Solid State Communications, Vo1.39, pp.253-256.

Pergamon Press Ltd. 1981. Printed in Great Britain. 0038-1098/81/260253-04$02.00/O

AB INITIO CALCULATION OF TIGHT-BINDING PARAMETERS OF THE Ni-Ni, Ni-H, Ni-0, Ni-S BONDS

Y. Boudeville and J. Rousseau-Violet

Institut de Recherche sur la Catalyse, C.N.R.S., 69626 Villeurbanne Cedex, France

and

F. Cyrot-Lackmann

Groupe des Transitions de Phase, C.N.R.S., 166 X, 38042 Grenoble Cedex, France

and

S.N. Khanna

Laboratoire de Physique Experimentale, Ecole Polytechnique Fed&-ale de Lausanne

1007 Lausanne, Switzerland

Calculations of the tight-binding parameters of the Ni-Ni, Ni-H, Ni-0 and Ni-S bonds have been carried out using wavefunctions and effective potentials obtained from an atomic self consistent Hartree- Fock - Slater calculation. Results for the Ni-Ni bond agree with the available para- meters obtained through interpolation schemes. A study has also been made, of the variations of parameters with bond length and the para- meters are shown to decrease rapidly with increasing distance.

I. Introduction

The moments method and the continued frac- tion technique developed within the framework of tight-binding limit(l) has proved to be an efficient way to a treatment of wide range of problems including electronic structure of dis- ordered metals, alloys, surfaces, chemisorption etc..One of the ingredients in the application of this method is the knowledge of tight-binding parameters(l). For pure metals and some compounds these have been obtained from the interpolation schemes(2) which use the band-structure calcu- lations carried out using other methods like APW, KKR, etc..However, most of the applications like chemisorption, catalysis, etc., involve coupling between atoms for which corresponding system has not been studied by other methods, and one is faced with the problem of tight- binding parameters.

The parameters have to be calculated in a way that they could transfer information i.e. we want to apply parameters from a given bond, such as Ni-H, to various situations such as compounds, chemisorption, etc.. We are thus seeking a good insight into electronic struc- ture with atomic interactions which will work in a systematic and realistic way. Tight-binding parameters being short-range ones, we shall use two or three atoms descriptions (first or next neighbours) to determine parameters, from the best atomic SCF results.

Let us consider, for example, the case of d-bands, for a transition metal. Following Slater and Koster(3), we can write the tight-

binding parameters in terms of two center inte- grals, namely dda, dd-rr and dd& defined by

ddX = <iXlVIjX> (1)

Where i and j are the site index, j being a nearest neighbour of i, and liX> designates an atomic orbital. A = IS, IT, or 6 refers, as usual, to the angular momentum component around the axis joining i and j. If we assume the tight- binding limit to be rigorously true, the above parameters could be obtained from the atomic orbitals liX> and an effective ionic potential Vi. This potential is usually chosen as that of the neutral atom in the spirit of the Wigner- Seitz a roximation to the exchange-correlation hole(4-8.

The aim of this work is to propose a study of these parameters in the above limit, starting from atomic orbitals and potentials obtained from an atomic self-consistent Hartree- Fock- Slater calculation(6). We present here results of calculations of the tight-binding parameters for interactions between neighboring nickel atoms and neighboring nickel and, respectively, hydrogen, sulfur, oxygen atoms. Variation with distance of separation of neighbors is also discussed. The density of states of bulk Ni is calculated and compared to that based on para- meters obtained from interpolation schemes.

II. Details of calculations and results

The present calculations of tight-binding parameters involve atomic properties of the

253

254 TIGHT-BINDING PARAMETERS OF THE Ni-Ni, Ni-H, Ni-0, Ni-S BONDS Vol. 39, No. 2

TABLE I

Ni-Ni

work V(l) - 0.0438 0.0275

work V(*) - 0.0422 0.0248

ref. 8 - 0.038 0.017

9 (o = 1) - 0.0428 0.0186

lO(a = 1) - 0.0277 0.0118

10 (o = 2/3) - 0.0341 0.0144

11 (CY = Z/3) - 0.0260 0.0121

12 (a = 0.85) - 0.041 0.038

- 0.0053 1.6 8.2

- 0.0045 1.7 9.4

- 0.0017 2.2 22

- 0.0022 2.3 19.5

- 0.0015 2.3 18.5

- 0.0018 2.3 18

- 0.0014 2.1 18.6

- 0.0063 1.1 6.4

metal derived from an atomic self-consistent Hartree- Fock- Slater method analogous to that of Herman and Skillmann(6) starting from a 3d8 4s* configuration for atomic Ni. Exchange and correlation effects are included through a Slater type exchange potential pl/3 with a weight factor "cl". We have taken for "cl" a value obtained by a minimization of the diffe- rence in the HFS energy and the HF energy, the Schwarz value of "X," method; i.e. for Ni, a = 0.70896 to be compared to 1 in the original Slater's approximation or to 2/3, value sugges- ted by Kohn and Sham and Dirac.

The SCF results of these atomic calcula- tions are numerical functions and have been used as such, as their representation through a multi-Slater basis (STO) was leading to a too large error. The two (and three) center inte- grals have been calculated using elliptical coordinates and Gauss- Legendre integration scheme in three dimensions (A, ii, $) with nume- rical errors estimated to be less than 0.5 %. Two different effective atomic potentials have been used. Both are the potential of the ionized atom inside the atomic cell, to treat in some way correlation effects in the spirit of the Wigner- Seitz approximation to the exchange- correlation bole(5). But, if the first poten- tial v(1) keeps the long range part of the ionic potential, the second potential V(2) is taken as that of the neutral atom outside the atomic cell to take into account the screening by s and d electrons.

Table I shows that the values obtained through interpolation scheme are widely spread, even though similar techniques (APW or KKR)have been used for the determination of band struc- ture. This comes from slight changes in the use of atomic structure calculations or in the ex- change potential which can induce large discre- pancies in the tight-binding parameters, i.e. in the width of the d band. But the overall shape of the d band is quite insensitive, as shown in fig. 1, which compares two densities of states obtained with our first set of parameters and a set of parameters scaled to the same width witha

ratio * of 2.2 and g of 18. One can thus

reasonably assume that the tight-binding para- meters are transferable quantities which can be scaled to describe various cases of pure transi- tion metals with a canonical shape of the densi- ty of states such as that given by fig. 1. Let US simply remark here that the ratios weobtained

for the integrals #- or $& are close to those

determined through the resonance theory(4),which are respectively of 1.5 and 6.

We have checked that the three center inte- grals < ixlVjlku > are small compared to the two center ones : they are at most of the order of dd6 when the potential is centered on a site j which is nearest neighbour of i and k. For exam- ple we obtained for an equilateral trianglestruc- ture ij k, with a neutral potential V(2), the values

( 1 - 0.0073 (Ryd.) 1 - 0.0097 (Ryd.) 1 - 0.0026 (Ryd.)

I I Table I gives the result of Slater- Koster The crystalline field integrals, which depend

hopping parameters for Ni 8 kel with an inter- crucially on the long range part of the potential

atomic distance of 2.492 compared to results will be discussed in a forthcoming paper. obtained through interpolation schemes. Let us also point out that the direct over-

Vol. 39, No. 2 TIGHT-BINDING PARAMETERS OF THE Ni-Ni, Ni-H, Ni-0, Ni-S BONDS 255

-0.2 - 0.1 0 0.1 0.2 E(Rvdd

Fig. 1 - Density of d-states in bulk FCC Ni based on present parameters (-), and those obtained through interpolation schemes (Ref. 1) and resealed to the same band width as given by the present parameters.

TABLE II

Ni - X d (8) 'dso Ni

"dso(Ryd.) X

"dso (Ryd.)

x= H 1.86 0.100 - 0.141 - 0.168

S 2.28 0.073 - 0.086 - 0.119

0 2.08 0.074 - 0.077 - 0.157

ligand orbitalsd Ni

ligand orbitals- 'dpa "d pa (Ryd.) "d(po 'dpii

Ni "dpn

VX dpn

x=

S I - 0.077 0 - 0.071 I

lap integrals S between atomic orbitals centered on nearest atoms is quite small :

So = 0.026 STI = - 0.022 s* = 0.005

We have done similar calculations for the nickel-hydrogen, oxygen and sulfur bonds.

Results are presented in table II and fig. 2 describes the variation of the dso (Ni-H) bond with the distance Ni-H. It shows that the de- crease with distance is very sharp and can be fitted by an exp (- XR) law with A s 2.9.

This variation would induce some uncertainty on the evaluation of the binding energy due to

the bad knowledge of the adsorption bond length. A precise calculation would need a minimization of the total energy as a function of the dis- tance.

The present method is likely to serve as an important method for the calculation of tight- binding parameters for chemisorption and other studies. We are in the process of using our parameters to carry out various electronic structure studies and these along with other details on parameters will be published in a coming paper.

Acknowledgments-The authors are grateful to Prof. M. Cyrot for useful discussions.

256 TIGHT-BINDING PARAMETERS OF THE Ni-Ni, Ni-H, Ni-0, Ni-S BONDS Vol. 39, No. 2

x , , W

Corresponding = x

-=Cakulated Values

1.5 2 3 R(A) 4

Fig. 2 - Variation of the dso bond with distance for Ni-H.

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