Introduction to Multiple Scattering Theory L aszl o Szunyogh - PDF document

Introduction to Multiple Scattering Theory L´ aszl´ o 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 3 1.1 Resolvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Observables and Green functions . . . . . . . . . . . . . . . . . 7 1.3 The T -operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Scaling transformation of the resolvents . . . . . . . . . . . . . . 11 1.5 The Lippmann-Schwinger equation . . . . . . . . . . . . . . . . 12 1.6 The optical theorem . . . . . . . . . . . . . . . . . . . . . . . . 13 1.7 The S -matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.8 Integrated density of states: the Lloyd formula . . . . . . . . . . 16 2 The Korringa-Kohn-Rostoker Green function method 18 2.1 Characterization of the potential . . . . . . . . . . . . . . . . . 18 2.2 Single-site and multi-site scattering: operator formalism . . . . . 19 2.3 The angular momentum representation . . . . . . . . . . . . . . 22 1

2.4 One-center expansion of the free-particle Green function . . . . 23 2.5 Single-site scattering . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6 Two-center expansion of the free-particle Green function . . . . 27 2.7 Multi-site scattering . . . . . . . . . . . . . . . . . . . . . . . . 28 2.8 Density of states, charge density, dispersion relation . . . . . . . 34 3 Generalization of multiple scattering theory 39 3.1 The embedding technique . . . . . . . . . . . . . . . . . . . . . 39 3.2 The Screened Korringa-Kohn-Rostoker method . . . . . . . . . 42 2

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

The resolvent of H is defined for any z ∈ ρ ( H ) as G ( z ) = ( z I − 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 d G ( 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 � | ϕ n �� ϕ n | | ϕ α ( ε ) �� ϕ α ( ε ) | � � G ( z ) = + dε . (8) z − ε n z − ε n α Relationship between the eigenvalues of H and the singularities of G ( z ): discrete spectrum of H − → poles of first order of G ( z ) . (9) continuous spectrum of H − → branch cuts of G ( z ) Therefore, at the real axis the so-called up- and down-side limits of G ( z ) are introduced, G ± ( ε ) = lim δ → +0 G ( ε ± iδ ) ( ε ∈ R ) , (10) having the following relationship, G ± ( ε ) = G ∓ ( ε ) † . (11) In particular, for ε ∈ ρ ( H ) ∩ R G + ( ε ) = G − ( ε ) = G ( ε ) . (12) 4

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

The Green function of free particles � d 3 k e i� k ( � r − � r ′ ) 1 r ′ ) = G 0 ( z ; � r,� z − k 2 , (19) (2 π ) 3 which can be evaluated as follows, � ∞ � 1 1 k e i� d 2 ˆ r ′ ) r ′ ) = dk k 2 k ( � r − � G 0 ( z ; � r,� (2 π ) 3 z − k 2 0 �� ∞ � ∞ � dk k e ik | � r − � r ′ | dk k e − ik | � r − � r ′ | i = − z − k 2 − . 8 π 2 | � r ′ | z − k 2 r − � −∞ −∞ 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 � � ke ik | � r − � r ′ | ke − ik | � r − � r ′ | 1 r ′ ) = G 0 ( z ; � r,� Res( ( p − k ) ( p + k ) , p ) − Res( ( p − k ) ( p + k ) , − p ) r ′ | 4 π | � r − � = − e ip | � r − � r ′ | r ′ | , (20) 4 π | � r − � while by choosing p ∈ C , Im p < 0, such that z = p 2 , one obtains � � ke ik | � r − � r ′ | ke − ik | � r − � r ′ | 1 r ′ ) = G 0 ( z ; � r,� Res( ( p − k ) ( p + k ) , − p ) − Res( ( p − k ) ( p + k ) , p ) r ′ | 4 π | � r − � = − e − ip | � r − � r ′ | r ′ | . (21) 4 π | � r − � Clearly, independent of the choice of the square-root of z ( p 1 = − p 2 ) the ex- r ′ ) is unique. In particular, pression of G 0 ( z ; � r,� r ′ ) = − e ± ip | � r − � r ′ | , p = √ ε ) , G ± 0 ( ε ; � r,� ( ε > 0 (22) r ′ | 4 π | � r − � and r ′ ) = − e − p | � r − � r ′ | √ G + r ′ ) = G − 0 ( ε ; � r,� 0 ( ε ; � r,� ( ε < 0 , p = − ε ) . (23) r ′ | 4 π | � r − � 6

1.2 Observables and Green functions In a system of independent fermions, the measured value of a one-particle ob- servable, say A , is given by A = Tr ( f ( H ) A ) , where A is the Hermitian operator related to A and, the Fermi-Dirac (density) operator is defined by � I + e β ( H− µ I ) � − 1 f ( H ) = , 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 ( ǫ ) � , (24) n α � 1 + e β ( ε − µ ) � where f ( ε ) = 1 / . Recalling Eq. (8) one can write, � f ( z ) � ϕ n | A | ϕ n � f ( z ) � ϕ α ( ε ) | A | ϕ α ( ε ) � � � f ( z ) Tr ( A G ( z )) = + dε , z − ε n z − ε n 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 ) if a is within the contour �  − 1 dz g ( z ) z − a = , 2 πi  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, 1 f ( z ) ≃ k B T for z ≃ z k , z k = µ + i (2 k + 1) πk B T ( k ∈ Z ) , z − z k have also to be taken into account, � A = − 1 � dz f ( z ) Tr ( AG ( z )) − k B T Tr ( AG ( z k )) , (25) 2 πi k C 7