Born-Oppenheimer approximation:
Potential Energy Surface

Empirical observations of molecular spectroscopy show that the total energy of a molecule can be viewed as the sum of several approximately non-interacting parts. Born-Oppenheimer approximation shows how the electronic motions can be approximately separated from the nuclear motions.

Let us define a molecule by a geometry structure determined by the nuclear positions. If the nuclei have fixed positions, the nuclear kinetic term vanishes $\hat T_N=0$ and the nuclear repulsion term $\hat V_{NN}$ becomes a constant. The Hamiltonian expression of equation 1.2 has a shortened form that we label as electronic Hamiltonian

$$\hat H^{elec}=\hat T_e+\hat V_{Ne}+\hat V_{ee}$$ (2.5)

The solutions of the electronic Hamiltonian are the electronic wavefunctions that will have to be solved for every nuclear configuration $R_K$

$$\hat H^{elec}\vert\Psi^{elec}_{R_K}\rangle=E^{elec}_{R_K} \vert\Psi^{elec}_{R_K}\rangle$$ (2.6)

The $\hat V_{nn}$ term is not usually included in the electronic Hamiltonian since it is only a constant at a given nuclear configuration. However we can define the potential energy adding the term $\hat V_{nn}$ to the electronic energy.

$$U_{R_K}=E^{elec}_{R_K}+\hat V_{NN}$$ (2.7)

The solutions { $\vert\Psi^{elec}_{R_K}\rangle$} is a complete set of functions of the n-electrons space. So, the total wavefunction of the system should belong to a full space created from the tensorial product between the nuclear space and electronic space: $\mathbb{E}_{tot}=\mathbb{E}_{elec}\otimes\mathbb{E}_{nucl}$.

$$\vert \Psi \rangle = \sum_p \sum_q C_{pq} \quad \vert\chi^{nucl} (_{R_K})_p\rangle \vert\Psi^{elec} (_{R_{K'}})_q\rangle$$ (2.8)

where $\vert\chi^{nucl}(_{R_K})_p\rangle$ is the wavefunction that describes the nuclear motion.

Although it is not a basis of the whole space, in the Born-Oppenheimer expansion the crossed terms are avoided and the wavefunction has the form

$$\vert\Psi\rangle = \sum_p C_{p} \quad \vert\chi^{nucl} (_{R_K})_p\rangle \vert\Psi^{elec} (_{R_{K}})_p\rangle$$ (2.9)

If the total wavefunction has the form of equation 1.9 when it is introduced in the total Schrödinger equation 1.4 we have

$$(\hat H^{elec}+\hat V_{NN}+\hat T_N)\sum_p C_{p} \quad \vert\chi^... ...angle = E\sum_p C_{p} \quad \vert\chi^{nucl}_p\rangle \vert\Psi^{elec}_p\rangle$$ (2.10)

where the subscripts $R_K$ are omitted for clarity. Multiplying this equation by $\langle\Psi^{elec}_q\vert$ we find the following coupled differential equations

$$\sum_p C_p \quad\Bigl\{U_p\delta_{pq} + \langle \Psi^{elec}_q\ver... ...i^{elec}_p\rangle\Bigl\} \vert\chi^{nucl}_p\rangle = E\vert\chi_q^{nucl}\rangle$$ (2.11)

it means that the electronic states are coupled through the nuclear kinetic operator. Several steps are needed to proceed. The chain rule must be applied to the expression $\hat T_N\vert\Psi^{elec}_p\rangle\vert\chi^{nucl}_p\rangle$ and the stationary state must be assumed. Finally, to uncouple the above equations we have to assume that the nuclear kinetic operator is diagonal under the electronic representation

$$\langle \Psi^{elec}_q\vert\hat T_N\vert\Psi^{elec}_p\rangle = \delta_{qp}\langle \Psi^{elec}_q\vert\hat T_N\vert\Psi^{elec}_p\rangle$$ (2.12)

it means that the nuclear motion only involves one electronic state. This is the Born-Oppenheimer adiabatic approximation. Sometimes this approximation is not good and the nuclear motion has to include the participation of several electronic states[20,21]. In any case, the assumptions considered so far permit to write the whole Schrödinger equation as

$$(\hat T_N+U^{(c)}_q)\vert\chi^{nucl}\rangle_q=E\vert\chi^{nucl}\rangle_q$$ (2.13)

where

$$U^{(c)}_q= U_q + \langle \Psi^{elec}_q\vert\hat T_N\vert\Psi^{elec}_q\rangle$$ (2.14)

The final approximation is to consider that even the diagonal correction can be neglected and therfore $U^{(c)}\approx U$. This last step is usually valid [22].

In conclusion, the nuclei move on a Potential Energy Surface (PES) $U$, where $U$ comes from the solution of electronic Schrödinger equation 1.6, and $E$ is the total energy of the system. The PES is a concept that will be used all along this thesis and we must keep in mind the level of approximations we have assumed to achieve such concept.