M. MAIDS and B. MAFFEO: Cluster Self-Consistent Calculations of Ionic Solids 541

phys. stat. sol. (b) 132, 541 (1985)

Subject classification: 13.1 and 13.4; 22

Departamento de Fisica, Pontificia Universidade Catdlica do Rio de Janeirol) ( a ) and Serpro - Servipo Federal de Processamento de Dados, Rio de Janeiro2) ( b )

Non-Equivalence Effects in Cluster Self -Consistent Calculations of Ionic Solids 3,

BY MARIA MATOS (a) and B. MAFFEO (b)

A simple model is used to analyze actual cluster calculations and an interpretation of the existence of central levels well separated from band levels is proposed for INDO and X, methods. The analysis shows that the convergence to the infinite solid through the use of a point charge boundary condition cannot in general eliminate this symmetry non-equivalence effect, which is related to the approximate treatment of two-electron interactions inherent to each kind of cluster method.

On utilise un modhle simple pour analyser des ri.sultats obtenus parmis des calculs d’structure de dans l’approximation d’agglomhrat cristallin. On propose une interprBtation pour I’existence de nivaux centrals separes en Bnergie des nivaux de bande dans les mBthodes qui utilizent l’aproxi- mation X, et dans la metode INDO. L’analyse montre que la convergence vers le solide infini parmis des conditions de frontihre (charge externes) ne peux pas en gi.neral eliminer I’effet de non equivalence lequhl est associi. aux interactions de deux electrons.

1. Introduction

Self-consistent methods of molecular theory have been widely used, in the last two decades, in the study of pure and defective ionic and covalent crystals. An important question was raised concerning the finiteness of the physical system, imposed by these methods. Due to the presence of the defect the crystal loses translational sym- metry. The physical system is generally represented by a crystal piece, the molecular cluster (MC), which preserves the local point symmetry. However, a comparative study of perfect and imperfect systems, by using a given MC model, is necessary to estimate the influence of the defect over the crystal. To this point the MC approxima- tion has the disadvantage of, when representing the perfect crystal, providing dif- ferent potentials for central and surface atoms. This symmetry non-equivalence is responsible for different charge populations in chemically equivalent atoms and for the appearance of excess levels inside the gap or in the valence interband region. Surface effects in cluster calculations have been reported in CNDO (complete neglect of differential overlap) calculations of ionic solids [l to 31, in partly covalent solid calculations by using the Mulliken-Rudenberg method [4], and in statistical exchange approximation (X,) calculations in ionic solids [5, 61. The basic difference observed between HF-LCAO and X, calculations concerns the relative positioning of the excess level. While in the former this level appears upward shifted from the group of molec- ular orbital levels associated to a given crystalline band, in the latter it is invariably located below this group.

I) R. Marquds de SBo Vicente, 225, GBvea, Rio de Janeiro, RJ, Brazil, CEP 22451. 2, Rua da Lapa, 236, Rio de Janeiro, R.J., Brazil, CEP 20000. 3, Work partially suported by Brazilian agencies FINEP and CNPq.

35 physiea (b) 132/2

542 M. MATOS and B. MAFFEO

it is in general believed that the convergence to the infinite solid can be achieved through the use of appropriate boundary conditions which intend to describe as realistically as possible the crystal potential inside the cluster. Brescansin and Ferreira [S] introduced boundary conditions in the calculation of pure NaCl which modified the cluster potential in a self-consistent way. For ionic crystals more commonly used boundary conditions consist of embedding the cluster in the electrostatic field of external point charges which is included as a one-electron term in the effective Hamil- tonian. They have the advantage of neutralizing the whole system and in some cases giving also very good Madelung energies inside the cluster [2,5, 71. Even though these boundary conditions improve considerably the cluster potential they are unsuccessful in eliminating non-equivalence effects. Boundary conditions which take into account the non-point nature of ions outside the cluster were introduced in CNDO [3] and X, [5 ] calculations. i n [5] the surface effects were not, completely eliminated.

Another different approach consists of constructing clusters of atoms which contain large unit cells (LUC) [4, 8, 91. The LUC is naturally neutral and takes into account the translational crystalline symmetry. I n this approximation the problem of surface states does not arise and the energy levels can be directly related to crystalline states. It presents however a disadvantage in that the LUC model defects are distributed periodically in the crystal and has been criticized in regard of the study of (relatively to the crystal) charged defects [lo]. Thus the problem of non-equivalence effects in MC calculations is still worth investigating.

Although considerable work has been devoted to this subject a connection between non-equivalence effect calculations and the mathematical equations of the approxi- mate molecular methods has not yet been done. I n this paper we analyze some aspects regarding this matter. We present in Section 2 the results of calculations on the LiF crystal by using the HF-LCAO semiempirical INDO (intermediate neglect of differential overlap) method, in which the relative position of band and central levels is the reserve of that obtained by the X, methods in this and similar highly ionic com- pounds. We also discuss the role of external point charges in the boundary condition with respect to non-equivalence effects in X, and INDO methods. I n Section 3 we report a brief analysis trying to explain the results of Section 2 and discuss the role of two-electron parameters y in semiempirical methods.

2. INDO Calculation of LiF 2.2 Isolated cluster

The isolated cluster used to represent the perfect LiF crystal is constituted by 27 ions centered on an anionic site. The semiempirical self-consistent iNDO method [ 111 was employed to perform the electronic structure calculation. This method uses a minimum STO basis, 2s and 2p Slater functions of Li and F, and the Fock operator technique, in which several one- and two-center integrals are replaced by empirical parameters.

I n our calculation the Iepulsion integrals y were parametrized according to the Weiss-Mataga-Nishimoto formula (spectroscopic approximation) [12] and the reso- nance integral parameters ,t?AB, where A and B indicate any two different atoms, were calculated according to ,t?AB = - m, the values of PA and PB are empirically chosen and listed in Table 1.

Tab le 1 Empirical parameters ,t? (in eV) (from [ll])

Li - 1.75 F -39.0

Non-Equivalence Effects in Cluster Self-Consistent Calculations 543

Fig. 1. a) The ground state electronic structure of LiF 27-ion cluster calculated by using INDO method. b) The ground state electronic structure of LiF %-ion cluster plus boundary

E & @ - L i + conditions, by using INDO method

- zp-F, -2p-fb

-2s-F- L

2s-5

Fig. l a shows the 2s and 2p fluorine bands and the central ion levels, B',, located above the corre- sponding band levels, F,. The energy differences AE, between central and band levels, are a measure of the symmetry non-equivalence of this cluster cal- culation. b

2.2 Cluste+ submitted to a boundary condition

The boundary condition used in our calculation is constituted by an ensemble of 316 plus and minus integer and fractional point charges, located a t the positions of nearest absent ions, which preserves the local cluster symmetry [7]. The exact electro- static potential due to these point charges is easily introduced into the Fock operators as an additional one-electron term. In order to appreciate how well this set is able to simulate the infinite crystal outside the cluster, the electrostatic potential of a sym- metric system of 343 point charges was computed on locations which correspond to the cluster sites. The exact values of. the crystalline Madelung energy were obtained for all these sites thus showing that the set chosen provides a very good simulation of the crystal outside the cluster.

Fig. 1 b shows the result of the INDO self-consistent calculation. All levels appear upward shifted and the energy differences A& have increased. This latter result could be anticipated since the electrostatic potential energy due to the set of point charges computed at the center of the cluster is greater than its value a t the outer fluorine ion sites. Thus the incorporation of the boundary condition not only did not reduce the effects of the symmetry non-equivalence but has instead increased them. It is worth- while mentioning that an equivalent calculation was performed for NaF and the same qualitative results were obtained.

I n similar calculations involving ionic crystals performed by making use of X, methods, the qualitative results are the reverse of the present ones: with no point charges the central levels appear well below the corresponding bands and then the incorporation of a point-charge boundary condition reduces the values of A& 151.

It is then clear that a point-charge boundary condition, designed to correct sym- metry non-equivalence effects, is only effective in reducing the value of AE if the central levels appear below the corresponding bands and that whether this situation should occur or not depends upon the method used. To understand better this problem the distinct features of these two cluster methods are analysed in the next section. 35'

544 M. MATOS and B. MAFFEO

3. Discussion

An important distinction concerning the self-consistent methods referred previously is related to the exchange terms. DV and MSX, methods use the local exchange X, approximation while INDO preserves the non-local nature of this interaction. Since two-electron integrals are strongly dependent on electronic charge distribution, central and band levels are expectedly affected in different ways when an X, or an INDO method is used and this distinct feature could in principle explain the opposite results obtained for As. To verify if this is true we will now develop an argument based on a simplified model which retains the essential features of each method while avoid- ing unnecessary mathematical complications. In this analysis we shall be considering INDO and DV methods as both use the LCAO formalism.

Let us consider a closed shell of N chemically equivalent atoms with one atomic orbital (AO) per atom. We will denote y c the molecular orbital (MO) associated to the central .level sc, supposedly well separated from the band levels E;, associated to MO’s y\ (i = 1, ... , N - 1). The total eigenfunction is given by the single deter- minant

The INDO one-electron energies are given by N-1

j=1 ~c = hc + Jcc + 2Jci - Kci + C’ (2Jcj - Kcj) ,

N - 1 - - ~i = hi + Jii + 2Jci - Kci + C‘ (2Jij - Kij) ,

j=1

and the corresponding expressions for X, by N-I

j=1

- ~c z= hc + J c c + 2Jci + C’ 2Jcj -

In (2) and (3) the subscript i refers to the band MO whose energy is nearest to that of the central level and the prime indicates that the subscripts i ((Za) and (3a)) and c ( (2 b) and (3 b)) are not included in the summation. h are one-electron integrals,

and J , K , and Vx , are two-electron Coulomb and exchange integrals

Non-Equivalence Effects in Cluster Self-consistent Calculations 545

I n (2) to (5) all energies are in atomic units and y is a given MO. The first term in (4a) refers to the kinetic energy operator and the second to the electrostatic interac- tion with all core and external point charges. In (5), a is a properly chosen constant and p the total electronic charge density.

Regardless of the method used, X, or INDO, one common feature results: yc is a localized (typically nonbonding) MO, predominantly described by the AO centered on the central atom, while yi are delocalized MO’s. Consequently we will assume the following explicit forms :

Yc = Yc )

N - 1 yi = C Ciaya ,

u = l

where {qc, yU (a = 1, ... , N - 1 ) } is the AO basis set.

(4a) and the two-electron (4b) to (5b) interactions present in (2) and (3). We will now proceed with the analysis by considering separately the one-electron

3.1 One-electron interactions ( h )

By substituting (6) into (4a) and taking into account normalization of MO’s the one- electron energies h can be separated in purely atomic ( U ) , electron-other nuclei (V) , and resonance (Ha& interaction terms:

hi = ub + Vb + C C g C i B H u B . US

U , V, and Hub are given, in atomic units, by

The prime in (8b) indicates that the nucleus which the atomic orbital Y belongs to is not to be included in the summation; the subscripts c and b refer to AO’s centered on central and border cluster atoms.

U , do not differ from Ub since they correspond to atoms of the same kind. The electron-other nuclei attractive terms, V, are directly evaluated in X, methods (DV and MS) and through quite precise parametrization in the INDO method. I n the absence of external point charges these terms would lead to Vc < V i since the central atom has, inside the cluster, more atoms surrounding i t than border atoms have. This could explain results of the X, calculation where A& < 0, but not of INDO calcula-

546 M. MAWS and B. MAFFEO

tions, where A& > 0. Of course these terms change when point charge boundary condi- tions are included but we shall disregard them since we are interested in determining interactions predominantly responsible for the existence of non-equivalence effects.

Resonance terms (8c) correspond, in a molecular system [ll], to one-electron energies of overlap charge densities which place nonbonding orbitals between bonding and antibonding MO’s. Thus HaB is not responsible for the relative position of central (nonbonding) levels, above (INDO) or below (X,) the corresponding band (bonding and antibonding) levels. I n actual INDO calculation this term is evaluated through proper paranietrization (see Section 2). We have repeated the calculation presented in Section 2 with a parameter choice which makes it equal to zero and no significant modification of the observed symmetry non-equivalence resulted showing that resonance terms have negligible influence in explaining the effect.

From the arguments above, the resonance interaction of a molecular orbital whose energy is located close to the middle of the band, say Vk, is zero. This leads to h, = hk for that particular band level. Let us denote by As’ the difference in energy between the central level and V k . This quantity is related to A& in the following way:

(9) 6 2

A&‘ = A& -,

where the plus (minus) sign corresponds to INDO (X,) method and 6 is the bandwidth (6 > 0). Equation (9) shows that A&’ is as convenient a quantity to describe non- equivalence as A& is, since the sign of A&’ is the same as that of A&. I n the foregoing analysis we shall be concerned with As’, to express symmetry non-equivalence effects.

3.2 Two-electron interactions (J, K, V x m )

According to the preceding analysis A&’ can be written as

N-1 X-1

j = 1 j=1 A&‘ = [ J c c + c’ (2Jcj - Kcj)] - [ J k k f c’ (2Jkj - Kkj)] (10a)

Let us now consider some terms of (10). The exchange K,j can be written in terms of the AO’s as follows:

where ( i c l c j ) is the Mulliken notation for two-electron integrals. The INDO method incorporates the zero differential overlap (ZDO) approximation

which in the case of our model consists of neglecting all terms ( i j lkl) when i + j or L =+ 1. As a consequence, Kcj = 0 since j =+ c for all j . I n an actual calculation the INDO method takes into account some monatomic terms such as (mnlmn) which do not occur here because our model assumes a single A 0 per atom; these mon- atomic terms are not relevant for the present discussion since they will be the same for all chemically equivalent cluster atoms. We will further assume that ZDO is a generally good approximation for highly ionic systems and impose it when discussing X, equations, although these do not use this approximation in actual calculations.

Non-Equivalence Effects in Cluster Self-consistent Calculations 547

The exchange terms Vx, can also be written in terms of AO’s. By considering ZDO approximations which imply that overlap charges are small, we get

Actual X, calculations attribute very nearly the same electronic charges to all cheniically equivalent ions [1,5]. Therefore, @(I) may be considered to assume the same values in (12a) and (12b). Since 1y8(1)I2 = [yC(1)l2 (same AO’s) one gets, from normalization of the MO’s, V:, = TI&,.

Equations (10) then become N - 1 N - 1 3 - 1

j = 1 j=1 j=1 As’ = [ J c e + C’ 2Jcj1 - [ J k t + C’ 2JkjI + C’ Kkj 7 (13a)

Equations (13) express an interesting feature : the symmetry non-equivalence effect in X, methods is strictly due to electronic repulsive interactions while in the INDO method i t is due to exchange terms as well. Even though actual X, and INDO cal- culations can attribute different values to the several terms in (13a) and (13b) AE‘ (INDO) retains a positive contribution from the last term of (13a) which is not present in (13 b) (X,). I n fact, actual X, and INDO calculation result in a good descrip- tion of the ionicities of ionic cyrstals, so it can be inferred that the terms in brackets assume nearly the same values in (13a) and (13b). These equations assert that the position of the central levels in INDO calculations are upward shifted as compared to X, calculations.

We have disregarded, in our model analysis, all cationic characteristics. This is consistent with the results of actual calculations, in particular that for NaF referred to in Section 2, where admixtures of cationic AO’s in anionic level eigenfunctions are small and with the fact that as long as an anion is a t the center of the cluster only anionic levels significantly show the symmetry non-equivalence effect considered. With a cation centered cluster the opposite would be true and the preceding analysis would also apply.

One crucial assumption of our analysis concerns the localized distribution of elec- tronic charge associated to central levels E,. In fact, no admixture of yc and y , AO’s in the description of the yc and yi MO’s was allowed in the model and this is a good approximation to results of actual calculations.

Another important consequence of the localization of yc concerns the ZDO ap- proximation used in the INDO method. This approximation imposes, as far as ex- change terms are concerned, a different treatment of localized and delocalized eigen- functions. As a consequence, exchange terms like Kcj in (10) are made equal t o zero thus “forbidding” that central-band MO exchange interactions provide an energy lowering of central levels. To check this argument we have performed a calculation on LiF using the NDDO semiempirical method [ l l ] , which retains some diatomic exchange integrals. Fig. 2 shows the resulting occupied level scheme and one observes that the symmetry non-equivalence effect is now different from that of Fig. 1. I n particular, referring to the F 2p-levels, the central level is below the band levels as is obtained in X, calculations.

548 WIA MATOS and R. MAFFEO

Oi i )% F2p-levels

F, 158%) 1 ='% FZs-levels

F, (39%) I --

Fig. 2 Fig. 3

Fig. 2. The ground state electronic structure of LiF 27-ion cluster obtained by using NDDO method

Fig. 3. Ground state LiF electronic structure obtained by using non-spectroscopic INDO param- etrization with the same MC and boundary condition as in Fig. 1 b

For the INDO method (13a) the present analysis suggests that non-equivalence effects should depend on the parametrization of two-electron integrals y. Another very commonly used approximation in semiempirical methods [ll] is to calculate y directly from s-type Slater orbitals,

y A A = (n(x/.a) = ( s A s A / s A s A )

= (am//?/?) = (SASA/SBSB)

(x t atom A , (x t atom A, /? t atom B . (14)

We have then performed a calculation on LiF by using the same conditions as in Section 2i except for y integrals which have been obtained from (14). I n Fig. 3 the interesting results of this calculation are shown. It can be observed that non-equiv- alence effects were eliminated, with no central levels appearing outside the bands.

Inspection of the MO coefficients of calculations of Fig. 2 and 3 indicates that central-level eigenfunctions are less localized than in the case of the INDO calculation reported in Section 2, where the same parametrization and boundary condition were used.

4. Conclusions Using a simplified model for actual calculations we were able to interpret results related to a particular effect due to the symmetry non-equivalence which exists when cruster methods are used in the study of perfect crystals.

Non-Equivalence Effects in Cluster Self-consistent Calculations 549

It was shown that, in view of the approximations inherent in INDO and X, methods, the effect will not disappear and may even increase through the use of a boundary condition defined by external point charges. Moreover, i t was possible to indicate that the treatment of the two-electron interactions, in a given method, is at the origin of the effect. It seems, in general, that boundary conditions, intended to eliminate the effect, should modify, in a self-consistent way, this treatment. It is worthwhile mentioning that in the work by Brescansin and Ferreira IS], who used a self-consistent boundary condition, the effect was eliminated.

I n this paper we have shown that with a suitable choice of empirical parameters y in semiempirical calculations (non-spectroscopic INDO), the effect of non-equivalence is eliminated, without disturbing the description of important physical properties of ionic crystals such as ionicities and band gaps. This procedure seems convenient since it allows the use of external point charge boundary conditions which, as was men- tioned in Section 2, incorporate important electrostatic contributions from the infinite crystal, while being computationally simple.

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(Received June 3,1985)