# Physical interpretation of transition density matrix

I am studying natural transition orbitals (NTOs) and I am following the Martin's article . 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.

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.