Spectral moment theoretical expressions

Theoretical models of correlation functions and line shapes have been proposed which satisfy the principle of detailed balance [35, 36, 41, 232]. These profiles, along with a number of extensions that were later added [69, 295, 47, 48], describe the known profiles well. Especially the BC and K0 functions, Eqs. 5.105 and 5.108, model multipole- and overlap-induced lines of the rototranslational bands closely. These three-parameter functions are simple analytical expressions that are readily computable, even on computers of small capacity (pocket calculators) the parameters can be computed from the lowest three spectral moments, see Chapter 5. 

A better approach actually exists. There is considerable empirical and theoretical evidence that the various spectral functions, g(v), of binary systems are indeed closely modeled by a combination of the BC and K0 profiles, Eqs. 5.105 and 5.108. These are functionals of reduced spectral moments, Mo, Mi/Mo, and M2/M0, which in the classical limit may be expressed in terms of reduced temperature, to the extent that interaction potentials are describable by reduced potentials. 

Theoretical attempts to deal with complexes of more than two atoms (molecules) are scarce. One notable exception is the intercollisional process [303, 304, 306], which models the existing correlations of subsequent collisions. Intercollisional effects are well known in collision-induced absorption, but in OILS not much experimental evidence seems to exist. Three-body spectral moment expressions have been obtained under the assumptions of pairwise interactions [198, 200, 208, 209, 212, 218, 340, 422] see also references in Part II. Multiple scattering will depolarize light and has been considered in several depolarization studies of simple fluids [273, 274, 290, 376]. 

The approach to the evaluation of vibrational spectra described above is based on classical simulations for which quantum corrections are possible. The incorporation of quantum effects directly in simulations of large molecular systems is one of the most challenging areas in theoretical chemistry today. The development of quantum simulation methods is particularly important in the area of molecular spectroscopy for which quantum effects can be important and where the goal is to use simulations to help understand the structural and dynamical origins of changes in spectral lineshapes with environmental variables such as the temperature. The direct evaluation of quantum time- correlation functions for anharmonic systems is extremely difficult. Our initial approach to the evaluation of finite temperature anharmonic effects on vibrational lineshapes is derived from the fact that the moments of the vibrational lineshape spectrum can be expressed as functions of expectation values of positional and momentum operators. These expectation values can be evaluated using extremely efficient quantum Monte-Carlo techniques. The main points are summarized below. 

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