On the performance of density functional methods for describing atomic populations, dipole moments and infrared intensities

1 March 1996

ELSEVIER

CHEMICAL PHYSICS LETTERS Chemical Physics Lette~ 250 (1996) 393-401

On the performance of density functional methods for describing atomic populations, dipole moments and infrared intensities Frank De Proft a, Jan M.L. Martin b,c, Paul Geerlings a.* a Eenheid Algemene Chemie, Vrije Universiteit Brussel, Faculteit Wetenschappen, Pleinlaan 2, B-1050 Brussel, Belgium b Limburgs Universitair Centrum, Institute for Materials Science (IMO), Department SBG, Universitaire Campus, B-3590 Diepenbeek, Belgium University of Antwerp (UIA), Institute for Materials Science, Department of Chemistry, Universiteitsplein 1, B-2610 Wilrijk, Belgium

Received 20 November 1995; in final form 15 January 1996

Abstract

Atomic populations according to the Mulliken, electrostatic, natural population, and atomic polar tensor (APT) definitions were evaluated for first- and second-row compounds using different correlated ab initio techniques, DFT methods, and basis sets. All definitions except MuUiken exhibit modest basis set sensitivity. B3LYP predicts partial charges in agreement with high-level ab initio results. Exact-exchange corrections are more important than gradient corrections for this property. B3LYP with at least sdpf basis sets usually predicts dipole moments and infrared intensities in agreement with more accurate calculations, while semiquantitative IR intensities are obtained even with the modest cc-pVDZ basis set.

I. Introduction

There is a growing interest in the quantum chemical literature in density functional theory (DFT) methods [ I ]. The advantage of these methods is that they incorporate electron correlation at an affordable cost. Standards for the non-specialist using DFT methods (as they appear to be established in the case of ab initio quantum chemistry [ 2 - 4 ] ) are still not available at the present time. Answers should be sought to questions such as "For a given property, what quality can be expected for a certain exchange-correlation potential ?" and "What is the basis set dependence of the calculated results?". Of utmost importance when studying molecular properties and reactivity is the correct description of

* Corresponding author.

molecular charge distributions [5]. Although some DFT calculations of charge distributions (see e.g. Ref. [ 6 ] ) have been published, the performance of the different DFT methods in their determination has not been investigated in a comprehensive way. (Limited comparisons have been made in the scope of more general studies [7,8]: neither of these studies include hybrid functionals [9] or comparison with accurate conventional electron correlation methods.) Due to its conceptual simplicity, the use of atomic populations in reporting the molecular charge distribution is still popular among chemists. We have therefore investigated the performance of different DFT based methods for calculated atomic populations, according to different definitions, by comparison with high-level conventional ab initio methods. In this work, the following population analyses were considered, besides the one due to Mulliken [ 10] : the

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electrostatic potential based CHELPG method [ 11 ], the natural population analysis [ 12,13] (NPA), and the atomic polar tensor [ 14] (APT) method. In the CHELPG method, atomic charges are fitted to the electrostatic potential under the constraint that the molecular dipole moment should be conserved. NPA is a molecular orbital based analysis which attempts a partitioning of space, as in Bader's topological atoms in molecules (AIM) approach [15]. Cioslowski's APT analysis, based on dipole moment derivatives, is the only method that can be related to observable quantities. It remains relatively unexplored, however, due to the computational demands for larger molecular systems and the fact, noted by Cioslowski in his original paper [ 14], that the APT charges are relatively sensitive to electron correlation, especially for systems involving multiple bonds. In addition to the atomic charges, dipole moments and infrared intensities were calculated at the DFT level, and compared with high level ab initio calculations and experiment.

2. Methods

All calculations have been carried out using the GAUSSIAN 94 package [ 16] running on the Cray YMP/116 at the Brussels Free Universities Computer Center. The following basis sets have been considered: (a) the Pople group 6-31G* basis set [ 17 ], because of its widespread use in the applied quantum chemistry literature; (b) Dunning's cc-pVDZ (correlation consistent polarized valence double zeta) basis set [ 18 ], which is a [3s2pld/2slp] contraction of a (9s4pld/4slp) primitive set; and (c) Dunning's cc-pVTZ (correlation consistent polarized valence triple zeta) basis set, which constitutes a [ 4s3p2d 1f/3s2p 1d ] contraction of a ( 10s5p2d 1f/5s2p 1d) primitive set. It has previously been shown [ 19] that the cc-pVDZ basis set appears to give superior results to the 6-31 G* one for geometries and harmonic frequencies, principally because all its exponents are optimized for electron correlation. The cc-pVTZ basis set, on average, affords accuracy of better than 10 cm -1 in harmonic frequencies using advanced electron correlation methods, and generally computes band distances with small and systematic errors [ 20,4].

Among wavefunction based methods, SCF, MP2 (second-order M~ller-Plesset perturbation theory) [ 21 ], and QCISD (quadratic configuration interaction with all single and double substitutions [22], actually an approximate coupled cluster method) were considered. CCSD (coupled cluster with all single and double substitutions [23] ) and especially CCSD(T) (the same method augmented by a quasiperturbative account of triple excitations) [24] would have been preferable, but no analytical gradients (or densities) are available for these methods in GAUSSIAN 94. For CO2, N20 and SO2, the QCISD/cc-pVTZ charges could not be calculated due to disk space limitations. The following five density functional methods were considered: (a) the local density approximation (LDA) [25,26]; (b,c) the gradient corrected functionals BLYP (Becke-Lee-Yang-Parr [27,28] ) and BP86 (Becke-Perdew 1986) [27,29]; and (d,e) combinations of Becke's three-parameter hybrid exchange functional [9] (which includes the exact Hartree-Fock exchange based on Kohn-Sham orbitals [25] as one of its components) with the LYP correlation functional (B3LYP) and with the more recent Perdew-Wang functional [30] (B3PW91). The Mulliken, NPA, and CHELPG population analyses are all available within GAUSSIAN 94. APT populations were obtained from the GAUSSIAN 94 outputs using the utility program gar2ped 1. All charges were computed at the equilibrium geometries at the respective levels of theory.

3. Results and discussion

Computed populations at the QCISD level with all three basis sets can be found in Table 1. The most conspicuous feature of the results is the well-known extreme basis set dependence of the Mulliken charges. Not only do the absolute values oscillate wildly, sign changes occur in the cases of BH, CO, and PH3. No such behavior is observed for any of the other l gar2ped is a utility program for post-processing the archive records of GAUSSIAN 94 frequencycalculations. Features include isotopic shifts, potential energy distribution, APT population analysis, and vibrational mode animation. It was written by C. Van Alsenoy and J. M. L. Martin and is available by anonymous tip from the Computational Chemistry Archives at ftp:/ / fip.osc.edu/pub/chemistry.

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Table I Charge distributions at the QCISD level M ulliken

CHELPG

N PA

APT

6-31G* cc-pVDZ cc-pVTZ 6-31G* cc-pVDZ cc-pVTZ 6-31G* cc-pVDZ cc-pVTZ 6-31G* cc-pVDZ cc-pVTZ BH

H B C2 H2 H C CH~ H C CO C O CO2 O C H2CO O C H H20 O H HeS S H HCI H CI HCN H C N HF H F NH3 N H NNO N N O PH3 P H SO2 S O

-0.076 0.076 0.240 -0.240 0.159 -0.637 0.179 -0.179 -0.354 0.709 -0.339 0,076 0,131 -0.817 0.409 -0.217 0.109 0.226 -0.226 0.275 0.036 -0.311 0.490 -0.490 -0.929 0.310 -0.063 0.516 -0.453 -0.021 0.007 0.897 -0.448

-0.054 0.054 0.054 -0.054 0.041 -0.164 -0.002 0.002 -0.165 0.330 -0.200 0.159 0.021 -0.274 0.137 -0.144 0.072 0.155 -0.155 0.117 -0.023 -0.095 0.215 -0.215 -0.236 0.079 -0.046 0.308 -0.262 0.033 -0.011 0.764 -0.382

0.010 -0.010 0.190 -0.190 0.113 -0.452 -0.026 0.026

-0.242 0.115 0.063 -0.459 0.230 -0.215 0.108 0.158 -0.158 0.186 -0.104 -0.082 0.319 -0.319 -0.499 0.166

-0.119 0.040

0.121 -0.121 0.239 -0.239 0.088 -0.353 0.021 -0.021 -0.389 0.777 -0.410 0.431 -0.011 -0.777 0.388 -0.338 0.169 0.262 -0.262 0.186 0.162 -0.348 0.436 -0.436 -1.049 0.350 -0.289 0.620 -0.331 -0.227 0.076 0.605 -0.303

0.121 -0.121 0.228 -0.228 0.069 -0.278 -0.003 0.003 -0.371 0.742 -0.387 0.426 -0.019 -0.695 0.347 -0.293 0.146 0.239 -0.239 0.178 -0.153 -0.331 0.420 -0.420 -0.863 0.288 -0.306 0.643 -0.338 -0.160 0.053 0.592 -0.296

t h r e e d e f i n i t i o n s , e x c e p t for the C H E L P G c h a r g e s in C O w h i c h are n e a r zero. G e n e r a l l y , C H E L P G , NPA, a n d A P T all a p p e a r to e x h i b i t c o m p a r a b l e , m o d e s t , basis set s e n s i t i v i t y : it h a s p r e v i o u s l y b e e n n o t e d (e.g. Ref. [ 5 ] ) t h a t t h e s e d e f i n i t i o n s a p p e a r to b e fairly stable t o w a r d s b a s i s set v a r i a t i o n b e y o n d s p l i t - v a l e n c e quality. C o n t r a r y to w h a t w o u l d b e e x p e c t e d f r o m elect r o n e g a t i v i t y a r g u m e n t s , a n d to t h e N P A a n d A P T results, C H E L P G p r e d i c t s n e g a t i v e c h a r g e s for B in B H a n d o n P in PH3, O n the w h o l e , N P A a n d A P T a p p e a r to a s s i g n t h e s a m e polarity. A n e x c e p t i o n , h o w e v e r , is m e t h a n e ,

0.146 -0.146 0.232 -0.232 0.091 -0.363 0.006 -0.006

-0.410 0.426 -0.008 -0.697 0.348 -0.266 0.133 0.218 -0.218 0.187 0.167 -0.354 0.418 -0.418 -0.886 0.295

-0.151 0.050

-0.324 0.324 0.235 -0.235 0.222 -0.887 0.491 -0.491 -0.514 1.028 -0.489 0.226 0.131 -0.930 0.465 -0.273 0.137 0.269 -0.269 0.228 0.076 -0.304 0.539 -0.539 -I.092 0.364 -0.051 0.395 -0.344 0.061 -0.020 1.583 -0.791

-0.332 0.332 0.225 -0.225 0.201 -0.806 0.477 -0.477 -0.531 1.063 -0.483 0.278 0.102 -0.868 0.434 -0.284 0.142 0.270 -0.270 0.210 0.107 -0.317 0.516 -0.516 -I.012 0.337 -0.075 0.418 -0.343 0.038 -0.013 1.520 -0,760

-0.349 0.349 0.218 -0.218 0.189 -0.754 0.472 -0.472

-0.069 0.069 0.196 -0.196 -0.002 0.008 0.217 -0.217 -0.562 1.123 -0.472 -0.509 0.2969 0.639 0.088 -0.065 -0.885 -0.520 0.443 0.260 -0.240 -0.044 0.120 0.022 0.243 0.181 -0.243 -0.181 0.215 0.244 0.081 -0.056 -0.296 -0.189 0.531 0.378 -0.531 -0.378 -0.998 -0.424 0.333 0.141 -0.281 0.783 -0.502 0.089 0.362 -0.030 -0.121 1.038 -0.519

-0.068 0.068 0.195 -0.195 -0.008 0.032 0.203 -0.203 -0.562 1.123 -0.514 0.670 -0.078 -0.467 0.234 -0.071 0.036 0.192 -0.192 0.241 -0.071 -0.170 0.382 -0.382 -0.326 0.109 -0.283 0.79 I -0.509 0.330 -0.110 1.004 -0.502

-0.047 0.047 0.206 -0.206 -0.005 0.021 0.216 --0.216

-0.531 0.634 -0.051 -0.492 0.246 -0.091 0.046 0.189 -0.189 0.248 -0.060 -0.188 0.381 -0.381 -0.399 0.133

0.297 -0.099

w h e r e the h y d r o g e n s b e c o m e s l i g h t l y n e g a t i v e l y c h a r g e d , c o n t r a r y to the s i g n i f i c a n t p o s i t i v e c h a r g e seen in the N P A analysis. F o r a c e t y l e n e , for e x a m p l e , b o t h N P A a n d A P T find s i g n i f i c a n t p o s i t i v e c h a r g e s o n H: it s h o u l d b e n o t e d that the A P T r e s u l t is c o n sistent w i t h t h e o b s e r v a t i o n that CH4 h a s n o acidic character, c o n t r a r y to C2H2. F u r t h e r m o r e , it m a y b e r e m a r k e d that the s m a l l a b s o l u t e m a g n i t u d e o f the A P T c h a r g e s in CH4 is c o n s i s t e n t w i t h t h e fairly low i n f r a r e d i n t e n s i t i e s for the active b a n d s . A g o o d m e a s u r e for the s e n s i t i v i t y o f a p a r t i c u l a r c h a r g e d e f i n i t i o n t o w a r d s e l e c t r o n c o r r e l a t i o n is prov i d e d b y the d i f f e r e n c e q ( Q C I S D ) - q ( S C F ) . T h e s e

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Table 2 Correlation contributionto different charge definitionsat QCISD/cc-pVDZlevel q(QCISD)-q(SCF)

BH

C2H2 CH4 CO CO2 H2CO

H20 H2S HCI HCN

HF NH3 NNO

PH3 SO2

H B H C H C C O O C O C H O H S H H CI H C N H F N H N N O P H S O

Mulliken

CHELPG

NPA

APT

0.000 0.000 -0.005 0.005 0.005 -0.021 -0.084 0.084 0.087 -0.173 0.081 -0.079 -0.001 0.019 -0.009 0.016 -0.008 -0.018 0.018 -0.016 -0.032 0.048 -0.012 0.012 0.021 -0.007 0.008 --0.135 0.127 -0.018 0.006 -0.203 0.092

-0.068 0.068 -0.031 0.031 -0.006 0.025 -0.055 0.055 0.085 -0.170 0.074 -0.051 -0.011 0.057 -0.028 0.029 -0.015 -0.017 0.017 -0.034 -0.002 0.037 -0.028 0.028 0.079 --0.026 0.024 -0.123 0.099 0.033 -0.011 -0.122 0.061

0.018 - 0.018 -0.008 0.008 0.003 -0.011 -0.121 0.121 0.109 -0.217 0.097 -0.100 0.002 0.042 -0.021 0.010 -0.005 -0.016 0.016 -0.010 -0.039 0.049 -0.026 0.026 0.039 -0.013 0.003 -0.121 0.118 -0.032 0.010 -0.293 0.146

-0.048 0.048 -0.02 I 0.021 0.007 -0.028 -0.151 0.15 I 0.202 -0.406 0.169 -0.127 -0.021 0.095 -0.048 0.030 -0.015 -0.037 0.037 -0.031 -0.068 0.098 -0.053 0.053 0.096 -0.032 0.107 -0.345 0.238 -0.035 0.012 -0.405 0.203

differences are depicted in Table 2 for the cc-pVDZ basis set. In all cases, introducing electron correlation preserves polarity, except for the Mulliken charges in CO and HCN and the CHELPG charge in CO. The latter is related to the fact that electron correlation causes the sign of the dipole moment to change. In 13 out of 15 cases, APT exhibits the largest correlation effect on the partial charges. If we exclude APT from consideration, the largest dependency is seen in 8 out of 15 cases for CHELPG, and 5 out of 15 for NPA. Despite its almost pathological basis set dependence, Mulliken population analysis appears to be relatively

insensitive to electron correlation. The observation that APT appears to be much less sensitive to the basis set than to electron correlation is consistent with the fact that computed infrared intensities appear to be even more sensitive to electron correlation than to the basis set [ 31 ], although the difference is not as marked as for harmonic frequencies [20]. Since evidently the APT charges appear to exhibit the greatest correlation sensitivity, we shall consider them for the remainder of the discussion. APT charges computed at the SCF, MP2, and QCISD levels, as well as using five different

E De Proft et al./Chemical Physics Letters 250 (1996) 393-401

397

Table 3 Sensitivity of APT charges in cc-pVDZ basis to method

BH C2H2 CH4 CO CO2 H2 CO

H:O H2S HCI HCN

HF NH3 NNO

PH3 SO2

tt B H C H C C O O C O C H O H S H H CI H C N H F N H N N O P H S O

SCF

MP2

QCISD

LDA

BP86

BLYP

B3LYP

B3PW91

-0.020 0.020 0.216 -0.216 -0.015 0.061 0.354 -0.354 -0.764 1.529 -0.684 0.797 -0.057 -0.562 0.281 -0.101 0.051 0.229 -0.229 0.272 -0.003 -0.269 0.434 -0.434 -0.422 0.141 -0.390 1.136 -0.747 0.365 -0.122 1.409 -0.705

-0.039 0.039 0.201 -0.201 0.002 -0.009 0.105 -0.105 -0.495 0.990 -0.467 0.629 -0.081 -0.486 0.243 -0.104 0.052 0.212 -0.212 0.245 -0.113 -0.132 0.391 -0.391 -0.363 0.121 -0.259 0.673 --0.414 0.305 -0.102 0.769 --0.384

-0.068 0.068 0.195 -0.195 -0.008 0.032 0.203 -0.203 -0.562 1.123 -0.514 0.670 -0.078 -0.467 0.234 -0.071 0.036 0.192 -0.192 0.241 -0.071 -0.170 0.382 -0.382 -0.326 0.109 -0.283 0.791 -0.509 0.330 -0.110 1.004 -0.502

-0.055 0.055 0.209 -0.209 0.030 -0.122 0.177 -0.177 -0.491 0.982 -0.513 0.669 -0.078 -0.474 0.237 -0.133 0.066 0.218 -0.218 0.248 -0.083 -0.166 0.377 -0.377 -0.377 0.126 -0.288 0.728 -0.440 0.228 -0.076 0.943 -0.471

-0.066 0.066 0.196 -0.196 0.004 -0.018 0.185 -0.185 -0.482 0.965 -0.507 0.688 -0.091 -0.428 0.214 -0.097 0.048 0.199 -0.199 0.237 -0.081 -0.156 0.356 -0.356 -0.298 0.099 -0.275 0.703 -0.428 0.286 -0.095 0.920 -0.460

-0.060 0.060 0.189 -0.189 -0.005 0.022 0.178 -0.178 -0.475 0.951 -0.501 0.691 -0.095 -0.403 0.201 -0.069 0.034 0.185 -0.185 0.231 -0.076 -0.155 0.343 -0.343 -0.262 0.087 -0.274 0.695 -0.421 0.324 -0.108 0.903 -0.451

-0.047 0.047 0.199 -0,199 -0.001 0.002 0.215 -0.215 -0.548 1.096 -0.543 0.701 -0.079 -0.455 0.227 -0.087 0.044 0.200 -0.200 0.246 -0.061 -0.185 0.371 -0.371 -0.326 0.109 -0.312 0.814 -0.502 0.316 -0.105 1.022 -0.511

-0.052 0.052 0.203 -0.203 0.007 -0.029 0.22 I -0.22 I -0.555 1.1 I 0 -0.549 0.698 -0.075 -0.474 0.237 -0.11 I 0.055 0.212 -0.212 0.249 -0.062 -0.187 0.379 -0.379 -0.356 0.119 -0.316 0.825 -0.510 0.284 -0.095 1.038 -0.519

0.034

0.029

0.032

0.013

0.019

mean absolute deviation from QCISD 0.104

0.046

0.000

exchange-correlation functionals, are given in Table 3 for the cc-pVDZ basis set. It appears to be clear that the differences between the various density functionals, as well as between them and the QCISD results, are much smaller than the difference between SCF and QCISD. The largest correlation effects appear to occur in multiple-bond systems (with the exception of C2H2); in addition, MP2 appears to overestimate the correlation effect for all such systems (again except for C2H2). This is consistent with the well-known importance of electron cor-

relation in multiple bonds, as well as the general tendency of MP2 to overestimate the correlation energy for such systems. The mean absolute deviations between QCISD and the other methods are given in the bottom line of Table 3. From those, it is clear that all DFT methods will perform better than MP2. The difference between LDA and the simple gradient-corrected functionals (BLYP and BP86) does not appear to be significant; the two exact-exchange corrected methods (B 3LYP and B 3PW91 ), however, are clearly superior.

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Table 4 Basis set sensitivity of various charge definitions at the B3LYP level

BH C2 H2 CH4 CO CO2 H2CO

H20 H2 S HCI HCN

HF NH3 NNO

PH3 SO2

H B H C H C C O O C O C H O H S H H CI H C N H F N H N N O P H S O

CHELPG 6-31 G*

CHELPG cc-pVDZ

CHELPG cc-pVTZ

NPA 6-3 IG*

NPA cc-pVDZ

NPA cc-pVTZ

APT 6-31 G*

APT cc-pVDZ

APT cc-pVTZ

0.155 -0.155 0.241 -0.241 0.093 -0.370 0.007 -0.007 -0.37 t 0.742 -0.392 0.422 -0.015 -0.745 0.373 -0.342 0.171 0.261 -0.261 0.192 0.144 -0.336 0.420 -0.420 - 1.017 0.339 -0.315 0.625 -0.311 -0.231 0.077 0.559 -0.279

0.160 -0.160 0.233 -0.233 0.068 -0.271 -0.011 0.011 -0.355 0.710 -0.379 0.419 -0.020 -0.682 0.341 -0.293 0.146 0.238 -0.238 0.186 0.135 -0.321 0.413 -0.413 -0.849 0.283 -0.323 0.641 -0.318 -0.166 0.055 0.569 -0.284

0.169 -0.169 0.230 -0.230 0.081 -0.326 0.005 -0.005 -0.377 0.754 -0.414 0.448 -0.017 -0.697 0.349 -0.266 0.133 0.221 -0.221 0.186 0.170 -0.356 0.418 -0.418 -0.887 0.295 -0.326 0.652 -0.325 -0.142 0.048 0.564 -0.282

-0.331 0.332 0.240 -0.240 0.229 -0.915 0.505 -0.505 -0.509 1.017 -0.494 0.226 0.134 -0.934 0.467 -0.304 0.152 0.282 -0.282 0.231 0.077 -0.309 0.535 -0.535 - 1.109 0.370 -0.072 0.395 -0.322 0.014 -0.005 1.561 -0.781

-0.326 0.326 0.239 -0.239 0.217 -0.866 0.493 -0.493 -0.527 1.054 -0.492 0.260 0. I 16 -0.888 0.444 -0.331 0.165 0.290 -0.290 0.222 0.099 -0.321 0.524 -0.524 - 1.050 0.350 -0.091 0.417 -0.326 -0.040 0.013 1.513 -0.757

-0.348 0.348 0.227 -0.227 0.202 -0.809 0.485 -0.485 -0.490 0.979 -0.478 0.286 0.096 -0.909 0.454 -0.284 0.142 0.264 -0.264 0.222 0.078 -0.300 0.544 -0.544 - 1.034 0.345 -0.081 0,404 -0.322 0.017 -0.006 1.572 -0.786

-0.058 0.058 0.195 -0.195 0.000 0.001 0.223 -0.223 -0.543 1.086 -0.534 0.685 -0.075 -0.491 0.246 -0.057 0.029 0.191 -0.191 0.243 -0.045 -0.198 0.359 -0.359 -0.405 0.135 -0.318 0.818 -0.500 0.356 -0.119 1.033 -0.516

-0.047 0.047 0.199 -0.199 -0.001 0.002 0.215 -0.215 -0.548 1.096 -0.543 0.701 -0.079 -0.455 0.227 -0.087 0.044 0.200 -0.200 0.246 -0.061 -0.185 0.371 -0.371 -0.326 0.109 -0.312 0.814 -0.502 0.316 -0.105 1.022 -0.51 I

-0.032 0.032 0.212 -0.212 -0.003 0.010 0.226 -0.226 -0.574 1.147 -0.561 0.687 -0.063 -0.491 0.245 -0.093 0.047 0.192 -0.192 0.25 I -0.055 -0.196 0.379 -0.379 -0.404 0.135 -0.326 0.852 -0.526 0.303 -0.101 1.069 -0.535

O f these, B 3 L Y P a p p e a r s to b e d o i n g s l i g h t l y b e t t e r than B 3 P W 9 1 , but the s m a l l d i f f e r e n c e m a y n o t b e significant. T h e r e a p p e a r s to b e little to c h o o s e b e t w e e n BLYP and BP86. F o r t h e l i m i t e d c c - p V T Z r e s u l t s available, w e d o find a s i g n i f i c a n t d i f f e r e n c e b e t w e e n L D A a n d t h e g r a d i e n t - c o r r e c t e d f u n c t i o n a l s , but this is p r i m a r i l y d u e to t h e o m i s s i o n o f t h e m o r e ' d i f f i c u l t ' c u m u l e n i c species, for w h i c h t h e l a r g e r e r r o r s w i t h all f u n c t i o n a l s t e n d to b l u r t h e d i s t i n c t i o n b e t w e e n t h e m in m e a n abs o l u t e error. All o t h e r o b s e r v a t i o n s are s i m i l a r to t h o s e w i t h t h e c c - p V D Z b a s i s set: t h e p a r t i c u l a r l y s m a l l B 3 L Y P m e a n a b s o l u t e e r r o r o f 0 . 0 0 5 s h o u l d b e noted.

T h e fact that e x a c t - e x c h a n g e c o r r e c t i o n s are definitely m o r e i m p o r t a n t t h a n g r a d i e n t c o r r e c t i o n s , even for a c o r r e l a t i o n - s e n s i t i v e p r o p e r t y s u c h as A P T charges, is a c o n s e q u e n c e o f t h e fact that t h e e x c h a n g e e n e r g y is 1 - 2 o r d e r s o f m a g n i t u d e m o r e i m p o r t a n t t h a n the c o r r e l a t i o n e n e r g y ( c o m p a r e T a b l e s 8.3 a n d 8.4 in Ref. [ 1 ] ). N o t surprisingly, t h e c o n c l u s i o n s r e a c h e d for b a s i s set s e n s i t i v i t y at the B 3 L Y P level are s i m i l a r to t h o s e r e a c h e d at the Q C I S D level ( T a b l e 4 ) . A P T c h a n g e s a p p e a r to b e s l i g h t l y less b a s i s set s e n s i t i v e t h a n the N P A a n d C H E L P G ones, a l t h o u g h t h e r e is n o t m u c h to c h o o s e b e t w e e n t h e m . W h i c h p a r t i c u l a r c h a r g e def-

E De Proft et al./Chemical Physics Letters 250 (1996) 393-401

399

Table 5 Computed and experimental dipole moments (debye) QC1SD

HCN H20 H2CO NH~ CO b HF HCI SO2 PH3 H2S BH NNO

B3LYP

CCSD(T)

cc-pVTZ

6-3 IG*

cc-pVDZ

cc-pVTZ

TZ2Pf 131 I

Expta

2.990 1.925 2.329 1.617 0.113 1.817 1.181

2.905 2.095 2.186 1.912 0.060 1.860 1.468 1.776 0.962 1.427 1.285 0.161

2.795 1.938 2.071 1.684 0.180 1.818 1.338 2.092 0.623 1.323 0.964 0.020

2.981 1.919 2.280 1.586 0.125 1.831 1.206 2.013 0.526 1.187 1.002 0.037

2.990 1.920 2.341 1.603

2.984 1.854 2.332 1.471 0.110 1.826 1.109 1.633 0.574 0.97 1.270" 0.161

0.654 1.086 1.357

a All experimental data taken from Ref. 134]. h All calculated values show a C - O + orientation, in agreement with experiment. c Ref. t35[.

Table 6 Computed and observed harmonic frequencies to (cm - j ) and infrared intensities / (km mol - j ) Experimenta

HCN

CO,, CH4 C2H2 H20

H2CO

NH3

wI to2 to3 w2 to3 to3 ton to3 ~o5 tol to2 ~o:~ tot 092 ~o3 to4 to5 to6 tol to2 to3 ~o4

CCSD(T) / TZ2Pf 131 ]

QCISD/ cc-pVTZ

to

1

to

I

to

3444 725 2130 673 2397 3157 1367 3415 747 3832 1649 3943 2923b 1778 b 1539 t' 1194 b 2999b 1276 h 3478 110221 3597 1684

54 46 0.1 48 548 694-3,72-1-11,704-3 29-t-1,414-6 714-2 1755:5 2.2,2.2,2.2,2.5 53.6,63.9,66.6,71.9 44.6,48.2,39.8 75.54-7.1 74.04-5.3 11.2-t-1.0 6.54-0.6 87.6-1--8.0 9.94-1.0 7.64-0.9 1384-6 3.84-0.8 28.24-0.5

3470 791 2128 672 2391 3147 1351 3434 794 3835 1650 3944 2926 1776 1541 1197 2996 1272 3471 1076 3600 1679

64 71 0.06 59 634 64 32 81 182 4.7 69.5 48.4 59.4 74.5 10.6 4.4 108.4 12.0 2.3 147 3.8 31

3468 741 2162

3165 1356 3429 765 3872 1678 3975 2952 1817 1556.8 1213 3017 1287 3499 1110 3623 1700

a For experimental references, see Ref. [31 ] unless indicated otherwise. h Ref. 1361.

B3LYP/ 6-3 IG* 1 60.4 71.1 0.5

69.6 30.5 84.8 181.4 4.3 65.6 42.5 59.0 81.2 10.6 3.8 114.1 12.8 2.1 140.5 1.3 28.8

B3LYP/ cc-pVDZ

B3LYP/ cc-pVTZ

to

I

to

/

to

/

3481 766 2214 640 2436 3163 1374 3442 775 3726 1713 3848 2916 1850 1563 1198 2967 1280 3437 1132 3568 1727

52.5 71.0 2.0 61.4 545.8 79.2 46.2 76.5 162.2 1.7 75.8 19.4 55.9 98.7 6.4 1.4 164.7 12.6 0.8 156.1 1.0 30.6

3466 772 2200 655 2422 3145 1311 3428 773 3751 1659 3852 2866 1831 1515 1186 2919 1253 3418 1122 3532 1673

63.2 68.2 2.0 28.7 577.6 63,0 35.9 95,3 153,3 2.9 55.8 20.0 59.5 108.8 5.8 1.6 177.3 16.0 2.0 105.5 1.9 17.9

3450 762 2202 672 2417 3132 1343 3416 767 3801 1640 3901 2876 1825 1537 1203 2929 1268 3461 1068 3577 1677

60.7 72.6 1.3 63.7 629.2 73.1 37.5 89.1 192.4 3.2 69.5 40.8 69.2 107.1 9.9 3.2 145.1 12.8 2.4 146.7 1.1 33.6

400

E De Proft et al./Chemical Physics Letters 250 (1996) 393-401

inition is used (other than Mulliken and its variants) will therefore largely depend on personal taste and the particular application. In situations where conservation of the dipole moment is important (such as approximating a crystalline environment by a net of point charges) CHELPG is clearly to be preferred: one the other hand, NPA is the only orbital-based method which yields charges that approximate topological [ 15] ones. APT, however, is the only definition which can directly be related to an experimentally observable property (the infrared intensities). Due to the necessity of obtaining dipole moment derivatives, it is by far the costliest option (except possibly for topological charges) if o n l y partial charges are of interest: if the harmonic frequencies are required anyway, however, APT charges are trivially obtained along with the infrared intensities at zero extra cost. In Tables 5 and 6, we illustrate the performance of B3LYP for dipole moments and infrared intensities, respectively. (The generally good performance for frequencies has been discussed in Ref. [ 32] ). Computed dipole moments are generally in excellent agreement with high-level ab initio calculations if a basis set of spdf quality is used; the agreement deteriorates significantly if basis sets of split-valence plus polarization quality are used. Some circumspection is due in comparing computed and observed infrared intensities due to the sometimes significant uncertainties in the latter: agreement between high-level ab initio calculations and B3LYP/cc-pVTZ intensities can be described as very good, with the notable exception of the CO stretch and the antisymmetric CH stretch in formaldehyde. As for the smaller basis sets, even these appear to be satisfactory if only a semiquantitative spectral intensity pattern is required (as it is in most cases). This is encouraging for the study of the infrared spectra of larger carbon clusters [33], where B3LYP/cc-pVDZ at this time represents the best computationally feasible option.

4. C o n c l u s i o n s

We are in a position to assert the following: - B3LYP predicts atomic populations, be they CHELPG, NPA, or APT, in excellent agreement with high-level correlated ab initio treatments; - Of the various population analyses considered,

APT is by far the most sensitive towards electron correlation; - This becomes a moot point, however, since B3LYP reproduces essentially all the correlation effects on atomic charges; - If harmonic frequencies are required anyway, APT charges provide important additional insight in the molecular charge distribution at no extra cost; - B3LYP/cc-pVTZ dipole moments are in excellent agreement with experiment for first-row compounds; - B3LYP/cc-pVTZ infrared intensities are, with a few exceptions, in agreement with more accurate calculations, and semiquantitative agreement is achieved even at the B3LYP/cc-pVDZ level; - Except for Mulliken charges, all atomic population definitions exhibit similar (modest) basis set sensitivity. It should be stressed that only first- and second-row compounds were considered in the present paper, and that conclusions for them do not necessarily apply to, e.g., transition metal compounds and organometallics.

Acknowledgements

JM is an NFWO/FNRS (National Science Foundation of Belgium) Senior Research Associate ("Onderzoeksleider") and FDP an NFWO/FNRS Postdoctoral Research Fellow. The authors would like to thank the Vrije Universiteit Brussel for a generous computer time grant. This research was partially supported (JM) by the Prime Minister's Office for Science Policy Programming (DPWB) through program IUAP/48 (Characterization of Materials). References

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