Category:XAS

X-ray absorption spectroscopy (XAS) is a powerful method for investigating the chemical properties of materials. The absorption of the X-ray photons involves excitations of the core electrons into conduction bands. The main challenge for an accurate description of such excitations is the interactions between a positive charge left in place of the excited core electron, i.e., core hole, and the excited electron, which can form a bound state called exciton. There are two main approaches to ab initio modeling of XAS. The first one is based on DFT and is commonly known as the supercell core-hole (SCH) method . The second approach is based on the many-body perturbation theory and requires solving the Bethe-Salpeter equation (BSE) . In VASP both approaches rely on the frozen core (FC) approximation, where the core electrons are kept frozen in the configuration for which the PAW potential was generated. Although, a shortcoming of the PAW approach implemented in VASP, it nevertheless has been shown to hold very well for deep core states mostly due to the weak coupling between the core state and the valence states ^[1][2].

Theoretical background

Supercell core-hole (SCH)

The SCH approach is explained in detail on the following theory page. In this approach, a fully self-consistent electronic minimization is carried out in the presence of the core hole, which is created by removing a core electron^[1][2]. This approach is also called final state approximation The treatment of the excited electron can vary depending on the approximation. Within the excited electron and core-hole method (XCH), the electron is placed in the lowest conduction band, which can lead to sizable self-interaction effects if the lowest conduction state is strongly localized. Self-interaction in the lowest conduction state causes a delocalization and an upward shift of the state. Alternatively, the excited electron can be placed into the negative background charge to ensure that the cell is neutral. This approach is called full core-hole (FCH) method and it shows a systematic improvement over XCH for systems like Li-halides, where the lowest conduction state is strongly localized. A detailed comparison of these methods can be found in Ref. ^[1][2]. When the core hole is explicitly introduced in one of the atoms, i.e., a core electron is removed, it is necessary to eliminate the effective interaction of the core hole with its image across the periodic boundary. That requires using a large supercell so that this interaction is negligible.

After the self-consistent electronic minimization is converged in the presence of the core hole, the dielectric function is calculated using Fermi's golden rule

${\displaystyle \varepsilon_{\alpha \alpha}^{(2)}(\omega)= \frac{4 \pi^2 e^2 \hbar^2}{\Omega \omega^2 m_e^2} \sum_{\text{core}, c, \mathbf{k}} 2 w_{\mathbf{k}} |\left\langle\psi_{c \mathbf{k}}\right| i \nabla_\alpha-\mathbf{k}_\alpha\left|\psi_{\text{core}}\right\rangle|^2\delta\left(\varepsilon_{c \mathbf{k}}-\varepsilon_{\text{core}}-\omega\right) }$

In the initial state approximation, the electronic minimization in the presence of the core hole is not carried out, thus the electron-hole interaction is completely neglected.

Bethe-Salpeter equation

Within the BSE approach, the interaction between the core hole and the excited electron is explicitly included via the ladder diagrams in the polarizability ^[2]

${\displaystyle \bar{L}(\omega)=L_0(\omega)+L_0(\omega)(\bar{v}-W)\bar{L}(\omega) }$

This equation is solved in the transition space as an eigenvalue problem, whose eigenvalues are the transitions including the excitonic effects and the eigenvectors are the excitonic states in the transition space. The BSE dielectric function is found via

${\displaystyle \varepsilon_{\alpha \alpha}^{(2)}(\omega)= \frac{4 \pi^2 e^2 \hbar^2}{\Omega \omega^2 m_e^2} \sum_\lambda\left|\sum_{\text{core},c, \mathbf{k}} A_{\text{core}, c \mathbf{k}}^\lambda \left\langle\psi_{c \mathbf{k}}\right| i \nabla_\alpha-\mathbf{k}_\alpha\left|\psi_{\text{core}}\right\rangle \right|^2 \delta\left(\varepsilon^\lambda-\omega\right) }$

The BSE calculation based on the ${\displaystyle GW}$ calculation quasiparticles is considered to be the state of the art for XAS and is overall more reliable and accurate than the SCH methods.

Comparing BSE and SCH

The scaling of the BSE+GW approach is ${\displaystyle N^4-N^5}$ with the system size, which makes it very expensive computationally for large cells. The SCH approach on the other hand scales as ${\displaystyle N^3}$ with the system size which makes it much cheaper for complex systems with large cells. Nevertheless, the BSE+GW approach provides more insight into the physics of the excitons and allows for analyzing the exciton eigenvectors and wavefunction. Furthermore, some results indicate that the valence state excitations have to be included in the XAS calculations as well to reproduce the experimental results, which is only possible within the BSE formalism .

SCH has been shown to provide overall an accurate approximation for the deep core excitations ^[1], but can be less accurate for some systems ^[3]. So for accurate spectra it is sometimes necessary to use the BSE+GW approach.

How to

Practical guides on how to perform XAS calculation in different approximations

• XAS via supercell core-hole method: XAS supercell core-hole calculations
• XAS via Bethe-Salpeter equation: XAS BSE calculations

Tutorial

• Tutorial for SCH XAS calculations.
• Tutorial for BSE XAS calculations.