Vibrationally resolved transient absorption between ground and core-excited states of molecules

Here we present our work on developing an algorithm for approximating the vibrational structure of core excited states in ionic molecules and motivate its utility by calculating the transient absorption spectra of selected molecules to investigate how vibrational structure manifests in transitions to these states. In order to compute the hessian and gradient of the ionic system for energetically low-lying and core-excited states, we compute a series of single point electronic structure calculations along displaced geometries to probe the potential energy surfaces. We generate the displaced geometries along the directions of the normal modes plus the 5 or 6 translations/rotations of the neutral system with variable step size corresponding to the curvature of that mode and utilize a combination of finite difference and least squares regression to approximate the first/second derivatives. The hessian and gradient are computed for the energetically low-lying and core-excited states, which then allows for the determination of the normal modes, vibrational frequencies and equilibrium geometry. Lastly, the Franck-Condon factors are computed between the normal modes of each state, allowing for the simulation of x-ray transient absorption following ionization. The transient absorption spectra is then computed for selected molecules to showcase phenomena arising from vibrational structure in these core-excited states. We benchmark this approach for transient absorption by comparison to the fourier transform of the time dependent dipole moment to ensure validity.