Real-space Green's function approach for the Langreth cumulant

The effect of intrinsic excitations on the x-ray photoemission spectra (XPS) can be described by the core-hole Green’s function G_c(t). As shown by Langreth, due to the linked-cluster theorem, G_c(t) can be expressed exactly in cumulant form, G_c(t) = G_c⁰(t)exp[C(t)], where the independent particle core-hole Green’s function is G_c⁰=exp(iε_ct) and C(t) is the cumulant. In an interacting electron system, the cumulant C(t) is given to linear order by the density-response to a suddenly turned-on core-hole.  In previous work, a real-time time-dependent density functional theory approach (RT-TDDFT) has been developed for the cumulant. Here we develop a real-space Green's function approach for the Langreth cumulant C(t).   Our formulation starts from the observation that in frequency space, the cumulant kernel β(ω) has the same form as the expression for the atomic polarizability in the TDDFT approach of Zangwill and Soven, but with the core-hole potential V(r) replacing the dipole interaction.  As a consequence, one can immediately derive a real-space Green's function approach, in analogy with the BSE-TDDFT formulation of Ankudinov et al. (TDDFT-BSE). In particular, the screened dipole matrix elements of XAS are replaced by screened monopole transition elements in response to a suddenly turned-on core-hole: M_cL = <c|V_sc|L>, where the dynamically screened core-hole potential is V_sc(r) = ε^-1(ω) V_c(ω), and the dielectric matrix is ε = [1-K^Lχ⁰]. In contrast to XAS, V_c(r) and V_sc(r) are spherically symmetric, so the selection rules preserve angular momentum L=L_c. Thus, the calculations with the RSGF approach are similar to XAS, except for modified matrix elements and selection rules. Also in analogy with the XAS, one can define a fine structure in the cumulant kernel due to near-neighbor scattering.