How can I formally justify the description of an electronic excitation that includes only the relevant orbitals?

Question

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?

Answer 1

The answer here depends on what you mean by "formal procedure" and "physically valuable".

The potential flaw in your approach is the assumption that

$$\Psi=\cos(\Omega_R)\sigma_\alpha + \sin(\Omega_R)\sigma^*_\alpha e^{−i\omega t}$$

is a valid description. Formally, the wave function of a two-electron system cannot be split into two independent one-electron wave functions. This is perhaps more clear if you write out the full determinants in the wave functions, which are $$\Psi_g=\begin{vmatrix}\sigma_\alpha(1)&\sigma_\beta(1)\\\sigma_\alpha(2)&\sigma_\beta(2)\end{vmatrix}$$ and (for a singlet excited state) $$\Psi_e=\begin{vmatrix}\sigma_\alpha(1)&\sigma^*_\beta(1)\\\sigma_\alpha(2)&\sigma^*_\beta(2)\end{vmatrix}$$

where (1) and (2) are the two electrons and $\alpha$ and $\beta$ are spin states.

That is, each wave function necessarily consists of contributions from each possible electron-orbital pairing rather than just assigning one electron to each orbital. (For a triplet excited state, there are actually two additional wave functions to account for the spin possibilities, but that's not necessary to consider for this discussion.)

That said, almost all computational approaches to dealing with such systems apply this invalid approach in order to make the mathematics tractable. The Hartree-Fock method is a very common starting point. By adding various terms to correct for the indistinguishability and interactions of the electrons, the results of these methods can approach the results we would expect to get from a formally correct description of the system (ie solving the Schrodinger equation for a single multi-electron wave function).

Furthermore, treating the total electron density of a molecule (ie the square of the multi-electron wave-function) as the sum of densities of multiple discrete orbitals is a standard pedagogical approach, and predicting the properties and reactions of molecules based on a subset of these discrete orbitals (valence or frontier orbitals usually) works quite well at a qualitative and semi-quantitative level. Whether this pedagaogical value counts as "physically valuable" depends on what you mean by that.

I'll conclude by invoking George Box's maxim that "All models are wrong. Some are useful." While your animations are not defensible on theoretical physics grounds, that does not mean that they are not reasonable and useful representations for pedagogical purposes.