Adiabatic potential energy surfaces APES

We now wish to analyse the potential part of the Hamiltonian in equation (1) in order to understand the combination of vibrational and (pseudo)rotational motion of the system. To do this, it is very convenient to follow O Brien [26,27] and use the angular parametrization 

It is clearly seen that the lowest APES can occur for values a = 0, . The 

O Brien has applied a full adiabatic approximation [19] to the Hamiltonian (1) for the motion on the lowest APES. Using the parametrization (3), the kinetic energy part of equation (1) becomes 

Under the adiabatic approximation, the lowest energy eigenvalue from equation (4) at a = 0 is used in equation (1) and together with equation (5), it can be shown that 

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The adiabatic potential energy surfaces (APES) can be easily calculated in a standard way and the explicit form of the two APES is... 

Originally, the Jahn-Teller (JT) theorem was formulated as a prediction of a symmetry break for any non-linear polyatomic molecule with an orbital degeneracy in its ground state. However, as we understand it now, the real meaning of the JT theorem is a non-zero slope of the adiabatic potential energy surface (APES) at the point of electron degeneracy. This does not necessarily result in a lowering of... 

Calculations of the second-order RFs are much more complicated than first-order RFs as they generally involve coupling to an infinite set of excited vibronic states for which details are often unknown. They can therefore only be calculated exactly in a pure adiabatic case, such as that found in the T <%> e system [1] in which the electronic states are not mixed. In other cases, approximations have to be made. Even so, as excited states with higher energies are included, the energy denominator increases whilst the overlaps between states located in different wells in the lowest adiabatic potential energy surface (APES) decrease. We note also that numerical approaches often mask the underlying physics whilst analytical methods such as... 

A good visualization of the problem is obtained by the adiabatic potential energy surfaces (APES s), whose shape is determined by the Hamiltonian... 

APES Adiabatic potential energy surface nance... 

For JT problems of higher dimensions such as that for the T (e t2) problem, the adiabatic potential V is complicated and cannot be written down in an analytical form. However, in such problems, the least action path can be approximated by the minimum energy path (or path of steepest descent) on the adiabatic potential surface. It is the path for which the tangent to it is parallel to the gradient of the APES. 

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