IC/80/166
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
AW EFFICIENT KORRINGA-KOHN-ROSTOKER METHOD
FOR "COMPLEX" LATTICES
M. Yussouff
and INTERNATIONAL ATOMIC ENERGY R. Zeller AGENCY
UNITED NATIONS EDUCATIONAL SCIENTIFIC AND CULTURAL ORGANIZATION 19R0 MIRAMARE-TRIESTE
IC/90/166 The well known rapid convergence and precision of the KKR- Green's function method (Korringa 1947, Kohn and Rostocker 1954) International Atomic Energy Agency have made it useful for the calculation of the electronic struc- and United Rations Educational Scientific and Cultural Organization ture of perfect as well as disordered solids (Ham and Segall 1961,
IHTEBSATIOHAL CENTRE FOR THEORETICAL PHYSICS Ehrenrelch and Schwartz 1976). Although extensive calculations of this kind exist for cubic crystals (Moruzzi, Janak and Williams 1978), there are so far very few calculations for the equally important class of hexagonal close packed solids. This is due to the computational efforts involved in evaluating the KKR matrix AH EFFICIENT KORRIHGA-KOHH-ROSTOKER METHOD FOR "COMPLEX" LATTICES • elements ATT,{k,E) using the conventional KKR for 'complex' lattices, i.e. crystals with several atoms per unit cell- Some simplifications can be achieved (Bhokare and Yussouff 1972, 1974) for k" vectors International Centre for Theoretical Physics, Trieste, Italy, in symmetry directions but the renewed interest in simplifying the and computations (Segall and Yang 19 80) has arisen in connection with self-consistent calcualtions. Here we describe an efficient modi- R. Zeller Institut fur FeBtkorperforschung der Kernforschungsanlage, fication of the KKR method which is particularly useful for 'complex1 D-5170 Jiilieh, Federal Republic of Germany. lattices.
ABSTRACT In our approach, the transformed KKR matrix elements are calcu- We present a modification of the exact KKR-band structure method lated directly using reciprocal space summation. A new energy ex- which uses (a) a new energy expansion for structure constants «nd (b) only the reciprocal lattice summation. It is quite effi- pansion leads to a separation of the slowly convergent but energy cient and particularly useful for 'complex' lattices. Tbe/band struc- independent quantities which can be stored and the fast convergent ture of hexagonal-close-packed Beryllium at symmetry points is quantities which are evaluated in each cycle to any given accuracy. presented as an example of this method. We believe that the reduction of computing effort on modern compu-
MIRAMARE - TRIESTE ters need not proceed along the lines adopted a decade ago (like October 1980 computation of D (E,)c) by Davis (1971)} and it is primarily a question of optimizing the use of storage to get significant reduction • Submitted for publication. in time for self-consistent calculations. It may be remarked here that •• Present addrese: Instltut fur Festkorperforschung der Kernforschungsanlaee, D-5170 Jiilich, Federal Republic of Germany. symmetry properties of specific lattices (Davis 1971) can lead On leare of absence from Indian Institute of Technology, Kanpur 208016, India. to a further reduction In time but the
-2- programmes so written are difficult to generalize to oth°r (or linear combinations of them). The band structure problem is
lattices of lower symmetry. The significant advan- to look for the zeros of the KKR determinant which can also be
tages of the expansion procedures (William, Janak and Moruzzi transformed to equivalent determinants. For example, we multiply
1972, Segall and Vang 1960) are retained in our method and equation (1) with j t (per .) j (, {icr , , } and use the properties of j(
we present the example of the band structure of hexagonal close and nt functions to get modified KKR matrix elements packed Beryllium at some symmetry points.
The basic equations of the KKR method in suitable forms for
computation are given by Ham and Segall 1961. We start by adap- (3) ting these equations to the case of 'complex' lattices along the where lines of Segall 1957. Let there be n. muffin-tin-spheres in the (B.it) sTE (4) unit cell with centers at T., j = 1,...nb and corresponding radii
r ji J * If.-.n^. The spherically symmetric potentials inside the It is important to point out that there are no irregular so- muffin-tin-spheres yield regular solutions R'^ (r) of the radial lutions like nt(icr) in equation (3).
Schrodinger equation with angular momentum t (L denotes t, m to- The summation is only over the reciprocal lattice
gether) . Then the KKR matrix elements for energy E and wave vec- vectors but unlike the structure constants of Ham and Segall 1961, tor jc are given by the quantities S £L?' (E,ic) of equation (4) are absolutely, al-
1 though slowly (like kn~ ) convergent. However, it is unwise to re- 1 (k j j
4 "I convergent parts at any fixed energy E so that the energy dependent jj [ (D part converges rapidly. For this, one can use the Identity 9fnt(»r>-n(((r|9 tnB/ (r) Here -—.————; 5—- r=r . r 3; I r 4 _J (5) 2 (kn -E) (2)
to get an exact energy expansion of S •! f (E,k) as follows. 1/2 and (E) and are spherical Bessel and Neumann func- tions, »r s —, Vc 1B the volume of the unit cell, R = % +fc" with the reciprocal lattice vectors K*n and YL(*n> are spherical harmonics -it- -3- contribution leaves behind a residual term whOBe exact E dependence
and convergence is not as transparent as those of ^j^ CErEo,k) in 3 J 1 LL equation (7). In fact, we can further separate the dominant term Of
§??,(E,Em,K) and carry this process to any arbitrary term in the Here j i1 IIIJ O (knj LL1 (7) aeries if necessary. Finally, it is also possible to carry out an ex-
pansion of S^(E,it) or £>^,(E ,E,S) in linear combinations of
Chebyshev polynomials as suggested by Williams, Janak and Moruzzi One can again use equation (5) in equation (7) to obtain 1972, but we have not followed this procedure in view of the ease higher powers of (E-E ). In fact, this process can be carried to of storage optimization. infinity and then the expansion is the same as a Taylor expansion The well known electronic structure of Beryllium at symmetry of S ' ? •(E,ii) around the arbitrary energy E . But in the form points (Terrell 1966, Bhokare and Yussouff 1974) has been reproduced of equation (6) or any finite number of terms, the last term al- (see table I) as an illustration of our method. We get the tabulated ways exactly describes the energy dependence,and the convergence of values quickly through a simple program using the muffin-tin poten- ciprocal lattice summation increases rapidly. In practice, equation tial tabulated by Terrel and summing over 561 reciprocal lattice (6) is sufficient because 3 ;*} is highly convergent (like k ~ ). LL n vectors. This provides the numerical justifications of our conjec-
Once a sufficiently fine grid is chosen in the Brillouln zone, the tures about the efficiency of our method and the convergence of
quantities S ^J (E ,k) can be computed and stored for use in sub- various sums. However, there is no exact comparison between our results
sequent energy cycles. and those of Terrel. He used c = 6.77 a.u. and a = 4.32 a.u. but
The advantages of energy expansion procedures are connected with c/a = 1.633 (ideal ratio). This cannot be done In our method and we
the proper choice of E and the evaluation of S £.? (E ,E,ic). As use c/a - 1.5? along with his values for c and a.
discussed by Segall and Yang 198O, the singularities of the energy We now summarize the advantages of our method against the dis- denominator lead to the dominant energy dependent behavior of advantages of (a) storage requirements on computer and (b) the slow
c oij'ir. ft* Neither of these are serious because the S „, (E,k). in this sense, equation (6) represents the separation convergence of S££.(EQ,k). of the dominant term with the proper choice of present generation of computers can handle the storage with ease E . However, equation (6) is better than the procedure of Segall and the slow convergence can be treated by an Ewald procedure if and Yang in two ways. First, the structure factors considered by more accuracy is desired. The advantages are as follows: (1) The KKR Segall and Yang are not absolutely convergent in reciprocal space matrix elements are directly evaluated by absolutely convergent reci- and an Ewald procedure must be used whereas S ^A (E,£) ± equation n procal lattice summation only.(2) An energy expansion of the matrix
(6) is absolutely convergent. Second, their separation of dominant elements separates the dominant term and retains the energy dependence
exactly. (3) The dominant terms are slowly convergent but -5- -6- can be stored after initial calculation to any desired accuracy. REFERENCES (4) The residual term is always rapidly convergent and can be calculated .fast and accurately. (5) Storage of the time consuming Bhokare, V.V. and Yussouff, M., 1972, Lett. Nuovo Cimento, 5, 470, part Is very effective in reducing time for self-consistent cal- 1974, Nuovo Cimento 19B, 149 culations. (6) The positive (+E) and negative (-E) energies are Davis, H.L., 1971, in Computational Methods in Band Theory, 183 treated by the same simple formulae (in contrast to the conven- (Plenum, New York)* tional structure constants which have different formulae for the Ehrenreich, H. and Schwartz, L.M., 1976, Solid State Physics 21,149. two cases}. (7) No irregular solutions {Neumann functions) appear Ham, F.S. and Segall, B., 1961, Phys. Rev, V2i, 1786. in the formulae. (8) Simplicity of formulae leads to easy pro- Kohn, W. and Rostoker, N., 1954, Phys. Rev. jM, 1111. gramming. That is particularly useful for any complex lattice with Korringa, J., 1947, Physlca V3' 392 * low svometry. In short, it is an efficient method which is specially Moru2«l, V.L., Janak, J.F. and Williams, A.R., 1978* Calculated suitable for crystals with several atoms per unit cell. Electronic Properties of Metals (Pergamon, New York)- Segall, B., 1957, Phys. Rev. 105, 108. Segall, B. and Yang, T.W., 19S0, Phys. Bey. B21_, 3737. Acknowledgment: The authors are thankful to Professor P.H. Terrell, J.H., 1966, Phys. Rev. 1_49, 526. Dederichs for many useful discussions. One of us (M.Y.) would like Williams, A.it., Janak, J.F. and Moruzzi, V.L., 1972, Phys. Rev. B6,45O9. to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.
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