Calculating excited states in solids

One very important aspect of practical applications and theoretical developments in solids is the accurate study of excited electronic states. We are interested in accurately treating systems like colour centers in solids and excited state potential energy surfaces in chemical reactions.

In our group we aim to do this in the framework of the Equation of Motion coupled cluster theory.

Similar to the usual coupled cluster exponential ansatz, i.e.,

$$\left | \mathrm{CC} \right \rangle = \exp \{\hat{T}\} \left | 0 \right \rangle$$

where $ \left | 0 \right \rangle $ is a single reference starting state such as Hartree-Fock, the Equation of Motion Ansatz also involves an operator $ \hat{R} ^{(i)} $ to create the $ i$-th excited state on top of the Coupled Cluster ground state

$$\left | i \right \rangle = \hat{R} ^{(i)} \left | \mathrm{CC} \right \rangle$$

The operator $ \hat{R} ^{(i)} $ is very similar to the $ \hat{T} $ cluster operator,

$$\hat{T} = t _{0} + t ^{a} _{i} \hat{c} ^{\dagger} _{a} \hat{c} _{i} + t ^{ab} _{ij} \hat{c} ^{\dagger} _{a} \hat{c} ^{\dagger} _{b} \hat{c} _{j} \hat{c} _{i} + \ldots$$ $$\hat{R} ^{(i)} = r _{0} + r ^{a} _{i} \hat{c} ^{\dagger} _{a} \hat{c} _{i} + r ^{ab} _{ij} \hat{c} ^{\dagger} _{a} \hat{c} ^{\dagger} _{b} \hat{c} _{j} \hat{c} _{i} + \ldots$$

with $ r ^{ab\ldots} _{ij\ldots}$ being the amplitudes. These amplitudes are obtained by diagonalizing the similarity transformed Hamiltonian $ \bar{H} $:

$$\bar{H} = \exp \left \{ -\hat{T} \right \} \hat{H} \exp \left \{ \hat{T} \right \}$$

Of course since $ \hat{T} $ is not an hermitian operator, also $ \bar{H} $ is not hermitian, which means that left and right eigenvectors must be obtained, which even further increases the computational cost of the problem.

Our task is to develop further this method and to propose more cost effective implementations to diagonalize $ \bar{H} $.