Anharmonicity, semiclassical

It may be shown [8] that both semiclassical [83,84], and full quantum mechanical approaches [7,32,33,58,87] of anharmonic coupling have in common the assumption that the angular frequency of the fast mode depends linearly on the slow mode coordinate and thus may be written... 

Now, return to Fig. 14. The right and left bottom damped lineshapes (dealing respectively with quantum direct damping and semiclassical indirect relaxation) are looking similar. That shows that for some reasonable anharmonic coupling parameters and at room temperature, an increase in the damping produces approximately the same broadened features in the RY semiclassical model of indirect relaxation and in the RR quantum model of direct relaxation. Thus, one may ask if the RR quantum model of direct relaxation could lead to the same kind of prediction as the RY semiclassical model of indirect relaxation. 

Wu, G. (1991), The Semiclassical Fixed Point Structure of Three Coupled Anharmonic Oscillators Under SU(3) Algebra with Iz = 0, Chem. Phys. Letts. 179, 29. 

In Section IE, a theoretical approach of the quantum indirect damping of the H-bond bridge was exposed within the strong anharmonic coupling theory, with the aid of the adiabatic approximation. In Section III, this theory was shown to reduce to the Marechal and Witkowski and Rosch and Ratner quantum approaches. In Section IV, this quantum theory of indirect damping was shown to admit as an approximate semiclassical limit the approach of Robertson and Yarwood. 

Purely quantum studies of the fully coupled anharmonic (and sometimes nonrigid) rovibrational state densities have also been obtained with a variety of methods. The simplest to implement are spectroscopic perturbation theory based studies [121, 122, 124]. Related semiclassical perturbation treatments have been described by Miller and coworkers [172-174]. Vibrational self-consistent field (SCF) plus configuration interaction (Cl) calculations [175, 176] provide another useful alternative, for which interesting illustrative results have been presented by Christoffel and Bowman for the H + CO2 reaction [123] and by Isaacson for the H2 + OH reaction [121]. The MULTIMODE code provides a general procedure for implementing such SCF-CI calculations [177]. Numerous studies of the state densities for triatomic molecules have also been presented. 

N. Makri and K. Thompson (1998) Semiclassical influence functionals for quantum systems in anharmonic environments. Chem. Phys. Lett. 291, pp. 101-109 ibid. (1999) Influence functionals with semiclassical propagators in combined forwardbackward time. J. Chem. Phys. 110, pp. 1343-1353 N. Makri (1999) The Linear Response Approximation and Its Lowest Order Corrections An Influence Functional Approach. J. Phys. Chem. B 103, pp. 2823-2829... 

K. Thompson and N. Makri (1998) Semiclassical influence functionals for quantum systems in anharmonic environments. Chem. Phys. Lett. 291, p. 101... 

Figure Al.2.7. Trajectory of two coupled stretches, obtained by integrating Hamilton s equations for motion on a PES for the two modes. The system has stable anharmonic S5mimetric and antisymmetric stretch modes, like those illustrated in Figure Al.2.6. In this trajectory, semiclassically there is one quantum of energy in each mode, so the trajectory corresponds to a combination state with quantum numbers n, n =, 1]. The woven pattern shows that the trajectory is regular rather than chaotic, corresponding to motion in phase space on an invariant toms.

To calculate n E-E, the non-torsional transitional modes have been treated as vibrations as well as rotations [26]. The former approach is invalid when the transitional mode s barrier for rotation is low, while the latter is inappropriate when the transitional mode is a vibration. Harmonic frequencies for the transitional modes may be obtained from a semi-empirical model [23] or by performing an appropriate normal mode analysis as a function of the reaction path for the reaction s potential energy surface [26]. Semiclassical quantization may be used to determine anharmonic energy levels for the transitional modes [27]. 

Herman M F, Kluk E and Davis H L 1986 Comparison of the propagation of semiclassical frozen Gaussian wave functions with quantum propagation for a highly excited anharmonic oscillator J. Chem. Phys. 84 326... 

W. H. Miller, R. Hernandez, N. C. Handy, D. Jayatilaka, and A. Willetts, Ab initio calculation of anharmonic constants for a transition state, with application to semiclassical transition state tunneling probabilities, Chem. Phys. Lett. 172 62 (1990). 

These SCTST expressions, in both the microcanonical (Eq. (27)) and canonical (Eq. (31)) forms, include coupling between all the degrees of freedom in a uniform manner. For example, even at the perturbative level, Eq. (23), there is an anharmonic coupling between modes of the activated complex x, , k and k < F -1) and between the reaction coordinate and modes of the activated complex (xkJ, < F — 1). This is not a dynamically exact theory, however, because these actions variables are in general only locally good. For energies too far above or below the barrier V0 they may fail to exist. This semiclassical theory is thus still a transition state theory (i.e., dynamical approximation). 

The preceding normal-mode/rigid-rotor sampling assumes the vibrational-rotational levels for the polyatomic reactant are well described by separable normal modes and separability between rotation and vibration. However, if anharmonicities and mode-mode and vibration-rotation couplings are important, it may become necessary to go beyond this approximation and use the Einstein-Brillouin-Keller (EBK) semiclassical quantization conditions [32]... 

EBK) semiclassical quantization condition given by Eq. (2.72). In contrast to the RKR method for diatomics, a direct method has not been developed for determining potential energy surfaces from experimental anharmonic vibrational/rotational energy levels of polyatomic molecules. Methods which have been used are based on an analytic representation of the potential energy surface (Bowman and Gazdy, 1991). At low levels of excitation the surface may be represented as a sum of quadratic, cubic, and quartic normal mode coordinates (or internal coordinate) terms, that is,... 

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