Numerical propagation method

NUMERICAL PROPAGATION METHOD 4.1. Propagation in a local interaction picture 

The equation of motion for Av5 which involves only a commutator with a Hamiltonian, could be solved by expanding the DOp in terms of density amplitudes satisfying a Schrodinger-like equation. More generally, however, the equation for a reduced DOp would contain dissipative rates, and this would make it necessary to solve the equation directly for the DOp. We therefore develop the numerical propagation method for the general case. The computational procedure starts with a basis set of quantum states, arranged as a row matrix l3 ) = l )2.], taken here to be 

Dropping in what follows the subindex W in the matrix, so that Tw = r, the DM equation is of the form 

Oscillations in time of quantal states are usually much faster than those of the quasiclassical variables. Since both degrees of freedom are coupled, it is not efficient to solve their coupled differential equations by straightforward time step methods. Instead it is necessary to introduce propagation procedures suitable for coupled equations with very different time scales short for quantal states and long for quasiclassical motions. This situation is very similar to the one that arises when electronic and nuclear motions are coupled, in which case the nuclear positions and momenta are the quasiclassical variables, and quantal transitions lead to electronic rearrangement. The following treatment parallels the formulation introduced in our previous review on this subject [13]. Our procedure introduces a unitary transformation at every interval of a time sequence, to create a local interaction picture for propagation over time. 

The calculation of the density operators over time requires integration of the sets of coupled differential equations for the quasiclassical trajectories and for the density matrix in a chosen expansion basis set. The density matrix could arise from an expansion in many-electron states, or from the one-electron density operator in a basis set of orbitals for a given initial many-electron state a general case is considered here. The coupled equations are as before dP/df = —QH/dR, dR/dt = dH/dP coupled now to 

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The center of the wavepacket thus evolves along the trajectory defined by classical mechanics. This is in fact a general result for wavepackets in a hannonic potential, and follows from the Ehrenfest theorem [147] [see Eqs. (154,155) in Appendix C]. The equations of motion are straightforward to integrate, with the exception of the width matrix, Eq. (44). This equation is numerically unstable, and has been found to cause problems in practical applications using Morse potentials [148]. As a result, Heller inboduced the P-Z method as an alternative propagation method [24]. In this, the matrix A, is rewritten as a product of matrices... 

Section II discusses the real wave packet propagation method we have found useful for the description of several three- and four-atom problems. As with many other wave packet or time-dependent quantum mechanical methods, as well as iterative diagonalization procedures for time-independent problems, repeated actions of a Hamiltonian matrix on a vector represent the major computational bottleneck of the method. Section III discusses relevant issues concerning the efficient numerical representation of the wave packet and the action of the Hamiltonian matrix on a vector in four-atom dynamics problems. Similar considerations apply to problems with fewer or more atoms. Problems involving four or more atoms can be computationally very taxing. Modern (parallel) computer architectures can be exploited to reduce the physical time to solution and Section IV discusses some parallel algorithms we have developed. Section V presents our concluding remarks. 

The differences in estimation of these moments for each scenario are graphically illustrated in Figures A2.4, A2.5 and A2.6, where the CDFs obtained from each numerical variance propagation method are compared with the analytical results. The results of the analytical method are assumed to represent the true moments of the model output and, therefore, the true CDF. Mean and standard deviation of ln(x) are used in plotting the analytical CDF. The equations for the transformation from arithmetic moments to the moments of ln(x) are as follows ... 

Electronic correlation in extended systems remains a central problem despite impressive progress in recent years. For small systems a number of very powerful methods have reached a high degree of accuracy thanks to a combination of formal algebraic and numerical techniques. These include configuration interaction,1-5 propagator methods,2,4 5 many-body perturbation procedures,3-5 and coupled-cluster methods.4 For extended systems density functional methods6,7 dominate the scene. Certain forms of correlation are taken into account by such methods, but how and to what extent are still unclear.8... 

In many physical applications the Hamiltonian is explicitly time dependent. The common solution for propagation in these explicitly time-dependent problems is to use very small grid spacing in time, such that within each time step the Hamiltonian H(r) is almost stationary. Under these semistationary conditions a short-time propagation method in the time-energy phase space is employed. The drawback of this solution is that it is based on extrapolation therefore the errors accumulate. Moreover, time ordering errors add with the usual numerical dispersion errors (108). 

This chapter starts vdth an introduction to modeling of chromatographic separation processes, focusing on different models capable to describe the dynamics of front propagation phenomena in the columns and the plant peripherals. A short introduction into numerical solution methods as well as an overview regarding methods for the consistent determination of the free model parameters, especially those of the thermodynamic submodels, is given. Methods of different complexity and experimental effort are presented. Finally, it will be illustrated that appropriate models can simulate experimental data with rather high accuracy. This validation is demonstrated both for standard batch elution and for a more complex multicolumn operation mode. 

There is another feature of numerical techniques that is very important. This is the apparent diffusion or dispersion that occurs with no physical bases, but is rather a consequence of the numerical computation method. A related problem is the ability of the model to handle sharp changes in geometry, sudden flow changes, or other such changes. That is, are these perturbations damped as they are in the physical system, or do they propagate and induce numerical errors. Since most users of numerical methods will merely use the model, these questions are very important. If the model is not well cast and used carefully, it could produce quite erroneous results which nonetheless may be difficult to detect if the physical situation is complex. 

Much of the observed structure of the stratosphere can be understood in terms of elementary wave propagation, momentum and heat transport by waves, and in situ forcing by radiatively active trace gases. The dynamical aspects of these interactions can often be described satisfactorily in analytical terms, although detailed calculations of the stratospheric circulation require the use of numerical (computer) methods. In what follows, a brief introduction to stratospheric dynamic meteorology is presented. The conceptual development will be oriented toward the interpretation of the stratospheric observations discussed in the previous sections. 

Oddershede, J., Propagator methods, Adv. Chem. Phys. 69 201 (1987). A concise review of the principles and applications of Green s functions and related approaches with numerous references to the literature. 

We apply the split-operator method Eq. (3.7) to separately handle the potential and kinetic energy terms of the Hamiltonian. We further apply the split-operator scheme to separate the kinetic energy term into the two exponentially noncommutative parts Tr + Tr and Tg to obtain a numerical short-time propagation method. 

Within the split-operator scheme, for example, numerical short-time propagation would involve diagonalizing the interaction matrix (either together with the potential energy matrix or as a separate term) at each time step. We will see several other time-propagation methods in Sec. 5.2.1. 

We may apply the split-operator method (Sec. 3.2.1) to the three matrices lijV) V/), and Vo for a short-time numerical propagation scheme. Ionization is now described by population of the neutral state Xn R,t) transferring to the ionized state partial-wave components Xc,kjix R,t) over time through the interaction represented by the matrix Vo(i ,f)-... 

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