I am writing a small program to animate electronic transitions.
To do this, I am animating a Rabi cycle for a single electron as it moves from an occupied into a virtual orbital.
The on-resonance excitation process during the interaction of a molecule with a coherent laser is modelled by the Rabi formula: $$\Psi = \cos(\Omega_R)\Psi_g + \sin(\Omega_R) \Psi_e e^{-i\omega t},$$
where $\Omega_R$ is the Rabi frequency and $\omega = (E_e - E_g)/\hbar$.
As a minimal example, I will consider the $\ce{H2}$ molecule. The first electronic excitation is $\sigma \rightarrow \sigma^*$.
Formally, the wave function of the ground state is: $\Psi_g = \det|\sigma_{\alpha}\sigma_{\beta}|$, and the excited state is: $\Psi_e = \det|\sigma^*_{\alpha}\sigma_{\beta}|$.
Intuitively, I think that the description of the excitation process can be understood by considering only the wave function of an isolated electron jumping from a $\sigma_\alpha$ into a $\sigma_{\alpha}^{*}$ orbital, rather than wave function as a whole. In this case, I express the transition state as follows:
$$\Psi = \cos(\Omega_R) \sigma_{\alpha} + \sin(\Omega_R) \sigma^*_{\alpha} e^{-i\omega t}$$
from which an animation of the changing electron density can be constructed by computing isosurfaces of $|\Psi(t)^{2}|$, and assigning the color depending on the phase of the underlying wavefunction at that point:
In a more complicated molecule, the same simplification might correspond to expressing the excitation process as follows:
$$\Psi = \cos(\Omega_R) \psi_{\text{HOMO}(\alpha)} + \sin(\Omega_R) \psi_{\text{LUMO}(\alpha)} e^{-i\omega t}$$
Which, for an s-tetrazine derivative leads to the following animation for the first excited state (side and top view):
My issue is that I am not 100% certain about whether there is a formal procedure that allows me make the claim that these animations provide me with a physically valuable description of what happens during an excitation. It is clear to me that these animations do allow us to understand the symmetry rules and the direction of the transition dipole moment. I am looking for some help in finding the formal way of justifying this approach. Could the "1 particle reduced density matrix" formalism help me here?
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