Basis sets of gaussian and Slater-type atomic orbitals

Volume 7; number 5

., ,:

..:. .. :

1 December 19:

CHEMICAL PHYSICS LETTERS

‘,,,

.Y

B’dS&SETS

OF

GAUSSIAN

AND

SLATER-TYPE

ATOMIC

ORBITALS

D. M. SILVER * Cenlre Em-opPen CECAM.

dc CaIcrtl Alomiqur cl Mol&rrIaire. Bt?fin~~~d 506. 91 Cantpus d’Orsay. Fv-ance

Receivcrl 9 October 1970

The‘concept of constructing Slater-type and gaussian-type atoms.

moleculnr wivcfunctions using basis sets whose members atomic orbit:ds is ex:lmined through pilot SCF calculations

1. INTRODUCTION

df = [2(2flf+l”,

Being the exact solutions to the radial Schriidinger equation for the hydrogen atom, the associared Laguerre functions would seem an appropriate choice of basis for the linear expansion of wavefunctions for more complex atoms. To reduce the labor of integral evaluations involved with the use of these functions, without sacrificing the accuracy of a calculation. radially-nodeless variants. now called Slater-type orbitals (STO), were introduced [l]. The STO’s have the form

(1) where YI,,, are normalized spherical harmonics and the radial normalization constant, N. is given by N = (2c~)~+l/’ [ (2JJ)! I- ‘I’.

(2)

For molecules, where various STO’s are placed on different nuclear centers, ample flexibility is provided but the multi-center two-electron integrals encountered are extremely difficult. To circumvent this problem. gaussiantype orbitals (GTO) have been introduced 12.31 since corresponding GTO integrals are quite tractable. A general form for the GTO’s can be written as follows

where the radial by

normalization

constant

is given

* Present &dress: Ap&ied Physics Laboratory. The Johns Hopkins University, Silver Spring. Maryland 20910,. USA.

‘r(JJ

consist of both on some simple

+ #‘”

and lY is the standard gamma function_ Using i-restricted class of GTO’s for which the integrals are simplest. namely, those with the parameter ‘Vi - 1” of eq. (3) limited to the values “I+ 2i”, where i is an integer, Huzinaga [4] showed that by taking a sufficiently large number of these GTO’s. one could achieve the accuracy obtainable with a much smaller ST0 basis set. Furthermore. these calculations demonstrated that for a given value of quantum number It the higher orbitals (i+O) did not contribute substantially over using a set of the simplest (i = 0) orbitals. Consequently. published gaussian basis sets usually contain only 1s and 2p functions [5-lo]. The problem of choosing an appropriate molecular basis set to achieve a certain level of accuracy in a calculation has been to decide between the following two schemes: (i) using a number of STO’s and tackling the time consumin computational job of integral evaluation: or (ii) using a much larger number of 1s and 2p GTO’s and tackling the time consuming data handling jo involved with the concomitant very large nurnbel of integrals generated. Two further possibilities arise: (iii) allowing the basis set to contain a number of STO’s (presumably many fewer than needed in (i) above) plus a number of GTO’s (certainly many fewer than would be required in (ii) above) and in this way, striking a balance that inccrporates much of the good and only little of the bad features of both previous schemes; and (iv) using a pure CT0 basis set but relieving the restriction on the type of GTO to be used (in particular, to include the functions 2s, 3p, 4d. etc. where the parameter “rl - 1” in eq. (3) takes

CHEMICAL PHYSICS LETTERS

Volume 7. number. 5

the value “I+2i cl”). relying on the expectation that the added flexibility could substantially reduce the basis set size from that required in (ii) above. In what follows. these two ideas are discussed and some preliminary numerical examples are given.

2. COMBINED ST0 AND GTO BASIS SETS An arbitrary collection whose members consist of both Slater and gaussian-type orbitals forms a valid representation of Hilbert space since each function contained in such a basis set is square-integrable in configuration space and each overlap integral between any two such functions exists as a finite sca!ar. An electronic wavefunction can therefore be expressed in terms of a direct linear expansion of this ml..ture of orbital types in the customary fashion. Because of the radial cusp at the origin for STO’s. these functions would most suitably be placed at nuclear positions whereas the GTO’s could easily be allowed the additional freedom of “floating” into non-nuclear molecular positions as well. The question concerning the feasibility of evalualing multi-center two-electron repulsion integrals involving both GTO’s and STO’s in various combinations can be considered from the point of view of the gaussian transform technique [ll] which is one of the successful schemes used for the evaluation of pure ST0 integrals. In the case of the general multi-center two-electron exchange integral over STO’s, one introduces an integral transformation of the kind. e-u?-

_ -

L

g,,

_-1,'2

ds s- 3/2 e- (Y2 k”

e-SY

2 ’

(5)

for each of the four STO’s involved in order to reduce their exponential radial dependence to gaussian form. The original configuration space integrations can now be easily evaluated leaving only the four transform integrations of eq. (5) to I be handled [ 111. Clearly. for each GTO appear’ ing in the integrand of a mixed exchange integral. 1. one of the transform integrations is totally eliminated thereby considerably simplifying the evaluation process. With a combined basis set. one would have some of the very difficult pure ST0 integrals to evaluate. some of the very simple pure GTO integrals and some mixed STO-GTO integrals of various intermediate difficulties. Combined basis sets can be constructed for distinct purposes and to achieve different levels 512

1 December 1970

of accuracy. For example. a minimal basis set of STO’s can be doubled _by augmenting it with GTO’s instead of adding forther STO’s in order to attempt to‘obtain close to “double-zeta,” accuracy; The effort required can be estimated by Looking at the approximate number of various types of integrals involved. lrruncating to the nearest order of magnitude, with an n orbital ST0 basis; there are n4.‘8 integrals. Adding 11 GTO’s to this set leaves the_ number of pure ST0 integrals unchanged while adding an equal number of pure GTO integrals plus the following: n? ‘2 integrals involving 3 STO’s and 1 GTO; 3rr4.:4 integrals with two of each type; and ~14,2 integrals containing 1 ST0 and 3 GTO’s. By contrast, doubling the number of STO’s would produce 2n4 pure ST0 integrals. On the other hand, if less accuracy is required, a much smaller set of STO’s can be chosen to describe those portions of a wavefunction that are more strictly atomic in character. such as inner shell electrons. while GTO’s are used for describing the remainder of the electronic structure of the molecule. The concept of combined orbital basis sets is not new. In 1936. Knipp 1121 had made calculations using basis sets consisting of STO’s and two-center elliptic orbitals. Although further similar calculations have been performed [13. 141. the use of elliptic orbitgk is specific to diatomic molecules ind is not too useful for general multi-center molecular situations. Combined ST0 and GTO basis sets had been suggested by Allen [15) but no calculations have appeared to test his idea. Recently. Cook et al. [16,171 have used the phrase “mixed basis” to refer to their technique for the approximate evaluation OF multi-center pure ST0 integrals. In their scheme, each ST0 allpearing in a two-electron integral is approxim:. led by a linear combination of two GTO’s so that ihe original integral can be evaluated from a sum of 16 pure.GTO integrals.

Because of this usage of the terms, the present scheme is being referred to here as “combined c basis”.

3. NONRESTRICTED

PURE GTO BASIS SETS

The restriction to using only the simplest of the GTO’s, namely the orbitals Is, 2p, etc. (i.e. the parameter “11- 1” of eq. (3) being limited to the value “Z”). enables integrals to be evaluated with ease. However, a particular set of functions not examined previously [4.8], namely the GTO’s 2s. 3p, 4d. etc. (i.e. the parameter ‘n - 1” of eq. (3) having the value “I + 1”). have certain proper-

,vobne

5

7, number

CHEMICAL FHYSICS LETTERS

ties in the neighborhood of the origin that make their use in a pure GTO basis set appear desirable._ .. Consider a wavefunction having the form

(6) whe& 9 represents a GTO. the subscript i signifies a GTO of the simplest kind (1s. 2p. or 3d. etc.), the subscript i signifies a GTO of the next highest kind (2s. 3p, or 4d, etc.) of the same symmetry as the orbitals in the first set, and ni and bj are corresponding linear expansion coefficients. With respect to the one-e!ectron hamiltonian.

If++-

z,r.

the local energy, given by EL=H*

(7)

EL. for the wavefunction

-cy ip-2

[(22 + 3)(Yj - 2+21

-@tTa+FfjjBjle-4y2]-1 ,

(8)

where the symbols ni and @i represent orbital exponents and Ai and Bj denote the appropriate radial normalization constant given in eq. (4) for the two kinds of orbital. respectively. Without the higher GTO’s. all bj are zero and at the origin, the local energy contains a simple pole limEL=-limZ,?-+eL. 7--O

(9)

r-o

the constant EL’

elimination of the pole in its local energy function at the origin although in this case. the simplest orbitals (1s. 2p. etc.) are sufficient for this purpose. Although the higher GTO orbitals offer the capability of improving the local properties of a pure GTO wavefunction in the vicinity of the ori. gin, they could prove less useful in other regior of space. Moreover. since orbital exponents ant expansion coefficients are generally determined on the basis of energy minimization. the higher GTO’s may not necessarily be exploited to their fullest extent in the nuclear region due to compc ing factors. In addition, it should be understood that the introduction of these functions into a ba: set would seriously decrease the simplicity of GTO integral evaluations.

4. CALCULATIONS

= -zr+{-Ja@ie7 i

where

December 192

@ is

411

x{ CniAje i

1

EL is given by

[Eafii”i(21+3)], i

[FU$il-

The inclusion of the higher orbitals this singularity since in this case, lim EL = -1im Y-O

6.9-i. EL,

(IO) can remove (11)

Y-Q

with the constant

6 having the form

6 = 2 + [Cb.&(I+1)1 j

31

[+4i1.

(12)

The wavefunction parameters can be adjusted With respect to Z in order to make 6 identically zero: An-ST0 wavefunction of the form expressed in eq. (6) gives a similar result with respect to

The motivation for this study of basis set con position stems from the desire to be able to obtain compact and accurate wavefunctions for mo ecular systems with the least amount of computational effort. As a first step in this direction. simple SCF calculations have been performed or several ground state atoms. These atomic resul serve as a guide for further work on molecular basis sets. The simplest atom to evamine is helium. fol which optimization of non-linear parameters is easily performed. Table 1 presents wavefunctions and energy results for three classes of basis set: pure STO. pure GTO and combined basi: sets. The two-member combined basis recover: 86% of the energy improvement between the “sin gle” and “double-zeta” pure ST0 sets; i.e. it is 1.2 kcal mole above the “double-zeta” STO. Starting with one STO. it is seen that adding a i: orbital is more advantageous than adding a 2s or 3s orbital in both the pure ST0 and combined sit uations. In the case of the pure CT0 basis. the 2s GTO is to be preferred. However. both in the two and three function GTO basis sets. it appear that replacing the 2s CT0 with two 1s GTO’s can give slightly more accurate energy values. On the other hand. other properties may be better described by using the 2s function: for instance. pure 1s GTO basis sets always must give rise to nuclear cusp-values of zero whereas the Is + 2s + 1s GTO basis has a cusp of 1.5 (the exact value should be 2.0). From eq. (12), it is expected that expansion coefficients for the 1s and 2s functions in the pure GTO basis sets should be opposite in sign. 513

CHEMICAL PHYSICS.LETTERS

VO~UIX-7. nuinber 5 Table 1

SC-Fwxvciunctiok

‘.

for the hdiiuk Coefficient

set :I)

Exponent

4

1s G

0.7670

B

Is E

1.6875

‘1 .oooo

Is E

1.4517

0.8412

Is E

2.8951

0.1633

Basis

C

D

E

F

G

H

I

J

;c

L

Al

s

0

1s E

1.5543

2s E

2.2606

1s E

1.6247

3s E

3.5069

Is E

1.66Q5

1s G

0.9216

1s E

1 .653ti

2s G

1.5214

Is E

1.6645

3s G

2.3626

1’.0000

0 . ‘i”73 _. I

1.62.37 13.w3’~ I_

0.0603

I.9989

0.4002

Is G

0.3829

0.6572

Is G

0.4850

Ii.6767

2s G

0.6062

-4 .379.5 -

Is G

1.2329

- 1s2-14

0.1666 0.6992

3s G

0.46.55

-2X)669

-2.HS-Ki2

3.OSOl -3.4241 1.2699

Is G

38.4750

0.0237

Is G

5.7829

0.1546

Is G

I :2426

0.4695

Is G

0.2981

0.5138

:I) E ::nd G denote ST0 *and GTO functions. ly. b) Encrgics given in ntbmic units. c) Ref. [Ai_

514

-2.74707

1 .-I968 -0.5369

Is G

Is G

-2.85952

2.0520

1s G

2s G

5: @SCUSSION

-2.86005

.

-1.09G9

1 Dcdcmbo~ 1970 .

-2.66663

-2.66164

-2.15663

Is G

Is G

-2.6&f

1.0420

0 .9’1%

3s G

-2.64766

-2.8552-l

-0.0464

”

-2.76216

-2:3OOW

1.0714 -0.07i9

0.2832

I.0102

Enerbb)

1.1645 -0.1 iO0

0 . 5792 .-

O.G-156

.,

I.1013 -0.1099

4:09i8

1s G

.

_. :I

The.entrie‘s @j table*i-f_6i_ll&&$ coefficieqts’in the’c~seof:~e‘ls.Ic’Zs;:ds’+ 2s 2 1s &d ‘1s t 2s. + 3s :GTO ha&is _?i$s she_: theie t&-b& ‘ti tih&ce & problems_ariiing due id humeri&al differencing. The- results ‘given in table:1 have been’ h&dchecked to ins~r.e;thal~he^~~ergies reported are ,valid at least .to four decimals assuming, eightplace accuracy i” the integrals. -‘-For healiier. atoms, -this effect Could lead to a serious source . : ; :-,’ :-_ of difficulty; Table.2, displays combined basis set SCF calculations performed on the ground states of be‘ryllium. carbon and neon.’ In &ach,case. the “single-zeta” ST0 basis of Clementi atid Raimondi 118 J was chosen and a gaussian orbital was added with its orbital exponent chosen by scanning the energy as a function of this parameter and picking the optimal value. In the case of Be. a second 1s GTO has been added to the three-orbital set in the same fashion. Using the energy difference between “single-zeta” ST0 results [18] and the fully-optimized “double-zeta’! ST0 results of Clementi [ 19 1 as a standard of comparison,. the addition of one GTO gives ‘73% while two GTO’s give 89% of the energy gain for Be. In the case of carbon. the addition of a 2p GTO gives 478 of this knergy gain but adding an additional 1s GTO with orbital exponent in the range of 10 seems to bring this gain up to about 70%. For neon. a 70% gain is found with the addition of a single 2p GTO.

atom

1.2237 -0.2311

1s. G

2s G

:. _

-2:w14

-2.$5616 a respective-

‘.

The pilot calculations reported here indicate that a combined STO-GTO basis set is capable of producing a wavefunction whose accuracy is close to that obtained with a wavefunction constructed from a pure ST@ fully-optimiied “double-zeta” basis set- [l?]. In particular, the combined basis sets for C and Ne are of smaller size and for all the atoms but Hk. the non-linear ST0 parameters have. not been optimized at all. Although this combined basis set scheme is of limited interest for the treatment of atoms per se, these calculations do provide a positive indication of what &ghf be expected for molticules. The use of higher orbitali in pure GTO basis sets.is not quite as encouraging.’ For,a’given level of accuracy in energy for He, th.e 2s orbital offers slightly_ smiller basis set sizes as Well as bet&r. descriptions of, the atomic structure near the ducleus than is obtainable~with’ 1s GTO’s alone: Hqwever; one.en&nte~s ‘much more difficult integrals and the podsibility‘ of &m&ical y.. differencing pl;oblems alsol

-Volume

7. number

Table 2 for the beryllium.

SCF wnvclunctions

..

Abarn .,

Basis

set

a)

Coefficients 1s --

NC

E

3.6S-18

0.99iG

2s E

0.9560

0.0124

1s E

:3.6Y48

l.OtiS9

2s E

0.9560

0.0070

Is G

a.7500

Is E

3.6848

1 .Ofi!iB

2s E

0.95W

0.0069

Is G

-l.7500

Is G

2.

Is E

2s

1.0182

0.1767

-0.0079

0.9974

-0.2351

2s E

l.tiOY3

Q.0114

2p E

1.5fii9

1s E

5.6727 i

O.!)!)i4

-0 . 27% ..

2s E

1.6063

O.Ollrl

1.0247

2p E 2p G

1.5679 0.5000 0.91)i-L

2 . 879s-

0.0109

“p E

2.9792

1s E

O.fi4Bl

O.!Kil

0.0119

a) E and G denote ST0 and GTO functions. h) Energies g&n in atomic units. c) Ref. [IS].

-37.62239

c)

1.3510

9.6-121

1.7000

1.0247

-0.3651

1s E

2~ G

-14.5FUGO

1 .oooo

2s E

2.8792

-1.1.56YOB

-1 .oPOi

.5.6-i27

2.8792

-14 . .i567.1 C)

0.0077

0.03’i’j L.

2s E

-

0.1973

-0.0724

:

-.

z.gyb)

-1.0176

0.0000

sp E

E. 2P

-0.2944

-0.0721i

IO00

197

:mtI neon :Itoms

carbon

Exponent

‘, Is

1 December

PHYSICS LETTERS

ciimrIc~L

5

-37_fi6”4Y

-0.2:50 1.0294 I .oooo

.

-127.X122~)

-O.X.?9 1.0294 1 .+&IS

-

-.-

-0.4643

-128.3164

---

respectirc(y.

ACKNOWLEDGMENTS

[3j R. McWecncy. Nature I6G (19.50) 21. S. HuzinnEa. J. Chcm. Phys. 42 (lIlfi5) 1293. 151 S. Huzinaga and Y_Snkni. J. Chcm. Phys. 50 (196!1) 1371. [6l S. Huzin:lgn ant1 C. Arnau. J. Chem. Phys. 5’2 (1!)70) .>O.>~ _I_ . [ij J. L.Whitlcn. J.Chcm. Phys. 99 (1963) 349: 4-l (19GG) 359. [6] W. J. Hchrc. R. F.Stcxtrt :md J. A. Pople. J. Chcm. Phys. 31 (1969) 2657. 191 A. Veillard. ‘t’hcorct. Chim. A&I (Berlin) 19 (19Gst 405. [lO] A.J. H.Wachter.s. J. Chcm. Ph.vs. 55 (1970) 1033. 1111 d. Chum. Phrs. 43 fl965b _ LShnvitt :md M.Knrolus. [-l]

Gratitude is expressed to Dr. Carl Moser and the Centre National de la Recherche Scientifique for the interest and support which enabled this research to be conducted.

REFERENCES

[;I J. CSlater, 1.21S.F.Boys.

Phys.Rcv. 36 (1930) 57, Proc. Roy.Soc. A200 (1950) 542.

398.

[I21 J.K.Knipp.

J.Chcm. Phys. 4 (193(i) 300. 515

Volume

[I:31 F.

7. number

5

CHEMICAL-PHYSICS

T. Ormxnd and F.A. Matsen. J. Chem.Phvs. 29 : :(1959) 100. [1.&iJ. C.&owe and F. A. M&son. Phys. Rev. Al35 (1964) LIZi. 1151 L.C. Allen. J. Chem. Phys. 31 (1959) 736. [I61 D. B. Cook and P.Pslmieri. Mol.Phys. 17 (1969) 271.

516

LETTERS

5.

,-

,.l December

1976

.i171 D. B. Cook. P. C. Hollis-and R: McWeenei, Mel; Phrs:13 f19671 553: .’ -. . [IS) E.klemo&i_anh D. L.+!m&idj. -J. Chem.,Phye. (1963) 2686. ’ [191 E. Clementi. ‘J. Cheq. bhys; 40 (i964)‘1944. ‘_

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