Non-resonant x-ray emission spectroscopy (XES) is a widely used technique to study the valence electronic system, thus complementing x-ray absorption spectroscopy, which re-veals the conduction electronic structure. From a theoretical viewpoint, XES is a special case of RIXS, where the excitation energy is chosen high above the absorption edge, and the emission spectrum from the valence-core transitions is observed. In order to avoid the explicit calculation of the polarizability for high energies above the absorption edge, we em-ploy the fact that the excited electron is in a conduction state high above the Fermi energy.
Both the correlation between the excited electron and the core hole, and the correlation between the excited electron and the final valence hole, can be neglected in this case. Thus, we can employ the cross section in the independent particle approximation from Eq. 5.9 as Furthermore, we assume a fixed excitation energy resonant to a specific independent-particle excitation, i.e. ω1 = ϵck−ϵµk. Then, theδ-function becomesδ(ω − (ϵck−ϵvk)) = δ(ϵvk−ϵµk−ω2)and the cross section becomes a function of the emission energyω2alone.
As we have fixed the excitation energy to a specific core excitation, all terms depending on the conduction stateckyield only a constant factor to the cross section. Focussing on the dependence on the emission energy, we then write the cross section as
d2σ
Non-resonant X-ray Emission Spectroscopy 5.3
where we consider it only as a function of the emission energyω2. Equation 5.22 is identical to expressions in literature [273, 274] used to calculate XES spectra. Within our approxima-tion, there are no effects of electron-hole correlation on the x-ray emission spectra, and the spectrum is thus given by single-particle energy differences,ϵvk−ϵµk, and the transition matrix elementsPµvk, fully determined by the electronic band structure. As such, the XES yields direct insight into the valence electronic structure.
CHAPTER 6
Absorption and Non-resonant Scattering in exciting
In this chapter, we present the implementation of the BSE formalism inexciting [254].
In general, BSE calculations require the knowledge of the quasiparticle energies for the construction of Eq. 6.26 which can be obtained from theGW approach of MBPT. Existing all-electron implementations ofGW typically adopt a basis set that is optimized to represent products of all-electron wavefunctions, know asproduct basis[275–277]. Details on theGW implementation in theexcitingcode within the product basis representation can be found in Refs. [254, 277, 278]. For the BSE implementation, the plane-wave representation for the non-local operators is chosen, which provides a description of the optical properties [225]
at lower computational cost. Since momentum and plane-wave matrix elements are central quantities, they are discussed in detail in this section.
The work described in this chapter was developed in close collaboration with Benjamin Aurich [261], extending the existing implementation in theexciting[253, 279]. Benjamin Aurich implemented optical BSE calculations for finite momentum transfer and optical BSE calculations beyond the Tamm-Dancoff approximation, while we have combined the imple-mentations for the calculations of optical excitations with the one for core excitations [239].
This consistent implementation is the starting point for our code development to determine the RIXS cross section (compare Chapter 7). This chapter in large parts follows the corre-sponding sections in Ref. [359].
6.1 Linearized Augmented Plane-wave Basis
excitingemploys the (L)APW+lo basis set [232–235] for the Kohn-Sham equations Eq. 1.7 to compute valence and conduction states. These states subsequently enter the expressions of the matrix elements of the BSE Hamiltonian. To obtain the basis functions, the unit cell is divided into non-overlapping muffin-tin (MT) spheres centered at the atomic positions and theinterstitialregion between the spheres. Different functions are employed in the two regions in order to account for both the rapid variation of the Kohn-Sham wavefunctions close to the nuclei and the smoother behavior in the interstitial region. In the MT sphere
6 Absorption and Non-resonant Scattering inexciting
surrounding an atom α, the wavefunctions are expanded in atomic-like basis functions ulα(r)Ylm(rˆ), while plane waves e−i(G+k)r are used in the interstitial region. As such, the basis functionsϕk+Gare expressed as
ϕk+G(r) = Here,V0 is the unit-cell volume andAk+Glm,p are expansion coefficients that ensure that the basis functions are continuous at the boundaries of the MT spheres. The radial functions ulα,p(r) are obtained from the solutions of the radial Schrödinger equation using the spher-ically averaged Kohn-Sham potential. The indexp denotesp-th derivative with respect to the energy, i.e. ulα,p = ∂∂ϵpupαl . Forp = 0, we recover the APW+lo basis set [235], while the summation up top = 1 yields the LAPW+lo basis set [232, 233]. Higher-order derivatives of the radial functions can also be included. In order to increase the variational degrees of freedom in the MT spheres, local orbitals (LOs) [235]ϕν(r) are used to complement the basis. These basis function are expressed as
ϕν(r) = The local orbitals vanish outside of the MT spheres and the coefficientsBν,p ensure that they are continuous and smooth at the MT-sphere boundary. As the LOs are added for specific MT spheres and (lm)-channels, they allow for a systematic improvement of the basis. For a review on the family of (L)APW+lo basis sets, see Ref. [254]. The eigenstates ψikof the Kohn-Sham Hamiltonian in Eq. 1.9 are expressed in the LAPW+lo basis as
ψikKS(r) =X where the radial functions are defined as
ulik =X Ci(k+G) andCiνk are the expansion coefficients, obtained from the diagonalization of the Kohn-Sham Hamiltonian.
While the expansion in this basis is convenient for the extended valence and conduction states, the highly localized core statesψα,κ,MKS require a different treatment. These states are fully localized in the muffin-tin sphere of the atomα. As spin-orbit coupling can play
62
Linearized Augmented Plane-wave Basis 6.1
Figure 6.1:Schematics of a unit cell in the LAPW framework.
a dominant role for these states, they are obtained from the solution of the radial Dirac equation in the respective spherically symmetrized crystal potential. The spinor solutions ψα,κ,MKS of these equations can be written as
ψα,κ,MKS (r) = uα,κ(r)Ωκ,M(rˆ)
The spherical part of the core wavefunctionsψα,κ,MKS is given by the spin spherical harmonics ΩL,S,J,M(rˆ), defined as
As such, the core wavefunctionsψα,κKS in Eq. 6.5 are four-dimensional Dirac vectors
com-6 Absorption and Non-resonant Scattering inexciting
posed of the two-dimensional spinors of the large and small component. The radial func-tionsuα,κ(r) for the large component and−ivα,κ(r)for the small component, respectively, are given by the coupled radial Dirac equations
∂uα,κ
In the calculation of matrix elements between core states and conduction states, the small component is neglected, and we obtain the core wavefunctionψα,κ,MKS at an atomic siteα:
ψα,κ,MKS (r) =
More details about the treatment of core states in the (L)APW+lo basis can be found in Ref. [239].