**Adiabatic and diabatic representation** The adiabatic potential energy curve is the one that corresponds to the adiabatic state (wavefunction) which in turn is the eigenvector of the Born Oppenheimer Hamiltonian.

The merit of the adiabatic representation is in the graphic and intuitive way the PES can be understood.

For example, the solution to the full Hamiltonian will contain not only the components of nuclear and electronic wavefunctions, but also the wavefunctions that cannot be presented as a product of nuc. and el. wavefunctions.

In many cases these latter components are immaterial and one may apply the BO approx. In the BO approximation the energy surfaces (eigenvalues of H) are different for different electronic states.

However in certain cases there may be strong vibronic couplings between these adiabatic PESs arising from different electronic states. Vibronic stands for a vibrational-electronic coupling, i.e. nuclei move so fast that one cannot neglect their motion with respect to the electronic motion. In such cases the BO approximation breaks down which means the results are no longer reliable.

To track such bad cases one usually looks at the behavior of the values of the so-called non-adiabatic coupling (NAC) matrix elements <state1|d/dX|state2>, where d/dX is the gradient with respect to the nuclear coordinates. It turnes out that the NAC matrix elements increase at avoided crossings and jump abruptly at conical intersections. The value of NAC is connected with the probability of inter(electronic)state transition.

**Quote:**

Adiabatic theorem:

A quantum mechanical system subjected to gradually changing external conditions can adapt its functional form, while in the case of rapidly varying conditions there is no time for the functional form of the state to adapt so the probability density remains unchanged.

Adiabatic process:

Gradually changing conditions allow the system to adapt its configuration, hence the probability density is modified by the process. If the system starts in an eigenstate of the initial Hamiltonian, it will end in the corresponding eigenstate of the final Hamiltonian.

Diabatic process:

Rapidly changing conditions prevent the system from adapting its configuration during the process, hence the probability density remains unchanged. Typically there is no eigenstate of the final Hamiltonian with the same functional form as the initial state. The system ends in a linear combination of states that sum to reproduce the initial probability density.

Ref:

http://en.wikipedia.org/wiki/Adiabatic_theorem#Diabatic_vs._adiabatic_processes **Quote:**

A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum.

ref:

http://en.wikipedia.org/wiki/Adiabatic_approximation **Quote:**

Adiabatic means: 'at constant entropy'. Only change volume, not the state.

__See also:__ Non-adiabatic transition

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