Born-Oppenheimer approximation

schafspelz

I am confused with a couple of terms usually used in the context of non-radiative transitions. I believe that I understand the concept of diabatic and adiabatic states described in
http://en.wikipedia.org/wiki/Adiabatic_theorem. The basic finding is that the coupling terms in the Hamiltonian matrix (in the basis of the diabatic states) result in an avoided crossing.

I want to transfer this finding to the case of the Born-Oppenheimer approximation, which is said to break down in the region around a level-crossing. And this is actually the point where I come across my first problem. When neglecting the first and the off-diagonal elements (closely related to the non-adiabacity operator), do I get diabatic states (case A) or adiabatic states (case B)?

If the case A is valid, the situation as depicted below would seem logical.

Here we face a level crossing, which is regarded as the breakdown of the Born-Oppenheimer approximation. As soon as the off-diagonals elements taken into account again, the avoided crossing would be obtained then.


But from my literature search I get the impression that Born-Oppenheimer approximation leads to adiabatic states. But what is the breakdown of the Born-Oppenheimer approximation then? And what are the non-adiabatic transitions resulting from the non-adiabacity operator in the last figure?

I hope anybody can resolve my problems with this stuff!

DrDu

To follow your line of argumentation, it would be helpful if you were to write down the Hamiltonian you are talking about. Which one is the first element?

schafspelz

Thanks for your interest!

Just to make sure that there aren't any misunderstandings, I will also repeat the main definitions. As usual, the total wavefunction [itex]\Psi({\bf r},{\bf R})[/itex] expanded as a series of electronic wavefunctions [itex]\chi_k({\bf r};{\bf R})[/itex]
[itex]\Psi({\bf r},{\bf R}) = \sum\limits_k \chi_k({\bf r};{\bf R}) \eta_k({\bf R})[/itex].
The electronic Hamiltonian [itex]{\cal H}_\mathrm{e}[/itex] is expressed as
[itex]{\cal H}_\mathrm{e}=T_\mathrm{e}+V_\mathrm{ee}+V_\mathrm{en}+V_\mathrm{nn}[/itex],
which satisfy the electronic Schrödinger equation
[itex]{\cal H}_\mathrm{e} \chi_k({\bf r};{\bf R}) = E_k({\bf R}) \chi_k({\bf r};{\bf R})[/itex].
The full Hamiltonian is defined as
[itex]{\cal H}=T_\mathrm{n}+E_k({\bf R})[/itex],
whose matrix elements should be calculated in the basis of [itex]\eta_k({\bf R})[/itex]. Then the Hamiltonian reads
[itex] {\cal H} =
\left(\begin{array}{ccc}
T_\mathrm{n}+E_1({\bf R}) & 0 & 0 & \cdots\\
0 & T_\mathrm{n}+E_2({\bf R}) & 0 & \cdots\\
0 & 0 & T_\mathrm{n}+E_3({\bf R}) & \cdots\\
\vdots & \vdots & \vdots & \ddots
\end{array}\right) + \underbrace{\left(\begin{array}{ccc}
\tilde{H}_{11} & \tilde{H}_{12} & \tilde{H}_{13} & \cdots\\
\tilde{H}_{21} & \tilde{H}_{22} & \tilde{H}_{23} & \cdots\\
\tilde{H}_{31} & \tilde{H}_{32} & \tilde{H}_{33} \cdots\\
\vdots & \vdots & \vdots & \ddots
\end{array}\right)}_\text{Non-adiabacity operator} [/itex].
The non-adiabacity operator contains the following several elements.
[itex]
\tilde{H}_{ij} = -\frac{\hbar}{2M}\Big( \underbrace{2\left\langle\chi_i({\bf r};{\bf R})\left|\nabla_\mathrm{n}\right|\chi_j({\bf r};{\bf R})\right\rangle \nabla_\mathrm{n}}_\text{first order} + \underbrace{\left\langle\chi_i({\bf r};{\bf R})\left|\nabla_\mathrm{n}^2\right|\chi_j({\bf r};{\bf R})\right\rangle}_\text{second order} \Big)
[/itex]
If [itex]i\neq j[/itex], the elements of [itex]\tilde{H}_{ij}[/itex] only appear as off-diagonal elements and are neglected in the Born-Oppenheimer approximation. So my question is whether the [itex]E_k({\bf R})[/itex] is already the adiabatic potential energy surface with an avoided crossing OR does the avoided crossing occur only when the off-diagonals are accounted for (which seems to be analogous to the concept of the adiabatic theorem)?

DrDu

The ##E_k(R)## are already the Born-Oppenheimer PES which show avoided crossings. There is a minor distinction between the Born-Oppenheimer PES and the adiabatic ones: The latter include the diagonal part ##\tilde{H}_{ii}(R)##.
The diabatic states are obtained by looking for a unitary transformation which diagonalizes the non-adiabaticity operator for a subset of states (e.g. states 1 and 2).
Conceptually easier are the crude-adiabatic states for which the whole non-adiabatic matrix is diagonal. This is obtained by using electronic states ##\chi_k(r;R_0)## referring to one fixed nuclear position R_0.

There is a famous theorem by Wigner that even adiabatic (or BO-) PES will cross when the space of nuclear displacements is more than onedimensional. In two dimensions, this happens at a point and the PES can be shown to have the form of a conus whence one speaks of a conical intersection. Nonadiabatic couplings become very large there or singular. This is what is meant with the breakdown of the BO approximation.
The best known examples occur in Jahn-Teller systems although conical intersections have turned out to be important in almost any photochemical process.

schafspelz

Thanks for your help! You have given me the needed impetus so that I can go deeper into this stuff now.