Introduction to Multiple Scattering Theory

Transcription

1 Introduction to Multiple Scattering Theory László Szunyogh Department of Theoretical Physics, Budapest University of Technology and Economics, Budapest, Hungary and Center for Computational Materials Science, Vienna University of Technology, Vienna, Austria Contents 1 Formal Scattering Theory Resolvents Observables and Green functions The T -operator Scaling transformation of the resolvents The Lippmann-Schwinger equation The optical theorem The S-matrix Integrated density of states: the Lloyd formula The Korringa-Kohn-Rostoker Green function method Characterization of the potential Single-site and multi-site scattering: operator formalism The angular momentum representation

2 2.4 One-center expansion of the free-particle Green function Single-site scattering Two-center expansion of the free-particle Green function Multi-site scattering Density of states, charge density, dispersion relation Generalization of multiple scattering theory The embedding technique The Screened Korringa-Kohn-Rostoker method

3 1 Formal Scattering Theory 1.1 Resolvents Spectrum of Hermitean operator, H, over a Hilbert space, H Discrete spectrum Continuous spectrum Hϕ n = ε n ϕ n Hϕ α ε) = εϕ α ε) ϕ n H) ϕ α ε) = lim n χ n, χ n H) ϕ n ϕ m = δ nm ϕ α ε) ϕ α ε ) = δ αα δ ε ε ) The generalized) eigenfunctions form a complete set, ϕ n ϕ n + dε ϕ α ε) ϕ α ε) = I. 1) n α Notation: SpH), ϱh) = C \ SpH) Units: = 1, m = 1/2, e 2 = 2 a 0 = 2 /me 2 = 1, Ryd = 2 /2ma 2 0 = 1 The free-particle Hamiltonian, H 0 = p 2 =, 2) has no discrete spectrum over the Hilbert-space, H=L 2 R 3). However, H 0 ϕ p) = p 2 ϕ p), ϕ p; r) = 1 r, 3) 2π) 3/2ei p ϕ p) = lim n χ n p), χ n p; r) = 1 2π) 3/2 exp i p r r 2 /4n 2), and χ n p) L 2 R 3). Thus, the continuous spectrum of H 0 covers the set of non-negative numbers. 3

4 The resolvent of H is defined for any z ρ H) as G z) = zi H) 1. 4) It obviously satisfies, G z ) = G z), 5) therefore, it is Hermitean only for ε ρ H) R. From the relation, G z 1 ) G z 2 ) = z 2 z 1 ) G z 1 ) G z 2 ), 6) immediately follows that dg z) = G z) 2, 7) dz and, by noting that G z) is bounded, it can be concluded that the mapping z G z) is analytic for z ρ H). From Eq. 1) the spectral resolution of the resolvent can be written as G z) = n ϕ n ϕ n z ε n + dε α ϕ α ε) ϕ α ε) z ε. 8) Relationship between the eigenvalues of H and the singularities of G z): discrete spectrum of H poles of first order of G z) continuous spectrum of H branch cuts of G z). 9) Therefore, at the real axis the so-called up- and down-side limits of G z) are introduced, having the following relationship, G ± ε) = lim G ε ± iδ) ε R), 10) δ +0 G ± ε) = G ε). 11) In particular, for ε ρ H) R G + ε) = G ε) = G ε). 12) 4

5 Using the identity, lim δ +0 ) 1 1 ε ε ± iδ = P iπδε ε ), ε ε with P denoting the principal value distribution, Re G + ε) = Re G ε) = ) 1 ϕ n ϕ n P + ε ε n n dε α ϕ α ε ) ϕ α ε ) P 1 ) ε ε, 13) and Im G + ε) = Im G ε) = π ϕ n ϕ n δ ε ε n ) + n dε α ϕ α ε ) ϕ α ε ) δ ε ε ) ), 14) where the real and imaginary part of an operator, A is defined as Re A = 1 ) A + A and Im A = 1 ) A A. 15) 2 2i Generally, a given representation of the resolvent is called the Green function. On the basis of the eigenfunctions of H, G nn z) = ϕ n G z) ϕ n = m ϕ n ϕ m ϕ m ϕ n z ε m = δ nn 1 z ε n 16) and, similarly, G αα z; ε, ε ) = δ αα δ ε ε 1 ) z ε, 17) while in the coordinate real-space) representation, G z; r, r ) = m ϕ n r) ϕ n r ) z ε n + dε α ϕ α ε; r) ϕ α ε; r ) z ε. 18) The primary task of the Multiple Scattering Theory or Korringa-Kohn-Rostoker method) is to give a general expression for G r, r ; z). 5

6 The Green function of free particles G 0 z; r, r ) = 1 2π) 3 which can be evaluated as follows, G 0 z; r, r ) = 1 2π) 3 i = 8π 2 r r 0 dk k eik r r z k 2 d 3 k ei k r r ) z k 2, 19) dk k 2 1 z k 2 d 2ˆk e i k r r ) ) dk k e ik r r z k 2 Obviously, the first and the second integral in the last expression can be closed in the upper and the lower complex semiplane, respectively. Thus, by choosing p C, Im p > 0, such that z = p 2 yields [ G 0 z; r, r 1 ke ik r r ) = Res 4π r r p k) p + k), p) Res ke ik r r ] p k) p + k), p) = eip r r 4π r r, 20) while by choosing p C, Im p < 0, such that z = p 2, one obtains [ G 0 z; r, r 1 ke ik r r ) = Res 4π r r p k) p + k), p) Res ke ik r r ] p k) p + k), p) = e ip r r 4π r r. 21) Clearly, independent of the choice of the square-root of z p 1 = p 2 ) the expression of G 0 z; r, r ) is unique. In particular,. and G ± 0 ε; r, r ) = e±ip r r 4π r r G + 0 ε; r, r ) = G 0 ε; r, r ) = e p r r 4π r r ε > 0, p = ε), 22) ε < 0, p = ε). 23) 6

7 1.2 Observables and Green functions In a system of independent fermions, the measured value of a one-particle observable, say A, is given by A = T r f H) A), where A is the Hermitian operator related to A and, the Fermi-Dirac density) operator is defined by f H) = I + e βh µi) ) 1, with β = 1/k B T, T the temperature and µ the chemical potential. Evaluating the trace in the basis of the eigenstates of H, the above expression reduces to A = f ε n ) ϕ n A ϕ n + dε f ε) ϕ α ε) A ϕ α ɛ), 24) n α where f ε) = 1/ 1 + e βε µ)). Recalling Eq. 8) one can write, f z) T r A G z)) = f z) ϕ n A ϕ n + dε f z) ϕ α ε) A ϕ α ε) z ε n n z ε α, which, in order to relate to Eq. 24), has to be integrated over a contour in the complex plane, C comprising the spectrum of H. In here, Cauchy s theorem is used, i.e., for a closed contour oriented clock-wise, g a) 1 2πi dz g z) z a = if a is within the contour 0 if a is outside of the contour where it is supposed that the function g has no poles within the contour. Thus the poles of the Fermi-Dirac distribution, f z) k B T 1 z z k for z z k, z k = µ + i 2k + 1) πk B T k Z), have also to be taken into account, A = 1 dz f z) T r AG z)) k B T 2πi k C 7 T r AG z k )), 25),

8 where only the Matsubara poles, z k, within the contour C are considered in the sum of the rhs. By splitting the contour into symmetric) upper and lower parts and making use of Eq. 5), Eq. 25) can further be transformed into A = 1 π Im dz f z) T r AG z)) 2k B T Re T r AG z k )). 26) Im z k >0 By deforming the contour to the real axis, the familiar expressions, A = 1 π Im dε f ε) T r AG + ε) ) = 1 π Im dε f ε) T r AG ε) ), 27) can be deduced. In particular, the number of electrons can be calculated by taking A = I, N = 1 π Im dε f ε) T rg + ε) = where the density of states DOS) is defined by dε f ε) n ε), 28) n ε) = 1 π Im T rg+ ε) = 1 π Im T rg ε). 29) Thus, the discontinuity of the imaginary part of the Green function at the real axis is directly related to the density of states.. DOS of free particles From Eq. 22) one immediately can derive for ε > 0, Im G + 0 ε; r, r) = Im G 0 sin κr) ε; r, r) = lim R 0 4πR = κ 4π, therefore, the density of states normalized to a unit volume can be written as ε n 0 ε) = Θ ε) 4π, 30) 2 since for ε < 0 the Green function is real. 8

9 1.3 The T -operator Defining the Hamiltonian, H as a sum of the Hamiltonian of a reference system, H 0 and the operator of perturbation, V H = H 0 + V, 31) the corresponding resolvents are coupled by G z) = zi H 0 V) 1 = zi H 0 ) I G 0 z) V)) 1 = I G 0 z) V) 1 G 0 z), 32) or in terms of Dyson equations, G z) = G 0 z) + G z) VG 0 z) = G 0 z) + G 0 z) VG z), 33) which can be solved by successive iterations to yield the Born series, G z) = G 0 z) + G 0 z) VG 0 z) + G 0 z) VG 0 z) VG 0 z) ) By reformulating Eq. 34) as G z) = G 0 z) + G 0 z) V + VG 0 z) V +...) G 0 z), 35) it is worth to define the so-called T -operator, T z) = V + VG 0 z) V + VG 0 z) VG 0 z) V +..., 36) such that G z) = G 0 z) + G 0 z) T z) G 0 z). 37) 9

10 The subsequent relationships between the resolvents and the T -operator can easily be proved: T z) = V + VG z) V, 38) T z) = V + VG 0 z) T z) = V + T z) G 0 z) V, 39) G 0 z) T z) = G z) V, 40) as well as T z) G 0 z) = VG z). 41) Since V is Hermitean, similar to the resolvents the T -operator satisfies T z ) = T z), 42) and, in particular, for the side-limits T + ε) = T ε). 43) From Eqs. 38), 7), 40) and 41): dt z) dz dg z) = V V = VG z) G z) V dz = T z) G 0 z) 2 T z) dt z) dz = T z) dg 0 z) T z). 44) dz 10

11 1.4 Scaling transformation of the resolvents In general, we are not restricted to choosing the system of free particles as reference. Let us consider the Hamiltonian, H = H 0 + U, 45) with the corresponding perturbation, V = V U, 46) such that H = H 0 + V = H + V. 47) From Eqs ) immediately follows, that the resolvent G z) can be expressed in terms of the resolvent of the new reference system, G z) = zi H ) 1. 48) or G z) = G 0 z) + G 0 z) UG z), 49) and a new T -operator, T z) = V + V G z) T z), 50) as G z) = G z) + G z) T z) G z). 51) Eqs ) represent the formal background to the so-called Screened Korringa- Kohn-Rostoker SKKR) theory to be discussed later. 11

12 1.5 The Lippmann-Schwinger equation Suppose ϕ α ε) is a generalized eigenfunction of H 0, εi H 0 ) ϕ α ε) = 0. 52) Let us seek for a generalized eigenfunction of H, εi H 0 ) ψ α ε) = Vψ α ε), 53) in the form of ψ α ε) = ϕ α ε) + δψ α ε). 54) Clearly, for vanishing perturbation, V = 0, δψ α ε) = 0 is expected. By substituting Eq. 54) into 53) we get εi H 0 ) ϕ α ε) + δψ α ε)) = εi H 0 ) δψ α ε) = Vϕ α ε) + Vδψ α ε), 55) where we made use of Eq. 52), from which εi H) δψ α ε) = Vϕ α ε) 56) can be deduced. Since, over the spectrum of H, the inverse of εi H is defined through two different side-limits, two different solutions exist, ψ α ± ε) = ϕ α ε) + G ± ε) Vϕ α ε), 57) or by using Eq. 40) ψ α ± ε) = ϕ α ε) + G 0 ± ε) T ± ε) ϕ α ε). 58) Moreover, from Eqs. 39) and 58) it follows that Vψ α ± ε) = V + VG ± 0 ε) T ± ε) ) ϕ α ε) = T ± ε) ϕ α ε), 59) therefore, ψ ± α ε) = ϕ α ε) + G ± 0 ε) Vψ± α ε). 60) 12

13 1.6 The optical theorem In general, energy conservation requires to use only the so-called on-the-energy shell matrixelements of the T -operator, T αα ± ε) = ϕ α ε) T ± ε) ϕ α ε). 61) Eq. 43) implies the relationship, T + αα ε) = T α α ε). 62) Starting from Eq. 38) the matrixelements of T ± ε) can be expressed as T ± αα ε) = V αα ε) + ϕ α ε) VG ± ε) V ϕ α ε), 63) where V αα ε) = ϕ α ε) V ϕ α ε). 64) Taking the difference, T αα + ε) T αα ε) = ϕ α ε) VG + ε) G ε))v ϕ }{{} α ε), 65) 2i Im G + ε) the spectral resolution of Im G + ε), Eq. 14), can be used to give, T + αα ε) T αα ε) = 2πi β ϕ α ε) V ψ + β ε) ψ+ β ε) V ϕ α ε), 66) which by employing Eqs. 59) and 62) can be written as T + αα ε) T αα ε) = 2πi β T + αβ ε) T βα ε), 67) referred to as the generalized optical theorem. 13

14 Using matrix notation, T ± ε) = { T ± αα ε) }, Eq. 67) can be rewritten into the compact form, T + ε) T ε) = 2πi T + ε) T ε), 68) or T + ε) 1 T ε) 1 = 2πi I. 69) Multiplying the above equation with T + ε) from the left and with T ε) from the right, it also follows that T + ε) T ε) = 2πi T ε) T + ε), 70) consequently, the operators T + ε) and T ε) commute with each other, T + ε) T ε) = T ε) T + ε). 71) 1.7 The S-matrix In order to get more insight into the physical meaning of the optical theorem we introduce the so-called S-matrix as ψ α + ε) = ψα ε) S α α ε), 72) α or S α α ε) = ψ α ε) ψ + α ε), 73) which has to be unitary, since both sets of the functions, {ψ + α ε)} and {ψ α ε)}, should be complete. Making use of Eq. 59), the S-matrix can be expressed in terms of the T -matrices: Vψ + α ε) = α Vψ α ε) S α α ε) 14

15 T + ε) ϕ α ε) = α T ε) ϕ α ε) S α α ε) T + βα ε) = α T βα ε) S α α ε) T + ε) = T ε) S ε), thus, S ε) = T ε) 1 T + ε) = I 2πiT + ε), 74) where the last expression was derived from Eq. 69). Consequently, S ε) S ε) = I 2πiT + ε) ) I + 2πiT ε) ) = I + 2πiT ε) T + ε) 2πiT + ε) T ε)) = I, 75) }{{} =0 i.e., the S ε) matrix is unitary as required. Note that this directly follows also from Eq. 71). In a diagonal representation, the S-matrix obviously has the form, S αα ε) = δ αα e i2δ αε), 76) where δ α ε) R is called the generalized) phaseshift.. From 74), therefore, e i2δαε) = 1 2πiT αα + ε) T αα ± ε) = 1 π e±iδ αε) sin δ α ε). 77) 15

16 1.8 Integrated density of states: the Lloyd formula Substituting Eq. 37) into 29) yields n ε) = 1 π Im T r G 0 + ε) + G+ 0 ε) T + ε) G 0 + ε)) = n 0 ε) + δn ε), 78) with n 0 ε) = 1 π Im T r G + 0 ε)), 79) and δn ε) = 1 π Im T r G 0 + ε) T + ε) G 0 + ε)) = 1 ) G π Im T r 0 + ε)2 T + ε) = 1 dg + ) π Im T r 0 ε) T + ε), 80) dε where we made use of Eq. 7). Employing Eq. 44) one can derive, δn ε) = 1 π Im T r T + ε) 1 dt + ) ε) dε = d ) 1 dε π Im T r ln T + ε) = d ) 1 dε π Im ln det T + ε). 81) The integrated DOS, N ε) = ε dε n ε ), 82) can then be directly expressed as N ε) = N 0 ε) + δn ε), 83) 16

17 where N 0 ε) = ε dε n 0 ε ) 84) and δn ε) = 1 π Im ln det T + ε), 85) referred to as the Lloyd formula. Quite trivially, δn ε) = 1 π Im ln det T ε), 86) therefore, δn ε) = 1 ) 2π Im ln det T ε) 1 T + ε) = 1 Im ln det S ε). 87) 2π In diagonal representation, see Eq. 76), this reduces to δn ε) = 1 2π Im ln α e i2δ αε) = 1 π δ α ε), 88) α known as the famous Friedel sum-rule. The excess DOS caused by the perturbation can then be written as δn ε) = 1 π α d δ α ε) dε, 89) having sharp peaks at resonances, i.e., at rapid changes of the phaseshifts when crossing π 2. 17

18 2 The Korringa-Kohn-Rostoker Green function method 2.1 Characterization of the potential Let us divide the configurational space, Ω isomorphic to R 3 ) into disjunct domains, Ω n Ω = n N Ω n, Ω n Ω m = 0. The potential V can then be written as a sum of single-domain potentials, V n, V r) = n N V n r) 90) V r) for r Ω n V n r) =. 91) 0 for r Ω \ Ω n The center of a particular domain, Ω n defined by the position vector, R n is usually associated with the position of the atomic nucleus, regarded for simplicity to be point-like. For open systems or surfaces, the centers of the so-called empty spheres, describing the interstitial region or the vacuum, are, however, not related to singularities of the potential. In electronic structure calculations, first the atomic positions, R n are fixed and the domains cells) are most commonly determined in terms of the Wigner-Seitz construction. In order to present a simple derivation of the Multiple Scattering Theory MST), we shall restrict our discussion to single-cell potentials confined to spherical domains, V n r) = 0 for r n S n r n r R ) n, 92) where S n is usually termed as the muffin-tin radius. It is important to note that by preserving the formalism we are going to derive, MST is valid also for space-filling potentials. 18

19 2.2 Single-site and multi-site scattering: operator formalism The case when only a single potential, V n of type 91) is present, is referred to as the single-site scattering. The corresponding single-site T-operator, t n see Eq. 39)), t n = V n + V n G 0 t n, 93) can formally be expressed as t n = I V n G 0 ) 1 V n. 94) Note that we droped the energy argument of the corresponding operators.) Inserting Eq. 90) into Eq. 38) yields T = n V n + nm V n G 0 V m + nmk V n G 0 V m G 0 V k +... = n Q n, 95) where we introduced the operators, Q n = V n + V n G 0 V m + V n G 0 V m G 0 V k +... m mk = V n + V n G 0 V m + ) V m G 0 V k +... m k = V n + V n G 0 Q m. 96) m Separating the term n on the right-hand side of Eq. 96) and taking it to the left-hand side, we get I V n G 0 ) Q n = V n + V n G 0 Q m, 97) m n) which, by using Eq. 94), can be transformed to Q n = t n + t n G 0 Q m. 98) m n) 19

20 The above Dyson-equation can be solved by successive iterations resulting in Q n = t n + t n G 0 t m + t n G 0 t m G 0 t k + therefore, m n) m n) k m) j k) m n) k m) t n G 0 t m G 0 t k G 0 t j +..., 99) T = t n + t n G 0 1 δ nm ) t m + t n G 0 1 δ nm ) t m G 0 1 δ mk ) t k n n,m n,m,k + t n G 0 1 δ nm ) t m G 0 1 δ mk ) t k G 0 1 δ kj ) t j ) n,m,k,j The operator comprising all the scattering events between two particular sites is called the scattering path operator SPO), τ nm = t n δ nm + t n G 0 1 δ nm ) t m + k t n G 0 1 δ nk ) t k G 0 1 δ km ) t k + k,j t n G 0 1 δ nk ) t k G 0 1 δ kj ) t j G 0 1 δ jm ) t m ) Obviously, T = nm τ nm 102) and the following Dyson equations apply, τ nm = t n δ nm + k t n G 0 1 δ nk ) τ km, 103) or τ nm = t n δ nm + k τ nk G 0 1 δ km ) t m. 104) 20

21 Inserting 102) into Eq. 37) yields G = G 0 + nm G 0 τ nm G 0, 105) which will be our starting point to evaluate the configurational space Green function. It is useful to introduce also the structural resolvent operator as G nm = G 0 1 δ nm ) + k,j G 0 1 δ nk ) τ kj G 0 1 δ jm ). 106) The SPO, τ nm, is obviously related to G nm by see Eq. 101) τ nm = t n δ nm + t n G nm t m. 107) Comparing expressions 103) and 104) with 107), the following useful identities can be derived, and G 0 1 δ nk ) τ km = G nm t m, 108) k τ nk G 0 1 δ km ) = t n G nm. 109) k 21

22 2.3 The angular momentum representation The spherical harmonics, Y L ˆr), where L N stands for the composite index l, m), satisfy the orthogonality and completeness relations, d 2ˆr Y L ˆr) Y L ˆr) = δ LL, 110) and L Y L ˆr) Y L ˆr ) = δ ˆr ˆr ) 1 sin ϑ δ ϑ ϑ ) δ φ φ ), 111) respectively. Note also the identity, l Y L ˆr) Y L ˆr ) = m= l l m= l Y L ˆr) Y L ˆr ). 112) The well-known solutions of the Schrödinger equation for free particles are j L ε; r) j l pr) Y L ˆr) n L ε; r) n l pr) Y L ˆr) h ± L ε; r) h± l pr) Y L ˆr) p = ε ), 113) where j l x), n l x) and h ± l x) = j l x) ± i n l x) are the spherical Bessel, Neumann and Hankel-functions, respectively. An orthonormal, complete set of basisfunctions is formed by since, ϕ L ε; r) = ε1/4 π 1/2 j L ε; r) ε > 0, L N), 114) d 3 r ϕ L ε; r) ϕ L ε ; r) = δ LL δ ε ε ) 115) and 0 dε ϕ L ε; r) ϕ L ε; r ) = δ r r ). 116) L Using the spectral resolution of the resolvent the Green function of free particles can then be written as G 0 z; r, r ) = 0 dε ε j L ε; r) j L ε; r ). 117) π z ε L 22

23 2.4 One-center expansion of the free-particle Green function In terms of contour integrations, it can be shown that by choosing p = z such that Im p > 0, G 0 z; r, r ) = ip L j l pr < ) h + l pr >) Y L ˆr)Y L ˆr ), 118) where r < = min r, r ) and r > = max r, r ). Introducing the functions with complex energy arguments, f L z; r) f l pr) Y L ˆr) fl = j l, n l or h ± l, p2 = z, Im p > 0 ) 119) with the corresponding conjugation Eq. 118) can simply be written as f L z; r) f l pr) Y L ˆr), 120) G 0 z; r, r ) = ip L j L z; r < ) h + L z; r >) = ip L h + L z; r >) j L z; r < ), called the one-center expansion of the free-particle Green function. 121) For later purposes it is useful to introduce a vector notation for the functions f L z; r), f z; r) [ f 1 z; r), f 2 z; r), f 3 z; r), ], 122) and the respective adjungate vector, f z; r) f 1 z; r) f 2 z; r) f 3 z; r) Eq. 121) can then be compactly written as.. 123) G 0 z; r, r ) = ip j z; r < ) h + z; r > ) = ip h + z; r > ) j z; r < ). 124) 23

24 2.5 Single-site scattering Let us first note that, according to Eqs. 38) and 91), t n z) is zero outside Ω n, t n z; r, r ) = 0 for r n S n or r n S n. 125) Because of traditional reasons, in the Lippmann-Schwinger equation the functions j L ε; r), rather than those in Eq. 114), are considered as eigenfunctions of the free-particle Hamiltonian H 0, RL n ε; r n ) = j L ε; r n ) + d 3 x n x n S n d 3 y n G 0 ε; r n, x n ) t n ε; x n, y n ) j L ε; y n ), y n S n 126) where we shifted the origin of our coordinate system to R n. For r n > S n the one-center expansion of the free-particle Green function, Eq. 121) can be used to yield RL n ε; r n ) = j L ε; r n ) ip h + L ε; r n ) t n L L ε), L 127) where the matrixelements of t n ε) single-site t-matrix) are defined as t n L L ε) = d 3 x n x n S n d 3 y n j L ε; x n ) t n ε; x n, y n ) j L ε; y n ). y n S n 128) By using the shorthand notations 122) and 123), the t-matrix, t n ε) {t n L L ε)}, 129) can be expressed as t n ε) = d 3 x n x n S n d 3 y n j ε; x n ) t n ε; x n, y n ) j ε; y n ) y n S n 130) and the normalization 127) as R n ε; r n ) = j ε; r n ) ip h + ε; r n ) t n ε) r n > S n ). 131) 24

25 Note that the functions R n L ε; r n) are regular at the origin r n = 0). Frequently, a different kind of regular scattering wavefunctions, Z n ε; r n ) = R n ε; r n ) t n ε) 1, 132) are used, which for r n > S n are normalized as Z n ε; r n ) = j ε; r n ) t n ε) 1 ip h + ε; r n ) ) = j ε; r n ) t n ε) 1 ipi + p n ε; r n ). 133) Because of the definition 128) the optical theorem, Eq. 68), takes the form, [ t n ε) 1 t n ε) ] 1 = 2ipI, 134) implying that the reactance matrix, 1 K n ε) = t n ε) 1 ipi) 135) is Hermitean. Clearly, Eq. 133) can be written as Z n ε; r n ) = j ε; r n ) K n ε) 1 + p n ε; r n ). 136) As we shall see, in the explicit expression of the Green function, scattering solutions that are irregular at the origin and normalized for r n > S n as H n ε; r n ) = ip h + ε; r n ) 137) or J n ε; r n ) = j ε; r n ), 138) occur. From the boundary conditions, Eqs. 131) and 137), it is straightforward to show that J n ε; r n ) = R n ε; r n ) H n ε; r n ) t n ε). 139) 25

26 From Eq. 135) the inverse of the t-matrix can be expressed as t n ε) 1 = K n ε) 1 + ipi, 140) which has to be inserted into Eq. 74), [ S n ε) = t n ε) ] 1 t n ε) ) ) 1 = K n ε) 1 ipi K n ε) 1 + ipi = I ipk n ε)) I + ipk n ε)) ) From the above equation it is obvious that, since K n ε) is a Hermitean matrix, S n ε) is unitary as stated in Section 1.7. For a spherical potential, the t-matrix is diagonal, therefore, see also Eq. 77)) t n L L ε) = δ LL t n l ε), 142) where with δ n l t n l ε) = 1 p eiδn l ε) sin δ n l ε), 143) ε) the l-like partial phaseshifts and the reactance matrixelement, K n l ε) = 1 p tan δn l ε), 144) is indeed real. By matching the regular solution at the muffin tin radius the familiar text-book formula tan δ n l ε) = Ln l ε) jn l ps n) pj n l ps n ) L n l ε) nn l ps n) pn n l ps n), 145) with L n l ε) = d dr ln ϕn l ε; r) r=s n, can be obtained. 26

27 2.6 Two-center expansion of the free-particle Green function In order to define matrixelements of the SPO, Eq. 101), and the structural resolvent, Eq. 106), an expansion of the free-particle Green function, G 0 ε; r, r ) in terms of the spherical Bessel functions centered around two different sites is needed, i.e., for n m and r n < S n, r m < S m, 146) G 0 ε; r n + R n, r m + R ) m = j L ε; r n ) G nm 0,LL ε) j L ε; r m), 147) LL or G 0 ε; r n + R n, r m + R m ) = j ε; r n ) G nm 0 ε) j ε; r m), 148) where the expansion coefficients G nm 0 ε) = { G nm 0,LL ε)} 149) are referred to as the real-space, free or bare) structure constants. The explicit expression for G nm 0,LL ε) reads, G nm 0,LL ε) = 4πpi i l l l h + L ε; R nm ) CLL L, 150) L L with the Gaunt coefficients, CLL L = d 2 k Y L ˆk) Y L ˆk) Y L ˆk) = d 2 k Y L ˆk)Y L ˆk)Y L ˆk). 151) Let us summarize the one- and the two-center expansion for the free-particle Green function as follows, G 0 ε; r n + R n, r m + R m ) = 152) = δ nm j ε; r < ) ip) h + ε; r > ) + 1 δ nm ) j ε; r n ) G nm 0 ε) j ε; r m). 27

28 2.7 Multi-site scattering Matrices in site-angular momentum representation Substituting Eq. 152) into Eq. 106) and performing the necessary integrations we obtain G nm ε; r n + R n, r m + R m ) = j ε; r n ) G nm ε) j ε; r m), 153) where we introduced the structural Green function matrix, G nm ε) = G nm 0 ε) 1 δ nm ) + and the matrix of the SPO, k n) j m) G nk 0 ε) τ kj ε) G jm 0 ε), 154) τ kj ε) = d 3 x k d 3 y j j ε; x k ) τ kj ε; x k, y j ) j ε; y j ). 155) x k S k y j S j Note that the SPO, τ kj ε; x k + R k, y j + R j ), is zero for x k > S k or y j > S j. It is straightforward to show that the operator equations 101), 103), 104) and 107) imply the matrix equations, τ nm ε) = δ nm t n ε) + t n ε) G nm 0 ε) 1 δ nm ) t m ε) 156) + k t n ε) G nk 0 ε) 1 δ nk ) t k ε) G km 0 ε) 1 δ km ) t m ε) +..., τ nm ε) = δ nm t n ε) + k t n ε) G nk 0 ε) 1 δ nk ) τ km ε), 157) τ nm ε) = δ nm t n ε) + k τ nk ε) G km 0 ε) 1 δ km ) t m ε), 158) and τ nm ε) = δ nm t n ε) + t n ε) G nm ε) t m ε). 159) 28

29 It is very useful to introduce the matrices in site-angular momentum representation, and t ε) = {t n ε) δ nm } = {t n LL ε) δ nm}, 160) G 0 ε) = {G nm 0 ε) 1 δ nm )} = { G nm 0,LL ε) 1 δ nm) }, 161) τ ε) = {τ nm ε)} = {τll nm ε)}, 162) G ε) = {G nm ε)} = {G nm LL ε)}, 163) since Eqs. 154), as well as ) can be rewritten as G ε) = G 0 ε) + G 0 ε) τ ε) G 0 ε), 164) τ ε) = t ε) + t ε) G 0 ε) t ε) + t ε) G 0 ε) t ε) G 0 ε) t ε) +..., 165) τ ε) = t ε) + t ε) G 0 ε) τ ε), 166) τ ε) = t ε) + τ ε) G 0 ε) t ε), 167) and τ ε) = t ε) + t ε) G ε) t ε), 168) respectively. In particular, the Dyson equation 166) or 167) can be solved in terms of the following matrix inversion, 1 τ ε) = t ε) 1 G 0 ε)). 169) By comparing Eqs. 165) and 168) we find, from which the equation, G ε) = G 0 ε) + G 0 ε) t ε) G 0 ε) + G 0 ε) t ε) G 0 ε) t ε) G 0 ε) +... = G 0 ε) + G ε) t ε) G 0 ε), 170) G ε) = G 0 ε) I t ε) G 0 ε)) 1 171) can be obtained. Eqs. 169) and 171) are called the fundamental equations of the Multiple Scattering Theory. 29

30 Evaluation of the Green function In this section it will be shown that the Green function, G appears to have the form, G ε; r n + R n, r m + R ) m = ε; r n + R n, r m + R m ), = R n ε; r n ) G nm ε) R m ε; r m) + δ nm R n ε; r < ) H n ε; r > ), 172) with r < = min r n, r n), r > = max r n, r n) and the scattering solutions, R n ε; r n ) and H n ε; r n ), as normalized according to Eqs. 131) and 137), respectively. For a particular pair of sites, n and m, we split the operator expression 105) as G ε) = G nm) ε) + G nm) ε) + G n m) ε) + G n m) ε), 173) where see also Eqs ) G nm) ε) = G 0 ε) τ nm ε) G 0 ε) = δ nm G 0 ε) t n ε) G 0 ε) + G 0 ε) t n ε) G nm ε) t m ε) G 0 ε), 174) G nm) ε) = G 0 ε) τ im ε) G 0 ε) = G nm ε) t m ε) G 0 ε), 175) i n) G n m) ε) = G 0 ε) τ nj ε) G 0 ε) = G 0 ε) t n ε) G nm ε), 176) j m) G n m) ε) = G 0 ε) τ ij ε) G 0 ε) + G 0 ε) = G nm ε) + δ nm G 0 ε). 177) i n) j m) First, for the case of r n > S n, r m > S m, we evaluate the configurational space representation of the above operators. By making use of the expressions 152) as well as 153), the corresponding integrations can easily be performed yielding, 30

31 G nm) ε; r n + R n, r m + R m ) = δ nm ip) 2 h + ε; r n ) t n ε) h + ε; r n) + ip) 2 h + ε; r n ) t n ε) G nm ε) t m ε) h + ε; r m), 178) G nm) ε; r n + R n, r m + R m ) = ip) j ε; r n ) G nm ε) t m ε) h + ε; r m), 179) G n m) ε; r n + R n, r m + R m ) = ip) h + ε; r n ) t n ε) G nm ε) j ε; r m), 180) G n m) ε; r n + R n, r m + R ) m = j ε; r n ) G nm ε j ε; r m) 181) + δ nm j ε; r n ) ip) h + ε; r n) Θ r n r n ) ) + ip) h + ε; r n ) j ε; r n) Θ r n r n). After adding all the contributions, we find G ε; r n + R n, r m + R ) m = 182) = [ j ε; r n ) ip h + ε; r n ) t n ε) ] G nm ε) [ j ε; r m) ip t m ε) h + ε; r m) ] [j + δ nm ε; rn ) ip h + ε; r n ) t n ε) ] ip) h + ε; r n) Θ r n r n ) + ip) h + ε; r n ) [ j ε; r n) ip t n ε) h + ε; r n) ] ) Θ rn r n). For real potentials, V in fact, for time-reversal invariant Hamiltonians), t n ε) turns to be symmetric. Note that this is still valid if an effective exchange) field, interacting only with the spin of the system, is present. Moreover, making use of the definition of the t-matrix, Eq. 128) and the identity 112), it is easy to prove that h + ε; r n ) t n ε) h + ε; r n) = h + ε; r n) t n ε) h + ε; r n ), 183) 31

32 such that we arrive at G ε; r n + R n, r m + R ) m = [ j ε; rn ) ip h + ε; r n ) t n ε) ] G nm ε) [ j ε; r m) ip h + ε; r m) t m ε) ] [ + δ nm j ε; r< ) ip h + ε; r < ) t n ε) ] ip) h + ε; r > ). 184) Obviously, outside the muffin-tin region, the Green function exhibits the form of 172). Since, however, the Green function satisfies the equation, + V r) ε) G ε; r, r ) = δ r r ), 185) in all space, it can readily been continued inside the muffin-tins by keeping the functions R n ε; r n ) and H n ε; r n ) solutions of the Schrödinger equation related to the muffin-tin indexed by n. An alternative expression of the Green function can be obtained when we replace the regular scattering solutions R n ε; r n ) by Z n ε; r n ) see Eq. 132)) in Eq. 172). Namely, R n ε; r n ) G nm ε) R m ε; r m ) = = Z n ε; r n ) t n ε) G nm ε) t n ε) Z m ε; r m) = Z n ε; r n ) τ nm ε) Z m ε; r m ) δ nm Z n ε; r n ) t n ε) Z n ε; r n), where we exploited Eq. 159), and R n ε; r < ) H n ε; r > ) = Z n ε; r < ) t n ε) H n ε; r > ). 186) Adding the above two expressions and utilizing Eqs. 132) and 139) we obtain G ε; r n + R n, r m + R ) m = = Z n ε; r n ) τ nm ε) Z m ε; r m) δ nm Z n ε; r < ) J n ε; r > ). 187) 32

33 In case of δ 0 it can be shown that, Im G + ε; r n + R n, r n + R n ) = Z n ε; r n ) Im τ nn ε) Z n ε; r n ), 188) where Im τ nn ε) = 1 2i τ nn ε) τ nn ε) +). 189) Namely, G ε + i0; r n + R n, r n + R ) n = Z n ε + i0; r n ) τ nn ε + i0) Z n ε + i0; r n ) and G Z n ε + i0; r n ) J n ε + i0; r n ), 190) ε + i0; r n + R n, r n + R n ) = Z n ε i0; r n ) τ nn ε i0) Z n ε i0; r n ) Z n ε i0; r n ) J n ε i0; r n ), 191) K n ε + i0) = K n ε i0) Z n ε + i0; r n ) = Z n ε i0; r n ), J n ε + i0; r n ) =J n ε i0; r n ), therefore, G ε + i0; r n + R n, r n + R ) n G ε + i0; r n + R n, r n + R n ) = Z n ε + i0; r n ) [τ nn ε + i0) τ nn ε i0)] Z n ε + i0; r n ), 192) which immediately leads to Eq. 188). Let us, finally, write the Green functions 172) and 187) by explicitly indicating the summations with respect to the angular momentum indices, G ε; r n + R n, r m + R ) m = 193) = RL n ε; r n ) G nm LL ε) Rm L ε; r m) + δ nm RL n ε; r < ) HL n ε; r > ) LL, and G ε; r n + R n, r m + R ) m = 194) = ZL n ε; r n ) τll nm L ε; r m) δ nm ZL n ε; r < ) JL n ε; r > ) LL. 33 L L

34 2.8 Density of states, charge density, dispersion relation In this section we give some expressions for quantities to be usually computed in electronic structure calculations of solids. The density of states defined in Eq. 29) can be written as n ε) = 1 d 3 r Im G + ε; r, r) π = 1 d 3 r i Im G ε; + r i + R π i, r i + ) R i, 195) Ω i i therefore, the local DOS LDOS), referenced to individual cells, can straightforwardly be defined as n i ε) = 1 d 3 r i Im G ε; + r i + R π i, r i + ) R i Ω i, 196) such that n ε) = i n i ε). 197) Inserting expression 188) into Eq. 196) yields n i ε) = 1 d 3 r i Z i ε; r i ) Im τ ii ε) Z i ε; r i ) π Ω i = 1 π tr F i ε) Im τ ii ε) ), 198) where now the trace has to be taken in angular momentum space and the matrix comprising the integrals of the scattering solutions, F i ε) is defined as F i ε) = d 3 r i Z i ε; r i ) Z i ε; r i ), 199) Ω i or FLL i ε) = d Ω 3 r i ZL i ε; r i ) ZL i ε; r i). 200) i 34

35 The case of spherical integration domains, i.e., the atomic sphere approximation ASA) Expanding the scattering solutions in terms of the spherical harmonics, ZL i ε; r i ) = Y L ˆr i ) ZL i L ε; r i ), 201) L Eq. 200) can be evaluated as follows, FLL i ε) = = L,L = L L,L S ASA i 0 S ASA i 0 Ω i d 3 r i Z i LL ε; r i) Y L ˆr i ) Z i L L ε; r i) Y L ˆr i ) ri 2 dr i ZLL i ε; r i) Z i L L i)) ε; r d 2ˆr i Y L ˆr i ) Y L ˆr i ) } {{ } δ L L r 2 i dr i Z i LL ε; r i) Z i L L ε; r i). 202) In the diagonal spherical symmetric) case this further reduces to S ASA FLL i ε) = δ i LL ri 2 dr i Zl i ε; r i ) 2, 203) } 0 {{ } Fl iε) and the LDOS can be determined by the simple expression, n i ε) = 1 π Fl i ε) l l m= l Im τ ii lm,lm ε). 204) Quite naturally, the partial local DOS, n i l ε), can also be defined as such that n i l ε) = 1 π F i l ε) l m= l Im τ ii lm,lm ε), 205) n i ε) = l n i l ε). 206) 35

36 The charge or particle) density, ρ r) = 1 π εf dε Im G + ε; r, r), 207) plays an essential role in electronic structure calculations based on the Density Functional Theory. Here we are restricted to T = 0, and the Fermi energy is denoted by ε F. Similar to the DOS, the charge density can also be decomposed into local components, ρ r) = i ρ i r i ), 208) where ρ i r i ) = 1 π εf LL dε Z i L ε; r i ) [ Im τ ii ε) ] LL Z i L ε; r i), 209) which for spherically symmetric potentials reduces to ρ i r i ) = 1 π LL εf dε Z i l ε; r i ) Z i l ε; r i) Im τ ii LL ε) Y L ˆr i ) Y L ˆr i ). 210) Using the expansion for the product of spherical harmonics, Y L ˆr i ) Y L ˆr i ) = L C L L L Y L ˆr i), 211) C L L L being the Gaunt coefficients as defined in Eq. 151), the local charge density can be decomposed into multipole components, ρ i r i ) = L ρ i L r i ) Y L ˆr i ), 212) where ρ i L r i ) = 1 π εf CLL L dε Z l i ε; r i) Z l i ε; r i) Im τ L ii L ε). 213) L L 36

37 In case of three-dimensional translation symmetry for brevity, let us discuss the case of a simple lattice only), the structure constants can be evaluated in terms of lattice Fourier transformation, G nn 0 ε) = 1 d 3 k G Ω 0 ε; ) k e i k R n R n ), 214) BZ BZ where BZ denotes the Brillouin zone of volume Ω BZ and the reciprocal space structure constants, G 0 ε; ) k = Rn G n0 0 ε) e i k R n, 215) can directly be calculated in terms of the so-called Ewald summation for slowly converging series. It is straightforward to show that the lattice Fourier transform of the SPO-matrix, τ ε; ) k = Rn τ n0 ε) e i k R n, 216) can be calculated as τ ε; ) k = [t ε) 1 G 0 ε; 1 k)], 217) where t ε) is the single-site t-matrix of the cells being now uniform for the entire system. The site-diagonal part of the SPO-matrix, τ 00 ε) = 1 d 3 k τ ε; Ω ) k, 218) BZ which enters the local DOS, n ε) = 1 π L BZ F l ε) Im τ 00 LL ε), 219) has poles at the eigenvalues of the Hamiltonian of the system, ε α k), where α labels the bands. From Eq. 217) it is obvious that these poles can be found by the condition, 37

38 det t ε) 1 G 0 ε; )) k = 0, 220) referred to as the KKR-equation which determines the dispersion relation in the bulk case. It should be noted that Eq. 220) represents a formally exact tool for searching the spectrum of a 3D translational invariant effective) one-electron system. The penalty one has to pay for this exactness is the non-linearity of Eq. 220), i.e., ε α k) can not be determined from a simple generalized) eigenvalue problem of a Hamiltonian matrix as in the case of linearized band structure methods. A useful visualization of the dispersion relation can be obtained by decomposing the DOS into k-dependent contributions, where the quantity, n ε) = 1 Ω BZ BZ d 3 k Aε; k), 221) Aε; k) = 1 π L F l ε) Im τll 00 ε; ) k, 222) called the Bloch spectral function, has δ-like sharp peaks at ε = ε α k). Obviously, a constant-energy surface in the Brillouin zone, in particular, the Fermi surface can be constructed by connecting the peak positions of Aε; k) for the given energy, ε. 38

39 3 Generalization of multiple scattering theory Referring to section 1.4, in the final section of this course we show how an arbitrary reference system can be used within the KKR Green function method. There are essentially two practical ways for doing this, depending on which of the fundamental equations, Eq. 169) or 171), is considered as starting point. It should be noted that both techniques are restricted to the case when the scatterers in the different systems are of the same geometrical arrangements, i.e., the position vectors R i are not changed. In order to handle structural relaxations, different techniques have to be used, the discussion of which extends the scope of this course. 3.1 The embedding technique We start with Eq. 169) which for the system of interest marked by index s) and the reference system r) reads as τ s ε) = t s ε) 1 G 0 ε)) 1, 223) and τ r ε) = t r ε) 1 G 0 ε)) 1, 224) respectively. Performing simple algebraic manipulations, ) 1 τ s ε) = t r ε) 1 t ε) 1 G 0 ε) ) 1 ) ) 1 = I t ε) 1 t r ε) 1 G 0 ε) t r ε) 1 G 0 ε), 225) where t ε) 1 = t r ε) 1 t s ε) 1, 226) it is straightforward to express τ s ε) in terms of τ r ε), 39

40 1 τ s ε) = τ r ε) I t ε) 1 τ ε)) r. 227) Therefore, once the t-matrices, t r ε) and t s ε), as well as the SPO matrix of the reference system, τ r ε), are known, the SPO matrix, τ s ε) can directly be calculated based on Eq. 227) and then the Green function of the system can be obtained from Eq. 194). A particularly important application of the embedding technique refers to the case when only a finite number say, a cluster) of scatterers differ from each other in the two systems, labelled by s and r. In this case, namely, t r t i ε) 1 i ε) 1 t s i ε) 1 if i C = 0 if i / C, 228) where C denotes the set of indices of the sites in the cluster. Rewriting formally Eq. 227), τ s ε) = τ r ε) + τ r ε) t ε) 1 τ r ε) + τ r ε) t ε) 1 τ r ε) t ε) 1 τ r ε) +..., 229) it is easy to express the projection of τ s ε) onto the cluster, τ s C ε), [τ s C ε)] ij = τ s,ij ε) for i, j C, 230) as τ s C ε) = τ r C ε) + τ r C ε) t C ε) 1 τ r C ε) + τ r C ε) t C ε) 1 τ r C ε) t C ε) 1 τ r C ε) +..., 231) where τ r C ε) is the corresponding projection of the SPO matrix of the reference system and t C ε) 1 comprises the nonvanishing t 1 i 228)). Eq. 231) can then be written in a form similar to 227), 40 matrices see Eq.

41 1 τ s C ε) = τ r C ε) I t C ε) 1 τ r C ε)), 232) with the obvious difference that now one has to deal with matrices of dimension of N l max + 1) 2, where N denotes the number of sites in the cluster and l max is the angular momentum cut-off in the calculations. Specifically, for a single impurity at site i the above equation reduces to 1 τ s,ii ε) = τ r,ii ε) I t i ε) 1 τ ε)) r,ii. 233) Within the single-site coherent potential approximation for randomly disordered substitutional alloys, the reference system is associated with the translational invariant) effective medium in which a particular constituent of the alloy labelled by α) is immersed via τ α,00 ε) = τ c,00 ε) I t α ε) 1 τ c,00 ε)) ) Without going into details, the condition for self-consistently determining the t-matrix of the effective system, t c ε), reads τ c,00 ε) = α c α τ α,00 ε), 235) where c α is the concentration of component α α c α = 1), while τ c,00 ε) is calculated by using Eqs. 217) and 218) when replacing t ε) by t c ε). 41

42 3.2 The Screened Korringa-Kohn-Rostoker method In principal, an equivalent technique can be developed by focusing on the structural Green function matrix of the system see Eq. 171)), G s ε) = G 0 ε) I t s ε) G 0 ε)) 1, 236) which by splitting the t-matrix into two parts, t s ε) = t r ε) + t ε), 237) with t ε) = t s ε) t r ε), 238) can be related to the structural Green function matrix of the reference system, G r ε) = G 0 ε) I t r ε) G 0 ε)) 1, 239) as follows, G s ε) = G 0 ε) I t r ε) G 0 ε) t ε) G 0 ε)) 1 [ = G 0 ε) I t ε) G 0 ε) I t r ε) G 0 ε)) 1] I t r ε) G 0 ε)) = G 0 ε) I t r ε) G 0 ε)) 1 I t ε) G 0 ε) I t r ε) G 0 ε)) 1) 1 ) 1 G s ε) = G r ε) I t ε) G r ε)) ) The equivalence of Eqs. 227) and 240) can be proved directly! Within the Screened Korringa-Kohn-Rostoker method, a reference system is used to obtain G r,ij ε) localized in real space. Obviously, a system formed by a uniform distribution of constant repulsive potentials typically, V r =2-3 42

43 Ryd) meets the above request, since the valence band of typical metals lies below V r, thereby, no states of the reference system can be found in this region. This implies that, similar to G 0 ε; r, r ) for negative energies see Eq. 23)), the Green function of the reference system exponentially decays in real space. For that reason, the matrices G r,ij ε) are referred to as the screened structure constants, used to define the SPO in the screened representation, 1 τ sr) ε) = [ t ε)] 1 G ε)) r. 241) Obviously, once a real-space cut-off for G r,ij ε) is used, G r,ij ε) 0 for R ij > D, 242) sparse matrix techniques can be used to solve Eq. 241). Such a method has been developed to treat surfaces and interfaces of solids, where the translational invariance is broken normal to the planes and, therefore, a lattice Fourier transform does not apply with respect to this direction. The screened SPO is obviously related to the structural Green function matrix of the system by τ sr) ε) = t ε) + t ε) G s ε) t ε), 243) and a similar expression holds for the physical representation of the SPO, τ s ε) = t s ε) + t s ε) G s ε) t s ε), 244) Thus the transformation of the SPO from the screened to the physical presentation can be written as τ s ε) = t s ε) [ t ε)] 1 τ sr) ε) [ t ε)] 1 t s ε) t s ε) [ t ε)] 1 t s ε) + t s ε), 245) which does not imply any further numerical complication since the transformation is site-diagonal. 43

44 Literature Gonis, A.: Green functions for ordered and disordered systems North-Holland, Amsterdam) 1992 Messiah, A.: Quantum mechanics North-Holland, Amsterdam) 1969 Newton, R.G.: Scattering theory of waves and particles McGraw-Hill, New York) 1966 Taylor, J.R.: Scattering theory John Wiley and Sons, New York) 1972 Weinberger, P.: Electron scattering theory of ordered and disordered matter Clarendon, Oxford)