru
barriers are closely related to differences in protonproton repulsion energy between the eclipsed and staggered configuration^.^^^'^ In a treatment of the 42 singlet resonance bond structures of an H3CCH fragment, Eyring, Grant, and Hecht found that the H. * . H interaction alone gave a barrier of 2.88 kcal./ mole in ethane, and they conclude that this interaction may be the best explanation of the barrier.I2 (1) On leave of absence from IstitUtO Superiore di SanitA, Rome, Italy, 1962-1963. (2) E. B. Wilson, Jr., in “Advances in Chemical Physics,” 1’01. 11. ed. by I. Prigogine, Interscience Publishers, Inc., New York, 9.Y., 1959, p. 367. (3) W. G. Penney, Proc. Roy. Soc. (London), A144, 166 (1934). (4) H. Eyring, J . Am. Chem. Soc., 54,3191 (1932). (5) E. Blade and G. E. Kimball, J . Chem. Phy-s., 18, 630 (1950). (6) E . Gorin, J. Walter, and H. Eyring, J . Am. Chem. SOC.,61,1876 (1939). (7) L. Pauling, Proc. Natl. Acad. Sci. U . S . , 44,211 (1958). (8) H. Eyring, G. H. Stewart, and R. P. Smith, ibid., 44, 259 (1958). (9) G. M. Harris and F. E. Harris, J . Chem. Phys., 31, 1450 (1959). (10) XI. Karplus, ibid., 33, 316 (1960). (11) H. G. Hecht, D. M , Grant, and H. Eyring, Mol. Phys., 3, 577 (1960). (12) H. Eyring, D. h,I. Grant, and H. Hecht, J . Chem. Educ., 39,466 (1962). (13) H. G. Hecht, Theor. Chim. Acta, 1, 133 (1963). (14) E. A. Magnusson and H. Shull, “Proceedings of the Inter-
national Symposium on Molecular Structure and Spectroscopy,” Science Council of Japan, Tokyo, 1962, p. (2405. (15) E. A. Mason and N. M. Kreevoy, J . Am. Chem. Soc., 77, 5808 (1955). (16) K. E. Howlett, J . Chem. Soc., 4353 (1957); 1055 (1960). (17) V. Illagnasco, Nuouo Cimento, 24, 425 (1962). (18) W.L. Clinton, J . Chem. Phys., 33, 632 (1960). (19) M. Karplus and R. G. Parr, ibid., 38, 1547 (1963).
S O N B O P i D E D INTERACTIONS A S D THE INTERNAL
ROTaTION BARRIER
It is clear at this point that there is a wide divergence of opinion on the role of nonbonded interactions in internal rotation. ‘We present here the results of a reinvestigation of this problem from several different points of view.
Magic For mula Nonbonded interaction energies may be calculated from overlap integrals, exchange integrals, and atomic ionization potentials by means of Mulliken’s “magic Pedley has recently used this formula to calculate the properties of a number of hydrocarbon molecules and radicals. 21 For ethane in the eclipsed configuration, the magic formula gives 0.5604 e.v. for the total energy of repulsion between hydrogen atoms attached to different carbon atoms. I n the staggered configuration, the same quantity is 0.5025 e.v., so that the contribution of hydrogen-hydrogen repulsions to the internal rotation barrier is 0.0579 e.v., or 1.34 kcal./mole. Details of the calculations are shown in Table I. Coefficients in the magic formula mere taken from the = 0.70 column of Table IV of ref. 20. Interatomic distances were computed for a structure with tetrahedral bond angles, a C-C distance of 1.54 A., and a C-H distance of 1.10 A.2z ’The parameters for this simple structure are close to the averages of recent experimental determinations. 2 3 , 2 4 JJ
~~
~~
Table I : Nonbonded €1. . .H Interactions from Magic Formula Dihedral angle, deg.
Distance, a u.
Overlap integrala
e.v.
Coefficient
0 60
4.295 4.720
120 180
0.1342 0.0756 0.0263 0.0163
3 (eclipsed) 6 (staggered) 6 (eclipsed)
5.815
0.1560 0.1171 0.0690 0.0543
5.474
1293
staggered and eclipsed configurations. Details of the calculation are shown in Table 11. The carbon atom SCF atomic orbitals of Shull,2jas slightly modified by Higuchi, 26 were used in these calculations.27 Integrals were evaluated on the IB3I 7090 computer a t the Computer Center of the University of California, Berkeley, with the Diatomic 3lolecular Integral Program. 28
Table I1 : Nonbonded C . . . H Interactions from Magic Formula Dihedral angle, deg. 0 60 120
180
K’nhI2, e.v.
Coefficient
0 01733 0 01769 0 01300 0 00795
6 (eclipsed) 12 (staggered) 12 (eclipsed) 6 (staggered)
Thus the magic formula result for the internal rotation barrier in ethane is 1.34 kcal./mole, about half the experimental value. Application of the same method to methylsildne and methylgermane gives 0.49 and 0.41 kcal./mole, respectively; the corresponding experimental values are 1.715 and 1.27 k ~ a l . / m o l e . ~ ~ Another semi-empirical formula, ‘/2AX21/( 1 - fP), with A = 0.65 and I = 13.60 e.v., has been used by Mulliken to calculate nonbonded H . . . H r e p ~ l s i o n s . ~ ~ Application of this formula gives 1.15 kcal./mole for the contribution of nonbonded H . . . H repulsions to the ethane barrier.
Yl2,b
3 (staggered)
a For 1s atomic orbitals. * This is the nonbonded repulsion energy of a pair of hydrogen atoms a t the distance tabulated. The factor of is sometimes incorporated into Y (see ref. 20 and 21).
Because of symmetry properties the contributions to the barrier of other nonbonded interactions must vanish ide~itically.~,~ We have verified this for the C . . .H and C . . * C interactions using the magic formula. All of the nonbonded C . . . C interaction terms are the same in the eclipsed and staggered configurations. The only nonbonded C . . . H interaction term dependent on the dihedral angle is KITh/2,which contributes 0.2600 e.v. to the atomization energies of both the
(20) R. S. Mulliken, J . Phys. Chem., 56, 295 (1952). (21) J. B. Pedley, Trans. Faraday Soc.. 57, 1492 (1961); 58, 23 (1962). (22) K. S. Pitzer and E. Catalano, J . Am. Chem. Soc., 78, 4844 (1956). (23) “Tables of Interatomic Distances and Configurations in Molecules and Ions,” ed. by L. E . Sutton, Special Publication No. 11, Chemical Society, London, 1958, p. h1135. (24) 13’. J . Lafferty and I?. K. I’lyler, J . Chem. Phys., 37, 2688 (1962). (25) €1. Shull, i b i d . , 20, 1095 (1952). (26) J. Higuchi, ibid., 2 8 , 527 (1958) (27) Some of the integrals are quite different when Slater BO’S are used. I t was found by Alulliken (ref. 20) that use of the more accurate SCE’ AO’s in the magic forniula is essential. (28) A. C. Switendick and F. J. Corhntb, “311 DIAT, Diatomic Molecular Integral Program, for 704, Fortran Coded,” Massachusetts Institute of Terhnology Computation Center, Cambridge, Maas., Feb. 1, 1960, adapted to the 7090 by E. B. Moore. Jr., Boeing Srientific Research Laboratories, Seattle, Wash., and G . Wiederhold, Universit,y of California Computer Center. Berkeley, Calif. (29) J . E. Griffiths, J . Chcm. Phys., 38, 2879 (1963). (30) R. S. Mulliken, J . Am. Chem. Soc., 7 2 , 4493 (1950); 7 7 , 88i (1955).
Volume 68, Number 6
J u n e , 1964
MAURIZIO CIGNITTIAND THOMAS L. ALLEN
1294
0
Valence-Bond Method For the conventional valence-bond structure of ethane ("perfect-pairing approximation") , nonbonded hydrogen-hydrogen repulsion energies may be calculated as Ehh
=
&hh
-
'/ZJhh
(1)
where &hh is the Coulomb or direct integral and J h h is the exchange i n t e g ~ - a l . ~The * ~ ~two integrals may be evaluated from the potential energy of a hydrogen molecule in its lowest l 2 1 and 32 states. Neglecting S2 with respect to unity, valence-bond theory gives these as
E('2) =
&hh
f
Jhh
(2)
E(32)=
&hh
-
Jhh
(3)
By combining these equations, the nonbonded repulsion energy may be expressed directly in terms of the potential energy functions32 Ehh
=
'/4E('Z)
+ 3/4E(32)
(4)
The different methods for evaluating the integrals a t the appropriate distances have led to a wide variety of results. Eyring originally used the Sugiura integrals plus an additional term for London force^.^ More recent investigations have used potential energy functions for the l21 state (experimental Morse or 32 ~ t a t e ' ~ , ' ~the ; exchange energy has or 1000j015-17 of the total energy. been taken as 850j012,13 In some calculations the factor of '/z in eq. 1 has been dropped on empirical grounds. l5-I7 The most accurate '21 and 321 potential energy functions for the Hz molecule have been obtained in recent theoretical i n v e s t i g a t i o n ~ , ~and ~ - ~in ~ an analysis of spectroscopic data by the Rydberg-Klein-Rees meth0d.~7 Figure 1 shows the ' 2 functions and the experimental Morse curve for the range of internuclear distance appropriate to the internal rotation problem. The Morse function was calculated from data listed by Herzberg.38 The functions of Hirschfelder and L i r ~ n e t tand ~~ Harris and Taylor36 are almost identical in this range and are shown as a single curve. (These are the W I P values from Table I1 of Hirschfelder and Linnett; they also list a most likely value of 10.0 kcal./mole a t 4.0 a.u.) The results of Tobias and Vanderslicea7 and the function of Dalgarno and Lynn34agree very closely and are shown as one curve. Although there is some difference between the various theoretical functions, Fig. 1 shows that t,he Morse curve is in error throughout this range by a factor of about two. By combining eq. 2 and 3, the following expression
+
The Journal of Physical Chemistry
3
+
d
c
E
2
6
c! .x
-w v
4 '
9
12
1
I
15 4
5
R,a.u.
7
6
Figure 1. Potential energy functions for the ''2state of the HPmolecule. M is the Morse curve, DL-TI' is from ref. 34 and 37, HL-HT is from ref. 33, Table I1 ( W I P), and ref. 36, and KR is from ref. 35.
+ +
is obtained for the fraction of exchange energy in the binding energy of the I 2 state Jhh ~- E('2)
E('2)
- E('21) 2E('21)
(5)
This quantity has been calculated from the potential energy functions of Hirschfelder and Linnetta3 and Dalgarno and Lynn.34 The results, shown in Table 111, are in striking contrast to calculations based on the Sugiura integrals, where the Coulomb energy is about 14Y0 a t 4.0 a.u., becoming even smaller as the internuclear distance increases. 3a (31) F. J. Adrian, J . Chem. Phys., 28, 608 (1958). (32) C . A. Coulson and D. Stocker, ililol. Phys., 2, 397 (1959). (33) J. 0. Hirschfelder and J. W. Linnett, J . Chem. Phys., 18, 130 (1950). (34) A. Dalgarno and N. Lynn, Proc. Phys. SOC. (London), A69, 821 (1956). (35) W. Kolos and C. C . 3. Roothaan, Rev. Mod. Phys., 32, 219 (1960). (36) F. E . Harris and H. S. Taylor, J . Chem. Phye., 38, 2691 (1963). (37) I. Tobias and J. T. Vanderslice, ibid.,35, 1852 (1961). (38) G. Herzberg, "Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules," 2nd Ed., D. Van Nostrand Co., Inc.. New York, N. Y . , 1950. (39) J. 0. Hirschfelder, J . Chem. Phys., 9, 645 (1941).
NONBONDED INTERACTIONS
AND THE INTERNAL
Table I11 : Contributions of Energy Terms to the Hz Binding Energy
Distance, a.u.
---Exchange Hirschfelder and IinnetV
4.0 4.5 5.0 6.0 7.0 8.0 10.0 12.0
energy, %-Dalgarno and Lynn
75.8 75.2 74.4 69.5 57.6 38.3 6.6 0.6
61.5
... 55.9 49.1 38.4 24.8 4.1 0.0
+ +
London enorgy,b %
... ... ... (25.7)~ 37.0 54.1 84.9 94.1
+
D a t a from Table I1 (W I P) and Table V (W P) of ref. 33. * London energy from equation of Pauling and Beach (ref. 40); total energy from Dalgarno and Lynn (ref. 34). c For values of R much less than 7 a.u., this calculation of the London energy is not significant (ref. 40). Q
The principal source of the discrepancy is the London energy, which has been known for some time to be greater than the exchange energy at large internuclear distances.*O Data for the contribution of this term are listed in the last column of Table 111. The London energy depends on electronic correlation, which is not adequately represented by the Heitler-London wave function. The proportion of the total binding energy attributed to the exchange term is critical in calculations of nonbonded repulsions. From the preceding equations Ehh
(2 - 3x)E(l2)/2
(6)
where x denotes the fraction of exchange energy. It is usual to assume that x is independent of distance in the relevant range, and the data in Table I11 show that this is approximately true. If E, designates E(I2) at thie internuclear distance corresponding to the dihedral angle 4, as shown in Table I, then the contribution of nonbonded repulsions to the ethane barrier is (2 - 3x)(3Eo - f%o
-t 6E1-20 - 3E180)/2 (7)
This is a sensitive function of x.
1295
ROTATION BARRIER
Calculations in which
x is assumed to be 0.85 will give values 45% less than those with x = 1.00; if x is taken as 0.75, the calculated barrier is only one-fourth as large. Below x = the eclipsed configuration becomes the more stable. The valence-bond method most compatible with the fundamental eq. 1-3 is the application of eq. 4, along with some set of mutually consistent potential functions. When these functions are based on the integrals which result from the Heitler-London wave function, and a London energy term is also included, the method is that which led to the earliest estirrate of the barrier,
0.36 k ~ a l . / m o l e . ~Alternatively, one may use the potential functions due to Hirschfelder and Linnett or Dalgarno and Lynn, which are found to give barrier heights of 0.42 and 0.07 kcal./mole, respectively. (In the latter case the eclipsed configuration is the more stable.) The H. . . H potential function of Bartel141is based on the results of valence-bond calculations at longer distances, and one of the semi-empirical formulas of MullikenaO at shorter distances; it also includes a London energy term. The barrier height calculated from this function is 0.41 kcal./mole. Intermolecular H . .H Interactions 3
Studies of intermolecular phenomena (such as molecular scattering, second virial coefficients, and gas viscosity) provide data on nonbonded interactions. Vanderslice and Mason found that the following function was consistent with data on the H-H2 and HzHz systems42
V(H.. .H)
=
30.21e-a*013R e.v.; 0.8 A.
5 R 5 2.1 A.
(This function is 1/2E(3Z),where E ( a Z ) was obtained by fitting an exponential function43to the results of theoretical calculations.) If the above function is assumed to be valid a t the longer distances involved in internal rotation, the resulting barrier is 0.45 kcal./mole. A much different function was obtained by Amdur, Longmire, and Mason from studies of the CH4-CH4 intera~tion~~ V(H . .H)
=
1.44/R6.I8e.v.; 2.09
A. 5
R
5 2.77 A.
The two functions are compared in Fig. 2. This latter function gives a barrier height of only 0.11 kcal./mole. (A short extrapolation beyond the recommended range was required for the longest H . . .H distances.) Similar results have been obtained by Hill.45 He derived different potential energy curves for the nonbonded interactions of hydrogen bonded to hydrogen and hydrogen bonded to carbon. The barrier heights calculated from these functions were 0.744 and 0.066 kcal./mole, respectively. The various theoretical and semi-empirical methods used thus far to derive nonbonded H . * . H potential functions do not distinguish between H bonded to H L. Pauling and J. Y . Beach, Phys. Rev., 47, 686 (1935). L. S. Bartell, J . Chem. Phys., 32, 827 (1960). J. T. Vanderslice and E. A. Mason, i b i d . , 33, 492 (1960). R. J. Fallon, E. A. Mason, and J. T. Vanderslice, Astrophys. *I., 131, 12 (1960). 'ILongmire, . and E. A. Mason, J . Chem. Phys., (44) I. Amdur, &S. 35, 895 (1961). (45) T. L. Hill, ibid., 16, 399 (1948). (40) (41) (42) (43)
Volume 68, Number 6 June, 1984
MAURIZIO CIGNITTIAND THOMAS L. ALLEN
1296
*c
.
2.0
Figure 2.
2.4
R , A.
2.8
3.2
Potential energy functions for nonbonded H
..H
repulsions. VM is from ref. 42, and ALM is from ref. 44.
and H bonded to C. Therefore, a fundamental deficiency in these methods is indicated by the marked difference between the experimental potential functions. Intermolecular repulsions are often represented by an inverse power law of the type Xr-", where n is between 9 and 12.46 Wilson assumes an inversetenth-power potential for intramolecular H * . H repulsions, and shows on this basis that nonbonded interactions cannot be the sole source of the barrier in both ethane and m e t h y l ~ i l a n e . ~However, ~ this function appears to decrease too rapidly with increasing distance to represent intramolecular interactions between nonbonded hydrogen atoms. The Amdur, Longmire, and Mason function for nonbonded H . . ' H interactions is inversely proportional to the 6.18 power of the dist a n ~ e , 4an~ empirical function of Aston, Isserow, Szasz, and Kennedy has an inverse fifth-power potential,48 and the proton-proton repulsions, which must be included in any complete treatment of these molec ~ l e s , l are ~ ~ 1of~course inverse first-power. +
H . * . H Interactions in Molecular Vibrations Conventional Urey-Bradley analyses of vibrational spectra indicate anomalously low intramolecular repulsions between nonbonded hydrogen atoms. Bartell and Kuchitsu have recently used a modified UreyBradley analysis in studying the vibrational spectra of The Journal of Physical Chemistry
simple hydrides.49 They find that nonbonded H * . H repulsive forces in these molecules are compatible with the nonbonded potential functions of Bartell, 41 Hill,45 and Vanderslice and Mason.42 (These functions, and the corresponding contributions of nonbonded repulsioris to the ethane barrier, have been considered above.) Another modified Urey-Bradley analysis, the hybridorbital force field, gives a similar value for the nonbonded repulsive force constant in ammonia.50 On the other hand, a modification which considers the lone pair of electrons leads to small values for noiibonded interactions in the group V h~drides.5~ The vibrational amplitude of deuterium is smaller than that of hydrogen, and it has been suggested that the lack of sensitivity of the ethane barrier to deuterium substitution is therefore evidence that steric repulsion between hydrogen atoms is not the principal cause of the barrier.52 Using Bartell's model of steric isotope effects,53the difference between the rotational barriers in ethane and deuterioethane is calculated to be about 0.031 kcal./mole. As mentioned above, the ethane barrier calculated from Bartell's H - . . H potential function is 0.41 kcal./mole. Thus, the steric effect of deuterium substitution is estimated to be only about 8% of the steric contribution to the barrier height.
Conclusions Quantitative agreement among various methods of calculation is lacking, but this is not surprising considering the present state of quantum chemistry with regard to molecules as complex as ethane. The semiempirical formulas of rVIulliken indicate fairly large contributions (ca. 50%) of nonbonded H . I3 repulsions to the internal rotation barrier in ethane. Valence-bond calculations consistent with the valence-bond method, and potential functions derived from data 011 intermolecular interactions, show relatively small contributions (ca. 15% or less) from nonbonded interactions. A number of valence-bond calculations giving much larger contributions (ca. 60-100%) have been reported, but they all involve one or more of the following : (a) use of the Morse function; (b) assumption of an unrealistically high proportion (85-100%) of exchange energy; (c) 8
(46) J. E. Lennard-Jones, Physica, 4 , 941 (1937). (47) E. B. Wilson, Jr., Proc.Natl. Acad. Sci. U . S., 43, 816 (1957). (48) J. G. Aston, S. Isserow, G. J. Szasz, and R. M. Kennedy, J . Chem. Phys.,12, 336 (1944). (49) L. S. Bartell and K. Kuchitsu, ibid., 37, 691 (1962). (50) W. T. King, ibid., 36, 165 (1962). (51) M. Pariseau, E, Wu, and J. Overend, ibid., 39, 217 (1963). (52) W. F. Libby, ibid., 35, 1527 (1961). (53) L. S. Bartell, J . Am. Chem. Soc., 83, 3567 (1961); I o w a State J . Sci., 36, 137 (1961); J . Chem. Phys., 36, 3495 (1962).
1297
NONBONDED INTERACTIONS AND THE INTERNAL ROTATION BARRIER
omission of the coefficient l / Z in eq. 1. Proton-proton repulsion energies consistently give very large contributions (ea. 170%) to the barrier in ethane and similar molecules.l9 It should be emphasized that there exist several nonbonded potential functions which give good barrier heights for ethane and a number of other molecules. 15-19,48 Although the theoretical significance of these functions is not completely clear a t present, their existence suggests that further work on the role of nonbonded interactions in internal rotation may be fruitful. One factor not previously considered in this connection is the role of hybridization by the exciteld 2s and 2p orbitals in expanding and polarizing the hydrogen atoms, which has proved to be so important in recent studies of the hydrogen b ~ n d . ~ ~ The - ~ ' effect of these excited orbitals will be to increase the overlap between nonbonded hydrogens in the same molecule,and hence to increase the nonbonded repulsion energy above that calculated using 1s orbitals only for the hydrogen atoms. Polarization will increase intramolecular repulsions, where the hydrogens form bonds in the same general direction, and decrease intermolecular repulsions whey the bonds are in the opposite direction. Thus the appropriate potential function for intramolecular H . H interactions will be larger in magnitude for a given internuclear distance than the potential function for intermolecular H * . H interactions. It will also have 1
9
an angular dependence somewhat similar to that introduced empirically by Howlett.16 Different potential functions for H bonded to H and H bonded to C should result from differences in the extent to which the hydrogen 2s and 2p orbitals participate in different molecules. In the hydrogen molecule, the coefficient of the 2pa orbital is about 0.10 (where the coefficient of the 1s orbital is taken as unity) .58 The corresponding value in Hz+ is 0.1605 (or 0.145 if different effective nuclear charges are used for the 1s and 2p orbitals).59 The coefficients of the hydrogen 2pa and 2pn orbitals on the same basis are, respectively, 0.088 and 0.031 in HzO. and 0.107 and 0.002 in HF.56 These rather large coefficients indicate that excited hydrogen orbitals may be of considerable importance in the theory of nonbonded H . . . H interactions.
Acknowledgments. The authors are indebted to Gio Wiederhold and Emmett B. Moore, Jr., for adaptation of the computer program. The financial assistance of a research grant from the National Science Foundation is gratefully acknowledged. (54) I. Fischer-Hjalmars and R. Grahn, Acta Chem. Scand., 12, 584 (1958). (55) L. Paoloni, J . Chem. Phys., 30, 1045 (1959). (56) R. Grahn, Arkiv Fysik, 15, 257 (1959). (57) E. Clernenti and A. D. McLean, J . Chem. Phys., 36,745 (1962). (58) N. Roaen, Phys. Rev., 38, 2099 (1931). (59) B. N. Dickinson, J . Chem. Phys., 1, 317 (1933).
Volume 68, Number 6
June, 196.4
