J. Phys. Chem. 1990, 94, 5586-5589
5586
however, RBD and UBD differ, the former giving superior results at the highest correction level. Acknowledgment. E.S.R. thanks the donors of the Petroleum Research Fund, administered by the American Chemical Society, for financial support. 6. Appendix
As mentioned in the text, the evaluation of E&(II) can be simplified by using a series of intermediate arrays. Initially, It is convenient to define the intermediate products yl-y8: y 1 (b,e) = C 4 be
( A .1)
y2(j,m) =
(A.2)
ija
iab
a$ba$,
y3(j,b,m,c) = Ca$bayk
(A.3)
y4(ij,k,n) = Ca$bagb,
(A.4)
y5(a,b,cf) = CGbaf
(-4.5)
y6(k,l,eJ) = Ca%cdI(ej7
(A.6)
y7(c,d,m,n) = Cagf(mnllk1)
('4.7)
y8(k,e,m,e) = Caif(mdll1e)
(A.8)
ia
ab ij
cd
kl Id
Note that many of these arrays are already available during the course of the evaluation of the lower order energies. These arrays can then be used to define further intermediate arrays y9-yI4: y9(k,l7eJ)= -%Cy1(b,e)d&+ Y4Cy30',b,k,e)ay( A . 9 ) b
lb
Ab Initio Prediction of the Geometry, Vibration Properties, PolarlzablNties, and First Hyperpolarlrabilttles of Phosphaethynet John E. Bloor* and Jianguo Yu Chemistry Department, The University of Tennessee, Knoxville, Tennessee 37996- I600 (Received: November 10, 1989; In Final Form: January 30, 1990)
Results of ab initio calculations are presented, using the GAUSSIAN86 code, on the geometry, harmonic vibration frequencies, vibration intensities, the dipole moment, the dipole polarizability tensors, and the first hyperpolarizability tensors for phosphaethyne. It is shown that if a large enough basis set is used, satisfactory results are obtained for the geometry and frequencies by including electron correlation at the MP2 level. However, with a doubler quality (DZZP) basis better results for the geometry and frequencies are obtained by using the CCD or CISD methods. Multiple sets of polarization functions are shown to be necessary to get accurate values of the dipole moment and the polarizabilities. In particular, it is shown that whereas two carefully chosen sets of D polarization functions can give reasonable results for the dipole polarizability tensors, this is not the case for the first hyperpolarizability tensors. The latter require a set of more than four sets of diffuse D functions before stable results are obtained.
Introduction During the past three decades a number of very sophisticated but "user friendly" computer programs have been developed which enable the nonspecialist to perform a b initio calculations on chemically interesting systems. Among the most successful and most easily available programs in this respect have been the Presented at the International Conferencein Honor of Professor John A. Pople, Oct 16-19, 1989. Center for Computational Quantum Chemistry, University of Georgia, Athens, GA.
0022-3654/90/2094-5586$02.50/0
programs developed by J. A. Pople and collaborators.' In applying such programs to large systems, it is necessary to strike a balance between the need to get reliable and meaningful results on the one hand and computational resources
GAUSSIAN set of
(1) Frisch, M. J.; Binkley, J. S.;Schlcgel, H. B.; Raghavachari, K.; Melius, C. F.; Martin, R. L.; Stewart, J. J. P.; Bobrowicz, F. W.; Rohlfing, C. M.; Kahn, L. R.; &frees, D. J.; Sccger, R.; Whiteside, R. A,; Fox, D. J.; Fleuder, E. M.; Pople, J. A. Gaussian86 (IBM 3090 version); Carnegie-Mellon Quantum Chemistry Publishing Unit: Pittsburgh, PA, 1984.
0 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94,No. 14, I990 5587
Ab Initio Calculations on Phosphaethyne TABLE I: HCP Geometry (in angstroms) for Different Basis Sets HF MPZb CCD type CP CH CP CH CP CH Expa 1.540 1.069 1.540 1.069 1.540 1.069 D95 1.548 1.061 1.591 1.085 D95(D) 1.524 1.066 1.558 1.082 D95(2D) 1.514 1.069 1.546 1.085 D95(2D,P) 1.514 1.063 1.563 1.074 1.547 1.074 D95(2DF,P) 1.556 1.073 1.539 1.073 D95(2D,P) TZ2P TZ4Pd
'Percentage error =
-
(albeorwaXp)/w,,, X
r e q d for wI errof 1278.3 1386.6 a 1445.9 13 1464.8 I5 1465.3 15
100. bReference IO.
i . 5 ~ 1.068~ 1.548 1.544
1.069 1.066
1 . 5 5 4 1.082' "From ref 6. bMP3 and MP4 results for the D95(2DF,P) basis in angstroms were 1.537 and 1.071 and 1.561 and 1.077, respectively, for the CP and CH bonds. CThe Dunning/Huzinaga 5S3P set was used; H and C polarization functions as in ref 2. For phosphorus the two D exponents were 10.75 and 0.25. dThe Dunning/Huzinaga 5S3P set with four sets of energy-optimized D polarization functions given by Feller et aI.* For the hydrogen, 3P polarization functions of exponents 4.15, 1.06, and 0.27 were added to the 3 s set. eCISD. 'CEPA-l(B) (ref 6).
on the other. Compromise always has to be made between the size of basis set used and the amount of electron correlation that is necessary to include to give reliable results for the particular problem under investigation. It is now well established that for single-bonded molecules containing atoms of the second period (first long row) of the periodic table, geometries can be obtained within 0.003 A and harmonic frequencies can be obtained without scaling to within 2%.U The situation is less clear for molecules containing multiple bonds,* although it has been suggested that a triple-{ basis with two sets of polarization of functions (TZ2P) will give bond lengths within 0.005 A and frequencies within 2% at the MP2 level. However, very little systematic work of a similar nature has been done on molecules containing atoms from the second and higher period^.^ In previous work4J it was found that, for the hydrides of these atoms, good results can be obtained at the MP2 level with a 6-3 11G* basis set. For sulfur dioxide, which contains multiple bonds, it was found to be much more difficult to get good molecular geometries by using Mdler-Plesset perturbation (MP) theory! However, for this molecule, the CCD and CISD methods did give satisfactory results for both the geometry and the vibration frequencie~.~ In the present paper we present the results of applying different methods available in the GAUSSIAN 86 program' to the calculation of the geometry, the harmonic frequencies, the dipole moment, and the infrared and Raman intensities of another multiple-bonded molecule containing a third-period atom, namely phosphaethyne (HCP). We also present results for the nonlinear optical properties (dipole polarizabilities and first hyperpolarizability) using multiple diffuse functions on all three atoms at the experimental geometry. Previously, Botschwina6and Botschwina and Sebald' have reported CEPA-1 calculations on this molecule, but in calculating the vibration frequencies they used a theoretical potential energy function corrected by reference to the experimental geometry and fundamental vibration frequencies. Results and Discussion Geometry. calculation^^^ on HCN at the Hartree-Fwk (HF) level using a standard's8 D95 Dunning/Huzinaga double-{ set consisting of a 4s2p contraction of a 9s5p primitive set gives bond (2) Handy, C.; Gaw, J. F.; Simandiras, E. D. J . Chem. Soc., Faraday Trans. 2 1987,83, 1517. (3) Amos, R. D.; Gaw, J. F.; Handy, N. C.; Carter, S. J . Chem. Soc., Faraday Trans. 2 1988, 84, 1247. (4) Bloor, J. E. Inr. J . Quantum Chem., Symp. 1989, 23, 187. ( 5 ) Bloor, J. E. Unpublished results. ( 6 ) Botschwina, P. Chem. Phys. 1983, 81, 73. (7) Botschwina, P.; Sebald, P. J . Mol. Specrrosc. 1988, 100, 23. (8) Feller, D.; Boyle, C. M.; Davidson, E. R. J . Chem. Phys. 1987, 86, 3424.
TABLE 11: Hartree-Fock Harmonic Vibrational F HCP w, errof w, erroP Expb 3216.9 674.70 HF/D95 3589.5 12 783.6 16 HF/D95(D) 3565.7 11 808.6 20 HF/D95(2D) 3442.6 7 774.0 15 HF/D95(2D,P) 3529.0 IO 792.2 17
TABLE III: Effects of including Correlation and of Functions on HCP Vibration Frequencies w1 errof w, erroP HF 3529.0 IO 792.2 17 3382.3 5 MP2 633.0 -6 2 3380.0 5 MP2(F) 690.5 7 3424.5 6 MP3(F) 721.0 5 a MP2 TZ2P(F)b 3377.2 730.9 5 3 MP2 TZ4P(FIb 3382.1 692.3 7 2 3453.1 CISD 688.6 5 3391.0 CCD 650.1 -4 6 712.7 5 3393 CCD(F)
" Percentage error in all cases. in Table I.
including F wI
1465.3 1239.7 1252.0 1376.8 1276.4 1263.5 1375.5 1325.2 1341
error" 15 3 -2
a
9 -1
a
4 5
Basis sets as described in footnotes
distances considerably shorter than the experimental values. Similar HF/D95 results for HCP (Table I) are fortuitously in reasonable agreement with experiment. Also, in both cases, the agreement with experiment is decreased by the addition of polarization functions. Applying electron correlation a t the MP2 level lengthens the bonds in HCP so that even for the MP2/ D95(2DF,P) calculation the bond distance is larger than experiment by 0.01 A for the triple bond and 0.004 A for the CH bond. However, at the coupled cluster (CCD) level, this basis set gave the triple bond too short by only -0.001 A and the C-H distance too long by +0.004 A. At the CISD level, the triple bond was too short by 0.005 A and the C-H bond differed from experiment by only 0.001 A. When correlation is treated a t the MP3 level the agreement with experiment is much improved (Table I), but on going to the MP4STDQ level the bonds again lengthen to 1.0561 and 1.077 8, for the triple and single bond. When the Dunning TZ (5s,3p) contraction was used with a set of 2D polarization functions on both the carbon and phosphorus, then, as predicted by Amos et a1.: the bond distances (row 8 of Table I) are in better agreement with experiment. Finally, in row 9 are the results of a very large basis set calculation using a set of energy-optimized 4D type polarization functions on the carbon and the phosphorus and a set of 3P polarization functions on hydrogen.* Both the geometry and, as discussed in the next section, the frequencies obtained by this set are excellent, but with a basis set this large it becomes impractical to perform vibration frequency calculations on molecules much larger than HCP. Vibration Frequencies. Pulay et a1.9 have previously shown that, in order to correctly obtain the dominant contribution (the nuclear core-core terms) to the quadratic stretching force constants, it is necessary to use a geometry close to the experimental geometry. They* reported results at a standard geometry, but the vibration frequencies reported in Table I1 were calculated for each method at the geometry predicted by that method. The HF results demonstrate that unless the geometry is given correctly, one cannot expect to improve the vibration frequencies by improving the basis sets. There is considerable improvement on including electron correlation (Table 111). However, the good agreement with experiment for the MP2 calculations is somewhat fortuitous because for bond distances that are too long there is an artificial lowering of the vibration frequencies which are normally too high. The net result is that the frequencies from the MP3 calculations, even though the predicted geometry at this level is considerably (9) Pulay, P.; M,.Jung-Goo; Boggs, J. E. J. Chem. Phys. 1983.79, 3382. (10) Strey, G.; Mills, I. M. Mol. Phys. 1973, 26, 129.
Bloor and Yu
5588 The Journal of Physical Chemistry, Vol. 94, No. 14, 1990 TABLE IV SIllrltirHy to M a k l of Infrud (I, im km/mol) Vibration Freauencv .ad R.nmn Intensib (S, in A4/rmu) of HCP
TABLE V Dipole Pduivbilitim ud First Hy)crpohrizrMfitka (8)
of HCP
~
method' HFlD95 HF/D95(D) H F/D95( 2D) HF/D95(2D,P) MP2/D95(2D,P) MP2/D95(2DF,P) MP2ITZ2P MP2/TZ4PF CEPA- 1 (ED)
11
12
13
s,
s2
s 3
calc
22.7 16.9 24.7 21.9 19.1 22.2 16.7 15.6 9.2
131.7 102.7 95.1 91.8 75.4 74.0 84.4 74.8
0.9 0.4 1.5 1.4 0.1 0.3 0.06 6.73 OS3
0.75 0.75 0.75 0.75
0.30 0.24 0.18 0.18
0.30 0.32 0.33 0.31
1
'Numbering system as for the frequencies in Tables I1 and 111.
better than the MP2 geometry, all have a greater error than the error for the MP2 results, obtained with the same basis set. For a doubler basis the best results are found for the CCD calculation although the error is still slightly greater than 5%. When a TZ2P contraction is used (row 5), the two stretching frequencies change very little but the bending frequency increases, making the agreement with experiment worse. On adding more polarization functions the agreement with experiment was much improved (row 6), but this set is too large to be used for larger molecules. At least part of the remaining error is due to the fact that experimental frequencies,' uncorrected for anharmonicity, are used for the comparison in Tables I1 and 111. The effect of including sets of F polarization functions on the carbon and the phosphorus was investigated at the MP2 and CCD levels. As was found for acetylene by Simandiras et the addition of these functions is important for the bending frequency which changed by up to 60 cm-' with virtually no corresponding change in the multiple bond stretch and only small changes in the C-H stretch. Vibrational Intensities. Infrared intensities are often very sensitive to changes in the basis sets and to the inclusion of electron correlation.I+ The results of the present calculations are given in Table IV. For the I, and I3 there are major changes on adding polarization functions to the heavy atoms. Correlation at the MP2 level changes I2 further so that the final result is reduced to 56% of the intensity calculated at the HF/D95 level. I3 however changes in a unpredictable erratic manner, as has been found for other molecules with low inten~ities."~ The Raman intensities are available only at the Hartree-Fock level (Table IV). The inclusion of polarization functions does not affect the SIand S3values, but the S2intensity decreases by 60% on the addition of 2Ds on the heavy atom and a P set on the hydrogen. Unfortunately, there are no experimental intensity measurements on this system so that it is not possible to comment on whether the values are reasonable or not. Dipole Moment. The dipole moment is a very sensitive index as to whether enough polarization functions have been added to provide a satisfactory electron distribution. Earlier calculations12 on phosphaethyne using relatively small basis sets predicted a dipole moment considerably larger than experiment.I2 For example, the value from a 6-31G** (0.619 D) is 1'/, times too large compared to that of experiment (0.390 D)." The H F D95 and D95(D) calculations give a similar high value, and it is not until at least two sets of D polarization functions have been added that the result is reasonable. By the time a multiple set of D functions have been added, as is required for the polarizabilities (e.g., a HF/D95++6D calculation gave a value of 0.3698 D), a result is obtained which is in error by only 5%. The necessity for more than one set of D polarization functions for calculating a reasonable dipole moment for molecules containing third-period atoms was found for other molecules, e.g., for H2S$ S02,4and HCl.5 Nonlinear Optical Properties. Although there is a very great contemporary interest in understanding how molecules respond ( I 1) Simandiras, E. D.; Rice, J. E.: Lee,J. J.; Amos, R. D.; Handy, N. C. J. Chem. Phys. 1988.88, 3187. ( 1 2) Thomson. C.; Ellam, P. Theor. Chim. Acta 1982, 62, 8 1. (13) Tyler, J. K.J. Chem. Phys. 1964, 40, 1170.
2 3 4 5 6
basisb D95++ (73/39) 3DF. 2PD (135/101) 6D, 3P2D (166j132) TZ, 6D, 4P2D (161/122) TZ, 4D, 4P2D (141/102) D95++2DC
%x.
18.88 30.26 31.14 31.06 30.33 27.51
ayy
8,
8'YY
33.96 -19.84 48.18 71.23 29.67 47.68 45.46 10.69 47.95 44.77 9.83 47.98 47.70 +12.79 -15.15 25.11 -2.95 47.72
'All quantities are in atomic units. The X axis is the internuclear axis. bThe first number inside the parentheses is the number of Gaussian functions used; the second number is the number of contracted functions. "Performed with a set of P and a set of D diffuse functions both with exponents 0.05 as used in ref 15 on benzene.
to external fields, there have been few ab initio attempts to calculate the molecular parameters describing these phenomena." Also, most of the calculations on the effect of electric fields have been confined to atoms or to small molecules containing atoms of the first two periods (i.e., first-row atoms) of the periodic table.14J5 As part of a systematic studys of the nonlinear optical properties of molecules containing third-period atoms, we have used the standard coupled Hartree-Fock method implemented in G A U S S I A N ~and ~ ~ other programs" to calculate the dipole polarizabilities (an and sty) and the first hyperpolarizability (Ox=, tensors for HCP using a number of different basis sets. These results are summarized in Table V. In performing similar calculations on the hydrogen halide^,^ it was found that, in order to get reliable results, multiple D functions (more than three) had to be added on both the heavy atoms and the hydrogens in addition to diffuse S and P functions. As a result of this experience, two large basis sets were chosen for the HCP calculations. Also, a number of calculationsdesigned to demonstrate the sensitivity of nonlinear optical parameters to the basis set were performed. The first calculation reported in Table V gives the result for a standard D95++ basis set without D type polarization functions. In the second calculation a standard1 3Df,2PD set of polarization functions was added to the D95++ set. There was a marked change in amand an enormous change in both the first hyperpolarizabilitytensors. In the third calculation the D95++ set was supplemented by a set of 6D functions on each heavy atom and a set of 2D functions on the hydrogen. The 6D functions were chosen by taking a normal d-function exponent of 0.74 on both the carbon and phosphorus and generating from it five other functions each with an exponent reduced by 0.5 of the value of its predecessor (i.e., a reduction factor of 0.5 was used). The fourth calculation on Table V was performed using the Dunning/Huzinaga T Z basis to which was added the same set of supplementary functions used for the D95++ calculations. The results are very similar except for the component. Calculation 5 is part of a study in which we are trying to develop a smaller set of diffuse functions. A set of four D functions in which the exponents were generated from the inner D exponent by applying a reduction factor of 0.25 was used. The results for the dipolar polarizabilities were virtually unchanged, but the first hyperpolarizabilities changed considerably and even the sign changed for PxYy Finally, in calculation 6 a small set of P and D functions (one of each) similar to those recently used by Perrin et a1.16 for calculationson benzene were tried. Although the am polarizability was reduced by 13%,this basis set is probably adequate to obtain approximate polarizabilities. However, neither of the first hyperpolarizability tensors bears much resemblance to the values obtained with the larger basis sets. These results are typical of those we have found for other molecules' and demonstrate clearly that, although reasonable dipole polarizabilities can be obtained by using only a set of one or two carefully chosen D diffuse functions, a much larger set is needed before the values
a,)
a,,
(14) Dykstra, C. E. Ab Initio Calculations of the Structures and Prop erties OJ Molecules; Elscvier Science: New York, 1989. ( 1 5 ) Liu, S.-Y.; Dykstra. C. E. S . J. Phys. Chem. 1987, 91, 1749. (16) Perrin, E.: Prasad, P. N.; Mougenot. P.; Dupuis, M. J. Chem. Phys. 1989, 91, 4728.
J. Phys. Chem. 1990,94, 5589-5592
of the first hyperpolarizabilities become reliable. The values for the dipole polarizabilities for HCP are as expected considerably larger by a factor of about 2 compared to those for HCN.5J3*i7 However, the very high value obtained for the high values for the /3 tensor compared to the corresponding HCN values were unexpected and suggest that further calculations on molecules containing heavier atoms than the first two periods, i.e., second row and higher, will yield interesting results.
Conclusions The geometry optimization results support the conclusion of Amos et al.' that accurate geometries can be obtained at the MP2 level if the basis set is made large enough. However, we also find that equivalent results may obtained using the coupled cluster CCD level but with a smaller basis set. Both methods also produce the experimental vibration frequencies with an accuracy of less than 5%. The results also support the conclusions of Simundiras ~~
(17)
~~
Jam",
C. J.; Fowler, P. W. J . Chem. Phys. 1986,85, 3432.
5589
et a1.I0 on the importance of including a set off basis functions in order to get accurate values for bending frequencies in bonded systems. The calculations on the polarizabilities and the first hyperpolarizabilities show that although the dipolar polarizabilities can be predicted with an accuracy of 1&15%, using only two sets of carefully chosen diffuse functions, the first hyperpolarizability tensor is far more sensitive to the number of diffuse functions and reliable values can only be obtained by using a set of more than 4D sets of functions. The exponents of these D functions can be obtained from the innermost D function by applying a reduction factor of 0.5 for successive exponents. Acknowledgment. This research was supported by the Naval Air Systems Command through Contract MDA 9003-86-C-0245. The author thanks the very supportive University of Tennessee Computing Center staff, especially Sue Smith for implementing the GAUSSIANM program on the IBM 3090/200 and for the allocation of an extensive amount of computer time and of disk space. Registry No. Phosphaethyne, 6829-52-3.
Ab Initio Calculations on the Geometries and Stablllties of Acetylene Complexest Jianguo Yu,* Sbujun Su, and John E. Bloor Department of Chemistry, The University of Tennessee, Knoxville, Tennessee 37996- I600 (Received: November IO, 1989; In Final Form: January 22, 1990)
Structures of four different molecular complexes of acetylene, (C2H2I2,(C2H2)3, (C2H2)+ and (CZHZ)~. have been studied at both the Hartree-Fock and the M0ller-Plesset perturbation (MP2) levels. For the dimer a T-shaped structure of Cb symmetry is found to be the most stable, for (C2H2)3 a "twisted" C3, structure is favored, and for (C2H2),two structures have the same interaction energies. The one with S, symmetry seems to be more experimentally favored. For ( C Z H ~ a) ~ , nonplanar structure is predicted to be most stable. The results show that all the acetylene complexes larger than the dimer which were studied show a preference for multiple T-type hydrogen bonding.
I. Introduction The structure and properties of weakly bound molecular complexes have gained more and more attention in recent years, both theoretical and experimental. Complexes of acetylene are one of the key systems in determining the geometries and stabilities of these complexes. For the acetylene dimer, the simplest acetylene cluster, Pendley and Ewing' reported five bands using long-path low-temperature Fourier transform infrared spectroscopy. Although their spectra do not allow a complete determination of the complex's structure, the band shapes are consistent with the staggered parallel structure (S-shaped) proposed by Sakai et al.* This structure is depicted in Figure la. Miller et aL3 discussed several possible interpretations of the three main peaks observed in the infrared spectrum of the dimer and suggested that there exists more than one stable dimer structure. A second paper by Fischer et al.' showed that the bands in the IR spectra of acetylene have rather different pressure dependencies and may be associated with different cluster sizes. Recently, Bryant et al? provided the evidence for two dimer structures and assigned one to the S-shaped one (Figure la) with C , symmetry. Although the free-jet infrared absorption spectroscopy study of Ohshima et a1.6 was interpreted as evidence for an acetylene dimer complex, which is a hydrogen-bonded T-shaped , symmetry (Figure lb), the radio-frequency, structure with C microwave, and IR spectra obtained by Muenter et a1.7 were used Pttscntcd at the International Conference in Honor of Professor John A. Pople, Oct 16-19, !989, Center for Computational Quantum Chemistry, University of Georgia, Athens, GA.
0022-3654/90/2094-5589$02.50/0
to conclude that the T-shaped acetylene dimer appears not to have C, symmetry. Fraser-8 presented model calculations to interpret the large C-H stretching vibrational dependencies of the interconversion tunneling splittings and the corresponding infrared vibrational tunneling state selection rules in the acetylene dimer. For the larger sizes of acetylene cluster, Prichard et al? proposed a trimer geometry with D3,, or C3,symmetry (see Figure 2a.b) based on the infrared vibration rotation spectra. Bryant et a1.I0 conducted a rotational analysis for the infrared spectra of the acetylene tetramer and found that a puckered ring structure of S, symmetry (see Figure 3a) is consistent with the experimental data. There have been a number of previous theoretical investigations of acetylene clusters; Sakai et aL2calculated the intermolecular
-
(1) Pendlcy, R. D.; Ewing, G. F. J. Chem. Phys. 1983, 78, 3531. (2) Sakai, K.; Koide, A,; Kihara, T. Chcm. Phys. L r r . 1977, 47, 416. (3) Miller, R. E.; Vohralik, P. F.; Watts, R. 0. J. Chem. Phys. 1984,80,
5453.
(4) Fischer, G.; Miller, R.E.; Vohralik, P. F.; Watts, R.0.J. Chem.Phys. 1985, 83, 1471. (5) Bryant, G. W.; Eggers, D. F.;Watts, R. 0. J. Chem. Soc., Faraday Trans. 2 1988,84, 1443. (6) Oshima, Y.;Matsumoto, Y.;Takami, M.Chem. Phys. Lett. 1988, 147, 1.
(7) Prichard, D. G.; Nandi, R.N.; Mucnter, J. S. J . Chem. Phys. 1988, 89, 115. ( 8 ) Fraser, G. T. J. Chcm. Phys. 1989, 90,2097. ( 9 ) Prichard, D.; Mucnter, J. S.; Howard, B. J. Chem. Phys. Leu. 1987, 135. 9. (IO) Bryant, G. W.; Eggers, D. F.; Watt, R. 0. Chem. Phys. Lrr. 1988, IS!, 309.
0 1990 American Chemical Society
