Wave functions, nonadiabatic quantum

The approximations defining minimal END, that is, direct nonadiabatic dynamics with classical nuclei and quantum electrons described by a single complex determinantal wave function constructed from nonoithogonal spin... 

The surface-hopping trajectories obtained in the adiabatic representation of the QCLE contain nonadiabatic transitions between potential surfaces including both single adiabatic potential surfaces and the mean of two adiabatic surfaces. This picture is qualitatively different from surface-hopping schemes [2,56] which make the ansatz that classical coordinates follow some trajectory, R(t), while the quantum subsystem wave function, expanded in the adiabatic basis, is evolved according to the time dependent Schrodinger equation. The potential surfaces that the classical trajectories evolve along correspond to one of the adiabatic surfaces used in the expansion of the subsystem wavefunction, while the subsystem evolution is carried out coherently and may develop into linear combinations of these states. In such schemes, the environment does not experience the force associated with the true quantum state of the subsystem and decoherence by the environment is not automatically taken into account. Nonetheless, these methods have provided com-... 

As a reflection of these properties, direct information on Tad is not required in the semi-classical analytical theory, as demonstrated in the previous section. That information is replaced by the analytical continuation of the adiabatic potentials into the complex R-plane (see Eq. (24)). In order to carry out the quantum mechanical numerical calculations, however, we always stay on the real R-axis and we require explicit information on the nonadiabatic couplings. Even in the diabatic representation, which is often employed because of its convenience, nonadiabatic couplings are necessary to obtain the diabatic couplings. The quantum mechanical calculations are usually made by solving the coupled differential equations derived from an expansion of the total wave function in terms of the electronic wave functions. 

As explained in the Introduction, one needs to distinguish the following kinds of surface hopping (SH) methods (i) Semiclassical theories based on a connection ansatz of the WKB wave function, " (ii) stochastic implementations of a given deterministic multistate differential equation, e.g. the quantum-classical Liouville equation, and (iii) quasiclassical models such as the well-known SH schemes of Tully and others. " In this chapter, we focus on the latter type of SH method, which has turned out to be the most popular approach to describe nonadiabatic dynamics at conical intersections. 

The ultrafast initial decay of the population of the diabatic S2 state is illustrated in Fig. 16 for the first 30 fs. Since the norm of the semiclassical wave function is only approximately conserved, the semiclassical results are displayed as rough data (dashed line) and normalized data (dotted line) [i.e. pnorm P2/ Pi + P2)]. The normalized results for the population are seen to match the quantum reference data quantitatively. It should be emphasized that the deviation of the norm shown in Fig. 16 is not a numerical problem, but rather confirms the common wisdom that a two-level system as well as its bosonic representation is a prime example of a quantum system and therefore difficult to describe within a semiclassical theory. Nevertheless, besides the well-known problem of norm conservation, the semiclassical mapping approach clearly reproduces the nonadiabatic quantum dynamics of the system. It is noted that the semiclassical results displayed in Fig. 16 have been obtained without using filtering techniques. Due to the highly chaotic classical dynamics of the system, therefore, a very large number of trajectories ( 2 x 10 ) is needed to achieve convergence, even over... 

To obtain a first impression of the nonadiabatic wave-packet dynamics of the three-mode two-state model. Fig. 34 shows the quantum-mechanical probability density P (cp, f) = ( (f) / ) (p)(cp ( / (f)) of the system, plotted as a function of time t and the isomerization coordinate cp. To clearly show the... 

The simplest model is the following the diabatic potentials are constant with V2 - Vx = A > 0 and the diabatic coupling is V e R where A = 2V0. Recently, Osherov and Voronin obtained the quantum mechanically exact analytical solution for this model in terms of the Meijer function (38). In the adiabatic representation this system presents a three-channel problem at E > V2 > Vu since there is no repulsive wall at R Rx in the lower adiabatic potential. They have obtained the analytical expression of a 3 X 3 transition matrix. Adding a repulsive potential wall at R Rx for the lower adiabatic channel and using the semiclassical idea of independent events of nonadiabatic transition at Rx and adiabatic wave propagation elsewhere, they derived the overall inelastic nonadiabatic transition probability Pl2 as follows ... 

The elechonic shucture of isolated molecular systems is most naturally described by using Gaussian type atomic orbitals (AO s) as basis functions in contrast to plane waves, which represent the natural choice in extended periodic systems. Here we present the approach for the calculation of the nonadiabatic couplings using KS orbitals expandedin terms of localized Gaussian atomic basis sets. This formulation is particularly convenient since it can be coupled with commonly used quantum chemical DFT codes. 

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