Algebraic approach to electronic spectroscopy and dynamics

Abstract

Lie algebra, Zassenhaus, and parameter differentiation techniques are utilized to break up the exponential of a bilinear Hamiltonian operator into a product of noncommuting exponential operators by the virtue of the theory of Wei and Norman [J. Math. Phys. 4, 575 (1963); Proc. Am. Math. Soc., 15, 327 (1964)]. There are about three different ways to find the Zassenhaus exponents, namely, binomial expansion, Suzuki formula, and q-exponential transformation. A fourth, and most reliable method, is provided. Since linearly displaced and distorted (curvature change upon excitation/emission) Hamiltonian and spin-boson Hamiltonian may be classified as bilinear Hamiltonians, the presented algebraic algorithm (exponential operator disentanglement exploiting six-dimensional Lie algebra case) should be useful in spin-boson problems. The linearly displaced and distorted Hamiltonian exponential is only treated here. While the spin-boson model is used here only as a demonstration of the idea, the herein approach is more general and powerful than the specific example treated. The optical linear dipole moment correlation function is algebraically derived using the above mentioned methods and coherent states. Coherent states are eigenvectors of the bosonic lowering operator a and not of the raising operator a(+). While exp(a(+)) translates coherent states, exp(a(+)a(+)) operation on coherent states has always been a challenge, as a(+) has no eigenvectors. Three approaches, and the results, of that operation are provided. Linear absorption spectra are derived, calculated, and discussed. The linear dipole moment correlation function for the pure quadratic coupling case is expressed in terms of Legendre polynomials to better show the even vibronic transitions in the absorption spectrum. Comparison of the present line shapes to those calculated by other methods is provided. Franck-Condon factors for both linear and quadratic couplings are exactly accounted for by the herein calculated linear absorption spectra. This new methodology should easily pave the way to calculating the four-point correlation function, F(tau(1),tau(2),tau(3),tau(4)), of which the optical nonlinear response function may be procured, as evaluating F(tau(1),tau(2),tau(3),tau(4)) is only evaluating the optical linear dipole moment correlation function iteratively over different time intervals, which should allow calculating various optical nonlinear temporal/spectral signals.

Zero‐phonon line profiles in Markovian and non‐Markovian schemes in high‐temperature systems: Applications to glassy water and ethanol

“…Electronic dephasing problem is of fundamental importance in spectroscopy and dynamics of condensed systems, and as such accurate treatment should be of paramount interest, especially as T falls in the intermediate range. In light of this, Toutounji [55] recently presented a detailed, rigorous algebraic approach to fully tackle this sort of problem in which the time evolution operator was disentangled into individual noncommuting exponential operators, using Wei and Norman theory as they form a Lie algebra set [56]. This methodology can readily be utilized to treat spin-boson Hamiltonians, from which nonlinear spectral signals may be calculated.…”

Analytic dynamics of the Morse oscillator derived by semiclassical closures

“…It is natural to consider the time evolution of these ladder operators, much like the time evolution of the raising and lowering operators of the harmonic oscillator [25][26][27] or spin system. 28,29 By considering equations of motion ͑EOM͒ for the ladder operators one can expect to extend stationary quantum mechanics to quantum dynamics and to obtain useful analytic expressions in time domain. Compared to the alternative representations of the Morse dynamics, 24,30,31 the SU͑2͒ operators provide a concise representation that limits the number of EOM.…”

Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach

“…Besides, unfamiliar expressions involving operator algebra which often lead to infinite series might crop up,while using coherent states, that is not easily summable to a finite value. 7,8 The main focus of this article is to present a new approach for evaluating time-dependent variables that are essential components to spectroscopy and quantum dynamics, e.g. linear/nonlinear dipole moment time/frequency correlation function, position correlation function, quantum solvation, wavepacket dynamics, scattering, etc.…”