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We use the Born-Oppenheimer approximation in both Hartree-Fock method and DFT. What are the problems where we cannot use this approximations.

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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 "avoided crossing". The general term for phenomena when the BOA breaks down is referred to as "non-adiabatic processes". The entire field of photochemistry is filled with examples of non-adiabatic processes. Some examples include internal conversion and singlet fission. The former is even responsible for our vision!

From a quantum chemistry point of view, this is very common when one computes excited states (in order to model the above mentioned photochemical processes). You mentioned about Hartree-Fock and DFT. Well, they are single-reference methods to compute ground state energies. For excited state problems, when BOA is no longer valid, one will have to resort to multi-reference methods such as MRCI or EOM-CC. In addition to computing the excitation energies, these methods then allow to compute the "coupling" between different electronic states, precisely the terms that have been neglected in the derivation of BOA.

In short, most excited state problems such as vibronic spectroscopy, charge transfer, photochemistry, singlet fission, and many others will require going beyond Born-Oppenheimer approximation!

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    $\begingroup$ By EOM-CC, do you mean MR-EOM? $\endgroup$ – Nike Dattani May 11 '20 at 21:56
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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

$\hat{H} = \hat{T}_N + \hat{T}_e +\hat{V}_{ee} + \hat{V}_{NN} + \hat{V}_{eN}$

$(\hat{T}_N + \hat{T}_e +\hat{V}_{ee} + \hat{V}_{NN} + \hat{V}_{eN})\Psi = E\Psi $

where N are the nuclei and e are the electrons.

The BO approximation starts by asking what happens if we we freeze the nuclei at position $\mathbf{R}_0$ and take $\Phi(\mathbf{r})\chi(\mathbf{R}_0)$ as an ansatz of the total $\Psi$

This is mathematically much easier to solve because nuclear and electronic wavefunctions can be solved seperately. As we will see below, this is a good approximation in most situations.

First separating the electronic component we have

$(\hat{T}_e +\hat{V}_{ee} + \hat{V}_{NN} + \hat{V}_{eN})\Phi_k(\mathbf{r};\mathbf{R}_0) = E_{el,k}\Phi_k(\mathbf{r};\mathbf{R}_0) $

Combining this with the total Hamiltonian and the ansatz

$[\hat{T}_{N} + E_{el,k}]\Phi_k(\mathbf{r};\mathbf{R}_0)\chi(R) = E\Phi_k(\mathbf{r};\mathbf{R}_0)\chi(R)$

Rearranging,

$\frac{\hat{T}_{N}\Phi_k(\mathbf{r};\mathbf{R}_0)\chi(R)}{\Phi_k(\mathbf{r};\mathbf{R}_0)} +E_{el,k}\chi(R) = E\chi(R)$

Now here is the technical BO approximation::

$\hat{T}_{N}\Phi_k(\mathbf{r};\mathbf{R}_0)\chi(R)\approx \Phi_k(\mathbf{r};\mathbf{R}_0)\hat{T}_{N}\chi(R)$

Essentially this says that the kinetic energy of the nuclei doesn't affect the electronic states. Therefore, the nuclear wavefunction can be solved separately

$[\hat{T}_{N} + E_{el,k}]\chi(R) = E\chi(R)$

Thus under the BO approx., $\chi(\mathbf{R})$ Is solved by solving a nuclear Schrodinger equation where the potential is given by $E_{el}$


But why is the B.O. approximation justified?

The nuclear kinetic energy operators are derivatives with respect to the nuclei $\hat{T}_{N}\Phi_k(\mathbf{r};\mathbf{R}_0)\chi(R)=-\sum_{l=1}^{N_a}\frac{1}{2M_l}\nabla_l^2[\Phi_k(\mathbf{r};\mathbf{R}_0)\chi(R)]$

Now doing some painful differentiation this equals

$-\sum_{l=1}^{N_a}\frac{1}{2M_l} \Phi_k(\mathbf{r};\mathbf{R}_0) \nabla_l^2\chi(\mathbf{r}) - \sum_{l=1}^{N_a}\frac{1}{2M_l}\chi(\mathbf{r})\nabla_l^2\Phi_k(\mathbf{r};\mathbf{R}_0) - \sum_{l=1}^{N_a}\frac{1}{M_l}[\nabla_l\chi(\mathbf{R})][\nabla_l \Phi_k(\mathbf{r};\mathbf{R}_0) ]$

The BO approx. amounts to assuming that the last two terms on the right hand side are negligible because

$\frac{1}{M_l}[\nabla_l\chi(\mathbf{R})][\nabla_l \Phi_k(\mathbf{r};\mathbf{R}_0)]\approx \frac{1}{2M_l}\chi(\mathbf{R})\nabla_l^2 \Phi_k(\mathbf{r};\mathbf{R}_0) ≪ \frac{1}{2M_e}\chi(\mathbf{R})\nabla_e^2 \Phi_k(\mathbf{r};\mathbf{R}_0) $

which is in turn because $\nabla_r \Phi_k (r;R)≈∇_e Φ_k (r;R)$ and $m_e≪M_l$

In other words, because the mass of an electron is much less than the mass of a nuclei it moves slower and the nuclei can be considered stationary with respect to the motion of the electron. Whew!

However, this reveals something VERY interesting. As we can see, it tells us that we can expect breakdown of the BO approximation any time the nuclear derivative of the electronic wavefunction changes rapidly.

For example in a conical intersection!!!

As you can see the electronic wavefunction changes rapidly with nuclear coordinates and it actually has a discontinuity.

enter image description here

Polarons are another example which have breakdown of the BO approximation and that's simply because the nuclei and electron are so far apart that the electrons cannot respond fast enough to the nuclei.

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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 dynamics).

In some cases, a (relatively) easy fix is possible - DBOC (diagonal Born-Oppenheimer correction), which is to take into account first-order correction only. Otherwise, non-adiabatic effects need to be taken into account (e.g. here is such a calculation for water).

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The Born-Oppenheimer approximation comprises two different approximations:

  1. Adiabatic separation of electron and nuclear coordinates

  2. 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 theory (DFPT).

The semi-classical approximation (2) can also cause some issues, particularly for light nuclei where quantum delocalisation, zero-point energy etc. are an issue. These quantum effects can be added back, for example using path-integral dynamics (esp. suitable for bosons), DFPT or vibrational SCF, or the light nuclei can be treated in an explicit quantum mechanical treatment (e.g. the double adiabatic separation). As well as systems with light nuclei, there are also experimental techniques where these issues are vital, e.g. muon spectroscopy, where the muons can be treated as very low-mass nuclei.

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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 charge distribution in space. This can play a role in chemical properties related to spin effects as the dipole moment of the nuclei are much different than what Born-Oppenheimer would tell you.

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    $\begingroup$ Is it really the "Born-Oppenheimer" approximation that assumes that the nuclei are point charges? Or would that be the "point-nucleus" approximation? Is it really incompatible to have a finite-sized nucleus and still be Born-Oppenheimer, as long as the wavefunction is a product of the nuclear and electronic wavefunctions? $\endgroup$ – Nike Dattani May 4 '20 at 2:25
  • $\begingroup$ My understanding is that the Born-Oppenheimer approximation is that the point-nucleus is a part of it. I am by no means an expert. $\endgroup$ – Raz Jugovic May 4 '20 at 18:01

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