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Rigorous nonvariational) solution of the Schrodinger equation for a crystal1 I. Geguzin and L. I. Leont eva

Physics Research Institute Rostov-on-Don Un iversity(Submitted 2 March 1989)Zh. Eksp. Teor. Fiz. 96,1075-1086 (September 1989)

A formalism is developed for a rigorous (nonvariat ional) solution of the single-particleSchrodinger equation for a many-atom system described by a potential of arbitrary form. The

method can be used for arbitrary crystals with chemical bonds of various types. For muffin-tinpotentials the equations of the method reduce to the known equations of the Korringa-Kohn-Rostoker theory. The correctness of the method is verified by a numerical analysis of the three-dimensional periodical Mathieu potential for which exact analytic solutions are known.

1 FORMULATION OF PROBLEM

The theory of the electronic structure of crystals isbased on the single-particle Schrodinger equation (S E)

[ -V2+V(r)-E] ( r )=O. (1 )

The effective potential V( r) is invariant to displacements ofthe coordinate frame along arbitrary lattice translation vec-tors r,, :

V(r+R,) V(r)

The solutions $ ( r ) should satisfy the Bloch theorem

k r+Rn) =exp(ikR,) ,(r). (2 )

They are therefore classified by a wave vector k located inthe first Brillouin zone.

A number of methods, forming jointly the band theory,have been proposed for the solution of the boundary-valueproblem ( 1 , (2 ) . It is appropriate to gather these methodsinto two groups: variational methods [pseudopotential,orthogonalized plane waves (OP W) , linear combinations ofatomic orbitals (LC AO ), linear combinations of muffin-tin orbitals (LMTO) and joining methods [cell method,augmented plane waves (A PW ), Korringa-Kohn-Ros-toker (KKK R) Green's functions]. The variational meth-ods are based on representation of the sought function byan expansion

in some set of basis functions p hat satisfy boundary con-ditions ( the Bloch theorem) . The boundary-value problem( 1 , (2 ) is replaced by a variational problem of finding theextrema of an energy functional with respect to variation ofthe coefficients a k .The choice of the form of the basis func-tions (plane waves, orthogonalized plane waves, atomic or-bitals) determines the corresponding method (pseudopo-tential, APW, LCAO). It is important that the basisfunctions p are not solutions of the initial SE ( 1 . As aresult, given a certain potential V( r) , different methods leadto different solutions that do not tend to the solution of theinitial problem as the number of basis function is increased.Depending on the type of the chemical bond in the crystal

(simple metals, transition metals, covalent semiconductors,transition-metal oxides, etc. ), one method or another is cho-sen by starting from the physical requirement of a reasonable

approximation of the sought function , by one set of func-tions or another.

Each of the variational methods has thus a limited ap-plicability and yields, from the fundamental point of view,an approximate solution of the initial problem.

The joining methods are based on the requirement that

the function be continuous together with its first deriva-tives everywhere in the crystal. A rigorous formulation ofthese methods is possible only for a limited class of potentialsof the so-called muffin-tin (MT) form. The crystallineMT potential is represented by a superposition

of atomic MT potentials

each of which differs from zero only within its own MTsphere having a radius b .

Our aim here is to develop a method that leads to arigorous (nonvariational) solution of the problem ( 1 , ( 2 )for an effective crystal potential of arbitra ry form. The meth-od should be valid for crystals with various types of chemicalbond.

The need for restricting ourselves to the class of MTpotentials in the traditional formulation, and a criticism ofthe concept of multiple scattering, are analyzed in Sec. 2.The general formalism is considered in Sec. 3. A particularcase of an MT potential is analyzed in Sec. 4. The correctness

of the method is verified by a numerical analysis of a three-dimensional periodic Mathieu potential, for which an exactanalytic solution is known. The results of a numerical analy-sis are given in Sec. 5. The distinctive features of the Green'sfunction of a system of many centers are considered in theAppendix.

2 JOINING OF WAVE FUNCTIONS N THE CONTEXT OF THEMULTIPLE SCATTERING CONCEPT

An analysis of an attempt to obtain a rigorous solutionby joining wave functions is best carried out by using themultiple-scattering concept. We transform from the bound-ary-value problem ( 1 , 2 ) to the integral SE

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The integration here is over the main region of the crystal,containing N unit cells of volume f l . The Green's function(GF) satisfying the periodic boundary conditions is ex-pressed by a series in the reciprocal-lattice vectors k, :

exp [ k+&) (r-r )~ ~ ( r , r < ;) = - - Z

IkfKp12-E( 6 )

K P

On the basis of the multiple-scattering idea, we repre-

sent the crystal potential V as a superposition of the poten-tials v of the individual cells:

V (r) (r-R.) n ( 4 ) .

Equation 5 ) is then equivalent to the system

We shall assume that each of the potent ials v vanishes in athin layer of thickness S in the vicinity of the faces of the celln. (The quantity S must tend to zero in the final expres-sions.) Then, placing the point r, in a small S-vicinity of thesurface of cell m, we can use in the left-hand sides of Eqs. 8)the form of the solution k known for a zero potential. Onthe other hand, the functions k for all r; ~ f l , an be expand-ed in some complete system of functions x that are solu-tions of an SE with potential v at a certain energy E:

A proof of this statement is given in Ref. 2. By considering insuccession the S-vicinities of all the cells m 1 2 ..., N) , wereduce the system of integral equations ( 8 ) to an algebraicsystem for the unknown coefficients A n .

From the physical viewpoint this approach correspondsto the Rayleigh-Huygens self-consistency condition for dif-fraction of a wave by a many-center object. On the one hand,the wave field in a small &vicinity of each of the cells mshould be determined by the asymptotic behavior of a wavescattered by the potential v . On the other hand, this verysame field should be a superposition of the fields scattered byall cells. The self-consistency condition consists of equating(in the S-vicinity of each cell) the amplitudes of the wavescattered by the given cell to the amplitude of the superposi-tion of the waves scattered by all the cell.

The above concept of multiple scatter ing for a potentialV of arbitrary shape, proposed in Refs. 3 and 4, does notsolve, however, our problem. It is clear from the form of thesystem 8 ) that to makeit algebraicit is necessary to be able,at a definite stage, to eliminate the explicit dependence onthe actual position of the point r,,. This requires that theright-hand side of ( 8 ) have a fully defined funct ional form ofa dependence on r,, . At the same time, the properties of the

atomic potentialsv

dictate a transition to the concept ofan angular momentum IL ) lm . t is obvious in this casethat the possibility of obtaining an algebraic system is gov-

erned by the existence of a single-valued two-center (withrespect to the centers m and n ) expansion of the G F ( 6 ) inthe spherical harmonics Y

It turns ou t that the functional form of such an expan-sion depends on the relations between the quantities r rland R that enter in the arguments of the G F in (8 ) (seethe Appendix) . Let us consider neighboring cells m and nand place the point r, in a 6-vicinity of the surface of the cella . In the calculation of the right-hand sides of ( 8 ) thepoint r, , runs through the entire volume of the cell n, and insome part of the volume of the cell 0 we have the inequality

in some other part we have the inequality

and in other regions of the cell n are possible in principleother relations between r r; and R,,, . (The forms of theinequalities and the shapes of the corresponding regions de-pend on the crystal geometry.) This means that for eachsuch region inside the cell it is necessary to use a specific

two-center expansion of the GF-either (A 6) , or ( A9 ) , orsome other (see the Appendix) . Moreover, the choice of anyof the G F expansion depends on the position of the point r,The absence of a single two-center expansion for the G Fmakes it impossible to determine uniquely the functionalform of the dependence of the field superposition ( 8 ) on theposition of the point r, in the vicinity of the surface of thecell m . Consequently, to calculate the contributions fromneighboring cells in (8 ) , called the near-field contribution^,^it is necessary to evaluate the integrals in (8 ) with the totalGF (6 ) independently fo r each actual position of the pointr, . The presence of near-field contributions makes it impos-sible therefore to reduce the problem to algebraic within theframework of the multiple-scattering concept , making a nu-merical analysis practically impossible.

Changing to MT potentials (3 ) and ( 4 ) results in asubstantial simplification. The integration region in eachterm of the right-hand side of ( 8 ) is bounded by a spheref lyTinscribed in the cell n, since the potential outside thesphere is zero by definition. We introduce a thin layer ofthickness S n the inner side of the f l y T sphere and place thepoint r, in this layer. One and the same inequality ( 9 ) s thensatisfied for arbitrary m and and for any position of thepoint r, in this spherical layer. We can therefore use aunique expansion of the G F ( A6 ), and this determines fullythe functional form of the superposition of the fields (8 ) inthe vicinity of the surface of the MT sphere f l y T. This allowsthe system (8) to be reduced to algebraic and yields theknown equation of the KKR method of the band theory'(see Sec. 3).

It follows from this analysis that the near-field effectsshould make a solution of our problem impossible. Nonethe-less, many attempts were made to develop a formalism onthe basis of the multiple-scattering concept. A characteristicfeature of these theories is that the secular equation containst-matrix elements, or generalized phase shifts, which de-scribe scattering of an isolated unit cell n by the potential v

(7 ) . A bibliography of these approaches can be found in Ref.5. We note in addition that the frequently used model ofoverlapping MT spheres is also incorrect.

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3 GENERAL FORMALISM FOR A CRYSTAL POTENTIAL OFARBITRARY FORM

The key to the solution of the problem is to forgo themultiple-scattering concept in its traditional formulation.that is to say, forgo a representation of the potential in theform ( 7) and the attempt of determining the fields in thevicinities of the cell surfaces, i.e., in a region asymptotic tothe atomic potentials u . We stipulate instead satisfaction ofthe Rayleigh-Huygens self-consistency condition at an arbi-

trary point r, located inside a certain sphere Dm having aradius smaller than the difference of the distance R to thenearest cell n and than the radius of the sphere 0 drawnaround the cell n (see Fig. 1 . It is obvious then that theinequality ( 9 ) will be satisfied for all cell pairs m and n andfor all r,ED,,, This means that we can use the single-valuedexpansion (A 6) of the GF . In other words, the functionalform of the dependence of the superposition (8) is complete-ly defined inside the sphere Dm .

The derivation given below was obtained for a particu-lar case of a single-atom crystal. Generalization to the case ofa many-atom crystal is trivial. We note only that it is always

possible to break up the crystal space into polyhedra thatensure satisfaction of the inequality ( 8 ) inside spheres Dm.These polyhedra need not necessarily be unit cells in thecrystallographic sense, and no atomic positions need be asso-ciated with some of them.

With the aid of the Bloch theorem, we reduce the inte-gration over the principal region in (8) to integration overthe unit cell fl

Using Green s theorem, we change to integration over a sur-face

- * r ) VIGk(r, ; E) ] =0 r=D. (11)

Here do is the differential element of an area oriented alongthe outward normal to the cell surface.

We define a certain set of funct ionsx, satisfying, insidethe sphere 0 circumscribing the sphere, the SE ( 1) with atotal potential:

It is important that we seek regular solutions X, specified byboundary conditions at zero, and impose no requirements on

FIG. 1. Schematic representation of two neighboring unit cellsn,,, ndfL with the distanc e between centers equal to the lattice vector R ,,,,.Thesphere O ircumscribes t he celln , and the sphere D is tangent to th esphere 0 The radius vectorr,,, occupies arbitrary positions inD,, , whiler:, runs throu gh the entire celln,,

the asymptotic behavior of these functions. In other words,we forgo the representation of the total potential V as a su-perposition ( 7 ) of atomic potentials each of which vanishesoutside its own cell. The boundary conditions on the cellsurface will be satisfied only for the total function ob-tained by solving Eq. ( 11 ), simultaneously with finding theenergy eigenvalue. The classification of the solutions X, byvalues of the angular momentum L is determined by the be-havior of the solutions as r-0, inasmuch as in a small vicini-

ty of zero the potential V(r ) can be regarded as sphericallysymmetric. It is just this feature of the behavior of the crystalpotential near atomic nuclei which makes advantageous, asmentioned in Sec. 2, the transition to the angular-momen-tum representation.

Let us obtain equations for x,. The potential Vand thefunctionsx, can be expanded everywhere inside the circum-scribed circle 0 in spherical harmonics:

We determine the matrix elements of the potential:

The requirements that the function X, satisfy 12), be regu-lar a t zero, an d be classified by the values of L lead then to asystem of Volterra integral equat ions for ~ L L Ref. 6):

r=max r, r ), j l x ) x jl (x )

El (x ) xn , (x) (15)

Here j (x ) and n , (x) are Bessel and Neumann sphericalfunctions, respectively, and ?c = E I 2 . It is known that thefunctions X, ( 12) form a basis in the sense that the soughtsolution , can be expressed everywhere inside the cell fl bythe expansion2

We choose now an arbitrary point rd ) (see Fig. 1) and sub-stitute the expansions A6) and (16) in (1 1). Since thepoint r belongs to the spherical region D we can use the

completeness of the set Y Condition ( 11 takes then theform of a system of homogeneous algebraic equations in theunknown amplitudes A :

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To solve the system ( 15) of integral Volterra equations weuse the phase-function PF) method.' We define the PF:

1C L L , r )=6LL, il ( x r ) L L ~ ,r f r p L , f ~ ,r )d r l ,

L

(25)

Equation ( 15 ) takes then the form

and the P F ( 25) satisfy the system of coupled equations

dSLL, 1=dr x

T Lx r ) U L L , , ( r ) T x r )CL,rLrr )L

f i l , r ( X ~ ) S ~ . , ~ -r )

and the boundary conditions

In (26) and (27) we have 0 < r < c, where c a 6 / 2 is theradius of the inscribed sphere. The system (2 7) of first-order

equation was solved by the Runge-Kutta method. The PFobtained, C and S . allow us to calculate the p (26)and obtain on their basis the integrals ( 18).

To obtain symmetrized ( in accordance with the irredu-cible representations of the point T ) combinations of basisfunctions and symmetrized integrands in ( 18) we used thesystem of analytic symbolic programming.x The maximumvalues L of the angular momenta in the expansion (6) areequal to 10 for the representation I?, and to 7 for the repre-sentation TI,. The dimensionalities of the corresponding se-cular matrices are equal to 5 r ) and 6 ( r 15 ) . o calculatethe GF expansion coefficients B L L A71 we used Ewald'sm e t h ~ d . ~he solution of the secular equation ( 10) calls for

finding the zeros of the determinant as a function of energy.The calculations were performed for a certain set of val-

ues of the parameter U, of the model potential (2 3) for alattice parameter a = n. It is also convenient to change todimensionless energy units Ea2/4n2. n Table I the cal-culated energies of the bands l r , and l rI5 re comparedwith the exact analytic values.

Note that a relatively small number of basis functionssuffices for quite satisfactory accuracy. Thus, in a wide range

of the parameter U, the deviation of the calculated energyeigenvalue from the exact one does not exceed 0.003 (in di-mensionless unit s). The rate of convergence relative to thenumber of basis functions included in the expansion 6 ) anddetermined by the value of l,,, can be demonstrated withthe r, , ands as an example (Table 11). Owing to the vari-ational character of the method, the energy eigenvalues de-termined by a specified number of basis functions can behigher as well as lower than the limiting exact value. Theconvergence is therefore not uniform.

Nevertheless, it is clear even now that the required val-ues of I,,, are considerably larger than those customarilyused in the MT model. It is known tha t in the case of an MTpotential a value I,,, = 2 ensures a relative energy accuracy~ 0 . 0 0 1n the calculation of the occupied states of crystalsconsisting of atoms of the first half of the periodic table.Values I,, 3 to 4 are needed only for crystals containingatoms with f shells. The increase of I compared with theMT case is a natural price to pay for taking into account apotential with a more complicated relief.

Our numerical analysis of the Mathieu equation is thefirst ever. The approaches mentioned in Sec. 2, based on theconcept of multiple scattering, were tested only with the aidof the case of an empty lattice, i.e., for V r) = U,. Thispotential can be regarded as a particular case of a Mathieu

equation at U = 0. The results of application of the methodto the empty lattice case are given in Ref. 10.

6 ON LUSIONS

The formalism considered is intended for calculationsof the electron s truc ture of crystals described by an effectivepotential of arbitrary form. It yields a rigorous (nonvaria-tional) solution of the problem of finding Bloch functions.The method can be used for arbitrary crystals with differenttypes of chemical bond. For the part icular case of MT poten-tials the formalism reduces to the known method of theKK R band theory.

A similar generalization is possible also for the electron

TAB LE I. Energy eigenvalues of bands l r nd l r or the three-dimensionalperiodic Mathieu potential 2 3 ) U , 0 4; n .

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TABLE 11. Convergen ce of energy eigenvalues of T ymmetry.

Calculation 19I I

stru cture of molecules and clusters. In this case the formal- 1 z ,,, x p I i ( k + R ) (.- I e x p ( - i k R n )=--ism for MT potential reduces to known Slater-Johnson scat- ( 2 n ) 3=, Ik + K , ' - -Etered-waves method. 2

From the fundamental point of view, the method is pre-ferable because it yields a final solu tionof an b n t o single- (th e integral is taken over the Brillouin zon e). We introduceparticle problem. Fr om the physical point of view the meth- the vector q = k +k,, . We have then

uz

bd is needed in cases when t he potential-anisotro py effectsare so strong that th e M T approxim ation can lead to resultsthat ar e incorrect both quantitatively and qualitatively. Letus name som e of these problems.

It is known from molecular theory that theMT approx-imation leads to considerable erro rs in the determination ofthe total energy of a molecule. The meth od developed herecan be used to calculate the total-energy surfaces as func-tions of the nuclear coordinates, and hence determine thevibrational spectra of the molecules.

In the theory of the electron structure of crystals, thelargest anisotropy effects should be expected for covalentcrystals, and also for com poun ds with a chemical bond of thecoordination type, inclu ding complex o xides in th e class ofhigh-temperature superconductors. A t the present t ime themain results for covalent crystals were obtained by LCAO

1 e x p i q r-r ' -R,) ]q2-E

I I

m a l n ~ a x 3I

Using the expansion of a plane wave in real sphericalharmonics Y we get

' n ~ a x= Exact value

n = 8 z l - - ' Y L ( i )Y L - i f ) k,,) L , L r , )

where

( L ,L', L ) j y L ( q )Y,, (6)Y L , , .hap.and pseudop otential variational method s, and in the case of Th e integralshigh-temperature superconductors with the aid of the -LM TO m ethod with an MT potential. A proper allowance

i l 4 r ) j l , q r ' ) j l ., q R n )for the anisotropy of the crystal potential can lead to a In E-qz q2 a4chang e of the dispersion law of the electron s and of the shap eof the Fermi surface, and can consequently influence the c n be calculated by residue theo ry.1f r < r' andmodel premises based o n results of calculations in the M Tapproximation. r+r l

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BLL.k ,E ) = 4ni'- Z ( L , L', L ) D ~ , . ( ~ , E ) A71L

are expressed in terms of the so-called st ruc ture constan ts

=-i - ' ( i x )[ + h1 ( x R . ) Y L (R.) exp ( i k ~ . ) ]2n R +O

If, however, inequality A4) is not satisfied, other ex-pansions are valid. Fo r example, if

r+R,,