Time dependent self consistent field

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... 

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. 

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. 

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 ... 

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... 

Figure 10. Comparison of quantum-mechanical time-dependent self-consistent field (time-dependent Hartree) (dashed fine) and quantum path-integral (dots) calculations obtained for Model Va (upper panel) and Model Vb (lower panel), respectively. Shown is the time-dependent population probabihty P t) of the initially prepared diabatic electronic state.

Let us briefly mention some formal aspects of the above-introduced formalism, which have been discussed in detail by Blaizot and Marshalek [218]. First, it is noted that the both the Schwinger and the Holstein-Primakoff representations are not unitary transformations in the usual sense. Nevertheless, a transformation may be defined in terms of a formal mapping operator acting in the fermionic-bosonic product Hilbert space. Furthermore, the interrelation of the Schwinger representation and the Holstein-Primakoff representation has been investigated in the context of quantization of time-dependent self-consistent fields. It has been shown that the representations are related to each other by a nonunitary transformation. This lack of unitarity is a consequence of the nonexistence of a unitary polar decomposition of the creation and annihilation operators a and at [221] and the resulting difficulties in the definition of a proper phase operator in quantum optics [222]. 

Recently, the effects of static and dynamic structural fluctuations on the electron hole mobility in DNA were studied using a time-dependent self-consistent field method [33]. The motion of holes was coupled to fluctuations of two step parameters of a duplex, rise and twist (Fig. 1), namely the distances and the dihedral angles between base pairs, respectively. The hole mobility in an ideally ordered poly(G)-poly(C) duplex was found to be decreased by two orders of magnitude due to twisting of base pairs and static energy disorder. A hole mobility of 0.1 cm V s was predicted for a homogeneous system the mobility of natural duplexes is expected to be much lower [33]. In this context, one can mention several theoretical studies, based on band structure approaches, to estimate the electrical conductivity of DNA [85-87]. 

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,... 

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. 

Classical trajectory aitd classical time-dependent self-consistent field calculations of the dynamics of sequential dissociation processes of the type XI,(u)Y X+yu jY X+Y-M ju")... 

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

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

R.B. Gerber and R. Alimi, Quantum molecular dynamics by a perturbation-corrected time-dependent self-consistent-field method, Chem. Phys. Lett., 184 (1991) 69. 

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 ... 

Difficulties arise in the band structure treatment for quasilinear periodic chains because the scalar dipole interaction potential is neither periodic nor bounded. These difficulties are overcome in the approach presented in [115] by using the time-dependent vector potential, A, instead of the scalar potential. In that formulation the momentum operator p is replaced by tt =p + (e/c A while the corresponding quasi-momentum Ic becomes k = lc + (e/c)A. Then, a proper treatment of the time-dependence of k, leads to the time-dependent self-consistent field Hartree-Fock (TDHF) equation [115] ... 

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 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. 

---