Hylleraas-type wavefunction for lithium hydride

Volume 5 1, number 3

HYLLERAAS-TYPE

CHEMICAL PHYSICS LETTERS

WAVEFUNCTION

1 November 1977

FOR LITHIUM HYDRIDE

DC. CLARY and N.C. HANDY Department Cambridge,

of Theoretical UK

Chemistry.

University

Chemical Laboratory.

Received 9 June 1977 Revised manuscript received 22 July 1977

A variational Hylleraas-type wavefunction yielding an energy of -8.0630 hartree (=91’% of the correlation energy) is reported for the lithium hydride molecule at the equilibrium bond distance. The correlation of the electrons in both the core and valence etectron shells is explicitly accounted for by the inclusion of the interelectronic distance re in the wavefunction.

l_ Introduction

3/=X

(1)

Ci*i,

i

In a recent publication Clary [I] has given a detailed description of a technique for obtaining very accurate energies from compact wavefunctions for manyelectron diatomic molecules. These variational “Hylleraas-type” wavefunctions explicitly include the interelectronic distance rii_ Methods and formulae for calculating the cumbersome many-electron integrals that are required to produce the matrix elements of the secular equations for a many-electron diatomic rnolecule were described. Test calculations on He; and He2 were reported, these being to our knowledge the first variational Hylleraas-type calculations on any molecule with more than two electrons. In this study we demonstrate that the Hylleraas method can be readily applied to a four-electron heteronuclear diatomic molecule of considerable chemical interest - lithium hydride.

2.

Calculations

We briefly outline the method for the LiH calculation. More complete details are given elsewhere [ 1,2] . The wavefunction 9 is expanded as a linear combination of configurations of the form +, the coefficients of which are found by solving the secular equations

(2) The operator 0, makes the wavefunction antisymmetric to the exchange of any two electrons. x is the spin function written so that * is an eigenfunction of S2 x = $ (011 02 - P&)

(a304

- P3a4)

*

v takes the values 0 and l.@(r)

are non-orthogonal focal elliptical orbitals of the form

0(r)=~Pqqe-AEeB71,

5 = (ra + rb)fR >

(3)

con-

p,q=O,1,2,3;

a=@,-+,)/R

-

(4)

the distance between the nuclei a and b, was given the equilibrium value 3.015 bohr for the LiH calculation. One pair of optimum exponents ([A, B] defined in eq. (4)) were first found for both the core and valence orbitals of LiH from a small a-C1 calculation using orbitals defied in (4). The resultant expcnents [4.021,4.371] for the core and 11.318, - 1.2291 for the valence shell closely correspond to an ionic electronic configuration of the form Lii H-. To obtain a

R,

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CHEMICAL PHYSICS LETTERS

good energy from a o-C1 calculation it was also found necessary to use an extra set of valence exponents ~1.065,0..595] to represent the polarisation of one valence electron towards the Li atom. in extensive a-type CI calculation was then performed by using orbit& defined by these exponents and the values of p and 4 in (4)_ All configurations formed by single, double, triple and quadruple substitutions from the “ground” configura’tion *,

= exp[-4.021($,

I November 1977

Table 1 Variational calculations on LM Reference

Method

Bond distance (bohr)

Energy (hartree)

this work

ground config.

[31 this work 1111

Hartree-Fock U-CI separated-pair

3.015 3.015 3.015 3.015

valence-bond

3.060

[51

+ &) + 4_37l(r, 1-k q2)

t41

[31

- 1.3 18(E3 + .&) - 1.229(7~ f q4)]

(5)

were included in the configuration set. The basic orbitals are of a crude form (the ground configuration had an energy of -7.9320 hartree compared to the Hartree-Fock [3] value -7.9871 hartree) and we found that the particular energetic contribction from many configurations was small but still greater than 0.00001 hartree. Thus the accumulative effect of a larger number of such configurations was significant and we required 117 terms to get an energy of -8.03 16 hartree. Two configurations were generated by multiplying the ground configuration defined in (5) by an rii function to reptesent the core (r12) and valence (r34) correlation. The r12 term decreased the energy to -8.0499

hartree and upon adding the ~~~ valence term to the wavefunction we obtained a final energy of -8.0630 hartree (~9 1% of the correlation energy)_ An “Tag” variable was also introduced to represent the corevalence intershell correlation but the energetic effect of this configuration on a small expansion set was less than 0.0001 hartree and the term was discarded. The final wavefunction GF can be represented by +‘F = 9, + &&‘L2

+dr34)X

+D is the u-C1 wavefunction,

+‘Gl -

(6)

c and a! are constants_

3. -Discussion In table 1 we list some previously published values for the energy of LiH obtained using alternative variational techniques. The calculations of Browne and Matsen [4] and Brown and Shull [S] both utilized a confocal-elliptical non-orthogonal valence-bond formalism similar to ours but with no ‘;i- functions. Despite 484

this work

valence-bond

3.046

-7.9323 -7.9871 -8.03 16 -8.0542 -8.0556 -8.0561

CI

3.0147

-8.0606

“exact”

3.015 3.015

-8.0647 -8.0705

Hylleraas PNO-CI

[cl 1121

3.015

-8.0630

the fact that they used many more orbital exponents than we did and also applied Z-orbit&, their energies are well above our value. In fact, our result appears to be superior to the energies reported in all previous publications except that produced in the PNO-CI study of Meyer and Rosmus, whose value is only 0.0017 hartree below ours [6]. It will be-of much interest if cur method can be improved so that an energy can be obtained for LiH

to an accuracy approaching that achieved by Kokos and Wolniewicz in their Hylleraas-type calculations on H, [73 _ We have used crude “minimai basis” orbitals as a starting point in these calculations and previous CI calculations [3] suggest that we could have obtained a good o-C1 energy using far less configurations (and in particular far fewer “triple” and “quadruple” orbitals substitutions) if we had used superior orbitals of the form h=

CaiOi i

f

(7)

where the Oj could be confocalelliptical or Slatertype orbit&. SIater-type orbitals can be expressed as

a linear combination of confocal-elliptical orbitals. In our calculation the ‘ii variables were multiplied by the minimal basis “ground” configuration (5) which closely represents the ionic structure LifHand gives bias to this electronic distribution. However, the o-C1 calculation suggests that it is import%& also

a strong

to represent the structure with one valence-electron strongly polarised towards the Li atom. For this reason it is no surprise that our wavefunction yields a dipole

Volume 5 1, number 3

CHEMICAL PHYSICS LETTERS

1 November 1977

moment (6.078

debye) which is larger than the “exact” value of 5.83 debye [8] since the configurations containing the_rii variables are the leading terms in the configuration expansion. The dipole moment is, of course, critically dependent on the basis set used and previous calculations [3] have shown that a wavefunction which gives a good energy can yield a poor dipole moment. One approach that could clearly be followed to attempt to obtain more reliable values for properties such as the dipole moment and an improved value for the energy from a compact wavefunction would be to use terms of the form O&

A: A; x) 7

(8)

where &, hb could, for example, be good SCF orbitals. Indeed the Hylleraas-type calculation on the neon atom of Clary and Handy [9] pointed the way to calculations of this type. We were unable, however, to carry out calculations using configurations of the form (8) with the computational facilities available to us and we give a simplified discussion for the reason for this below. The alternative approach to using configurations of this form would be to generate a large number of configurations by combining the rii variables with the leading terms in our a-C1 expansion_ This would be a very expensive and computationally cumbersome procedure although it has been possible to perform analogous CI Hylleraas calculations using Slater type orbitals on small atoms such as beryllium [lo] since the techniques for calculating the many-electron integrals are far less complicated in the atomic case. The rate determining step in our Hylleraas-type calculation is the computation of the two-, three- and fourelectron diatomic integrals and our method of integral evaluation [l] , reduces the expressions for these integrals to sums of products of K functions and other basic functions.

X eHAaE eBarlPl(.$)

P; (q) .

81

Pi(t) is an associated Legendre polynomial. By separating .$and 9, K is expressed as a product of C and D functions

(10)

1 D? = fi,a

s

dr) r)qaeBas P,S (q) .

(11)

-1

The variable z refers to an integration point and since the most cumbersome many-electron integrals were reduced to a two-dimensional numerical integration z can take N(N + 1)/2 different values, N being the number of integration points in one dimension [2]. If SCF orbitals are used instead of minimal basis set orbitals then it is necessary to replace $Pa gqa eeAae eBaq h (9) by a linear combination

of many such functions. The K functions themselves would then be more expensive to evaluate but this is not important as each one is calculated and stored once only as the first step in the calculation. In particular the label “a” of the K function could just as easily refer to a product of two SCF orbitals as a product of two minimal basis orbitals. If all the required K functions could be stored in core it would be a relatively inexpensive procedure to compute matrix elements between configurations of the form (8). We would require at least 800 X lo3 bytes to perform such a tabulation for L,iH as the p and ~.r’indices would range from 0 to 16, “a” could have at least three different values and we would use 24 integration points, z thus having 300 different values. Only when we use a minimal basis set or orbit& of the restricted product form

(12) can the K functions be expressed as a single product of C and D functions_ Extensive sums over the C and D functions would be required for SCF orbitals. H INever, when using minimal basis set orbitals the C and D functions require only l/g’ of the storage space that would be needed for the K functions and can be tabulated. IBM 370/165 at Cambridge has a maximum storage capacity of 400 X LO3 bytes and it is for this reason that we can only perform Hylleraas-type calculations with minimal basis orbitals, not SCF orbitals. On the other hand, our discussion suggests that calculations using configurations of the form defined in (8) might

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well be the next advance in Hylleraas-type diatomic molecules.

studies of

Acknowledgement One of us (DCC) acknowledges

an SKC award.

References [l]

D-C. Clary, Variational Calculations on Many-Eiectron Diatomic Molecules using Hylleraas-type Wavefunctions, Mol. Phys. (1977),-to be published. [2] DC_ Clary, Ph.D. Thesis, University of Cambridge (1977), in preparation.

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1 November 1977

I31 C.F. Bender and E-R. Davidson, J_ Phys. Cbem. 70 (1966)

2675.

141 J-C. Browne and F.A. Matsen, Phys. Rev. 135 (1964)

A1227. [SI R.E. Brown and H. Shuil, Intern. J. Quantum Chem. 2 (1968) 663. PI W. Meyer and P. Rosmus, J. Chem. Phys. 63 (1975) 2356. r73 W. K&x and L. Wolniewicz, J: Chem. Phys. 41 (1964) 3663. 181 L. Wharton, L.P. Gold and W. Klemperer. J. Chem. Phys. 37 (1962) 2149. PI D.C. Clary and N.C. Handy, Phys. Rev. Al4 (1976) 1607. 1101J.S. Sims and S.A. Hagstrom, whys. Rev. A4 (1971) 908. 1111E.L. Mehler, K. Ruedenberg and D-M. Silver, J. Chem. Phys. 52 (1970) 1181. [I21 R. Valasco, Can. J. Phys. 35 (1957) 1024.