2.2 The Born-Oppenheimer approximation

The forces on both electrons and nuclei due to their electric charge are of the same order of magnitude, and so the changes which occur in their momenta as a result of these forces must also be the same. One might, therefore, assume that the actual momenta of the electrons and nuclei were of similar magnitude. In this case, since the nuclei are so much more massive than the electrons, they must accordingly have much smaller velocities. Thus it is plausible that on the typical time-scale of the nuclear motion, the electrons will very rapidly relax to the instantaneous ground-state configuration, so that in solving the time-independent Schrödinger equation resulting from the Hamiltonian in equation 2.12, we can assume that the nuclei are stationary and solve for the electronic ground-state first, and then calculate the energy of the system in that configuration and solve for the nuclear motion. This separation of electronic and nuclear motion is known as the Born-Oppenheimer approximation [6].

Following Ziman [7], we assume the following form of an eigenfunction for the Hamiltonian (2.12):

$${\tilde \Psi} \left( \{ {\bf r}_i \} , \{ {\bf r}_\alpha \} ... ... r}_\alpha \} \right) \Phi \left( \{ {\bf r}_\alpha \} \right)$$ (2.18)

and require that $\Psi \left( \{ {\bf r}_i \} ; \{ {\bf r}_\alpha \} \right)$ (which is a wave-function only of the $\{ {\bf r}_i \}$ with the $\{ {\bf r}_\alpha \}$ as parameters) satisfies the time-independent Schrödinger equation for the electrons in a static array of nuclei:

$$\left[ -\frac{1}{2} \sum_i \nabla_i^2 - \sum_i \sum_{\alpha}... ...ht) \Psi \left( \{ {\bf r}_i \} ; \{ {\bf r}_\alpha \} \right)$$ (2.19)

in which the dependence of the eigenvalues ${\cal E}_{\mathrm e}$ on the nuclear positions is acknowledged. Applying the full Hamiltonian (2.12) to the whole wave-function:

$${\hat H} {\tilde \Psi} \left( \{ {\bf r}_i \} , \{ {\bf r}_\alpha... ...rt} \right] {\tilde \Psi} \left( \{ {\bf r}_i \} , \{ {\bf r}_\alpha \} \right)$$

$$\quad= \Psi \left( \{ {\bf r}_i \} ; \{ {\bf r}_\alpha \} \right)... ... {\bf r}_{\gamma} \right\vert} \right] \Phi \left( \{ {\bf r}_\alpha \} \right)$$

$$\qquad - \sum_{\beta} \frac{1}{2 m_{\beta}} \Bigl[ 2 {\bf\nabla}_... ...bla_{\beta}^2 \Psi \left( \{ {\bf r}_i \} ; \{ {\bf r}_\alpha \} \right) \Bigr]$$ (2.20)

The energy ${\cal E}_{\mathrm e}\left( \{ {\bf r}_\alpha \} \right)$ is called the adiabatic contribution of the electrons to the energy of the system. The remaining non-adiabatic terms contribute very little to the energy, which can be demonstrated using time-independent perturbation theory [8]. The first order correction arising from the first non-adiabatic term in the last line of equation 2.20 is of the form:

$$~- \int \prod_j {\mathrm d}{\bf r}_j \prod_{\beta} {\mathrm d}{\b... ...{\bf\nabla}_{\gamma} \Psi \left( \{ {\bf r}_i \} ; \{ {\bf r}_\alpha \} \right)$$

$$= - \sum_{\gamma} \int \prod_{\beta} {\mathrm d}{\bf r}_{\beta} \Ph... ...la}_{\gamma} \Psi \left( \{ {\bf r}_i \} ; \{ {\bf r}_\alpha \} \right) \right]$$

and the term in square brackets can be rewritten

$$\int \prod_j {\mathrm d}{\bf r}_j \Psi^{\ast} \left( \{ {\bf r}_i... ...{\bf\nabla}_{\gamma} \Psi \left( \{ {\bf r}_i \} ; \{ {\bf r}_\alpha \} \right) = {\textstyle{1 \over 2}}{\bf\nabla}_{\gamma} \int \prod_j {\mathrm... ...t\vert \Psi \left( \{ {\bf r}_i \} ; \{ {\bf r}_\alpha \} \right) \right\vert^2$$

$$= {\textstyle{1 \over 2}} {\bf\nabla}_{\gamma} (1) = 0 ,$$ (2.21)

since the normalisation of the electronic wave-function does not change when the nuclei move, so that the first order contribution vanishes. The second-order shift due to this term does not vanish and gives rise to transitions between electronic states as the ions move, otherwise known as the electron-phonon interaction, which will modify the energy.

The second non-adiabatic term in the final term of equation 2.20 will be largest when the electrons labelled $i$ are tightly bound to the nuclei labelled $\alpha$ in which case $\Psi \left( \{ {\bf r}_i \} ; \{ {\bf r}_\alpha \} \right) = \Psi \left( \{ {\bf u}_{(i,\alpha)} \} \right)$ where ${\bf u}_{(i,\alpha)} = {\bf r}_i - {\bf r}_{\alpha}$ and the first order correction from this term is

$$~- \int \prod_j {\mathrm d}{\bf r}_j \prod_{\beta} {\mathrm d}{\b... ...\right) \nabla_{\gamma}^2 \Psi \left( \{ {\bf u}_{(i,\alpha)} \} \right) \Bigr]$$

$$= - \sum_{\gamma} \frac{1}{2 m_{\gamma}} \left[ \int \prod_{\beta} ... ...right) \nabla_{\gamma}^2 \Psi \left( \{ {\bf u}_{(i,\alpha)} \} \right) \right]$$

$$= - \sum_{(k,\gamma)} \frac{1}{m_{\gamma}} \int \prod_{(j,\beta)} {... ...ac{1}{2} \nabla_{(k,\gamma)}^2 \Psi \left( \{ {\bf u}_{(i,\alpha)} \} \right) ,$$ (2.22)

and this quantity is of the order of the electronic kinetic energy multiplied by the ratio of the electron and nuclear masses, typically a factor of the order of $10^{-4}$ or $10^{-5}$, so that the contributions from this term to all orders can be neglected.

We therefore neglect the non-adiabatic terms and note that equation 2.20 is satisfied if $\Phi \left( \{ {\bf r}_\alpha \} \right)$ obeys a Schrödinger equation of the form

$$\left[ - \sum_{\beta} \frac{1}{2 m_{\beta}} \nabla_{\beta}^2... ... \right) = {\cal E} \Phi \left( \{ {\bf r}_\alpha \} \right) .$$ (2.23)

This adiabatic principle is crucial because it allows us to separate the nuclear and electronic motion, leaving a residual electron-phonon interaction. From this point on it is assumed that the electrons respond instantaneously to the nuclear motion and always occupy the ground-state of that nuclear configuration. Varying the nuclear positions maps out a multi-dimensional ground-state potential energy surface, and the motion of the nuclei in this potential can then be solved. In practice Newtonian mechanics generally suffices for this part of the problem^2.7, and relaxation of the nuclear positions to the minimum-energy configuration or molecular dynamics [11,12] can be performed. These aspects go beyond the scope of this dissertation so that from now on it is assumed that a system with a fixed nuclear configuration is to be treated, so that the electronic energy ${\cal E}_{\mathrm e}$ is a constant and the electronic wave-function $\Psi \left( \{ {\bf r}_i \} \right)$ obeys the Schrödinger equation 2.19. The dependence of the electronic wave-function on the nuclear positions $\{ {\bf r}_{\alpha} \}$ is now suppressed.