Proc. Nat. Acad. Sci. USA Vol. 72, No. 3, pp. 854-858, March 1975

A New Approach to Empirical Intermolecular and Conformational Potential Energy Functions. II. Applications to Crystal Packing, Rotational Barriers, and Conformatilnal Analysis (electrons and nuclei/alkanes/amines/alcohols/carbohydrates) ANTONY W. BURG(ESS*t, LESTER L. SHIPMAN*, AND HAROLD A. SCHERAGA*t * Department of Chemistry, Cornell University, Ithaca, New York 14853; and t Department of Biophysics, Weisrnann Institute of Science, Rehovoth, Israel Contribued by Harqld A. Scheraga, November 7, 1974

ABSTRACT An empirical potential energy function have fixed bond lengths and bond angles; thus, alterations in based on the interactions of electrons and nuclei (EPEN) only from rotations around the single has been tested on molecules other than those used for its conformations arise parameterization. The results indicate that this 'energy bonds. The conformational and intermolecular energies were function is able to predict reliably the lowest energy con- evaluated by considering the coulombic, overlap, and attrac- formations, the potential energy differences between pon- tive terms of the EPEN algorithm (1). formations, rotational barrier heights, and dipole mo- The procedure for generating the positions of molecules in a ments for a series of alkanes, anines, alcohols, and carbo- hydrates. Crystal packing studies on n-hexane, n-octane, crystal has been described previously (2, 3). During the mini- methylamine, methanol, and a-D-glucose, using this same mization of the crystal lattice energy, the space group re- potential, indicate that it is also reliable for calculating mained invariant while the size and shape of the unit cell were intermolecular interaction energies and low-energy oriex4- changed until minimum crystal lattice energy was obtained. tations. The minimization procedure of Powell (4), as modified by the crystal packing studies. An empirical potential energy function based on the inter- Zangwill (5), was used for and the differences in energy actiQns of electrons and nuclei in molecules (EPEN) was de- When rotational barriers veloped earlier (1), by approximating intra- and inter-mo- between conformers were calculated, a Newton-Raphson 7) was used to locate lecular energies with the pairwise sum of coulombic interac- technique requiring second derivatives (6, tions between all point charge centers (i.e., electrons and the precise positions of the maxima and minima for all degrees in the conformation nuclei), an exponential repulsion to represent electron-electron of rotational freedom multidimensional overlap, and an R- (R = distance) attraction term to sim- space. ulate dispersion, many-bo4y, and other attractive energies. RESULTS AND DISCUSSIONS Parameters for EPEN were derived (1) by a least squares A. Rotational Barriers, Relative Conformational Energies, and fitting procedure using gas-phase and crystal data for ethane, Dipole Moments. The relative conformational energies and propane, n-butane, n-pentane, methanol, methylamine, dipole moments calculated using EPEN for several molecules water, and ammonia. In order to test EPEN, it is used here to are given in Table 1. Wherever possible the corresponding calculate the relative energies of stable conformers, rotational experimental values (8-28) are given for comparison. The barriers, and dipole moments, and to pack crystals of other dipole moments of all molecules are represented well byEPEN, alkanes, amines, and alcohols. The results of these calcula- and the calculated dipole moments for the most stable con- tions are presented and compared to experimental data. formers are given. The results for some molecules are discussed METHODS below. Alkanes. In the previous report (1), a satisfactory fit was to of a methyl group was taken The potential barrier rotation obtained fpr the rotational barriers Qf ethane and propane; as the energy difference between the eclipsed and staggered also the trans-gauche energy difference and trans -- gauche In most cases, the methyl rotational barrier of conformations. were The calculated stable conformer has been reported. The ter- barrier of n-butane predicted accurately. only the most rotational barriers of iso-butane and neo-pentane are within 1 (g) is used to describe the posi- minology trans (t) and gauche of the result. These calculated barriers tions of the hydroxyl proton and the amine lone-pair electrons kcal/mol experimental are both slightly higher than the experimental results, and relative to a C-C bond attached to the functional group. may be due partly to the rigid rotor approximation in the barrier for a particular fragment was When the rotational calculations for these crowded molecules. In neo-pentane, the all other were optimized to be in their calculated, fragments C-C bond length [1.54 A (11)] has been reported to be lowest energy conformations. slightly longer than for other alkanes, and is an indication that Where it was possible, the nuclear geometries of the mole- there is some distortion of the geometry of this molecule. cules studied were taken directly from experimental data or is than the standard of analogous molecules. The bonding and Most likely the CCH bond angle greater from the structure value of 111° used for the EPEN calculations, and larger lone-pair electrons were positioned by using the procedures values of this angle cause the calculated methyl rotational described previously (1). All molecules were considered to barriers to be closer to their experimental values. In the ab- for Abbreviation: EPEN, empirical potential using electrons and sence of experimental data on the'proton geometry neo- nuclei. pentane, the standard value for the CCHI bond angle was used 854 Downloaded by guest on September 24, 2021 Proc. Nat. Acad. Sci. USA 72 (1975) Applications of Empirical Potential 855 TABLE 1. Calculated (calc.) and experimental (expt.) rotational barriers, conformational energy differences, and dipole moments for some alkanes, amines, and alcohols Rotational barrier (kcal/mol) Dipole moment (debyt) Molecule Barrier type or AE Cale. Expt. Cale. Expt. iso-butanea CHr- 4.12 3.9 i 0.75b 0.02 0.130 neo-pentaned CH,- 5.32 4.3-4.7e 0.0 0. Of Ethylamine CH-(g) 3.56 1.407 1.23 -NH2 (g- g) 2.12 1. 6i -NH2 (g-t) 3.11 2.1i AE (g - t) -0.20 -0. 297i iso-propylamine CHF-(g) 4.11 4.4ji 2.7i 1.35 -NH2(g-t) 2.83 2.1i -NH2 (t-t) 3.59 2.1 AE(g-t) -0.75 --0.12 tert-butylamine CH,- 5.14 5.0j, 4. i, 2.7i 1.32 1.29-1.321 -NH2 3.42 2.5i Methanol CHs- 1.16 1.09m Ethanol CHr-(t) 3.53 3.3n 1.83 1. 73w -OH (t-g) 1.70 0 8q -OH (g _ g) 1.60 AE (t - g) -0.38 iso-propanol CHs-(t) 3.9 3.4' 1.80 1.698 -OH (t- g) 1.78 --OH(t t) 1.67 AE (t - g) -0.53 -0.28t tert-butanol CH,- 5.06 4.13u 1.79 1.67v -OH 1.86

a Geometry taken from ref. 8; b ref. 9; ° ref. 10; d geometry taken from ref. 11; e ref. 12; f ref. 13; g ref. 14; h ref. 15; i ref. 16; i ref. 17; these experimental barrier heights were calculated from observed fundamental torsional transitions under the assumption of threefold cosine-type potentials and, probably, under the additional assumption of uncoupled methyl rotation3 (although this is not made clear in ref. 17); since we found, during the course of our computations, that both of these assumptions are invalid, these experimental barrier heights must be considered to be approximations to the true barrier heights; k ref. 18; 1 ref. 19; m ref. 20; " ref. 21; P ref. 22; Qref. 23; r ref. 24; ' ref. 25; t ref. 26; uref. 27; v ref. 28.

for the calculations. However, EPEN reproduces the mono- served results for ethylamine (16). Experimental studies on tonic increase of the methyl rotational barrier in this alkane iso-propylamine (17, 18) indicate that the energy difference series (see also Table 1 of ref. 1). It should be noted that the between the all gauche and the gauche-trans conformations is rigid rotor approximation does not allow the gauche -- gauche approximately -0.12 kcal/mol in carbon tetrachloride solu- barrier for n-butane to be predicted accurately by EPEN. tion (18). EPEN gives -0.75 kcal/mol, a value very similar The calculated gauche-gauche barrier for n-butane was 14 to that (-0.67 kcal/mol) found by ab initio LCAO-SCF kcal/mole, while the experimental barrier is only 5-7 kcal/mol theory employing a 4-31 G extended gaussian basis set (32). (29, 30). A similar difficulty was found by Radom et al. (31, 32) The calculated methyl rotational barrier in tert-butylamine is when using linear combination of atomic orbitals self-con- within 1 kcal/mol of the experimental barrier, and the cal- sistent field (LCAO-SCF) molecular orbital theory (33) to culated amine barrier is also in agreement with the experi- study n-butane; a reasonable fit to the gauche-gauche barrier mental barrier of 2.5 (17). was obtained only after the CCC bond angle was increased Methanol and ethanol. The calculated conformational to 115.30. energy barrier to rotation in methanol (1.16 kcal/mol) is in Ethylamine. The most stable conformation was calcu- excellent agreement with the experimental value (1.09 kcal/ lated to have the NH2 lone-pair electrons gauche (g) to the mol; ref. 20). The conformation of ethanol calculated to have C-C bond. This is in agreement with the experimental result the lowest energy has the hydroxyl proton trans to the C-C (16). The magnitude of the calculated energy difference bond. Again, this is in agreement with the observed conforma- between the gauche and trans conformers (-0.20 kcal/mol) tion of ethanol (34). The calculated barrier for methyl rotation also agrees well with the experimental measurements (-0.297 (3.53 kcal/mol) is in good agreement with the experimental kcal/mol; ref. 16). The calculated barrier for the methyl value (3.3 kcal/mol; ref. 21); however, the barriers to rotation rotation in both the t and g conformers is within 0.2 kcal/mol of the hydroxyl group are nearly 1 kcal/mol higher than the of the experimental barrier. The experimental barrier for the observed value (0.8 kcal/mol; ref. 23). methyl rotation (3.74 kcal/mol, ref. 14) is probably a weighted iso-propanol and tert-butanol. The conformation of average of the barrier for the t and g conformers. iso-propanol calculated to have the lowest energy has the iso-propylamine and tert-butylamine. The confor- hydroxyl proton trans to one of the methyl groups; this mation of iso-propylamine with the lowest calculated energy conformation is 0.53 kcal/mol more stable than the form in has the NH2 lone-pair electrons gauche to both methyl groups. which this proton is gauche to both methyl groups. Although This is in agreement with the calculated (see above) and ob- both of these forms have been detected experimentally (35), Downloaded by guest on September 24, 2021 856 Chemistry: Burgess et al. Proc. Nat. Acad. Sci. USA 72 (1976) TABLE 2. A comparison of experimental and calculated crystal lattice constants and energies

LLatticeLLatticeLattice constantsb energy Molecule sizes a b c a , y (kcal/mol) n-hexane Calc. 10,7,5 4.16 4.45 8.91 98.0 82.5 109.3 -12.14 Expt.,O 4.17 4.70 8.57 96.6 87.2 105.0 - 13.2d n-octane Cale. 10, 7,3 4.19 4.69 11.21 94.5 82.5 105.4 -16.78 Expt.e - 4.22 4.79 11.02 94.7 84.3 105.8 -17.3f Methylamine Calc. 5,5,3 5.87 6.31 13.51 90 90 90 -10.60 Expt.g 5.73 6.11 13.51 90 90 90 -8.2h Methanol Calc. 5,5,7 5.86i 7.33 4.87 90 90 90 -13.04 Expt.- 6.43 7.24 4.67 90 90 90 N.A.k a-D-glucose Calc. 3,3,5 10.43 14.94 4.69 90 90 90 -64.37 Expt.' 10.36 14.84 4.97 90 90 90 N.A.k

* Total number of unit cells along the a, b, c axes, respectively; b a, b, c are in units of X and a, ,, y are in units of degrees; Oref. 37; d see § and ref 38; e ref. 36; f see § and ref. 38; g ref. 42; h ref. 45; i see discussion in text; j ref. 46; k not available; ' ref. 49. § Burgess, Shipman, and Scheraga, manuscript in preparation

relative intensity measurements of infrared bands have indi- is also of a similar accuracy as the methods used previously. cated that the trans, gauche conformer is 0.28 keal/mol more Thus, the EPEN energy function appears to be directly appli- stable than the gauche, gauche conformation (26). The calcu- cable to both the intramolecular and intermolecular interactions lated methyl rotational barrier in tert-butanol shows the same of hydrocarbons, and there is no need to adjust the parameters trend as the calculated barriers in neo-pentane and tert- with add-on functions such as those referred to as "intrinsic butylamine; it is nearly 1 kcal/mol higher than the experi- torsional contributions" (38, 40). mental value. Methylamine. The proton geometry was taken from the gas phase microwave structure (41), and was the same as that B. Crystal Packing. In order to test the usefulness of EPEN used in the previous report (1). Although the x-ray diffraction for simulating interactions between molecules, several crystal study on methylamine (42) reported the proton positions, it packing studies were undertaken. The molecules were chosen was decided to search for the minimum energy orientation to represent alkanes,L amines, alcohols, and carbohydrates. and conformation of the methylamine protons before the The crystal packing was simulated by allowing only the lattice crystal energy minimization was carried out. In a 3 X3 X3 constants to vary (3) and by assuming that the molecules unit cell crystal in which the lattice constants were fixed at the move as rigid bodies during the search for the lowest energy experimental values (42), the first molecule was rotated clock- of the crystal. wise (looking down the C-N bond from C to N) by an angle 0 The results of the crystal packing studies for n-hexane, around the C-N bond. Rotation of the molecule was defined n-octane, methylamine, methanol, and a-D-glucose are sum- with respect to the (reference) positive direction through the N marized in Table 2. Some of the individual crystals are dis- atom parallel to the b axis of the crystal (zero rotation, i.e., cussed in detail below. O = 0°, corresponded to an orientation in which the lone-pair n-Hexane and n-octane. Hydrogen atoms were placed made an angle of -120° with respect to the reference axis). according to the standard geometry described by Momany The methyl protons were rotated clockwise (looking down the et al. (3). The end methyl groups were kept in the staggered C-N bond from C to N) by an angle 4) around the C-N conformation with respect to the C-C bonds. The calculated bond. The value of O was defined with respect to the lone-pair lattice constants for n-octane are all within 2% of the experi- electrons on the NH2 (0 = 0° when a methyl proton eclipses mental values (36) and the calculated lattice energy is only the lone pair). The positions of the other molecules in the unit 0.5 keal/mol lower than experiment. Although the results for cell were generated using the crystal symmetry determined n-hexane are not as good as this, all of the calculated lattice experimentally (42). The total energy of the orientation (speci- constants are still within 5% of their corresponding experi- fied by 0) and conformation (specified by 4) was taken to be mental values (37) and the calculated lattice energy is only 1 the sum of the intramolecular and intermolecular interactions. kcal/mol different from the experimental value. When com- This is in contrast to procedures used by Momany et al. (3) pared to previously published crystal packing studies on (in which only the intermolecular interactions were con- hydrocarbons (in which the parameters of the potential func- sidered) and also to Hagler and Lifson (43, 44) (where only tion were adjusted to obtain the best fit to the crystal lattice intramolecular interactions were considered) when obtaining constants; see refs. 2, 3, 38, and 39), the results obtained by the conformation of the molecule in the crystal. The minimum EPEN are quite acceptable. The EPEN potential reproduces energy positions for the protons were obtained by a grid the experimental lattice energies more closely than most of the search over 0 and o. The resulting energy surface is shown in other studies (2, 3, 39) and is as good as the result obtained by Fig. 1. Although three minima were located, the lowest energy Warshel and Lifson (38). This is an especially encouraging was found to occur at = -10° and 4 = 600 (i.e., the methyl- result for EPEN since, whereas these crystals were used to amine molecule was in a *staggered conformation). This obtain the parameters of the energy function in the other orientation and conformation is almost exactly the same as studies, n-hexane and n-octane were not used to compute the that described experimentally by Atoji and Lipscomb (42), parameters for EPEN. The fit to the crystal lattice constants who report coordinates equivalent to 0 = -40 and 4 = 65°. Downloaded by guest on September 24, 2021 Proc. Nat. Acad. Sci. USA 72 (1975) Applications of Empirical Potential 857 a

70 III * H. .7- H '---)...... Ho H:3 H 5~~~~~~ ? HI ICEso6l0 l ASP

:H H

H' ... A H----- Hq ...... H--- C -180 -120 -60 0 60 120 I80 9(Orientation of Molecule, deg.) H* 2 H* FIG. 1. An energy surface, calculated using EPEN, for the FIG. 2. A view down the b-axis of the methanol crystal orientation (0) and conformation (O) of methylamine molecules (1630K, ref. 46) showing the positions of the hydrogen atoms. in the crystalline state (42). Solid lines represent contours of Each molecule is shown as a Newman projection. The dotted equal energy in kcal/mol, A represents the position of the lowest lines represent the C-H bonds and the solid lines the O-H calculated energy minimum on the surface, X represents the bonds. O is the angle between the proton H* and the hydroxyl position of other (higher energy) calculated minima, and 0 proton and 0 the angle between the c-axis and the proton Ho. represents the position found experimentally for the methylamine This arrangement, which was inferred by Tauer and Lipscomb molecule (42). (46), 0 = 00 and ti = 60°, is considered here as the reference, from which others may be generated by varying 0 and 4. Upon minimization of the lattice energy with respect to the lattice constants in a 5X5X3 unit cell crystal with 0 = -10° that the protons on the second and fourth molecules rotate in and o = 600, the calculated lattice constants were within 3% opposite directions around the b axis compared to the proton of the experimental values. The calculated lattice energy for rotations on the first and third molecules (i.e., 0A = -02 = methylamine (-10.60 kcal/mol) is lower than the preliminary 03 = -04). Arrangement B is produced when the protons on all estimate of the corresponding experimental value of -8.2 molecules in the unit cell are rotated in the same direction kcal/mol (45). around the b-axis (i.e., 01 = 02 = 03 = 04). Arrangement C may Methanol. The crystal structure of methanol (46) used be produced by rotating the protons on the third and fourth for this packing study was the higher temperature (1630K) molecules in the opposite direction to the rotation of the pro- form. The geometry for the protons was taken from the gas tons of the first and second molecules (i.e., Ol = 02 = -03 = phase structure (47), except that the methyl protons were -04). The arrangement of protons generated by the fourth set placed with C3 symmetry around the C-O bond. A reference of 0 angles (i.e., Ol = -02 = -03 = 04) leads to a net dipole orientation and conformation of the methanol molecules in the moment for the crystal structure, and, therefore, was not crystal was constructed by assuming that (i) the OH bond investigated in detail here. would be staggered with respect to the methyl protons and The total energy (including both inter- and intra-molecular (ii) the angle between the OH bond and the vector between interactions) was calculated during the 0,+, search for the low- hydrogen-bonded oxygen atoms was zero (46). The unit cell est energy orientation of the methanol molecule. diagram for this reference structure is shown in Fig. 2. Since When arrangement A is used for generating the proton the C-O bond is parallel to the b axis of the crystal, the orien- conformations and orientations, i.e., introducing the con- tation and conformation of the protons on each methanol straint 01 = -02 = 03 = -04, the grid search of all 0,4k space in molecule in the crystal were determined by a counterclockwise 100 intervals in a 3X3X3 unit cell crystal indicated that the rotation (about the b axis) of the molecules through an angle 0 orientation and conformation assumed by Tauer and Lipscomb relative to the reference orientation shown in Fig. 2, and an (46) (where the projection of the hydroxyl proton was posi- internal counterclockwise rotation through an angle 0 of the tioned along the ... *0 vector and the methyl protons were hydroxyl proton relative to a proton of the methyl group. staggered with respect to the hydroxyl proton, i.e., 0 = 00, Zero rotation of each molecule (i.e., 0A = 02 = 03 = 04 = °°) o = 600) was the most stable. When this orientation and con- was taken to be the reference arrangement of Fig. 2, and formation was used for the minimization of the crystal lattice variations in 0 were defined with respect to an axis parallel to constants, the final values were a = 6.08 A, b = 6.94 i, c = the +c direction, passing through the methyl carbon atom. 5.16 A, a = = y = 900. The lattice energy was -12.42 In the reference structure of Fig. 2, 0 = 600 for all four mole- kcal/mol. Although the a axis had contracted by 0.35 X this cules. Deviations from the reference orientation and confor- was not so surprising, since Tauer and Lipscomb had reported mation were made by changing 0 and 0 on each molecule. In that there is a large amount of thermal motion (about 0.32 X the energy minimization with respect to and 0, the following root mean square amplitude of vibration) along the a axis. At constraints were imposed: 0 was taken to be the same on all 1630K, the crystal is quite near its melting point (175.370K, molecules, as was the magnitude of 0. The direction of the 0 ref. 48) and appears to be quite expanded along the a axis. At a rotations for each molecule was varied, however, leading to slightly lower temperature than that of the structure deter- four different possible arrangements (A-D) for the directions mination, the methanol crystal undergoes a X transition of the 0 rotations, as described below. Arrangement A is (157.80K, ref. 48). An examination of the methanol crystal produced when the protons are generated using 0 and 0 such in which the protons were generated using arrangement B Downloaded by guest on September 24, 2021 858 Chemistry: Burgess et al. Proc. Nat. Acad. Sci. USA 72 (1975)

was performed in a similar way. Again a grid search was 12. Durig, J. R., Craven, S. M. & Bragin, J. (1970) J. Chem. performed in a 3 X 3 X 3 unit cell crystal to find the orienta- Phys. 52, 2046-2052. 13. Habgood, H. W. (1953) Diss. Abstr. 13, 296. tion and conformation which gave the lowest calculated 14. Li, Y. S. (1972) quoted as personal communication in Vi- energy; this was located at 0 = 350, 0 = 500 (and, by sym- brational Spectra and Structure, Durig, J. B. (Dekker, New metry, at 0 = -350, = 700). When the lowest energy York), Vol. 1, p. 168. orientation and conformation of methanol (0 = 350, 0 = 15. Barclay, G. A., Le Fbvre, R. J. W. & Smythe, B. M. (1950) 500) was used for the energy minimization of the cell lattice Trans. Faraday Soc. 46, 812-820. 16. Manocha, A. S., Tuazon, E. C. & Fateley, W. G. (1974) J. constants in a 5 X 5 X 7 unit cell crystal, the minimized Phys. Chem. 78, 803-807. lattice energy was -13.04 kcal/mol. As expected the a axis 17. Scott, D. W. (1971) J. Chem. Thermodyn. 3, 843-852. was again contracted but now the lattice constants involved 18. Krueger, P. J. & Jan, J. (1970) Can. J. Chem. 48, 3229-3235. in the hydrogen bonding network were within 0.2 X of the 19. Few, A. V. & Smith, J. W. (1949) J. Chem. Soc. London, 3057-3060. experimental values (Table 2). When the 0 angles were 20. Shull, H. & Hall, G. G. (1959) Nature 184, 1559-1560. generated using arrangement C, a single minimum was 21. Michielson-Effinger, J. (1964) Bull. Cl. Sci. Acad. Roy. found on the 0, 0 grid (0 = 100, 0 = 700). This minimum Belg. 50, 645-657. was 0.3 kcal/mol more stable than the reference orientation 22. McClellan, A. L. (1963) in Tables of Experimental Dipole Moments (W. H. Freeman & Co., San Francisco, Calif.), pp. and conformation (i.e., 0 = 00, = 600) but 0.4 kcal/mol 71-72. less stable than the minimum found using arrangement B. 23. Barrow, G. M. (1952) J. Chem. Phys. 20, 1739-1744. a-D-glucose. The positions of all atoms (including the 24. Schumann, S. C. & Aston, J. G. (1938) J. Chem. Phys. 6, protons) in a-D-glucose were taken directly from the neutron 485-488. diffraction study of Brown and Levy (49). The calculated 25. Stranathan, J. D. (1937) J. Chem. Phys. 5, 828-830. 26. Hirota, E. (1973) quoted as private communication by lattice energy was -64.37 kcal/mol. The calculated values for Lathan, W. A., Radom, L., Hehre, W. J. & Pople, J. A. the crystal lattice constants agree within 1% for the a and b (1973) J. Amer. Chem. Soc. 95, 699-703. axis and 5% for the c axis. 27. Beynon, E. T., Jr. & McKetta, J. J. (1963) J. Phys. Chem. 67, 2761-2765. CONCLUSIONS 28. McClellan, A. L. (1963) in Tables of Experimental Dipole Moments, (W. H. Freeman & Co., San Francisco, Calif.), The EPEN energy function derived earlier (1) was able to p. 125. simulate both intramolecular and intermolecular interactions 29. Ito, K. (1953) J. Amer. Chem. Soc. 75, 2430-2435. of the molecules studied here. There is no need for "intrinsic 30. Piercy, J. E. & Rao, M. G. S. (1967) J. Chem. Phys. 46, 3951-3959. torsional potentials" or "special hydrogen bond functions" to 31. Radom, L. & Pople, J. A. (1970) J. Amer. Chem. Soc. 92, account for the observed conformations or intermolecular 4786-4795. orientations of molecules. Despite the rigid rotor approxima- 32. Radom, L., Lathan, W. A., Hehre, W. J. & Pople, J. A. tion, the lowest energy conformations of these alkanes, (1973) J. Amer. Chem. Soc. 95, 693-698. and alcohols were and the rela- 33. Roothaan, C. C. J. (1951) Rev. Mod. Phys. 23, 69-89. amines, predicted by EPEN, 34. Sasada, Y., Takano, M. & Satoh, T. (1971) J. Mol. Spectrosc. tive energy differences between conformers agree closely with 38, 33-42. experiment. The rotational barriers and dipole moments were 35. Kondo, S. & Hirota, E. (1970) J. Mol. Spectrosc. 34, 97-107. well represented using EPEN, indicating the highly transfer- 36. Norman, N. & Mathisen, H. (1961) Acta Chem. Scand. 15, able nature of the fragments (1). 1755-1760. 37. Mathisen, H., Norman, N. & Pedersen, B. F. (1967) Acta 1. Shipman, L. L., Burgess, A. W. & Scheraga, H. A. (1975) Chem. Scand. 21, 127-135. Proc. Nat. Acad. Sci. USA 72, 543-547. 38. Warshel, A. & Lifson, S. (1970) J. Chem. Phys. 53, 582-594. 2. Williams, D. E. (1969) Acta Crystallogr. Sect. A 25, 464- 39. Ferro, D. R. & Hermans, J., Jr. (1972) Biopolymers 11, 105- 470. 117. 3. Momany, F. A., Carruthers, L. M., McGuire, R. F. & 40. Scheraga, H. A. (1968) Advan. Phys. Org. Chem. 6, 103-184. Scheraga, H. A. (1974) J. Phys. Chem. 78, 1595-1620. 41. Lide, D. R., Jr. (1957) J. Chem. Phys. 27, 343-352. 4. Powell, M. J. D. (1964) Comput. J. 7, 155-162 42. Atoji, A. & Lipscomb, W. N. (1953) Acta Crystallogr. 6, 5. Zangwill, W. I. (1967) Comput. J. 10, 293-296. 770-774. 6. Raphson, J. (1690) Analysis Equationum Universalis, cited 43. Hagler, A. T., Huler, E. & Lifson, S. (1974) J. Amer. Chem. by Cajori, T. (1919), in A History of Mathematics (McMillan, Soc. 96, 5319-5327. New York), p. 203. 44. Hagler, A. T. & Lifson, S. (1974) J. Amer. Chem. Soc. 96, 7. Macdonald, P. (1966) in Mathematics and Statistics for 5327-5335. Scientists and Engineers (Nostrand Co. Ltd., London), p. 45. Baglin, F. G., Bush, S. F. & Durig, J. R. (1969) Mol. Crystal- 75. logr. Liquid Cryst. 5, 331-352. 8. Lide, D. R., Jr. (1960) J. Chem. Phys. 33, 1519-1522. 46. Tauer, K. J. & Lipscomb, W. N. (1952) Acta Crystallogr. 5, 9. Lide, D. R., Jr. & Mann, D. E. (1958) J. Chem. Phys. 29, 606-612. 914-920. 47. Venkateswarlu, P. & Gordy, W. (1955) J. Chem. Phys. 23, 10. Maryott, A. A. & Birnbaum, G. (1956) J. Chem. Phys. 24, 1200-1202. 1022-1026. 48. Staveley, L. A. K. & Gupta, A. K. (1949) Trans. Faraday 11. Livingston, R. L., Lurie, C. & Rao, C. N. R. (1960) Nature Soc. 45, 50-61. 185, 458-459. 49. Brown, G. M. & Levy, H. A. (196.5) Science 147, 1038-1039. Downloaded by guest on September 24, 2021