Time-independent approaches

Baer M, Yahalom A and Engelman R 1998 Time-dependent and time-independent approaches to study effects of degenerate electronic states J. Chem. Phys. 109 6550... 

In the previous sections, we have seen that a resonance can be described through both a time-dependent approach and a time-independent approach. The goal of this section is to create a unified picture that joins both methodologies and explain how general wave packets evolve into pseudostationary decaying states. 

A procedure in this time-independent approach that corresponds to following the time evolution of the collision system would be to study the annihilation function x P p) defined in terms of the hyperspherical coordinates by... 

The pump-probe spectroscopic time-resolved study of autoionization processes in atoms and molecules uses an ultra-short (100-500 as) XUV pulses for the pump stage in conjunction with an intense (1012-1014 W/cm2), few-cycle IR pulse as probe. Traditional time-independent approaches are inadequate to interpret these kind of experiments. This is so because, on the... 

The treatment of poly-atomic systems - the time-independent approach 106... 

In Section 4.1 we will use the time-independent continuum basis 4//(Q E,0), defined in Section 2.5, to construct the wavepacket in the excited state and to derive (4.2). Numerical methods are discussed in Section 4.2 and quantum mechanical and semiclassical approximations based on the time-dependent theory are the topic of Section 4.3. Finally, a critical comparison of the time-dependent and the time-independent approaches concludes this chapter. 

One of the main assets of the time-dependent theory is the possibility of treating some degrees of freedom quantum mechanically and others classically. Such composite methods necessarily lead to time-dependent Hamiltonians which obviously exclude time-independent approaches. We briefly outline three approximations that are frequently used in molecular dynamics studies. To be consistent with the previous sections we consider the collinear triatomic molecule ABC with Jacobi coordinates R and r. 

The time-independent and time-dependent approaches merely provide different views of the dissociation process and different numerical tools for the calculation of photodissociation cross sections. The time-independent approach is a boundary value problem, i.e., the stationary wavefunction... 

In the time-independent approach one has to calculate all partial cross sections before the total cross section can be evaluated. The partial photodissociation cross sections contain all the desired information and the total cross section can be considered as a less interesting by-product. In the time-dependent approach, on the other hand, one usually first calculates the absorption spectrum by means of the Fourier transformation of the autocorrelation function. The final state distributions for any energy are, in principle, contained in the wavepacket and can be extracted if desired. The time-independent theory favors the state-resolved partial cross sections whereas the time-dependent theory emphasizes the spectrum, i.e., the total absorption cross section. If the spectrum is the main observable, the time-dependent technique is certainly the method of choice. 

In order to calculate the absorption spectrum in the time-independent approach one solves the time-independent Schrodinger equation for a series of total energies and evaluates the overlap of the total continuum wavefunction, defined in (2.70), with the bound wavefunction of the parent molecule, ( tot(E) Pio I o(Ei)). Any structures in the spectrum are thus related to the energy dependence of the stationary wavefunction "Jftot(E). As illustrated schematically in Figure 7.4 for the one-... 

The excited complex breaks apart very rapidly and only a minor fraction performs, on the average, one single internal vibration. Therefore, the total stationary wavefunction does not exhibit a clear change of its nodal structure when the energy is tuned from one peak to another (Weide and Schinke 1989). In the light of Section 7.4.1 we can argue that the direct part of the total wavefunction, S dir-, dominates and therefore obscures the more interesting indirect part, Sind- The superposition of the direct and the indirect parts makes it difficult to analyze diffuse structures in the time-independent approach. In contrast, the time-dependent theory allows, by means of the autocorrelation function, the separation of the direct and resonant contributions and it is therefore much better suited to examine diffuse structures. 

This is inherently impossible in the time-independent approach because the wavefunction contains the entire history of the wavepacket. The real understanding, however, is provided by classical mechanics. Plotting individual trajectories easily shows the type of internal motion leading to the recurrences which subsequently cause the diffuse structures in the energy domain. The next obvious step, finding the underlying periodic orbits, is rather straightforward. 

Lagana, A., Pack, R.T. and Parker, G.A. (1988) Faraday Disc. Chan. Soc. 84, 409. Honvault P. and Launay, J.M. (2001) A quantum-mechanical study of the dynamics of the 0( D)-1-H2 OH J- H insertion reaction, J. Chem. Phys. 114, 1057-1059. Jaquet, R. (2001) Quantum reactive scattering the time-independent approach. lY. Jakubetz (ed.), Lecture Notes in Chemistry 77, Methods in Reaction Dynamics, Springer-Verlag, Berlin, pp. 17-126. 

Time-independent approaches to quantum dynamics can be wxriational where the wavefunction for all coordinates is expanded in some basis set and the parameters optimized. The best knowm variational implementation is perhaps the S-matrix version of Kohn s variational prineiple which was introduced by Miller and Jansen op de Haar in 1987[1]. Another time-independent approach is the so called hyperspherical coordinate method. The name is unfortunate as hyperspher-ical coordinates may also be used in other contexts, for instance in time-dependent wavepacket calculations [2]. 

In summary, the rules for the construction and subsequent evaluation of diagrams corresponding to matrix elements in the CCSDT model have been given. The adaptation of certain features from time-dependent diagrams, not usually found in time-independent approaches, have been seen to clarify and/or expedite time-independent diagrams. 

The relationship between the time-dependent and time-independent approaches, given by a Fourier transform between G(e) and U t), was elucidated by Bloch [43]. Of course, in either approach, the final result is time-independent, since, after all, we are solving a time-independent or stationary Schrodinger equation. It would thus seem that the time-independent approach—as followed by Hugenholtz—would turn out to be more natural and simpler. This is indeed the case when we are primarily interested in the energy. However, in order to elucidate the PT structure of the exact wave function, the time-dependent approach is beneficial cf. Ref. [34]). 

In the same volume of the Sanibel proceedings as the Monkhorst paper was an attempt at a dilferent, time-independent approach to excited states in CC theory by Harris [134], that used the equation-of-motion (EOM) formalism of D.J. Rowe [135] (see also [136, 137]). Harris approach did not lead to viable equations because he proposed to write an excited state as an exponential, = exp(S)exp(7 )l0), where like T, S was an excitation operator, which just redefines T. (For some special cases such an operator can be used... 

When the whole information over all the oi)cn states is needed in a single fixed energy QM cafeufation, a method based on a time-independent approach is used. In this a.j)j)roach, the time variable is factored out and the stationary wavefunction is exj)anded in terms of a set of one-dimension-less functions of the bound coordinates. This expansion and the subsequent integration over all the bound variables leads to a set of coupled differential equations on the coordinate connecting reactants to products (reaction coordinate) [25]. 

The approaches to electron transfer described in previous sections are based on the assumptions (i) that electronic motion occurs much faster than nuclear motion at the transition state (i.e., that r ei t nu) and (ii) that stationary-state, or time-independent approaches to the electronic wave function provide appropriate descriptions of the reaction coordinate. These approaehes are not always valid in covalently linked complexes. 

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