TRANSITION METAL WAVE FUNCTIONS : A FIRST PRINCIPLES RESONANT TB APPROACH

HAL Id: jpa-00215062

https://hal.archives-ouvertes.fr/jpa-00215062

Submitted on 1 Jan 1972

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

TRANSITION METAL WAVE FUNCTIONS : A FIRST PRINCIPLES RESONANT TB APPROACH

D. Pettifor

To cite this version:

D. Pettifor. TRANSITION METAL WAVE FUNCTIONS : A FIRST PRINCIPLES RESO- NANT TB APPROACH. Journal de Physique Colloques, 1972, 33 (C3), pp.C3-191-C3-194.

�10.1051/jphyscol:1972328�. �jpa-00215062�

JOURNAL DE PHYSIQUE

Colloque C3, supplkment au no 5-6, Tome 33, Mai-Juin 1972, page C3-191

TRANSITION METAL WAVE FUNCTIONS : A FIRST PRINCIPLES RESONANT TB APPROACH

D. G. PETTIFOR

Dept. of Physics, Imperial College, London S. W. 7

RBsumB.

- Les ktats d des mBtaux de transition se comportent comme des ktats liBs ou rkso- nances, ce qui permet d'exprimer d'une manike particulikrement simple la variation de leurs fonc- tions d'onde radiale avec l'knergie. Les equations KKR (Korringa, Kohn et Rostoker) peuvent alors &re Bcrites directement dans une forme pseudo-TB (liaisons fortes), dans laquelle les sommes de Bloch sont construites sur les fonctions d'onde rksonnantes centrks sur chaque site du rBseau.

Ces fonctions rksonnantes ne dkcroissent pas exponentiellement comme dans le cas atomique mais oscillent sur toute l'etendue du cristal, ce qui fait apparaitre la possibilitk d'interfkrence construc- tive et d'hybridation. C'est pourquoi nous extrayons cette hybridation en mettant la fonction d'onde initiale sous une forme mixte NFE-TB.

Abstract. - The d-electrons in transition metals behave like virtual bound states or resonances, which allows the energy dependence of their radial wave functions to be expressed in a particularly simple manner. The KKR (Korringa, Kohn and Rostoker) equations can then be written directly in a pseudo - TB (tight-binding) form, in which the corresponding Bloch sums consist of the resonant wave functions centred on each lattice site. However, these resonant functions do not decay exponentially as in the atomic case, but instead oscillate sinusoidally throughout the crystal, which gives rise to the possibility of constructive interference or hybridization. We, therefore, separate out this hybridization by a transformation that takes the first principle wave function into the mixed NFE - ^{TB form.}

1. Introduction.

Although a lot of attention has been focussed on the band structure of the transition metals, very little work has been done on the corres- ponding wave functions. In particular, we now well understand the physical make up of the energy levels, since the intuitive model Hamiltonians of Hodges et al. [I] and Mueller [2], which were based on a hybrid nearly-free-electron tight-binding (H-NFE-TB) picture, have been formally justified by the first principles work of Heine [3], Hubbard 141, and Jacobs [5], who transformed the KKR (Korringa [6], Kohn and Rostoker

secular equation into this H-NFE-TB form, Moerover, their justification depended strongly on the fact that the transition metal d-electrons behave like virtual bound states or resonances [8], so that, within the muffin tin approximation, the I

2 phase shifts may be written

tan q2(e)

r(E) + tan q;(e) ,

Eo - E

(1)

where

and I' are the position and half-width of the resonance respectively. Thus, the scattering behaviour of the d-electrons is characterized by essentially only these two resonant parameters, in terms of which it is possible to derive analytic expressions for the TB overlap parameters ddv, ddn and dd6 and the hybridi- zation matrix elements of the intuitive model Hamil-

(*) Now

at

Department

Physics, University

Salaam, Tanzania.

tonian [9]. However, it still remained unclear as to how the corresponding first principle basis functions could be written in the TB form, since the resonance wave functions do not converge rapidly beyond the first few nearest neighbours, but instead oscillate sinusoidally throughout the crystal. Nevertheless, in the present paper we shall show that these oscillating tails are completely taken care of by the conduction basis functions, so that the first principle wave func- tion may indeed be written in the intuitive OPW-TB form.

The crux of the present work is the fact that the I

2 radial wave function, which is a solution of the single muffin tin Schrodinger equation, may be written in a particularly simple manner, following the method of Kapur and Peierls [lo] for treating resonant nuclear scattering, later adapted by Hubbard

to the present problem. We find, after removing the implicit energy dependence [ I l l , that the radial wave function may be written

provided that

where ri is the muffin tin radius, j, (and n,) are the spherical Bessel functions, and

It follows immediately from (2) that ~ ( r ) is the resonant wave

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972328

C3-192 D. G. PETTIFOR

function, which, because of the boundary conditions,

is given outside the muffin tin sphere by

q ( r )

- n2(rco r) for r > ^ri . (4) In figure 1 we compare the approximate form of the radial wave function with the exact solution for

FIG. I. - A comparison of the approximate radial wave function r R ~ ( r , ^E), given by equation (2), with the exact result (dashed) given by Wood [I21 for b.c.c. iron. The energies

8 = 0.41 _and 0.88 ryd. correspond to the bottom and top of the d-band respectively. rt and rw, are the inscribed and Wigner-

Seitz radii.

b. c. c. iron [12], which has the resonant parameters

E~ =

0.65 ryd. and r

0.061 ryd. We see that it yields an accurate representation of the energy depen- dence of the wave function over the entire width of the d-band, so that it will clearly serve as a very useful and important expression for the 1

2 radial wave function.

2. The pseudo-TB form of the KKR equations. - We shall show in this section that the I

2 part of the KKR equations may apparently be written in the TB form. The trial wave function

leads, upon variation of the KKR functional, to the following secular equation.

where

E,,,,,,(R,

4 ~ c T 2 i-r' n,(lc0 R) Ys(R) C~",~m,2m'

L

(7)

with

The resonant form of the d-phase shift and the direct- lattice (R) space representation of the structural Green's function have been used [Ill. This secular equation, which is exact except for our removal of the implicit energy dependence in E2,,2,t(R, E) has the same form as the TB equations, in which the E,,,,,, are the so-called energy integrals, tabulated by

Slater and Koster [13] in terms of the overlap para- meters dda, ddn and dd6. Moreover, by using the approximation (2) for the radial wave function, the trial wave function (5) becomes

x

Y2rn(r) c2,

(9) which, on application of the secular equation (6), may itself be written in the TB form, namely

where

~ m ( r - R)

(The operator P, projects out only the I

2 compo- nent of the expansion of q,(r - R) in terms of sphe- rical harmonics about the origin.) That is, the first principles wave function may be expanded in terms of TB Bloch sums of the resonant wave functions, as was originally suggested by Heine

Unfortunately, however, these resonant functions do not decay exponentially like in the atomic case, but instead oscillate sinusoidally throughout the crystal. This is very clearly reflected in the corres- ponding TB overlap parameter ddo, which is drawn in figure 2. The position of the first few nearest neigh-

FIG. 2. - A plot of ddo/T versus ko R in the absence of the Ewald split (full curve) and in the presence of the Ewald split with

B

B

The latter are drawn for the particular case of iron. We have also marked along the abscissa the positions corresponding to the first three nearest neighbours of f.c.c. iron.

bours in f. c. c. iron have also been marked along the abscissa, from which it is immediately apparent that ddo does not fall off rapidly with distance, but instead oscillates with an amplitude that decays only weakly.

In fact, it behaves asymptotically like cos

R) d d o w - 5r

n o

R

TRANSITION METAL WAVE FUNCTIONS : A FIRST PRINCIPLES RESONANT TB APPROACH C3-193

This leads to the possibility of constructive interfe- rence, since the structural phase factor k. R. in equa- tions (6) and (10) will be in phase with these resonant oscillations when

k2, leading to a divergence in both the TB Bloch sum and the corresponding matrix elements. This is equivalent to hybridization or mixing with the conduction electrons, because the influence of a given d-electron is now no longer confined to a particular neighbourhood, but may be felt throughout the entire crystal. We must, therefore, unscramble the TB and hybridization contributions by removing these singularities.

3. The H-NFE-TB transformation.

We clearly

require a mixed form for the wave function, in which the d-electrons are represented by TB Bloch sums in R-space and the conduction electrons by OPW's in K-space. We, therefore, follow Ham and Segall [I41 and write the structural Green's function as the sum of two parts, namely

where the individual terms are now dependent on the choice of the Ewald splitting parameter P. Then, substituting this into the KKR functional and perform- ing the H-NFE-TB transformation of equations (4.13) to (4.14) of 11 I], we arrive at an exact secular equation of the desired form, namely

where T is equivalent to the conventional TB d-block with two centre and three-centre integral contributions, H represents the hybridization matrix and v the weak pseudopotential terms. Moreover, if this secular equa- tion is used instead of the original KKR equation (6), then the trial wave function, including all 12 2 contributions, may be re-written in terms of energy independent basis functions as follows

which is accurate to first order in the width of the d-band provided that P satisfies the inequality

The conduction block basis functions are given by

where the Xi+K are essentially OPW-like (see equa- tion (4.26) of [l l]). The d-block basis functions have the expected TB Bloch form and are given by

where

and

Thus, the presence of the splitting parameter P modi- fies the effective atomic wave functions, since qm(r, P)

must now be consistent with the fact that the mean position of the TB d-band has been shifted upwards from the resonant level by the small amount d;(,f?) [l 11.

Further, on comparing equations (21) and (ll), we see that not only is q,"(r - R, P) expanded in terms of all the spherical harmonics about the origin in a manner similar to that suggested by Lowdin 1151, but it also dies away rapidly over the first few neighbours because the Ewald split has removed the oscillating tail from the energy integral, as is clearly illustrated in figure 2. Finally, these TB basis functions are consistent with the d-block elements in (15), because identical matrix elements are recovered if the

atomic

wave functions (20) and (21) are substituted into the conven- tional two-centre and three-centre integrals. Moreover, we have shown [16] that the resonant rather than atomic nature of the wave functions is very important if the virial theorem is to be satisfied.

We have, therefore, derived from first principles

wave functions of the physically intuitive OPW-TB

form. Moreover, they may be accurately orthonorma-

lized using the known energy derivatives of the matrix

elements [I 11. Thus we expect the computation of

electronic properties such as self-consistent potentials,

optical transition probabilities, and electron-phonon

coupling constants to be considerably simplified. In physical chemists, and, therefore, to discuss the addition, the physical nature of the TB basis functions change in nature of the atomic wave function as the allows us to take over the ideas of binding from the crystal is formed [16].

References

[I] HODGES (L.), EHRENREICH (H.), LANG (N. D.), Phys. [9] PETTIFOR (D. G.),

Phys. C., 1969,2, 1051.

Rev.. 1966.152. 505.

[lo]

(P. L.), PEIERLS (R. E.), PYOC. Roy. Soc., 1938, [2] MUELLER (F. M.), Phys. Rev., 1967,153,659. A 166, 277.

[3] HEINE (V.), Phys. Rev., 1967, 153, 673. [ l l ] PETTIFOR (D. G.), J. Phys. C., 1972,5,97.

[4] HUBBARD (J.), Proc. Phys. Soc., 1967,92,921. [I21 WOOD (J. H.), Phys, Rev., 1960, 117, 714.

[13] SLATER (J. C.), KOSTER (G. F.), Phys. Rev., 1954, 94, [5] JACOBS (R. L.),

Phys. C., 1968,1,492.

[6] KORRINGA (J.), Physica, 1967, 13, 392. [14] HAM (F. S.), SEGALL

Phys. Rev., 1961, 124, 1786.

[71 KOHN

ROSTOKER

Phys. Rev., 1954,94,1111. [15] LOWDIN (P.

Adv. Phys., 1956,5,1.

181 ZIMAN (J. M.), Proc. Phys. Soc., 1965, 86, 337. [I61 PETTIFOR (D. G.),

Phys. F., 1971,