Physical interpretation of transition density matrix

Question

I am studying natural transition orbitals (NTOs) and I am following the Martin's article [1]. I understand the mathematics, but I am a bit confused about what exactly is the transition density matrix defined for the electronic excitation of the ground state $\Psi_0$ to an excited state $\Psi_{ex}$ as follows:

$$ T_{ia} = \langle \Psi_{ex} | c_i^{\dagger} c_a | \Psi \rangle, $$

where $c, c^{\dagger}$ are destruction and creation operators. Here, it's assumed that we realized a SCF calculation and obtained a set of occupied and virtual orbitals $\psi_p$. The indices $i$ and $a$ represent occupied and virtual orbitals, respectively.

The article said that this is the relevant quantity associated with the electronic excitation $\Psi_0 \rightarrow \Psi_{ex}$, but I don't know what it means. Is the $\mathbf{T}$ matrix related to the probability of the excitation?

Reference

1. Martin, R. L. Natural Transition Orbitals. Chem. Phys. 2003, 118 (11), 4775–4777. DOI: 10.1063/1.1558471.

Answer 1

I like to see the transition density matrix as the expansion coefficients of an excited state into singly excited configurations. The expression $$ c_i^\dagger c_a|\Psi_0\rangle \equiv |\Psi^i_a\rangle $$ is basically a singly excited configuration. If we assume that $|\Psi_0\rangle $ is given a by single determinant with doubly occupied orbitals, then $|\Psi^i_a\rangle $ represents all single excited determinants when $a$ runs over occupied orbitals and $i$ over unoccupied orbitals. We can then expand an actual excited state $|\Psi_{ex}\rangle$ into the basis of singly excited configurations and the expansion coefficients are basically the elements of the transition density matrix.