Intensities and rates in the spectral domain without eigenvectors

A computational prediction of spectrum for polyatomic molecules typically requires the need to diagonalize an N 2 matrix, a computationally demanding effort at large values of N. Although matrix diagonalization techniques, such as the Lanczos diagonalization, are accurate, it is computationally cumbersome for molecules of even moderate size. The technique discussed in this paper allows for calculating peaks, peak widths, and other spectral features with relatively small computational effort. Matrix fluctuation dissipation theorem is a technique that allows one to calculate the eigenvalues (spectral amplitude) with superficial knowledge of the eigenfunctions. These eigenfunctions correspond to states that are formed due to anharmonic coupling between bright states and dark states of the molecule. Introduction Spectroscopy is of great interest to the physics, chemistry, and biology community. Many spectroscopic techniques provide insight on quantum properties of systems ranging from materials to biomolecules. Much of spectroscopy involves the excitation and emission of electrons to and from energy levels, such as from electronic or vibrational states, corresponding to intramolecular interactions. Intuitively, if an electron is moved into an excited state, one would expect to see an indefinitely thin peak corresponding to the specific frequency of the excitation and/or emission. A typical spectrum typically is, nevertheless, a broad distribution of intensities. This broadening of the spectral intensities related to leaking of population density from an excited energy states to states in the neighboring vicinity during vibrational spectroscopy is called Intramolecular Vibrational Relaxation. Intramolecular Vibrational Relaxation (IVR) In polyatomic molecules, short LASER pulses can populate vibrational states of higher energy. This population can redistribute itself to other energy levels of similar energy, in the vicinity of the initial state, without any external influence of the surrounding or molecular collisions called Intramolecular Vibrational Relaxation. IVR is the fundamental process by which the vibrational relaxation of an activated intermediate leads to stabilization of the product so it does not go back to the reactant state, thus playing an important role in many chemical and biochemical reaction rates, from unimolecular decomposition and isomerization to protein folding, to explaining data in molecular fluorescence spectra. It is hence important, in the field of chemistry, to have a detailed knowledge of this process. To obtain a general estimation of the frequency corresponding to peaks within the spectral domain, one can base calculations from the harmonic oscillator model as energy states. The energy of these states can be calculated using the Harmonic Hamiltonian. This would result in indefinitely thin peaks corresponding to individual energy states of the molecule; peak broadening, therefore, can be explained by basing calculations from an anharmonic oscillator model. Here we define the terms dark states and bright states for use in the future in terms of the harmonic model. Dark and Bright States To enumerate the process underlying IVR we first describe the nature of vibrational states in Figure 1: The Harmonic Model vs. Anharmonic Potential (Morse Potential in this case) polyatomic molecules. The vibrations in polyatomic molecules can be described by oscillations of nuclei of the equilibrium configuration in a Harmonic Potential also called the Normal Modes. Under this assumption each atom in the molecule moves with the same frequency and phase for a given Normal Mode and hence the Hamiltonian of the molecule under the harmonic potential consist of a potential energy and kinetic energy both of which are square terms. The Hamiltonian with the explicit treatment of states as in the framework of a Harmonic Model is the called the Zero Order Hamiltonian. These eigenstates, |i>, are the normal modes of vibration of the polyatomic molecule. Among the states |i>, one state, |0>, carries all of the oscillator strength or population. This is called the Zero Order Bright State (ZOBS). Only a few states are bright states due to severe selection rules (Frank-Condon etc.). As one goes from lower to higher energy the density of energy levels increases exponentially; there are hence a large number of energy levels in the vicinity of the Bright States that are isoenergetic but are sparsely populated as compared to the Bright States. These states are then called the Zero Order Dark States (ZODS). These bright and dark states, |i> are eigenstates of the zero order Hamiltonian H 0 with energy E 0 i.