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BornOppenheimer approximation 
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#1
Feb313, 07:56 AM

P: 5

I am confused with a couple of terms usually used in the context of nonradiative 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 BornOppenheimer approximation, which is said to break down in the region around a levelcrossing. And this is actually the point where I come across my first problem. When neglecting the first and the offdiagonal elements (closely related to the nonadiabacity 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 BornOppenheimer approximation. As soon as the offdiagonals elements taken into account again, the avoided crossing would be obtained then. But from my literature search I get the impression that BornOppenheimer approximation leads to adiabatic states. But what is the breakdown of the BornOppenheimer approximation then? And what are the nonadiabatic transitions resulting from the nonadiabacity operator in the last figure? I hope anybody can resolve my problems with this stuff! 


#2
Feb413, 01:30 AM

Sci Advisor
P: 3,633




#3
Feb413, 04:43 AM

P: 5

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{Nonadiabacity operator} [/itex]. The nonadiabacity 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 offdiagonal elements and are neglected in the BornOppenheimer 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 offdiagonals are accounted for (which seems to be analogous to the concept of the adiabatic theorem)? 


#4
Feb413, 06:35 AM

Sci Advisor
P: 3,633

BornOppenheimer approximation
The ##E_k(R)## are already the BornOppenheimer PES which show avoided crossings. There is a minor distinction between the BornOppenheimer 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 nonadiabaticity operator for a subset of states (e.g. states 1 and 2). Conceptually easier are the crudeadiabatic states for which the whole nonadiabatic 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 JahnTeller systems although conical intersections have turned out to be important in almost any photochemical process. 


#5
Feb413, 04:08 PM

P: 5

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



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