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THE USE OF CLUSTERS OF ATOMS IN THE CALCULATION OF ELECTRONIC STRUCTURES
To cite this version:
J. Keller. THE USE OF CLUSTERS OF ATOMS IN THE CALCULATION OF ELEC- TRONIC STRUCTURES. Journal de Physique Colloques, 1972, 33 (C3), pp.C3-241-C3-244.
JOURNAL DE PHYSIQUE Colloque C3, suppliment au no 5-6, Tome 33, Mai-Juin 1972, page C3-241
THE USE OF CLUSTERS OF' ATOMS
IN THE CALCULATION OF ELECTRONIC STRUCTURES
J. KELLER (*)
H. H. Wills Physics Laboratory, Royal Fort, Tyndall Avenue, Bristol, BS8 1TL
Rksumk. - Les techniques habituelles de diffusion multiple pour le calcul des structures elec- troniques utilisent les proprietks diffusantes de l'atome isole. Nous montrons ici que beaucoup d'avantages sont retires de l'utilisation de groupes d'atomes comme unites de base. On obtient ainsi, en utilisant des conditions aux limites approprikes, une bonne description de proprietes des cristaux mol6culaires et materiaux amorphes. Des resultats pour des semi-conducteurs, semi- metaux, m6taux de transitions et cristaux mol6culaires sont presentb.
-The usual multiple scattering techniques for the calculation of electronic structures involve the use of the scattering properties of the single atom. Here we show that if clusters of atoms are used as the basic units there are several advantages to be obtained. By considering the scattering properties of clusters of atoms with appropriate boundary conditions we can get a good description of some properties of molecular crystals and amorphous materials. Some results for semi-conductors, semi-metals, transition metals and molecular crystals are presented.
The cluster model as used by Johnson [I] and the groups at Massachusetts Institute of Technology and the University of Florida focus attention on discrete bound states, here I want to present a further part of the theory where the energy range of interest is in the continuum, above muffin-tin zero. The basic theory was developed independently a t the University of Bristol using a formalism due to Lloyd  and Klima and McGill  first applied it to a cluster of atoms in a model of amorphous semiconducting materials.
The usual multiple scattering techniques for the calculation of electronic structures use the scattering properties of single atoms in the medium. In this paper we show that in many cases clusters of atoms can be employed as convenient basic units, and several advantages are thereby obtained.
The clusters can be defined, for example, as chemical radicals, radical-ions or composite ions. In general they must form a representative part of the material.
Once we have defined the nature of the cluster we must use some specific boundary conditions t o repre- sent the properties of the rest of the material in which that cluster exists.
Formulation of the cluster method.
-The cluster method assumes that the one-electron Schrodinger equation. for a given set of atoms in a medium can be written as
where Vj are the potentials of N atoms in the cluster, V i are the potentials of the N' cc interstitions H used to represent large open regions of the interatomic space (as defined in KeIler  or as discussed by Ziman 
in these proceedings) and Vboundary is the fictitious potential which represents the effect of the rest of the medium on the cluster and defines the appropriate boundary conditions. This Hamiltonian will be used in connection with eq. (9) below.
The usual expression for the density of states of an infinite system of scatterers can be written as :
where No(E) is the free electron part (at positive energies). This result was transformed by Lloyd 
into the form :
where the determinant of the matrix D
(*I present address : ~~~~l~~ of chemistry, University of is in mixed angular momentum-position representa- Mexico. Mexico 20, D. F. tion, kL,SL'(Ri) is the K-matrix of the scatterer a t site
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972336
C3-242 J. KELLER and G + is a free-electron propagator. It can be factorized
(in A, only terms referring to cluster i appear, in Bij only the interaction terms between i and j, etc.) and the result expressed as :
The evaluation of the second term in eq. (8) requires the sum for all possible clusters in such a way as to give the correct average when the density of states per atom or per unit volume is evaluated.
The third term in the R. H. S. of eq. (8) can be approximated in several ways, but the idea of the clus- ter method is to write eq. (8) in the form
C B ~ a 1 [Abl
CBLcl . .
[BLJ [Bib] [A,]
. . .
where Pa is a probability factor for the occurrence of the cluster and %=(E) is the Friedel sum for a cluster of type a in the presence of the fictitious potential Vboundary
of eq. (1).
This fictitious potential can be taken to be the muffin-tin zero of the material (Keller , Keller and Jones ), or a dispersive medium outside the clus- ter , or represented by periodic boundary condi- tions  or fixed by a self-consistent argument (Keller and Smith ) similar to CPA  and the Anderson and McMillan [lo] technique.
The multiple scattering formalism using the muffin- tin approximation as introduced by Korringa 
has been successful in solid state physics and in mole- cular problems. In this approximation eq. (4) and (5) are easy to derive and use in numerical work. The potentials do not need to be spherically symmetrical inside the muffin-tins  (and interstitions), but then the actual computation of the single site K-matrix becomes very involved.
Using the muffin-tin approximation the equations for the K-matrix of the cluster, the T-matrix and the
Green's function matrix are
If K, T and G are given in angular momentum repre-
sentation all the other matrices are in mixed angular momentum-position representation
and the rest of the symbols have their usual meaning (for example as defined in ).
We should mention some of the physical quantities related to these functions. The reaction matrix can be used to calculate the density of states whilst from the T-matrix of the cluster, we can calculate band structures and scattering-cross sections. From the Green's function we can evaluate among other things the local density of density of states and map the electronic structure, not only as chemists do but also as a conti- nuous function of energy. The latter scheme is parti- cularly useful for understanding the formation of bonds and the characters of the energy bands. The computed electron density can be used as a basis for a self consistent calculation.
Application to condensed systems.
-The cluster method as defined by eq. (9) has been used in the study of some models of liquid transition metals, semi-metals and amorphous semi-conductors. It is currently being applied to the further study of these materials and also some alloys. The method is expected to be particularly useful for the investigation of impurities, vacancies and structural defects in condens- ed systems including those organic and inorganic materials which are composed of associations of more or less well defined molecular clusters. Under- standing of the electronic properties of those systems in which there is clustering or chemical bonding in an otherwise uniform medium is an intrinsically difficult problem, and one which is suitable for the application of this method.
Our systematic study of the scattering properties of clusters of atoms in semi-conductors with tetrahedrally coordinated random network structures (e. g. silicon and hypothetical amorphous diamond) began with calculations made for clusters of 1, 2, 6, 8, 10, 18 and 30 atoms in a series of different configurations. The boundary conditions assumed that the clusters were embedded in a potential equal to the interstitial potential in the cluster . (These results were ori- ginally presented at the Chelsea, London, conference in december 1970.)
For amorphous diamond there is a definite tendency to obtain three distinct zones, consisting of two bands separated by one region of low density states. This result confirms the original work of Klima and McGill.
In all cases the energy region with low density of
THE USE OF CLUSTERS OF ATOMS IN THE CALCULATION C3-243 states is at an energy range higher than that where an (iii) The environment was represented by a disper- energy gap is found in perfect crystal using the same sive medium outside the cluster.
muffin-tin potential. Results for the large (really amorphous) clusters confirm that the low density of states is not produced by the high symmetry of the small clusters.
An analysis of the behaviour of the density of states curve in the energy gap region as a function of the size of the cluster, shows a definite tendency towards a perfect gap in a very large cluster. The calculations also predict that this gap should be at the same energy as the gap in the crystal.
The eventual cancellation of the free electron density of states is a reasonable possibility. Similar results were also obtained for silicon and recently for germanium. Using the short range order of the cluster properly introduces the higher order correla- tion functions into the problem and this is obviously advantageous.
Unfortunately, this procedure of increasing the cluster size cannot be carried much further without enormous computational expense, and with clusters of only a few dozen atoms, the intercluster contribution from matrix B in (7) cannot be ignored. If we throw away these terms we must introduce some other device to eliminate spurious effects from the boundary of the cluster.
To get some idea of the importance of the boundary conditions calculations were made for three different extreme cases (Fig. 1) :
(i) The scattering properties of the environment of the cluster were neglected as in the previous calcula- tions.
(ii) The cluster was subject to periodic boundary conditions.
FIG. 1. - Schematic one-dimensional analogy of the potentials and boundary conditions used in the calculations a) the outside potential is equal to the average interstitial potential of the model.
6 ) the outside potential is dispersive in the studied energy range, taken to be equal to the atomic zero. c) The outside potential is a
periodic replica of the cluster.
The results of these calculations were reported in the Manchester Conference in January 1971. The main results were that in cases (ii) and (iii) the region of low density of states was transformed into a perfect gap region (see [ 6 ] ) .
Once the basic rules for the construction of the clusters have been given i. e. a fixed coordination number and a restricted range of values for the nearest neighbour distance and for the angles between the bonds, the topologically disordered network so obtain- ed represents the basic structure of an amorphous semi-conductor. We can construct more realistic structures containing << defects )) of two kinds : (i) topological defects in which the coordination number rule is not satisfied and (ii) geometrical defects which are produced when any of the nearest neigh- bour distances or bond angles lie outside their speci- fied ranges.
If the cluster has geometrical defects the energy gap region is substantially altered. Decreasing the density of the material destroyed the gap by making the bands overlap.
A particular case of interest is obtained by changing some angles of the <( bonds )) from the tetrahedral configuration by about 15
%.This corresponds to a highly disordered amorphous cluster in which the second to nearest neighbour distance of many atoms is drastically changed, and this new cluster gives rise to a larger number of states in the energy gap region.
The use of periodic boundary conditions show that the bands do not necessarily overlap but in any case, the gap will be much smaller.
On the basis of our numerical calculations we conclude that with the correct combinations of scatter- ing potentials in systems with appropriate short range order we can have sufJicient condition for the existence of an energy gap above muffin-tin zero even in the absence of long range order. In any material it is necessary to include in the analysis clusters containing enough information about the local order and to use the appropriate boundary conditions. This can be seen clearly in two complementary examples, i. e. the calculation of the density of states of a semi- metal, graphite and of the d-electron bands of the transition metals, iron, copper and nickel.
In the first case, graphite , the boundary condi- tions are very important as we have an electron density of states at the Fermi level of the order of the free electron value, In the second case, the height of the density of states in the d-band is more than an order of magnitude larger than the free electron value and meaningful results are obtained even for very small clusters . Our present work in the d-electron alloys confirm these results .
In recent weeks we have been working on a way of improving the analysis by a self-consistent argument
(23-244 J. KELLER roughly along the lines of the coherent potential approximation, but we have not yet had time to test the method numerically in detail.
Keller and Smith  have been considering the very tempting possibility of calculating the K-matrix (or T-matrix) of clusters and then using these results in a calculation for a cluster of clusters.
Unfortunately, it turns out that to have a result accurate to one part in the thousand we need a K-matrix of size (I,,,
+with I,, of the order of 7-9.
Even if I,, is kept low (say I,,, = 3) it takes 6 times longer to compute the density of states using this procedure than the time needed using the original Lloyd determinant. The main increase in computing time is due to the need for inverting large matrices.
A very attractive result of our analysis of the scatter- ing properties of clusters of atoms has been the clear indication that the lines joining the atoms, repre- sented usually as chemical bonds, are also the channels along which a wave of electrons scatter. The channelling of electrons in the direction of the bonds, which is clearly seen at energies 10 or 15 eV above the muffin- tin zero, (where scattering is less isotropic) provides supporting evidence for the picture of the bonds as regions of high electron density and we feel this concept will be important for the full understanding of the subtleties of the chemical bond. Indeed our calculations  show that the bonds themselves behave as strong electron scatterers.
Since the object of this conference is to look at perspectives this allows me to mention a particularly interesting type of cluster, which is the case of a radical or a molecular group in a solid (for example CCI,,
-CCI,, = CCI,, =CCI-). Using the T-matrix of the group as a single muffin-tin (Fig. 2) we have a better representation of the local potential, and in particular, of the local muffin-tin zero of that group as different from the crystalline muffin-tin zero. This simple device allows the use of KKR and multiple scattering techniques for molecular crystals and molecular problems and all the accompanying advan- tages of the non-overlapping muffin-tins. A general potential version of KKR or multiple scattering 
may be necessary but a full program for organic and inorganic chemistry is then open, without the necessity of introducing further complications into the basic
theory. A collection of T-matrices of radicals will substitute atomic muffin-tin wave functions as bases for calculation.
-a) a cluster of (< atoms )) XYZ with a local interstitial potential VI is placed in a medium where the interstitial potential is VIL, this object is replaced by b) a cluster t-matrix (zc) in the
interstitial local potential VI and this complex scatterer is finally replaced as c) a T-matrix (TM) in medium 11. TM will represent entire molecules or radicals or representative parts of the system.
Of course we do not need to restrict ourselves to clusters enclosed in spherical boundaries or use a cluster of spherical muffin-tins, but I want to empha- size the fact that we can go a long way ahead without giving up the simplicity of dividing the space into spherical regions of interest.
Finally I want to point out again that many experi- mental measurements on the condensed system deal with phenomena occurring in the region of energies where we have a continuous spectrum, and here our techni- ques are particularly useful. If we study a condensed system (as opposed to the case of isolated molecules where the same range of energies correspond to discrete states) quantum chemistry needs practical techniques to use in these cases.
On the other hand conventional Quantum Chemis- try usually deals with energy ranges corresponding to discrete states.
I would like to thank all my colleagues in the Theoretical Physics group of the University of Bristol for their encouragement and help and especially, R. Evans for reading the manuscript and Prefessor J. M. Ziman who suggested this type of approach 1161.
The Faculty of Chemistry of the University of Mexico granted leave of absence during the period of this research which was supported in part by the U. K.
Ministry of Aviation Supply.
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