Introduction

In these calculations, we wish to study the time evolution of nuclear motion propagating on different potential energy surfaces (PES), and the conditions  necessary for maintaining electronic coherence in the excited states of Thiophene. In particular, we wish to understand how exciting the electronic wave function to a superposition of two excited states affect the evolution of the nuclear wave function propagating on these pairs of PES, and how parallel these PES have to be in order for the nuclear motion to be similar for both states and for the electronic states to maintain its coherence.

The following calculations were performed for pairs of states (3, 8) and (3, 9), of normal modes 3, 7, 11, 12, 15, 16, 19, and 21 of thiophene. The remaining normal modes were excluded from the analysis due to negligible shifts in the location of the minima of the potential energy curves between the ground state and the excited states, meaning that vibrational motion would not be excited in these modes.

Extrapolation of Potentials

The potential energy curves along the various normal mode coordinates had an insufficient data range to contain our wave packet. Therefore, the curves were extrapolated by fitting them to a quadratic function. The following image carousels display the fittings to the potential energy curves for all modes of each pair of states (left for states 3 and 8, right for modes 3 and 9). The minimum energy was also subtracted from all potential energy curves since we are only interested in the relative phase advances at each coordinate, rather than an absolute phase difference (graphs are displayed with a vertical shift for visibility).

Determining the Ground State Wave Function

Since we are making the approximation that the states for the normal modes is roughly harmonic near their minima, we set the wave function for the ground state in each normal mode to be the ground state for the harmonic oscillator:

where m is the reduced mass for the nth normal mode, and w can be found with the quadratic fit of the ground state. The following figure shows the ground state for a couple of the modes.

Time Evolution of Nuclear Wave Functions

The split operator technique was used to propagate the initial wave function on these pairs of PES in time. The coherence is quantified by: