Wavefunction propagation

The quantity < f m > e has been termed Raman overlap (Myers and Mathies, 1987). It is the product of the modulus of a time-dependent Franck-Condon factor between the final state and the initial wavefunction propagated on the electronic surface, i. e. I < f m > I and a damping function e which decreases exponentially with time. 

In addition to the phase modulation an even more pronounced amplitude modulation is observed (Fig. 8.12, top). After the first pre-pulse (red), the oscillating dipole (blue) is damped simultaneously with the temporal decrease in overlap of the nuclear wavefunctions propagating on the X S+ and the A S+ surface (Fig. 8.12, second panel). Due to the difference in position and shape of both surfaces, the freely evolving nuclear wavepackets get out of phase. Their spatial overlap ai(t) aj(t) rlri(R,t) fj R,t))ii is reduced, which is again a decisive factor for the electron dynamics [Eq. (8.12)]. Its decrease stops the electron dynamics as observable in the damping of the electric dipole oscillation and in the loss of control for large subpulse separations [69]. In this sense K2 is an example for the third factor in Eq. (8.12), which determines the electron dynamics. This third factor can be regarded as time-dependent EC overlap. 

The development of water potentials is not a search for the ultimate potential to end all other potentials. Indeed, we have no simple set of criteria by which a potential model can be termed good or bad. The employment of explicit water models is a trade-off among research interests, reliability of results, and available computer resources. The choice of a molecular model of water sets the microscopical length scale on which the interactions need to be modeled. At this level of detail, the atoms and molecules still obey classical mechanics, and atom interactions can be described by potential energy functions. For most problems it is not necessary to describe the system in terms of wavefunctions, although recently the techniques for wavefunction propagation... 

In the absence of coupling to a bath the influence functional is equal to unity. In that limit the path integral variables are coupled only to their nearest neighbors in time, i.e is coupled only to j +i. This fact is a consequence of the Markovian nature of the dynamics for the quantum particle alone, as the wavefunction for the latter obeys the first-order SchrSdinger differential equation. Because of this structure, the path integral in the absence of influence functional interactions can be broken into a sequence of one-dimensional integrations which form the basis of iterative matrix multiplication schemes and their variants that are routinely used for wavefunction propagation of small molecules (see also Wave Packets). 

In words, equation (Al.6.89) is saying that the second-order wavefunction is obtained by propagating the initial wavefunction on the ground-state surface until time t", at which time it is excited up to the excited state, upon which it evolves until it is returned to the ground state at time t, where it propagates until time t. NRT stands for non-resonant tenn it is obtained by and cOj -f-> -cOg, and its physical interpretation is... 

It is possible to use full or limited configuration interaction wavefunctions to construct poles and residues of the electron propagator. However, in practical propagator calculations, generation of this intermediate information is avoided in favor of direct evaluation of electron binding energies and DOs. 

The initial wavefunction is then expanded as in Eq. (2), and the wavepacket is propagated using the split operator method42 ... 

In both the diatom-diatom and atom-triatom reactions, the energy-dependent scattering wavefunction is obtained by a Fourier transform of the propagated wavepacket ... 

In many chemical and even biological systems the use of an ab initio quantum dynamics method is either advantageous or mandatory. In particular, photochemical reactions may be most amenable to these methods because the dynamics of interest is often completed on a short (subpicosecond) timescale. The AIMS method has been developed to enable a realistic modeling of photochemical reactions, and in this review we have tried to provide a concise description of the method. We have highlighted (a) the obstacles that should be overcome whenever an ab initio quantum chemistry method is coupled to a quantum propagation method, (b) the wavefunction ansatz and fundamental... 

In its most general form, a quantum graph is defined in terms of a (finite) graph G together with a unitary propagator U it describes the dynamics of wavefunctions

[Pg.79]

So eq. (11.47) can be viewed as a diffusion equation in the spatial coordinates of the electrons with a diffusion coefficient D equal to j. The source and sink term S is related to the potential energy V. In regions of space where V is attractive (negative) the concentration of diffusing material (here the wavefunction) will accumulate and it will decrease where V is positive. It turns out that if we start from an initial trial wavefunction and propagate it forward in time using eq. (11.47),... 

In the self-consistent field linear response method [25,46,48] also known as random phase approximation (RPA) [49] or first order polarization propagator approximation [25,46], which is equivalent to the coupled Hartree-Fock theory [50], the reference state is approximated by the Hartree-Fock self-consistent field wavefunction < scf) and the set of operators /i j consists of single excitation and de-excitation operators with respect to orbital rotation operators [51],... 

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