TDSCF

For the case of intramolecular energy transfer from excited vibrational states, a mixed quantum-classical treatment was given by Gerber et al. already in 1982 [101]. These authors used a time-dependent self-consistent field (TDSCF) approximation. In the classical limit of TDSCF averages over wave functions are replaced by averages over bundles of trajectories, each obtained by SCF methods. 

With the above definitions, there is no additional overall phase factor to be included in (27). Eqs. (24)-(27) are the CSP approximation.Like TDSCF, CSP is a separable approximation, using a time-dependent mean potential for each degree of freedom. However, the effective potentials in CSP... 

The accuracy of the CSP approximation is, as test calculations for model. systems show, typically very similar to that of the TDSCF. The reason for this is that for atomic scale masses, the classical mean potentials are very similar to the quantum mechanical ones. CSP may deviate significantly from TDSCF in cases where, e.g., the dynamics is strongly influenced by classically forbidden regions of phase space. However, for simple tunneling cases it seems not hard to fix CSP, by running the classical trajectories slightly above the barrier. In any case, for typical systems the classical estimate for the mean potential functions works extremely well. 

It is obvious that CSP depends, as does TDSCF, on the choice of coordinates. As pointed out in Sec. 2.2, numerical convenience often limits the choice of the coordinates. CSP may, however, offer practical prospects for the choice of physically optimal modes. The deviation of the true potential from CSP separability is given by ... 

Fig. 1. Comparison between the CID-CSP, CSP, TDSCF, and the numerically exact autocorrelation functions.

Inserting the separation ansatz, i.e., U , results in two nonlinearly coupled single particle Schrodinger equations, the so-called time dependent self-consistent field (TDSCF) equations ... 

Approximation Property We assume that the classical wavefunction 4> is an approximate 5-function, i.e., for all times t G [0, T] the probability density 4> t) = 4> q,t) is concentrated near a location q t) with width, i.e., position uncertainty, 6 t). Then, the quality of the TDSCF approximation can be characterized as follows ... 

Thus, TDSCF is the better an approximation of full QD the sharper located the probability density (p remains in the course of the evolution. 

Theorem 2 (Thm. 4.2. in [6]). Let 4> initialli have width <5(0) < 5 and let e be small enough. Moreover, assume that caustics do not appear in time interval [0,rj. Then, the semiclassical wavefunctions and-ipQc approximate the TDSCF wavefunction ip up to an error of order 5 -F e, i.e.,... 

Remark The statement of Prop. 6 is also valid for the g-expectation (g) = ( ,g ) of the TDSCF solution. Consequently, TDSCF fails near the crossing, a fact, which emphasizes that the reason for this failure is connected to the separation step. 

To summarize, the results presented for five representative examples of nonadiabatic dynamics demonstrate the ability of the MFT method to account for a qualitative description of the dynamics in case of processes involving two electronic states. The origin of the problems to describe the correct long-time relaxation dynamics as well as multi-state processes will be discussed in more detail in Section VI. Despite these problems, it is surprising how this simplest MQC method can describe complex nonadiabatic dynamics. Other related approximate methods such as the quantum-mechanical TDSCF approximation have been found to completely fail to account for the long-time behavior of the electronic dynamics (see Fig. 10). This is because the standard Hartree ansatz in the TDSCF approach neglects all correlations between the dynamical DoF, whereas the ensemble average performed in the MFT treatment accounts for the static correlation of the problem. 

Another, purely quantum mechanical approximation is the so-called time-dependent self-consistent field (TDSCF) method. For general reviews see Kerman and Koonin (1976), Goeke and Reinhard (1982), and Negele (1982). For applications to molecular systems see, for example, Gerber and Ratner (1988a,b). In the TDSCF method the wavepacket is separated according to... 

Several improvements of the TDSCF approach have been proposed in the recent literature (Kucar, Meyer, and Cederbaum 1987 Makri and Miller 1987 Meyer, Kucar, and Cederbaum 1988 Kotler, Nitzan, and Kosloff 1988 Meyer, Manthe, and Cederbaum 1990 Campos-Martinez and Coalson 1990 Waldeck, Campos-Martinez, and Coalson 1991). 

The TDSCF approximation is a good starting point for a mixed quantum mechanical/classical treatment. Let us assume that R is the classical and r the quantum mechanical mode. Then, the wavefunction r(r t) describing the vibration of the fragment molecule is a solution of the time-dependent Schrodinger equation... 

Makri, N. and Miller, W.H. (1987). Time-dependent self-consistent field (TDSCF) approximation for a reaction coordinate coupled to a harmonic bath Single and multiple configuration treatments, J. Chem. Phys. 87, 5781-5787. 

We will restrict the further considerations to the case, where only one product in the expansion of the total wave function is relevant. Instead of the MCTDSCF approximation the solution is approximated by a single product function wherein these functions are determined in a self consistent way (time dependent SCF approximation, TDSCF). The situation is similar to that where there are several electronic degrees of freedom for a molecule, but where it has been demonstrated that the a batic Bom-Oppenheimer approximation works substantially well for the description of most spectroscopic and other properties of molecules. 

It has been demonstrated in an earlier paper that one can derive a mixed quantum-classical propagation scheme based on the TDSCF approximation [47]. The TDSCF scheme has been extensively used in many studies [48,27,49] and shall be resumed briefly in order to introduce the notations. 

The time-dependent self consistent field (TDSCF) approximation... 

An approximate solution of Eqn. (1) can be obtained within the TDSCF scheme as... 

The time dependent quantum dynamical method based on the cissumption of separability is so called TDSCF approach (also called the Time-Dependent Hai tree method) J3]. The goal is to find using the time-dependent variational principle the best single particle separable rej)ies( ntation of the multidimensional tinic -dependent wavefimetion. Thus, wc star with cxi)rcssing the total wavefnnetion as f) where the multiplication runs over all modes. For... 

The separation in Eqs. (4.7-4.S) and the use of the averaged potential in Eq. (4.8) follow from the general time-dependent self-consistent field (TDSCF) approach (Gerber et al. 1982 Gerber 1987 Makri and Miller 1987b). In this theory, a full wavefunction for two sets of coordinates, X and T, is approximated via... 

An important feature is that the TDSCF solution conserves energy, provided the original TDSE conserves energy. For the molecule-surface system, this means that energy exchange between the gas molecule and the solid is described correctly on the average. 

Other interesting treatments of the solid motion have been developed in which the motion of the solid s atoms is described by quantum mechanics [Billing and Cacciatore 1985, 1986]. This has been done for a harmonic solid in the context of treatment of the motion of the molecule by classical mechanics and use of a TDSCF formalism to couple the quantum and classical subsystems. The impetus for this approach is the fact that, if the entire solid is treated as a set of coupled harmonic oscillators, the quantum solution can be evaluated directly in an operator formalism. Then, the effect of solid atom motion can be incorporated as an added force on the gas molecule. Another advantage is the ability to treat the harmonic degrees of freedom of the solid and the harmonic electron -hole pair excitations on the same footing. The simplicity of such harmonic degrees of freedom can also be incorporated into the previously defined path-integral formalism in a simple manner to yield influence functionals (Feynman and Hibbs 1965). 

The second part of the chapter (Section III) deals with the time-dependent self-consistent-field (TDSCF) method for studying intramolecular vibrational energy transfer in time. The focus is both on methodological aspects and on the application to models of van der Waals cluster systems, which exhibit non-RRKM type of behavior. Both Sections II and III review recent results. However, some of the examples and the theoretical aspects are presented here for the first time. 

The SCF, or mean-field, approximation does not include the effect of energy transfer processes between the modes. The Cl approach incorporates such effects in a time-independent framework, but as was noted in the previous section this method loses much of the simplicity and insight provided by the SCF model. The most natural extension of the SCF approximation that also describes energy transfer among the coupled modes in the system, and treats this effect by a mean-field approach, is the time-dependent self-consistent-field (TDSCF), or time-dependent mean-field, approximation. 

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