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Within Born-Oppenheimer approximation, the motion of atomic nuclei are considered as being separated from the electrons movement. Although in general this approximation is valid for a high number of cases, for some particular problems like Vibro-electronic (Vibronic) interaction one should prepare its calculations beyond it.

The vibronic interaction have been deeply studied since the decade of 1980's and even earlier. This phenomena assumes that the electron transition (usually a intramolecular charge transfer in a mixed valence system), is mediated by the vibrations of the nuclei. In this regard, the contributions of Piepho, Krausz and Schatz and later by Piepho herself introduced a mathematical model for dealing with this problem in low nuclearity complexes.

My question is: Is there any methodology which can explicitly calculate the electro-vibrational levels in a molecule beyond Born-Oppenheimer approximation?

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This is a good question. In order to go beyond Born-Oppenheimer approximation, one has to first make a choice of electronic basis - diabatic or adiabatic - in order to define the vibronic coupling (vibrational-electronic coupling). The adiabatic choice seems to be more practical since it is a well-defined basis (adiabatic states are eigenstates of the electronic Hamiltonian) and most quantum chemistry program make use of them anyway. So the "only" thing remains to compute is the non-adibatic coupling vectors (NADVEC) between different electronic states. These calculations are now possible with programs such as MOLPRO, COLUMBUS, CFOUR and others. Adiabatic basis is often used in conjunction with "on-the-fly" molecular dynamics methods such as the popular Surface Hopping scheme.

On the other hand, most vibronic problems are actually studied using diabatic basis, and for good reason. While diabatic states are not unique (and NOT eigenstates of electronic Hamiltonian), they preserve the electronic character with the change in nuclear position and are therefore "smoothly varying". Not only this has better mathematical properties (potential energy surface is always differentiable, unlike in adiabatic basis, where PES may have "cusps"), but also it has more "chemical" meaning. Think of diabatic states as the valence bond structures (like resonance structure of benzene!).

A large body of work has been done to model vibronic problems in diabatic basis. The most popular approach is the "vibronic model Hamiltonian" proposed by Koppel-Domcke-Cederbaum (KDC) in 1984 [1]. It remains a classic and is an excellent place to start for all things vibronic. Even today the linear variant of vibronic models (Linear Vibronic Model or commonly known as LVC) is used widely to model many different kind of processes. The parameters of the vibronic models can be obtained in many different ways, and the process of extracting these parameters is known as "diabatization". Since diabatic states are not unique, many diabatization procedures have been proposed in the literature. However, this depends on the problem and is often also a matter of taste, and does not take away the power of vibronic model Hamiltonian ansatz!

PS: Vibronic model Hamiltonian is also referred to as KDC Hamiltonian for obvious reasons!

References:

  1. H. Köuppel, W. Domcke, L. S. Cederbaum, Multimode Molecular Dynamics Beyond the Born‐Oppenheimer Approximation, Advances in Chemical Physics, Volume 57, 1984. https://doi.org/10.1002/9780470142813.ch2
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    $\begingroup$ +1. Would you be able to add the KDC Hamiltonian to the Hamiltonian Zoo? The open-source, open-collaboration database of all Hamiltonians? $\endgroup$
    – Nike Dattani - No Free Time
    Commented May 13, 2020 at 3:50
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    $\begingroup$ Hi，Prateek, do you know any program that could do transformation from Adiabatic basis to diabatic basis? As mentioned in your answer all these commercial packages (e.g.MOLPRO, COLUMBUS, CFOUR) could only do Adiabatic basis. I know there are a few vibronic models for small molecules available in Dr Marcel Nooijen’s website and I think those models are all presented in diabatic basis and there is your PhD thesis as well. I am doing similar line of research at present and I am glad that you could provide detailed explanations on non-Adiabatic dynamics. $\endgroup$
    – Paulie Bao
    Commented May 13, 2020 at 14:03
  • $\begingroup$ @PaulieBao if this is a new question, it might be best for you to ask it as a question, so that anyone can answer, not just Prateek. It will also earn you more points. $\endgroup$
    – Nike Dattani - No Free Time
    Commented May 13, 2020 at 22:33
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    $\begingroup$ @PaulieBao I think Nike is right in saying that you can ask a new question. Also, you're more than welcome to contact me by e-mail as this could be a larger discussion. I'm afraid the answer isn't that simplistic here. $\endgroup$
    – Prateek Goel
    Commented May 14, 2020 at 0:00
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    $\begingroup$ Hi， Prateek ，you are absolutely right，the question is non-trivial and that is reason that enormous effect has been made over decades on this problem. I definitely want to contact as some point. I appreciate that your could provide a nice review literature on non-Adiabatic nuclear dynamics. I have just finished reading the introduction section of it and it provides sufficient background knowledge of the subject. I can‘t reveal details before publications but I want to say that we are making progress on this subject. Marcel has just went to the Sanibel conference to present my project in $\endgroup$
    – Paulie Bao
    Commented May 14, 2020 at 3:10

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