Beyond the adiabatic approximation

It is of importance to note that we shall consider, in the present section, that the fast and bending modes are subject to the same quantitative damping. Indeed, the damping parameter of the fast mode yG and that of the bending mode y will be supposed to be equal, so that we shall use in the following a single parameter, namely y (= yG = y5). This drastic restriction cannot be avoided when going beyond the adiabatic approximation. 

The Rosch and Ratner [58] lineshapes, which are obtained within the adiabatic approximation, are not modified when working beyond the adiabatic approximation, if the anharmonic coupling between the slow and fast modes has a low magnitude. 

A. Vardi, M. Shapiro, Two-photon dissociation /ionization beyond the adiabatic approximation, J. Chem. Phys. 104 (1996) 5490 Theory of radiative recombination with strong laser pulses and the formation of ultracold molecules via stimulated photorecombination of cold atoms, A. Vardi, D. Abrashkevich, E. Frishman, M. Shapiro, J. Chem. Phys. 107 (1997) 6166. 

E. Sjdqvist and O. Goscinski The Molecular Aharonov-Bohm efifect in Bound States Beyond the Adiabatic Approximation Chem. Phys. 186, 17 (1994). 

We are, therefore, beyond the adiabatic approximation (which requires a single vibronic state, a product function) and the very notion of the single potential eneigy hypersurface for the motion of the nuclei becomes irrelevant. In the adiabatic approximation, the electronic wave function is computed from Eq. (6.8) with the clamped nuclei Hamiltonian i.e., the electronic wave function does not depend on what the nuclei are doing, but only where they are. In other words, the electronic structure is determined [by finding a suitable R) through solution of the... 

Fig. 6.20. A strange situation An election is unable to follow the motion of the nuclei (we are beyond the adiabatic approximation, a non-adiabatic case), ia) Some molecular dipoles with a sufficiently large dipole moment may bind an extra electron (a cloud on the right), which in such a case is far from the dipole and is attracted by its pole. The positive pole plays a role of a pseudonucleus for the extra electron, (b) When the dipole starts to rotate (a state with a nonzero angular momentum), the electron follows the motion of the pole. This is, however, difficult for high angular momenta (the electron has not enough time to adapt its position right toward the pole), and it is even harder because the centrifugal force pushes the extra electron farther away.

Vardi, A. and Shapiro, M., Two-photon dissociation. Ionization beyond the adiabatic approximation, J. Chem. Phys., 104, 5990-5496, 1996. 

We have used non-Hermitian scattering theory to calculate the transition probability amphtude, within the framework of the complex adiabatic approach. It should be stressed that non-Hermitian quantum mechanics allows us to use complex adiabatic potential energy surfaces in cases where one has to go beyond the adiabatic approximation in Hermitian quantum mechanics [1,2]. In the adiabatic approximation, we assume that the motion in the y direction is much slower than in the x direction. This assumption is based on the geometry of the two-dimensional potential surface (see Fig. 5). 

Dependent Density-Functional Theory beyond the Adiabatic Approximation. 

They acknowledge that they may need to go beyond the adiabatic approximation and work with a frequency dependent coupling matrix. They state that Tn the exact theory it is precisely the complicated pole structure of K in the o>-representation which leads to the proper description of complex excitation processes absent in the adiabatic approximation. ... 

At present DFT calculations for frequency dependent properties are not competitive with conventional Hartree-Fock based ones as the methods are not fully developed. There is also much work still to be done to develop suitable functionals, probably current dependent ones, and the theory beyond the adiabatic approximation needs much more investigation. Nevertheless the initial results are promising, and the relative... 

The development of useful exchange-correlation kernels which go beyond the adiabatic approximation has been a slow process, but a recent application of one such kernel to semiconductors is encouraging [60]. I will give a brief overview of some of the principal work leading up to this recent application. The first attempt at a frequency-dependent exchange-correlation kernel was given by Gross and Kohn [61]. Almost a decade later, Dobson proved the harmonic potential theorem (HPT) which must be obeyed by... 

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