BRIXS: Calculating Resonant Inelastic X-Ray Scattering Spectra from Bethe-Salpeter Equation Calculations

This code determines resonant inelastic x-ray scattering (RIXS) spectra for solids, starting from Bethe-Salpeter Equation (BSE) calculations in an all-electron many-body framework.

How to Cite BRIXS

If you use BRIXS, I ask that you cite our initial publication, as well as Christian Vorwerk's PhD thesis.

Theoretical Background

In RIXS, an x-ray photon with energy $\omega_1$ is absorbed, which leads to the excitation of a core electron to a conduction band. Coherently, an x-ray photon with energy $\omega_2$ is emitted as a valence electron fills the core hole. The final state of the scattering process contains a valence hole and an excited electron in a conduction band. The energy loss $\omega_2-\omega_1$ is transferred to the electronic system. In RIXS experiments, the double-differential cross section (DDCS) $\mathrm{d}^2 \sigma/\mathrm{d}\Omega_2 \mathrm{d}\omega_2$ is measured, which describes the rate of scattering for photons with final energy $[\omega_2,\omega_2+\mathrm{d}\omega_2]$ in a solid angle range $[\Omega_2,\Omega_2+\mathrm{d}\Omega_2]$.

Following the derivation presented in Phys. Rev. 2, 042003(R) (2020), the RIXS double-differential cross section (DDCS) is defined as

$$\frac{\mathrm{d}^2 \sigma}{\mathrm{d} \Omega_2 \mathrm{d}\omega_2}= \alpha^4 \left( \frac{\omega_2}{\omega_1} \right) \mathrm{Im} \sum_{\lambda_o} \frac{|t^{(3)}_{\lambda_o}(\omega_1)|^2}{(\omega_1-\omega_2)-E^{\lambda_o}+\mathrm{i}\eta},$$

where $E^{\lambda_o}$ are the valence excitation energies of the system, and $\alpha$ is the fine-structure constant. The DDCS is an explicit function of the energy loss $\omega_2-\omega_1$, while the oscillator strength $t^{(3)}_{\lambda_o}(\omega_1)$ depends solely on the excitation energy $\omega_1$. The oscillator strenght is defined as

$$t^{(3)}_{\lambda_o}(\omega_1) = \sum_{\lambda_c} \frac{t^{(2)}_{\lambda_o, \lambda_c}t^{(1)}_{\lambda_c}}{\omega_1-E^{\lambda_c}+\mathrm{i}\eta},$$

where $E^{\lambda_c}$ are the core excitation energies of the system. The core excitation oscillator strength $t^{(1)}_{\lambda_c}$ is defined as

$$t^{(1)}_{\lambda_c}=\sum_{c \mu \mathbf{k}} X_{c \mu \mathbf{k}, \lambda_c}\left[ \mathbf{e}_1 \cdot \mathbf{P}_{c \mu \mathbf{k}} \right],$$

where $X_{c \mu \mathbf{k}, \lambda_c}$ is the BSE eigenstate corresponding to the excitation energy $E^{\lambda_c}$, $\mathbf{e}_1$ is the polarization vector of the incoming photon, and $\mathbf{P}_{c \mu \mathbf{k}}=\langle c \mathbf{k} | \hat{\mathbf{p}} | \mu \mathbf{k} \rangle$ are the momentum matrix elements.

The excitation pathway $t^{(2)}_{\lambda_o,\lambda_c}$ is defined as\

$$t^{(2)}_{\lambda_o,\lambda_c}=\sum_{c v \mathbf{k}} \sum_{\mu} X_{cv \mathbf{k}, \lambda_o}\left[ \mathbf{e}_2^* \cdot \mathbf{P}_{\mu v \mathbf{k}} \right] \left[ X_{c \mu \mathbf{k}, \lambda_c} \right]^*$$

Code Usage

To perform calculations with BRIXS, you first need to calculate two BSE calculations to obtain the core excitation eigenstates $X_{c \mu \mathbf{k}, \lambda_c}$ and the valence excitation ones $X_{c v \mathbf{k}, \lambda_o}$. Both calculations have to be performed on the same $\mathbf{k}$-grid, but the number of empty states in the calculations can differ.

NOTE: Here, we are interested in the eigenstates of the polarizability. In a typical BSE calculation, one is only interested in the dielectric function. As such, the eigenvectors from these calculations are typically obtained from a reduced BSE Hamiltonian, using the adjusted Coulomb potential $\bar{v}$ [compare Eqs. 6.57 and 6.58 in Christian Vorwerk's PhD thesis and the further discussion there]. Make sure that you calculate the eigenstates of the actual BSE Hamiltonian when you create eigenstates for the RIXS spectra.

To use the BRIXS code, first perform a valence- and core-level BSE calculation with the exciting code. To calculate the matrix elements of the polarizability, add the following part to the input.xml

chibar="true"
chibar0comp="1"

The first attribute triggers the calculation of the full polarizability at vanishing momentum transfer, while the second one defines for which the BSE is solved. The additional dependence occurs since we calculate the analytical limit for the exchange interaction, i.e.

$$V_{c \mu \mathbf{k}, c' \mu' \mathbf{k}'}(\mathbf{q}=0)= V_{c \mu \mathbf{k}, c' \mu' \mathbf{k}'}(\mathbf{q}=0, \mathbf{G}=0) + \sum_{\mathbf{G}} v_{\mathbf{G}}(\mathbf{q})M^*_{c \mu \mathbf{k}}(\mathbf{G},\mathbf{q}=0) M_{c' \mu' \mathbf{k}'}(\mathbf{G},\mathbf{q}=0),$$

where the first term in the sum is needed to tame the divergence of the Coulomb potential as $\lim_{q \rightarrow 0} v_0(\mathbf{q}) \propto 1/\mathbf{q}^2$, since

$$\lim_{q \rightarrow 0} M_{c \mu \mathbf{k}}(\mathbf{q},\mathbf{G}=0)= \lim_{q \rightarrow 0} \langle c \mathbf{k} | \mathrm{e}^{-\mathrm{i}\mathrm{q}\mathbf{r}} | \mu \mathbf{k} \rangle= -\mathrm{i}\mathbf{q}\langle c \mathbf{k} |\mathbf{r}| \mu \mathbf{k} \rangle + \mathcal{O}(q^2)= -\mathrm{i}\mathbf{q}\frac{\langle c \mathbf{k}|\mathbf{p}| \mu \mathbf{k} \rangle}{\epsilon_{c \mathbf{k}}-\epsilon_{\mu \mathbf{k}}}+\mathcal{O}(q^2)$$

This yields

$$V_{c \mu \mathbf{k}, c' \mu' \mathbf{k}'}(\mathbf{q}=0, \mathbf{G}=0) \approx \frac{1}{V} \mathrm{e}_q \frac{\langle c \mathbf{k} |\mathbf{p}| \mu \mathbf{k} \rangle^* \langle c' \mathbf{k}' |\mathbf{p}| \mu' \mathbf{k}' \rangle}{\left(\epsilon_{c \mathbf{k}}-\epsilon_{\mu \mathbf{k}} \right)\left( \epsilon_{c' \mathbf{k}'}-\epsilon_{v' \mathbf{k}'} \right)}$$

As we can see, this matrix element depends on the direction $\mathbf{e}_q$ of the momentum transfer as it goes to zero. Therefore, the BSE calculation depends on the component of the dielectric tensor that one wants to calculate.

NOTE: When one calculates the dielectric function as described above, only the component of the dielectric tensor specified by chibar0comp will agree with the corresponding component obtained from a calculation with chibar="false". The other components will differ and will be incorrect.

Once the BSE calculations are performed, the HDF5 files optical_output.h5, core_output.h5, and pmat.h5 have to be gathered in a directory together with the input file input.cfg. Then, the execution of rixs-pathway creates the file data.h5. Once this file is present, the execution of rixs-oscstr creates the final output in the file rixs.h5. A more detailed tutorial is provided in the file TUTORIAL.md