basis sets

Optimization of Gaussian-type basis sets for local spin density functional calculations. Part I. Boron through neon, optimization technique and validation

NATHALIE GODBOUT AND DENNIS R . SALAHUB'

De'partement de chimie, Universite' de Morzrre'ul, C .P . 6128 Succursc~le A , Morzrre'al (Que'bec). Carzadu H3C 357

A N D

JAN ANDZELM' AND ERICH WIMMER Cruy Reseurch Irzc., 655E Lone Ouk Drive, Eugnrz. MN 55121. U.S.A.

Received September 19, 199 1

This paper is dedicuted to Professor Sigeru Huzirlugu or1 the occasion of his 65th birthday

NATHALIE GODBOUT, DENNIS R. SALAHUB, JAN ANDZELM, and ERICH WIMMER. Can J. Chem. 70, 560 (1992). Gaussian-type orbital and auxiliary basis sets have been optimized for local spin density functional calculations. This

first paper deals with the atoms boron through neon. Subsequent papers will provide a list through xenon. The basis sets have been tested for their ability to give equilibrium geometries, bond dissociation energies, hydrogenation energies, and dipole moments. These results indicate that the present optimization technique yields reliable basis sets for molecular calculations.

Key words: Gaussian basis sets, density functional theory, boron-neon, geometries, energies of reactions.

NATHALIE GODBOUT, DENNIS R. SALAHUB, JAN ANDZELM et ERICH WIMMER. Can. J . Chem. 70,560 (1992). Des bases orbitalaires et auxiliaires de type gaussienne ont kt6 optimiskes pour des calculs de la fonctionelle de la densite

de spin locale. Ce premier article Ctudie les atomes de bore jusqu'au neon. Des articles subsequents fourniront des bases pour tous les atomes jusqu'au xenon. Les bases ont kte verifikes pour leurs capacitks a reproduire des gkomktries d'equilibre, knergies de dissociation de liaison, energies d'hydrogknation et moments dipolaires. Ces rksultats indiquent que la prksente technique d'optimisation produit des bases fiables pour les calculs molCculaires.

Mots cle's : Bases orbitalaires gaussienne, thkorie de la fonctionnelle de la densitk, bore-neon, geometries, energies de rkactions.

1. Introduction

1 .I General considerations Ever since their introduction, orbital basis sets have been

the subject of extensive efforts and discussions. Because of their critical role in determining the accuracy and because of computational cost, the subject of basis sets in molecular calculations continues to be of great interest. All computational approaches that solve the Schrodinger equation with the expansion method express the molecular wave functions in terms of a basis set that contains a finite number of basis functions. As mentioned in a review by Huzinaga (1), "it is this finiteness that is the cause of all the frustrations and fascinations about the basis set."

A number of reviews about basis sets have been pub- lished (2-7). The fact that there has been many a review article about the subject serves to show that the field is important and very much alive. There exists no one finite basis set for each element that is small enough to be used in molecular calculations with today's computers, yet is flexible enough to universally describe all properties. In fact, results of various degrees of accuracy are obtained depending on the choice of the basis set. The choice of the basis set is critical.

1.2 Basis set functions Several types of functions have been proposed. The two

more commonly used are the Slater-type functions (STF), introduced by Slater (8) in 1930, and the Gaussian-type functions (GTF) proposed by Boys (9) in 1950. The STO's are functions that represent the main features of the radial part

' ~ u t h o r to whom correspondence may be addressed. 'present address: Biosym Technologies, 10065 Barnes Canyon

Road, San Diego Calif. 92121, U.S.A.

of an atomic orbital. The problem in using these functions comes from the difficulty in calculating the integrals.

Boys showed that by taking GTF, complete systems of functions can be constructed appropriate to any molecule, and that the necessary integrals can be evaluated explicitly. Al- though they do not describe the atom as well as does the STF (a linear combination has to be used in order to represent an STO), they retain the advantage that the molecular integral formulas are straightforward to derive and to code.

The usefulness of GTF as basis functions for large-scale molecular calculations was put forward by Huzinaga (10) in 1965. He concluded that even if the number of GTF needed in the analytical Hartree-Fock expansion calculation would be larger than if STF were used, the number of GTF would not be prohibitively large. Currently, most molecular calculations are done using CGTF (Contracted Gaussian-Type Functions).

1.3 Basis sets of GTF Various methods to obtain expansions of S T 0 in terms of

GTO have already been proposed. A variational method (1 1) was developed but it suffered from the difficulty of achiev- ing true minimization because of the multiple minima problem which grew rapidly with the number of variational parameters. A least-squares method had also been proposed, Boys and Shavitt (12), but it also had the problem of multiple minima. Huzinaga ( lo , 13) proposed a method based on a least-squares-fit procedure that could avoid such a problem. In parallel, Pople and co-workers proposed the STO-nG method (14- 18).

A method based on least-squares fitting and atomic OP- timisation was proposed by Tavouktsoglou and Huzinaga (19). The emphasis was put on bonding valence orbitals in order to achieve accurate molecular results. In this method,

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GODBOUT ET AL. 56 1

the orbital exponents are optimized together with the contraction coefficients. While they remain at the 3G level (3 Gaussians to represent 1 STF), this was a new approach to their formation. This technique yielded very good molecular results for main group elements and succeeded with third row elements where the STO-3G basis sets had failed. This was mainly due to the fact that no least-squares fi t to the ST0 was done and therefore the basis set did not inherit the de- fects of the STO. A similar contraction scheme had been proposed by Clementi and co-workers (20) but was limited to the contraction coefficients.

This method was further improved by Tatewaki and Huzinaga (21) and applied to third row transition metals. The main problem with the Tavouktsoglou and Huzinaga method (19) was that, with the limited number of Gaussian functions used, primitive Gaussians were overspent in the inner core region of the atom. Pople's method was a success in that sense since it considered the electronic distribution through- out the atom. Huzinaga's new scheme was able to achieve a proper electron distribution. Valence shell orbital energies that were quite close to those of double-zeta quality were obtained using a minimal basis set with a modest representation of the core orbitals. The authors concluded that basis sets constructed in that manner should provide good molecular results.

These minimal basis sets could be improved further for molecular calculations by splitting the valence shell orbitals into two parts in order to better describe a molecular envi-

I ronment, as suggested by the work of Pople and co-workers (22) and Hehre and Latham (23) with the 4-31G basis sets. 1 The success of the method intitiated a series of articles by Huzinaga and co-workers in which basis sets were system- atically prepared for the atoms helium through cadmium (24- 29). A monograph by Huzinaga et al . (30) contains various basis sets developed in this method for all atoms of the pe- riodic table.

1.4 Basis sets in density functional theory 'The major drawback of the Hartree-Fock theory (HFT)

is that it neglects electron correlation, which is known to be important for many systems including transition metal com- plexes, organometallic compounds, and even relatively small highly correlated molecules like FOOF (3 1, 32), C2F2 (34), and CH3N0, and CH3NH2 (33). The HFT formally scales as N', N being the number of primitive basis functions, al- though some programs can perform as N3 or N2 due to the Gaussian-type decay of the localized basis sets and (or) the use of pseudospectral techniques (35, 36). When electron correlation is included, the scaling factor grows rapidly, making calculations on large molecular systems impossible unless some approximations are made to the integrals. There is growing evidence (33, 37-40) that density functional theory (DFT) (41-44) is a promising alternative to the Hartree-Fock approach (33). The DFT includes electron correlation and it scales formally as N3. With increasing system size, the computational effort scales between N2 and N3, while correlated Hartree-Fock based methods giving results of similar quality scale at least with the fifth power of N. It is believed (33) that these features of molecular DTF open opportunities for the study of large and complex molecular systems that are out of reach for other ab initio molecular orbital methods.

Various techniques are used to solve the Kohn-Sham equations. Those techniques can use various types of func-

tions. There have been implementations using Slater-type functions (45, 46), numerical functions (47-49), numerical functions generated from a muffin-tin potential (50, 51) plane-waves (52, 53), augmented plane-waves (54, 56), and Gaussian-type functions (57, 58). Basis set free solutions have also been introduced (59). The reason for using GTF in DFT, apart from the straight-forwardness of the integral evaluations, is that one can benefit from extensive Hartree- Fock experience.

The Gaussian-based implementation of the DFT equations relies on a variational, analytical approximation to the density. To achieve a variational fit to the density, the Coulomb energy term arising from the difference between the exact and the fitted density is minimized. This procedure was proposed by Dunlap et a l . (58). It leads to an analytical expression for the fitting coefficients involving Coulomb-type three-index two-electron integrals. It may be noted that, with Gaussians, the Coulomb terms may be evaluated exactly, if desired, by the usual methods involving N~ integrals.

The first implementation of DFT that used Gaussian functions (LCGTO-LSD, Linear Combination of Gaussian- Type Orbitals-Local Spin Density) employed extended uncontracted Hartree-Fock Gaussian basis sets, which inevitably limited the size of the systems that could be studied. Contracted basis sets were optimized in the LSD approximation for a series of transition metal atoms by Andzelm, Radzio, and Salahub (60) using Huzinaga's method (19, 21). It was actually found (60) that using contracted HF optimized basis sets in LSD calculations might not be appropriate since there are quite pronounced differences in the valence region between LSD and HF atoms that could in- fluence the molecular results. Furthermore, it was shown that Hartree-Fock optimized basis sets used in DFT calculations would likely create basis set superposition errors (BSSE). Hartree-Fock optimized basis sets such as 6-3 lG* (61, 62) can be used in DFT calculations to give good geometries, but accurate, and economical, predictions of reaction energies require the use of DFT optimized Gaussian- type basis sets (33).

1.5 Overview of the present paper In this first paper of a series, LSD optimized Cartesian

Gaussian basis sets for boron to neon are reported and their performance in molecular calculations is discussed. For that purpose, we have used a recent implementation of the density functional Gaussian-type orbital approach, DGauss (for Density-Gaussian) (33, 34). Subsequent papers will include basis sets for atoms through xenon. Equilibrium geometries, bond dissociation and hydrogenation energies, and electric dipole moment results obtained with the proposed basis sets will be presented and compared with experimental and ab initio results. It seems that the present technique of optimizing LSD basis sets yields high quality double-zeta- split-valence + polarization (DZVP) basis sets.

2. Theory

2 .I Atomic numerical calculation The optimization scheme of the contracted LSD-VWN

(Vosko, Wilk, and Nusair (63)) basis sets, based on the Huzinaga method (21) as implemented by Andzelm, has been described elsewhere (60). Only an abbreviated version will be given here. The atomic numerical calculations were done as a first iteration because they provide an exact solution to

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562 CAN. J . C H E M . \

the LSD equations and therefore a reference for the atomic calculation with an orbital basis set. The radial Schrodinger equation can be written in the following way:

where E, is the orbital energy, R,,,(r) is the radial function, and V,(r) is the total potential. There is one radial function at- tributed for each sub-shell. For example, for carbon, there are three radial functions: Rls(r), R2s(r), and R,,,(r) and they are obtained by solving [I].

2.2 Atomic calculation with an orbital basis set In the analytical SCF (Self-consistent Field) method, the

radial functions are approximated by a linear combination of basis functions denoted by s,(r),s,(r),. . .,sM,(r);p,(r),p,(r), . . . ,p (r);d,(r),d,(r),. . . ,dMd(r) for s, p , and d symmetry re-

Mp spectlvely.

The radial functions can then be written in the following way:

M,

[21 Rm(r) -. C cns.,~,(r) , = I

MP

[31 Rnp(r) = cnP,,~,(r) , = I

Md

[41 RAr) = cnd.,dl(r) 1 , = I 1

These radial functions can approach those obtained numerically if the basis functions and the length of the expansions M,, M,, and M, are well chosen. Theoretically there is an optimal form for these basis functions. It can be obtained by expressing it as a linear combination of simple functions, the primitives. The primitives chosen for this work, as mentioned in the introduction, are GTF,

The CGTF are then defined as K,,

[&I s,(r) = C d,~.k~gr(ar~.k;r) I= 1

KP'

P I p,(r) = dp,,kgp(ap,,k;r) k = 1

Kd,

[ 101 = C dd,,~d(ad,,,;r) k = 1

where K,,, d,,,,, and a,,, are the parameters to be varied. For

I example, substituting the s basis function in eq. [8], the ra-

I dial s function in eq. [2] becomes

where R,(r) are the approximate functions. Once the size of the K,, expansions has been chosen, the optimization of the atomic orbital basis set is done by optimization of the exponents and linear coefficients so as to minimize the total atomic energy.

For example, the procedure to optimize an orbital basis set

TABLE 1 . Coefficients and exponents for (41 / I* ) orbital ba-

sis sets of hydrogen

Hydrogen

for carbon can be described as follows. If the K2s(2)2p(2) electronic configuration is approximated using a (63/3) basis set, there are 6, 3, and 3 GTO used to represent the s , , s2, and p , CGTO (Contracted Gaussian-Type Orbital) of carbon respectively. First, the s,(r) basis function is varied so as to minimize the energy while keeping the s,(r) and p,(r) frozen, that is, the exponents and coefficients are not varied. The s,(r) function is then varied while the others are kept frozen. This process is repeated until consecutive calculations do not produce any improvement in the total energy within lo-' a.u. Most of the computing time was spent on the core orbitals as this is where most of the energy resides. The Hartree-Fock optimized basis sets (30) were used as the starting basis sets.

In ref. 60, a (4333/43/41+) expansion pattern was chosen for all the first row transition metal atoms. This choice was based on the orbital energies and molecular calculations as it fulfilled reasonable requirements of accuracy and economy. In the present work, three expansion patterns, which differ in the number of primitives used for the core orbitals, were optimized: (63/5) for boron through neon, (731 6) for boron through fluorine, and (83/7) for carbon and fluorine. The augmented number of primitive functions for the core orbitals is expected to reduce the BSSE. The orbital basis for hydrogen (Table 1) was derived from a least- squares fit to numerical LSD results followed by scaling by the usual factor (1.44, which corresponds to 1.2 for a hy- drogenic orbital (64)).

The basis sets are optimized as minimal basis sets but they are decontracted for molecular calculations. For example, the (63/5) set was split to a (621/41) set. To further improve the molecular results, a d polarization function was added and the basis set would become a (62 1 /4 1 / 1 *) set. The polarization functions were chosen from the previous experience in LCGTO-LSD calculations and from Pople's 6-31G* basis sets. This brings the basis sets to the quality of DZVP for the valence electrons.

3. Results and discussion 3.1 Basis sets and orbital energies

The LSD-VWN optimized orbital basis sets are shown in Tables 1-4. The basis sets found in these tables are as they were optimized, except for the d polarization functions, which was not optimized, and the hydrogen basis set that was decontracted as it was used. The electronic configurations of the atoms are ~ 2 ~ ~ 2 ~ " where n takes the value 1-6 when going from boron to neon.

The orbital energies as a function of the expansion pattern are shown in Table 5. The orbital energies calculated with the (63/5) basis sets are in very good agreement with

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TABLE 2. Coefficients and exponents for (63/5/1*) orbital basis sets for boron through neon

Boron Carbon Nitrogen

0.4000000 1 .OOOOOOOO 0.6000000 1 .OOOOOOOO 0.7000000 1 .OOOOOOOO

Oxygen Fluorine Neon

the ones calculated numerically and the ones with the (731 6) are in even better agreement. They are all calculated higher in energy than the numerical ones. This is also the case for the (73/6) set with the exception of the 1s & orbitals for boron, carbon, nitrogen, and fluorine, which are lower in energy. The average difference between the orbital energies calculated with an expansion and numerically (2) along with the standard deviations are shown in Table 5. The errors in the orbital energies for the (63/5) set are in the tenth of an eV range for both valence and core levels. We would therefore expect a good valence representation for most pur- poses, and also a small BSSE. The errors are even smaller for the (73/6) set. The (83/7) set, for fluorine, leads to some further improvement but probably not enough to justify the additional expense. Finally, the total energy of the system decreases with the size of the expansion as a better description of the atoms is provided. With these results in mind, we expect that these basis sets will perform well in molecular calculations.

3.2 Auxiliary sets Besides the orbital basis sets, fitting basis sets have to be

introduced to approximate the electron density and the exchange-correlation potential and energy. Sambe and Felton (57) first proposed fitting the electron density and the ex-

change-correlation potential, thereby obtaining a method that scaled as N~ rather than M. The fit to the electron density was later made variational by the work of Dunlap et al. (58). The electron density and exchange-correlation potential are ex- panded in auxiliary sets of Gaussian-type basis functions. The expansion of the s-, p-, and d-type uncontracted Gaussians provides a flexible enough representation of the atomic density provided the variational method of fitting is chosen. As for the exchange-correlation terms, the determination of the expansion coefficients has to be carried out numerically using a grid. The fit to the electron density is done analyti- cally.

These auxiliary sets are derived from the orbital basis sets, according to a procedure similar to that developed by Dunlap. The smallest exponent of the orbital set is used to start the expansion for the electron density set. Since the charge density is the sum of the squares of the molecular orbitals, this small exponent is multiplied by two. It was found that an economical expansion could be built, to represent the density well, by using an even-tempered expansion for the lowest rz - 1 exponents, and then breaking the even-tempered sequence for the tightest exponent. The auxiliary set is built using s-type functions and blocks of s-type, p-type, and d-type functions with shared exponents. This facilitates the calculation of molecular integrals. For carbon-like atoms,

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564 CAN. J . CHEM. VOL. 70. 1992

m 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 v i 0 0 0 0 0 0 e 2 8 0 0 0 m o w o r - O O O N o c n ~ m b ~ w m m m 0 , N W r - , t - m o m % s % " \ o : 8 m NO^ b ~ m m N W N - ~ ~ o r - O v i O 0 \ - N t - \ o m 8 a g 1 , ~ r ; ' D S $ z % % $ H 9 9 9 - ? t N 9'1'1 9 9 ? - ? N 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -

0 0 0 om2 o o o o t - w 8 P 8 , o o ~ z:, , o a w x z 8 g z E G 3 r ; E o w w N - r - \ o m

o c n w ,m3 g omz t -www m m r - 3Fz t -cn2 b ? =!=!=!? ?=!N I ? ? ? - ? * * N t - g m = b Z O O \ O z b - o o 0 z m o m ' D m 0

m o o t - w m m N b W r - c n O t- o m m r - N o \ o - Z N m o o o m O m m 9 9 9 ? ? b b 0 0 0 0 0 0 ~

0 0 0 0 0 ~ 0 o m - o o o o w - 8 8 8 8 8 g t - % K Z g O F g - Z Z o O O O I W m m m m m W W ~ - m w o w ~ c n t - t - e m c n ~ m s.gq ; q $ $ R S 8

r - o o m - m - d o o b -

t - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 t - m o ,,,8888 E 8 a 2 E 8 8 8 4 m o w \ ~ o m \ o z o o w \ o m - w m cnwbommo ~ \ o w z ~ e m dEG2"ocn $ S % : z o z 2 : 882:!"8 lr,Fa 8 % 3 G g z 9 9 9 - ' ? 6 . N 9'1'1 9 9 ? ? b N 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -

0 0 0 o m r - o o o - 8 8 8 8 s ; m g W C O W O 1 t - ~ 2 % W v i N -1%%;.lr,0f d W W m 0 0 g ,,mZKi3G' d H d W 0 6 c d ? - ? \ . = ! ? ? ? t ? S zz!g68 2 N - ' t - ~ b - m o o g E Z r n N

the use of three s-type and three p-type functions having the same exponents in addition to four s-type functions was found to work well. In our notation, this corresponds to a (4,3;4,3) auxiliary set where the semi-colon separates the charge density from the exchange correlation. The resulting set is shown in Table 6. The same pattern is used for the exchange-correlation set. It is built from the electron density set by dividing the exponents by 3 since the exchange correlation goes essentially as (58).

The auxiliary sets shown in Tables 6 and 7 were made to be used with the (63/5) and the (73/6) orbital sets respectively. Each of the sets in Table 7 have one more p- and d-type function. Typically, the calculations were performed with triple-zeta p- and d-type fitting sets, i.e., (4,3;4,3). It was found (Dixon in ref. 33) that quadruple-zeta sets do not change molecular results significantly. 3.3 Equilibrium geometries

The LSD optimized basis sets presented in this paper have already been successfully used in LSD (and NLSD, Non- LSD) calculations by several authors to study fluorinated methanes (Dixon et al. in ref. 33), a methylamine complex (Hill et al. in ref. 33), nitro compounds (Redington and Andzelm in ref. 33), organic molecules (Andzelm in ref. 33), and many small organic compounds (34) as a part of the validation of the DGauss code to study various properties such as equilibrium geometries, vibrational frequencies, energetics of different types of reactions, and electric dipole moments. For completeness, some of these results will also be shown in the present paper.

The (63/5) basis set pattern was chosen for the present calculations as we initially hoped it would become the de- fault pattern for DGauss. From other studies, this basis set seemed, at least on paper, to be a good compromise between economy and quality. In ref. 33, Dixon et al. tested these LSD basis sets as a function of the number of primitive functions for the core orbitals and the expansion pattern (DG- 1 to DG-9 in their notation) for the study of isodesmic reactions for fluoromethanes. Hartree-Fock optimized 6-31G** and 6-31 1G** basis sets were also included in the study. Their finding was that all the levels predict about the same bond distances but that the LSD bonds calculated with the Hartree-Fock basis sets come out shorter, but still they are respectable. This is in agreement with the statement by Andzelm (in ref. 33) that Hartree-Fock basis sets used in DFT calculations lead to accurate molecular geometries for this class of molecules (see Redington in ref. 33).

Andzelm and Wimmer (34) used the CH3NH, molecule to study the sensitivity of calculated geometric, vibrational, and energetic properties on the computational level. They found that the calculated bond lengths differ by 0.004 A or less when the different levels are compared. The largest variation in bond angles is 1.4" (for HCN). Overall they concluded that there is relatively little sensitivity of the geometric variables on the choice of the computational levels. For small molecules and fragments containing C, N, 0 , and F atoms with the (621/41/1*) orbital basis sets and (41/1*) set for hydrogen, they found that the C-C single bond lengths are typically 0.01 -0.02 A too short whereas the C-C double bond and c-C aromatic bond are within a few thousandths of an A of experiment. As for the C-H bond length, it is usually too long by about 0.01-0.02 A and this trend is even more pronounced for the G H , N-H, and

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GODBOUT ET AL

TABLE 4. Coefficients and exponents for (83/7/1*) orbital basis sets for carbon and fluorine

Carbon Fluorine

TABLE 5. Total energy (hartree) and orbital energies (eV) for atoms boron through xenon as a function of the number of primitives in the expansion pattern

Atom Basis -TE - € I S T - € I S 4 - E ~ S T - E ~ S J - 2 ~ T - ~ 2 p J

Boron (63/5) (73/6) Nurn."

Carbon (63/5) (73/6) Num.

Nitrogen (63/5) (73/6) Num.

Oxygen (63/5) (73/6) Nurn.

Fluorine (63/5) (73/6) (83/7) Num.

Neon (63/5) Num.

1 u (63/5) (73/6)

"Num. refers to the numerical calculation.

H-F distances. For C-0, C-N, C-F, N-0, and N-F bonds, they find agreement within 0.02 A or less. The bond angles deviate from experiment by less than lo. The homo- nuclear diatomics are fairly well described.

The orbital basis set effect, i.e., the effect of the number of primitive basis functions and decontraction pattern, on the geometrical parameters can be seen in Tables 8 and 1 1. The basis set notation is given at the end of Table 8. The orbital basis sets are given on the left side of the semi-colon and the auxiliary sets on the right side. The notation used in ref. 34

is also indicated. From Table 8, we conclude that the C-H bond distance in CH, and in the CH, radical does not vary significantly with the basis sets. The standard deviation is 0.002 A for both systems. In the case of the C-C, C-N, and C-0 bonds, the addition of a d polarization function affects the bond lengths more than in the case of the C-H bond. The average bond lengths and the standard deviations when considering the results from the five basis sets studied for the C-C, C-N, and C-0 bond lengths respectivdy are 1.518 + 0.005, 1.450 + 0.003, and 1.418 * 0.016 A.

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CAN. J. CHEM. VOL. 70. 1992

TABLE 6. Exponents for the (4,3;4,3) auxiliary sets for boron through neon, for use with the (63/5) orbital set

Boron Carbon Nitrogen Oxygen Fluorine Neon

TABLE 7. Exponents for the (4,4;4,4) auxiliary sets for boron through neon, for use with the (73/6) orbital set

Boron Carbon Nitrogen Oxygen Fluorine Neon

If we exclude the results obtained with the basis set without the polarization function, on the ground that polarization functions are necessary, these results bccome 1.5 17 t 0.005, 1.45 1 + 0.002, and 1.41 1 + 0.002 A and all the standard deviations are in the thousandths of an A. These results reinforce the previous conclusion that the geometrical parameters are not significantly affected by the orbital and auxiliary basis sets chosen for this study.

The equilibrium geometries for a series of compounds containing first-row atoms are shown in Tables 9 and 10. These geometries were obtained using the LSD (621 /41/ 1 *) orbital basis sets for carbon-like atoms and (41) orbital basis set for hydrogen. The triple-zeta auxiliary sets were built as previously described. The equilibrium geometries for ethane, ethene, and ethyne are shown in Table 9. The results show that this bond is very well described using our basis sets. The differences with the experimental values are of 0.014, 0.003, and 0.014 A for the triple, double, and single bonds respectively. The carbon-hydrogen bond is well reproduced, being too long by 0.01-0.02 A.

Table 10 shows the calculated equilibrium geometries for first-row heteroatom compounds. Again, we find our results to be in excellent agreement with the experimental data. With the somewhat limited number of compounds studied, we have derived some statistics using the average difference between the calculated and experimental bond lengths and angles (z). For the C-F(1), C-N(2), C-0(3), C-H(8), N-H(2), and 0-H(l) bond types we find the values of ( G ) of 0.005, 0.010, 0.009, 0.011, 0.013, and 0.01 1 A respectively, the numbers in parentheses being the number of values from which the values of ( Z ) were tabu- lated. p n average, all the bond lengths are calculated within 0.02 A and in many cases within 0.01 A of experiment, which is respectable. The highest deviation for the bond angles is 2.3" but most (six out of ten) are predicted within 1" of experiment. The average deviation regardless of the types of angles is 1". These values are in agreement with the previous statement that LSD can predict bond angl~s within 2.0" of experiment and bond lengths within 0.02 A. Out of the 11 angles calculated, four are calculated higher than the ex-

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TABLE 8. C-X(X = C,N,O) bond lengths as a function of the basis set

Basis CH3-CH3 CH3-NH2 CH3-OH

The following notation reads as: orbital set for C-like atoms, orbital set for H; auxiliary set for C-like atoms, auxiliary set for H

A: (621/41),(41);(4,3;4,3),(4;4) or DZV" B: (621/41/1*),(41); (4,3;4,3),(4;4) or DZVP" C: (621/41/1*),(41/1); (4,3;4,3),(3,1;3,1) or

DZVPP" D: (721/51/1*),(41/1); (4,4;4,4),(3,1;3,1) or

DZVPT E: (71 11/4I1/l*).(41/1); (4,4;4,4),(3,1;3,1) or

TZVY "Notation of ref. 34.

TABLE 9. Calculated and experimental equilibrium geometries for a series of organic molecules using the (62 1 /4 1 / 1 *) orbital basis

set for carbon and (4 1 / 1 *) for hydrogen

Point Geometrical Molecule group parameter Explt.

perimental value, six are smaller, and one is exact. As for the calculated bond distances, most overestimate the corre- sponding experimental value. In conclusion, the equilibrium geometries that we showed, calculated using the proposed LSD basis sets, provide structures that compare favorably with MP2 results, obtained with 6-31G* basis sets, and experiment.

3.4 Bond dissociation energies The bond dissociation energies, and the hydrogenation

energies, are calculated by subtracting the sum of the energies of the reactants from those of the products. Fully optimized geometries were used. The non-local corrections to the energy are calculated as a perturbation at the end of the LSD calculation using the functional forms proposed by Becke for exchange (65) and Perdew for correlation (66) and using the LSD equilibrium geometries.

The results from the study of the CH, -+ CH,' + H' reaction are in Table 1 1 . At the MP4(6-31G**//6-31G*) level, a 6-3 1G** MP4 single-point calculation using a Hartree-Fock 6-31G* equilibrium geometry, a 3 kcal/mol difference with experiment still remains. Increasing the size of the basis set (6-31 lG**) at the same level of theory does not change the accuracy. The LSD(VWN) results overestimated the dissociation energy by an average of 11.9 kcal/mol. This result is not suprising since it is known that the binding energy is overestimated in the LSD approximation. The use of non- local corrections, Becke-Perdew potential, greatly im-

TABLE 10. Equilibrium geometries for a series of organic molecules using the (62 1 /4 1 / 1 *) orbital basis set for carbon-like atoms

and the (41/1*) set for hydrogen

Geometrical Molecule parameter LSD

1.175 1.378 1.106

109.5 1.274 1.035 1.110 1.106

125.4 118.2 11 1.0

1.212 1.123

115.9 1.45 1 1.114 1.105 1.025

107.1 116.2 106.2

1.410 0.974 1.103 1.110

106.8 108.4 108.6

Exptl

TABLE 11. Calculated and experimental" CH bond length (A) and dissociation energy, D, (kcal/mol), of methane

Basis De RC-H ZPE sets/

method VWN BP CH, CH, CH, CH,

A B C D E H F ~ M P ~ ~ M P ~ ~ M P ~ ~ MP4' GVB CCCI Exptl.

"Hartree-Fock, non-DFT, and experimental values from ref. 61 b6-3 1G**//6-31G*. '6-3 1 lG**.

proves the D,, which is then only overestimated by 0.9 kcal/mol. This conclusion has been drawn by other authors (33). The D, is basis set independent as is the geometry and the zero point energy. For the D,, this can be seen by the small value of 0.2 kcal/mol of the standard devia-

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TABLE 12. Bond dissociation energies (kcal/mol) for the dissociation of a series of molecules

CH,-CH, CH3-NH? CH,-OH Basis sets/

method LSD NLSD LSD NLSD LSD NLSD

A B C D E HF' M P ~ " M P ~ " Expt.

"Orbital basis set: 6-3 1G**//6-3 IG*. "Orbital basis set: 6-31G**//6-3lG*.

tion at the LSD and NLSD level. Since the decontraction pattern and the number of core primitive functions does not affect the D, significantly, it is not certain that larger or more decontracted basis sets may improve the results.

Andzelm and Wimmer (34) computed the bond energies for the successive removal of H atoms from methane with a (62 1 /21/ 1 *) orbital basis set for carbon and (4 1 / 1 *) set for hydrogen. They found that the CH-H bond is overestimated at the LSD level by as much as 15 kcal/mol. The NLSD calculations bring results to within 2 kcal/mol of experiment. The C-C bond dissociation energies were also studied and the same conclusions as for the C-H bond were obtained in terms of the need to include non-local corrections to the energy. None of the correlated Hartree-Fock based methods came as close to experiment as the NLSD approach.

The bond dissociation energies involving the atoms C, N, 0 , and F in a reaction of the type A-B -+ A + B represent a difficult problem for the ab itzitio approach. The Hartree- Fock approach gives poor agreement with experiment whereas MP2 often brings agreement within a few kilocalo- ries per mole. Table 12 shows the dissociation energies for the C-C, C-N, and C-0 bonds for the CH,-CH,, CH3- NH,, and CH3-OH molecules as a function of the basis sets at the LSD and NLSD levels. The average D, and the standard deviations for these bonds in the order given above are 115.8 2 0.8, 114.2 ? 1.1, and 121.3 ? 3.7 kcal/mol. At the NLSD level, these values are 94.8 + 0.9, 90.4 + 0.6, and 96.5 + 3.4 kcal/mol. The deviations are larger than in the case of the C-H bonds, the highest being for the dissociation of the C-0 bond. The NLSD level is necessary to bring agreement to within 3 kcal/mol of experiment.

These results are preliminary and it would be dangerous to draw too general conclusions from them, but they are very encouraging since reasonable D,, within 1-2 kcal/mol of the experimental results, could be obtained and this at a lesser cost than that of perturbation theory. Certainly, more results are needed to complement the present ones but one conclusion that seems unavoidable is that non-local corrections are needed if a close agreement with experiment is to be expected.

3.5 Hydrogenation reactions The LSD and NLSD hydrogenation energies for a series

of reactions are found in Table 13 along with the Hartree-

Fock and Moller-Plesset results as compiled by Hehre et al. (61). In ref. 61 it is said that, at the Hartree-Fock level, the 6-3 lG* and 6-3 1G** sets are needed to bring the agreement to within 3-5 kcal/mol of experiment.

The calculation of energies of hydrogenation is quite new in D R . When looking at the LSD and NLSD results in Table 13, what is first surprising is that the inclusion of a d polarization function to the DZV (double-zeta-split-valence) basis set slightly deteriorates the agreement with experiment. In general, a polarization function improves the results. The average energies of hydrogenation and the standard deviations for the CH3-CH,, CH3-NH,, and CH3-OH molecules at the LSD level are respectively 17.8 ? 0.7, 24.7 + 1.4, and 27.5 2 1.7. At the NLSD level, these values are 18.9 + 0.6, 25.7 2 1.2, and 27.7 2 1.8. The standard deviations indicate a basis set sensitivity of 1-2 kcal/mol. Even if the basis sets cause fluctuations in the energies, the extent of this behavior is restrained to within 2 kcal/mol, which is reasonable. Further, no large discrepancies with experiment are found. At the LSD level, the agreement with experi- ments is on average within 1-3 kcal/mol. The non-local corrections to the density bring the agreement to within 2 kcal/mol or less. The correction is not as large as was the case with dissociation energies. With a few exceptions, the energies are underestimated.

Again, the present LSD optimized basis sets provide quality results. In their study Andzelm and Wimmer (34) found that there were some serious deficiencies for reactions involving triple bonds while the hydrogenation of the single bonds is fairly well described at the LSD level. The NLSD level performs as well as the MP2 with 6-31G**// 6-3 lG* orbital basis sets and is closer to the MP4 approach with the same basis sets. They also found that the agreement between NLSD and the experimental values corrected for zero-point energies is quite reasonable for many of the reactions but significant discrepancies of up to about 10% do exist.

3.6 Electric dipole moments The calculation of the electric dipole moments provides

an insight into the overall distribution of charge, which may be important to know when structures and reactivities of molecules are to be evaluated. This one-electron property can be measured experimentally so that comparisons with theory can be made. Generally, this comparison is limited to the

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TABLE 13. Calculated and experimental energies of hydrogenation reactions (kcal/mol)"

CH3CH3 + Hz CH,NH2 + Hz CH30H + Hz Method Basis set -+ 2CH, -+ CH, + NH, -+ CH., + HzO

MP2 MP3 MP4SDQ MP4 LDS

NLSD

Expt.

"All expt., Hartree-Fock, and Moller-Plesset results as compiled by W. Hehre in ref. 61.

magnitude of the vector as a lot of uncertainties often exist in the experimental directions. The dipole moments were calculated by computing the expectation value of the dipole moment operator, p.

Again we turn to the Hartree-Fock experience to give us an idea of the basis set effect (61) on the electric dipole moments. For diatomic and small polyatomic molecules, the 6-3 lG* model is the most successful of the theoretical models as it provides a reasonable description of the magnitude of the dipoles for a variety of different compounds. In going from STO-3G to 6-31G*, the mean absolute deviation difference is 0.35 D and from 3-21G* to 6-31G* it is 0.04 D. From this we may conclude that once a somewhat high quality of basis set is reached, the calculated dipole moment is not affected as much by the choice of the basis set. This conclusion can also be applied to compounds containing heteroatoms. Still, significant errors remain if we con- sider that the typical range of values for dipole moments is 0 to about 5 D. Near the Hartree-Fock limit, the magni- tudes of the electric dipole moments are frequently in error by a few tenths of a debye. From this comparative study, we expect that our basis sets will provide at least as good dipole moments as does the 6-3 lG* level.

The LSD dipole moments of two compounds as a function of the basis sets are shown in Table 14 along with the Hartree-Fock values. They were calculated with the equilibrium geometry as calculated by each of the basis sets. We do not feel that this will prevent us from having a discussion in terms of the basis set pattern as no major differences are found between the geometries. Looking at the LSD results as a function of the basis set, it is apparent that it is necessary to include a d polarization function to get agreement with experiment within 0.2 D or less. Disregarding the results without the d function, the highest deviation from experiment is of 0.193 D (CH3NH2) and the lowest one is of 0.046 D (CH30H). The best agreement is found with the basis set C. In all cases the dipole moments are overestimated. On average, the dipole moments for CH3NH2 and

TABLE 14. Dipole moments (debye) for heteratom-containing compounds

Method Basis set CH,-NH? CH,-OH

HF STO-3G//STO-3G 3-2 lG//STO-3G 3-2 lG*//STO-3G 6-3 lG*//STO-3G

LSD A B C D E

Expt .

CH30H, when considering all the basis sets studied, are 1.34 + 0.26 and 1.89 t 0.2 D respectively. If we neglect the results obtained with basis set A from our statistics, these values become 1.46 + 0.04 and 1.80 2 0.04 D. The important conclusion is that the deviations become smaller.

The effect of electron correlation can be tentatively ac- counted for as our basis sets are of similar size as the 6-3 lG* Hartree-Fock set. If that comparison can be made, the correlation has lowered the dipole moment by 0.2 D. The NLSD dipole moments are not shown here but other preliminary results tend to show that the correction should not be large (0.1-0.2 D) (67). In conclusion, for the two systems studied, the basis sets provide reasonable results but a more lengthy study is needed to determine trends and account for the 0.2 D (or less) difference with experiment that remains. 3.7 Final conclusions

We have proposed all-electron LSD optimized Gaussian basis sets for the atoms boron through neon. The optimization procedure used, based on the Tatewaki and Huzinaga method, provides high quality valence orbitals as judged by comparison with the energies and orbitals obtained by the numerical solution of the Kohn-Sham equations. We be-

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570 CAN. J . CHEM. VOL. 70. 1992

lieve that these sets will cause only small BSSE since they have a good representation of the core orbitals.

The equilibrium geometries, bond dissociation and hydrogenation energies, and electric dipole moments have been calculated with the newly optimized basis sets. The results that we have shown make us believe that we have basis sets that can provide information with a reliable accuracy. The equilikrium geomem parameters are predicted at least within 0.02 A and l o of experiment.

The bond dissociation energies for one homolytic dissociation process could be predicted within 1 kcal/mol of experiment and other results obtained using the proposed basis sets are quite encouraging. They are as good as the MP2 and MP4 results, with the 6-31G**//6-31G* orbital basis sets, in reproducing the experimental values. Such an agreement is obtained, though, only if non-local corrections to the energies are included. The hydrogenation energies can be calculated within 1-3 kcal/mol and 2 kcal/mol of experiment at the LSD and NLSD levels respectively using the proposed basis sets. The electric dipole moments were calculated within 0 .2 D of the experimental values. These conclusions are based on very preliminary studies. At first glance, the basis sets that we have proposed give results that are as good as those of calculations with other correlated methods and this can be an indication that they are ade- quate.

These basis sets were also used in other studies to calcu- late harmonic vibrational frequencies and energies of isodesmic reactions. Dixon (in ref. 33) found a much better agreement with experiment when using LSD optimized basis sets in LSD calculations than when using Hartree-Fock optimized basis sets for isodesmic reactions. He studied several patterns and sizes and concluded that the results are sensitive to the number of contracted functions in the core orbitals. The basis set with six primitives for the 1s orbital gives an error of about 4 kcaljmol whereas an improved treatment of the core by just one additional function decreases this error by more than 1 kcal/mol.

The vibrational frequencies in DFT are currently calculated with the second derivatives being obtained by numerical differentiation of the gradient at the equilibrium geometries, since the analytical second derivatives are not available yet. Andzelm (in ref. 33) calculated the vibrational frequencies, using the basis sets proposed in this paper, for a series of simple organic molecules containing C, N, 0, H, and F using the basis sets found in this paper. He found .a good agreement between the LSD and experimental data. Typically, the high energy frequencies are overestimated while the low frequencies are smaller than the correspond- ing experimental frequencies. The average percentage deviations in the calculated frequencies from the measured harmonic values are 4.5% at the DZVPP/LSD level. Andzelm and Wimmer (34), again using the present basis sets, calculated vibrational frequencies consistently closer to experiment than those obtained with the Hartree-Fock theory. The LSD results were at least as close to experiment as those obtained from the MP2 theory with similar size and quality of orbital basis sets.

Acknowledgments W e would like to thank Cray Research Inc. for their sum-

mer intern program which has rendered this work possible. W e also acknowledge the contribution of D. Dixon for

helpful discussions. Financial support from the Natural Sci- ences and Engineering Research Council of Canada and la Fondation pour la Formation des Chercheurs et 1'Aide a la Recherche is also acknowledged.

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