The Born-Oppenheimer Approximation

The Born-Oppenheimer Approximation is the assumption that the electronic motion and the nuclear motion in molecules can be separated. It leads to a molecular wave function in terms of electron positions and nuclear positions .

This involves the following assumptions

The electronic wavefunction depends upon the nuclear positions but not upon their velocities, i.e., the nuclear motion is so much slower than electron motion that they can be considered to be fixed.
The nuclear motion (e.g., rotation, vibration) sees a smeared out potential from the speedy electrons.

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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 .

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

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

(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:

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

(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:

		$\displaystyle {\hat H} {\tilde \Psi} \left( \{ {\bf r}_i \} , \{ {\bf r}_\alpha... ...rt} \right] {\tilde \Psi} \left( \{ {\bf r}_i \} , \{ {\bf r}_\alpha \} \right)$
		$\displaystyle \quad= \Psi \left( \{ {\bf r}_i \} ; \{ {\bf r}_\alpha \} \right)... ... {\bf r}_{\gamma} \right\vert} \right] \Phi \left( \{ {\bf r}_\alpha \} \right)$
		$\displaystyle \qquad - \sum_{\beta} \frac{1}{2 m_{\beta}} \Bigl[ 2 {\bf\nabla}_... ...bla_{\beta}^2 \Psi \left( \{ {\bf r}_i \} ; \{ {\bf r}_\alpha \} \right) \Bigr]$	(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:

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

and the term in square brackets can be rewritten

$\displaystyle \int \prod_j {\mathrm d}{\bf r}_j \Psi^{\ast} \left( \{ {\bf r}_i... ...{\bf\nabla}_{\gamma} \Psi \left( \{ {\bf r}_i \} ; \{ {\bf r}_\alpha \} \right)$	$\textstyle =$	$\displaystyle {\textstyle{1 \over 2}}{\bf\nabla}_{\gamma} \int \prod_j {\mathrm... ...t\vert \Psi \left( \{ {\bf r}_i \} ; \{ {\bf r}_\alpha \} \right) \right\vert^2$
	$\textstyle =$	$\displaystyle {\textstyle{1 \over 2}} {\bf\nabla}_{\gamma} (1) = 0 ,$	(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 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

	$\displaystyle ~- \int \prod_j {\mathrm d}{\bf r}_j \prod_{\beta} {\mathrm d}{\b... ...\right) \nabla_{\gamma}^2 \Psi \left( \{ {\bf u}_{(i,\alpha)} \} \right) \Bigr]$
$\textstyle =$	$\displaystyle - \sum_{\gamma} \frac{1}{2 m_{\gamma}} \left[ \int \prod_{\beta} ... ...right) \nabla_{\gamma}^2 \Psi \left( \{ {\bf u}_{(i,\alpha)} \} \right) \right]$
$\textstyle =$	$\displaystyle - \sum_{(k,\gamma)} \frac{1}{m_{\gamma}} \int \prod_{(j,\beta)} {... ...ac{1}{2} \nabla_{(k,\gamma)}^2 \Psi \left( \{ {\bf u}_{(i,\alpha)} \} \right) ,$	(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

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

(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.

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The Born-Oppenheimer Approximation

The next few lectures will treat the problem of quantum chemistry, a subfield of quantum mechanics also known as molecular quantum mechanics. The idea of quantum chemistry is to use only the simple facts molecules and, indeed, all of ordinary matter, can be viewed as composed only of positively charged nuclei and negatively charged electrons. This universal description is then subject to a quantum mechanical treatment from which the properties of the system are derived or computed.

We begin our discussion of AIMD by considering a system of nuclei described by coordinates, ${\bf R}_1,...,{\bf R}_N \equiv {\bf R}$ , momenta, ${\bf P}_1,...,{\bf P}_N \equiv {\bf P}$ , and masses , and electrons described by coordinates, ${\bf r}_1,...,{\bf r}_{N_e} \equiv {\bf r}$ , momenta, ${\bf p}_1,...,{\bf p}_{N_e}\equiv {\bf p}$ , and spin variables, $s_1,...,s_{N_e}\equiv s$ . The Hamiltonian of the system is given by

$\displaystyle H$	$\textstyle =$	$\displaystyle \sum_{I=1}^N \frac{{\bf P}_I^2}{2M_I} + \sum_{i=1}^{N_e}\frac{{\b... ...f R}_I-{\bf R}_J\vert} - \sum_{i,I}\frac{Z_Ie^2}{\vert{\bf R}_I-{\bf r}_i\vert}$

	$\textstyle \equiv$	$\displaystyle T_{\rm N} + T_{\rm e} + V_{\rm ee}({\bf r}) + V_{\rm NN}({\bf R}) + V_{\rm eN}({\bf r},{\bf R})$

where

is the mass of the electron, and

is the charge on the

th nucleus. In the second line, $T_{\rm N}$ , $T_{\rm e}$ , $V_{\rm ee}$ , $V_{\rm NN}$ , and $V_{\rm eN}$ represent the nuclear and electron kinetic energy operators and electron-electron, electron-nuclear, and nuclear-nuclear interaction potential operators, respectively. Note that this Hamiltonian is universal in that it describes all of everyday matter from biological macromolecules such as proteins, enzymes and nucleic acids, to metals and semiconductors to synthetic materials such as plastics. Thus, if we could solve for the eigenvalues and eigenfunctions of this Hamiltonian, we could, predict any property we wished of a given system. This fact lead the physicist P. A. M. Dirac to comment that all of chemistry is a solved problem, at least, in principle. Of course, the problem cannot actually be solved exactly, so approximation methods are needed and it is in the development of approximation methods and an analysis of their accuracy that quantum chemistry is focused.

If we sought to solve the complete quantum mechanical problem, we start by seeking the eigenfunctions and eigenvalues of this Hamiltonian, which will be given by solution of the time-independent Schrödinger equation

$\begin{displaymath} \left[ T_{\rm N} + T_{\rm e} + V_{\rm ee}({\bf r}) + V_{\rm... ...,{\bf R})\right]\Psi({\bf x},{\bf R}) = E\Psi({\bf x},{\bf R}) \end{displaymath}$

(1)

where ${\bf x}\equiv ({\bf r},s)$ denotes the full collection of electron position and spin variables, and $\Psi({\bf x},{\bf R})$ is an eigenfunction of

with eigenvalue

. Clearly, an exact solution of Eq. (1) is not possible and approximations must be made. We first invoke the Born-Oppenheimer approximation by recognizing that, in a dynamical sense, there is a strong separation of time scales between the electronic and nuclear motion, since the electrons are lighter than the nuclei by three orders of magnitude. In terms of Eq. (1), this can be exploited by assuming a quasi-separable ansatz of the form

$\begin{displaymath} \Psi({\bf x},{\bf R}) = \phi({\bf x},{\bf R})\chi({\bf R}) \end{displaymath}$

(2)

where $\chi({\bf R})$ is a nuclear wave function and $\phi({\bf x},{\bf R})$ is an electronic wave function that depends parametrically on the nuclear positions.

The Born-Oppenheimer (named for its original inventors, Max Born and Robert Oppenheimer) is based on the fact that nuclei are several thousand times heavier than electrons. The proton, itself, is approximately 2000 times more massive than an electron. In a dynamical sense, the electrons can be regarded as particles that follow the nuclear motion adiabatically, meaning that they are ``dragged'' along with the nuclei without requiring a finite relaxation time. This, of course, is an approximation, since there could be non-adiabatic effects that do not allow the electrons to follow in this ``instantaneous'' manner, however, in many systems, the adiabatic separation between electrons and nuclei is an excellent approximation. Another consequence of the mass difference between electrons and nuclei is that the nuclear components of the wave function are spatially more localized than the electronic component of the wave function. In the classical limit, the nuclear are fully localized about single points representing classical point particles. Thus, it follows that the nuclear wave function rises more steeply than the electronic wave function, which means that $\nabla_I\chi({\bf r})\gg\nabla_I\phi({\bf x},{\bf R})$ , from which we may approximate

$\displaystyle T_{\rm N}$			(3)
$\displaystyle psi({\bf x},{\bf r})\chi({\bf R})$	$\textstyle =$	$\displaystyle -{\hbar^2 \over 2}\sum_{I=1}^N {1 \over M_I} \left[\psi({\bf x},{... ...f R}) + 2\nabla_I\psi({\bf x},{\bf R})\cdot\nabla_I\phi({\bf x},{\bf R})\right]$
	$\textstyle \approx$	$\displaystyle -{\hbar^2 \over 2}\sum_{I=1}^N {1 \over M_I} \psi({\bf x},{\bf R})\nabla_I^2\chi({\bf R})$	(4)

Substitution of Eq. (2) into Eq. (1) and using the approximation that the nuclear wave function $\chi({\bf R})$ is more localized than the electronic wave function, i.e. $\nabla_I\chi({\bf R}) \gg \nabla_I \phi({\bf x},{\bf R})$ , yields

$\begin{displaymath} \frac{\left[T_{\rm e} + V_{\rm ee}({\bf r}) + V_{\rm eN}({\b... ...m N} + V_{\rm NN}({\bf R})\right]\chi({\bf R})}{\chi({\bf R})} \end{displaymath}$

(5)

From the above, it is clear that the left side can only be a function of ${\bf R}$ alone. Let this function be denoted, $\varepsilon({\bf R})$ . Thus,

$\displaystyle \frac{\left[T_{\rm e} + V_{\rm ee}({\bf r}) + V_{\rm eN}({\bf r},{\bf R})\right] \phi({\bf x},{\bf R})} {\phi({\bf x},{\bf R})}$	$\textstyle =$	$\displaystyle \varepsilon({\bf R})$

$\displaystyle \left[T_{\rm e} + V_{\rm ee}({\bf r}) + V_{\rm eN}({\bf r},{\bf R})\right] \phi({\bf x},{\bf R})$	$\textstyle =$	$\displaystyle \varepsilon({\bf R})\phi({\bf x},{\bf R})$	(6)

Eq. (6) is an electronic eigenvalue equation for an electronic Hamiltonian, $H_{\rm e}({\bf R}) = T_{\rm e} + V_{\rm ee}({\bf r}) + V_{\rm eN}({\bf r},{\bf R})$ which will yield a set of normalized eigenfunctions, $\phi_n({\bf x},{\bf R})$ and eigenvalues, $\varepsilon_n({\bf R})$ , which depend parametrically on the nuclear positions, ${\bf R}$ . For each solution, there will be a nuclear eigenvalue equation:

$\begin{displaymath} \left[T_{\rm N} + V_{\rm NN}({\bf R}) + \varepsilon_n({\bf R})\right]\chi({\bf R}) = E\chi({\bf R}) \end{displaymath}$

(7)

Moreover, each electronic eigenvalue, $\varepsilon_n({\bf R})$ , will give rise to an electronic surface, and these surfaces are known as Born-Oppenheimer surfaces. Thus, the full internuclear potential for each electronic surface is given by $V_{\rm NN}({\bf R})+\varepsilon_n({\bf R})$ . On each Born-Oppenheimer surface, the nuclear eigenvalue problem can be solved, which yields a set of levels (rotational and vibrational in the nuclear motion). This is illustrated in the figure below:

The Born-Oppenheimer surfaces are surfaces on which the nuclear dynamics is described by a time-dependent Schrödinger equation for the time-dependent nuclear wave function $X({\bf R},t)$ :

$\begin{displaymath} \left[T_{\rm N} + V_{\rm NN}({\bf R}) + \varepsilon_n({\bf R... ...]X({\bf R},t) = i\hbar\frac{\partial}{\partial t}X({\bf R},t) \end{displaymath}$

(8)

will evolve. The physical interpretation of Eq. (8) is that the electrons respond instantaneously to the nuclear motion, therefore, it is sufficient to obtain a set of instantaneous electronic eigenvalues and eigenfunctions at each nuclear configuration, ${\bf R}$ (hence the parametric dependence of $\phi_n({\bf x},{\bf R})$ and $\varepsilon_n({\bf R})$ on ${\bf R}$ ). The eigenvalues, in turn, give a family of (uncoupled) potential surfaces on which the nuclear wave function can evolve. Of course, these surfaces can (and often do) become coupled by so called non-adiabatic effects, contained in the terms that have been neglected in the above derivation.

An important assumption of the Born-Oppenheimer approximation is that there are no excitations of the electrons among the various surfaces. Such excitations constitute non-adiabatic effects which are, therefore, neglected. As an example of a conditions in which this approximation is valid, consider a system at temperature . If the electrons are in their ground state $\varepsilon_0({\bf R})$ , then, if $\varepsilon_1({\bf R})$ denotes the first excitates state, there will be no excitations to this state if

$\begin{displaymath} \vert\varepsilon_1({\bf R})-\varepsilon_0({\bf R})\vert \gg kT \end{displaymath}$

(9)

for all nuclear configurations. Without complete knowledge of these surfaces, it is not possible to know a priori if this condition will be satisfied, and there could be regions where the surfaces approach each other with an energy spacing close to

. If the system visits such nuclear configurations, then the Born-Oppenheimer approximation will break down.

In many cases, non-adiabatic effects can be neglected, and we may consider motion only on the ground electronic surface described by:

$\displaystyle \left[T_{\rm e} + V_{\rm ee}({\bf r}) + V_{\rm eN}({\bf r},{\bf R})\right] \phi_0({\bf x},{\bf R})$	$\textstyle =$	$\displaystyle \varepsilon_0({\bf R})\phi_0({\bf x},{\bf R})$

$\displaystyle \left[T_{\rm N} + \varepsilon_0({\bf R}) + V_{\rm NN}({\bf R})\right] X({\bf R},t)$	$\textstyle =$	$\displaystyle i\hbar\frac{\partial}{\partial t}X({\bf R},t)$	(10)

Moreover, if nuclear quantum effects can be neglected, then we may arrive at classical nuclear evolution by assuming $X({\bf R},t)$ is of the form

$\begin{displaymath} X({\bf R},t) = A({\bf R},t)e^{iS({\bf R},t)/\hbar} \end{displaymath}$

(11)

and neglecting all terms involving $\hbar$ , which yields an approximate equation for $S({\bf R},t)$ :

$\begin{displaymath} H_{\rm N}(\nabla_1S,...,\nabla_NS,{\bf R}_1,...,{\bf R}_N) + \frac{\partial S}{\partial t} = 0 \end{displaymath}$

(12)

which is just the classical Hamiltonian-Jacobi equation with

$\begin{displaymath} H_{\rm N}({\bf P}_1,...,{\bf P}_N,{\bf R}_1,...,{\bf R}_N) =... ...f P}_I^2}{2M_I} + V_{\rm NN}({\bf R}) + \varepsilon_0({\bf R}) \end{displaymath}$

(13)

denoting the classical nuclear Hamiltonian. The Hamilton-Jacobi equation is equivalent to classical motion on the ground-state surface, $E_0({\bf R}) = \varepsilon_0({\bf R}) + V_{\rm NN}({\bf R})$ given by

$\displaystyle \dot{{\bf R}}_I$	$\textstyle =$	$\displaystyle {{\bf P}_I \over M_I}$

$\displaystyle \dot{{\bf P}}_I$	$\textstyle =$	$\displaystyle -\nabla_I E_0({\bf R})$	(14)

Note that the force $-\nabla_I E_0({\bf R})$ contains a term from the nuclear-nuclear repulsion and a term from the derivative of the electronic eigenvalue, $\varepsilon_0({\bf R})$ . Because of the Hellman-Feynman theorem, the latter can be expressed as

$\begin{displaymath} \nabla_I \varepsilon_0({\bf R}) = \langle \phi_0({\bf R})\vert\nabla_I H_{\rm e}({\bf R})\vert\phi_0({\bf R})\rangle \end{displaymath}$

(15)

Equations (14) and Eq. (15) form the theoretical basis of the AIMD approach. The practical implementation of the AIMD method requires an algorithm for the numerical solution of Eq. (14) with forces obtained from Eq. (15) at each step of the calculation. Moreover, since an exact solution for the ground state electronic wave function, $\vert\phi_0({\bf R})\rangle$ and eigenvalue, $\varepsilon_0({\bf R})$ are not available, in general, it is necessary to introduce an approximation scheme for obtaining these quantities. This is the topic of the next section

Master of Science in Chemistry

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Friday, April 1, 2011