Multiple scattering theory (MST) provides an efficient technique for solving the wave equation for the special case of muffin-tin potentials. Here MST is extended to treat space filling non-muffin tin potentials and its validity, accuracy and efficiency are tested by application of the two dimensional empty lattice test. For this test it is found that the traditional formulation of MST does not coverage as the number of partial waves is increased. A simple modification of MST, however, allows this problem to be solved exactly and efficiently. 15 refs., 3 tabs

Brown, R.G. (Duke Univ., Durham, NC (USA). Dept. of Physics)

Butler, W.H. (Oak Ridge National Lab., TN (USA))

Nesbet, R.K. (IBM Research Div., San Jose, CA (USA). Almaden Research Center)

UNT Digital Library

Multiple Scattering Theory for Space Filling Potentials\V. H. Butler', R. G. Brown**, and R. K. Nesbet"* x'Metals and Ceramics Division, Oak Ridge National Laboratory. "Qak Ridge. TN37S31-6114 V""Duke University Physics Department. Durham, NC 27706""IBM Research Division. Almaden Research Center, 650 Harry Road, San Jose CA9 3 1 2 0 - 6 0 9 9 COHP-9004S6-64ABSTRACT D E 9° 0 1 2 ' 4 0Multiple scattering theory (MST) provides an efficient technique for solving the waveequation for the special case of muffin-tin potentials. Here MST is extended to treat spacefilling non- muffin tin potentials and its validity, accuracy and efficiency are tested byapplication of the two dimensional empty lattice test. For this test it is found that thetraditional formulation of MST does not converge as the number of partial waves is increased.A simple modification of MST, however, allows this problem to be solved exactly andefficiently.INTRODUCTIONMultiple Scattering Theory (MST) is a powerful tool for treating problems involvingthe interaction of waves with matter. When it is applied to the problem of calculatingthe electronic structure of materials it yields the Korringa Kohn Rostoker band theorytechnique[l,2] for treating periodic crystals and the Scattered Wave Method[3,4] for treatingclusters of atoms. It is the extremely efficient set of basis functions provided by multiplescattering theory and the variational nature of the formalism that are responsible for therapid convergence and small set of secular equations that characterize these methods. Thepopular and efficient linear combination of muffin-tin orbitals technique for band structurecalculations can be viewed (at its simplest level) as an approximation to MST[5].Multiple scattering theory is sometimes called the Green function method because itgreatly facilitates the calculation of the single particle Green function for the system underconsideration. The Green function, which is extremely useful for calculating the effect onthe system of a perturbation such as an impurity or an applied field is generated by theinverse of the MST secular matrix. Another important feature of multiple scattering theoryis its ability to separate the potential aspect of a problem from the structural aspects. It isthis feature together with its facility for generating the Green function that makes MST sosuitable for calculating the electronic structure of alloys.The primary limitation of MST has been its restriction to potentials of muffin-tin form.A muffin-tin potential is one which vanishes outside a set of non-overlapping spheres. Withineach muffin-tin, the potential is generally assumed to be spherically symmetrical about thecenter of the sphere but this restriction is not difficult to remove. The requirement thatthe potential vanish outside of non-overlapping spherical regions has, however, proven tobe very troublesome. If this restriction could be removed, it would be possible to applyMST to problems which are currently of great interest in solid state physics and materialsscience.After many false starts and a large amount of work by a number of authors over thepast fifteen years, a consensus seems to be emerging that MST is not, in fact, restricted tomuffin-tin scatterers and is rigorously correct at least in principle, for space filling poten-tials. The calculations presented here strengthen this consensus. There are, however, someDISTRIBUTION OF THIS DOCUMENT IS UNLIMITED 'subtleties associated with this increased generality which are just now becoming apparent.Fortunately, once these subtleties are understood, they are relatively easy to overcome. Wedemonstrate here that MST for space filling scatterers works very well indeed for the strin-gent test of a square lattice of space filling potentials each of which is a step function shapedlike a square.MULTIPLE SCATTERING THEORYMultiple Scattering theory for space filling scatterers can be derived from scatteringtheory[6]. from an identity in integral equation theory[7]. from the Kohn Yariational principle[S],from a variational principle applied to the reaction matrix[9], and from wave matching argu-ments. The last method is perhaps the least rigorous, but in our opinion, the most intuitive.The short derivation sketched below is intended to be applicable to one, two, or three di-mensions. Table I shows how the equations derived in this section should be modified forvarious dimensionalities[10,ll]Table I: Translation Table for One, Two and Three Dimensionsgeneric symbol one dimension two dimensions three dimensionsW) cos(* - £TT/2) W) W)H((z) i-U" Jt{z) + iNt{z) ht{z)YIL X)/=O,I _ lJfc=o,oo 2Zm=-i,i Z)/=o,oo 12m=-e,eYL(r) sign(xY/V2In order to derive the basic equations of MST, we imagine that space is divided intocells which surround each atom. Generally we will call that volume or space which is closerto the center of atom n than to any other atom, cell n. The potential inside this cell can bewritten as V(rn) where r n = r — R n is a vector originating from Rn , the center of cell n.We further imagine that there is a region of infinitesimally small volume surrounding eachpotential where the potential is constant. We will call this volume region II to distinguishit from the region where the potential is equal to the true crystal potential which we callregion I. This is not really an approximation since region II has zero volume. One wayto view multiple scattering theory is that it allows one to solve the Schrodinger equation(SE) in a piecewise fashion, finding solutions for one cell (or scatterer) at a time, and thenusing the superposition principle to fit the pieces together in such a manner that all of theboundary conditions are satisfied.Within region In the total wave function can be written in terms of functions whichsatisfy the Schrodinger equation and behave like regular bessel functions at the origin,9m{E,r) = Y,cl(E)R<(E,rn)YL(rn). (1)LIn the part of region II surrounding atom n the wave function can be expanded in termsof wave functions which satisfy the free electron SE. These are J*(KX) which is a solutionof the radial part of the SE for energy E = K2 that is regular at the origin (va-yiug as rl),and Hi(Kr) which is a solution that satisfies outgoing wave boundary conditions at largedistances.¥//„(£, r) = J^[al(E)Jt(Krn) + bl(E)He(Krn)]YL(fn). (2)LIt is clear that Eq,(2) cannot be a solution for very large values of rn because Jt(z) doesnpt'behave properly at infinity. However, a multi-center expansion based on outgoing waveswhcih behave properly at large distances can be valid throughout region II,*;/(£, P) = T,T.bUE)Ht[Krn)YL[rn). (3)n LThe two representations for the wave function in region II, * / / „ and $ / / will agree if (inthe vicinity of cell n ) ,= ]T Y.bL'^)HL.(Krn,)YLI(fn.). (4)This equation provides a means of eliminating the coefficients a£ in favor of the coefficientsrn)YL{rn) (5)b\. This is facilitated by an addition theorem of the formLwhich can be used in Eq.(4) to obtain«l(E) = E E bt.(E)gL.L(EyRn - Rn-)- (6)In addition we need to ensure that the wave function "inside" potential n, $/„ matcheson smoothly to the potential "outside", ^/;n. A necessary condition for this matching isthat the following surface integral over the surface of cell n vanishf dSh • [*nnV^ /n - Vr /7ntf /n] = 0. (7)JilnThis relation is a straight-forward generalization of the step in the derivation of muffin-tinMST in which the logarithmic derivatives of the radial wave functions inside the muffin-tin are matched to the logarithmic derivatives of those outside in order to determine thescattering phase shifts or t-matrix. It can also be derived from a variational principle.The MST equations are obtained by using Eq.(l) for \P/n and Eq.(2) for $//„ in Eq.(7)which yieldsE E l - a 2 ^ L ' + ̂ £ L , ] ^ = 0 (8)L L>whereCiv = - / dSh • [Ht(Krn)YL(fn)V(Rt(rn)YL,(rn)) - V(Ht(Krn)YL(rn))R('(rn)YL,(rn)} (9)•/fin= SLL, - f dvHt(Krn)YL(rn)V(rn)Rl,(rn)YL.(rn)(10)andSh' = I dSn • [J((Krn)YL(KMRt{rn)YL,{rn)) - V(Jt(Krn)YL(?n))Rt>(rn)YL,(rn)} (11)Jlln= ( dvJt(Krn)YL{rn)V(ra)Rt.(rn)YL.(rn). (12)•/finUse of Eq.(6) in Eq.(8) gives the extended MST equations&u - ClL,6nM' = 0. (13)V n' L"We shall see below that although Eq.(5) converges if the ratio z = r n / |R n — Rn»| is lessthan unity, it may converge very slowly if z is not small enough. For muffin-tin scatterers zcan be no larger than 1/2, the maximum value being obtained for a point at the muffin tinradius of atom n and 7?' being a touching muffin-tin. For square scatterers z may be as largeas l / \ /2. In this case it may be better to avoid the use of the addition theorem (Eq. 5).Its use is not necessary since Eq.(4) is almost as convenient for eliminating the coefficientsa£ in favor of the coefficients b\. Thus the extended MST equations, Eq.(13) can also bewritten (in a notation which omits the angular momentum indices) asY. [9nn'Sn> - Cnn> - Cn6nn,}cn' = 0 (14)where n + 8 labels the nearest neighbors to cell n and whereC-li, = - / dvHt(Kr'n)YL(rnl)V(rn)Rt>(rn)YL,(fn). (15)TWO DIMENSIONAL EMPTY LATTICE TESTFaulkner[ll] has pointed out that the two dimensional empty lattice test provides astringent and effective test of a multiple scattering theory which has been generalized totreat non-muffin tin potentials. In this test the total crystal potential is represented asthe sum of "atomic" potentials each of which is shaped like a square (or other space fillingpolygon) and has depth Vo. Since the total crystal potential is uniform, the exact bandstructure and wave functions are known trivially. On the other hand MST will be stronglychallenged to construct these free electron wave functions from "atomic" states of squareatoms.The formalism of the last section did not describe how the radial wave functions, Re(Kr),used to expand <&/„ are obtained. There is, in fact, a small controversy over how they shouldbe calculated. The basic procedure for solving for the wave function in a partial wave basisfor a non-spherical scatterer was laid out by Williams and Morgan[l2,13] who derived acoupled set of differential equations for the wave function (here generalized for use in ddimensions where d = I,2or3),vLi?) d-l j 1 \V^Tr M i t \a r L"l<f>VL = M*r)CvL{r) + Ht,(nr)SL.L{r)VUL{T) = j drYv{r)V{r)YL{r)l J drRt{r)YL{r) = YdYL.{r)<t>L>L{r)u(16)with initial conditionsCX<L(0) = 6VLSL-L{0) = 0 (17)In the original formulation[12] V(r) is a truncated potential that vanishes when r liesoutside the cell at the origin which we denote by fi0V(r) = / VW if r G n ° 1K) \ 0 otherwiseBrown and Ciftan[7], however, have suggested that the potential should not be truncatedand that the coupled difFerential equations (Eqs.16) be integrated out to the radius of thebounding sphere using the full untruncated potential \ ' ( r) , with integrals being performedover the truncated potentials to determine CfL and SVL only after the radial wave functionhas been determined.ClFL = 8vL- f Ht.(Kr)YL.{r)V{r)Rt{r)YL(r)dr (19)andSfFL= Mnr)YL.(r)V(r)Rt[r)YL{f)dr. (20)JnoRecently. Nesbet has shown formally that both procedures should lead to the same wavefunction within a particular cell provided both procedures converge[l4].RESULTSWe have solved the coupled channel equations both ways. Of course, the Brown-Ciftanprocedure is trivial in this instance since the total potential is a constant. In general, forphysical potentials we expect that the Brown-Ciftan version of the equations(16- 20) will beeasier to solve and will converge faster because of the absence of the singular step function.In fact we have found that for the present test case, the Williams-Morgan procedure doesnot converge. Although the divergence is rather mild and does not become apparent untilextremely large numbers of partial waves are included in the calculation, we are confidentthat the observed divergence is real and not merely a numerical artifact. A detailed studyof this problem will be reported elsewhere[15].The eigenvalues found in the empty lattice test using Eq.(13) and the Brown-Ciftanprescription for the wave function for the second lowest energy level with full square sym-metry are given in Table II. The columns correspond to different values of lmax which is thenumber of different radial wave functions used in the calculation. It is traditional in MSTto work with square matrices so that the upper limit in the sum on L" used to multiplythe structure constant matrix gn,» by the sine matrix SL»V would be equal to the upperlimits on L and L'. Surprisgly, this procedure does not converge for the two dimensionalempty lattice test as can be seen by comparing the first entries in each column of Table II.Non-muffin tin MST only converges for this test if the internal sum on L" is taken quitehigh. In fact, the cut-off of this internal sum (if) must be taken higher for higher valuesof (-max-, the cut-off used for the outer sums. To obtain an energy within 1% of the exactenergy for imax = 12, for example, it was necessary to use an internal cut-off, if, of 56.The necessity for the large internal sum cut-off arises from the slow convergence of theexpansion of an "outgoing" wave in terms of "incoming" waves centered on an adjacent site(Eq. 5) in situations where the distance from the center of the cell to a corner is a largefraction of the separation between centers. If that expansion is avoided for near neighbors,(Eq.14) the convergence is greatly improved as is shown in Table III.CONCLUSIONSOur results are consistent with the following propositions: (1) MST is exact for spacefilling scatterers. (2) In solving for the wave functions for a single scatterer, abrupt trun-cation of the potential can lead to convergence problems in the partial wave expansions. Atechnique which avoids this truncation such as that of Brown and Ciftan is preferable to theoriginal prescription of Williams and Morgan. (3) If the potential is strong in the corner of aspace filling cell in two or three dimensions the internal anguiar momentum sum associatedTable II: Energy calculated using Eq.(13) of the second lowest state with full square sym-metry for an "empty lattice" filled with potentials of side a = - and depth V = —5 as afunction of £max and the internal L" sum cut-off, Cf. The exact energy is E = — 1.0 1204S121620-.30614-.27744-.27889-.27878-.27879-.27879-1.55582-.S7705-1.05816-.99265-1.01194-2.405S-.7S965-1.55215-.89194-.49045-.75225-.64S3724 -.27879 -1.00615 -1.08408 -.78061Table III: Energy calculated using Eq.(14) of second lowest state with full square symmetryfor an "empty lattice" filled with potentials of side a = ir and depth V = —5 as a functionof C i i and the internal i sum cut-off, if. The exact energy is E = —1. Note the vastimprovement in convergence with respect to if compared to Table II.If 0 8 1204812162024-.27302-.27871-.27879-.27879-.27879-.27879-.27879-.99309-1.00671-1.00738-1.00744-1.00744-1.00744-.998254-.999752-.999959-.999987-.999990-.999528-.999910-.999984-.999997with the expansion of an irregular (J£TL) function in terms of regular (J^) functions centeredon an adjacent site will converge slowly and more partial waves must be included in thissum than appear in the secular equation. (4) This difficulty can be avoided by reformulatingthe MST equations so that the HL functions are used directly without being re- expandedabout neighboring sites.ACKNOWLEDGMENTSResearch sponsored by the Division of Materials Sciences, Office of Basic Energy Sci-ences, U.S. Department of Energy, under Contract No. DE-AC05-84OR21400 with MartinMarietta Energy Systems, Inc. Helpful discussions with A. B. Chen, A. Gonis, D. M.Nicholson, and C. Y. Yeh are gratefully acknowledged.References[1] J. Korringa, Physica 13, 392 (1947).[2] W. Kohn and N. Rostoker, Phys. Rev. 94, 1111 (1954).[3} ,L. Eyges, Phys. Rev. l l l 6S3, (195S).[4] K. H. Johnson, J. Chem. Phys. 45 3085, (1966).[5] H. L. Skriver, The LMTO Method (Springer- Verlag, New York, 1984).[6] A. Gonis, Phys. Rev. B 33,5914 (19S6).[7] R. G. Brown and M. Ciftan, Phys. Rev. B 27, 4564, (1983), Phys. Rev. B 32, 1343(1985), Phys. Rev. B 33,7937 (1986).[8] C. Yeh, A. B. Chen, D. M. Nicholson, and W. H. Butler, (to be published)[9] R. K. Nesbet, Phys. Rev. B 30 4230, (1984).[10] W. H. Butler, Phys. Rev. B 14,468 (1976).[11] J. S. Faulkner, Phys. Rev. B 38, 1686 (1988).[12] A. R. Williams and J. van W. Morgan, J. Phys. C 5, L293, (1972)[13] A. R. Williams and J. van W. Morgan, J. Phys. C 7, 37, (1974).[14] R. K. Nesbet Phys. Rev. B (accepted for publication)(1989).[15] R. G. Brown and W. H. Butler (to be published)

Multiple scattering theory for space filling potentials

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Multiple scattering theory for space filling potentials

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