Quasi-classical state

QC quasi-classical (state). A state with alternating... 

Fig. 2. Example of an HBDC molecular rotor chemically bound to a polyene chain holding a metal cluster. As in Fig. 1, X-X is the rotation axis. This molecule must be prepared in a quasi-classical state to ensure lifting of the cluster load using the polymer chain as a string...

In the case of coherent laser light, the pulses are characterized by well-defined phase relationships and slowly varying amplitudes (Haken, 1970). Such quasi-classical light pulses have spectral and temporal distributions that are also strictly related by a Fourier transformation, and are hence usually refered to as Fourier-transform-limited. They are required in the typical experiments of coherent optical spectroscopy, such as optical nutation, free induction decay, or photon echoes (Brewer, 1977). Here, the theoretical treatments generally adopt a semiclassical procedure, using a density matrix or Bloch formalism to describe the molecular system subject to a pulsed or continuous classical optical field, which generates a macroscopic sample polarization. In principle, a fully quantal description is possible if one represents the state of the field by the coherent or quasi-classical state vectors (Glauber, 1965 Freed and Villaeys, 1978). For our purpose, however. 

Figure 3 Energy curves for H2 as a function of intemuclear distance. The curves displayed, from top to bottom, correspond to the triplet state, vFt, the quasi-classical state, S qc, the HL state, hl, and the exact (full Cl) curve, exact-...

P.C. Hiberty, D. Danovich, A. Shurki, S.S. Shaik, Why benzene possess a symmetry— a quasi-classical state approach for probing Jt-bonding and delocalization energies,... 

Classical dynamics is studied as a special case by analyzing the Ehrenfest theorem, coherent states (16) and systems with quasi classical dynamics like the rigid rotor for molecules (17) and the oscillator (18) for various particle systems and for EM field in a laser. 

A complete description of the method requires a procedure for selecting the initial conditions. At t 0, initial values for the complex basis set coefficients and the parameters that define the nuclear basis set (position, momentum, and nuclear phase) must be provided. Typically at the beginning of the simulation only one electronic state is populated, and the wavefunction on this state is modeled as a sum over discrete trajectories. The size of initial basis set (N/it = 0)) is clearly important, and this point will be discussed later. Once the initial basis set size is chosen, the parameters of each nuclear basis function must be chosen. In most of our calculations, these parameters were drawn randomly from the appropriate Wigner distribution [65], but the earliest work used a quasi-classical procedure [39,66,67], At this point, the complex amplitudes are determined by projection of the AIMS wavefunction on the target initial state (T 1)... 

Figure 8. Li + H2 Ground-state population as a function of time for a representative initial basis function (solid line) and the average over 25 (different) initial basis functions sampled (using a quasi-classical Monte Carlo procedure) from the Lit2/j) + H2(v — 0, j — 0) initial state at an impact parameter of 2 bohr. Individual nonadiabatic events for each basis function are completed in less than a femtosecond (solid line) and due to the sloped nature of the conical intersection (see Fig. 7), there is considerable up-funneling (i.e., back-transfer) of population from the ground to the excited electronic state. (Figure adapted from Ref. 140.)...

In this chapter different aspects of data processing and reconciliation in a dynamic environment were briefly discussed. Application of the least square formulation in a recursive way was shown to lead to the classical Kalman filter formulation. A simpler situation, assuming quasi-steady-state behavior of the process, allows application of these ideas to practical problems, without the need of a complete dynamic model of the process. 

That is, the classical DoF propagate according to a mean-field potential, the value of which is weighted by the instantaneous populations of the different quantum states. A MFT calculation thus consists of the self-consistent solution of the time-dependent Schrodinger equation (28) for the quantum DoF and Newton s equation (32) for the classical DoF. To represent the initial state (15) of the molecular system, the electronic DoF dk Q) as well as the nuclear DoF xj Q) and Pj 0) are sampled from a quasi-classical phase-space distribution [23, 24, 26]. 

There is, however, an important conceptional difference between the two approaches. On the quasi-classical level, this difference simply manifests itself in the initial conditions chosen for the electronic DoF. Let us consider an electronic two-level system that is initially assumed to be in the electronic state vl/]). In the mean-field formulation, the initial conditions are action-angle variables [cf. Eq. (18)], the electronic initial distribution in (90), is given by pgj = 6 Ni — 1)6 N2). In the mapping formalism, on the other hand, the initial electronic state vl/]) is represented by the first oscillator being in its first excited state and second oscillator being in its ground state [cf. Eq. (91)]. This corresponds to the... 

Figure 19. Time-dependent (a) diabatic and (b) adiabatic electronic excited-state populations and (c) vibrational mean positions as obtained for Model 1. Shown are results of the mean-field trajectory method (dotted lines), the quasi-classical mapping approach (thin full lines), and exact quantum calculations (thick full lines).

Figure 24. Diabatic (left) and adiabatic (right) population probabilities of the C (full line), B (dotted line), and X (dashed line) electronic states as obtained for Model II, which represents a three-state five-mode model of the benzene cation. Shown are (A) exact quantum calculations of Ref. 180 mean-field trajectory results [panels (B),(E)] and quasi-classical mapping results including the full [panels (C),(F)] and 60% [panels (D),(G)] of the electronic zero-point energy, respectively.

Figure 3.5. 2D dissociation probability S0 (= S) as a function of translational energy and vibrational state v for H2 (D2) dissociation on a PES similar to (but not identical) to that of Figure 3.4(a). (a) Quantum dissociation probabilities plotted logarithmically, (b) Dotted lines are results of quasi-classical dynamics and solid lines are from quantum dynamics. From Ref. [222]. 

Figure 3.28. N2 vibrational state distribution in associative desorption from Ru(0001). (a) Observed in experiment. From Ref. [126]. (b) From 3D (Z, R, q) first principles quasi-classical dynamics, with the solid triangles pointing upward being adiabatic dynamics and the squares from molecular dynamics with electronic frictions also from DFT. Based on the PES and frictions of Ref. [68]. The open triangles pointing downward are the results of 6D first principles adiabatic quasi-classical dynamics from Ref. [253].

The first two sections of Chapter 5 give a practical introduction to dynamic models and their numerical solution. In addition to some classical methods, an efficient procedure is presented for solving systems of stiff differential equations frequently encountered in chemistry and biology. Sensitivity analysis of dynamic models and their reduction based on quasy-steady-state approximation are discussed. The second central problem of this chapter is estimating parameters in ordinary differential equations. An efficient short-cut method designed specifically for PC s is presented and applied to parameter estimation, numerical deconvolution and input determination. Application examples concern enzyme kinetics and pharmacokinetic compartmental modelling. 

Existence and uniqueness of the particular solution of (5.1) for an initial value y° can be shown under very mild assumptions. For example, it is sufficient to assume that the function f is differentiable and its derivatives are bounded. Except for a few simple equations, however, the general solution cannot be obtained by analytical methods and we must seek numerical alternatives. Starting with the known point (tD,y°), all numerical methods generate a sequence (tj y1), (t2,y2),. .., (t. y1), approximating the points of the particular solution through (tQ,y°). The choice of the method is large and we shall be content to outline a few popular types. One of them will deal with stiff differential equations that are very difficult to solve by classical methods. Related topics we discuss are sensitivity analysis and quasi steady state approximation. 

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