A new methodology for dealing with time-dependent quantities in anharmonic molecules I: theory

Abstract

Morse oscillator coherent states tool has not witnessed much use or applications in quantum dynamics and non-equilibrium statistical mechanics due to the lack of rules of operations and hence the inability to deal with the Morse time evolution operator while still in the coherent states representation. This paper is about developing these needed operation rules of Morse coherent states and applying them to evaluate time correlation functions and thus be able to look at dynamics. This paper further provides two different approaches to dealing with the Morse oscillator propagator while still in coherent states representation, avoiding resorting to eigenstate representation or any other path integral techniques, such as initial value representation approach. Two approaches (one exact and another approximate) are developed to show how to handle anharmonic time evolution operator when acting on Morse coherent states, and calculate time correlation functions analytically. The Morse oscillator partition function is reproduced using the two approaches so as to test the correctness and applicability of the herein methodology. Additionally, the autocorrelation function is derived for a Morse wavepacket pumped on to the excited state, which is represented by a displaced Morse oscillator. A unitary transformation is utilized in order to move from the ground-state nuclear Hamiltonian to excited-state Hamiltonian. Evaluating this autocorrelation function using Morse coherent states representation without the alluded-to-unitary transformation in the future may become possible as more progress is made on this novel approach. Absorption spectra are calculated. It is noticed in those spectra that while the zero-phonon line (ZPL) does not seem to be measurably affected by anharmonicity and only the phonon-side band (PSB) is shifted and broadened in case of weak linear electron–phonon coupling [Huang–Rhys factor (S) is less than unity], both the ZPL and PSB are noticeably affected in case S > 1. It is further observed that a Voigt profile (a product of Lorenzian and Gaussian in the time domain) predominates the spectrum in case of an appreciable anharmonicity in the molecule. It is inferred that anharmonicity can be critically significant even at low temperature. It is finally concluded that quadratic exponential in the Morse oscillator energy eigenvalue that emerges presents itself as a Gaussian envelope in the autocorrelation function, leading to broadening, whereas the red shift is attributed only to the anharmonicity.