HAL Id: jpa-00215605
https://hal.archives-ouvertes.fr/jpa-00215605
Submitted on 1 Jan 1974
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.
STUDIES ON DISORDERED SYSTEMS USING CLUSTERS OF MUFFIN-TIN POTENTIALS
J. Faulkner, G. Painter, W. Butler, W. Coghlan
To cite this version:
J. Faulkner, G. Painter, W. Butler, W. Coghlan. STUDIES ON DISORDERED SYSTEMS USING CLUSTERS OF MUFFIN-TIN POTENTIALS. Journal de Physique Colloques, 1974, 35 (C4), pp.C4- 85-C4-87. �10.1051/jphyscol:1974414�. �jpa-00215605�
JOURNAL DE PHYSIQUE Colloque C4, suppl4ment au no 5, Tome 35, Mai 1974, page C4-85
STUDIES ON DISORDERED SYSTEMS USING CLUSTERS OF MUFFIN-TIN POTENTIALS
(")J. S. FAULKNER, G. S. PAINTER, W. H. BUTLER and W. A. COGHLAN Metals and Ceramics Division
Oak Ridge National Laboratory Oak Ridge, Tennessee, 37830, USA
Rbumk. - Des calculs d'Ctats electroniques d'amas d'atomes ont Bte effectues ces dernibres annQs dans le but de reprbenter les etats de solides ordonnes ou dksordonnes. Nous proposons une condition aux limites pour l'amas qui representerait mieux l'effet de I'environnement des clusters que celles qui ont Cte utilisees jusqu'a present.
Abstract. - Calculations of the electronic states of clusters of atoms have been carried out in recent years for the purpose of representing the states of ordered or disordered solids. We propose a boundary condition to be put on the cluster that should simulate the effect of the system outside the cluster better than the ones presently in use.
During the past few years a number of uses for multiple-scattering-cluster calculations have been found for studying electronic states in condensed matter.
For example, Johnson [l] and Keller [2] described such calculations at a conference at Menton in Septem- ber of 1971.
We will write out the general equation for the one- electron Green's function in the neighborhood of a cluster and use this as a basis for a new boundary condition that we propose for use when the cluster is part of a solid.
The first step in a cluster calculation is to generate a local one-electron potential function for a given array of atoms. This potential is then approximated by writing it as a sum of non-overlapping spherically symmetric potentials within spheres centered on each atom with the potential in the region between the spheres being a constant. This muffin-tin potential facilitates the use of multiple-scattering theory [3, 41 on the problem because the t-matrices that enter are diagonal in the angular momentum representation.
Boundary conditions can be applied to the cluster using a diagonal t-matrix if the cluster is circumscribed by a sphere and the potential in the region outside the sphere is symmetric about the center of the cluster.
Taking the origin of the coordinate system to be the center of the cluster, the potential for the system des- cribed above may be written as
v(r) = vc(r)
+
v ~ ( r ) (1) whereVc(d =
x
%(I r - Rn I) (2)11
(*) Research sponsored by the U. S. Atomic Energy Commis- sion under contract with the Union Carbide Corporation.
is the sum of the atomic muffin-tin potentials and V,(r) is the potential in the region outside. The inco- ming wave on a scatterer can be written in the region between scatterers as
where r, = r - R, andf,(arn) = jl(arn) if the scatterer is an atom while,f,(arn) = h,(ar,) if it is the boundary.
Similarly the outgoing wave from a scatterer can be written
where gl(arn) = h,(ocr,) if the scatterer is an atom and gl(arJ = j,(ar,) if it is the boundary. The equations of multiple scattering theory say that the incoming wave on a scatterer is the sum of the outgoing waves from all of the other scatterers while at the same time the outgoing wave from a scatterer can be found from the incoming wave since the elements of the t-matrix.
t!, are known. These relations can hold simultaneously for all the scatterers only if the coefficients for the outgoing waves satisfy the set of homogeneous equa- tions
where the g ~ ~ l r m l depend on energy and describe pro- pagation from one scatterer to another.
In the work of Johnson and others the potential outside the cluster is chosen in such a way that the electrons are bound to the cluster for all the states of interest. The elements of the scattering matrix from the boundary, t:, thus appear in eq. (5) and the desired
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974414
C4-86 J. S. FAULKNER, G. S. PAINTER, W. H. BUTLER AND W. A. COGHLAN eigenvalues for the system are obtained as the zeros
of the determinant of the coefficients of the bFm. This boundary condition is sensible for an approximate treatment of molecules, but although it has been used for the case when the cluster is part of a solid it clearly gives a discrete spectrum when the eigenvalues should be in continuous bands. The group at the University of Bristol normally assumes that the potential outside the cluster is zero so that the t? do not appear in eq. (5).
Of course the determinant of the coefficients has no zeros for energies above the muffin-tin zero under this assumption so they treat the problem with scattering theory. A formula due to Lloyd [5] which is a gene- ralization of Friedel's formula [6] allows the integrated density of states to be found from the logarithm of the determinant of the coefficients. This approach leads to continuous bands of eigenvalues in this energy range, but the assumption that the potential outside the cluster just has the value of the muffin- tin zero is not very good when the cluster is part of a solid.
For the case where the cluster is a part of a solid that has been singled out for treatment we wish to find a boundary condition that will represent the effect of the rest of the solid better than the ones described above but will still be relatively easy to use since the advantage of the cluster approach is its relative simpli- city. In other words, if we are treating a cluster of copper atoms we want to simulate the effect of an infinite set of copper atoms outside [7].
A possibility that is suggested by the experience of the last few years is to surround the cluster with a complex, energy-dependent potential which is spatially homogeneous. This potential will be deter- mined by the requirement that an electron propagating through it would not be scattered when it impinges on the cluster. Since we are only trying to find an improved boundary condition and not an exact one we will not require that the scattering be zero but only minimized.
The first problem in carrying out this scheme is to write down the Green's function for a cluster embedded in a complex medium. Thus the potential V,(r) in eq. (1) is now
where rb is the radius of a sphere which circumscribes the cluster. It is convenient to define a Green's func- tion for the case when there are no atoms in the cluster. In abstract operator notation this is
where K is the kinetic energy operator and Z approa- ches the energy E from above. The Green's function for the system including the atoms is then
where Rc is the scattering operator
Rc = Vc(1
+
GH R 3 , (9) and Vc is the potential of the muffin-tins given in the position representation in eq. (2). The scattering operator for the muffin-tins for Z = 0 can be written Tc = Vc(l - Go Vc)-' , (10) where Go = (2 - K)-' is the free-electron propaga- tion. Taking the inverse T;' and multiplying it by Tc we see the unit operator can be writtenOperating this unit operator on Rc from eq. (9) yields Rc = Tc
+
Tc(GH - Go) Re-
(12) Now G, can be writtenGH = Go f Go tB Go , (13) where
tB = VH(l
+
Go tB) . (14) Inserting eq. (13) into eq. (12) and iterating leads to leads to an expression for Rc in terms of scattering operators defined relative to the muffin-tin zeroWe now wish to find the Green's function for the total system in the position representation, G(r, r'), for the case where r and r' are outside the sphere that separate the cluster from the medium,
G(r, r') = GH(r, r')
+
+ J
G ,1
R ~ I ,1
, r f d 32 . (16)One can show by standard methods that for r and r' in the region outside the sphere
where p = (E - ,Z)lI2 and r > rf. In this expression
where the square bracket means
Since R,(r, r') is zero unless r and r' are inside the sphere, we need an expression for GH(r, rf) with one argument inside and the other outside the sphere for use in the integral in (16). It can be shown that the expression is
STUDIES ON DISORDERED SYSTEMS USING CLUSTERS OF MUFFIN-TIN POTENTIALS C4-87
when r is outside and r' inside the sphere and a similar The notation in eq. (29) means we form matrices with expression for the other case. In eq. (20) angular momentum subscripts, carry out the indicated matrix operations, and then take the Im, 1' m' element.
D1 = { i x r x h ~ ( p r b ) , jl(arb)l
1
-.
(21) We note that We then arrive at+(- ip)'
x
xm&) hl(pr)qn,l,m,
hr(pr') Y L , ~ ' ) , This, along with the results in eq. (261, (281, and (29)1 ~ 1 ' ~ ' can be inserted in eq. (23) to obtain
(22)
where [hi(Brb), hl(arb)] -
, "
*BB - 1 "BB
x ( [(tB)- - G
1
G )lmrrn' andx r bI a r b ~
)
. ( 3 1 )Rlrnrm, = ( - ia12
1
jl(ar) Y j z c ) Rc(r, r t ) xFrom eq. (22) it can be seen that if TI,,,,,, = 0,
X ~ ~ ~ 6 ' ) j , ( a + ) d3r d3r' . (24) G(r, r') would be the Green's function for a system To evaluate the quantity in eq. (24) we go back to the
expansion of R, in eq. (15) and put all the operators in the position representation. Using
A
Go@, r') = - ia
x
Xm(r) j,(ar) h,(ar') Yj;Gr) (25)lm
for r' > r, and
1
hl(ar)~ ; 6 )
tp(r7 r f ) q r m 8 $ ' ) h1,(cw1) d3r d3r1 == t; 81, a m i n , (26)
which can be shown for the case considered here to reduce to
B [hl(prb)7 h , ( ~ r b ) l iat, =
[hl(prb), jdarb)] ' along with definition
/\BB
Girn~,m, = (- ia)
J
j,(ar) q ; G ) Tc(r, r ' ) x leads to"BB B -1 "BB
R r n
+
{ (1 G t ) G i n , (29)that just had the potential C everywhere. Thus our problem is the calculate T,rn,,,r for a given energy and various values of C. The values of Z that minimize Tlml~,,l, for various E yield the C(E) that we seek.
We assume that the cluster is large and spherical so that the off-diagonal elements of T,,,,,, can be ignored.
This can, of course, be checked in the calculation.
Averaging over configuration of atoms in the cluster will enhance this.
The only time consuming part of the calculation is the evaluation of
31",?,,,.
AS can be seen from eq. (28) this is just the element of the scattering matrix for a system of muffin-tins with zero potential outside the cluster. These matrix elements can be calculated with a slight modification of the programs used in the ordinary cluster calculations described by eq. (5).Once the
Ggrmr
are calculated the scattering matrix of eq. (31) can be evaluated for many values of C rather easily because C only appears in t:, tr, and the square brackets. Averaging is done by calculating theA
matrix elements G!:~,, for several configurations of atoms and replacing the matrix elements in eq. (31) with their weighted average.
We have programmed these equations and have run tests on small clusters without averaging. There seems to be no numerical difficulty.
References
[I] JOHNSON, K. H., J. Physique 33 (1972) C3-195. [6] FRIEDEL, J., Adv. Phys. 3 (1954) 446.
[2] KELLER, J., J. Physique 33 (1972) C3-241. [7] KELLER, J., has considered the problem of defining an [3] Lord RAYLEIGH, Phil. Mag., 34 (1892) 481. outside potential to simulate the effect of the rest of the solid and shows calculations done with a complex [4] GOLDBERGER, M. L., WATSON, K. M., CoIlisio~z theory energy-dependent medium in the proceedings of this (John Wiley and Sons, Inc., New York) 1964. conference. The potential we propose fits this descrip- [5] LLOYD, P., Proc. Phys. Soc. 90 (1967) 207. tion, but it is defined differently.
