Time dependent mean field approximation

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. 

The result, Eq. (2.40) is known as the time-dependent mean field or time-dependent Hartree approximation. In this approximation each system is moving in the average field of the other system. 

Phonons and librons in herringbone and pinwheel commensurate and incommensurate N2 monolayers on graphite were investigated based on quantum-mechanical mean-field and time-dependent Hartree methods for the ground state [19]. The latter method includes on a systematic basis rotation-translation coupling which is neglected within the mean-field approximation, and it is able to treat motions with larger (but still finite) amplitudes around... 

Chemical oscillators are described on the basis of nonlinear dynamics, in that the underlying kinetic equations under steady-state conditions are nonlinear. If one assumes that the spatial distribution of the reaction species is uniform, then these variables will only depend on time, and mathematical description in the mean field approximation for the concentration variables c,- is achieved by a set of coupled (nonlinear) ordinary differential equations (ODEs). This will be the approach applied in this chapter. 

The main features of SF, such as excitation intensity dependence, emission pulse shortening, and time delay, can be described within a simplified semiclassical approach, which uses Maxwell-Bloch equations while neglecting the dipole-dipole interaction [113,114]. It was shown by Bonifacio and Lugatio [113] that in a mean field approximation the system of noninteracting emitters is described by the damped pendulum equations with two driving terms, as given below ... 

The theoretical framework for all regimes is the (time-dependent) local-density approximation (TDLDA) which is much discussed also in Chapter 1 of this book. We thus will present here only a short discussion of the essential ingredients and compare it with the analog mean-field models in nuclear physics. This is done as a starter in the next section. 

In fact, the PBE approximation is a mean field approximation valid for dilute systems. Moreover, this approximation does not account for internal structure and differences in the spatial configuration of clusters. For the known functional dependence of km,u time evolution of the cluster population (<) can be calculated... 

To describe the functioning of the lEMs, theory from the field of charged membranes must be adapted for MCDI to describe the voltage-current relationship and the degree of transport of the colons. This implies that (in contrast to most membrane processes) the theory must be made dynamic (time dependent) because it has to include the fact that across the membrane the salt concentrations on either side of the membrane can be very different, and change in time. This means that approximate, phenomenological approaches based on (constant values for) transport (or transference) numbers or permselectivities are inappropriate, and that instead a microscopic theory must be used. An appropriate theory includes as input parameters the membrane ion diffusion coefficient and a membrane charge density X. 

Now that the MM induced dipoles [ps] are influenced by the QM electric field, while the wavefiinction vP in the QM region depends on j, an iterative procedure must be used in the determination of the MM induced dipoles and QM wavefunction to ensure the convergence of the total energy of the system. It may be noted that the use of equation (14) implies a mean field approximation. This procedure leads to a significant increase in the computational time. As a result, there has been only a limited number of applications reported in the literature, which include explicit treatment of the MM polarization in hybrid QM/MM calculations. ... 

Fluctuations dominate for T > For typical values fiq (350-F500) MeV and for Tc > (50 A- 70) MeV in the weak coupling limit from (26), (22) we estimate Tq< (0.6 A- 0.8)TC. If we took into account the suppression factor / of the mean field term oc e A /T, a decrease of the mass m due to the fluctuation contribution (cf. (11)), and the pseudo-Goldstone contribution (25), we would get still smaller value of T < (< 0.5TC). We see that fluctuations start to contribute at temperatures when one can still use approximate expressions (22), (20) valid in the low temperature limit. Thus the time (frequency) dependence of the fluctuating fields is important in case of CSC. 

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... 

Abstract. In this chapter we discuss approaches to solving quantum dynamics in the condensed phase based on the quantum-classical Liouville method. Several representations of the quantum-classical Liouville equation (QCLE) of motion have been investigated and subsequently simulated. We discuss the benefits and limitations of these approaches. By making further approximations to the QCLE, we show that standard approaches to this problem, i.e., mean-field and surface-hopping methods, can be derived. The computation of transport coefficients, such as chemical rate constants, represent an important class of problems where the QCL method is applicable. We present a general quantum-classical expression for a time-dependent transport coefficient which incorporates the full system s initial quantum equilibrium structure. As an example of the formalism, the computation of a reaction rate coefficient for a simple reactive model is presented. These results are compared to illuminate the similarities and differences between various approaches discussed in this chapter. 

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 ... 

The shortcoming of the mean field method is that it admits no correlation between the motions of the individual particles. This correlation can be introduced by means of the random phase approximation (RPA) or time-dependent Hartree (TDH) method. In order to formulate this method, we introduce excitation operators (Ep) which replace f) p by when applied to the mean field ground state of the crystal when applied to any other state, they yield zero. Then, we write the Hamiltonian as a quadratic form in the excitation operators (Ep)+ and their Hermi-tean conjugates Ep... 

A mean field theory has recently been developed to describe polymer blend confined in a thin film (Sect. 3.2.1). This theory includes both surface fields exerted by two external interfaces bounding thin film. A clear picture of this situation is obtained within a Cahn plot, topologically equivalent to the profile s phase portrait d( >/dz vs < >. It predicts two equilibrium morphologies for blends with separated coexisting phases a bilayer structure for antisymmetric surfaces (each attracting different blend component, Fig. 32) and two-dimensional domains for symmetric surfaces (Fig. 31), both observed [94,114,115,117] experimentally. Four finite size effects are predicted by the theory and observed in pioneer experiments [92,121,130,172,220] (see Sect. 3.2.2) focused on (i) surface segregation (ii) the shape of an intrinsic bilayer profile (iii) coexistence conditions (iv) interfacial width. The size effects (i)-(iii) are closely related, while (i) and (ii) are expected to occur for film thickness D smaller than 6-10 times the value of the intrinsic (mean field) interfacial width w. This cross-over D/w ratio is an approximate evaluation, as the exact value depends strongly on the... 

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