Adiabatic approximation, geometrical

II electronic states, 634-640 theoretical background, 625-626 triatomic molecules, 611-615 pragmatic models, 620-621 Ab initio multiple spawning (AIMS) conical intersection location, 491-492 direct molecular dynamics, 411-414 theoretical background, 360-361 Adiabatic approximation geometric phase theory ... 

M. Lombardi What is not needed is the validity of the adiabatic approximation, that is, that there is no transition between adiabatic states. But the geometric phase is defined by following states along a path in parameter space (here nuclear coordinates) with some continuity condition. In the diabatic representation, there is no change of basis at all and thus the geometric phase is identically zero. Do not confuse adiabatic basis (which is required) and adiabatic approximation (which may not be valid). 

A diatomic in the adiabatic approximation, the origin of the coordinate system is in the geometric center of the molecule (at R/2). The nuclei vibrate in the potential ... 

It is well known that the Born-Oppenheimer adiabatic approximation establishes geometrical shape of a molecule. The atoms (atomic groups) constituting a molecule are imagined to be placed in the vertices of certain three-dimensional (3D) shape as illustrated in Figure 9.1. 

The approximation in which the couplings between counterpropagating beams are neglected can be regarded as an extension of the geometrical-optics approximation to liquid crystals. It is different from the adiabatic approximation in which no coupling is considered at all. 

To our knowledge no generalization of the adiabatic or geometrical-optics approximation has been presented for the case of two or three-dimensional spatial variations of the director. This problem arises for instance in the interpretation of diffraction from liquid crystal gratings or self-focusing. We believe that an extension of Ong s method to these more complex situations would be very useful. 

A basic means of modelling approximate reaction paths is the adiabatic mapping or coordinate driving approach [123,149]. The energy of the system is calculated by minimizing the energy at a series of fixed (or restrained, e.g. by harmonic forces) values of a reaction coordinate, which may be the distance between two atoms, for example. More extensive and complex combinations of geometrical variables can be chosen. This approach is only valid if one... 

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