55 1
INTERNAL ROTATION IN MOLECULES tegrable. it yields
For an Eckart function, V(y) = l/cosh2 (y),
while for a gaussian barrier,
P(y)
=
it yields
where erfc(y) is the well known error function.1s Analogously simple expressions for the phase shifts are readily obtained for other model barriers. Using the above results, the "WKB-improved" permeability calculations would proceed precisely as before except that eq A2 replaces eq LQL 5 as the source of the starting values of the wave function used in initiating the numerical integration, and eq A6 replaces Of course a subroutine eq LQL 7 as the source of K @ ) . would have to be added to the computer program to supply the values of 6+(y) and &(y) appropriate to the asymptotic form of the given potential. Testing
this approach with Eckart barriers of different sizes (for which exact ~(z) values are known analytically), it was found that the inclusion of these corrections allows a given accuracy in K(E)to be achieved using a value of 2 [see eq A l l one order of magnitude larger than before. For example, K(E) could be obtained while LQL accurate to 1 X 10" using 2 = 1 X had found that 2 = 1 X was required. One additional improvement in the present computer program over that used by LQL is the addition of a capability for changing the integration mesh in the course of a calculation. Including this and the edgeeffect corrections may lead to a considerable saving in computation time. However, this would probably be relatively more important for potentials which die off at long range with an inverse-power form, than for the exponential-tailed potentials considered here. Annotated FORTRAN listings of the computer programs used in the present calculations are available from the first author (R. J. L.)
The Role of Nonbonded Intramolecular Forces in Barriers to Internal Rotation in Molecules with Two Equivalent Methyl Groups by Kenneth C. Ingham Department of Biophysics, Michigan State University, East Lansing, Michigan 48833 (Received September 9, 1971) Publication costs borne completely by The Journal of Physical Chemistry
It is suggested that the often large barrier difference between molecules with two equivalent methyl groups and their corresponding one-top analogs can be readily explained on the basis of a nonbonded repulsion between the methyl groups. The repulsion arises primarily from an interaction between the protons of one methyl group and the carbon atom of the other. This dominance of the Ha C interactions offers a simple explanation of the general success of the independent oscillator approximation in most of these two-top molecules.
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A recent study of hindered internal rotation in oxylene generated interest in the role of nonbonded (van der Waals) forces in barriers to internal rotation in two-top molecules.1,2 A survey of available data revealed that molecules containing two equivalent neighboring methyl groups often have rotational barriers whose magnitudes differ substantially from those of their corresponding one-top analogs, for example, dimethyl ether 2)s. methanol (see Table I). Yet, the independent oscillator approximation, which assumes that each methyl group oscillates in a potential well whose shape is independent of the orientation of the other methyl group, is'generally a fairly good approximation in these molecules. These observations could
be made compatible by postulating a strong repulsive interaction between the hydrogen atoms of one methyl group and the carbon atom of the other. In order to check this hypothesis a variety of nonbonded potential functions were tested for their ability to explain the diflerence in barrier between the two-top molecules and their one-top analogs. The calculation is similar to that used by other^^,^ t o explain the effect (1) K. C. Ingham, Ph.D. Thesis, University of Colorado, 1970. (2) K. C. Ingham and S. J. Strickler, J. Chem. Phys., 53,4313 (1970). (3) J. L. DeCoen, G. Elefante, A. M. Liquori, and A. Damiani, Nature (London),216,910 (1967). (4) R. Scott and H. Scheraga, J. Chem. Phys., 42,2209 (1966).
The Journal of Physical Chemhtry, Vol. Y6,No. 4, 197.8
KENNETH C. INGHAM
552
Table I: Barriers to Internal Rotation in Molecules Containing Two Equivalent Methyl Groups and their One-Top Analogs" -Compound
c -
One-top
Two-top
Methanoldve Methanethiolh Methylselenoli Ethane I s m Methylsilaneg Methylgermane# Methyldiazirine" ** PropyleneW Propylene Methylamine'" Acetaldehydedd Propylene oxideou Tolueneii
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Vh-----------
One-top
Two-top
Two-top, (dad)
1.07 1.24 0.91 2.93 1.70 1.24 0.77 1.98 1.98 2.0 1.15 2.56 0.013
2.72 2.11 1.50 3.3-3.5 1.65 1.18 1.13 2.21 0.73 3.3-3.6 0.78 1.61 2.0
2.84 2.00 1.40 3.49 1.78 1.27 1.05 2.44 0.4$ 2.80 1.60 1.53 1.57'
c
Dimethyl etherf Dimethyl sulfide" Dimethyl selenidek Propane"-" Dimethylsilane' Dimethylgermane$ Dimethyldiazirine" Isobutene" cis-Butene" Dimethylamines*bb~c' Acetoneessff ~is-2~3-Epoxybutane~~~ O-Xylenec'if,''
a All barriers in kcal/mol. a Using 127.8' for the C=C-CHs bond angles. 'Using X-ray data for durene taken from J. M. Robertson, Proc. Roy. SOC.Ser. A., 141, 594 (1933). P. Venkateswarlu, H. D. Edwards, and W. Gordy, J . Chem. Phys., 23,1195 (1955). p. Venkateswarlu and W. Gordy, ibid., 23,1200 (1955). P. H. Kasai and R. J. Myers, ibid., 30, 1096 (1959). U.Blukis, P.H. Kasai, and R. J. Myers, ibid., 38,2753 (1963). T. Kojima, J . Phys. SOC.Jap., 15,1284 (1960). L. Pierce and M. Hayashi, J . Chem. Phys., 35,479 (1961). A. B. Harvey and M. K. Wilson, ibid., 45, 678 (1966). J. F. Beecher, J . Mol. Spectrosc., 21, 414 (1966). 8. Weiss and G. E. Leroi, J . Chem. Phys., 48,962 (1968). L. S.Bartell and H. K. Higginbotham, ibid., 42,851 (1965). A E. Hirota, c. Matsumura, and Y . Morino, Bull. Chem.SOC.Jap., 40, 1124 (1967). J. R. Hoyland, J . Chem. Phys., 49,1908 (1968). D. R. Lide, Jr., ibid., 33,1514 (1960). R. W.Kilb and L. Pierce, ibid., 27,108 (1957). ' L. Pierce, ibid., 34,498 (1961). v. w.Laurie, ibid., 30,1210 (1959). E. C. Thomas and V. W. Laurie, ibid., 50,3512 (1969). L. Scharpen, J. Wollrab, D. Ames, and J. Merritt, ibid., 50,2063 (1969). 'J. Wollrab, L. Scharpen, D. Ames, and J. Merritt, ibid., 49,2405 (1968). E. Hirota, ibid., 45, 1984 (1966). D. Lide and D. Christensen, ibid., 35,1374(1961). V. W. Laurie, ibid., 34,1516 (1961). T. N.Sarachman, ibid., 49,3146 (1968). K. Tamagake, M.Tsuboi, and A. Hirakawa, ibid., 48, 5536 (1968). " W. G.Fateley and F. A. Miller, Spectrochim. Acta, 18, 977 (1962). '' K. D. Moeller, A. R. Demeo, D. R. Smith, and H. L. London, J . Chem. Phys., 47,2609 (1967). dd R.Kilb, C.c. Lin, end E. B.Wilson, Jr., ibid., 26, 1965 (1957). *' J. D. Swalen and C. C. Costain, ibid., 31,1562(1959). R. Nelson and L. Pierce, J . Mol. Spectrosc.,18, 344 (1965). uu J. D. Swalen and D. Herschbach, J . Chem. Phys., 27, 100 (1957). hh M. L. Sage, ibid., 36, 142 (1961). H. D. Rudolph, H. Dreizler, A. Jaeschke, and P. Wendling, 2. Naturforsch. A , 22, 940 (1967). j' K. 5.Pitzer and D. w.Scott, J . Amer. Chem. SOC., 65,803 (1943). J.J. Rush, J . Chem. Phys., 47,3936 (1967).
'
(L
f'
li
''
of halogen substitution on the barrier in ethane, and i s discussed in detail in ref 1. One first calculates the nonbosded contribution in the one-top molecule. The difference between this and the measured barrier gives the so-called "axial" contribution, which is assumed to be a property of the type of bond and the particular atoms which it joins. This axial component is expected to be the same in the two-top molecules as in the onetop analogue. One then evaluates the nonbonded contribution in the two-top molecule, which when combined with the axial contribution gives the total calculated barrier. When this method was executed using a variety of nonbonded functions from the literature, the results were uniformly poor. The calculated barrier diflerence was always less than that observed, reflecting a need for a stronger Ha C repulsion. A stronger He . C repulsion makes sense for the following reason. Most of the functions in the literature are evaluated emperically from data on gas viscosity, molecular scattering, and other data pertaining to intermolecular forces. Now, two molecules such as methane approaching each other in the gas phase have unlimited rotational freedom to assume a mutual orientation which minimizes their repulsion. However, two methyl groups rotating in a
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The Journal of Physical Chemistry, Vol. 76, No. 4, 1972
molecule have a fixed mutual orientation and thus stronger interaction might be expected. The He .H and H* . C functions used by DeCoen, et al.,3 which are of the Buckingham oir "6-exp" type, were used as a starting point, U = a(exp -bR) cRV6. Then the b parameter of the He .C function was adjusted to give the best overall agreement for the pairs of molecules in Table I. The results shown there were obtained using a = 31,400, b = 3.85, and c = 121 for the Ha * Cfunction and a = 6600, b = 4.08, and c = 49.2 for the He H function (in units of kcal/mol with R in Angstroms). The general agreement in Table I is encouraging considering the simplicity of the model and in every case except acetone supports the postulation of a strong Ha C interaction. Acetone seems to constitute a special case and has been discussed recently by Lowe.6 The interatomic distances used in this calculation were determined graphically and have an estimated uncertainty of &0.01 A. This does not include uncertainties in the experimental geometries and equilibrium configurations which were taken from the references in Table I. Also, the methyl groups were as-
-
-
(5) J. P.Lowe,J. Chem. Phys., 51,832 (1969).
553
THEORETICAL STUDY OF HYPERFINE COUPLING CONSTANTS sumed to be tet!ahedral with carbon-hydrogen bond lengths of 1.09 A. Actually, some of the molecules have asymmetric methyl groups while in others the methyl rotate about axes which are not collinear . groups with the axial bonds? All of these effects will lend unAs accurate certainty to the structural and barrier data become available, this ap-
proach should lead to improved intramolecular nonbonded potential functions. Such functions are important for predicting the conformations of proteins and other polymers.’ (6) J. Wollrab and V. W. Laurie, J. Chem. Phys., 48,5058 (1968). (7) H. A. Scheraga, J. J. Leach, R. A. Scott, G. Nemethy, Discuss. Faraday SOC., 40,268 (1965).
A Theoretical Study of Hyperfine Coupling Constants of Some u
Radicals Based on the INDO Method by M. F. Chiu, B. C. Gilbert,* and B. T. Sutcliffe Department of Chemistry, University of York, Heslington, York, United Kingdom Y O 1 600. (Received July 9, 1971) Publication costs borne completely by The Journal of Physical Chemistry
A study based on the INDO method has been made of a variety of Q radicals. The desirability of minimizing the energy with respect to molecular geometry for a number of small radicals (e.g., .CONH2, H2C=N., H2C= CH) is discussed. INDO calculations have also been performed for a series of iminoxy radicals (RaC=NO.), the appropriate geometry around the radical center being chosen as that which minimizes the energy for the smallest radical in the series. An alternative method of determining the proportionality constants for relating esr hyperfine splittings to spin density matrix elements is suggested. Spin density distributions in the space around some of the radicals studied are expressed in the form of spin density contour maps.
1. Introduction Organic radicals exhibit a wide variation in their isotropic esr hyperfine splittings, but it is possible, broadly speaking, to divide them into two classes. These are the 9 radicals, typified by aromatic anions and cations, by semiquinones, and by methy1,I and the so-called u radicals, such as phenyl.2 From a theoretical point of view it is possible to rationalize the observed hyperfine splittings in many 9 radicals by means of calculations made within the context of the so-called “9-electron approximation,” e.g., a Huckel Moaor a Pariser-Parr-Pople3 110 calculation. I n these approaches, certain 140 coefficients are related to proton splittings by means of a McConnell-type relationship. For u radicals, however, calculations made within the T-electron approximation are inadequate to account for the observed splittings. In a pioneering VB calculation on the vinyl r a d i ~ a l Barplus ,~ and Adrian6 were able to obtain good agreement with experiment and were able to interpret their calculations in terms of an unpaired electron in a hybrid orbital in the molecular plane a t the radical center. This result suggests that for a calculational method to be successful for u radicals it must take
into account the electronic “core” of the molecule, which is specifically excluded from consideration in Telectron methods. With the development of the INDO procedure by Pople, Beveridge, and Dobosh,’ a semiempirical method has become available which can, to some extent, allow the presence of an electronic “core.” The details of this method are well known and the relevant parts are summarized in the Appendix to this paper. Pople, Beveridge, and Dobosh (PBD)* and Beveridge and Dobosh (BD)9 have already carried out calculations (1) R. W. Fessenden and R. H. Schuler, J . Chem. Phys., 42, 3670 (1965). (2) J. E. Bennett, B. Mile, and A. Thomas, Proc. Roy. SOC.Ser. A , 293, 246 (1966). (3) See, for instance, R. G. Parr, “Quantum Theory of Molecular Electronic Structure,” Benjamin, New Y o r k , N. Y . , 1964. (4) H. M. McConnell, J . C h m . Phys., 28, 1188 (1958). (5) (a) R. W. Fessenden, J . Phys. Chem., 71, 74 (1967); (b) R. W. Fessenden and R. H. Schuler, J . Chem. Phys., 39, 2147 (1963). (6) M.Karplus and F. J. Adrian, ibid., 41, 56 (1964). (7) J. A. Pople, D. L. Beveridge, and P. A. Dobosh, ibid., 47, 2026 (1967). (8) J. A. Pople, D. L. Beveridge, and P. A. Dobosh, J . Amer. Chem. Soc., 90, 4201 (1968). (9) D. L. Beveridge and P. A. Dobosh, J . Chem. Phys., 48, 5532 (1968).
The Journal of Physical Chemistry, Vol. 76, No. 4 , 1973
