Adibatic Approx.(i.e., Born-Oppenheimer)

[R Van Camp]

I was educated as a physical chemist so this subject is not unknown to me. However, I am currently taking a condensed matter physics course and this was presented primarily as the "adiabatic approximation." Can someone elaborate on this notion of adiabicity (i.e., S remains fixed) and how this is an alternative means of stating the B-O approximation. I'll monitor this discussion and add comments where it seems useful for me to do so.

I do not recall this being presented as an adiabatic approximation when I learned it graduate school the first time around.

Thanks,

Rick

 

[alxm]

Welcome to PF.
Well, as you probably know, the B-O approximation is the assumption that the electronic motion is decoupled from that of the nuclei, allowing you to separate the molecular Hamiltonian into electronic and nuclear parts, where the electronic part is dependent on the nuclear locations alone.

In classical thermodynamics, an adiabatic process is one that doesn't exchange any heat with its surroundings. In quantum mechanics, an "adiabatic" process is one where the energetic state of the system does not change during a changing potential - no transitions occur. If it was in the ground state of the original Hamiltonian, it will be in the ground state of the new Hamiltonian. This holds if the change in potential is sufficiently slow.

A diabatic process then, is one where the potential changes rapidly, and so your wave function doesn't have time to change and thus no longer represents the same state in the new potential, but likely is a superposition of states. (e.g. think of instantly widening a particle-in-a-box)

So the BO-approximation is an adiabatic approximation - saying that the nuclear motion is adiabatic as far as the electrons are concerned. The motion of the nuclei changes the potential sufficiently slowly to not alter the electronic state. No kinetic energy is transferred from the nuclei to the electrons.

 

[R Van Camp]

alxm,

Thank you for your reply and explanation. However, I remain scratching my head regarding this because my undergraduate degree is best described as chemical physics. However, I stopped this early to begin graduate school during the fall so I was awarded a BA in chemistry. Nonetheless, I had a semester year of quantum mechanics, E&M, and a graduate quantum chemistry course and too much of this sounds new.

Can you elaborate further?

Have experimentalists been able to demonstrate this slowly varying change of a potential field, such that the eigenfunctions and eigenstates remain as they were before the change in potential? Have you ever seen this material presented in a heuristic manner? I would be very interested in reading through it on my own if yes. That said, I suspect this may be theoretical constructs. This must also involves statistical mechanics regarding the population distributions.

Thanks,

Rick

 

[alxm]

It's not really very difficult to conceptialize; If I pull a tablecloth slowly, the objects on top of it will not move relative the tablecloth. If I pull it quickly, they will.

Have you tried searching for 'adiabatic theorem'? In any case, yes, it's been rigorously proven mathematically. It's covered in most textbooks (e.g. Griffiths "Introduction to QM" has a whole chapter on the adiabatic theorem).

Has it been experimentally verified? Of course. To begin with you have the aforementioned Born-Oppenheimer approximation. Which is more than just an approximation to the Schrödinger equation; just to begin with it provides a theoretical justification for the fact that what constitutes a molecule is determined only by its nuclei and their positions. If there was significant coupling between nuclear and electronic motion, chemistry would be fundamentally different.

 

[R Van Camp]

alxm,

Thank you; I appreciate your efforts. I do know about Griffiths but do not yet own a copy; that discrepancy will be resolved soon. I'll also perform the search for adiabatic theorem to see what results it brings.

Rick

 

[DrDu]

The Born Oppenheimer approximation in sensu stricto refers to the method developed in the paper by Born and Oppenheimer in 1927 (?) and refers to a perturbation series in the ratio of the mass of the electrons and the nuclei. The adiabatic approximation was formulated by Born based on the variational principle. The original article is available as an appendix in the book by Born and Huang, "Dynamical theory of crystal lattices".

 

[R Van Camp]

DrDu,

Thanks for the reference. I look into it.

Rick

 

[DrDu]

I re-read your first post again and noted some misconception. The term "adiabatic" in the context of molecular motion and in the context of thermodynamics are not related to each other. "Adiabasis" is greek and means "no stepping through" or no trespassing (also to be seen on greek airports!). In molecular physics it refers to the electron making no transition from one potential energy surface to another while in thermodynamics it refers to no heat entering or leaving the system.

 

[genneth]

I re-read your first post again and noted some misconception. The term "adiabatic" in the context of molecular motion and in the context of thermodynamics are not related to each other. "Adiabasis" is greek and means "no stepping through" or no trespassing (also to be seen on greek airports!). In molecular physics it refers to the electron making no transition from one potential energy surface to another while in thermodynamics it refers to no heat entering or leaving the system.

Actually, they are not entirely disconnected. Entropy can be calculated as a sum of p log p, where p is the probability of being in a certain microstate. Adiabatic changes do not change p, and thus do not change the entropy. The connection between entropic change and heat exchange is then as usual in thermodynamics.

 

[DrDu]

Actually, they are not entirely disconnected. Entropy can be calculated as a sum of p log p, where p is the probability of being in a certain microstate. Adiabatic changes do not change p, and thus do not change the entropy. The connection between entropic change and heat exchange is then as usual in thermodynamics.

True, but "adiabatic" refers there to the change of the occupation probability of the microstates with the external parameters (like volume or field).
In molecular physics, it refers to the change of the electronic wavefunction with the nuclear motion (with itself is a dynamical quantum mechanical variable).

 

[alxm]

Sure, but the Adiabatic Theorem is general and applicable to all QM. The B-O approximation is just one example of where this kind of 'adiabicity' comes into play.

 

[R Van Camp]

DrDu,

Thanks again for the reference to Born & Huang. I purchased a used copy and you are correct; the subject matter receives a thorough treatment.

Rick

 

[DrDu]

You are welcome, Rick!
Maybe it might be interesting for you that on an advanced level of solid state physics, namely in the context of Greens functions, the adiabatic approximation is used in the form of "Migdal's theorem", especially in metallic systems. It is shortly introduced e.g. in Fetter and Walecka. However, I am unaware of any careful discussion of its (serious) limitations.