Introduction

In linear response TD-DFTB, the transition energies and oscillator strengths are obtained by solving the so-called Casida equation:

\[\boldsymbol{\Omega} \mathbf{F}_I = \omega_{I}^2 \mathbf{F}_I\]

\(\boldsymbol{\Omega}\) is called the response matrix and its elements depend on the occupied and virtual Kohn-Sham orbitals (\(\phi_{i\sigma}\) and \(\phi_{a\sigma}\), respectively) and their energy difference, \(\omega_{ia\sigma} = \epsilon_{a\sigma} - \epsilon_{i\sigma}\):

\[\Omega_{ia\sigma,jb\tau} = \delta_{ij} \delta_{ab} \delta_{\sigma \tau} \omega_{jb\tau}^2 + 2 \sqrt{\omega_{ia\sigma} \omega_{jb\tau}} K_{ia\sigma, jb\tau}\]

The matrix \(\mathbf{K}\) is obtained as the first derivative of the DFTB Hamiltonian with respect to the ground-state electron density matrix.

The eigenvalues of this problem are the square of the excitation energies, while from the eigenvectors, \(\mathbf{F}_I\), one can derive excited state properties such as the excited state total spin and the oscillator strengths of the transitions. Specifically, the latter property can be obtained using:

\[f^I = \frac{2}{3} \sum_{k=1}^3 \left| \sum_{ia\sigma} d^k_{ia\sigma} \sqrt{ \omega_{ia\sigma}} F_{ia\sigma}^I \right|^2\]

where \(d^k_{ia\sigma}\) are the k-th component of the elements of the dipole matrix.

For close-shell systems, it is possible to unitary-transform the Casida equation into two independent eigenvalue problems for singlet and triplet transitions. This reduces the dimension of the equations and allows us to study excited states with different multiplicities, independently.

It is important to mention that due to the minimal basis set employed within DFTB, only valence excited states can be computed with this formalism.