NEW METHODS IN THE THEORY OF MOLECULES AND SOLIDS

HAL Id: jpa-00215035

https://hal.archives-ouvertes.fr/jpa-00215035

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.

NEW METHODS IN THE THEORY OF MOLECULES AND SOLIDS

J. Slater

To cite this version:

J. Slater. NEW METHODS IN THE THEORY OF MOLECULES AND SOLIDS. Journal de Physique

Colloques, 1972, 33 (C3), pp.C3-1-C3-5. �10.1051/jphyscol:1972301�. �jpa-00215035�

JOURNAL DE PHYSIQUE Colloque C3, supplkment au no

5-6,

Tome

33,

Mai-Juin

1972,

page C3-1

NEW METHODS IN THE THEORY OF MOLECULES AND SOLIDS ^(*)

3. C . SLATER

University of Florida, Gainesville, Florida

U.

S.

A.

RtsumB.

-

On suggkre une methode susceptible de traiter aussi bien les molCcules complexes que les cristaux, travail conjoint de I'auteur et d'un certain nombre de collkgues dont K. H. John- son, J. W. D. Connolly, J. B. Conklin, Jr., K. Schwarz et F. C. Smith, Jr. Elleest basee sur l'utili- sation de l'kchange statistique, propose en 1951 par I'auteur, multiplie par un facteur a. Ce para- mktre a pour un atome isole est choisi de sorte qu'une expression convenablement choisie de l'energie totale soit tgale a l'energie Hartree-Fock. Schwarz a montre que cela conduit a des valeurs de a allant d'environ 0,77 pour un atome a deux electrons a environ 0,70 pour un atome a qua- rante electrons. On traite ensuite la molecule ou le cristal a l'aide d'un potentiel ⁽⁽ muffin-tin >>.

A l'interieur des sphkres atomiques, on utilise le mCme a que pour l'atome isole, et dans la region interatomique une moyenne convenable des valeurs pour les differents atomes. Pour rhoudre l'Cquation de Schrodinger dans un tel potentiel, on utilise la methode de la << diffusion multiple ^>) proposee par Johnson et Smith. Cette methode est aisement rendue self-consistante et Johnson decrira des applications aux ions (S04)--, (C104)-- et (Mn04)-. Connolly et Johnson I'ont appli- quQ a la mol6cule SFG. Toutes ces applications conduisent a des orbitales moleculaires et des energies Ctonnamrnent bonnes, apparemment plus precises que les meilleures valeurs existantes, en un temps calcul quelque trois ordres de grandeurs plus petit que par la methode LCAO. Pour le calcul des energies d'excitation, on utilise ce qu'on appdle un ittat de transition, dans lequel le nombre d'occupation des orbitales est mi-chemin entre ceux des etats initiaux et finaux. On peut aussi appliquer ces techniques dans le cadre de la methode APW utile pour les cristaux p6riodiques ; Conklin et Schwarz ont fait beaucoup &applications qui seront decrites a des composks tels que T i c et ScN. L'expression de l'energie totale trouvee avec l'echange statistique montre une dkpen- dance en fonction du parametre du reseau qui est en accord Btroit avec l'experience, comme T. M. Hattox l'a montre pour I'energie de cohesion du vanadium et I?. Averill pour 1'6nergie de cohesion du cesium. Les applications possibles de la methode aux problkmes magnetiques sont decrites ci-aprks.

Abstract. - A method is suggested for handling the theory of both complex molecules and crystals, the joint work of the author and a number of colleagues, including K. H. Johnson, J. W.

D. Connolly, J. B. Conklin, Jr., K. Schwarz, and F. C. Smith, Jr. It is based on the use of the statistical exchange, as proposed in 1951 by the author, multiplied by a factor a. This parameter a, for an isolated atom, is chosen so that a suitably defined expression for total energy, involving the statistical exchange, precisely equals the Hartree-Fock energy. Schwarz has shown that this leads to values of cr going from about 0.77 for a two-electron atom to about 0.70 for a forty-electron atom.

One next treats the molecule or crystal by use of what is sometimes called a << muffin-tin ^B potential, spherically symmetrical within a sphere surrounding the atom, constant in the volume between the atoms. Within the atomic spheres one uses the same a as for the isolated atom, and in the inter- atomic region one uses a suitable mean of the values for the various atoms. For solving Schrodin- ger's equation in such a potential, for a molecule or atomic cluster, one uses a multiple-scattered- wave method proposed by Johnson and Smith. It is easy to make this into a self-consistent-field method, and Johnson will describe applications of the method to the (S04)--, (C104)-, and (Mn04)- ions. Connolly and Johnson have applied it to the SF6 molecule. All of these applications lead to surprisingly good molecular orbitals and energy levels, apparently more accurate than the best existing values, with a computing time some three orders of magnitude less than by the LCAO method. In calculating the excitation energies, one uses what we call a transition state, in which the occupation numbers of the orbitals are half way between those for the initial and final states. One can also apply these methods with the APW (Augmented Plane Wave) method, which is useful for periodic crystals, and Conklin and Schwarz have made many applications, which will be described, to such compounds as T i c and ScN. The expression for total energy found by use of the statistical exchange shows a dependence on lattice spacing which is closely in agreement with experiment, as T. M. Hattox has shown in calculations on the cohesive energy of vanadium, and F. Averill on the cohesive energy of cesium. Possible applications of the method to magnetic problems are described in the following talk.

1. The

Xu

Potential Method.

-

As mentioned in well as another paper by the author, will be found

in

the Abstract, the work described in this paper has been the present volume, and these contain many references the joint work of numerous colleagues. Papers by to additional work. We refer the reader t o these papers,

J. B.

Conklin,

K. H.

Johnson, and

K.

Schwarz, as rather than duplicating

the bibliographies found

there

in the present paper.

(*) Assisted by the National Science Foundation. The methods which we are

describing are the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972301

SLATER

outgrowth of techniques which have been found very

useful in the solid-state field for well over a decade, but whose application to molecular problems is much more recent. There are two main features of the techniques. First, the exchange-self-interaction terms in the Hartree-Fock method are replaced by a local function of the charge density, proportional to the 113 power of the density, containing a parameter a to be determined as described in the present section. This has a great practical advantage, resulting in much simpler computational methods. At the same time it has a great theoretical advantage, in that it automati- cally leads to an expression for total energy which goes properly to the energies of the separated atoms, in the limit of large internuclear distances. This means that it includes not only exchange, but also the long- range correlation which can be achieved in the Hartree- Fock method only by very extensive linear cornbina- tions of determinantal functions.

The other main feature of the techniques is the so-called

<<

muffin-tin

^)>

approximation which is made in the one-electron equations of the self-consistent- field problem. In this approximation, one surrounds each nucleus by a sphere, generally large enough so that the spheres of neighboring atoms touch. Inside each of these spheres, the potential function, which is very nearly spherically symmetrical, is replaced by its spherical average. Outside the spheres, the potential function, which is nearly constant, is replaced by its constant average value. These approximations lead to a potential function so simple that one can easily set up exact methods for solving Schrodinger's equation for the molecular orbitals in such a field, in terms of easily managed functions. In many cases, the errors arising from the use of the

^<(

muffin-tin

^)>

potential are small enough so that one can neglect them and still get useful results. If this is not the case, it is not hard to use the resulting functions as the starting point for a further calculation which removes the approxi- mations by perturbation methods.

We shall come to these methods of solving the Schro- dinger equation in the Section 3, and now pass on to the p1I3 potential, where p is the electronic charge density. We start by writing the spin-orbitals ui. In terms of them, the charge density is

where ni is the occupation number of the ith spin- orbital. This quantity must lie within the range from zero to unity, on account of the exclusion principle, and for some purposes we shall wish to allow it to be treated as a continuous variable through this range.

Some of the spin-orbitals will correspond to spin-up, some to spin-down, and we shall wish to write

We then write the total energy of the system in the form

<

^{E X a}

>

⁼ ( i ) ni

J

u x l )

f l ui(l) dv, +

+ internuclear potential energy (3) Here f, is the one-electron operator representing the kinetic energy of the first electron, plus its potential energy in the field of the nuclei. Hence the first term of eq. (3) represents the total kinetic energy plus the potential energy of interaction of electrons and nuclei.

The next two terms represent the electron-electron repulsions, and the last term is the nuclear-nuclear repulsion.

The electron-electron interactions are different for electrons of the two spins, so that they are written as two separate terms, each of the form 112 times the charge density times the effective potential energy acting on the electrons, integrated over the volume.

The first term in the effective potential energy is

1 p(2) g12 dv,, where g,, is the Coulomb interaction between electrons I and 2. Thus this is simply the electrostatic potential energy of all electrons (including electron 1) acting on an electron 1. This is corrected by the exchange-correlation term, Uxa T (1) or Uxa 4 (1) as the case may be. This must remove the self-interac- tion energy, incorrectly included in the first term in the effective potential energy, and in addition, must include all exchange effects. The exact energy of a quantum-mechanical system can be written in the form of eq. (3), by use of the first- and second-order density matrices. The only approximation which we make is that of replacing the exchange-correlation term which would rigorously hold, by an approximate value proportional to the 113 power of the charge density. This approximation is

with a similar formula for U,, 4 (1). In these formulas we are using Rydbergs as units of energy.

The functional dependence of the exchange-correla-

tion term on the density p is easily explained on

dimensional grounds. We have mentioned that the

main purpose of this term is to take care of the self-

interaction energy. A given electron is really not acted

on by all electrons (including itself) but by all electrons

minus one. If we assume that this one missing electron

is removed from a hole centered on the electron in

question, with a constant charge density equal to the

charge density of electrons of the same spin as the

electron is question, at its local position, we find a

formula identical with eq.

(4),

with a value of

a of

0.87. A somewhat more diffused charge distribution

NEW METHODS IN THE THEORY OF MOLECULES AND SOLIDS C3-3

for this electron would reduce the value of

a.

The

parameter

a

can be chosen so as to make the value of total energy given by eq. (3) exactly equal to the correct energy. We discuss this method of choice, which is the one which we prefer, in a later paragraph.

Before coming to this point, we should state the method used to derive the one-electron equations for the spin-orbitals ui. We vary the spin-orbitals ui, in eq. (3), keeping the occupation numbers ni and the value of

a

fixed, to make the total energy of eq. (3) stationary. Then we find that we have the equations

where - V: is the kinetic energy (using Rydbergs as units of energy), V , is the total Coulomb energy acting on an electron, arising from all nuclei and all electrons including itself, and

If the ul s are determined by solving these equations, we can prove an important result

:

the kinetic and potential energies derived from them, by eq. (3), satisfy the virial theorem, and also the Hellmann- Feynman theorem. (There are small restrictions on these statements if

a

is chosen to be different on diffe- rent atoms of a molecule, as we shall mention later.) These theorems are satisfied, for any constant value of a.

We thus see that if we are dealing with an isolated atom, and choose the value of

a

so that the total energy of eq. (3) will exactly equal the correct energy, we shall also have the kinetic energy equal to the correct kinetic energy, since by the virial theorem it equals the negative of the total energy. The correctness of the kinetic energy is a very sensitive test of the correctness of the spin-orbitals, and when this value of

^a

is chosen, the spin-orbitals prove to be very close to Hartree-Fock orbitals, for an isolated atom repre- sented by a single determinant. As a practical method of approximately achieving this result, we choose the a so that the energy of eq. (3) equals the Hartree-Fock energy of the average of all multiplets associated with the ground-state configuration. Schwarz has made such calculations, and finds that

a

goes from 0.77 for a two-electron atom, rather rapidly and smoothly down as the atomic number increases, achieving a value of about 0.70 in the middle of the periodic table.

With this choice of

a,

the resulting charge density, Coulomb potential energy, and exchange-correlation potentials Uxa (1) and Uxa 1 (I), obtained by the Xa method for an isolated atom, all prove to be very close to the Hartree-Fock values. Studies by Connolly and Schwarz have shown that the spin-orbitals so computed have about the same accuracy, in compari- son with correct Hartree-Fock orbitals, as the double- zeta orbitals of Clementi.

2.

Further properties of the

Xu

method. The tran- sition state. -

One respect in which the

Xa

method

differs in principle from the Hartree-Fock method is in the significance of the one-electron energies

ci.

In the Hartree-Fock method, it is well known that the eigen- values ci are given by differences of many-electron energies in atom and ion

:

EiHF =

< EHF (n,

=

1) >

^-

< E H F (ni

=

0) > (7) which is the Hartree-Fock energy with the ith spin- orbital occupied, minus that with it empty, in which the spin-orbitals are assumed not to be modified when the electron is removed. This is the essence of Koop- mans' theorem. In the Xa method, on the other hand, the eigenvalues prove to be derivatives of the total energy with respect to occupation numbers

:

Since the total energy is not a linear function of the occupation numbers, the results of eq. (7) and (8) will not be equal to each other. The differences increase as we go to the inner orbitals of an atom, and they represent very sizable quantities for the K and L elec- trons of heavy atoms.

An optical excitation energy is given approximately, but not exactly, by the Hartree-Fock eigenvalue of eq. (7). The reason why it is not exact is that really we should calculate the energy of the ion in eq. (7) by independently varying the spin-orbitals, to take account of what we may call the relaxation of the orbitals as an electron is removed. We must proceed differently in the

^Xa

method. If we regard <

^EXa

>

as a continuous function of the

nl s, we could of

course integrate the derivative of eq. (8) from the values ni

⁼

0 to ni

⁼

1, and get a correct value, including the relaxation effect. It is easy to show, however, that practically the same result can be secured much more simply, by using what we call the transition state. Namely, if we set the occupation numbers ni equal to the average values which they have between the initial and final states of the transi- tion, and carry out a self-consistent calculation in this transition state, the difference between the eigen- values

^E~~~

corresponding t o the initial and final states of the transition, computed for the transition state, is equal to the correct excitation, including rela- xation, except for very small third-order terms. This theorem proves to be very convenient to use in prac- tical calculation, and to be very accurate.

We have, then, a convenient procedure for finding the energies of both ground state and excited states of a system. We first solve eq. (5) for the ground-state spin-orbitals and occupation numbers. Then we use the transition state technique to get the energy diffe- rences between the ground state and various excited states.

Next we consider the ground state, and in particular

the correct occupation numbers

ni to use in this

state. Here the one-electron energies of eq. (8) prove

to have a very desirable property. Namely, if we shift

C3-4 J. C . SLATER

an infinitesimal amount of electronic charge from a

spin-orbital with higher one-electron energy

ziXa

to one with lower energy, we can prove directly from eq. (8) that we reduce the total energy of the system.

Hence to get the lowest state we merely must fill all spin-orbitals whose energy

ziXa

is below a certain Fermi energy, and empty all those above the Fermi energy, and no further reduction of energy is possible.

In fact, eq. ( 8 ) is precisely the condition required to make the Fermi statistics rigourously justified, a property which is not true when we use the Hartree- Fock energies

E i H F

for one-electron energies.

There is a further property which eq. (8) has

:

it can serve to give a rigorous definition to the electro- negativity of an atom. If two atoms are brought into contact, electrons will necessarily flow from the atom with higher

ciXa

in the highest of its occupied levels, into another atom which has a partially occupied level of lower AS this flow continues, the energy levels of the two atoms will be modified on account of the electrostatic charges on them, the topmost partially occupied levels of the two will reach equality with each other and with the Fermi energy, and this will correspond to the ionicity which they will achieve.

Each atom in this equilibrium state may well have a fractional occupation number in its outermost filled shell, and this is consistent with the Fermi statistics, which says that energy levels exactly at the Fermi level can have any occupation number between zero and unity. In this way we get very interesting interre- lations between the Xci method, Fermi statistics, and electronegativity, which have not been possible with previous methods of treatment.

3. Solution of the one-electron problem. ^-

The solution of the one-electron equation, eq.

(5),

is of course trivially simple for an isolated atom, where

u i

can be written as the product of a spherical harmo- nic of angles, and a radial function whose values can be found by numerical integration of the radial equa- tion. For the real self-consistent-field problem encoun- tered in a molecule or a solid, an exact solution is quite difficult, but the muffin-tin

⁾⁾

approximation which we have mentioned earlier leads to a problem which is close to the correct problem, and which is easily solved. For the interior of the spherical regions surrounding the atoms, we have the same spherical symmetry as with an isolated atom, and we can write the general solution of Schrodinger's equation as a linear combination of products of spherical harmonics of angle, times radial functions computed for an energy parameter which is the eigenvalue

ci

of the problem, which satisfy the boundary condition of being regular at the nucleus. Such a linear combination has an infinite number of arbitrary constants, namely the coefficients of the various functions u,,,,(r,

8, cp)

corresponding to different values of the quantum numbers I, m,, but in practice we need only a few such functions.

There are a number of techniques used for solving the

⁽⁽

muffin-tin

^D

problem, and they differ from each other only in the way of approximating the solution outside the atomic spheres. The earlier methods worked out were for periodic crystals. Here the two leading methods are the Augmented Plane Wave (APW) method, in which the solution outside the spheres is expanded as a linear combination of plane waves, and the Korringa-Kohn-Rostoker (KKR) method, in which it is expanded as a linear combination of scat- tered wavelets emerging from the various atomic spheres. For highly excited wave functions, the two methods are about equally convenient, and careful intercomparisons over a period of years have shown that they lead to identical results. For the lower ener- gies, however, particularly for the occupied orbitals in the ground state, the KKR method tends to be more convenient. The reason is that in this case the scattered wavelets fall off exponentially as one goes away from the atoms (on account of the radial depen- dence of the solutions of the wave equation for nega- tive energies), and this leads to a rapid convergence, nor many wavelets being required to produce a good representation of the wave function.

No matter which of the methods is used, one must satisfy boundary conditions at the surfaces of the spheres, to achieve continuity of the wave function and its normal derivative on the surface. This condition is applied in well-worked-out computer programs for both methods. The conditions can be satisfied only for discrete energies, for a wave function in the crystal corresponding to a given crystal momentum k, and the results of these calculations lead to the energy bands and wave functions for the crystalline problem.

For a large molecule, or for a cluster of atoms inside a disordered solid, the analog of the KKR method is more practical. This method has been developed and programmed by Johnson and Smith, and results of calculations using it, for a number of large molecules and radicals, will be described by Johnson in his paper in this volume. It seems capable of being extend- ed to more complicated molecules, and furthermore, it seems likely that a fruitful method of studying disor- dered solids and their relations to regular crystals may be to apply the cluster method to larger and larger clusters.

4. Some extensions of the method. -

One can think of applications of the general methods described in this paper to a variety of problems. First, the expres- sion for total energy can be studied as a function of the positions of the nuclei, and information can be derived regarding elastic constants, cohesive energy, and related problems. Before this can be done, we must know how to choose the a parameter in a crystal.

The method which has been found to be satisfactory is to use inside each of the atomic spheres of the

c<

muffin-tin

^D

method the

a

found for the corres-

ponding isolated atom, and for the region between the

NEW METHODS IN THE THEORY OF MOLECULES AND SOLIDS C3-5

spheres a suitable average of the values for the consti-

tuent atoms. This involves small discontinuities of

a

on the surfaces of the spheres, and as was intimated in Sec. 1, this can lead to small deviations from the correctness of the virial and Hellmann-Feynman theorems, deviations which would be met in a mole- cule or crystal composed of different types of atoms.

For such a problem as a crystalline metal, these difficulties will not be encountered, and Conklin will describe calculations by Averill and Hattox on crystals of metallic cesium and vanadium respectively, which are giving good agreement between the total energy and the experimental values. The programs for com- puting the total energy are much more difficult than those for the one-electron eigenfunctions and eigen- values, since they demand an accuracy of something like Ry out of a total energy of the order of

lo4

Ry, but this difficulty has been successfully over- come in our programs for crystalline calculations.

Work is still going on jointly between the groups at Gainesville and MIT to work out and test similar computer programs for the isolated molecules, and so

far we have not been able to work with the total energy of polyatomic molecules, but the difficulties will be overcome shortly.

Another application of the method is to the transi- tion state. Here Johnson will report interesting results for clusters, in which the method is working very satisfactorily. For a crystal, the important transitions are localized transitions, as in an x-ray transition, in which an electron on a given atom goes from an inner orbital to a higher orbital. The transition state then has a localized atom with a different occupation num- bers of electrons from the rest of the crystal or mole- cule. The one-electron equation will then have loca- lized solutions with eigenfunctions near this localized atom, as well as other solutions which avoid this atom.

It is the localized orbitals, which can be easily computed

by use of Johnson and Smith's cluster method, which

correspond to the large oscillator strengths, and

which must be considered. These orbitals will come

into the theory of the exciton, and also into the theory

of magnetic excited states, which will be described

further in the other paper by the author in this volume.