15416
J. Phys. Chem. 1996, 100, 15416-15420
Potential Function for Tellurium Hexafluoride Molecules in the Solid Kurtis E. Kinney and Lawrence S. Bartell* Department of Chemistry, The UniVersity of Michigan, Ann Arbor, Michigan 48109 ReceiVed: May 15, 1996; In Final Form: July 8, 1996X
How accurately various model intermolecular interaction functions are able to reproduce the detailed structure of orthorhombic TeF6 has been investigated by molecular packing analyses. Experimental structure parameters were extrapolated to the vibrationless limit. Model potential functions were six- and seven-site pairwiseadditive atom-atom functions including partial charges. Lennard-Jones and Buckingham functions worked equally well. Scoles combining rules gave more consistent results than the conventional Lorenz-Berthelot and Good-Hope rules. Partial charges of approximately 0.21 e- on fluorine atoms gave the most accurate account of the experimental structure parameters. Potential functions based solely on interactions between fluorine sites were distinctly inferior to functions explicitly incorporating tellurium atoms. Although not required to do so by physical principles, the optimal partial charges for intermolecular interactions agreed well with charges derived for individual molecules by the Rappe-Goddard charge equilibration method.
A significant outcome of the molecular dynamics (MD) approach has been the information it has yielded about homogeneous nucleation in clusters.1 Clusters were chosen for the MD simulations rather than bulk systems to avoid the strong influence periodic boundary conditions impose on transformations. The success of several simplified potential functions has shown that full accuracy in the functions applied is not essential for qualitative success in modeling phase changes. Nevertheless, it is prudent to construct the best functions possible given the form of the model functions adopted. Simplified functions are necessary at the present stage of computer technology because of the heavy computational demands made by the simulations. The hexafluorides have proven to be informative subjects in studies of phase changes,2-12 with SF6 and TeF6 having served most frequently as trial systems. A number of investigations2-7 have used model functions based on the fluorine sites only. Others8-12 have included contributions from the central atom as well, even though it is buried within the molecule and out of direct contact with the surrounding molecules. In a few cases partial charges have been assigned to the atoms.9,12 In no cases have three-body or higher forces been included or molecular flexibilities or physically accurate repulsive forces. Neither have ion-induced dipole interactions been taken into account. How much detail must be included before simulations can be considered to be realistic has not been considered in detail. Nevertheless, despite the simplifications, results of simulations have already been pleasingly informative about the molecular behavior during phase changes. The aim of the present study is to investigate whether the ability of model functions to reproduce observable properties of matter depends sensitively upon the exact form of the potential function and whether simple adjustments of potential parameters might lead to significant improvements. It is apparent that such thermodynamic properties as the molar volume and the heat of sublimation at a given temperature are trivial to fit by a simple adjustment of size and energy parameters. On the other hand, structural properties depend upon more specific details of interactions than a simple scaling of size and energy. Therefore, as an initial step, it seemed worthwhile to select as an observable the crystal structure of orthorhombic TeF6.13 This structure is the most precisely X
Abstract published in AdVance ACS Abstracts, September 1, 1996.
S0022-3654(96)01404-9 CCC: $12.00
determined of the low-symmetry structures of the heavier hexafluorides and, being characterized by five independent structural parameters besides the scale factor, is a structure offering a significant touchstone for discriminating between various model fields. We seek, first, to find how well the crystal structure can be accounted for by various model potential functions, paying particular attention to the electrostatic contributions needed to give the best representation. Then, after establishing the optimal partial charges, we assess them by comparing them with charges suggested by several other popular schemes. Procedure Potential Functions. In the following treatment, all intermolecular functions are taken to be pairwise-additive atomatom interactions. Three-body interactions are neglected except insofar as their effects are included in an average way by the optimized pair interactions. Partial charges are incorporated in the model functions using the customary approximation in which no account is taken of the effective dielectric constant of the medium between the molecules or the polarization induced in contacting atoms. Molecules are taken to be rigid octahedra, with bond lengths of 1.915 Å.14 In discriminating between various model fields, the exact scale factors in the fitting of cell volumes and the heat of sublimation are of minor importance because, as mentioned above, any physically plausible potential function can be scaled to fit these parameters. What was considered to be more useful in diagnoses was the set of structure parameters independent of the scale factors. The forms of potential functions tested did not cover a wide range of possibilities. We only examined Lennard-Jones and Buckingham interactions, each over a range of partial charges for electrostatically neutral molecules. In one test the central atom was excluded. When partial charges were included in this test, the void at the center was given a positive charge to compensate for the negative charges on the fluorines. Three different combining rules governing the Te - - F interactions were investigated for a given set of Lennard-Jones Te - - Te and F - - F interactions. They included the commonly applied Lorentz-Berthelot rule of taking a geometric mean for the and arithmetic mean for the σ parameters, the Good-Hope rule15 © 1996 American Chemical Society
Potential Function for TeF6 in the Solid
J. Phys. Chem., Vol. 100, No. 38, 1996 15417
TABLE 1: Seven-Site Potential Parameters for TeF6 Molecules Based on Various Functional Forms and Combining Rules potential function LJG LJL LJS LJF exp-6a exp-6b
atom pair
A, kJ/mol‚Å6
B, kJ/mol‚Å12
Te-Te F-F Te-F Te-Te F-F Te-F Te-Te F-F Te-F Te-Te F-F Te-F Te-Te F-F Te-F Te-Te F-F Te-F
36923.3 566.5 4573.1 36923.3 566.5 5089.6 34502.5 529.3 4273.3 0.0 1279.8 0.0 66273 979 8055 66273 844 7479
232 333 000 367 303 9 237 250 232 333 000 367 303 11 441 115 216 283 260 341 798 8 597 980 0 699 088 0
B, kJ/mol
C, Å-1
combining rule
Good-Hopea Lorentz-Berthelotb Scolesc Te not includedd 472 950 64 539 174 710 472 950 363 725 414 757
2.41 3.60 3.00 2.41 4.16 3.29
a Geometric mean for both and σ. b Geometric mean for , arithmetic mean for σ. c See eqs 1-4 of text. d LJF is a six-site model related to one proposed by Pawley2 containing interactions only between flourines but modified to include a charged interaction site at the molecular center.
using the geometric mean for both, and the Scoles rule.16 Scoles introduced his rule to represent interatomic forces in collisions between single, unlike atoms (or, sometimes, atomic groups treated as a single interaction center) rather than to reproduce atom-atom components in pairwise-additive functions for compound molecules. Moreover, the interatomic functions to which the rules were applied were not simple Lennard-Jones functions. Nevertheless, it seemed worthwhile to find how the Scoles rules performed when they were adapted to LennardJones functions and compared with results of conventional combining rules. Accordingly, we take the Scoles parameters for unlike atoms to be
The LJF field used the parameters of Pawley and co-workers who modeled the potential energy between SF6 molecules solely in terms of interactions between fluorine sites.2-7 Values of the parameters for all of the fields, exclusive of the charges, are listed in Table 1. Parameters are defined by the expressions
σs12 ) (σ11 + σ22)/2Yσ
for Lennard-Jones functions. The Aij, Bij parameters for the Lennard-Jones functions are related to σ and by the relations
(1)
Uij ) Bij exp(-Cijrij) - Aijr-6 for Buckingham functions and
Uij ) Bijr-12 - Aijr-6
and
s12 ) (1122)1/2Y
(2)
where
Yσ ) 1 - 0.6 (1 - X)
3
(6)
Aij ) 4ijσij6
(7)
Bij ) 4ijσij12
(8)
and
(3)
and
Y ) 1 - {(1 - X5/2)/[1 + (1/3)X5/2]}3
(5)
(4)
with X e 1 being the ratio of ionization potentials of the two atoms involved. The potential functions resulting are designated as LJL, LJG, or LJS for the Lorentz-Berthelot, Good-Hope, or Scoles Lennard-Jones variants, LJF for the function neglecting the central atom, and exp-6a or exp-6b for the Buckingham functions based on the Caillat17 or Williams18 atomic potential parameters. The exp-6a parameters are those developed by Caillat17 in his determination of the structure of monoclinic TeF6 from potential energy minimization together with diffraction data from clusters. The LJL and LJG parameters were adapted from Caillat’s parameters17 and adjusted to account for the molar volume and heat of sublimation of cubic TeF6 in MC runs excluding partial charges. In the cubic and liquid phases partial charges have much less influence than they do in the orthorhombic crystal. The exp-6b field adopted the fluorine parameters of Williams and Houpt18 and Caillat’s tellurium parameters.
Calculation of Crystal Structure Parameters. Molecular packing analyses were performed with a modified version of the program PCK83 written by Williams19 and subsequently checked by (and confirmed by) the updated commercial program mpa.20 In the packing analyses, (rigid) molecules are placed in the crystal lattice at preliminary positions and orientations. The program then minimizes the potential energy with respect to variations in the lattice parameters and the positions and orientations of the molecules. In the case of orthorhombic TeF6, space group Pnma, Z ) 4, the crystal structure13 is characterized (apart from the symmetry relations of the space group) by the lattice parameters a, b, and c, the x and z coordinates of a reference Te atom, and the molecular tilt about the y axis. Because potential energy minimizations disregard the effects of molecular motions, which tend to increase molar volumes, the parameters used to test how well the model fields represent experiment were the ratios a/b, a/c, and b/c (of which only two are independent), u ) x/a, w ) z/c, and the tilt θ. Computations were carried out for each model potential function by assigning partial charges to fluorines ranging from 0 to 0.40 e- in increments of 0.025 e-. An opposite positive charge was assigned to the molecular center to enforce electrostatic neutrality.
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Extrapolation of Results to Vibrationless Crystal. Results are compared with experimental structure parameters derived from neutron diffraction powder patterns acquired at 15 and 190 K.13 Because effects of anharmonicity make the structure parameters temperature-dependent, an attempt was made to extrapolate experimental results to the limit corresponding to a vibrationless crystal (devoid of thermal and zero-point vibrations). Several approximate approaches suggest themselves, of which we shall describe only the simplest and least sensitive to details. Corrections lack rigor because they do not include an analysis of the sensitivity of individual structure parameters to the distribution of individual lattice frequencies (and, hence, to temperature), but they are likely to be adequate since the corrections from the 15 K parameters are small. For this reason our sketch of the procedure will be cursory. We assume that shifts in all structure parameters increase linearly in proportion to the mean-square amplitudes of molecular vibrations, as expected21 from Ehrenfest’s relations.22 Denoting a representative parameter as ξ (cf. a/b, u, θ, ...), we estimate the corrections from the relation
ξ00 ) ξ15 + [ξ190 - ξ15]f
(9)
f ) [V00 - V15]/[V190 - V15]
(10)
where
in which the subscripts represent temperature or, for 00, the absence of molecular motion. All quantities are known except for ξ00 and V00. The problem then reduces to the inference of the cell volume V00 for the vibrationless crystal. This might be done by carrying out classical MD runs from the region where quantum and classical volumes are the same (hence giving a calibration point for the volume scale factor) to absolute zero where the classical volume is less than that for the system with zero-point vibrations. Instead, for simplicity, we take advantage of the close relation between effects of molecular motion on coefficients of thermal expansion and the effects on heat capacity first noted by Gru¨neisen in 1908.23 Indeed, Gru¨neisen’s early observation that the thermal coefficient of expansion is proportional to Cp over a large range of temperature turns out to be more accurate for many simple systems than his better known and much more complex relation formulated later.24 Since the relation is expected to hold both for quantum and for classical amplitudes of vibration, it is simple to extrapolate volumes from 190 K where quantum effects are negligible to absolute zero as illustrated by Hovick,25 but using the easily estimated classical heat capacity. The result is that the factor f of eq 9 is approximately 0.18. This factor has been used to give the vibrationless limit of all structure parameters treated in the next section. Results Structure parameters derived from the various model potential functions varied significantly with the assigned partial charge. Calculated values are shown in Figures 1-3, where they are compared with the results derived from neutron diffraction. Experimental quantities obtained at 190 and 15 K are represented by discontinuous horizontal lines and our estimated vibrationless values are given by solid horizontal lines. In the case of the molecular tilt illustrated in Figure 2b, the effect of molecular motion is too small to be discerned. Most of the model functions are able to account for each of the structure parameters at some assumed charge, and the figures suggest a rough consensus of the magnitude of the charge effective in governing the structure. For an individual potential function
Figure 1. (a) Lattice parameter ratio a/b vs charge on fluorines. (b) Ratio a/c vs partial charge. The LJG potential functions are represented by circles, LJL by squares, LJS by diamonds, LJF by crosses, exp-6a by plus signs, and exp-6b by triangles. Wide horizontal dashes correspond to 190 K, narrow, to 15 K, and the solid line to the vibrationless limit. Standard deviations in the positions of the horizontal lines inferred from the reported uncertainties in lattice constants13 are no greater that the line thicknesses. Effective charges are inferred from the points at which the calculated curves cross the line corresponding to the vibrationless limit.
Figure 2. (a) Lattice parameter ratio b/c vs partial charge on fluorines. For the meaning of the symbols, see Figure 1. (b) Tilt of the reference TeF6 molecule about the y axis vs partial charge.
and structure parameter, this estimate is derived from the point at which the calculated curve crosses the experimental vibrationless line and is listed in Table 2. The experimental parameter least well represented over the range of model potential functions is the b/c ratio. Since the ratio b/c is not independent of the set [a/b, a/c], only two ratios need be retained. We arbitrarily excluded results for b/c from our final estimate of charge. Obviously, a physically accurate function would fit all parameters. Discussion Form of Interaction Potential. The range of variability of the model potential functions examined in this study is too narrow to permit any very definitive conclusions to be drawn
Potential Function for TeF6 in the Solid
J. Phys. Chem., Vol. 100, No. 38, 1996 15419
Figure 3. (a) Reduced coordinate u ) x/a for the reference tellurium atom vs partial charge. (b) The reduced coordinate w ) z/c vs partial charge. For the meaning of the symbols, see Figure 1. Standard deviations in the positions of the 190 K lines are approximately the heights of the plotted markers. For the 15 K lines, uncertainties are half as great.
about the optimal form of functions constructed from atomatom representations. All of the variants were able in molecular dynamics computations to reproduce the known phases and, where tested, the spontaneous phase transitions in TeF6, and all gave an account of the specific orthorhombic crystal structure of sufficient accuracy to be regarded as satisfactory in many contemporary studies. Nevertheless, it is clear that the performance of most forms tested is considerably improved by the inclusion of appropriate partial charges. Even though it is generally believed that Buckingham functions give atomic repulsions that are substantially better than those implied by Lennard-Jones functions, no appreciable advantage in the fitting of the orthorhombic structure was shown by the Buckingham form. It is likely that the Buckingham function would give a more accurate compressibility and coefficient of thermal expansion. Even though the forces governing the packing arrangements of molecules must be mediated primarily by atomic contacts, a decisive advantage was seen in the functions with partial charges explicitly including the (buried) central atom as well as the fluorines. On the other hand, if charges are neglected, the potential function based only on fluorines gives a better fit than the others. The combining rule most successful in reproducing the five retained structure parameters by a single value of the charge was neither of the most commonly applied ones. It was the Scoles rule.16 This rule also accounts better for the trajectories of unlike colliding atoms. How general its advantage is in other applications is unknown. Magnitude of Partial Charges. According to Table 2, the partial charges best accounting for the experimental structure
are approximately 0.21 e- on fluorines and a compensating charge on the tellurium. What this effective charge means physically is unclear because no effect of polarizability on results has been included in the treatment. Therefore, there is no reason to expect that the charge optimized for a medium considered to have a dielectric constant of unity should agree with an estimate of charge based on any other scheme for assigning charges. Be this as it may, it is still of interest to compare the charges derived to fit the orthorhombic structure with those estimated by other procedures. Alternative schemes for the estimation of charge include the “charge equilibration” method of Rappe and Goddard,26 ab initio quantum computations, and the inference from the electronegativity difference between Te and F. Details of carrying out such estimates are given elsewhere.27 The partial charges on the fluorines of SF6, SeF6, and TeF6 calculated by the charge equilibration method using the proprietary program Biograph/Polygraph28 are 0.177, 0.184, and 0.237 e-, respectively. The result for TeF6 is close to that derived in the foregoing from the structure parameters. LewisLangmuir charges based on Allen’s prescription29 depend somewhat upon the convention for assigning electronegativities. If we arbitrarily take the average of the charges implied by the scales of Pauling,30 Mulliken,31 Allred and Rochow,32 Allen,29 and Nagle,33 we obtain 0.240, 0.268, and 0.327 e- per fluorine in SF6, SeF6, and TeF6. These values are too high because they are not corrected for the high valence state of the central atom and the fact that five other fluorine atoms are competing with a given fluorine for charge. The relative charges in the series, however, agree well with those from the charge equilibration method. An “electric potential derived” charge computed from a high-level quantum calculation as outlined by Williams34 would make a worthwhile and physically significant comparison. Such a charge is not yet available. Mulliken overlap populations, whose physical significance is known to be marginal, are 0.43 or 0.35 e- on the fluorines in TeF6 when computed via HF/3-21G or HF/6-311G(2d) basis sets. These charges are also excessive. Concluding Remarks Although the effect of partial charges on the dynamics of phase changes has not yet been examined, it has been found in molecular dynamics simulations that the associated Coulombic interactions can have a substantial effect on calculated transition temperatures.9,12 It has also been shown that partial charges can play a major role in the way molecules pack together.35 Therefore, it is not surprising that the details of a crystal structure are able to provide a useful estimate of the effective partial atomic charges for potential functions of molecules in condensed phases. In the present example of TeF6, the charges are of the same magnitude as charges derived from independent estimates based on single molecules. While there is no requirement for the agreement to be close, at least the charges are not unreasonable. The present results suggest a way of making the
TABLE 2: Partial Charges Derived from Crystal Packing Analysisa
a
c
potential functionb
a/b
a/c
b/c
u
w
θ
avc,d
std devd
LJG LJL LJS LJF exp-6a exp-6b
0.154 0.187 0.227 NV 0.238 0.240
0.040 0.116 0.192 NV 0.182 0.160
0.33 0.33 0.37 0.31 NV NV
0.190 0.202 0.202 0.210 0.176 0.188
0.186 0.202 0.216 0.127 0.212 0.217
0.167 0.193 0.233 0.062 0.243 0.236
0.147 0.180 0.214 0.13 0.210 0.208
062 0.036 0.017 large 0.031 0.034
Parameters for which the computed curves do not intersect the experimental value and, hence, yield no value, are listed as NV. b See Table 1. Over all parameters except for b/c. d Disregards b/c.
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potential functions more realistic for general members of the family of hexafluorides by incorporating partial charges based on the packing of molecules in the solid. Because the relative charges yielded by the Lewis-Langmuir-Allen procedure (which is readily applied) closely parallel those found by the charge equilibration method, which charges, in turn, agree closely with the present results, it is plausible to propose that charges for other members of the AF6 series can be estimated from the simple proportion
q(AF6) ) [qen(AF6)/qen(TeF6)]q(TeF6)
(11)
where the subscript en refers to the electronegativity method. In view of the fact that the hexafluorides have provided a series of informative results on stable and metastable phases, and on nucleation and nucleation rates in phase changes in condensed matter, it is agreeable to envisage continued molecular dynamics simulations to follow the molecular behavior in such processes in detail. The present results suggest one way of improving the basis of such a program of study. Acknowledgment. This research was supported by a grant from the National Science Foundation. We gratefully acknowledge the award of a Regent’s Fellowship (to K.K.) during part of this research. We thank Dr. T. S. Dibble for carrying out the ab initio computations on TeF6 and Professor D. C. Martin for his help in applying the charge equilibration method. References and Notes (1) See, for example: Bartell, L. S. J. Phys. Chem. 1995, 99, 1080. (2) Pawley, G. S. Mol. Phys. 1981, 43, 1321. (3) Powell, B. M.; Dove, M. T.; Pawley, G. S.; Bartell, L. S. Mol. Phys. 1987, 62, 1127. (4) Boyer, L. L.; Pawley, G. S. J. Comp. Phys. 1988, 78, 405. (5) Fuchs, A. H.; Pawley, G. S. J. Phys. Fr. 1988, 49, 41. (6) Beniere, F. M.; Fuchs. A. H.; de Feraudy, M. F.; Torchet, G. Mol. Phys. 1992, 76, 1071. Beniere, F. M.; Boutin, A.; Simon, J.-M.; Fuchs. A. H.; de Feraudy, M. F.; Torchet, G. J. Phys. Chem. 1995, 97, 10472.
(7) Rousseau, B.; Boutin, A.; Fuchs, A. H.; Craven, C. J. Mol. Phys. 1992, 76, 1079. Boutin, A.; Rousseau, B.; Fuchs, A. H. Chem. Phys. Lett. 1994, 218, 122. (8) Bartell, L. S.; Xu, S. J. Phys. Chem. 1991, 95, 8939. (9) Xu, S.; Bartell, L. S. J. Phys. Chem. 1993, 97, 13550. (10) Bartell, L. S.; Xu, S. J. Phys. Chem. 1994, 98, 6688. (11) Bartell, L. S.; Xu, S. J. Phys. Chem. 1995, 99, 10446. (12) Kinney, K. E.; Bartell, L. S. J. Phys. Chem., in press. (13) Bartell, L. S.; Powell, B. M. Mol. Phys. 1992, 75, 689. (14) Gundersen, G; Hedberg, K; Strand, T. G. J. Chem. Phys. 1978, 68, 3548. (15) Good, R. J.; Hope, C. J. J. Chem. Phys. 1970, 53, 540. (16) Scoles, G. International Journal of Quantum Chemistry: Quantum Chemistry Symposium; John Wiley & Sons, Inc.: New York, 1990; Vol. 24, p 475. (17) Bartell, L. S.; Valente, E. J.; Caillat, J. C. J. Phys. Chem. 1987, 91, 2489. (18) Williams, D. E.; Houpt, D. J. Acta Crystallogr. 1986, B42, 286. (19) Williams, D. E. Quant. Chem. Progr. Exch.; Program 481; Indiana University; Bloomington, IN 47405; 1983. (20) Williams, D. E. Program mpa/mpg; Department of Chemistry; University of Louisville; Louisville, KY 40292. (21) Bartell, L. S. J. Chem. Phys. 1963, 38, 827. Bartell, L. S.; Fitzwater, S. J. J. Chem. Phys. 1977, 67, 4168. (22) Ehrenfest, P. Z. Phys. 1927, 45, 455. (23) Gru¨neisen, E. Ann. Phys. 1908, 26, 211; 1910, 33, 65. (24) Gru¨neisen, E. In Handbuch der Physik; Geiger, H.; Scheel, K., Eds.; Springer Verlag: Berlin, 1926, Vol. 10, p 1. (25) Hovick, J. W.; Bartell, L. S. J. Mol. Struct. 1995, 346, 231. (26) Rappe, A. K.; Goddard, W. A., III J. Phys. Chem. 1991, 95, 3358. Ding, H.; Karasawa, N.; Goddard, W. A. III Chem. Phys. Let. 1992, 193, 197. (27) Kinney, K. E. Ph.D. Thesis, University of Michigan, Ann Arbor, MI, 1995. Note that there are errors in tabulated quantities and in the labeling of figures in the thesis. (28) Biograph Molecular simulations Inc., Waltham, Mass. (29) Allen, L. C. J. Am. Chem. Soc. 1989, 111, 9003. Allen, L. C.; Knight, E. T. J. Mol. Struct. 1992, 261, 313. (30) Pauling, L. J. Am. Chem. Soc. 1932, 54, 3570; Nature of the Chemical Bond; Cornell Univ. Press: Ithaca, NY, 1939. (31) Mulliken, R. S. J. Chem. Phys. 1934, 2, 782. Pearson, R. G. Inorg. Chem. 1988, 27, 734. (32) Allred, A. L.; Rochow, E. G. J. Inorg. Nucl. Chem. 1958, 5, 264. Little, E. J.; Jones, M. M. J. Chem. Ed. 1960, 37, 231. (33) Nagle, J. K. J. Chem. Soc. 1990, 112, 4741. (34) Williams, D. E. ReV. Comp. Chem. 1991, 2, 219. (35) Williams, D. E.; Starr, T. H. J. Comput. Chem. 1977, 1, 173.
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