ELECTRONIC STRUCTURE OF THE GROUND STATE OF THE COVALENT AND IONIC SOLIDS : USE OF LOCALIZED ORBITALS. APPLICATION TO THE EQUATION OF STATE OF THE DIAMOND

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ELECTRONIC STRUCTURE OF THE GROUND STATE OF THE COVALENT AND IONIC SOLIDS : USE OF LOCALIZED ORBITALS. APPLICATION TO

THE EQUATION OF STATE OF THE DIAMOND

J. Gelard, Ph. Durand

To cite this version:

J. Gelard, Ph. Durand. ELECTRONIC STRUCTURE OF THE GROUND STATE OF THE COVA-

LENT AND IONIC SOLIDS : USE OF LOCALIZED ORBITALS. APPLICATION TO THE EQUA-

TION OF STATE OF THE DIAMOND. Journal de Physique Colloques, 1972, 33 (C3), pp.C3-127

-C3-130. �10.1051/jphyscol:1972318�. �jpa-00215052�

JOURNAL DE PHYSIQUE

Colloque C3, suppliment au no 5-6, Tome 33, Mai-Juin 1972, page C3-127

ELECTRONIC STRUCTURE OF THl3 GROUND STATE OF THE COVALENT AND IONIC SOLIDS : USE OF LOCALIZED ORBITALS.

APPLICATION TO THE EQUATION OF STATE OF THE DIAMOND

J. GELARD and PH. DURAND

Laboratoire de Physique Quantique, Universitt Paul-Sabatier, 31-Toulouse

RBsumB. - Dans la determination de la structure electronique des molkcules ou des solides, on trouve deux grandes categories de mkthodes baskes sur I'emploi :

- soit d'orbitales molkculaires dBlocalisQs dans les molecules ou de bandes dans les solides,

- soit d'orbitales localis6es.

Nous utilisons cette seconde catkgorie de methodes pour etablir, dans le cas des solides ioniques ou covalents, l'expression de l'Bnergie totale de leur Btat fondamental. Cette methode est appliquee au calcul de l'equation d'8tat du diamant.

Abstract. - In the determination of the electronic structure of molecuIes or solids, we can find two great categories of methods based on the use :

- whether of molecular orbitals delocalized in molecules or bands in solids,

- or of localized orbitals.

We use the second category of methods to set up, in the case of ionic or covalent solids, the expression of the total energy of the ground state. This method is applied to the calculation of the state equation of the diamond.

I. Introduction. - T o determine the electronic structure of molecules or of solids in their ground state, we find two great categories of methods ;

- delocalized methods : molecular orbitals in molecuIes and bands in solids ;

- localized methods, mostly used in chemistry : valence-bond method. By this localized method Lowdin has studied the cohesive energy of ionic crystals [I].

We were interested in the localized systems rare gas (crystals, covalent crystals, ionic crystals) because we think that such a study finds its place in the trend of the researches undertaken nowadays, which tends to mix the ideas and methods of chemistry and those of the solid state physics. Phillips's work shows, for instance, the interest of applying the concept of chemical bond to the study of the structure and of the cohesive energy of the crystals [2]. In spite of nume- rical difficulties, the reduced quantity of work carried out in that localized method about the solids is surpris- ing to us when we compare with what has been done about the molecular structures all the more as the problems about the solids are much easier to solve because of the symmetry of the crystals.

11. Expression of the electronic energy of a crystal.

- Consider a ionic or covalent crystal in which the electrons can be localized by pairs.

Let us associate with each pair of electrons a loca- lized orbital cp.

For a crystal of N = 2 n electrons, therefore, we

the wave function of the ground state of the system is Slater's determinant :

The total energy E = E, -I- Een 4- Eee + ^{Ehn can be}

decomposed into kinetic energjr E,, potential energy of Coulomb interaction of the electrons with the nuclei E,,, energy of Coulomb interaction of electrons E,,, and energy of repulsion among the nuclei. The various terms of energy have been given by Lowdin [3].

In atomic units, we have :

" Za Z m : number of nuclei of the crystal

Erin ⁼ zb ^{2 ( Za :} atomic number of nucleusa

have to consider n localized orbitals p i of any kind (not orthogonal). In the one-electron approximation,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972318

C3-128 5. GELARD AND PH. DURAND However, owing to the term (pq ( rs), the number of

bielectronic integrals increases like n4, so to reduce the calculation time we had rather apply Mulliken's approximation :

which makes the number of integrals proportional to n2. This approximation merely modifies Een and Eee [4l.

These expressions are easy enough to make possible a complete and very quick calculation, not semi-empiric, of the energy of a crystal from a basis of localized orbitals reasonably chosen. Thanks to the symmetry of the crystal the calculation of the total energy is reduced to that of the energy of an atom (or of several atoms or ions) interacting with all the other atoms or ions of the crystal. Let us study the case of an atom to which no localized orbitals are associated, the expres- sion of energy by atom is given by :

E = Ec + ^Ee, + ^Een + ^{En, with :} (we suppose the crystal is infinite) n, m -- ^cc

" 5

En, = 5 ^2, C -

b = 1 Rob

where

= + 3 when only one electron of the bond belongs to the atom and + ¹ in the other cases.

Instead of E,,, we can take :

because these two expressions of energy lead to the same result.

If, instead of one atom, we are led to consider ma atoms or ions, we introduce in the expression of En, and EL,, a sum from a = 1 to ma.

Another simplification occurs because the sums (on p and a) which extend infinitely can be limited to the atoms near the considered atom.

In effect, when we get far from this atom, the contribution of the other atoms quickly tends to zero.

Moreover the calculation can be even more simplified, for by reason of the symmetries, it is possible to deduce the elements Tpq of the matrix T (the inverse of S), which occurs in the sum, from the inversion of a much smaller matrix.

111. Example : the diamond. - To give an example, we have studied the equation of state of a covalent crystal : the diamond. The electronic structure of this crystal is simple enough for the complete use of the method. The core-electrons of the atom of C are described by a Slater atomic orbital of the type 1 s : qc ⁼ Cte e-5r. In truth, in calculation, for merely numerical reasons, we reproduce this Slater orbital by two Gaussian orbitals centered on the atom of carbon [ 5 ] .

The bonding-orbitals C-C are spherical Gaussian orbitals centered in the middle of the bond C-C by reason of symmetry :

q c - , = Cte e-ar2

We only consider Gaussian spherical orbitals for which

the various integrals of energy are known analytically.

ELECTRONIC STRUCTURE OF THE GROUND STATE OF THE COVALENT C3-129

Moreover, these reduced basis have been used sucessfully to calculate the electronic structure of the molecules.

Each atom of carbon has four nearest neighbours and twelve second nearest neighbours. In this work, we have limited the summations on p to the 4 nearest atoms.

The calculation, therefore, makes occur 5 core- orbitals and sixteen bonding-orbitals.

The energy by atom is calculated for various values of the distance-between two atoms of carbon. For each value of this distance, we determine the coeffi- cients I: et a minimizing the energy.

The curve we obtain is represented on the sketch 3.

It is noticeable that the minimum of the curve corresponds to d c - , = 1,548 A, what practically corresponds to the experimental value, d c - , = 1,544 A

at O°K.

The equation of state is deduced from these results by relation

p = - - . dE dv

On the sketch 4 we have represented the pression as a function of v/v, (v : volume of the crystal - vo : volume of the crystal in equilibrium).

IV. Results. - The comparison with the experi- ment lets appear a notable difference with the curve obtained by Bridgman ; for P = 0,l Mbar, we find 6 = 1,010 whereas the experiment gives the following result 6 = 1,018 (6 = vo/v). The origin of this diffe- rence, that we are studying, is probably due, by order of importance :

a) To Mulliken approximation,

b) to the use of a very reduced localized base, c) to the one-electron approximation model, d ) to the mere consideration of the 4 nearest atoms.

r In spite of the imperfection of the presented model

0,733 1,548 d,,en A we can follow in a quantitative way, the variation of the parameters a and 5 during the compression

FIG. 3. (Table I).

Variation of the parameters of the core-orbitals Is, and of the Gaussian spherical orbital of valence in function of the distance d c - , between 2 neighboring atoms of carbon.

(5, = 5,672 7, best coejficient for the orbital Is of the isolated atom of carbon.) dc-c (4

-

1,603

1,548

1,542

1,352

1,272

1,098

0,917

C3-130 J. GELARD AND PH. DURAND

We notice that these coefficients increase with the with the experimental curve is not very satisfying we compression (6 = voJv), what corresponds to an increas think 'that it is necessary to study in details simple sing of the average kinetic energy of the electrons. models of crystals as the one we have presented.

When the necessary improvements are carried-out V. Conclusion. - On such a very simple model as we think that it will be possible, by means of simple the diamond, the use of localized orbitals permitted to and merely theoretical calculations, to obtain with calculate the internuclear distance C-C very precisely. a great precision the main quantities of the ground-

Although for the equation of state, the concordance state.

References

[I] LOWDIN (P. O.), Ark. Mat. Astronom Fysik, 1947, [4] NICOLAS (G.) et DURAND (Ph.), C. R. Acad. Sci. Paris,

A

1971. 272. 1482.

"-

[5]

( S f , J. Chem. Phys., 1965, 42, 4, 1293.

i21

''), Rev. of Modern Physics, lg70, 42. r61 FROST (A. A.),

Chem. Phvs., 1967, 47, 370'7.

[31

(P. 0.1, J. Chem. ~ h y s . , 1950, 18, 3, 365. i7j

(P.' w.), ~ndeavour; 1951, 63.