The harmonic frequencies of benzene. A case for atomic natural orbital basis sets

29 August 1997

CHEMICAL PHYSICS LETTERS

ELSEVIER

Chemical Physics Letters 275 (1997) 414-422

The harmonic frequencies of benzene. A case for atomic natural orbital basis sets Jan M.L. Martin a, Peter R. Taylor b, Timothy J. Lee c Department of Organic Chemistry, Kimmelman Building, Room 262, Weizmann Institute of Science, 76100 Rehocot, Israel b San Diego Supercomputer Center and Department ofChemistr3' and Biochemistry, University of California, San Diego. P.O. Box 85608, San Diego, CA 92186-5608, USA c MS230-3, NASA Ames Research Center, Moffett Field, CA 94035-1000, USA

Received 20 May 1997; in final form 24 June 1997

Abstract The geometry and harmonic force field of benzene have been computed at the CCSD(T) level with basis sets of ~pdf quality. Two out-of-plane modes, o94 and to5 (and to a lesser extent wt7), exhibit a pathological basis set dependence due to basis set superposition error. Using an atomic natural orbital (ANO) basis set of [ 4 s 3 p 2 d l f / 4 s 2 p ] quality, the best available experimentally derived harmonic frequencies can be reproduced with an RMS error of 6 cm-~ without any empirical corrections. We strongly recommend the use of ANO basis sets for accurate frequency calculations on unsaturated and aromatic systems. Our best estimate for the equilibrium geometry is r e ( C C ) = 1.392(2), r e ( C H ) = 1.081 ,~. © 1997 Published by Elsevier Science B.V.

1. Introduction

Starting with the venerable work of Wilson [1], the high symmetry and semirigidity of benzene have attracted many theoretical and experimental attempts to determine at least the quadratic force field of this medium-sized molecule. A detailed review of the work until 1991 has been given in the landmark paper of Goodman, Ozkabak and Thakur (GOT) [2] and will not be repeated here. GOT present what at that point was probably the most accurate set of quadratic force constants available, as well as a set of experimentally derived harmonic frequencies which will be denoted tOob~ in this Letter. More recent computational studies of the harmonic frequencies include inter alia the large-basis set MP2 calculations of Handy et al. (HMAAML) [3] and the nonlocal density functional [4] work of

Handy, Murray and Amos (HMA) [5]. To the authors' knowledge, the highest level of theory applied hitherto is represented by CCSD/[4s3p2dlf/3s2p] (coupled cluster with all single and double excitations [6]) calculations on the in-plane frequencies by Brenner, Senekowitsch and Wyatt (BSW) [7]. On another front, Maslen et al. (MHAJ) [8] calculated a complete S C F / D Z P quartic force field and combined the computed anharmonicities with the observed fundamentals to obtain a new set of harmonic frequencies, which they denoted toest. The t-.Oobs and west sets differ by as much as 20 cm-~ for several modes, owing to the fact that anharmonicity in some modes was entirely neglected (for want of sufficient data) in GOT and is possibly overestimated at the SCF level - - the disappointing quality of computed S C F / D Z P anharmonicities for ethylene compared to large-scale calculations [9] suggests that

0009-2614/97/$17.00 © 1997 Published by Elsevier Science B.V. All rights reserved. PII S 0 0 0 9 - 2 6 1 4 ( 9 7 ) 0 0 7 3 5 - 5

J.M.L. Martin et al. / Chemical PhysicsLetters 275 (1997) 414-422 the MHAJ anharmonicities themselves should probably be treated with some caution. H M A A M L tried to select between the ~0obs and ¢.Oest of individual modes based on their computed MP2 results and density functional results reported in HMA, while H M A simply suggested the average of both sets, way = (¢Oobs + ¢Oest)/2, as the most reliable set. Clearly, a number of questions remain unanswered. Extensive experience (e.g. Refs. [9,10])has shown that CCSD(T), i.e. the CCSD method [6] augmented with a quasiperturbative estimate for connected triple excitations [11], can predict harmonics and even fundamental frequencies of most semirigid molecules to within better than 10 cm -~ as long as a basis set of at least spdf quality is used. While determining a full quartic force field of benzene with such a basis set at the CCSD(T) level would be an arduous undertaking at the present state of computer technology, the harmonic frequencies can definitely be obtained at this level, and may shed light on what level of theory would be appropriate for a full anharmonic treatment. This is the subject of this Letter.

2. Methods The large basis set CCSD(T) [6,11,12] calculations were carried out using the MOLPRO 96 ~ ab initio package running on the Cray C90 at San Diego Supercomputer Center, and on a DEC Alpha 5 0 0 / 5 0 0 workstation and an SGI Origin 2000 minisupercomputer at the Weizmann Institute. The symmetry of the largest nondegenerate subgroup was exploited throughout in the electronic structure calculations. Two correlation consistent basis sets due to Dunning [ 13] were used, namely cc-pVDZ and cc-pVTZ (correlation-consistent polarized valence double and triple zeta, respectively). The former is a [ 3 s p 2 p l d / 2 s l p ] contraction of a ( 9 s 4 p l d / 4 s l p ) primitive set (114 contracted basis functions), the latter a [ 4 s 3 p 2 d l f / 3 s 2 p l d] contraction (264 basis

I MOLPRO is a package of ab initio programs written by H.-J. Werner and P.J. Knowles,with contributionsfrom J. AlmlSf,R.D. Amos, A. Berning, M.J.O. Deegan, F. Eckert, S.T. Elbert, C. Hampel, R. Lindh, W. Meyer, A. Nicklass, K. Peterson, R. Pitzer, A.J. Stone, P.R. Taylor, M.E. Mura, P. Pulay, M. Schiitz, H. Stoll, T. Thorsteinsson,and D.L. Cooper.

415

functions) of a ( l O s 5 p 2 d l f / 5 s 2 p l d) primitive set. Furthermore, we considered the latter basis set with the d functions on hydrogen removed (leaving 234 contracted basis functions), which we will denote cc-pVTZ'. In addition, calculations were carried out with an atomic natural orbital (ANO) basis set [14] of comparable contracted size, [ 4 s 3 p 2 d l f / 4 s 2 p ] (240 contracted basis functions), as the cc-pVTZ' basis set, but which is based on a much larger ( 1 3 s 8 p 6 d 4 f / 8 s 6 p ) primitive set. This basis set will be denoted ANO4321'. The force field itself was determined by central differences, with a step size of 0.01 ,~ or radian, in symmetry coordinates. The generation of displacements and the calculation of the harmonic frequencies from the symmetry coordinate force constants were carried out with the aid of the utility program INTDER 2. The symmetry coordinates are unscaled variants of those given by Whiffen [15]. In order to verify the correctness of our procedure for calculating the harmonic frequencies by finite differences, C C S D ( T ) / c c - p V D Z frequency calculations were also carried out using the ACES II package 3 with the built-in finite difference frequency calculation routine, both using energies only and using analytical gradients. These procedures in turn were checked at the H F / c c - p V D Z level against frequencies from an analytical Hessian obtained using G A U S S I A N 94 4. Discrepancies between the

2 INTDER is a general coordinatetransformationprogram written by Wesley D. Allen. 3 J.F. Stanton, J. Gauss, J.D. Watts, W.J. Lauderdale, D.E. Bernholdt, and R.J. Bartlett, ACES H, an ab initio program system (Quantum Theory Project, University of Florida, Gainesville,FL, 1992). This package includes:J. Alml~Sfand P.R. Taylor, MOLECULE, a vectorized Gaussianintegralprogram; and T. Helgaker, P. JCrgensen, H.J. Aa. Jensen, J. Olsen and P.R. Taylor, ABACUS, a Gaussian integral derivative and molecular properties program. 4 M.J. Frisch, G.W. Trucks, H.B. Schlegel, P.M.W. Gill, B.G. Johnson, M.A. Robb, LR. Cheeseman, T. Keith, G.A. Petersson, J.A. Montgomery, K. Raghavachari, M.A. AI-Laham, V.G. Zakrzewski, J.V. Ortiz, J.B. Foresman,J. Cioslowski,B.B. Stefanov, A. Nanayakkara, M. Challacombe, C.Y. Peng, P.Y. Ayala, W. Chen, M.W. Wong, J.L. Andres, E.S. Replogle, R. Gomperts, R.L. Martin, D.J. Fox, J.S. Binkley, D.J. DeFrees, J. Baker, J.P. Stewart, M. Head-Gordon, C. Gonzalez and J.A. Pople, Gaussian 94, Revision D.4, Gaussian, Inc., PittsburghPA, 1995.

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J.M.L. Martin et al. / Chemical Physics Letters 275 (1997) 414-422

different derivative evaluation procedures were found to be of the order of 0.1 cm -1 or less: it was therefore assumed that our finite differentiation procedure for obtaining the large basis set CCSD(T) results was reliable. SCF and MP2 calculations were carried out using Gaussian 94 running on a DEC Alpha 5 0 0 / 5 0 0 at the Weizmann Institute of Science, using analytical second derivatives.

3. Results and discussion

3.1. Harmonic frequencies Computed harmonic frequencies for benzene at different levels of theory are given in Table 1, together with the ogob~ (GOT), oge~t (MHAJ) and ogav = (Wob~ + oge~t)/2 (HMA) sets of experimentally derived data. Agreement between Way and the present C C S D ( T ) / c c - p V T Z ' results is good. Only for five modes are there differences larger than 10 cm ~: o94 (37 c m - l ) , o95 (28 c m - l ) , o913 (16 cm ~), o9J4 (14 cm - l ) and o917 (15 c m - l ) . Of these, w13 was noted by Maslen et al. (MHAJ) [8] to be part of a 3-way resonance /313 ~ /38 "~ /319 ~ /320' leading them to assign "an uncertainty of[about] 10 cm -j to o920 and an even bigger uncertainty to o913", o9J4 is the 'Kekul6 mode' which corresponds to stretching towards one of two limiting localized structures and the other three are out-of plane modes which involve 'twisting' HCCH moieties in one way or another. (There is quite good agreement for the other out-ofplane modes.) We shall discuss first some methodological observations and then consider these particular problem modes in detail. The basis set sensitivity at the CCSD(T) level, measured as the d i f f e r e n c e b e t w e e n the C C S D ( T ) / c c - p V D Z and C C S D ( T ) / c c - p V T Z ' results, is informative. All four CH stretching frequencies systematically drop by about 9 cm -~ upon enlarging the basis set: interestingly, the totally symmetric CC stretch remains virtually unchanged. Most of the other frequencies are affected less, with five notable exceptions: o93 ( + 25 c m - ~), 0)4 ( q- 52 cm-1), W5 ( + 2 5 c m - l ) , o912 ( + 17 cm 1) and o917

( + 19 c m - l ) , w 3 corresponds to distortion towards a 'Catherine wheel' type structure: appreciable basis set sensitivity for that mode ( + 25 c m - 1 between the same two basis sets) was previously noted by Martin et al. [16] using the B3LYP (Becke 3-parameterLee-Yang-Parr) [17,18] hybrid density functional method, wj2 corresponds to an alternating-angle distortion of the carbon ring into D3h symmetry: again an appreciable effect was found at the B3LYP level as well ( + 17 cm 1 for the same basis sets). By way of comparison, at the SCF level we see that the CH stretches decrease systematically by 23 cm-~ upon enlarging the basis set from cc-pVDZ to cc-pVTZ. The only other mode for which a fairly sizable effect is seen is o93 ( + 2 1 cm 1): the shifts for o94, o95, o912 and w~7 are only + 5 , + 1 1 , + 1 0 and + 10 cm -1, respectively. Particularly the large shift seen for o94 at the CCSD(T) level, therefore, appears to be largely a correlation effect. By comparison of the C C S D ( T ) / c c - p V D Z and M P 2 / c c - p V D Z results, we obtain insight into the effect of higher order correlation effects. As expected, there is a substantial difference for the stretching modes and good agreement for the bending modes. The mode for which by far the largest difference is seen, - 132 cm - j , is the Kekul6 vibration ogH: since correctly describing the curvature along this vibration is essentially a two-reference problem at large amplitude, it is not surprising that a low-order perturbation theory method would fail (see also Ref. [3]). MP2 has a known tendency (e.g. Ref. [19], where erratic completely symmetric structures were systematically found for cyclic carbon clusters) to overestimate delocalization effects, and therefore a motion going towards a localized structure would go more steeply uphill on an MP2 surface than on the true surface, which is consistent with our observation. Conversely, SCF would underestimate these effects and would go less steeply uphill, presumably partially canceling with the general tendency of SCF to overestimate vibrational frequencies: in the present case, the net result is excellent agreement between SCF and CCSD(T) for this mode. (It was pointed out before [20] that in cases with moderately strong nondynamical correlation effects, SCF is probably a more robust treatment than low-order perturbation theory.) The power of the CCSD(T) method is once more attested to by the fairly small

.LM.L. Martin et al. / Chemical Physics Letters 275 (1997) 414-422

difference between the CCSD(T) and observed frequencies for this mode. Interestingly, at the B3LYP level, this is another mode which exhibits a signifi-

417

cant basis set effect upon going from cc-pVDZ to cc-pVTZ [ 16]. We then have three modes for which we see large

Table 1 Computed and experimentally derived harmonic frequencies ( c m - I ) for benzene

`ol `o2 `o3 094 `o_s `o6 `o7 `os `o9 `o10 `oil `or2 ,ol3 o914 `ol5 `ol6 `ol7 `ol8 °)19 `o20

`ol `o2 `o3 `o4 `o5 `o6 `o7 `o8 `o9 `olo `oil `ol2 `ol3 `o14 `ol5 `oL6 `o17 `o]8 0919

atg air a2g b2g bo R e2g e2g e2g e2g elg a2u bh, bu, b2u b2u e2u e2u elu el, el. r(CC) (f) r(CH) (f)

alg alg a2~ b2g b2g e2R e2g e2g e2g elg azu blu blu bzu b2u e2u e~u eno etu

HF/ cc-pVDZ this work

HF/ cc-pVTZ this work

MP2/ cc-pVDZ this work

CCSD(T)/ cc-pVDZ this work

CCSD(T)/ cc-pVTZ' this work

CCSD/ cc-pVTZ' [7]

CCSD(T)/ ANO4321' this work

09,,v HMA [5] (b)

1081.0 3371.5 1477.5 771.2 1116.7 660.8 3340.3 1787.7 1273.1 948.5 752.4 1085.8 3328.5 1341.4 1188.6 449.7 1090.1 1130.9 1627.2 3359.5 1.3886

1072.6 3348.2 1498.1 776.0 1127.3 662.9 3317.4 1774.9 1281.7 955.9 758.5 1095.4 3305.5 1336.7 1171.1 451.6 1100.1 1129.6 1635.0 3336.6 1.3827

1017.0 3246.7 1358.9 633.9 962.0 605.6 3220.3 1649.7 1190.7 859.0 686.0 1007.0 3209.3 1472.5 1163.2 400.9 953.7 1061.4 1505.0 3236.8 1.4057

1038 1494 3182 1.393

1002.8 3209.9 1379.9 708.8 1008.8 610.8 3183.1 1637.2 1194.4 865.1 687.2 1019.7 3173.1 1326.1 1163.1 405.8 984.8 1055.5 1509.4 3199.7 1.3967

1001 3198 1378 712 1000 610 3182 1623 1185 856 680 1016 3173 1313 1158 402 978 1048 1503 3186 (e)

1.0734

1.0952

1004.0 3202.7 1378.6 677.3 972.3 606.9 3174.4 1636.3 1189.6 858.5 685.6 1011.7 3163.0 1326.5 1158.1 401.9 962.6 1053.7 1505.4 3192.0 1.3976 (a) 1.394(1) 1.0840 (a) 1.082(1)

1012 3228 1391

1.0822

1003.9 3212.1 1353.6 624.9 947.5 602.8 3183.8 1640.5 1181.0 847.2 676.6 994.6 3173.0 1340.5 1151.6 394.6 943.7 1050.0 1499.9 3201.3 1.4107 1.386(3) 1.0978 1.084(5)

1.082

1.0834

(e)

613 3204 1672 1207

1025 3189 1304 1166

MP2/ 6-311G * * [3]

MP2/ TZ2Pf [3]

BLYP/ TZ2Pf [51

B3LYP/ cc-pVDZ [16]

B3LYP/ cc-pVTZ [ 16]

`oob~ GOT [2] (c)

`oe~,t MHAJ [8] (d)

1015 3240 1367 413 903 610 3215 1645 1199 842 678 1009 3204 1451 1173 389 908 1063 1509

1018 3242 1374 684 996 608 3217 1637 1195 865 683 1039 3218 1461 1178 404 979 1074 1515

985 3116 1351 703 980 608 3093 1573 1169 836 663 1005 3083 1302 1152 399 958 1028 1471

1019 3200 1365 723 1022 618 3173 1646 1187 866 691 1014 3163 1358 1163 414 987 1060 1507

1015 3192 1390 727 1021 624 3167 1637 1201 867 690 1031 3157 1335 1177 414 988 1062 1519

994 3191 1367 707 990 608 3174 1607 1178 847 674 1010 3174 1309 1150 398 967 1038 1494

1008 3208 1390 718 1011 613 3191 1639 1192 866 686 1024 3172 1318 1167 407 989 1058 1512

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J.M.L. Martin et al. / Chemical Physics Letters 275 (1997) 414-422

Table 1 (continued)

to20

etu r(CC) (f) r(CH) (f)

MP2/ 6-311G * *

MP2/ TZ2Pf

BLYP/ TZ2Pf

B3LYP/ cc-pVDZ

B3LYP/ cc-pVTZ

Wob~ GOT [2]

West MHAJ [8]

[3]

[31

[51

[ 16]

[ 16]

(c)

(d)

3230 1.398

3238 1.3896

3181

3191

1.0804

3189 1.3986 1.391(2) 1.0927 1.080(2)

3182 1.3908

1.086

3107 1.401 1.390(3) 1.088 1.079(4)

1.0820

(a) Full CCSD(T)/cc-pVTZ geometry r(CC)= 1.3975, r(CH)= 1.0831 ,~. Empirically corrected: 1.394(1), 1.082(1) ,~. (b) Average of values in GOT [2] and MHAJ [8]. (c) Derived from expt. overtones and combination bands [2]. (d) Derived from expt. fundamentals and SCF/DZP anharmonicities [8]. (e) Expt. r m structure: rm(CC)= 1.3902(3), rm(CH)= 1.0862(15) ,~. Full CCSD(T)/cc-pVTZ geometry r(CC)= 1.3975, r(CH)= 1.0831 ,~. (f) With empirical corrections [33,16]. CCSD(T)/cc-pVTZ corrections with d functions used for the CCSD(T)/cc-pVTZ' calculations.

basis set effects at the CCSD(T) level but not at the B3LYP level: o94, o95 and o9J7. Note that for these three modes, Handy et al. [3] found enormous frequency shifts between M P 2 / 6 - 3 1 I G * * and M P 2 / T Z 2 P f : + 2 7 3 , + 9 3 and + 7 1 cm J. They related this to a similar problem with the bending frequencies of acetylene, for which Simandiras et al. [21] found that it was necessary to use a basis set including f functions to obtain even fair agreement with experiment. Lee et al. [22] and Ahem et al. [23] saw similar effects on the o98 twisting mode of C 2 H 4. Later (e.g. Ref. [24]) an explanation was advanced in terms of symmetry discontinuities between linear and nonlinear geometries for basis sets truncated at finite angular momentum, in general, and at d functions, in particular. However, Dateo and Lee [25] saw a similar effect for the o98 and o99 modes of C 3 H 2 , which has too low a molecular symmetry (C2~,) for this kind of symmetry discontinuity to occur with an spd basis set. Furthermore, we see here that using correlation consistent basis sets of spd versus spdf quality instead cuts the basis set effect by a factor of three or more: we had similar experiences elsewhere with ethylene [9] and with acetylene [26]. Moreover, for acetylene Almltif and co-workers [27] demonstrated a large basis set superposition error by calculating the bending frequencies of an N 2 molecule (isoelectronic with C2H 2) with two hydrogen 'ghost' atoms (i.e. hydrogen basis sets centered on fictitious atoms). Rather than zero or near-zero values, they found a fairly large unphysical ~-~ bending frequency, which

clearly suggests a basis set superposition error issue. Since atomic natural orbital (ANO) basis sets [14] exhibit the smallest basis set superposition error of all commonly used basis sets of similar size, one would therefore expect the anomalous basis set dependence to be reduced even further when switching from correlation consistent to A N O basis sets, which was indeed observed [26,27]. This becomes particularly clear when considering the anharmonicities for the bending modes of HCCH, which have the wrong sign even with a cc-pVQZ basis set but appear to be at least reasonable with an ANO [5s4p3d2flg] basis set [26]. We have therefore repeated our frequency calculations with the ANO4321' basis set. And as is seen in Table 1, this completely resolves the issue. All 20 symmetry-unique values in the o9av set are now reproduced with a mean absolute error of 6.5 cm -~, with all three of the 'problem' frequencies lying closer than 10 cm-~ to the corresponding ogav values. Since the CPU time in CCSD(T) force field calculations on small molecules is dominated by the integral evaluation time and switching from a correlation consistent to an equivalent ANO basis set can easily increase integral evaluation times by an order of magnitude, the former are generally preferred [28]. Also in the present case, integral evaluation for a calculation in D2n symmetry on the DEC Alpha 5 0 0 / 5 0 0 took 53 min. with the ANO4321' basis set, compared to 5.5 min. with the cc-pVTZ' basis set. However, for a case with as many correlated elec-

419

J.M.L Martin et al. / Chemical Physics Letters 275 (1997) 414-422

trons as C 6 H 6, the C C S D and particularly (T) steps d o m i n a t e the calculation to such an extent that the total C P U t i m e for the s a m e point was 394 min. with the A N O 4 3 2 1 ' basis set, only 22% l o n g e r than with the c c - p V T Z ' basis set (322 min.). Interestingly, the pathological basis set dependence noted a b o v e is absent f r o m the B 3 L Y P results. This clearly illustrates that basis set c o n v e r g e n c e b e h a v i o r in density functional and c o n v e n t i o n a l electron correlation m e t h o d s is qualitatively different. Orbital-based m e t h o d s for treating correlation conv e r g e s l o w e r since they must describe the two-electron correlation cusp using products o f one-electron functions; in density functional m e t h o d s there is no such requirement. F o r the in-plane m o d e s other than C H stretches, O h n o and S h i n o h a r a [29] f o u n d that a simple 8p a r a m e t e r e m p i r i c a l force field c o u l d reproduce the 0)ob~ set with an R M S error o f only 18.6 c m -~, w h i l e e v e n C C S D / c c - p V T Z ' ( B S W ) cannot do better than 33.9 c m i. The present C C S D ( T ) / A N O 4 3 2 1 ' results actually a c h i e v e a m e a n absolute error o f 14.4 c m -~, w h i c h serves to underscore the importance o f triple excitations for vibrational f r e q u e n c y calcula-

tions, particularly w h e n multiple bonds are involved. If the c o m p a r i s o n is instead m a d e with the 0)av set, the R M S difference drops to 6.4 c m - 1 for the present C C S D ( T ) / A N O 4 3 2 1 ' results: including the C H stretches as well leaves this value essentially unc h a n g e d at 6.5 c m ~ (the R M S error o v e r the c o m plete set is 6.5 c m - l . ) W e therefore feel justified in saying that the present quadratic force field is the best n o n e m p i r i c a l one available. If we c o m p a r e with the O)est set instead, the m e a n absolute error actually slightly drops to 4.1 c m -~. For almost all the individual vibrations, the c o m puted values are closer to the 0)est values o f M H A J than to the O)obs values of G O T , e x c e p t for w 3, 0)6, 0) 7 w h e r e it is in b e t w e e n both and 0)4 where it is closest to G O T . H o w e v e r , since to4 is the worst o f the ' p r o b l e m ' m o d e s and further basis set enhancements are likely to m o v e w 4 up by an order o f 10 c m - l (as was the case in the I r m o d e in C 2 H 2 [26]) we w o u l d argue that the M H A J 0)est set is probably the best available e x p e r i m e n t a l l y d e r i v e d set overall. The consideration o f deuterated species provides further support. For C6D6 ( T a b l e 2), the C C S D ( T ) / A N O 4 3 2 1 ' f r e q u e n c i e s agree as well with

Table 2 Computed and experimentally derived harmonic frequencies (cm- ~) for benzene-d 6

o91 o92 o93 o94 o95 o96 o97 o98 o99 o910 o911 o912 o913 o914 o915 wt6 co17 o918 o919 o920

alg atg a2g b2g b2g e2g e2g e 2g e2g e lg a2, b lu b lu b2u b~u e2u e2u elu elu e i~

CCSD(T)/ cc-pVDZ

CCSD(T)/ cc-pVTZ

CCSD(T)/ ANO4321'

ogobs GOT

o9~, MHAJ

og,v HMA

956.5 2384.8 1053.1 577.9 724.7 574.0 2350.0 1598.5 862.8 659.1 496.8 956.0 2335.2 1328.9 821.7 345.6 762.3 1352.8 816.1 2372.2

957.0 2376.8 1072.3 601.0 775.1 577.7 2343.6 1592.9 870.1 667.8 503.4 971.7 2329.6 1312.9 827.7 350.8 780.1 1353.3 821.8 2364.6

956.0 2382.0 1073.1 608.3 831.5 581.6 2349.8 1593.3 873.6 672.9 504.6 979.2 2337.5 1312.1 831.5 353.0 800.8 1354.6 824.7 2370.0

947 2362 -599 829 581 2332 1564 867 660 496 970 2344 1286 828 345 787 1341 814 2346

960 2384 1076 609 839 584 2353 1594 869 670 501 981 2366 1301 828 351 799 1358 824 2375

954 2373 1076 604 834 583 2343 1579 868 665 499 976 2355 1294 828 348 793 1350 819 2361

J.M.L. Martin et al. / Chemical Physics Letters 275 (1997) 414-422

420

the ¢..Oav or we~t as for C 6 H 6 (mean absolute deviations 6.8 and 4.6 cm -~, respectively), except for o~3. However, as for C 6 H 6 , P13 was not observed directly but obtained from combination bands and MHAJ note the same 3-way resonance as discussed above for C 6 H 6 and estimated a resonance perturbation, based on their S C F / D Z P quartic force field, of about 7 cm i downwards. If o~13 is eliminated, the mean absolute deviations drop to 6.3 cm ~ relative

to ¢Oav, and 3.6 c m - J with respect to ~%~t. Again, it appears that the west set is the better one and definitely superior to the ~Oob~ set (12.2 cm 1).

3.2. Quadratic force

constants

Our computed quadratic force fields, expressed in the c o m m o n l y used symmetry coordinate set of Whiffen [15], are given in Table 3, compared with

Table 3 Computed and experimentally derived quadratic force constants ('aJ/,~) and reference geometries (,~). The coordinate system of Whiffen [15] has been used

A ig

Fl,l F2, t

F2, 2 A2g B 2g

E2g

F3, 3

F4,4 Fs. 4 Fs. 5 F6,6

]77,6

EIg A 2u B lu

B 2u

E2u

Elu

F~, 6 F9, 6 FT. 7 Fs, 7 /79,7 Fs, 8 Fg, 8 F9, 9 Fi0,10 Fi i,i i FI2,12 FI3A2 Ft 3,L3 FI 4,14 Fi5.14 Fts.i 5 FI6.16 FI7,16 FI 7,17 FIsA8 Fig.i 8 F2o, t 8

FI9.19 F20,19

CCSD(T)/ cc-pVDZ

CCSD(T)/ cc-pVTZ'

CCSD(T)/ ANO4321'

this work

this work

this work

7.777 0.081 5.614 0.860 0.196 0.263 0.510 0.632 -0.128 0.306 -0.138 5.577 0.057 - 0.037 7.074 -- 0.405 0.882 0.337 0.251 0.632 --0.196 5,575 4.237 0.315 0.810 0.163 -0,176 0.411 7.656 0.213 0.141 0.917

7.773 0.114 5.586 0.894 0.205 0.264 0.526 0.644 -0.124 0.310 -0.144 5.543 0.069 - 0.032 7.024 - 0.423 0.898 0.347 0.258 0.654 --0.190 5.535 4.142 0.323 0.824 0.166 -0.177 0.425 7.624 0.222 0.165 0,935

7.755 0.112 5.612 0.894 0.209 0.263 0.545 0.651 -0.124 0.314 -0.143 5.573 0.072 - 0.033 7.029 - 0.427 0.905 0.351 0.259 0.664 --0.189 5,569 4.142 0.329 0.830 0.168 -0,178 0.438 7.612 0.226 0.167 0.943

0.000

0,005

0.004

F2o.2o

5.594

5.564

5.592

r(CC) r(CH)

1.4107 1.0978

1.3976 1.0840

1.3967 1.0834

Expt. derived GOT [2]

MP2/ 6-311G * * GOT [2]

7.616 0.157 5.554 0.877 0.202 0.249 0.519 0.644 -0.136 0.308 -0.140 5.510 0.054 - 0.066 6.690 - 0.398 0.895 0.337 0.252 0.658 --0.237 5.571 3.939 0.298 0.828 0.160 -0.168 0.420 7.380 0,209 0.572 0,926 0.151 5.568

7.955 0.088 5.714 0.877 0.194 0.286 0.484 0.649 -0.122 0.321 -0.141 5.683 0.062 0.036 7.217 -- 0.449 0.900 0.333 0.252 0.650 --0.190 5.680 5.039 0.342 0.827 0.164 -0.178 0.392 7.801 0.235 0.150 0.934 0.003 5.697

1.397 1.084

1.398 1.086

J.M.L. Martin et al. / Chemical Physics Letters 275 (1997) 414-422

the GOT set. It should be noted that the Whiffen bend, out-of-plane and torsion coordinates are scaled with r(CC) or r(CH), as appropriate, to have a J / 4 units throughout. Interestingly, we see here that F4.4 and Fs. 5 do not change quite as drastically between the basis sets as do the harmonic frequencies themselves: the bond distance scaling apparently absorbs most of the change. The most conspicuous differences between the presently computed and GOT force constants are seen in FIg,18 and F20.~8, for both of which previous calculations (e.g. [30]) yield values close to the present work. Other significant differences involve the couplings Fg.7, F2. l and FI3A2, all of which are calculated to be substantially weaker than in the G O T force field, with concomitant changes in some of the diagonal force constants (e.g. FI,1).

3.3. Geometry. Pliva, Johns and Goodman [31] (PJG) determined an r m geometry (as defined by Watson [32]) of benzene: in general, r m distances should be close to r e distances for nonhydrogen atoms, but may not be for hydrogen atoms. We were able to carry out a CCSD(T) geometry optimization with the full ccpVTZ basis set. Martin [33] proposed empirical correction terms of - 0 . 0 0 1 8 A for a single and - 0 . 0 0 5 7 4 for a double bond at the C C S D ( T ) / c c - p V T Z level, leading to mean absolute errors over the reference molecules in Ref. [33] of less than 0.001 A. Assuming that the correction term for the aromatic CC bonds would be the average of those for single and double bonds, the correction term would be - 0 . 0 0 3 8 4 ; we thus obtain an estimated r e geometry of r e ( C C ) = 1.3937(10), r e ( C H ) = 1.0813(10) A, compared to rm(CC) = 1.3902(3), rm(CH) = 1.0862(15) A. We concur with H M A that the experimental CH bond length is definitely too long. Concerning re(CC), we note that the single bond correction in Ref. [33] is heavily biased towards AH (A = C, N, O, F) bonds and therefore is probably too small, since about half the residual error in bond distances for {H,C,N,O,F} molecules comes from core correlation [34], which is more important for C - C bonds than for C - H bonds. Furthermore, rather than being halfway between a C - C single bond and a C = C double bond, re(CC) in benzene is much o

o

421

closer to a typical C = C double bond. To a rough first approximation then, it is probably better to use simply the C = C correction from Ref. [33]. This would yield re(CC)= 1.3918(20) A, where the increased error margin reflects the uncertainty in the correction. Alternatively, because of the high symmetry we can easily check what the effect of changing re(CH) on re(CC) would be with constant rotational constants (where we approximate r m ~ re). We then find that shortening r(CH) by 0.005 A would be consistent with an extension of r(CC) by 0.0006 A, which is quite consistent with our best predicted geometry. Coincidentally, the latter is in essentially perfect agreement with the C C S D / c c - p V T Z ' geometry of BSW [7], because of an error compensation between neglect of further valence basis set extension and core correlation, on the one hand, and neglect of triple excitations, on the other hand.

4. Conclusions We have computed the geometry and harmonic frequencies of benzene at the CCSD(T) level with basis sets of spdf quality. Due almost certainly to basis set superposition error, to4, to5 and, to a lesser extent, w17 exhibit a pathological basis set dependence which causes large errors in these frequencies at the C C S D ( T ) / c c - p V T Z ' level. Switching to an atomic natural orbital (ANO) basis set solves the problem, leading to a mean absolute deviation from experimentally derived values of 6.5 c m - i compared to the tOav and 4.1 cm 1 compared to the toest sets. Because the CPU time for systems with a large number of valence electrons is dominated by the CCSD and particularly the (T) steps, rather than integral evaluation, the overall extra cost is only about 20%. We strongly recommend the use of A N O basis sets for accurate frequency calculations on unsaturated and aromatic systems. Of the available experimentally derived sets of harmonic frequencies for benzene, the toe~t set of MHAJ is probably the most reliable. Our calculations further suggest that the experimental rm(CH) is as much as 0.005 4 longer than the true re(CH) and that the rm(CC) is probably only slightly shorter than the true re(CC). Our best esti-

422

J.M.L. Martin et al. / Chemical Physics Letters 275 (1997) 414-422

mate for the equilibrium geometry is re(CC)= 1.392(2), re(fH)= 1.081 A. Acknowledgements JMLM is a Yigal Allon Fellow and an Honorary Research Associate ("Onderzoeksleider in eremandaat") of the National Science Foundation of Belgium ( N F W O / F N R S ) and thanks Dr. Peter Stern and Nava Shaya for technical assistance with the SGI Origin 2000. This research was supported by the National Science Foundation (USA) through Cooperative Agreement DASC-8902825 and by Grant No. CHE-9320718 (PRT); and by a grant of computer time from SDSC. The DEC Alpha workstation at the Weizmann Institute was purchased with USAID (United States Agency for International Development) funds. References [1] E.B. Wilson, Phys. Rev. 45 (1934) 706. [2] L. Goodman, A.G. Ozkabak, S.N. Thakur, J. Phys. Chem. 95 (1991) 9044. [3] N.C. Handy, P.E. Maslen, R.D. Amos, J.S. Andrews, C.W. Murray, G.J. Laming, Chem. Phys. Lett. 197 (1992) 506. [4] N.C. Handy, in: Lecture Notes in Quantum Chemistry II, ed. B.O. Roos, Lecture Notes in Chemistry, Vol. 64 (Springer, Berlin, 1994) and references therein. [5] N.C. Handy, C.W. Murray, R.D. Amos, J. Phys. Chem. 97 (1993) 4392. [6] G.D. Purvis III, R.J. Bartlett, J. Chem. Phys. 76 (1982) 1910. [7] L.J. Brenner, J. Senekowitsch, R.E. Wyatt, Chem. Phys. Lett. 215 (1993) 63. [8] P.E. Maslen, N.C. Handy, R.D. Amos, D. Jayatilaka, J. Chem. Phys. 97 (1992) 4233.

[9] J.M.L. Martin, T.J. Lee, P.R. Taylor, J.P. Francois, J. Chem. Phys. 103 (1996) 2589. [10] T.J. Lee, G.E. Scuseria, in: Quantum mechanical electronic structure calculations with chemical accuracy, Ed. S.R. Langhoff (Kluwer, Dordrecht, The Netherlands, 1995). [11] K. Raghavachari, G.W. Trucks, J.A. Pople, M. Head-Gordon, Chem. Phys. Lett. 157 (1989) 479. [12] C. Hampel, K.A. Peterson, H.J. Werner, Chem. Phys. Lett. 190 (1992) 1. [13] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007. [14] J. AlmlSf, P.R. Taylor, J. Chem. Phys. 86 (1987) 4070. [15] D.H. Whiffen, Phil. Trans. Royal Soc. London A 248 (1955) 131. [16] J.M.L. Martin, J. E1-Yazal, J.P. Francois, Mol. Phys. 86 (1995) 1437. [17] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [18] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [19] J.M.L. Martin, P.R. Taylor, J. Phys. Chem. 100 (1996) 6047. [20] J.M.L. Martin, J.P. Francois, R. Gijbels, J. Chem. Phys. 93 (1990) 8850. [21] E.D. Simandiras, J.E. Rice, T.J. Lee, R.D. Amos, N.C. Handy, J. Chem. Phys. 88 (1988) 3187. [22] T.J. Lee, W.D. Allen, H.F. Schaefer III, J. Chem. Phys. 87 (1987) 7062. [23] A.M. Ahern, R.L. Garrell, K.D. Jordan, J. Phys. Chem. 92 (1988) 6228. [24] W.H. Green, D. Jayatilaka, A. Willetts, R.D. Amos, N.C. Handy, J. Chem. Phys. 93 (1990) 4965. [25] C.E. Dateo, T.J. Lee, Spectrochim. Acta A, in press. [26] T.J. Lee, J.M.L. Martin, P.R. Taylor, to be published. [27] J. Alml~Sf, H. Sellers, A.D. McLean, S. S~eb~, P.R. Taylor, unpublished results. [28] J.M.L. Martin, J. Chem. Phys. 97 (1992) 5012. [29] K. Ohno, H. Shinohara, J. Phys. Chem. 98 (1994) 10063. [30] P. Pulay, G. Fogarasi, J.E. Boggs, J. Chem. Phys. 74 (1981) 3999. [31] J. Pliva, J.W.C. Johns, L. Goodman, J. Mol. Spectrosc. 148 (1991) 427. [32] J.K.G. Watson, J. Mol. Spectrosc. 48 (1973) 479. [33] J.M.L. Martin, J. Chem. Phys. 100 (1994) 8186. [34] J.M.L. Martin, Chem. Phys. Lett. 232 (1995) 343.