Molecular electric properties: an assessment of recently developed functionals

16 January 1999

Chemical Physics Letters 299 Ž1999. 465–472

Molecular electric properties: an assessment of recently developed functionals Aron J. Cohen ) , Yuthana Tantirungrotechai Department of Chemistry, UniÕersity of Cambridge, Lensfield Road, Cambridge, CB2 1EW, UK Received 17 August 1998; in final form 23 November 1998

Abstract We investigate the performance of the recently proposed exchange-correlation functionals, B97, B97-1 and HCTH, on the prediction of electrical properties. The molecular dipole and quadrupole moments and the molecular dipole polarizabilities are computed for the set of first- and second-row molecules. We compare the predictions of these new functionals with those of SCF, MP2, BD and the DFT method with the BLYP and B3LYP functionals. The new functionals perform well compared with well-established functionals. The B97 and B97-1 functionals predict the dipole moment of CO in good agreement with the experimental value. The distributed multipole analysis of CO obtained by different techniques is also reported. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction Recently there has been interest in applying density functional theory ŽDFT. to chemical problems. This semi-empirical method is based on the idea that the electron density determines the Hamiltonian, the total energy and all other properties of the system in its ground state w1x. Kohn and Sham w2x showed that one can write down self-consistent equations for non-interacting electrons in a similar way to Hartree–Fock theory. The computational cost of the DFT calculation, even when electron correlation is treated, is therefore not as expensive as conventional high-level ab-initio methods such as configuration interaction ŽCI. or coupled-cluster ŽCC. methods.

)

Corresponding author.

The true exchange-correlation functional Ž Exc . is not yet known and an approximate functional is used to represent Exc . We know that the generalized gradient approximation ŽGGA. which includes the gradient of the density in the formulation is minimally required for application to chemical problems. Examples of GGA functionals are BLYP which includes the Becke 88 correction w3x to the Dirac exchange functional and the Lee–Yang–Parr correlation functional w4x, and B3LYP which includes a fraction of the exact exchange w5x. Recently Becke has published a series of papers suggesting a systematic way to improve an approximate exchange-correlation functional. In Part V of the series w6x, Becke introduces a new GGA functional called B97. The gradient correction factor g in the new functional is a polynomial series of u, g s Ý mis0 c i u i , where u s g s 2r1 q g s 2 , s and g are the reduced spin-density gradient and the fitted pa-

0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 1 3 1 7 - 7

466

A.J. Cohen, Y. Tantirungrotechair Chemical Physics Letters 299 (1999) 465–472

rameter from atomic calculations w7x. An exact exchange term EXHF is also included in the EXC expression. The polynomial and exact exchange coefficients were determined by a least-squares fit to the G-2 thermochemical data set of Pople and co-workers w8x. The fitting of the data was not done in a selfconsistent manner but was performed ‘post-LSDA’. Namely, the densities used were LSDA densities, and all gradient-corrected energies were derived from these. To obtain the best results, the g series was truncated at m s 2. This functional, called B97, performs well on energy and geometry predictions compared to B3LYP and other functionals w6x. In a series of papers by Handy and Tozer, the authors suggest that a GGA functional form, not including exact exchange, be fitted not only to energetic data but also to the shape of the exchange correlation potential. This idea was extended by using the same functional form as the B97 functional w9x, though they did not include exact exchange in the fitting procedure. They fit the functional form to the total energies of first-row atoms, ionisation potentials, atomisation energies, molecular gradients and the exchange-correlation potentials computed from high quality ab-initio electron densities. Their functional, denoted HCTH, has all the g terms up to m s 4. They also refit the B97 functional with the energetic data in a self-consistent way. This refitted B97 functional is named B97-1 w9x. All the above functionals have been obtained by fitting to energies and geometries; it is, therefore, of interest to examine the performance of these functionals on non-energetic properties. In this Letter, we investigate molecular electric properties. The permanent multipole moments reflect the molecular charge distribution, and should indicate the quality of the ground state electron density obtained. The molecular dipole polarizability is in turn related to the transition moment from the electronic ground state to the excited states. This quantity reflects how easily the excited states can be mixed into the ground state when the system is perturbed by an external electric field. McDowell et al. reported the calculation of molecular polarizabilities using DFT with the LDA and BLYP functionals and ab-initio methods with SCF, MP2 and BDŽT. wavefunctions w10x. The LDA

and BLYP results are systematically too high, this is probably due to the incorrect asymptotic behaviour of the functionals. A good description of polarizability should indicate a good asymptotic behaviour of approximate functionals.

2. Computational details We calculate the electric dipole Ž ma ., quadrupole moment ŽQab . and electric dipole polarizability Ž aab . for a set of molecules using wavefunctions from Hartree–Fock ŽSCF., second-order Møller– Plesset perturbation theory ŽMP2., the Brueckner coupled-cluster technique upto double excitations ŽBD. and density functional theory with BLYP, B3LYP, HCTH, B97 and B97-1 functionals. We use experimental geometries for all the molecules considered. The equilibrium bond distance of diatomic molecules are taken from Ref. w11x. In other cases, the experimental geometries are tabulated in Ref. w12x. The basis sets used in our calculation are due to Sadlej w13–15x. The dipole and quadrupole moments are obtained from the density matrix in the SCF and DFT cases. Due to the non-variational nature of MP2 and BD wavefunctions, these are calculated from the gradient density matrix. The SCF, MP2, BLYP and B3LYP polarizabilities are obtained analytically. The BD, B97, B97-1 and HCTH polarizabilities are calculated using the finite-field approximation. The electric field used in most cases is 0.001 au. For some BD calculations, we have done another calculation using twice the original fields as a check. The isotropic polarizability of trans butadiene ŽC 4 H 6 . has been calculated as it is a prototype for conjugated long chain molecules Že.g. polyacetylene.. A good estimation for a will yield a better description of the hyperpolarizability and hence of the non-linear optical properties. The absolute mean error and the relative mean error are reported. These errors are computed using the quoted experimental errors as a mean. In the case of more than one quoted experimental values, we use the average of these values as a mean. All the calculations used the CADPAC program w16x.

A.J. Cohen, Y. Tantirungrotechair Chemical Physics Letters 299 (1999) 465–472

3. Results and discussion The molecular dipole moment and quadrupole moment for our set of molecules are reported in Tables 1 and 2, respectively. Only the independent components of the quadrupole moment with respect to the centre of mass are reported due to its traceless definition. For the purpose of comparison, we separate the DFT results into two groups according to the inclusion of exact exchange. We therefore compare results of HCTH with BLYP, and B97 and B97-1 with B3LYP. Considering the results of conventional ab-initio methods, the SCF dipole and quadrupole moments are, as is well known, too high compared with experimental values. This reflects the overpronounced polarity found in the SCF charge distribution. Both properties decrease as we include the effects of electron correlation, as in MP2, BD. The BLYP values are generally too low compared to experiment or MP2 and BD results. On improving the functionals, the results of HCTH are generally greater than that of BLYP. This suggests an im-

467

provement of the electron density which affects the total ground state energy w9x. The same trend occurs in the B97, B97-1 and B3LYP comparison. The performance of B97-1 functionals is slightly improved upon Becke’s original B97 functionals. The hybrid functionals results, B3LYP, B97 and B97-1, are greater than the non-hybrids results. It can be seen from results in Tables 1 and 2 that the multipole moments improve upon the inclusion of exact exchange. This stresses the importance of exact exchange on the electron density. The absolute mean error of the new functionals, B97 and B97-1, with respect to the experimental values is in between MP2 and BD. The error analysis of each of the methods on the quadrupole moments is rather less clear cut. Since the multipole moments from DFT are generally too low, the improvement of these properties should be indicated by the increase of their values. From inspection of the trends in the molecular quadrupole moments, we see an increase in these properties when the new functionals are used. Judging from its trend and the absolute and the relative error with

Table 1 The dipole moment along the molecular axis. All values are in au. The linear and C 3v molecules lie in on the z axis of the coordinate system. The C 2v symmetry molecules lie in the xz plane with the C 2 symmetry axis coinciding with the z axis of the coordinate system. CO H 2O H2S HCl HF LiH LiF NH 3 PH 3 SO 2 Abs. mean error a

SCF

MP2

BD

BLYP

HCTH

B3LYP

B97

B97-1

Experiment

y0.0989 a 0.7805 0.4359 0.4746 0.7565 2.3598 2.5461 0.6364 0.2811 0.7846

0.1208 0.7297 0.4038 0.4490 0.7078 2.3358 2.4770 0.5994 0.2435 0.6068

0.0398 0.7306 0.3911 0.4401 0.7080 2.3101 2.4964 0.5990 0.2284 0.6789

0.0759 0.7090 0.3803 0.4246 0.6897 2.1920 2.3815 0.5805 0.2320 0.6167

0.0783 0.7072 0.3991 0.4335 0.6892 2.2234 2.4110 0.5793 0.2637 0.5862

0.0389 0.7307 0.3966 0.4391 0.7090 2.2417 2.4280 0.5981 0.24587 0.6576

0.0458 0.7251 0.4003 0.4397 0.7034 2.2534 2.4350 0.5949 0.2563 0.6365

0.0431 0.7263 0.4009 0.4407 0.7046 2.2563 2.4367 0.5958 0.2558 0.6413

0.0432 b 0.7268 c 0.401e 0.441e 0.707 f 2.3143 g 2.4727 h 0.603 i 0.2258 d 0.6400 k

0.0181

0.0083

0.0361

0.0355

0.0177

0.0158

0.0145

0.0675

Negative value means the dipole vector points away from the O atom. Ref. w12x. c Ref. w20x. d Ref. w21x. e Ref. w15x. f Ref. w22x. g Ref. w13x. h Ref. w23x. i Estimated ab-initio values w24x. k Ref. w25x. b

0.383 " 0.002 d ,

468 Table 2 The molecular quadrupole moment with respect to the center of mass. Unless stated otherwise, the reported values are of Qz z . For molecule with a C n axis, n) 3, Qx x and Qy y are equal to y Qz z r2. All molecules align their symmetry axis along the z axis. The C 2 H 4 molecules lies in the xz plane with the C–C bond along z axis. All values are in au

Abs. mean error Rel. mean error a b c d e

Ref. w26,27x. Ref. w28x. Ref. w21x. Ref. w29x. Ref. w30x.

SCF

MP2

BD

BLYP

HCTH

B3LYP

B97

B97-1

Experiment

1.2763 1.5091 2.4916 y1.5364 y3.8082 2.3600 0.6143 0.4387 1.8954 y0.0994 2.1523 0.7996 2.8570 1.7421 10.5798 y3.3694 4.5862 y0.9274 y2.1270 y1.7265 y4.3572 0.9132

1.1881 1.2369 2.5000 y1.5262 y3.1174 2.0998 0.8708 0.4123 1.9409 y0.0972 2.1055 0.8018 2.7837 1.7389 10.2418 y3.2782 4.6998 y1.2255 y2.2175 y1.6392 y3.4673 0.5540

1.1547 1.1892 2.4106 y1.4733 y3.3102 2.1776 0.8036 0.3941 1.8994 y0.0927 2.0489 0.7822 2.7246 1.7178 10.7364 y3.1735 4.6682 y1.1399 y2.1564 y1.5731 y3.7969 0.7285

1.0612 1.0376 2.3691 y1.6140 y3.3332 1.9387 0.6631 0.3258 1.8396 y0.1208 2.0217 0.7798 2.6647 1.6386 8.6445 y2.8556 4.6939 y1.2227 y2.1237 y1.5681 y3.6074 0.6296

1.1242 1.0832 2.4015 y1.5534 y3.1422 2.0345 0.7201 0.3465 1.8497 y0.1120 2.0159 0.7902 2.6756 1.6529 9.3474 y2.8918 4.6901 y1.1653 y2.1315 y1.5618 y3.4168 0.5535

1.1248 1.1505 2.4148 y1.5813 y3.4161 2.0955 0.6517 0.3561 1.8514 y0.1161 2.0468 0.7855 2.7129 1.6643 9.2687 y2.9895 4.6626 y1.1516 y2.1139 y1.5892 y3.7719 0.7172

1.1505 1.1790 2.4418 y1.5439 y3.3040 2.1688 0.6827 0.3655 1.8524 y0.1127 2.0416 0.7867 2.711 1.6648 9.6613 y2.9951 4.6714 y1.1146 y2.1154 y1.5838 y3.6461 0.6688

1.1542 1.1932 2.4537 y1.5408 y3.3119 2.1957 0.6827 0.3684 1.8529 y0.1128 2.0460 0.7877 2.7167 1.6664 9.6737 y3.0052 4.671 y1.1070 y2.1140 y1.5859 y3.6615 0.6791

1.204 a 1.204 a

0.17 0.08

0.15 0.09

0.10 0.06

0.19 0.13

0.19 0.12

0.14 0.10

0.135 0.09

0.128 0.09

y1.44"0.03 b y3.34"0.11a 2.675"0.133 a 0.44"0.02 b 1.96"0.02 b y0.10"0.02 b

2.78"0.09 c ,

2.78"0.02 d 1.75"0.02 a

y1.09"0.06 b y2.42"0.04 b y1.56"0.7 d y3.94 e 0.97 e

A.J. Cohen, Y. Tantirungrotechair Chemical Physics Letters 299 (1999) 465–472

C 2 H 4 Ž xx . C 2 H 4 Ž zz . Cl 2 CO CO 2 CS 2 F2 H2 H 2 OŽ xx . H 2 OŽ zz . H 2 SŽ xx . H 2 SŽ zz . HCl HF Li 2 LiH LiF N2 NH 3 PH 3 SO 2 Ž xx . SO 2 Ž zz .

A.J. Cohen, Y. Tantirungrotechair Chemical Physics Letters 299 (1999) 465–472

469

Table 3 The distributed multipole analysis of CO Žin au. from different methods

q ŽO. q ŽC. m z ŽO. m z ŽC. Qz z ŽO. Qz z ŽC.

SCF

MP2

BD

BLYP

HCTH

B3LYP

B97

B97-1

y0.2921 0.2921 y0.1789 0.7029 0.6730 y0.3594

y0.0255 0.0255 y0.4864 0.6615 1.0737 y0.1156

y0.0516 0.0516 y0.4860 0.6358 1.1062 y0.1755

y0.0252 0.0252 y0.5998 0.7292 1.2336 0.0092

y0.1052 0.1052 y0.4011 0.7036 0.8714 y0.0456

y0.0895 0.0895 y0.4897 0.7195 1.0818 y0.0731

y0.1037 0.1037 y0.4141 0.6810 0.9285 y0.1234

y0.1005 0.1005 y0.4204 0.6777 0.9439 y0.1301

respect to the experimental values, the quality of DFT results for the permanent multipole moments is in the order B97-1 ; B97 ) B3LYP ) HCTH ) BLYP. We believe that the systematic under-prediction of the permanent multipole moment comes from the too-diffuse nature of the electron density produced by DFT.

In the case of CO, it has been known that SCF predicts the wrong direction of the dipole moment. This direction is corrected upon the inclusion of electron correlation. However the BD dipole moment is still too low. DFT results, on the other hand, are much more encouraging. The B3LYP result is comparable to that of BD. The new functionals, B97 and

Table 4 The average electric dipole polarizability, a , in au SCF

MP2

BD

BLYP

HCTH

B3LYP

B97

B97-1

Experiment

C2 H4 CH 4 Cl 2 CO CO 2 CS 2 F2 H2 H 2O H2S HCl HF Li 2 LiH LiF N2 NH 3 PH 3 SO 2 trans-C 4 H 6

28.13 15.91 29.89 12.23 15.81 55.07 8.58 5.12 8.51 23.77 16.67 4.88 202.30 24.044 7.46 11.42 12.94 29.93 23.75 57.21

27.56 16.54 30.56 13.09 17.82 57.15 8.22 5.10 9.80 24.70 17.37 5.67 203.99 26.20 9.87 11.45 14.42 30.69 26.18 55.14

26.99 15.07 29.13 12.92 17.84 64.51 8.23 5.00 9.37 23.52 16.85 5.56 213.21 28.65 9.31 11.82 13.78 30.30 26.58

29.09 17.59 31.97 13.66 17.92 56.35 8.96 5.53 10.64 26.07 18.54 6.26 197.90 34.10 12.32 12.27 15.62 32.13 26.75 59.32

28.26 17.11 30.89 13.33 17.40 54.73 8.69 5.37 10.26 25.17 17.84 6.02 207.31 33.63 11.82 11.94 15.08 31.31 25.97 57.88

28.47 17.03 31.16 13.18 17.31 55.61 8.69 5.39 9.96 25.24 17.90 5.83 195.44 30.38 10.63 11.92 14.73 31.35 25.75 58.03

28.21 16.90 30.89 13.08 17.19 55.09 8.62 5.32 9.85 24.98 17.73 5.77 199.92 30.07 10.57 11.84 14.57 31.01 25.55 57.64

28.24 16.89 30.91 13.06 17.17 55.15 8.62 5.31 9.82 24.98 17.73 5.75 198.96 29.74 10.46 11.83 14.54 31.01 25.52 57.69

27.70 a 17.27 a 30.35a 13.08 b 17.51a 59.79, 55.28 b 8.38 a 5.41, 5.428 b 9.64 a 24.71c 17.39 a 5.6 a 216 " 20 d 25.9137, 24.834

Abs. mean error Rel. mean error

1.76

0.95

1.29

2.25

1.38

1.79

1.50

1.53

0.06

0.02

0.05

0.07

0.04

0.034

0.03

0.029

a b c d e

Ref. w10x. Ref. w26,27x. Ref. w21x. Ref. w31x. Ref. w32x.

11.74 b 14.56 a 30.9 " 1e , 32.03 a 25.61a

A.J. Cohen, Y. Tantirungrotechair Chemical Physics Letters 299 (1999) 465–472

470

B97-1, both show surprisingly good agreement with experiment. Due to the accuracy of the result for the CO dipole moment, we therefore decided to investigate further into how charge is partitioned in this molecule. There are several schemes proposed to partition the molecular charge distribution, such as Mulliken population analysis, Stone’s distributed multipole analysis w17x or Bader’s atom in molecules w18x. We choose Stone’s distributed multipole analysis ŽDMA. scheme due to its availability in CADPAC. The DMA of CO is reported in Table 3. DMA is essentially a more systematic way to partition the charge distribution in basis-set space w17x. There is a large charge accumulation at C and O in SCF confirming the overpronounced polarity. The electron correlation reduces this charge excess such that in MP2 and BLYP there is almost no charge left at C and O. Inclusion of electron correlation causes the

distributed dipole moment at oxygen to be enhanced by a factor of 2.3–3.2 whilst there is only a slight change of the distributed dipole moment at carbon. The new functionals, B97, B97-1 and HCTH, yield about the same distributed charge but different distributed dipole moments. Our results for polarizabilities are reported in Tables 4 and 5. We choose to report the average and the anisotropy of the polarizability instead of its individual components. We do not attempt to evaluate the higher order polarizabilities due to the demanding basis sets needed. Nevertheless, we would expect the results to have a similar trend to that of dipole polarizabilities. Our results for SCF and MP2 are comparable to those reported by McDowell et al. w10x. Our coupled cluster results differ from theirs as no triple excitations are included in our calculations. Our BD result is generally lower than MP2. This is the same trend

Table 5 The anisotropy dipole polarizability, D a a , in au SCF

MP2

BLYP

HCTH

B3LYP

B97

B97-1

Experiment

C2 H4 Cl 2 CO CO 2 CS 2 F2 H2 H 2O H2S

13.05 18.30 3.37 12.04 56.39 9.01 2.16 1.00 0.83

10.47 16.56 3.91 14.93 62.03 4.91 2.21 0.40 1.53

10.83 16.78 4.01 14.37 62.18 5.75 2.23 0.55 0.78

11.60 16.46 3.77 13.96 56.15 5.78 2.59 0.17 1.53

11.58 16.24 3.75 13.65 55.42 5.62 2.46 0.28 1.22

11.79 16.75 3.67 13.59 56.10 6.32 2.44 0.52 0.98

11.73 16.60 3.67 13.46 55.62 6.21 2.37 0.50 0.91

11.73 16.63 3.66 13.44 55.62 6.26 2.36 0.52 0.90

11.4 b 17.53 b

HCl HF Li 2 LiH LiF N2 NH 3 PH 3 SO 2

1.87 1.28 72.56 3.25 0.52 5.37 0.52 0.95 12.36

1.67 1.10 97.07 3.04 0.82 4.44 1.93 1.68 13.25

1.68 1.17 140.33 3.00 0.79 4.82 1.64 1.43 13.64

1.33 1.16 98.20 0.39 1.05 5.04 2.68 2.34 12.90

1.43 1.12 102.05 1.72 1.17 5.06 2.53 2.08 12.75

1.52 1.20 91.53 1.26 0.78 5.04 1.91 1.89 12.76

1.53 1.19 94.76 2.12 0.86 5.03 1.88 1.79 12.65

1.54 1.19 93.83 2.13 0.85 5.02 1.83 1.77 12.65

1.19

0.52

0.37

1.00

1.02

0.84

0.90

0.90

0.19

0.19

0.07

0.27

0.21

0.11

0.11

0.10

Abs. mean error Rel. mean error a b c d

BD

14.17 " 0.07 d ,13.3 b 63.84 c ,62.42 c 2.12 "0.03 d 0.66 b 0.65 b , 0.67 c , 0.68 c 1.45 b , 1.96 c 1.33 b

4.70 " 0.04 d 1.94 b 13.0 b

The polarizability anisotropy is defined as a 3 y Ž a 1 q a 2 .r2, where a 3 0 a 2 0 a 1 are the principal polarizabilities. Ref. w10x. Refs. w26,27x. Ref. w33x.

A.J. Cohen, Y. Tantirungrotechair Chemical Physics Letters 299 (1999) 465–472

as in McDowell et al. w10x. As already reported by these authors, the BLYP functional gives rather too large polarizabilities. The HCTH functional gives an improved result but is still not satisfactory. This again implies the rather diffuse nature of the electron density in the outer region. One could also view this in terms of the energy gap in the denominator of the polarizability, aab A 1rŽ Ej y Eo . f 1rŽ I y A.. The ionisation energy, I, computed by DFT is known to be too small while the opposite is true for SCF. Therefore the DFT polarizability is too large and the SCF one is too low. By including exact exchange, as in B3LYP, and especially B97 and B97-1, the polarizability decreases significantly. The anisotropy of the dipole polarizability has a larger variation than, but nevertheless follows the same trend as, the other properties considered. This means that it is more difficult to obtain good results. The experimental data for the anisotropies are much less certain due to the experimental difficulty. There is generally insufficient information to deduce static values. The comparison with the experimental values is much more difficult. The relative error for BLYP and HCTH is rather large compared to both SCF and MP2. The result of B3LYP and B97 cases are in between that of MP2 and BD. The new functionals again yield improved results over the popular functionals. After this work had been completed, it came to our attention that Hartmut and Becke have refined the B97 functional by refitting the functional form to the training sets of G-2 quality w19x. We do not investigate the performance of these new functionals, but expect that the new functionals perform slightly better than the original B97 but of the same quality as the B97-1 functional of Hamprecht et al. w9x as reported in this Letter.

4. Conclusions Our results indicate that calculations of molecular properties are improved by the use of the new functional forms. Notably HCTH is an improvement over BLYP, and B97 and B97-1 are improvements over B3LYP. The general trend of performance is in the order B97-1; B97 ) B3LYP) HCTH) BLYP. We compare the result using the experimental values as a

471

standard. The improvement of the new functionals should be contributed to a more flexible functional form. B97, for example, contains 10 parameters compared to three parameters in B3LYP. This flexibility should improve the description of the electron density. The errors calculated are dependent on the different experimental values selected. The molecular properties obtained from experiment tend to be the rovibrational-averaged ones. Moreover the response quantities such as the polarizabilities are more likely to be the frequency-dependent properties. It is our belief that the fairest way to compute the error is to compare the DFT results with the converged ab-initio value.

Acknowledgements AJC and YT would like to thank Dr. D.J. Tozer and P. Fischer for checking some of the polarizability calculations and to Professor N.C. Handy and G.K.-L. Chan for some helpful discussions. AJC acknowledges receipt of an EPSRC scholarship. YT acknowledges receipt of a Thai Government scholarship.

References w1x R.G. Parr, W. Yang, Density Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1989. w2x W. Kohn, L.J. Sham, Phys. Rev. A 140 Ž1965. 1133. w3x A.D. Becke, Phys. Rev. A. 38 Ž1988. 3098. w4x C.T. Lee, W.T. Yang, R.G. Parr, Phys. Rev. B 37 Ž1988. 785. w5x A.D. Becke, J. Chem. Phys. 98 Ž1993. 5648. w6x A.D. Becke, J. Chem. Phys. 107 Ž1997. 8554. w7x A.D. Becke, J. Chem. Phys. 84 Ž1986. 4524. w8x L.A. Curtiss, K. Raghavachari, G.W. Trucks, J.A. Pople, J. Chem. Phys. 94 Ž1991. 7221. w9x F.A. Hamprecht, A.J. Cohen, D.J. Tozer, N.C. Handy, J. Chem. Phys. 109 Ž1998. 6264. w10x S.A.C. McDowell, R.D. Amos, N.C. Handy, Chem. Phys. Lett. 235 Ž1995. 1. w11x K.P. Huber, G. Herzberg, Constants of Diatomic Molecules, Van Nostrand Reinhold, New York, 1979. w12x D.R. Lide ŽEd.., Handbook of Chemistry and Physics, CRC Press, New York, 1994. w13x A.J. Sadlej, M. Urban, Theochem-J. Molec. Struc. 80 Ž1991. 147.

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A.J. Cohen, Y. Tantirungrotechair Chemical Physics Letters 299 (1999) 465–472

w14x A.J. Sadlej, Collec. Czech. Chem. Comm. 53 Ž1988. 1995. w15x A.J. Sadlej, Theoret. Chim. Acta 79 Ž1991. 123. w16x CADPAC: The Cambridge Analytic Derivatives Package Issue 6, Cambridge, 1995. A suite of quantum chemistry programs developed by R.D. Amos with contributions from I.L. Alberts, J.S. Andrews, S.M. Colwell, N.C. Handy, D. Jayatilaka, P.J. Knowles, R. Kobayashi, K.E. Laidig, G. Laming, A.M. Lee, P.E. Maslen, C.W. Murray, J.E. Rice, E.D. Simandiras, A.J. Stone, M.-D. Su, D.J. Tozer. w17x A.J. Stone, Chem. Phys. Lett. 83 Ž1981. 233. w18x R.F.W. Bader, Atoms in Molecules: the Quantum Theory, Oxford University Press, UK, 1991. w19x L. Hartmut, A.D. Becke, J. Chem. Phys. 108 Ž1998. 9624. w20x S.A. Clough, Y. Beers, G.P. Klein, L.S. Rothman, J. Chem. Phys. 59 Ž1973. 2254. w21x A.J. Russell, M.A. Spackman, Mol. Phys. 90 Ž1997. 251. w22x J.S. Muenter, W. Klemperer, J. Chem. Phys. 52 Ž1970. 6033.

w23x F.J. Lovas, E. Tiemann, J. Phys. Chem. Ref. Data 3 Ž1974. 609. w24x A. Halkier, P.R. Taylor, Chem. Phys. Lett. 285 Ž1998. 133. w25x D. Patel, D. Margolese, T.R. Dyke, J. Chem. Phys. 70 Ž1979. 2740. w26x M.A. Spackman, J. Phys. Chem. 93 Ž1989. 7594. w27x M.A. Spackman, J. Chem. Phys. 94 Ž1991. 1288. w28x K. Wolinski, A.J. Sadlej, G. Karlstrom, ´ ¨ Mol. Phys. 72 Ž1991. 425. w29x H.M. Kelly, P.W. Fowler, Chem. Phys. Lett. 206 Ž1993. 568. w30x A.W. Ellenbroek, A. Dymanus, Chem. Phys. Lett. 42 Ž1976. 303. w31x V. Tranovsky, B. Bunimovicz, L. Vuskovic, ˇ ´ B. Stumpf, B. Bederson, J. Chem. Phys. 98 Ž1993. 3894. w32x J. Dougherty, M.A. Spackman, Mol. Phys. 82 Ž1994. 193. w33x A.D. Buckingham, in: C.T. O’Konski ŽEd.., Molecular Electro-Optics, vol. 1, Dekker, New York, 1976, p. 27.