# Derivation of time-dependent transition probability for $2\mathrm{e}-2\mathrm{h}$ generation from $1\mathrm{e}-1\mathrm{h}$ state in the presence of external electromagnetic field

## TL;DR

This paper derives a comprehensive, diagrammatic-based expression for the time-dependent transition probability of 2e-2h states from 1e-1h states under electromagnetic influence, enabling more efficient computations in many-electron systems.

## Contribution

It presents a novel derivation of the time-dependent transition probability without approximations, factoring it into pre-computable components for improved efficiency.

## Key findings

- Derived a complete set of diagrams for transition probability
- Factored transition probability into time-independent and time-dependent parts
- Facilitates efficient computation of electronic state transitions

## Abstract

In this work, we investigate the effect of electromagnetic (EM) field on the generation of 2e-2h states from 1e-2h states. One of the fundamental ways by which electromagnetic (EM) waves interact with matter is by the generation of excited electronic states. The interaction of EM field with atoms and molecules is given by the field-dependent Hamiltonian. Excited states are intrinsically transient in nature because they are not stationary states of the field-dependent Hamiltonian. Consequently, the time-dependent dynamics of excited states depend strongly on the external electromagnetic field. Starting with the 1e-1h excitation in a general many-electron system, the system was propagated in time using time-dependent perturbation theory (TDPT). The expression for time-dependent transition probability of $(1\mathrm{e}-1\mathrm{h}) \rightarrow (2\mathrm{e}-2\mathrm{h})$ was evaluated for a given time $t$ up to second-order in TDPT using diagrammatic techniques. The derivation does not assume any a priori approximations to the electron-electron correlation operator and presents the derivation of a complete set of contributing diagrams associated with the full configuration interaction wave function. The result from this work show that the calculation of time-dependent transition probability can be factored into a time-independent and time-dependent components. This is a significant outcome for efficient computation of the time-dependent transition probability because it allows for pre-computation of time-independent components before the start of the calculations.