Time-dependent theory

Lee S-Y and Heller E J 1979 Time-dependent theory of Raman scattering J. Chem. Rhys. 71 4777... 

Many experimental techniques now provide details of dynamical events on short timescales. Time-dependent theory, such as END, offer the capabilities to obtain information about the details of the transition from initial-to-final states in reactive processes. The assumptions of time-dependent perturbation theory coupled with Fermi s Golden Rule, namely, that there are well-defined (unperturbed) initial and final states and that these are occupied for times, which are long compared to the transition time, no longer necessarily apply. Therefore, truly dynamical methods become very appealing and the results from such theoretical methods can be shown as movies or time lapse photography. 

Time-Independent NGSO Equations from Time-Dependent Theory... 

To properly describe electronic rearrangement and its dependence on both nuclear positions and velocities, it is necessary to develop a time-dependent theory of the electronic dynamics in molecular systems. A very useful approximation in this regard is the time-dependent Hartree-Fock approximation (34). Its combination with the eikonal treatment has been called the Eik/TDHF approximation, and has been implemented for ion-atom collisions.(21, 35-37) Approximations can be systematically developed from time-dependent variational principles.(38-41) These can be stated for wavefunctions and lead to differential equations for time-dependent parameters present in trial wavefunctions. 

Time-Dependent Theory of the Partial Differential Cross Section... 

Time-independent and time-dependent theories are not really separate disciplines. This should be clear from the work of Kouri [188,189] and Althorpe [136,158], who use time-independent wavepacket techniques. These are easily derived from the more natural time-dependent versions by Fourier transforming the propagator over time. This is equivalent to transforming from the time domain to the energy domain at the beginning rather than the end of the calculation. 

While it is not clear how the constant frequency low field dielectric relaxation measurements mentioned above should be applied to reactions in liquids, save for a complete time-dependent theory of liquids, these effects are very significant. At short times (<10ps) the effective Onsager distance may be 20 nm, even in methanol or ethanol, but over the next two or three decades of time reduce to more nearly 2 nm. Such a change can reduce the rate of reaction much more rapidly than that which occurs by decay of the transient time dependence discussed in the previous sub-section. 

A more tractable theory based on the probability that a reactant pair will react at a time t (pass from reactants to products) is that due to Szabo et al. [282]. If the survival probability of a geminate pair of reactants initially formed with separation r0 is p (r0, t) at time t, the average lifetime of the pair is /dr0 p(r0, t)t and this is longer for larger initial separation distances. It provides a convenient and approximate description of the rate at which a reactant pair can disappear, but it does so without the need of a full time-dependent solution of the appropriate equations. Nevertheless, as a means of comparing time-dependent theory and experiment in order to measure the value of unknown parameters, it cannot be regarded as satisfactory. 

R.F. Sekerka. Application of the time-dependent theory of interface stability to an isothermal phase transformation. J. Chem. Phys. Solids, 28(6) 983—994, 1967. 

R.F. Sekerka. A time-dependent theory of stability of a planar interface during dilute binary alloy solidification. In H.S. Peiser, editor, Crystal Growth, pages 691-702. Pergamon Press, Oxford, 1967. 

The eigenvectors are positive and negative combinations of 4 and Ta, respectively. f d ( Fa) is a wave function where the mobile electron is at the donor (acceptor). In a time dependent theory, the wave function is set equal to at t=0. The electronic wave packet moves across the barrier to the acceptor. Landau-Zener theory [23,24] provides the probability for crossing to the upper surface at the avoided crossing ... 

The theoretical background that will be needed to calculate the excited state distortions from electronic emission and absorption spectra is discussed in this section. We will use the time-dependent theory because it provides both a powerful quantitative calculational method and an intuitive physical picture [7-11]. In this section we will concentrate on the physical picture and on the ramifications of the theory. 

Fig. la c. Illustration of the time dependent theory of emission spectroscopy for one-dimensional harmonic potential energy surfaces, a schematic view of the emission transition, b time dependence of the overlap < (f> t) >, e calculated emission spectrum... 

In the specific cases of the spectra of the metal compounds discussed in this chapter, a third mode is observed with no evidence of coupling to the Qx and Qy coordinates. In the framework of the time-dependent theory, the total autocorrelation function is the product of < 4> 4>(t) > of the coupled coordinates discussed above and the <4> 4>(t)> from a separate calculation for the third mode. 

The geometry changes which transition metal complexes undergo when excited electronic states are populated are determined by using a combination of electronic emission and absorption spectroscopy, pre-resonance Raman spectroscopy, excited state Raman spectroscopy, and time-dependent theory of molecular spectroscopy. 

Excited State Distortions of W(C0)5pyridine and W(C0)5piperldlne from Time-Dependent Theory, Pre-resonance Raman Spectroscopy, and Electronic Spectroscopy... 

The bond length changes determined from pre-resonance Raman spectra, electronic spectra and time dependent theory provide a detailed picture of the results of bonding changes caused by populating excited electronic states. There is a direct but not linear correlation between bond length changes and the... 

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. 

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. 

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. 

Heather, R.W. and Metiu, H. (1989). Time-dependent theory of Raman scattering for systems with several excited electronic states Application to a H3" model system, J. Chem. Phys. 90, 6903-6915. 

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