# Evaluating the Impact of the Tamm–Dancoff Approximation on X-ray Spectrum Calculations

## TL;DR

This paper evaluates how the Tamm–Dancoff approximation affects X-ray absorption and emission spectra calculations, finding small discrepancies and proposing methods to improve accuracy.

## Contribution

The study introduces a core–valence separation scheme for X-ray emission spectra and clarifies the impact of the Tamm–Dancoff approximation on X-ray calculations.

## Key findings

- TDA leads to small discrepancies in excitation energies and intensities for X-ray spectra.
- A core–valence separation approach for XES yields good agreement with full-space calculations.
- Discretized continuum contributions complicate TDDFT and TDA calculations for XAS and XES.

## Abstract

The impact of the
Tamm–Dancoff approximation (TDA) for time-dependent
density functional theory (TDDFT) calculations of X-ray absorption
and X-ray emission spectra (XAS and XES) is investigated, showing
small discrepancies in the excitation energies and intensities. Through
explicit diagonalization of the TDDFT Hessian, XES was considered
by using full TDDFT with a core-hole reference state. This has previously
not been possible with most TDDFT implementations as a result of the
presence of negative eigenvalues. Furthermore, a core–valence
separation (CVS) scheme for XES is presented, in which only elements
including the core-hole are considered, resulting in a small Hessian
with the dimension of the number of remaining occupied orbitals of
the same spin as the core-hole (CH). The resulting spectra are in
surprisingly good agreement with the full-space counterpart, illustrating
the weak coupling between the valence–valence and valence–CH
transitions. Complications resulting from contributions from the discretized
continuum are discussed, which can occur for TDDFT calculations of
XAS and XES and for TDA calculations of XAS. In conclusion, we recommend
that TDA be used when calculating X-ray emission spectra, and either
CVS-TDA or CVS-TDDFT can be used for X-ray absorption spectra.