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The adiabatic approximation for solid state systems is rather radical. I was wondering in which cases it breaks down.

As it is based on the idea of the nuclii being much heavier than the electrons I would imagine there would be problems for very light atoms like hydrogen. Also there could occur problems for "heavy electrons" due to the strong curvature is the dispersion relation.

So these two scenarios problems for the adiabatic approximation? Are there more cases it breaks down? An furthermore, could you give explicit examples of cases where it breaks down, best with a explanation?

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    $\begingroup$ Much is known about this issue. See, for instance, en.wikipedia.org/wiki/Adiabatic_theorem and en.wikipedia.org/wiki/Avoided_crossing $\endgroup$
    – Urgje
    Feb 10 '14 at 11:10
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    $\begingroup$ I presume you are talking about using the adiabatic approximation for phonons? Its a rather general term.. could you be more specific? $\endgroup$ Feb 13 '14 at 23:20
  • $\begingroup$ What you say sounds more Born-Oppenheimer than adiabatic to me. Also, I do not think the “heavy electrons” are relevant here, since the effective mass is a renormalized parameter. $\endgroup$
    – xebtl
    Sep 10 '15 at 9:04
  • $\begingroup$ @xebtl The BO approximation is an adiabatic approximation. The BO approximation essentially says that as the nuclei move, the electrons are always in the ground state for the given nuclear configuration. This behavior can be related rigorously to the limit where the electrons are much lighter than the nuclei. Also, I agree that the effective-mass picture isn't relevant here, since once you use the effective mass picture is a free-particle description of electrons. You've eliminated the nuclei, so an adiabatic approximation is ill-defined. $\endgroup$
    – Ian
    Aug 1 '16 at 2:23
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(Regarding crystals and their vibrations...) Usually, when calculating the vibrational frequencies, one first calculates the second order energy derivatives within the Born-Oppenheimer approximation, where the vibrational frequency is proportional to the curvature of the potential (curvature being the second derivative of the potential with respect to moving nuclei, adiabatically, along their normal mode coordinate). Quasi-harmonic and harmonic approximations stop at the 2nd order adiabatic derivative.

Calculations of anharmonic vibrations usually don't feature the calculation of adiabatic derivatives of the potential beyond fourth order, because the magnitude of the error in the Born-Oppenheimer approximation becomes comparable to the magnitude of higher order derivatives. These are still anharmonic methods, because they include derivatives beyond second order. That higher than fourth order derivatives are not considered accurate in the adiabatic approximation gives one a hazy sense of how reliable it is.

It is rare for the Born-Oppenheimer approximation to "break down" in the description of solids at low temperatures. At high temperatures, it is also rare to observe this, because one has much larger sources of error in their calculation. You are correct that smaller nuclei are expected to show more variation from adiabatic behaviour, but this effect could be washed out in vibrational calculations due to anharmonicity caused by a non-quadratic potential energy surface. Hydrogen vibrates violently, for instance, and does not display motion as simply harmonic as its heavier counterparts. This is not (primarily due to) a breakdown of the Born-Oppenheimer approximation however, but a result of $F=ma$ giving more acceleration to smaller masses, and thus pushing the light atoms over the purely harmonic region of motion.

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  • $\begingroup$ Just like to add that the most spectacular place where the BO approximation fails is with ordinary superconductivity. It was shown very early on that the condensation energy lies in the kinetic energy of the nuclei, not electrons! $\endgroup$
    – KF Gauss
    Apr 18 '17 at 15:44
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Even in hydrogen the ratio between the masses of an electron and the proton is about 1:1800 and solis state systems are mostly composed of heavier elements. I guess it depends on your notion of accuraccy if the approximation is valid. I guess using myons or tauons instead of electrons would skip this validity, but I don't think that's the point of your question.

From what I remember from my studies the adiabatic approximation is working fine for molecules in ground states, but for exalted states of molecules or cathiones it can lead to poor results. So for solid state systems I'd have a look at high eneryg quasiparticels like phonons, magnons and the like.

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