# When is the adiabatic approximation for solid state systems valid?

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?

• I presume you are talking about using the adiabatic approximation for phonons? Its a rather general term.. could you be more specific? Feb 13 '14 at 23:20
• 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. Sep 10 '15 at 9:04
• @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.
– Ian
Aug 1 '16 at 2:23

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.