18

TL;DR conical intersections, and polarons. Or any other case when the velocity of the nuclei is faster than the electrons can respond nearly instantaneous The long answer requires a lot of mathematics! The Mathematical derivation The total nuclear and electronic Hamiltonian can be written as $\hat{H} = \hat{T}_N + \hat{T}_e +\hat{V}_{ee} + \hat{V}_{NN} ...


18

Adding to the answer given by Cody Aldaz, there are many situations in chemistry when the Born-Oppenheimer approximation (BOA) breaks down, conical intersection is just one of them! In fact, the electronic states do not necessarily have to cross/intersect each other for BOA to be invalid. A classic example of this is given by what is famously known as "...


17

Geometry optimization corresponds to a system in equilibrium. It is the "average" position of a molecule vibrating in a well. However, there are many cases where the system is "non-equilibrium" which are important to model. For example, photochemistry can involve important non-equilibrium processes. In photochemistry, relaxation to the ground-state ...


12

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 ...


9

The Born-Oppenheimer approximation comprises two different approximations: Adiabatic separation of electron and nuclear coordinates Semi-classical approximation for nuclei Previous answers have already addressed (1). I would also like to add that although it neglects electron-phonon coupling, this can be added back in using density functional perturbation ...


9

In addition to the classic examples of where non-adiabatic effects are important, the Born-Oppenheimer approximation cannot be taken for granted in the electronic structure computations of small chemical systems, where high-accuracy is needed (e.g. isotope dependence of molecular properties, high-resolution rovibrational spectroscopy, and quantum nuclear ...


8

The first answer is most relevant to materials modeling, but I also want to chip in that Born-Oppenheimer is more than just a kinetic assumption, it also assumes nuclei are point charges. This is especially important for actinides and super heavy elements where nuclei become not only much larger relative to the electron cloud but also more ovoid with unequal ...


7

The Hamiltonian Just as for vibrations we have the harmonic oscillator approximation, for rotations we often use the rigid rotor approximation, where bond lengths are fixed. Recall the rigid-rotor Hamiltonian (in this case the kinetic energy operator) for a diatomic, which is often written as follows: $$\tag{1} \hat{H} = \hat{T} = \frac{J_x^2}{2I_x} + \frac{...


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