As derived in PRB 73, 035120 (2006) in appendix, the Green's function moments are,
G=M0iω+M1(iω)2+M2(iω)3+⋯and can be computed from
Mk=(−1)k+1[dkdτkG(0+)−dkdτkG(0−)]This follows from integration by parts of G(iω)=∫β0G(τ)eiωτdτ.
We thus have
M0=G(O−)−G(0+)=δαβM1=−⟨[H,ψα]ψ†β⟩−⟨ψ†β[H,ψα]⟩=−⟨{[H,ψα],ψ†β}⟩M2=⟨[H,[H,ψα]]ψ†β⟩+⟨ψ†β[H,[H,ψα]]⟩=⟨{[H,ψα],[ψ†β,H]}⟩M3=−⟨{[H,[H,ψα]],[ψ†β,H]}⟩where we took into account that H commutes with e−βH and hence H can be moved from the first to the last place in the brackets.
First we compute
[H,ψα]=−εαiψi−Uαijkψ†iψjψk−Vαkck[ψ†α,H]=−ψ†iεiα−Uαijkψ†kψ†jψi−c†kVkαwhere H is the general Anderson impurity Hamiltonian.
M1 is thus
M1αβ=εαβ+(Uαijβ−Uαiβj)nijM2 becomes
M2αβ=εαiεiβ+VαkVkβ+(Uαijk−Uαikj)εkβ⟨ψ†iψj⟩+εαk(Uβjik−Uβjki)⟨ψ†iψj⟩+(Uαijk−Uαikj)(Uβlpk−Uβlkp)⟨ψ†pψ†iψjψl⟩+UαkjlUβkip⟨ψ†pψ†iψjψl⟩+(Uαijk−Uαikj)Uβljk⟨ψ†iψl⟩To calculate the moments of the self-energy, we need to use
G=1iω−ε−Σ∞−Siω+⋯−Δ=1iω+M1(iω)2+M2(iω)3+⋯=1iω+ε+Σ∞(iω)2+S+(ε+Σ∞)2+(Δ∗(iω))(iω)3+⋯We thus see
Σ∞=M1−εS=M2−(M1)2−V2Finally, we obtain
Σαβ(∞)=(Uαijβ−Uαiβj)nijand
Sαβ=(Uαijk−Uαikj)(Uβlpk−Uβlkp)(⟨ψ†pψ†iψjψl⟩−⟨ψ†pψl⟩⟨ψ†iψj⟩)+UαkjlUβkip⟨ψ†pψ†iψjψl⟩+(Uαijk−Uαikj)Uβljk⟨ψ†iψl⟩