2544
J . Phys. Chem. 1985, 89, 2544-2549
Internuclear Probability Distributions of Anharmonically Vibrating Polyatomic Molecules John F. Stanton and Lawrence S. Bartell* Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109 (Received: January 25, 1985) Intramolecular radial distribution functions of molecules consist of peaks that are significantly skewed from the Gaussian shape characteristic of idealized multidimensional harmonic oscillators. It is shown how to characterize the radial distribution peaks in terms of their first three moments derived from the anharmonic force field by a path integral technique. Illustrative calculations are presented for the geminal nonbonded peaks of a series of octa- and tetrahalides of main-group elements that have recently been found to conform surprisingly well to a simple repelling points-on-a-sphere model of bending anharmonicity. Comparisons with experiment, electron diffraction and spectroscopic, are made where data are available. Implications are briefly discussed.
I. Introduction TABLE I: Nonzero Etements" of Transformation Tensor of Eq 2 and 3 Sufficient To Characterize a Representative Nonbonded Distance of Electrons diffracted from molecules carry information about an AXs Molecule with Oh Symmetry stretching and bending anharmonicity, the manifestation of which may become plainly evident when molecules are thermally excited. iik ijk T;'k T/k ijk This can be helpful in studies of potential energy surfaces because 1 4242 6'12/ 12R 4x4y5z -1/8R 5z2a5z 3'I2/6R it reveals properties of polyatomic molecules that, so far, have been 1 5z5z 1/2R 6'I2/24R 4 x 5 ~ 6 ~1/8R 5z3x4y difficult to characterize by spectroscopy. Indeed recent diffraction 1 6262 1/4R 6'12/12R 4x3y5z 6 ~ 5 x 6 ~1/8R investigation^'-^ uncovered a simple but powerful predictive 2a4z4z -3'12/ 12R 4y3x5z 1/4R 6 ~ 5 x 6 ~1/8R 4y4x5z -1/8R 2a5z5z -3'I2/24R principle5-' governing bending potential constants that has since 6 ~ 4 x 5 ~1/8R 2a6z6z -3'12/ 12R 421 42 -6'I2/6R 3'12/6R 6z2a6z been confirmed by molecular orbital computations.' On the other 4z2a4z 3x4y5z -1/8R 3'12/6R 6zl 6z -6'f2/6R hand, this sensitivity of diffraction features to heretofore poorly 5 ~ 3 x 6 ~1/2R 3 x 5 ~ 6 -1/8R ~ 6 ~ 3 x 5 ~114R understood aspects of molecular vibrations imposes a penalty, as 5zl 52 -6'I2/6R 3y4x5z -1/8R well. Unless the anharmonicity is properly handled, the structural information that is conventionally sought in electron diffraction "All stretch-stretch-stretch elements are zero as are 1 3242, 203z4z, 321 4z, 3z2a4z, 3x3y5z, 3y3x5z, 4z2a3z, 5z4x4z, 5x6y6z, 5 ~ 4 x 6 ~R . refinements may be significantly distorted. Even for such simple represents the equilibrium bond lenght. and rigid molecules as SiF4and SF6,it was found for hot samples that intensity residuals increased by a factor of 2 and derived handling warmer molecules, however. We hereby extend a recent geminal internuclear distances shifted by over 0.01 A (when treatmentIs of tetrahedral molecules based on this approach to standard deviations were only 0.0003 A) when anharmonicity was octahedral molecules and, in the process, add an additional renegle~ted.~ finement to the tetrahedral treatment. Because the distributions For the above reasons it is worthwhile to get a better underfor bonded and trans nonbonded distances are quite well understanding of methods for treating the problem and of the magstood1,2,8-11 and are not sensitive to bending anharmonicity, we nitudes involved. Prior analyses'**-]'of the Morse-like stretching shall concentrate on geminal nonbonded distribution functions. anharmonicity have been quite successful in accounting for the anharmonic stretching effects encountered for covalent bonds, and 11. Treatment useful predictive models have been advanced. Bending anharA . Anharmonic Force Field. The natural express10n'~for the monicity and distributions for geminal nonbonded distances have potential energy is in terms of curvilinear coordinates Si (identified only recently been attacked seriously. The first problem, namely hereafter by tildes) whereas computations are more conveniently the establishment of anharmonic potential functions, has been out with the rectilinear symmetry or normal coordinates partially solved by the introduction of realistic m o d e l ~ ' * ~ ~ ~carried J~ Si or Qi (identified by plain symbols). In the following we shall augmented by quantum calculations. The second problem, that consider effects only through cubic terms, writing the potential of deriving the relationship between the potential energy function energy as and intensities of diffracted electrons, has long been solved for modestly excited small molecules where p e r t u r b a t i ~ n ~ or ~~'~J~ variationalt4 probability densities corresponding to individual vibrational states can be averaged over a Boltzmann population. An application of path integral techniques due to FeynmanI5 and Miller16 has been shownI7J8 to be much more promising for (1) Goates, S. R.; Bartell, L. S. J. Chem. Phys. 1982, 77, 1866. (2) Goates, S.R.; Bartell, S . L. J . Chem. Phys. 1982, 77, 1874. (3) Bartell, L. S.;Vance, W. N.; Goates, S. R. J . Chem. Phys. 1984,80, 3923.
(4) Bartell, L. S.; Stanton, J. F. J. Chem. Phys. 1984, 81, 3792. (5) Bartell, L. S . J . Mol. Struct. 1982, 84, 117. (6) Bartell, L. S. Croat. Chem. Acta 1984, 57, 927. (7) Bartell, L. S.;Barshad, Y . Z . J. A m . Chem. SOC.1984, 106, 7700. (8) Bartell, L. S. J. Chem. Phys. 1955, 23, 1219. (9) Kuchitsu, K.; Bartell, L. S. J . Chem. Phys. 1961, 35, 1945. (10) Kuchitsu, K.; Morino, Y . Bull. Chem. SOC.Jpn. 1965, 38, 805, 814. (11) Bartell, L. S. J . Mol. Struct. 1981, 63, 259. (12) Kuchitsu, K.; Bartell, L. S. J. Chem. Phys. 1962, 36, 2460, 2470. Jpn. 1967, 40, 498, 505. See also: (13) Kuchitsu, K. Bull. Chem. SOC. Reitan, A. Acta Chem. Scand. 1958, 12,785; ET. Nor. Vidensk. Selsk., Skr. 1958. No. 2. (14) Hilderbrandt, R. L.; Kohl, D. A. J . Mol. Struct.: THEOCHEM. 1981, 85, 25. Kohl, D. A.; Hilderbrandt, R. L. Ibid. 1981, 85, 325. (15) Feynman, R. P.; Hibbs, A. R. "Quantum Mechanics and Path Integrals"; McGraw-Hill: New York, 1965; pp 273-279. (16) Miller, W. H. J. Chem. Phys. 1971, 55, 3146.
0022-3654/85/2089-2544$01.50/0
Coordinates
si and Si are related by the nonlinear transformation si
=
si
+ '/zcxT/ksjsk-k ... J k
(2)
whence, by ( l a ) and (1b), the cubic constants for rectilinear coordinates can be derived from the natural A j k via2'
(17) Spiridonov, V. P.; Gershikov, A. G.;Butayev, B. S. J . Mol. Struci. 1979, 51, 137; 1979, 52, 53. (18) Bartell, L. S . J. Mol. Struct. 1984, 116, 279. (19) Hoy, A. R.; Mills, I. M.; Strey, G.Mol. Phys. 1972, 24, 1265.
@ 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 12. 1985 2545
Internuclear Probability Distributions found with the aid of the linear transfor-
Cubic constants mation
si = WQ, I
(4)
a relation for a one-dimensional oscillator that reduces to Feynman’s expression if the potential energy is quadratic. The probability density p can conveniently be expressed in terms of an effective potential energy Pffexplicitly given by Mil1erl6 as a function of the temperature and the actual potential energy V, as p
In the treatment of geminal nonbonded distributions it is sufficient to consider a single (representative) distance and to relate it to the above deformation coordinates. Elements of the T tensor sufficient for this purpose are listed in Table I for octahedral AX6 molecules and in ref 18 for tetrahedral molecules. No spectroscopic analyses of the complete cubic force field of any octahedral molecules have yet been carried out. For illustrative calculations, to be presented in a later section, we invoke for most anharmonic potential constants the so-called KBFF, an anharmonic Urey-Bradley model force fieldI2 shown to yield an excellent representation of effects of stretching anharmonicity. For bend-bend-bend constants, the most important contributors to the effects we treat, we adopt values obtained by molecular orbital calculations or by application of the “POS” model6,’ which gives very similar results. Explicit equations and tabulations for POS force constants are tabulated in ref 7 . N o prior listing of the KBFF constants for octahedral molecules has been published. In the notation of ref 1 and 12 the constants are
7111 = (-3areK + 4F3)/61/2re JlZ2 = (-3areK - 3F’+ 3 F + F3)/6’l2re 7133 = (-3areK - 2F’+ 2 F + 2F3)/6lJ2re 7222= (-3ar& + [-9F’+ 9 F - F3]/2)/121/2re 7233 = (-3areK + [-7F’+ 7 F + F3]/2)/31/2re 7144 = (-3F’- F + F3)/24‘I2re JIs5 = (-F’+ F + F3)/24‘l2re 7166 = (F’+ F + F3)/24’l2re 7244 = ( 6 F + F3)/192II2re JZs5 = (8F’7266
= (-8F’-
7134
= (-5F’
7234
= (-5F’ 7335
F - 2F3)/192’I2re
+ F3)/192’/’re + 5 F + 2F3)/48’I2re + 5 F + 2F3)/961/2re 3F
= ( 4 F + F3)/4re
= (5F’- F3)/8r3 = (2F’- F - F3)/8r3
7345
7356
Sign conventions for the symmetry coordinates (see Appendix) are those of Heenad] and differ from those in the POS of ref 7 . Equations 6i, 6j, 6k, 6n, 60, and 6p were derived in this research, while the others had been worked out previously.22 B. Probability Density. FeynmanI5 showed that an exact density matrix appropriate for an ensemble of multidimensional quantum harmonic oscillators in thermal equilibrium can be derived easily by his path integral technique. Miller16 has proposed Bartell, L.S.;Fitzwater, S. J. J. Chem. Phys. 1977, 67, 4168. Heenan, R. K. Ph.D. Thesis, University of Reading, U.K., 1979. Heenan, R. K.; Bartell, L.S., unpublished research. Note that the derivations of eq 6a-6p involve tedious exansions with imposed redundancy conditions that have not been subjected to rigorous checks. The major contributions involving F3and the Morse a constant are comparativelysimple to calculate, but it is possible that errors in coefficients of minor terms have gone unnoticed. Such errors would be unlikely to alter significantly the results sought here, however. We publish to document what we have used, not to establish the equations.
= N exp(-Pff/kT)
(7)
where N is a normalization function that depends mildly upon the coordinates of the system if Vis not quadratic. The expression can be applied to anharmonic quantum oscillators for which the expression, while no longer exact, is still substantially more accurate than that corresponding to the classical density Pcl
= Ncl exp(-V/kT)
(8)
As shown in the next section, the parameter of greatest importance in characterizing the effect of anharmonic motion on a distribution peak is the index of skewness
6 = ( ( x - (x))3)/((x - ( x ) ) z ) z
(9)
where x is the displacement from the equilibrium internuclear distance. This parameter is zero, of course, for a harmonic oscillator. When calculated via eq 7 incorporating Miller’s Pff, results for a Morse anharmonic oscillator become satisfactory for k T 1 hv/2. A temperature 4-fold higher is required for comparable accuracy if the classical distribution corresponding to eq 8 is applied. Extension of Miller’s formula for Pffto multidimensional oscillators coupled by anharmonic terms was recently carried out, approximately, by an inductive approach in ref 18. It has since been checked by a perturbation computation which gave quite similar results. For simplicity we adopt the published approximations’* relating effective force constants C#Ii; and C#Iilke to the actual force constants 4iiand 4ijk, for any combination of i, j , k. When effective force constants are substituted for the true force constants in eq I C and the resultant V(QI, Q3N-6)is used in eq 7, a serviceable probability density p(Ql, .-,Q3,,,+) is obtained. In carrying out calculations we make the additional aproximation of neglecting the dependence of N upon the normal coordinates. A further simplification is made by separating Pffinto the sum of Vheff and Vaeff,the harmonic and anharmonic components, whence -e,
p = i=
N eXp(-Vheff/kT) exp(-l/,eff/kT)
N exp(-Vheff/kT)(l - V,eff/kT)
(10)
regarding the anharmonicity as a perturbation. While it is expected that eq 7 becomes more accurate, the higher the temperature, the simplification of N , the truncation of V2ff beyond cubic terms, and the above further truncation of the expansion of exp(-Vaeff/kT) prevent eq 10 from attaining the correct asymptotic limit at extremely high temperature. Because k T increases roughly as the mean square of the normal coordinates, and Vtff increases as the cube of displacements, V,eff/kT becomes a larger perturbation at increasing temperature. Nevertheless, in calculations carried out upon Morse oscillators with frequencies, amplitudes of vibration, and asymmetries of distribution comparable to those of the molecules treated in the next section, the approximations made are excellent.23 Over the entire temperature range considered in this paper the errors introduced are smaller than those associated with the uncertainties in the anharmonic potential functions. Moreover, the calculations involved are many orders of magnitude simpler than the thermal averaging over individually calculated single-state probability densities. C. Shape of Radial Distribution Peak. Although a procedure has been described for reducing a multidimensional probability density to the one-dimensional radial distribution function corresponding to the electron diffraction o b ~ e r v a b l ea, ~considerably ~ more tractable method can be applied.18 The position of a (23) Bartell, L.S. J . Phys. Chem., in press.
2546
Stanton and Bartell
The Journal of Physical Chemistry, Vol. 89, No. 12, 1985 is then made via the L matrix, with
TABLE II: Coefficients 6, and Bu of Eq 13 Contributing to (x12) for AX, and (x,,~) for AX6 Molecules' i
1 2a 32 42 1 2a 3x 3Y 4x 4Y 52 6x 6v
Td
Oh
Yij
Substitution of eq 14 into eq 12 yields expressions for (x") for any desired n in terms of the harmonic oscillator mean-square amplitudes ( e m 2 )which, in turn, are given by
I12
((x
= (hv,/2dmm) coth (hvm/2kT) From ( x ) , (x2), and (x3) can be calculated ( r ) = re + ( x )
8-112
-'/4 -i/d
OijlB
1,2 = (x2) -
4 4 4 1 1 8 4 -4 4 2 2 2 2 .-2 -2 -2
ii =
1 1 -4 1 1 -4 1 1 -1 -1 -1
CP,Si + '/,CCPipisj i
i
J
(11)
(12)
(13)
where Bi and &, derivable from simple considerations of geometry, are listed in Table I1 for octahedral and tetrahedral molecules. A transformation to
CriQi + !hCCrijQiQj i J I
(24) Kacner, M. A.; Bartell, L. S. J . Chem. Phys. 1979, 71, 192.
(19)
+ 2(x)3)/1,4
(20a) (20b)
(14)
= dlg4/6
(21)
Explicit equations for such quantities are listed for AX4 molecules in ref 18 where, however, the y,, of eq 14 were discarded. It is logically inconsistent to neglect the ylJ in the treatment, however, and corrected values of the d parameter are given in the next section. It is found that the ylJcontributions to tetrahedral molecules are small (zero for off-diagonal elements). In the tabulation of ref 18, corrections to d for CF4 and SiF4should be +0.07 and +0.02 A-1, respectively. Corrections for hexafluorides are larger. Even though the procedure for carrying out the computations outlined above is straightforward, the resultant expressions are too cumbersome and susceptible to errors in recording and proofreading to warrant listing here. Therefore, alternatively, we reproduce in the supplementary material the computer codes written for the AX4 and A x 6 cases. These document in complete detail the expressions encountered and the method of their application.
111. Illustrative Calculations For purposes of illustration the foregoing approach was applied to a series of tetrafluorides, tetrachlorides, and hexafluorides of the main-group elements S, Se, and Te. Although all of these cases have been analyzed by both electron diffraction and vibrational spectroscopy, only CF4, SiF,, and SF6among them have been studied at sufficient vibrational excitation to enable an experimental measurement of d for the geminal radial distribution peak. In none of the examples are a large enough set of cubic constants available in the literature to allow a meaningful calculation of ci. We nevertheless can construct reasonable representations of the anharmonic force fields in which we have considerable confidence in the general magnitudes of the most crucial components, the bend-bend-bend interactions. Furthermore, we can formulate plausible uncertainties for the associated values of 6. Cubic constants corresponding to the KBFF of eq 6a-6p for octahedral molecules and ref 18 for tetrahedral molecules were evaluated by estimating the anharmonic Urey-Bradley potential constants listed in Table I11 from the known quadratic cons t a n t ~ by ~ a~ procedure ~ ~ ~ - closely ~ ~ following that outlined previously.l Two sets of constants, I and 11, were constructed. In ~
x =
- 3(x2)(x)
K
It is enough to consider a single one of the equivalent nonbonded distances and expand the associated displacement in the series x
(18)
where A3 is the standard coefficient of skewness of the distribution peak. The electron diffraction frequency modulation parameter K13 is related to d and 1, via
-1
..., Q3jv-fi) dQi...dQ3jv+
((x3)
= '43/4
-2
- (x))2)
(x)2
(17)
and
and the standard coefficient of skewness which is just 1, times 6 of eq 9. As before, the coordinate x represents the displacement ( r - re) from equilibrium for the distances of interest, namely the geminal nonbonded X-X distance in AX6 or AX4. Moments (x") are readily obtained by the elementary integrals ( x " ) = Sx"p(Qi,
= (kT/$mme)
(Qm2)
-l/4
distribution peak is given by its first moment while the nearly Gaussian shape is adequately characterized by the second and third moments. These are easily related to the mean-square amplitude parameter 3
(16)
m n
'Numbering as defined in Appendix. Constant B represents [ ( 2 / 3 ) 1 / 2 / 1 6 r e ]for AX4 and [ 2 1 / 2 / 3 2 r , ]for AX6.
1,2
= C CPmnLm'Li
and
'I2
2626 3X3X 3Y 3Y 4X4X 4Y 4Y 2b2b 3X3X 3x3y 3Y3Y 3X4X 3x4y 3y4x 3Y 4Y 3x6~ 3x6~ 3y6x 3Y6Y 4X4X 4x4y 4Y 4Y 4x6~ 4x6~ 4y6x 4Y 6Y 5X5X 5Y 5Y 6x6~ 6x6~ 6Y 6Y
Oh
(15)
m
(2/3)'/' 113 (2/3)'12 6-112 3-112 -6-112
ij
TA
= CPmLm'
161
~~~
~~
(25) Clark,R. J. M.; Rippon, D. M. J. Mol. Specrrosc. 1972, 44, 479. (26) McJhwell, R. S.; Aldridge, J. P.; Holland, R. F. J. Phys. Chem. 1976, 80, 1203. (27) Konigen, F.; Muller, A.; Selig, H. Mol. Phys. 1977, 34, 1629. (28) Abramowitz, S.; Levin, I. W. J . Chem. Phys. 1966, 44, 3353.
Internuclear Probability Distributions
The Journal of Physical Chemistry, Vol. 89, No. 12, 1985 2541
TABLE 111: Constants’ for Anharmonic Force Fields of Hexafluorides molecule set re a K F’ F SF, SeF,
TeF, SF,
I1 I I1 I I1 I
1.555 1.555 1.680 1.680 1.811 1.811
1.8 1.8 1.9 1.9 2.1 2.1
6.845 6.845 5.61 5.61 5.50 5.50
3.63 4.47 3.96 4.63 4.59 4.83
-0.125 -0.072 -0.058 -0.036 -0.028 -0.018
-f335
TABLE V: Selected Results‘ for Mean Displacements and Mean-Square Displacements from Equilibrium of Geminal Nonbonded Distances and for Contributionsbto the Asymmetry Constant ri
F,
1.032 0.559 0.518 0.291 0.267 0.147
-8.51 -4.56 -4.62 -2.545 -2.58 -1.38
5
71 34 7144 f445 I1 -1.043 -1.203 -0.705 -0.305 exptb 0.26 (160) -1.49 (107) -0.41 (150) -0.53 (22)
’re in A , a in
mdyn/A2.
f , , - F , in mdyn/A, cubic constants in Reference 21 converted to mdyn/A*, uncertainties .&-I,
30.
SF6
’
F6
set I1
ref 21
111 122 133 144 155 166 134 222 233 244 255 266 234 335 345 356 445 456 566
-92 -6 1 -137 -20 -30 -5 -6 -28 -153 -1 27 -18 -3 -12 13 1 -23 -2 1 42
-60 -72 -1 10 -32 1 13 0 -56 -168 -4 2 -5 20 -1 5 -24 4 -32 13 7
29.0 24.0 set IIa 29.0 expt 30.3 (5)
SeF6 set I1 -87 -67 -99 -8 -16 4 -9 -40 -1 23 -10 14 -19 -17 -16 4 -1 -17 -16 35
TeF6 set I1 -9 1 -8 1 -101 5 -4 14 -5 -54 -136 -14 4 -2 1 -1 3 -1 2 -5 -8 -14 -13 28
138
0.29
-0.21
2.15
1.95 (46) 1.72 2.52‘ 2.29 2.75 ( 1 7 )
138
0.29
-0.21
31.6 26.8
220
0.17
-0.33 2.64
1.70 (66) 1.54
30.8 26.3
352
0.08
-0.44
3.10
1.48 (88) 1.39
20.1 16.4 20.3 (30)
109
0.31
-0.63
1.04
1.98 (68) 1.32 2.0 (4)
9.7 7.9 9.7 (5)
233
0.15
-1.22
1.44
0.76 (36) 0.54 0.72 (13)
SeF, set I1
set I TeF6 set I1
set I cF4
set I1 set I
expt SiF4
TABLE I V Cubic Constants btjk ijk
set I1 set I
set I1 set I expt
‘x in A, Ci in A-’. All results at 1700 K except for SiF, where the temperature is 900 K. Pure bend cubic constants from POS model, n = 4, ref 7 except for SF6 set IIa where constants are from EHMO calculations normalized to experimental quadratic field. Stretch and stretch-bend cubic constants, KBFF. Uncertainties (in parentheses) as discussed in text. Experiment, ref 1 and 4, where temperature was ~ measured ~ and ( x ) , ~was inferred from (x2), where ( x ) ~ -( x )was calculated from set 11. Uncertainties represent 3a. “Morse” contributions are from pure stretch cubic constants and include F3 as well as the Morse a. Bend contributions are from the curvilinear pure bend cubic constants. eThe principal difference between sets I1 and IIa is not that I1 is based on the POS model and IIa on a molecular orbital calculation. It is that set IIa corresponds to an EHMO (frozen H,,) calculation found to correlate with n = 6 in the POS model whereas set I1 used n = 4, a value found to correlate with ab initio MO calculations with self-consistentmatrix elements.
’In cm-I.
the first, set I, values of the nonbonded force constants F’, F, and F3 were calculated directly for fluorides from a fluorine-fluorine potential function due to W ~ e h l e raugmented ,~~ slightly by assuming that each fluorine bears 0.15 excess electrons. For chlorides a Cl-Cl interaction was constr~cted.’~In set 11, alternatively, the ratios F’:F and EF3 were set at the values computed from the nonbonded functions and the absolute values were adjusted to fit the quadratic constant, f34. In each case the remaining KBFF constants were adjusted to fit the known quadratic constants in accordance with the governing relations of ref 9 and 31. Values of the Morse potential constant a associated with bond stretching (not to be confused with the skew parameter a) were taken from the tabulgtiots of-Herschbach and La~r!e.~* Pure bend cubic constants fU5, f456rj-566 for octahedral and f222,f2- for tetrahedral molecules were estimated as outlined in ref 18 with the aid of the POS model ( n = 4). Bond lengths for tetrahedral molecules were taken from ref 25 while those from octahedral molecules were from individual s t ~ d i e s ’ ~corrected, -~~ approximately, to an re basis.
-
(29) Woehler, S. W.; Bartell, L. S., unpublished research described in: Bartell, L. S.; Doun, S . K.; Goates, S.R. J. Chem. Phys. 1979, 70, 4585. (30) By applying Slater’s rules updated by the orbital exponents of Clementi, E.; Raimondi, D. L. J . Chem. Phys. 1963, 38,2686, a CI-CI potential energy function was derived from the Ar-Ar function of Hill, T. L. J. Chem. Phys. 1948, 16, 359. The result is V&r) = 788 exp(-3.247r) - 15.8r4 with r in A, Vin aJ. (31) Kim,H.; Souder, P. A.; Claassen, H. H. J. Mol. Specfrosc.1968.26, 46.
(32) Herschbach, D. R.; Laurie, V. W. J . Chem. Phys. 1961, 35, 458. (33) Bartell, L. S.; Doun, S. K. J . Mol. Srrucr. 1978, 43, 245. (34) Bartell, L. S.; Jin, A. J . Mol. Srrucr. 1984, 118, 47. (35) Gundersen, G.; Hedberg, K.;Strand, T. G. J . Chem. Phys. 1978,68, 3548.
Cybic constants c $ were ~ ~ derived for octahedral molecules from theflrk values by eq 3-5. Results are listed in Table IV where they are compared, in the case of SF6, with Heenan’s values2’ The only anharmonic components that went into Heenan’s cubic constants, however, besides the nonlinear transformations between internal and normal-coofdinates, were_the Morse a, and experimental estimates 0ffi34, f144,f335, and fU5,listed in-TableJII. As a check of our program we introduced Heenan’sfJ andf;rk, and reproduced his exactly. Effective cubic constants 4,,:, designed to yield a quantum probability density via eq 7 , were then calculated by the expressions of ref 18. Finally, the skew parameter ci of eq 20 was calculated. Results for ci and some of its components are listed in Table V for molecules at experimentally studied temperatures. Values of ci calculated for a series of tetrahedral and octahedral molecules at various temperatures are given in Table VI. Some idea of the uncertainty in ci corresponding to the uncertainty in the estimated force field can be formed by comparing the results of calculations based on sets I and 11. In ref 1 sets I1 were preferred in accounting for thermal expansions of molecules. Our crude method for assessing the uncertainties was to assign 20% errors to the Morse parameter and to the bending anhannonicity and 50% to the KBFF remainder. Results are given in parentheses in Table V. IV. Discussion
Prior investigations have shown that the KBFF gives a good account of effects of vibrational motions on mean bond lengths. While this success hardly demonstrates that the same type of model field will also reproduce asymmetries of nonbonded distribution peaks, the present results give some reason for optimism. Since the pure stretch and stretch-bend contributions are often modest, an imperfect modeling of these may suffice. The pure bend anharmonicity is important. Even though it had not been adequately characterized for polyatomic molecules before the
2548
Stanton and Bartell
The Journal of Physical Chemistry, Vol. 89, No. 12, 1985
TABLE VI: Asymmetry Constants . i and rms Amplitudes of Vibration I , Calculated for Geminal Nonbonded Distances at Various Tempertures' temv. K set 400 900 1400 2000 1.94 1.80 2.00 I1 1.39 CF, 1.30 0.96 1.21 1.33 I 0.93 1.11 0.77 101, 0.57 0.80 0.79 0.76 I1 0.64 SiF, 0.57 0.56 0.54 1 0.46 1.38 1.12 0.79 1.64 101, 0.64 0.62 0.54 0.65 I1 GeF, 0.36 0.35 0.34 0.29 I 1.72 2.05 0.96 1.39 101, 1.79 1.73 1.77 1.51 I1 CCl, 1.88 1.91 1.60 1.84 I 1.64 1.11 1.37 0.77 101, 0.92 0.93 0.93 I1 0.86 SiCI, 0.57 0.57 0.58 I 0.54 2.19 1.48 1.84 1.oo 101, 0.79 0.80 0.76 0.80 GeCI, I1 0.40 0.40 0.40 I 0.38 2.52 1.70 2.1 1 101, 1.01 1.19 1.21 1.18 1.20 I1 SnCI, 0.80 0.80 11' 0.81 0.78 0.21 0.21 I 0.20 0.19 2.08 2.59 3.09 1.40 101, 1.81 2.08 0.84 1.51 I1 SF6 1.41 1.63 1.80 0.82 I 1.22 0.86 1.os 0.64 101, 1.77 0.97 1.63 1.45 SeF, I1 1.49 1.59 0.93 I 1.35 1.56 1.33 1.08 0.78 101, 1.52 1.45 1.36 1.06 TeF, I1 1.41 1.36 I 1.01 1.29 1.69 1.37 2.03 0.96 101, " 6 in k', I , in A. Pure bena cubic constants according to POS model, ref 7 with n = 4; other cubic constants according to KBFF, set I1 or set I; SnCI4 set 11' adjusts set I1 constants as discussed in text.
present series of studies of hot molecules, it has emerged as a highly systematic, readily predictable quantity. The most complete experimental information available on cubic constants for an octahedral molecule, SF6, is due to Heenan.21 4 s ca_nbe seen in Table 111, the observed stretch-bend c o n ~ t a n t s f ifi44, ~ ~ , and f 3 3 5 , with uncertainties comparable to or larger than the values, are not incompatible witJh the KBFF constants. The one pure bend constant measured, f445, is better established and agrees within experimental error with our model value. Heenan's &,,, for 32SF6, in Table IV, being nonzero in most cases by virtue of the nonlinear transformation between internal coordinates and normal coordinates, are on the whole not strikingly different from those of our model field. Indeed, they predict a thermal expansion coefficient for the S-F bond length that is only slightly less than that for ours. Yet they yield a value for B(FFds) less than 1/5 the observed and presently calculated values. This is because of the incomplete incorporation of information about bending anharmonicity in Heenan's field. Values of the skew parameter B and some of its principal components are listed in Table V for the geminal F-F distributions in SF6, SeF6, TeF6, CF4, and SiF4. Some understanding of the trends can be gained by noting that the mean-square breadth IFF2 of the distribution is the sum of ib2 due to bending motions and I?, due to stretching. All of the bonds in the series are quite stiff to stretching deformations. As the lengths of the bonds increase, then, the total mean-square amplitudes increase primarily because 12 increases. Within the context of the physical pictures sketched h ref 5 and 11, it is simple to see, as the central atom gets larger, 1 ,iereby increasing the i2/12ratio, why the contribution rib& from pure bendJ,k must increase while the contributions heansfand iM(from the nonlinear transformations and stretching anharmonicity) must decrease algebraically. It is less evident which contributions prevail. Clearly, the general magnitude of ri tends to be dominated by the bending anharmonicity, a quantity which has received little
TABLE VII: Coordinates of Atoms in AX, atom X Y A 0 0 XI 0 0 x2 0 0 x3 R 0 x4 -R 0 x5 0 R x6 0 -R
Z
0 R
-R 0 0 0 0
, R represents the equilibrium bond length.
TABLE VIII: Stretching Symmetry Coordinates for AX6
internal coordinates" species
coord
r, AI, SI 1 S2" 2m E, S2b 0 P s3, Ti" S3X 0 S3Y 0 " 1 = 6-1/2,m = 12-II2, n =
r, 1 2m 0
-P 0 0
rl
rp
1
-m
1
-m
n n 0 0 P - P 0 0
r5
1 -m -n 0 0
r6 1 -m
-n 0 0
P - P
p = 2-'/*, R is the equilibrium bond
length. attention in the vibrational spectroscopy of polyatomic molecules. That the present approach properly models the radial distribution peak shape is corroborated by the fair agreement between the calculated and observed second and third moments of the peaks. (Experimental second moments are not listed explicitly in Table V because, of necessity, they agree with the calculated moments at a given T. Experimental temperature was inferred from (x*). For the results listed in Table VI it is fair to ask whether the temperatures, which range from 400 to 2000 K, are high enough for eq 7 to be valid. Some inference can be drawn from the fact that d calculated for a Morse oscillator via eq 7 approaches to within 5% of the perturbation value at a temperature T* = hv/2k and rapidly improves as T increases. It is plausible to introduce T*,and T*b corresponding, respectively, to the highest stretching and bending frequencies of a polyatomic molecule. Presumably, T*b is a more realistic lower limit for the geminal nonbonded 6 value, which is often not sensitive to stretching anharmonicity, while T*,may be reasonable for bond distributions. Only for C F 4 ( P b= 455) and SF,(T*b = 446) is T*b greater than 400 K, and only for C F 4 ( P , = 924) is P,in excess of 900 K. In m a t cases (cases where the Urey-Bradley quadratic constant f34 calculated from fundamental frequencies agrees more or less with t h e h 4 derived with the aid of auxiliary information such as isotope shifts and Coriolis coupling constants) the POS pure bending cubic constants (adopted) are of the same magnitude as the corresponding KBFF constants (not adopted), and ri values for sets I and I1 are of roughly comparable magnitudes. In these cases set I1 is preferred because its KBFF components gave a better account of the experimental bond lengths and asymmetries in those hot molecules for which data exist. As the atomic number of the central atom increases, however, the "Urey-Bradley character" of the experimental force field appears to deteriorate and the Urey-Bradley value forf,, drops below that of the general valence force field.25 The discrepancy becomes acute for the GeF4 and SnC14force fields tabulated in ref 25 where the general valence constants f34 seem anomalously high in comparison with the other examples. In these cases the largeh4 are responsible for anomalously large values of F and F3 (the KBFF constants) and of d in set 11. A later more extensive analysis21 of GeF, by Heenan yielded a value of fM entirely consistent with the others in the series and much closer to the Urey-Bradley value; therefore Heenan's field was adopted for GeF4. Because of the possibility that the anomalous result for SnC14 is due to a similar experimental artifact, a field 11' was constructed by adjustingfN to 0.099 mdyn/A, a value in line with the trends of the other tetrachlorides. As discussed elsewhere6 and reinforced by the smallness of the set I constants, the POS valence-shell repulsions and optimal KBFF
The Journal of Physical Chemistry, Vol. 89, No. 12, 1985 2549
Internuclear Probability Distributions TABLE IX: Bending Symmetry Coordinates for AXs
internal coordinatesa species
coord
a13
a15
a14
a16
"23
a25
a24
a26
a35
a54
ff46
ff62
TI"
Sb
-4
-4 4 0
-4 0 4
4 -4
9 0 -9 0
0 -9 -4
0 -9 4
0
n
S5Y
n
0 4
0 4 -9 -n 0 0 0 4 4
0
s5x
4 4 0 0 0
4
S4Y s5,
-4 -4 0 0
s4x
T2, T2"
s6x
-4 4
'6~
0
S6z
0 -4 0
0 0
-n
-4 -4
0 4 0
0
9
-n
0 -4
0 0 0
0
-n 4 4 0
-n
0 -4 0 -4
n 4 -4 0
0 4 0
n 0 -4 0 4
n 0 0
0 -4
4
4 4 n
0 0 0 4 -4
-n
0 0
0 -4 -9
'n = R / 2 , 4 = R/S1l2.
frequencies could materially augment the analysis of anharmonicity. It is likely, however, that information in this area will prove to be easier to extract from quantum calculations than from experiment. As information accumulates on radial distribution peak shapes it may become possible to estimate ii values without carrying out calculations of the present sort. Values encountered to date range from small values to over 2 A-l. If the associated skewness is not taken into account in diffraction analyses, derived internuclear A or, sometimes, distances may be distorted by the order of A or better, it is more. Since precisions can approach 3 X of practical interest to acquire a better understanding of this manifestation of anharmonicity. Even if proficiency in this application does not come soon, however, the observations of distribution shapes to date have served a useful purpose. They have called attention to a previously unrecognized regularity in the potential energy surfaces of
TABLE X Coordinatesa of Atoms in AX,
atom
X
Y
z
A
0 a -a a -a
0 a -a -a a
0 a
XI x2 x3
x4
a
-a -a
'a = R/3Il2. TABLE XI: Stretching Symmetry Coordinates for AXI
internal coordinates' species
coord
AI T2
rl
rl
r2
SI
S
S
S
s3z
S
S
S
s3x
S3Y
S
-S
-S
-S
r A
S
-S
S
-S
S
-S
a s = ]I2. TABLE XII: Bending Svmmetrv Coordinates for AX,
internal coordinates" species E
coord
s2* SZb
T2
aI3
2t 0
-t
S4z
u
n 0
SPx
0
u
0
0
s, at
a12
aI4
a24
a34
-t
-t
-t
2t
-n
-n
n
0
0 0 u
a2,
0 0 -u
0 -u
0
-
u
0 0
= R/12'I2, n = R/2, u = R/2Il2.
repulsions seem to be stronger than simple atom-atom nonbonded repulsions. A spectroscopist, recently commenting on the limited progress in spectroscopic investigations of anharmonicity, remarked that what is needed is a reasonable way to model other aspects of anharmonicity in the way one can model stretching anharmonicity by a Morse oscillator model. Whether or not the KBFF is adequate for other components than stretch, there is reason for optimism that the POS model of ref 6 and 7 gives a fair representation of bending. Furthermore, it appears that simultaneous refinement of electron diffraction data together with spectroscopic
Acknowledgment. This research was supported by the National Science Foundation under Grant No. CHE-7926480. We thank Dr. Richard Heenan for permission to cite his unpublished data. We gratefully acknowledge a generous allocation of computing time from the University of Michigan Computing Center.
Appendix For trivial historic reasons the conventions adopted for the symmetry coordinates of octahedral molecules followed those of the Reading School (cf. Heenan, ref 21), while those for tetrahedral molecules followed, instead, the Japanese school (cf. ref 12). To avoid confusion we list the conventions explicitly in Tables VII-XII. Registry No. SF,, 2551-62-4; SeF6,7783-79-1; TeF6, 7783-80-4; CF4, 75-73-0; SiF4, 7783-61-1; GeF,, 7783-58-6; CCI,, 56-23-5; SKI4, 10026-04-7; GeCI,, 10038-98-9; SnCI4, 7646-78-8.
Supplementary Material Available: Detailed outlines in the form of computer codes for the transformations of cubic force constants from bases of curvilinear to rectilinear symmetry coordinates and normal coordinates, and the calculations of the moments and skewness of distributions characterizing geminal nonbonded atoms X-X in AX4 and AX6 molecules (23 pages). Ordering information is given on any current masthead page.
