Time dependent methods

This section is divided into two sections the first concerned with time-dependent methods for describing the evolution of wavepackets and the second concerned with time-independent methods for solving the time independent Sclirodinger equation. The methods described are designed to be representative of what is in use. 

Thachuk M and Schatz G C 1992 Time dependent methods for calculating thermal rate coefficients using flux correlation functions J. Chem. Phys. 97 7297-313... 

It is also possible to detennine the resonant Raman intensities via a time-dependent method [16]. It has the... 

Iterative approaches, including time-dependent methods, are especially successfiil for very large-scale calculations because they generally involve the action of a very localized operator (the Hamiltonian) on a fiinction defined on a grid. The effort increases relatively mildly with the problem size, since it is proportional to the number of points used to describe the wavefiinction (and not to the cube of the number of basis sets, as is the case for methods involving matrix diagonalization). Present computational power allows calculations... 

This time-dependent method allows one to nieely eoimeet the theoretieal and experimental observations. As mentioned earlier, the eorrelation fiinotion and its generalizations yield the speetra for a large number of other photospeetroseopy proeesses, sueh as Raman proeesses [ ], as well as moleeular seattering [73, 74],... 

Development of laser technology over the last decade or so has permitted spectroscopy to probe short-time events. Instead of having to resort to the study of reactants and products and their energetics and shuctures, one is now able to follow reactants as they travel toward products. Fast pulsed lasers provide snapshots of entire molecular processes [5] demanding similar capabilities of the theory. Thus, explicitly time-dependent methods become suitable theoretical tools. 

RPA, and CPHF. Time-dependent Hartree-Fock (TDFIF) is the Flartree-Fock approximation for the time-dependent Schrodinger equation. CPFIF stands for coupled perturbed Flartree-Fock. The random-phase approximation (RPA) is also an equivalent formulation. There have also been time-dependent MCSCF formulations using the time-dependent gauge invariant approach (TDGI) that is equivalent to multiconfiguration RPA. All of the time-dependent methods go to the static calculation results in the v = 0 limit. 

SIMULATIONS, TIME-DEPENDENT METHODS AND SOLVATION MODELS... 

K.C. Kulander, Time-Dependent Methods for Quantum Dynamics (A Thematic Issue of Computer Physics Communications Comp. Phys. Commun. 63, 1(1991). 

The frequency dependence is taken into accoimt through a mixed time-dependent method which introduces a dipole-moment factor (i.e. a polynomial of first degree in the electronic coordinates ) in a SCF-CI (Self Consistent Field with Configuration Interaction) method (3). The dipolar factor, ensuring the gauge invariance, partly simulates the molecular basis set effects and the influence of the continuum states. A part of these effects is explicitly taken into account in an extrapolation procedure which permits to circumvent the sequels of the truncation of the infinite sum-over- states. 

In the adiabatic bend approximation (ABA) for the same reaction,18 the three radial coordinates are explicitly treated while an adiabatic approximation was used for the three angles. These reduced dimensional studies are dynamically approximate in nature, but nevertheless can provide important information characterizing polyatomic reactions, and they have been reviewed extensively by Clary,19 and Bowman and Schatz.20 However, quantitative determination of reaction probabilities, cross-sections and thermal reaction rates, and their relation to the internal states of the reactants would require explicit treatment of five or the full six degrees-of-freedom in these four-atom reactions, which TI methods could not handle. Other approximate quantum approaches such as the negative imaginary potential method16,21 and mixed classical and quantum time-dependent method have also been used.22... 

See K. C. Kulander (ed.), Time-dependent methods for quantum dynamics, Com,put. Phys. Commun. 63 (1991). 

Zapalac G.H., A Time-Dependent Method for Characterizing the Diffusion of 222Rn in Concrete, Health Physics 45 377-383 (1983). 

Fig. 2 Electronic conduction of a benzene ring between two conducting electrodes. These calculations are performed by the time-dependent method presented here (solid line) and by the ESQC method (dashed line). The electrodes are connected either in ortho (left column) or meta (right column) position. Two regimes are investigated tunneling with v = —0.25 ev (upper row), pseudoballistic with v = —2 ev (lower row). The vertical dashed lines represent the energy of the benzene s molecular orbitals...

Fig. 4 Electronic conduction between four identical electrodes via a naphthalene calculated with the time-dependent method...

These equations enable us to compute aU the possible photofragmentation cross sections. An example of the use of these equations applied to the photodissociation of HOBr may be found in Ref. 84, and similar applications to electronically nonadiabatic photofragmentation of HF, DF, and HCl can be found in Refs. 76, 97, and 96. Time-dependent methods have been used most recently to compute vector correlations and alignment parameters [98,99]... 

On the other hand, it is pleasing to see that the organic chemist s standard evidence for structural identification such as NMR and IR spectral data can be computed quite accurately UV/vis spectra can also be computed with time-dependent methods, which, however, cannot yet be used to optimise the excited states as the Franck-Condon principle is a common assumption in electron spectroscopy, this is no serious drawbackfor computing... 

A.K. Kazansky, P. Selles, L. Malegat, Hyperspherical time-dependent method with semi-classical outgoing waves for double photoionization of helium, Phys. Rev. A 68 (2003) 052701. 

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