A time correlation function theory for the fifth order Raman response function with applications to liquid CS2

Abstract

A new theory for the fifth order Raman response function, R(5)(t1,t2), is presented. Using this result, R(5)(t1,t2) is shown to have a classical limit given by a combination of time derivatives of the real and imaginary parts of a two time correlation function (TCF) of the polarizability. In contrast with one time correlation functions, no exact analytic relationship exists between the real and imaginary parts of the quantum mechanical TCF that would allow the classical limit to be written in terms of classical TCF’s. Writing the nonlinear response function in terms of classical TCF’s would allow R(5)(t1,t2) to be calculated with minimal computational effort, in contrast to existing (exact) classical formulations. However, a simple approximate relationship is shown to exist between the real and imaginary parts of the two time TCF for a harmonic system with nonlinear polarizability. In the spirit of quantum correction, this relationship is used to write the exact classical response function in terms of classical TCF’s. The resulting TCF expression is then calculated from (fully anharmonic) molecular dynamics calculations supplemented by a suitable spectroscopic (polarizability) model. The approximate expression is demonstrated to have correct limiting behaviors and leads to a two-dimensional spectrum for ambient carbon disulfide in excellent agreement with existing experimental and theoretical work. The proposed approach makes the calculation of fifth order response functions practical for a wide variety of chemically interesting systems.

How a solute-pump/solvent-probe spectroscopy can reveal structural dynamics: Polarizability response spectra as a two-dimensional solvation spectroscopy

“…[48][49][50][51][52] Their development retreats to the quantum mechanical analogue of Eq. (A1) (with timedependent quantum mechanical operators and commutators in the place of classical dynamical variables and Poisson brackets) expressed in terms of real and imaginary parts of quantum time correlation functions, builds in a relationship between those parts valid for harmonic systems, and then identifies the real part of the quantum time correlation function with its classical analogue.…”

“…Some of these have made use of generalized-Langevin-equation 40,41 and modecoupling 42-47 frameworks, but there has also been a novel formulation in which the classical results have been predicted based on approximations to the quantum mechanical analogue. [48][49][50][51][52] We propose yet another approach here. What turned out to be key for our purposes was the realization that for the kinds of problems of interest in solvation spectroscopy, the time scales T and t are usually going to be widely separated.…”

“…In their approximation, when A(t), B(t), and C(t) are all equal to δV(t), Eq. (A1) becomes 48,51 R DRSK (0, T , T + t) = − 1 2 β 2 ∂ 2 ∂T 2 − 2 ∂ 2 ∂T ∂t δV (0) δV (T ) δV (T + t) = β 2 δV (0) δV (T ) δV (T + t)…”

A classical iterative theory based on the Langevin equation for two-dimensional nonlinear terahertz spectroscopy

“…However, most of the currently existing methods are based on the calculation of multi-time correlation functions or Poison brackets [2][3][4][5][6][7][8][9][10]12], which are usually more complex than the iterative method presented here. We propose this classical iterative theory for 2DTS because the photon energy is much lower than the room temperature in the THz spectral region and the intra-and inter-molecular vibrational modes have classical nature in this region in most cases.…”

“…In 1993, Tanimura and Mukamel proposed the two-dimensional Raman spectroscopy (2DRS) [1] technique, which is suitable for investigating such dynamics in molecular liquids. It revealed information unavailable from linear spectroscopies and has triggered a considerable amount of theoretical [2][3][4][5][6][7][8][9][10][11][12] and experimental [13][14][15][16] investigations. However, as pointed out by Blank [17] and Cho et al [18], the fifth-order Raman signal is contaminated by the cascaded third-order signal so that it is difficult to obtain the pure fifth-order Raman signal.…”