Supercell core-hole theory

In the Super-cell core-hole method (SCH) a chosen core electron is removed from the core leaving behind a positive charge. Since one wants to simulate the excitation of this electron into energetically higher lying states, one electron is added to valence/conduction bands resembling the final state of the excitation process (this is also referred to as the final state approximation). With this setup a fully self-consistent electronic minimization is carried out. The core hole is perfectly screened by the other electrons in metals so there should be no difference between core-hole calculations and regular calculations. In semiconductors and insulators, this screening is very weak and a very strong attraction between the electrons and hole occurs, which results in not only a lowering of the excited states compared to the valence states, but also a very strong change of the valence/conduction band structure. Hence the relaxation of the valence/conduction electrons is the main effect in core-hole calculations. Fortunately the relaxation of the core states in core-hole calculations is negligible. This makes the implementation into a PAW framework smooth, since no on-the fly recalculation of PAW potentials is needed in every step of the electronic self-consistent cycle.

Very important is that a sufficiently large super cell is used to minimize the interaction of neighboring core holes within periodic boundary conditions. Hence the computational expense within this methods comes mainly from the use of large super cells. Nevertheless this method is usually still computationally cheaper than the BSE method for core electrons.

Excited electron treatment

Besides adding the excited core electron to the valence/conduction bands it can be also placed into the backround as a homogeneous negative background charge. If the electron is added to the valence/conduction bands the method is often referred to as excited electron and core-hole (XCH) method^[1]. When placing the electron in the background the method is often referred as full core-hole (FCH) method^[2]. The two methods are compared in great detail in Ref.^[3]. There XCH and FCH calculations were compared to BSE+GW calulations, which are considered overall as more accurate. For some materials with not that strong localization of the excited electron (e.g. diamond and hexagonal-BN) both the XCH and FCH give very good results compared to GW+BSE. For materials with strong localization (such as e.g. Li-Halides) XCH fails to reproduce the GW+BSE results, while FCH give a very good approximation to GW+BSE. This is attributed to self-interaction errors of semilocal DFT functionals. The self-interaction error has two effects. First, it shifts the lowest conduction bands up. Second, it delocalizes the bands. If the excited core electronis put into the lowest conduction band, it fully experiences these effects. If it is put into the background it is not exposed to this effect. The self-interaction errors are unphysical and

Dielectric function used in the SCH method

Since the wave length of the electromagnetic waves in absorption spectroscopy is usually much larger than the characteristic momentum in solids, we start from the transversal expression for the imaginary part of the dielectric function in the long wavelength limit ( $\mathbf{q}=0$ ) which is directly proportional to the absorption spectrum

${\displaystyle \epsilon_{\alpha \beta}^{(2)} (\omega,\mathbf{q}=0) = \frac{4 \pi^{2} e^{2} \hbar^{4}}{\Omega \omega^{2} m_{e}^{2}} \sum\limits_{c,v,\mathbf{k}} 2w_{\mathbf{k}} \delta(\varepsilon_{c\mathbf{k}}-\varepsilon_{v\mathbf{k}}-\omega) \times M_{\alpha}^{v\rightarrow c} {M_{\beta}^{v\rightarrow c}}^{*} }$

where $M$ and $\varepsilon$ denote momentum matrix elements and orbital energies. Here we consider excitations only between valence ( $v$ ) and conduction ( $c$ ) bands. The components of the dielectric tensor are indexed by the Cartesian indices $\alpha$ and $\beta$ . $\Omega$ , $e$ and $m_{e}$ denote the unit cell volume, electron charge and mass of the electron, respectively. In the PAW method the all-electron orbitals $|\psi_{n\bold{k}}\rangle$ are given by a linear transformation of the pseudo orbitals $|\tilde{\psi}_{n\bold{k}}\rangle$ :

${\displaystyle |\psi_{n\bold{k}}\rangle = |\tilde{\psi}_{n\bold{k}}\rangle + \sum\limits_{i} (|\phi_{i}\rangle - |\tilde{\phi}_{i}\rangle) \langle \tilde{p}_{i} |\tilde{\psi}_{n\bold{k}}\rangle. }$

The pseudo orbitals depend on the band index $n$ and crystal momentum $\bold{k}$ . $|\phi_{i}\rangle$ , $|\tilde{\phi}_{i}\rangle$ and $|\tilde{p}_{i}\rangle$ are all-electron partial waves, pseudo partial waves and the projectors, respectively. The index $i$ is a shorthand for the atomic site and other indices enumerating these quantities at each site (such as angular and magnetic quantum numbers). In the PAW formalism, the matrix elements are given by

${\displaystyle M_{\alpha}^{v\rightarrow c} = \langle \psi_{c \mathbf{k}} |i \nabla_{\alpha} - \mathbf{k}_{\alpha}| \psi_{v\mathbf{k}} \rangle = \langle \tilde{\psi}_{c\mathbf{k}}| i \nabla_{\alpha}-\mathbf{k}_{\alpha}|\tilde{\psi}_{v\mathbf{k}} \rangle + \sum\limits_{ij} \langle \tilde{\psi}_{c\mathbf{k}} | \tilde{p}_{i} \rangle \langle \tilde{p}_{j} | \tilde{\psi}_{v\mathbf{k}} \rangle i \left(\langle \phi_{i}| \nabla_{\alpha}| \phi_{j} \rangle - \langle \tilde{\phi}_{i}| \nabla_{\alpha}| \tilde{\phi}_{j} \rangle \right). }$

where the one-center terms are calculated within the PAW sphere for each atom.

Since in X-ray absorption spectroscopy, one only considers one core hole at a single site, we can from now on restrict the equations to a single site. The index $i$ then only enumerates the main quantum number, the angular and the magnetic quantum numbers. As usual in the PAW method, using the completeness relation, $\sum_{i} |\tilde{p}_{i}\rangle \langle \tilde{\phi}_{i}| = 1$ , the first and third term in the previous equation cancel each other leading to the following simplified matrix elements ${\displaystyle M^{\mathrm{core}\rightarrow c\mathbf{k}}_{\alpha}=\sum\limits_{i}\langle \tilde{\psi}_{c\mathbf{k}} | \tilde{p}_{i} \rangle \langle\phi_{i}| \nabla_{\alpha}| \phi_{\mathrm{core}}\rangle. }$

Also the summation over bands can be limited to the conduction bands $c$

${\displaystyle \epsilon_{\alpha \beta}^{(2)} (\omega) = \frac{4 \pi^{2} e^{2} \hbar^{4}}{\Omega \omega^{2} m_{e}^{2}} \sum\limits_{c,\mathbf{k}} 2w_{\mathbf{k}}\delta(\varepsilon_{c\mathbf{k}}-\varepsilon_{\mathrm{core}}-\omega) \times M^{\mathrm{core}\rightarrow c\mathbf{k}}_\alpha {M^{\mathrm{core}\rightarrow c\mathbf{k}}_\beta}^*. }$

References

[hetenyi:jcp:2004-1] B. Hetényi, F. De Angelis, P. Giannozzi and R. Car, Calculation of near-edge x-ray-absorption fine structure at finite temperatures: spectral signatures of hydrogen bond breaking in liquid water , J. Chem. Phys. 120, 8632 (2004).

[Prendergasst:prl:2006-2] D. Prendergasst and G. Galli, X-Ray Absorption Spectra of Water from First Principles Calculations, Phys. Rev. Lett. 96, 215502 (2006).

[unzog:prb:2022-3] M. Unzog, A. Tal, G. Kresse, X-ray absorption using the projector augmented-wave method and the Bethe-Salpeter equation, Phys. Rev. B 106, 155133 (2022).

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