Semi-classical description of nuclear dynamics in x-ray emission of water

In this article we present a semi-classical approximation to the Kramers-Heisenberg formula for calculating x-ray emission (XES) spectra, including vibrational effects. We compare the method to the quantum Kramers-Heisenberg formula for a test system consisting of a model water dimer where the hydrogen-bond donor is core-ionized and obtain excellent agreement. In the semi-classical approach we average spectra from classical trajectories where the core-hole-induced dynamics is performed with initial conditions sampling the quantum zero-point position and momentum probability distributions in the O-H vibration. We find very similar time-evolution of the squared quantum wave packet compared to the probability distribution under classical dynamics until the proton interacts with the next water. We compare our semi-classical approach with other methods to compute the XES spectra of water that have been used in the past and conclude that our approach gives superior results while requiring the same computational effort.