J. Phys. Chem. 1995, 99, 16883-16902
16883
Ab Initio Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density Functional Force Fields: A Comparison of Local, Nonlocal, and Hybrid Density Functionals F. J. Devlin, J. W. Finley, and P. J. Stephens* Department of Chemistry, Universiry of Southem California, Los Angeles, Califomiu 90089-0482
M. J. Frisch Lorentzian, Inc., 140 Washington Avenue, North Haven, Connecticut 06473 Received: June 8, 1 9 9 9
W e report predictions of the unpolarized vibrational absorption and vibrational circular dichroism spectra of 10 chiral molecules based on harmonic force fields calculated ab initio using density functional theory. The molecules are 1, oxirane (ethylene oxide)-truns-2,3-d2; 2, methyloxirane (propylene oxide); 3, trans-2,3dimethyloxirane; 4, trans-2,3-dimethyloxirane-2-d1; 5, tmns-2,3-dimethyloxirane-truns-2,3-d~; 6, methylthiirane (propylene sulfide); 7, trans-2,3-dimethylthiirane; 8, trans-2,3-dimethylthiirane-truns-2,3-d2; 9, cyclopropanetruns-l,2-d2; 10, ~yclopropane-anti-l,2,3-d3-I-'~C. Large TZ2P basis sets are used to minimize basis set error. Three density functionals are employed: (1) the local spin density (LSDA) functional, (2) the BeckeLee-Yang-Parr (BLYP) nonlocal functional, and (3) the Becke 3-Lee-Yang-Parr (B3LYP) hybrid functional. Predicted spectra are sensitive to the choice of density functional. Spectra predicted using the BSLYP functional are in the best agreement with experimental spectra while spectra predicted using the LSDA functional are in the worst agreement. Spectra predicted using the BLYP functional are more similar to those obtained using the B3LYP functional. We conclude that the relative accuracies of the functionals are B3LYP > BLYP >> LSDA.
Introduction We report predictions of unpolarized vibrational absorption and vibrational circular dichroism (VCD) spectra based on ab initio harmonic force fields calculated using density functional theory (DFT).' Spectra are predicted for 10 small, chiral molecules. Three density functionals are employed. Our goal is to evaluate the relative and absolute accuracies of DFT harmonic force fields calculated using these three functionals by comparison of the vibrational spectra predicted thence to experimental spectra. DFT is increasingly used in the ab initio calculation of molecular properties. DFT codes are increasing in versatility, efficiency, and availability. DFT calculations exhibit an attractive ratio of accuracy to computational cost. In particular, DFT is increasingly used in calculating harmonic force fields and vibrational frequencies. The introduction of analytical gradient techniques2permitted numerically accurate calculations; very recently, the introduction of analytical second-derivative techniques3has substantially improved the efficiency of calculations. DFT frequencies are much more accurate than SCF frequencies and are comparable in accuracy to MP2 frequenc i e s . ' ~ However, ~ from a computational standpoint DFT calculations are substantially less demanding than MP2 calculations. DFT can obviously be expected to become increasingly the method of choice. The accuracy of DFT calculations depends, of course, on the density functional adopted. At this time, many density functionals are available, varying substantially in sophistication and accuracy. Broadly, they can be grouped into three classes: (1) local, ( 2 ) nonlocal, and (3) hybrid. Local functionals for the
* Author to whom correspondence should be addressed. @
Abstract published in Advance ACS Abstracts, November
1, 1995.
exchange and correlation functionals were the first to be used. Nonlocal (or gradient) corrections were then added. Most recently, a number of functionals have been introduced in which some percentage of exact (Hartree-Fock) exchange is admixed. In the most sophisticated of these "hybrid" functionals, based on the adiabatic connection method of Becke? the values of three weighting factors are determined by optimizing the fit of predicted properties to experiment. Many studies have examined the accuracies of molecular properties predicted using functionals in the first two classes. So far, studies of hybrid functionals are much fewer in number.6 The principal goal of this work is to compare the accuracies of the harmonic force fields predicted via DFT using three different density functionals for a set of small organic molecules. The density functionals are (1) a local functional, the local spin density (LSDA) functional; ( 2 ) a nonlocal functional, the Becke-Lee-Yang-Parr (BLYP) functional; (3) a hybrid functional, the Becke 3-Lee-Yang-Parr (B3LYP) functional. They span the range of functionals used to date. The LSDA and BLYP functionals have been widely used; the B3LYP functional has been introduced only recently. We compare both vibrational frequencies and intensities predicted using these functionals to experimental frequencies and intensities. Harmonic force fields are most often evaluated by comparison of predicted vibrational frequencies to experiment. However, vibrational intensities, through their dependence on vibrational coordinates, are also a sensitive function of the force field. Comparison of vibrational intensities to experiment thus provides an additional critique of the force field. The molecules we have chosen for study are a set of small, chiral organic molecules, for which, in addition to unpolarized absorption spectra, VCD spectra' have been reported. In evaluating the accuracies of the DFT harmonic force fields
0022-3654/95/2099-16883$09.00/0 0 1995 American Chemical Society
16884 J. Phys. Chem., Vol. 99, No. 46, 1995 calculated for these molecules, we compare not only predicted unpolarized absorption intensities but also VCD intensities to experiment. As will be evident, the combined use of unpolarized absorption and VCD spectra substantially enhances the reliability of the conclusions resulting from the comparison of theory and experiment. The molecules studied are 1, oxirane (ethylene oxide)-trans2,3-d2; 2, methyloxirane (propylene oxide); 3, truns-2,3dimethyloxirane; 4, truns-2,3-dimethyloxirane-2-d~ ; 5, truns2,3-dimethyloxirane-truns-2,3-dz; 6, methylthiirane (propylene sulfide); 7, truns-2,3-dimethylthiirane;8, truns-2,3-dimethylthiirane-truns-2,3-d2; 9, cyclopropane-truns-l,2-d~; 10, cyclopropane-~nti-1,2,3-d3-l-’~C. All calculations are carried out at the TZ2P basis set level-3s2p for H, 5s4p2d for C and 0, and 9s6p2d for S-in order to minimize basis set error. We recently compared experimental unpolarized absorption and VCD spectra for the chiral molecule 4-methyl-2-oxetanone to spectra predicted from DFT harmonic force fields calculated using the LSDA, BLYP, and B3LYP density functionak8 The hybrid B3LYP functional gave superior agreement between theory and experiment. The work reported here extends this preliminary study to a much larger number of molecules and substantiates its preliminary conclusion.
Devlin et al. where (I$)’ is evaluated with the origin at 0 and En&p(n)is the electronic component of the AFT of nucleus 1 in the momentum representation.’ Large TZ2P basis sets-3s2p for H, 5s4p2d for C and 0, and 9s6p2d for S-were used in all calculations. These basis sets were defined in detail previously’’ for H, C, and 0; for S , the exponents of the d functions are 0.90 and 0.30. Vibrational frequencies, dipole strengths, and rotational strengths were calculated from the harmonic force fields, APTs, and AATs. The dipole strength, D,of the ith fundamental is given byI2
~ ( 0 -1); = CI(OICuel)PIl)j12 where
hwi is the excitation energy of the ith fundamental, and the S J , ~ . ~ matrix defines normal coordinates Qi in terms of Cartesian displacement coordinates
Methods D l T harmonic force fields were calculated ub initio using analytical derivative methods, implemented in GAUSSIAN 92/ DFTS9 Three density functionals were used: (1) local spin density approximation, LSDA; (2) Becke-Lee-Yang-Parr, BLYP; (3) Becke 3-Lee-Yang-Parr, B3LYP. These functionals are standard options in GAUSSIAN 92” and have been defined in detail previously.8 The “fine” grid of GAUSSIAN 92/DFTs was used. (1= nucleus; a,p = DFT atomic polar tensors (APTS), x , y , 2 ) were calculated simultaneously with the harmonic force fields. “Semi-DFY atomic axial tensors (AATs) were calculated using the distributed origin with origins at nuclei in which
(4)
B
i
The rotational strength of the ith fundamental is given by1*
where
The use of the distributed origin gauge in calculating AATs guarantees origin-independent rotational strengths.l0 Unpolarized absorption and circular dichroism spectra were derived from calculated frequencies, dipole strengths, and rotational strengths viaI6.l7 (9)
where (M$)O is the AAT of nucleus 1 with respect to origin 0, R: is the equilibrium position of nucleus A, and I$ is the electronic atomic axial tensor of nucleus A defined
where &,bGlaXA,is the derivative of the electronic wave function of the ground state G with respect to the Cartesian nuclear displacement Xn,; alyGlaHp is the derivative of the electronic wave function of G in the presence of the perturbation H‘ = -@‘&JpHp with respect to Hp; ,Zkagis the electronic magnetic moment operator. The distributed electronic AAT, (I$)1, is evaluated with the origin at the equilibrium position of nucleus A. (I$)A tensors are calculated at the SCF level of approximation since, at this time, DFT methods for the calculation of I$ tensors have not been developed. Analytical derivative technique^'^ implemented in the CADPAC package (version 5)14 were used. (I&)A tensors were obtained using the equation”.l5
A@) = 4 whereJ(F;,V) is a normalized line-shape function. In this work, Lorentzian line shapes are employed throughout; y is the halfwidth at half-height of fi. Ideally, in predicting spectra experimental line widths would be used. Of the molecules studied here, only in one case have line widths been reported (for 2”). Consequently, all predicted spectra are obtained using arbitrarily chosen, constant line widths. In the mid-IR spectral region we use y = 6 cm-I. In C-H and C-D stretching regions we use y = 10 cm-’. These values are “typical” of experimental line widths in these regions for CS2 and CC4 solutions of molecules having no specific solvent interaction (e.g., hydrogen bonding). In comparing the peak intensities of bands in predicted and experimental spectra, it must be remembered that differences originate in part from differences in line widths.
Results and Discussion Frequencies, dipole strengths, and rotational strengths calculated for 1-10 on the basis of LSDA, BLYP, and B3LYP
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Figure 1. Experimental and predicted unpolarized absorption and VCD spectra of 1. Experimental spectra (a, e) are for C2C14 solutions.IE Calculated spectra are for B3LYP (b, f), BLYP (c, g), and LSDA (d, h) density functionals; y = 10 cm-I for all bands. VCD spectra are for (2S,3S)-1.
Figure 2. Experimental and predicted unpolarized absorption and VCD spectra of 1. Experimental spectra (a, e) are for CS2 solutions.'E Calculated spectra are for BSLYP (b, f), BLYP (c, g), and LSDA (d, h) density functionals; y = 6 cm-'. VCD spectra are for (2S,3S)-l. Note that absorption at 1034 and 1117 cm-I is attributed to impurities.'E
DFT harmonic force fields are given in Tables 1- 10. Absorption and VCD spectra obtained thence are compared to experimental spectra in Figures 1- 14. The unpolarized absorption and VCD spectra predicted for a given molecule vary with the density functional used. The variation in frequencies directly reflects the variation in the harmonic force field. The variations in absorption and VCD intensities reflect variations in both the harmonic force field and AF'Ts and AATs. Calculation and experiment can be compared at the qualitative level and at the quantitative level. The qualitative pattern of predicted frequencies and intensities, most clearly exhibited in predicted spectra, can be compared to the experimental spectra. The quantitative accuracy of predicted frequencies, dipole strengths, and rotational strengths can be assessed by comparison to values extracted through analysis of experimental spectra. Before theory and experiment can be compared, assignment of the experimental spectra must be in place. In this work, our calculations are carried out at the harmonic level and are therefore restricted to fundamental transitions. Comparison of predictions to experiment requires that fundamental transitions have been assigned, Le., distinguished from the overtone and combination transitions also present in the experimental spectra.
We therefore discuss first the assignment of the unpolarized absorption and VCD spectra of 1-10, Initially, we assign the spectra of 1-10 on the basis of the B3LYP calculations. We then examine the possibility that the assignments so arrived at might be in error, and improved, by consideration of the BLYP and LSDA calculations. With optimum assignments of the fundamentals of 1-10 in hand, the comparison of calculation and experiment then proceeds straightforwardly. Assignment of Fundamentals from B3LYP Calculations. We seek to make a one-to-one correspondence between the calculated fundamentals of each molecule and the observed bands in its absorption and VCD spectra. The assignment is that optimally reproducing simultaneously the distribution of frequencies, the pattern of relative absorption intensities, and the pattern of relative VCD intensities, including their signs. Oxirane-trans-2,3-&, 1. Absorption and VCD spectra of 1 over the frequency ranges 2950-3100 and 2150-2300 cm-' in C2C4 solution and 900-1350 cm-' in CS;! solution have been reported by Freedman et al.Is These spectra are reproduced in Figures 1 and 2. Note that absorption at 1034 and 1117 cm-' is attributed by Freedman et al. to impurities.
16886 J. Phys. Chem., Vol. 99, No. 46, 1995
Devlin et al.
TABLE 1: Calculated and Experimental Frequencies, Dipole Strengths, and Rotational Strengths of Oxirane-tran~3J-d~ (1)" LSDA BLYP B3LYP EXPb mode V D R V D R V D R V D R 15 3068 27.0 15.8 3058 47.3 20.6 3141 41.7 11.4 14 3063 7.6 -19.9 3052 8.5 -22.5 3135 7.5 53 c8.9 13 2257 9.3 10.1 2247 9.5 11.5 2310 8.8 10.8 2252 5.7 12.1 12 2245 24.4 -7.3 2238 40.3 -9.8 2299 36.4 -9.3 2232 27 - 10.4 11 1393 21.5 -9.7 1379 9.7 -9.3 1428 12.4 -9.5 (1397) (12) (-15) IO 1300 2.7 -1.6 1330 0.2 -0.8 1366 0.5 -1.2 (1339) (2.3) (-2.5) 9 1236 27.0 17.1 1206 24.6 12.1 1254 28.4 13.4 1226 29.6 24.1 8 1082 0.9 -3.8 1093 1.8 -5.4 1136 0.3 -2.3 1109 -4.9 7 1074 7.9 13.6 1082 5.4 7.5 1125 6.1 8.8 1102 8.6 11.1 6 961 121.3 -42.3 936 35.5 -21.1 975 67.9 -30.0 948 54 -29.0 5 894 16.6 -7.0 892 9.4 -3.1 928 11.8 -2.7 914 6.3 -6.2 4 881 73.0 14.9 853 141.9 4.1 893 128.1 6.9 (885) (-+5) 3 873 43.6 2.7 766 44.8 1.0 831 42.5 0.7 (817) (+) 2 733 90.8 11.5 730 123.4 9.2 759 119.0 10.8 (754) 1 641 1.4 -0.7 648 1.2 0.0 667 1.3 0.3 (673) a Frequencies 5 in cm-I: dipole strengths D in esu2cm2;rotational strengths R in esu' cm2. Rotational strengths are for the (2S,3S) enantiomer. From Freedman et a1.;I8in C2C14 solution for C-H and C-D stretching regions and in CS2 solution otherwise. Values in parentheses are from gas-phase data. Dipole and rotational strengths were obtained from solution data by curve fitting. Rotational strengths were not normalized to 100% enantiomeric excess (ee); ee values for (2S,3S)-1 and (2R,3R)-1 were estimated to be 94%.
47:: E)
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wavenum bers Figure 4. Experimental and predicted unpolarized absorption and VCD spectra of 2. Experimental spectra (a, e) are for CC4 (a, > 1325 cm-l; e, > 1300 cm-I) and CS2 (a, 1350 cm-I; e, > 1310 cm-I) and CS2 (a, 1300 cm-I) and CS2 ( < 1300 cm-') solutions.25 Calculated spectra are for B3LYP (b, f), BLYP (c, g), and LSDA (d, h) density functionals; y = 6 cm-I for all bands. VCD spectra are for (2R,3R)-8. Note that solvent absorption was not adequately subtracted in a: the CS2 absorption at -855 cm-I (indicated) is particularly prominent.
trans-2,3-Dimethylthiirane-trans-2,3-d2,8. There are no noteworthy differences. The BLYP calculations fully support the assignment of fundamentals arrived at from the B3LYP calculations. Cyclopropane-trans-I,2-d2, 9. There are no noteworthy qualitative differences between the BLYP and B3LYP calculations for 9. The BLYP calculations fully support the assignment of fundamentals arrived at from the B3LYP calculations. Cyclopropane-~nh'-I,2~J~-I-'~C, 10. In the C-H and C-D stretching regions there are no qualitative differences between the BLYP and B3LYP calculations. The BLYP calculations fully support the assignment of fundamentals 1621 arrived at from the B3LYP calculations. Assignment of Fundamentals from LSDA Calculations. Frequencies, dipole strengths, and rotational strengths calculated using the LSDA functional for 1-10 are given in Tables 1- 10. The absorption and VCD spectra obtained thence are given in Figures 1- 14. Overall, the LSDA calculations differ substantially from the B3LYP and BLYP calculations. We now discuss the differences in detail. Oxirane-fruns-2J-d2, 1. The absorption and VCD spectra predicted using the LSDA functional are very simi!ar, qualitatively, to the B3LYP spectra except for modes 3-5, whose frequency distribution is substantially different. The span for
wavenum bers Figure 12. Experimental and predicted unpolarized absorption and VCD spectra of 9. Experimental spectra (a, e) are for CzC14 solutions.26 Calculated spectra are for B3LYP (b, f), BLYP (c, g). and LSDA (d, h) density functionals; y = 10 cm-' for all bands. VCD spectra are for (1S.28-9.
these modes is reduced from 97 cm-' with B3LYP and 126 cm-' with BLYP to 21 cm-I with LSDA. The LSDA pattern of frequencies is in substantially worse agreement with that of the experimental bands at 817, 885, and 914 cm-' than is the B3LYP pattem of frequencies. However, there does not appear to be an alternative assignment providing better agreement of the LSDA calculations with experiment. Methyloxirane, 2. The absorption and VCD spectra predicted using the LSDA functional differ substantially from the B3LYP spectra. Modes 5-8 exhibit a different frequency pattern, modes 6 and 7 being much closer to modes 5 and 8, respectively, than is the case with B3LYP. The relative absorption intensities of modes 7 and 8 are reversed; the VCD of mode 5 is changed in sign. Similarly, modes 15-17 exhibit a different spacing, and the VCD of mode 17 is altered in sign. Likewise, the frequency pattem of modes 20-23 is substantially changed. With the assignment of fundamentals arrived at from the B3LYP calculations, the LSDA spectra are in worse agreement with experiment than are the B3LYP spectra, especially with respect to the pattem of frequencies. In the region above 1400 cm-', the LSDA calculations alone would probably lead to a different assignment of modes 16 and 17. Elsewhere, including the complex C-H stretching region, alternative assignments in better agreement with experiment are not apparent. truns-2,3-Dimethyloxirane,3. The absorption and VCD spectra predicted using the LSDA functional differ significantly
Ab Initio Calculation of Absorption and VCD Spectra 1600
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wavenumbers Figure 13. Experimental and predicted unpolarized absorption and VCD spectra of 9. Experimental spectra (a, e) are for CS2 solutions.26 Calculated spectra are for B3LYP (b, f), BLYP (c, g), and LSDA (d, h) density functionals; y = 6 cm-' for all bands. VCD spectra are for (1S,2s)-9.
from the B3LYP spectra. The frequency distribution of modes 7-10 is substantially altered. While modes 11 and 12 are similarly spaced, their VCD intensities are reversed in sign. While the pattern of absorption intensities of modes 13-16 is not qualitatively different, the pattern of VCD intensities is completely different. Substantial differences also occur for modes 18-25. Mode 18 is reduced in absorption intensity. Mode 25 is much more separated from modes 21-24 and has greater absorption and VCD intensity. The LSDA spectra are in substantially worse agreement with experiment, assuming the assignment of fundamentals arrived at from the B3LYP calculations. However, direct comparison of the LSDA and experimental spectra does not provide an alternative assignment for any fundamental which would provide better agreement between theory and experiment. auns-2d-DimethyIo~~~2-d1,4. The absorption and VCD spectra predicted using the LSDA functional differ significantly from the B3LYP spectra. The frequency distribution of modes 7-9 is substantially different, as are their relative absorption and VCD intensities. The sign of the VCD of mode 10 is changed. The frequency spread of modes 15-17 is reduced from 62 to 22 cm-', and the relative absorption and VCD intensities are quite different. The biggest differences are for modes 19-25. Neither absorption nor VCD spectra resemble
the B3LYP spectra. Assuming the assignment of fundamentals arrived at from the B3LYP calculations, the LSDA spectra are in much worse agreement with experiment than are the B3LYP spectra. If the experimental spectra were assigned from the LSDA calculations alone, it is likely that the assignment of modes 8 and 9, 15 and 16, and 21 would be different. The 793 cm-' band would be more naturally assigned to mode 9 with mode 8 being assigned to either 793 or 719 cm-I. Modes 15 and 16 would be assigned to the 1131 and 1109 cm-' bands, respectively-Le., to an experimental ordering opposite to that predicted. The shoulder at 1418 cm-I would be assigned to mode 22, not 21, while the bands peaking at 1448 cm-' would be assigned only to modes 23 and 24, instead of 22-24. However, even with such reassignments the agreement between theory and experiment would remain significantly worse than that for the B3LYP calculations. trans-2,3-Dimethyloxirane-trans-2,3-&,5. The absorption and VCD spectra predicted using the LSDA functional differ significantly from the B3LYP spectra. The largest differences involve modes 15-18 and 19-25. In the case of modes 1518, the pattern of frequencies and absorption intensities is not much altered from the B3LYP calculations. However, the VCD intensities of modes 16 and 17 are substantially different; in the case of mode 17 the sign is changed. In the case of modes 19-25 the differences in the distribution of frequencies, absorption intensities, and VCD intensities are all substantial. The largest differences are for modes 21 and 25. The absorption and VCD intensities of mode 21 are much reduced at LSDA while the opposite is the case for mode 25. The LSDA calculations also place mode 7 much closer to mode 8 than do the B3LYP calculations. On the basis of the assignment of fundamentals arrived at from the B3LYP calculations,the LSDA calculations, where qualitatively different from the B3LYP calculations, are in worse agreement with experiment. Agreement would be improved if modes 7 and 8 were both assigned to the 769 cm-' band, attributing the 718 cm-' band to a nonfundamental, and if the 1403 cm-' band were also assigned as a nonfundamental, leaving the location of mode 21 undefined. However, there is no apparent reassignment that improves the agreement for modes 16 and 17 or for mode 25. Methylthiirane, 6. The absorption and VCD spectra predicted using the LSDA functional differ significantly from the B3LYP spectra. The largest differences are for modes 6-8 and 12-15. While the absorptions of modes 6-8 are qualitatively identical, their VCD is are very different. Modes 12 and 13 are more separated, and the signs of their VCD are inverted. The relative absorption intensities of modes 14 and 15 are altered, and the VCD intensities are very different. The LSDA calculations are in worse agreement with experiment assuming the assignment based on the B3LYP calculations. However, an assignment in better agreement with the LSDA calculations is not apparent. trans-2,3-Dimethylthiirane,7. The absorption and VCD spectra predicted using the LSDA functional differ significantly from the B3LYP spectra. The largest differences are for modes 10- 17,20, and 21. The LSDA spectra are in worse agreement with experiment assuming the assignment based on the B3LYP calculations. The LSDA calculations alone would probably lead to the assignment of modes 13 and 14 to the bands at 1057 and 1088 cm-' and mode 15 to the band at 1154 cm-I. truns-2,3-Dimethylthiirane-truns-2,3-d~, 8. The absorption and VCD spectra predicted using the LSDA functional differ significantly from the B3LYP spectra. The differences are most noticeable for modes 14-18 and 19-21. The LSDA spectra are in worse agreement with experiment assuming the assignment based on the B3LYP calculations. However, there does
16898 J. Phys. Chem., Vol. 99, No. 46, I995
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TABLE 9: Calculated and Experimental Frequencies, Dipole Strengths, and Rotational Strengths of Cyclopropane-trans-Z~-~~ (9)n mode 21 20 19 18 17 16 15 14 13 12 11 10 9 ' 8 7 6 5 4 3 2
V
3161 3123 3119 3075 2299 2289 1428 1328 1264 1196 1114 1071 1042 1014 944 933 872 768 719 612 60 1
LSDA D 2.7 7.4 1.8 9.4 1.2 4.7 6.1 9.7 17.0
BLYP -
R
3.3 -15.2 10.9 1.o
-3.5 3.4 2.2 - 12.3 18.2
V
D
3138 3103 3099 3061 2282 2275 1457 1342 1301 1170 1132 1085 1050 1030 937 879 829 766 726 620 608
14.1 19.3 3.8 20.0 3.1 15.2 1.9 1.8 4.8 0.2 4.8 10.3 11.9 24.3 6.6 6.8 76.6 80.3 13.5 5.3 1.7
B3LYP -
R 6.3 -25.1 17.4 1.6 -6.6 6.4 0.8 -6.7 8.6 3.5 -23.3 -4.7 23.0 -5.0 6.5 2.6 3.4 - 10.6 1.9 0.2 0.0
EXPb
V
D
R
3214 3177 3173 3133 2338 2329 1498 1381 1333 1208 1165 1119 1081 1069 974 921 864 791 744 638 625
12.4 17.6 3.2 18.3 2.8 13.4 2.2 2.4 5.8 0.2 4.8 9.7 11.6 21.9 3.0 10.9 86.2
5.7 -23.8 16.2 2.0 -6.2 5.9 1.0 -7.6 9.8 3.3 -23.6 -4.4 23.3 -5.6 4.8 3.4 4.9 -11.1 1.9 0.1 0.0
V
3076 304 1 3035 3012 2270 2257
D
R
19 25
4.8
-9.9 4.4 1.4 -3.7 3.3
21 15
1338 8.4 -7.7 1290 6.8 10.2 3.O 1180 0.0 -0.6 1134 13 -42.6 9.3 -35.1 1087 21 -13.3 7.8 -9.5 1052 23 28.2 27.4 39.4 1037 32.8 38 -15.9 -6.8 942 15 5.5 0.5 -0.8 13.6 2.0 (909) 98.4 15.6 (857) 80.0 79.6 -13.4 (786) 12.3 11.9 0.9 (736) 7.3 5.3 0.3 (632) 1 1.7 3.1 0.0 (618) Units as in Table 1. Rotational strengths are for the (1S,2S) enantiomer. From Freedman et al.;2bin C2C14 solution for C-H and C-D stretching modes and in CS2 solution otherwise. Values in parentheses are from gas-phase data. Dipole and rotational strengths were obtained from solution data by fitting using Lorentzian band shapes. Rotational strengths were not normalized to 100% ee; ee values for (1R,2R)-9 and (1S,2S)-9 were '98%.
TABLE 10: Calculated and Experimental Frequencies, Dipole Strengths, and Rotational Strengths of C yclopropane-unti-I ,&?-&-I
mode 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5
4 3 2 1
J3C (1O)n
V
LSDA D
3125 3119 31 10 2300 2289 2278 1354 1269 1257 1175 1057 1052 1027 932 926 878 812 132 670 583 568
8.5 1.9 5.1 0.4 4.7 3.9 0.2 14.5 17.3 3.4 0.0 40.5 22.4 7.3 4.8 8.7 96.9 77.6 8.6 0.9 8.2
BLYP R 6.0 -11.0 4.9 1.8 -3.4 1.5 1.2 -4.7 3.7 -2.1 0.3 -1.9 3.7 3.7 -2.8 0.5 -1.7 0.0
0.1 0.0 0.0
-
Y
D
3106 3099 3090 2283 2275 2263 1354 1306 1294 1151 1071 1070 1033 92 1 890 846 795 734 678 590 575
27.0 3.9 11.5 1.4 15.5 10.7 0.1 3.7 4.9 3.4 2.7 22.4 13.0 0.5 0.5
11.9 89.2 76.4 5.8 0.3 6.5
B3LYP R
8.8 - 17.3 8.5
3.9 -6.0 2.0 1.2 -2.5 1.7 -2.9 0.0 1.1 1 .5
1.4 -0.7 1 .0 -1.9 0.1 0.1 0.0 0.0
-
EXPb
V
D
R
3180 3173 3164 2339 2329 2318 1397 1339 1326 1188 1106 1103 1064 955 929 884 824 757 694 607 592
24.2 3.3 10.7
8.6 -15.6 7.0 3.4 -5.7 2.2 1.2 -2.8 1.9 -2.9 -0.3 1.2) 1.5 2.0 -1.2 1.1 -1.9 0.1 0.1 0.0 0.0
1.1
13.6 9.6 0.1 4.7 5.9 3.4 5.2 19.0 12.6 1.5 0.3 14.6 92.9 76.1 5.0 0.3 6.5
V
3054 3041 3025 227 1 2258 2245 1348 1302 1290 1163 1074 1037
R
1.54 -3.23 1.84 1.32 -2.14 0.65
Units as in Table 1. Rotational strengths are for the (2S,3S) enantiomer. From Freedman et al.;?' in c2c14 solution for C-H and C-D stretching regions and in CS? solution otherwise. Rotational strengths were obtained by fitting using Lorentzian band shapes. Rotational strengths were not normalized to 100% ee; ee was not specified. not appear to be an alternative assignment in better agreement with the LSDA calculations. Cyclopropane-h.ans-ZJ-&, 9. The largest qualitative differences from the B3LYP calculations are for modes 7 and 12. The LSDA calculations are in worse agreement with experiment for these modes. There is no alternative assignment available to improve the agreement. Cy~lopropane-anti-ZJJ-d3-Z-~~C, 10. In the C-H and C-D stretching regions there are no qualitative differences from the B3LYP calculations. The LSDA calculations fully support the assignment of fundamentals 16-21 arrived at from the B3LYP calculations. To summarize, the B3LYP calculations most closely reproduce the pattern of frequencies and intensities found in the
experimental absorption and VCD spectra of 1-10, and the assignment of fundamentals is most reliably based on these calculations. The BLYP calculations are, overall, quite similar to the B3LYP calculations, allowing for the systematic shift to lower frequency on replacing B3LYP by BLYP. Where BLYP and B3LYP calculations do differ substantially, the BLYP calculations are in inferior agreement with experiment, even allowing for the possibility of alternative assignments of fundamentals. The LSDA calculations are much more different from the B3LYP calculations and in much worse agreement with experiment, no matter what alternative assignments of experimental bands are contemplated. Comparison of Calculated and Experimental Frequencies, Dipole Strengths, and Rotational Strengths. Having estab-
Ab Initio Calculation of Absorption and VCD Spectra
J. Phys. Chem., Vol. 99, No. 46, 1995 16899
TABLE 11: Percentage Deviations of Calculated from Experimental Freauencie# _ _ _ _ LSDA BLYP B3LYP molecule mean range mean range mean range
50 55]
~
2.1 2.5 2.2 1.8
1 2 3 4 5 6 7
1.8
8 9
3.0 2.1 1.9 1.9
10
1.8
setl-10
2.1
a
-4.816.9 -3.916.0 -7.316.4 -3.315.6 -3.514.3 -3.1111.9 -3.310.9 -4.410.1 -3.212.8 -2.612.8 -7.311 1.9
2.0 1.9 1.9 1.5 1.5 2.2 1.1 1.2 1.3 0.8 1.6
-6.211.3 -4.612.0 -4.710.6 -4.710.8 -4.710.6 -6.116.2 -3.110.7 -3.210.9 -3.312.1 - 1.012.1 -6.216.2
2.1 2.2 1.7 2.3 2.2 2.4 2.2 2.3 2.7 3.2 2.4
-0.914.0 - 1.315.0 -2.113.7 0.813.2 0.713.2 -2.019.5 0.113.3 0.813.4 0.614.5 2.114.6 -2.119.5
45
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Means are of absolute percentage deviations. 3200
3000
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2.
2200
I
.1
4
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0
0
500
1000
1500 V
2000
2500
3000
3500
exp
Figure 15. Percentage deviations of calculated and experimental frequencies: a, B3LYP; b, BLYP; c, LSDA. Filled and open symbols are for molecules 1, 2, 9, 10 and 3-8, respectively.
18
450 7
4
C
17
20
21
A
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-0 8
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0 0
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200
0
150
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21/20/10
100 50
0 3200
3000
2400
2200
wavenum bers Figure 14. Experimental and predicted unpolarized absorption and VCD spectra of 10. Experimental spectra (a, e) are for C2C14 solution^.^' Calculated spectra are for B3LYP (b, 0, BLYP (c, g), and LSDA (d, h) density functionals; y = 10 cm-' for all bands. VCD spectra are for (2S,3S)-10. lished the assignments of the fundamentals of 1-10 to the extent made possible by the experimental spectra and arrived at a qualitative evaluation of the relative accuracies of the three density functionals, we tum now to the quantitative comparison of calculation and experiment. In addition to frequencies, dipole strengths and rotational strengths have been reported for 1-10. We consider these three experimental parameters in turn.
0
i
I
I
I
I
I
0
25
50
75
100
125
Dexp
Figure 16. Calculated and experimental dipole strengths: a, B3LYP; b, BLYP; c, LSDA. Filled and open symbols are for molecules 1, 2, 9, 10 and 3-8, respectively. Mode 5 of 2 is off-scale and not included. (1) Frequencies. In the course of assigning the spectra of 1-10, we have noted qualitative differences in the frequency predictions of the B3LYP, BLYP, and LSDA functionals. BLYP frequencies are always lower than B3LYP frequencies, the shifts being fairly uniform. LSDA frequencies are generally lower than B3LYP frequencies, but the shifts are significantly less uniform. Thus, the patteras of frequencies resulting from
Devlin et al.
16900 J. Phys. Chem., Vol. 99, No. 46, 1995 C
”01
/
0 0
a
/ A
A
-100
P. -60
I
-40
I
I
-20
I
I
0
I
I
20
I
I
40
I
1
60
I
I
I
80
Rexp
Figure 17. Calculated and experimental rotational strengths: a, B3LYP; b, BLYP, c, LSDA. Filled and open symbols are for molecules 1, 2, 9, 10 and 3-8, respectively.
B3LYP and BLYP calculations are more similar than those resulting from B3LYP and LSDA calculations. In comparison to experimental frequencies, B3LYP frequencies are almost always too high while BLYP and LSDA frequencies are mostly too low. BLYP and LSDA frequencies are generally closer to experimental frequencies. However, where significantly different, the patterns of frequencies given by BLYP and LSDA are in worse agreement with experiment than those given by B3LYP. A quantitative comparison of calculated and experimental frequencies is provided in Table 11 and Figure 15. Percentage deviations for the fundamentals of 1-10 are plotted in Figure 15. The qualitative differences in the B3LYP, BLYP, and LSDA calculations are immediately apparent: the B3LYP deviations are overall more positive than the BLYP and LSDA deviations; the scatter in the B3LYP and BLYP deviations is similar, while that of the LSDA deviations is significantly greater. The mean absolute percentage deviations, together with minimum and maximum deviations, are listed in Table 11. With B3LYP, the mean absolute percentage deviations for 1-10 lie in the range 1.7-3.2%, the global mean being 2.4%. With BLYP and LSDA, the ranges are 0.8-2.2% and 1.8-3.0%, respectively; the global means are 1.6% and 2.1%. With B3LYP, across the set 1-10, the deviations vary from a low of -2.1% to a high of 9.5%. With BLYP this range is -6.2 to +6.2%. With LSDA it is -7.3 to +11.9%. Thus, the global scatter is 11.6% for B3LYP, 12.4% for BLYP, and 19.2% for LSDA. On average, the BLYP and LSDA functionals reproduce observed frequencies better than does the B3LYP functional. On the other hand, the errors in the B3LYP calculations exhibit a narrower distribution. These apparently contradictory results can be reconciled when the contributions of anharmonicity are recognized. Observed frequencies are lower than “true” harmonic frequencies, due to anharmonicity. The shift is typically a few percent.2x As noted above, B3LYP frequencies are
systematically higher than BLYP and LSDA frequencies and than observed frequencies. If the latter were to be corrected for anharmonicity, the mean deviation of B3LYP frequencies would decrease, while for BLYP and LSDA frequencies the opposite would be the case. The mean deviation for B3LYP would then become lowest. This expectation is supported by our prior results for the set of molecules HF, H20, NH3, C h , C2H2, C 2 h , C2H.5, CO, H2C0, HCN, and CH3F.29 When TZ2P harmonic frequencies are compared to observed frequencies, the mean absolute percentage deviations are 2.3%, 1.4%, and 3.2% for LSDA, BLYP, and B3LYP, respectively. However, when calculated harmonic frequencies are compared to experimental harmonic frequencies, obtained from observed frequencies by correcting for anharmonicity, the mean absolute percentage deviations are 3.6%, 3.3%, and 1.2%. Thus, the B3LYP harmonic frequencies are in fact the most accurate. Since we do not know the magnitudes of anharmonicity shifts for 1-10, the deviations of calculated frequencies from experiment cannot be explicitly corrected for anharmonicity. It is reasonable to assume that the frequencies are of comparable accuracy to those for the small molecules studied earlier. (2) Dipole Strengths. Experimental dipole strengths reported for l-10’7.’8.20.25.26 are given in Tables 1-10. Dipole strengths were generally obtained by fitting of spectra using Lorentzian band shapes. Their accuracy is difficult to assess since (with the exception of 217) fits were not documented. Greater uncertainty is obviously to be expected in the case of overlapping bands. Dipole strengths calculated using the B3LYP, BLYP, and LSDA functionals are plotted against experimental dipole strengths in Figure 16. The B3LYP and BLYP calculations exhibit a comparable correlation with experiment; the LSDA calculations are clearly inferior. Even for the B3LYP and BLYP functionals there is considerable scatter in the plots. As shown in Figure 16, the scatter is much greater for the set of molecules 3-8 than for the set 1, 2, 9, and 10. We therefore conclude that a considerable part of the scatter is attributable to experimental errors. As was noted above, solvent absorption was not carefully subtracted in the case of 3-8. It would not be surprising if the dipole strengths for these molecules are less accurate. Indeed, Pickard et aLzo remark that “the absorption intensities determined by the curve-fitting procedure are somewhat approximate”; Polavarapu et a1.22reported “uncertainties in the measurement of the pathlength of the cells”; and Polavarapu et al.25 state that “the absorption intensities are approximate due to some arbitrariness in the curve-fitting procedure”. Given the undefined magnitude of the uncertainties in the experimental dipole strengths-which, of course, even if smaller must also exist for 1, 2, 9, and 10-it is not possible at this time to quantitate the accuracies of the calculations more precisely. (3) Rotational Strengths. Experimental rotational strengths are given in Tables 1-10. reported for 1-10’7~’s~20.24-27 Rotational strengths were obtained either by fitting of spectra using Lorentzian band shapes or from integrated intensities. As with dipole strengths, their accuracies are hard to determine. Not only were fits not documented (with the exception of 217) but, in addition, VCD spectra exhibit much lower signal-tonoise ratios than do absorption spectra and, also, are subject to artifacts. Greater uncertainty is obviously to be expected in the case of rotational strengths of overlapping bands obtained by fitting. Rotational strengths calculated using the B3LYP, BLYP, and LSDA functionals are plotted against experimental rotational strengths in Figure 17. Correlation with experiment is com-
Ab Initio Calculation of Absorption and VCD Spectra parable for B3LYP and BLYP calculations and better than that for the LSDA calculations. For the B3LYP and BLYP calculations the scatter in the rotational strength plots is substantially less than that in the corresponding dipole strength plots. In addition, the scatter does not appear to be greater for the molecules 3-8. Given the intrinsically lower accuracy of experimental rotational strengths in comparison to dipole strengths, this (unexpected) finding supports the conclusion that the scatter in the dipole strength plots for 3-8 is due to special experimental problems. Given the undefined magnitude of the uncertainties in the experimental rotational strengths, it is not possible at this time to quantitate the accuracies of the calculations more precisely. In summary, the quantitative comparison of calculated and experimental frequencies, dipole strengths, and rotational strengths supports the following conclusions arrived at from the qualitative comparison of calculated and experimental absorption and VCD spectra: (1) the B3LYP functional is the most accurate; (2) the LSDA functional is the least accurate; (3) the BLYP functional is closer in accuracy to the B3LYP functional than to the LSDA functional. The errors in B3LYP frequencies are comparable to, or even less than, the contributions of anharmonicity. Comparison to Prior Work. The fundamental modes of 1-10 have been assigned previously. In the case of 1 Freedman et al.I8 used the scaled SCF force field of Lowe et aL30 Their assignments were conf-med by the more accurate MP2 calculaand, tions of Stephens et al.31332In the case of 2, Lowe et subsequently, Kawiecki et al.l69l7used scaled SCF force fields. Their assignments were confirmed by the more accurate MP2 calculations of Stephens et al.32 In the cases of 3-5, 6, and 7-8, Pickard et aL20 Polavarapu et al.? and Polavarapu et al.25 used unscaled SCF force fields. For 6, their assignments were confirmed by the more accurate MP2 calculations of Amos et al.33 In the cases of both 9 and 10, Freedman et al.26.27used the empirical valence force field of Duncan and Bums.34 Their assignments were confirmed by the MP2 calculations of Stephens et al.32 The assignments arrived at in this work based on the B3LYP/ DFT calculations are fully consistent with the prior assignments of 1, 2, 6, 9, and 10. For 3-5, 7, and 8 there are a number of differences, which are detailed in Tables 3-5, 7, and 8. The deficiencies of unscaled SCF force fields are well-known, and it is not surprising that much more accurate calculations lead to changes in assignments based thereon. Prior work’ 8.20-23.25-27,30 has also discussed the relative contributions of their intemal coordinates to the normal modes of 1-10. The normal modes resulting from our B3LYP/DFT force fields are in many cases substantially different from those resulting from the force fields used previously; overall, differences are most pronounced for unscaled SCF force fields. Unfortunately, space does not permit a more detailed discussion of these differences here. There have been many prior calculations of harmonic frequencies using DFT. Almost all have used local (such as LSDA) or nonlocal (such as BLYP) functionals. To date, relatively few have used hybrid functionals (such as B3LYP). The conclusion that the BLYP functional is more accurate than the LSDA functional is well-precedented, The even higher accuracy of the B3LYP functional was anticipated from the results of Becke5 in predicting thermochemical quantities but had not been explicitly documented when this work began. The work reported here, and previously in ref 8, is the first to combine the study of frequencies and intensities, both absorption and VCD, in the evaluation of DFT force fields.
J. Phys. Chem., Vol. 99, No. 46, 1995 16901
Conclusion The accuracy of DFT harmonic force fields increases with the accuracy of the density functional employed. In this work we have examined the relative accuracies of three functionals: LSDA, BLYP, and B3LYP. Harmonic frequencies, calculated using these functionals, deviate from observed fundamental frequencies by -2% on average for all three functionals. However, observed frequencies differ from true harmonic frequencies by several percent, on average, also. It is thus not immediately possible to differentiate between the functionals. Closer examination does reveal differences, however. The range of deviations of calculated frequencies from experimental frequencies is greater with the LSDA than with the BLYP and B3LYP functionals, indicating a lower accuracy. In addition, the BLYP and B3LYP frequencies generally deviate in opposite directions. When anharmonicity is taken into account, it becomes clear that B3LYP frequencies are the more accurate. The examination of unpolarized absorption and circular dichroism intensities adds a new dimension to the evaluation of the DFT harmonic force fields. Intensities are a sensitive function of normal coordinates, which, in turn,sensitively reflect the force field. Absorption and VCD intensities predicted using the BLYP and B3LYP functionals are quite similar, while LSDA intensities differ much more. The BLYP and B3LYP intensities agree much better with experimental intensities, confirming the superiority of the BLYP and B3LYP functionals over the LSDA functional. Statistically, the BLYP and B3LYP intensities are in equally good agreement with experiment. However, across the set of 10 molecules studied here, a number of groups of bands exhibit substantially different absorption and/or VCD intensity pattems. In all such cases, the B3LYP calculations are in better agreement with experiment. Again, we are led to conclude that the B3LYP functional is more accurate than the BLYP functional. Our conclusions are identical to those arrived at in our prior study of 4-methyl-2-0xetanone.~ The differences in spectra predicted from the three functionals were more pronounced for this molecule than for 1-10. This is not surprising, given the greater density of states of the same symmetry in this larger, asymmetrical molecule. We can expect that study of the spectra of even larger molecules will amplify the differences between the functionals even further. Our work is limited to three of the many density functionals found in the literature. In the future we expect to study additional functionals, especially those which promise to be of even greater accuracy than B3LYP. In the meantime, it is reasonable to assume that our results with the LSDA, BLYP, and B3LYP functionals are typical of other local, nonlocal, and hybrid functionals, respectively. Clearly, at this time the use of hybrid functionals such as B3LYP is to be preferred in DFT calculations of vibrational spectra. While the computational demands of DFT calculations do increase with the sophistication of the density functional, the extra load of hybrid functional calculations over local and nonlocal functional calculations is not severe and will generally be more than compensated by the increased accuracy achieved.
Acknowledgment. We are grateful to NSF, NM, NATO, and the San Diego Supercomputer Center for support of our work on VCD over the years. We are grateful to Professors Freedman and Nafie for allowing us to reproduce their unpublished data on 9 in ref 32 and in this paper. References and Notes (1) Ziegler,
T.Chem. Rev. 1991, 91, 651.
16902 J. Phys. Chem., Vol. 99, No. 46, 1995 (2) Versluis, L.; Ziegler, T. J . Chem. Phys. 1988, 88, 322. Fournier, R.; Andzelm, J.; Salahub, D. R. J. Chem. Phys. 1989, 90, 6371. (3) Johnson, B. G.; Frisch, M. J. Chem. Phys. Lerr. 1993, 216, 133; J. Chem. Phys. 1994, 100, 7429. Komomicki, A.; Fitzgerald, G. J. Chem. Phys. 1993, 98, 1398. (4) Johnson, B. G.; Gill, P. M. W.; Pople, J. A. J . Chem. Phys. 1993, 98, 5612. ( 5 ) Becke, A. D. J . Chem. Phys. 1993, 98, 1372, 5648. (6) See e.g.: Baker, J.; Muir, M.; Andzelm, J. J. Chem. Phys. 1995, 102, 2063. (7) Stephens, P. J.; Lowe, M. A. Annu. Rev. Phys. Chem. 1985, 36, 213. (8) Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. J. Phys. Chem. 1994, 98, 11623. (9) GAUSSIAN 92/DFT, Frisch, M. J., et al., 1993. (10) Stephens, P. J. J. Phys. Chem. 1987, 91, 1712. (11) Stephens, P. J.; Jalkanen, K. J.; Amos, R. D.; Lazzeretti, P.; Zanasi, R. J . Phys. Chem. 1990, 94, 1811. (12) Stephens, P. J. J . Phys. Chem. 1985, 89, 748. (13) Amos, R. D.; Handy, N. C.; Jalkanen, K. J.; Stephens, P. J. Chem. Phys. Lett. 1987, 133, 21. (14) CADPAC, Version 5 , Amos, R.D., 1993. (15) Amos, R. D.; Jalkanen, K. J.; Stephens, P. J. J . Phys. Chem. 1988, 92, 5571. (16) Kawiecki, R. W.; Devlin, F. J.; Stephens, P. J.; Amos, R. D.; Handy, N. C. Chem. Phys. Lett. 1988, 145, 411. (17) Kawiecki, R. W.; Devlin, F. J.; Stephens, P. J.; Amos, R. D. J . Phys. Chem. 1991, 95, 9817. (18) Freedman, T. B.; Spencer, K. M.; Ragunathan, N.; Nafie, L. A,; Moore, J. A.; Schwab, J. M. Can. J . Chem. 1991, 69, 1619. (19) Fateley, W. G.; Miller, F. A. Specfrochim. Acta 1963, 19, 611.
Devlin et al. (20) Pickard, S. T.; Smith, H. E.; Polavarapu, P. L.; Black, T. M.: Rauk, A.; Yang, D. J . Am. Chem. SOC. 1992, 114, 6850. (21) Black, T. M.; Bose, P. K.; Polavarapu, P. L.; Baron, L. D.; Hecht, L. J . Am. Chem. SOC. 1990, 112, 1479. (22) Polavarapu, P. L.; Hess, B. A,; Schaad, L. J.; Henderson, D. 0.; Fontana, L. P.; Smith, H. E.; Nafie. L. A,: Freedman. T. B.; Zuk, W. M. J . Chem. Phys. 1987, 86, 1140. (23) Polavarapu, P. L.; Bose. P. K.; Pickard, S. T. J . Am. Chem. SOC. 1991, 113, 43. (24) Yang, D.; Rauk, A. J . Chem. Phys. 1992, 97, 6517. (25) Polavarapu, P. L.; Pickard. S . T.; Smith, H. E.; Black, T. M.; Rauk, A.; Yang, D. J . Am. Chem. SOC. 1991, 113, 9747. (26) Ffeedman, T. B.; Ragunathan, N.; Spencer, K. M.; Cianciosi, S. J.; Baldwin, J. E. Unpublished work. (27) Freedman, T. B.; Cianciosi, S. J.; Ragunathan, N.; Baldwin. J. E.; Nafie, L. A. J. Am. Chem. SOC. 1991, 113, 8298. (28) Hehre, W. J.; Radom, L.; Schleyer, P. R.; Pople, J. A. Ab Iniria Molecular Orbitul Theon.; Wiley: New York, 1986; pp 232-245, Tables 6.39-6.41. (29) Finley, J. W.; Stephens, P. J. THEOCHEM, in press. (30) Lowe, M. A.; Alper, J. S.; Kawiecki, R. W.; Stephens, P. J. J . Phys. Chem. 1986, 90, 41. (31) Stephens, P. J.; Jalkanen. K. J.: Devlin. F. J.: Chabalowski. C. F. J . Phys. Chem. 1993, 97, 6107. (32) Stephens, P. J.; Chabalowski, C. F.; Jalkanen, K. J.; Devlin, F. J. Chem. Phys. Lett. 1994, 225, 247. (33) Amos, R. D.; Handy, N. C.; Palmieri, P. J . Chem. Phys. 1990, 93. 5796. (34) Duncan, J. L.; Bums. G. R. J . Mol. Spectrosc. 1969, 30. 253. JP95 1612M
