Excitation energies at the SCF-levelΒΆ
In this tutorial we will look at the theory behind the calculation of excitation energies at the SCF-level, that is, either by time-dependent Hartree-Fock (TD-HF) or TD-DFT. The implementations in DIRAC are presented in [Bast2009].
Excitation energies \(\omega_m\) are found by solving the generalized eigenvalue problem
\[\left(E_0^{[2]}-\hbar\lambda S^{[2]}\right)X_m=0\]
where the matrices \(E_0^{[2]}\) and \(S^{[2]}\) are the electronic Hessian and generalized metric, respectively. The eigenvector have a paired structure
\[\left(\lambda_{+;m}=+\left|\omega_m\right|, X_{+;m} = \right)\]
\[\begin{split}\left(\begin{matrix} a & b \\
c & d \end{matrix}\right)\end{split}\]
\[\begin{split}a & = (x + y)^2 \\
& = x^2 + 2xy + y^2\end{split}\]