Time dependent self consistent field approximation

Gerber R B, Buch V and Ratner M A 1982 Simplified time-dependent self consistent field approximation for intramolecular dynamics Chem. Phys. Lett. 91 173... 

Peskin U and Steinberg M 1998 A temperature-dependent Schrodinger equation based on a time-dependent self consistent field approximation J. Chem. Phys. 109 704... 

Kosloff R and Hammerich A D 1991 Nonadiabatic reactive routes and the applicability of multi configuration time dependent self consistent field approximations Feredey Discuss. Chem. Soc. 91 239-47... 

Gerber, R.B., Buch, V., Ratner, M.A. Time-dependent self-consistent field approximation for intramolecular energy transfer. I. Formulation and application to dissociation of van der Waals molecules. J. Chem. Phys. 77 (1982) 3022-3030. 

For the H + H2 —H2 + H reaction, for example, the reactive flux requires time evolution of only -25 fs. The requirement of only short-time dynamics does indeed suggest the utility of a variety of approximations, such as the time-dependent self-consistent field approximation. 

The concept of tunneling has recently been invoked to explain the mechanism of photodissociation of matrix-isolated molecules. Previously, photodissociation was customarily accounted for by the fact that transla-tionally hot photofragments escape from the cage and stabilize in separate matrix sites, thereby avoiding recombination. Using the time-dependent self-consistent field approximation for molecular dynamics simulations,... 

The multi configuration time dependent self consistent field approximation (MCTDSCF)... 

R.B. Gerber, V. Buch and M.A. Ratner, Time-dependent self-consistent field approximation for intramolecular energy transfer. I. Formulation and application to dissociation of van der Waals molecules, J. Chem. Phys., 77 (1982), 3022 M.A. Ratner and R.B. Gerber, Excited vibrational states of polyatomic molcecules the semiclassical self-consistent field approach, J. Phys. Chem., 90 (1986) 20 R.B. Gerber and M.A. Ratner, Mean-field models for molecular states and dynamics new developments, J. Phys. Chem., 92 (1988) 3252 ... 

Much more enigmatically, the first-order time-dependent self-consistent field approximation is also widely called the Random Phase Approximation (RPA). This terminology is entrenched and so, although the name time-dependent self-consistent perturbation theory is more descriptive and preferable to both time-dependent Hartree-Fock and RPA all are used more or less interchangeably. The evolution of the concept and phrase random phase approximation is sketched in Appendix 27.A to this chapter. 

Kotler Z, Nitzan A, Kosloff R (1988) Multiconfiguration time-dependent self-consistent field approximation for curve crossing in presence of abath. A Fast Fourier Transform study. Chem Phys Lett 153 483... 

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. 

The goal of this chapter is twofold. First we wish to critically compare—from both a conceptional and a practical point of view—various classical and mixed quantum-classical strategies to describe non-Born-Oppenheimer dynamics. To this end. Section II introduces five multidimensional model problems, each representing a specific challenge for a classical description. Allowing for exact quantum-mechanical reference calculations, aU models have been used as benchmark problems to study approximate descriptions. In what follows, Section III describes in some detail the mean-field trajectory method and also discusses its connection to time-dependent self-consistent-field schemes. The surface-hopping method is considered in Section IV, which discusses various motivations of the ansatz as well as several variants of the implementation. Section V gives a brief account on the quantum-classical Liouville description and considers the possibility of an exact stochastic realization of its equation of motion. 

The classical-path approximation introduced above is common to most MQC formulations and describes the reaction of the quantum DoF to the dynamics of the classical DoF. The back-reaction of the quantum DoF onto the dynamics of the classical DoF, on the other hand, may be described in different ways. In the mean-field trajectory (MFT) method (which is sometimes also called Ehrenfest model, self-consistent classical-path method, or semiclassical time-dependent self-consistent-field method) considered in this section, the classical force F = pj acting on the nuclear DoF xj is given as an average over the quantum DoF... 

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... 

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. 

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

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... 

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. 

R. Alimi, R. B. Gerber, A. D. Hammerich, R. Kosloff, and M. A. Ratner,/. Chem. Phys., 93, 6484 (1990). Validity of Time-Dependent Self-Consistent-Field (TDSCF) Approximations for Unimolecular Dynamics A Test for Photodissociation of the Xe-HI Cluster. 

AIMD = ab initio molecular dynamics B-LYP = Becke-Lee-Yang-Parr CCSD = coupled cluster single double excitations CVC = core-valence correlation ECP = effective core potential DF = density functional GDA = gradient corrected density approximation MCLR = multiconfigurational linear response MP2 = M0ller-Plesset second-order (MRD)CI = multi-reference double-excitation configuration interaction RPA = random phase approximation TD-MCSCF = time-dependent multiconfigurational self-consistent field TD-SCF = time-dependent self-consistent field. 

The MFT equation of motion (25) can be derived in many ways, including the WKB approximation, the eikonal method, a (semi)classical time-dependent self-consistent field ansatz, density-matrix approaches, and the classical limit of algebraic quantization. Depending on the specific approach used, slightly different MFT schemes may result. For example, the classical force can be described either by the average of the quantum force as in Eq. (25) or by the derivative of the average quantum potential. 

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