Semiclassical approaches

The present paper is organized as follows In a first step, the derivation of QCMD and related models is reviewed in the framework of the semiclassical approach, 2. This approach, however, does not reveal the close connection between the QCMD and BO models. For establishing this connection, the BO model is shown to be the adiabatic limit of both, QD and QCMD, 3. Since the BO model is well-known to fail at energy level crossings, we have to discuss the influence of such crossings on QCMD-like models, too. This is done by the means of a relatively simple test system for a specific type of such a crossing where non-adiabatic excitations take place, 4. Here, all models so far discussed fail. Finally, we suggest a modification of the QCMD system to overcome this failure. 

The semiclassical approach to QCMD, as introduced in [10], derives the QCMD equations within two steps. First, a separation step makes a tensor ansatz for the full wavefunction separating the coordinates x and q ... 

Fig. 2. The BO model is the adiabatic limit of full QD if energy level crossings do not appear. QCMD is connected to QD by the semiclassical approach if no caustics are present. Its adiabatic limit is again the BO solution, this time if the Hamiltonian H is smoothly diagonalizable. Thus, QCMD may be justified indirectly by the adiabatic limit excluding energy level crossings and other discontinuities of the spectral decomposition.

On the other hand, in the semiclassical approaches [83-85,92-95], one uses Eqs. (126) and (127) and assumes that the fast mode obeys quantum mechanics whereas the slow one obeys classical mechanics. 

The minimization of this functional, which includes second order gradient corrections leads to the relativistic analogous of the Thomas-Fermi-Dirac-Weizsacker model and constitutes the state of the art in relativistic semiclassical approaches for many-electron systems. 

We want to mention here that the application of the near-nuclear corrections could have been performed with the complete relativistic functional, and we have utilized the semi-relativistic expressions just for simplicity and for testing them. For not large Z, the remaining errors above mentioned should be addressed to limitations of the semiclassical approach and of the procedure utilized for the near-nuclear corrections, rather than to the truncation of the expansion in powers of... 

Error bars of the experimental data were not specified. However, Hunt provided measurements at two different ortho- to para-H2 concentration ratios, 3 1 (solid squares and dots) and 1 1 (open squares and circles). According to the theory developed above, variation of this ratio should not affect the results. While symmetry considerations of the interacting H2 molecules are important at low temperatures (T < 40 K), the semiclassical approach does not distinguish para- and ortho-H2 we think that the differences of the data taken at different ortho-para ratios may, in essence, reflect the uncertainties of the measurement. We note that earlier works by Chisholm and Welsh [121] and by Hare and Welsh [175] gave values... 

Spin-spin coupling is observed in the NMR spectra of both solids and liquids, but its origin is quite different for the two phases. Solids will be considered first, using a qualitative, semiclassical approach, rather than the correct quantum-mechanical treatment. 

The key feature to making a semiclassical approach practical is to avoid having to deal explicitly with the double-ended boundary conditions in Eq. (3.3) [16-20]. (The initial condition x(X, p( 0) = X is obviously easy to deal with.) To do this, one uses the standard coordinate space representation of Eq. (2.5),... 

As a first test [22] of the semiclassical approach described above we have computed the transmission probability through the lie kart potential barrier,... 

This property may not be possessed by many other approximate methods based on, e.g., mean field or semiclassical approaches. Also, in low dimensional systems, the above property is not true for CMD, so to apply CMD to such systems is not consistent with spirit of the method (though perhaps still useful for testing purposes). 

L. Demeio and B. Shizgal, /. Chem. Phys., 98, 5713 (1993). Time Dependent Nucleation. II. A Semiclassical Approach. 

On the contrary, the semiclassical approach in the problem of the optical absorption is restricted to a great extent and the adequate description of the phonon-assisted optical bands with a complicated structure caused by the dynamic JTE cannot be done in the framework of this approach [13]. An expressive example is represented by the two-humped absorption band of A —> E <8> e transition. The dip of absorption curve for A —> E <8> e transition to zero has no physical meaning because of the invalidity of the semiclassical approximation for this spectral range due to essentially quantum nature of the density of the vibronic states in the conical intersection of the adiabatic surface. This result is peculiar for the resonance (optical) phenomena in JT systems full discussion of the condition of the applicability of the adiabatic approximation is given in Ref. [13]. 

Our group has coupled the MST (PCM) method to Metropolis Monte Carlo sampling algorithms (MC-MST [78]). Within this approach cavitation and van der Waals terms are computed as in normal MST, while a semiclassical approach [79, 80] is used to compute the electrostatic component of solvation (see Equation (4.37)). Solute-solute energy terms are computed using a classical force field and Metropolis is then applied to the effective energy shown in Equation (4.38). 

Equation (153) is the semiclassical limit of the quantum approach of indirect damping. Now, the question may arise as to how Eq. (153) may be viewed from the classical theory of relaxation in order to make a connection with the semiclassical approach of Robertson and Yarwood, which used the classical theory of Brownian motion. 

As it appears, the classical spectral density (174) is very similar to the semiclassical approached (144) obtained above and it is equivalent to that given by Eq. (151). 

This leads to the starting equation of Robertson and Yarwood in their semiclassical approach of indirect damping ... 

Within the semiclassical approach, each wavepacket is propagated by running a large number of classical trajectories. In order to calculate the correlation function, a double summation with respect to these large number of trajectories should be performed at each time step, which is, unfortunately, also very demanding computationally. 

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