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Adiabatic and diabatic states in the Born-Oppenheimer approximation = MACRO-m3c-diab

Consider the forces between two slowly moving atoms or molecules in a collision. According to the Born-Oppenheimer approximation, the motion of the nuclei can be separated from the motion of the electrons. In order to describe the forces, defined by the states of the valence electrons, the states have to be given in an appropriate representation: adiabatic or a diabatic.

Born-Oppenheimer approximation

The motion of a diatomic system is described by the Schroedinger equation. This yields the formal set of exact coupled equations for the nuclear wave functions $\chi_i(R)$ given in equation 0.11.

$\begin{displaymath}[T_R + T''_{ii} + V_{ii}(R) - E]\chi_i(R) = - \Sigma_{i \neq j} (V_{ij} + T'_{ij} + T''_{ij}) \chi_j(R), \end{displaymath}$ where T_R is the nuclear kinetic energy operator in the Hamiltonian H = T_R + H_e and where
$\begin{displaymath}V_{jk}(R) = < \phi_j \vert H_e \vert \phi_k > \end{displaymath}$ are the electronic matrix elements of H_e, and

$\displaystyle T'_{ij} = -2 ( \hbar^2/2M) < \phi_i \vert \nabla_R \vert \phi_j > . \nabla_R$
$\displaystyle T''_{ij} = -2 ( \hbar^2/2M) < \phi_i \vert \nabla_R^2 \vert \phi_j >.$

The Born-Oppenheimer approximation allows for a substantial simplification of the equations of motion of a molecular system. The Born-Oppenheimer approximation is probably the most powerful tool to describe forces acting between two atoms. This approximation, which is also known as the adiabatic approximation, is based on the enormous mass difference between light and therefore very mobile, electrons, and the heavy slow nuclei. The ratio M/m of nuclear to electronic mass runs from a minimum value of 2000 for H to 30,000 for atmospheric gasses and over 400.000 for the heaviest atoms.

The physical consequence of that large mass ratio, is that, to first order, the motion of the nuclei does not influence the highly quantized motion of the electrons. In other words, the interactions between the atoms are described by potentials, which actually are the expectation values of the energy of electronic states. The potentials, as well as the wave functions of the electrons, depend parametrically upon the internuclear separation of the two atoms.

The Born-Oppenheimer approximation can be expected to be valid in the region of nuclear velocity below $v << e^2/\hbar$ , if the following assumptions hold: firstly the rotational velocity is not much larger than the radial, such that we can concentrate on the radial component of $\nabla_R$ , and secondly the electronic states do no vary greatly over distance much smaller than the atomic unit of distance a₀, i.e.

$\begin{displaymath}\vert \nabla \phi_i \vert < a_0^{-1} \vert\phi_i\vert. \end{displaymath}$

[Arguments for the validity of the approximation in this region are given by O'Malley (link type: `detailed in/argued in/external'; target: O'Malley)]

The mathematical counterpart of the qualitative difference between the electronic motion $(\phi_i)$ and the nuclear motion $(\chi_i)$ is the recognition that under all reasonable conditions the T' and T'' terms in equation 0.11 are totally negligible compared with the other terms in the equation. If one accordingly makes the Born-Oppenheimer approximation to neglect T' and T'', equation 0.11 becomes
$\begin{displaymath}[T_R + V_{ii}(R) - E]\chi_i(R) = - \Sigma_{i \neq j} (V_{ij} \chi_j(R)). \end{displaymath}$

Adiabatic representation

In the adiabatic representation, or more precisely the stationary adiabatic representation, of the electronic states, the electronic functions $\phi_i^{ad}$ of the system are defined as the stationary eigenvalues of the electronic Hamiltonian H_e:

$\begin{displaymath}H_e \phi_i^{ad} = V_{ii}^{ad}(R) \phi_i^{ad} \end{displaymath}$ or equivalently in matrix form
$\begin{displaymath}V_{ij}^{ad} = < \phi_i^{ad} \vert H_e \vert \phi_j^{ad} > = V_{ii}^{ad} \delta_{ij}. \end{displaymath}$ This is the diagonal representation of H_e in stationary states, called adiabatic states. With this representation in the adiabatic states, equation 0.11 for the nuclear motion reduces to
$\begin{displaymath}[T_R + V_{ii}^{ad}(R) - E]\chi_i(R) = 0. \end{displaymath}$ The essential element of equation 0.19 is that the nuclear functions $\chi_i$ are totally uncoupled, so that the states, i, are permanent. In other words, the equation describes a one-state problem.

A further interesting property of the stationary adiabatic representation is the famous non-crossing rule of von Neumann and Wigner , which states that two potential curves, V_ii^ad(R) and V_jj^ad(R) may not cross if they have the same symmetry (spin, parity, angular momentum).

Diabatic representation

The diabatic representation is used to describe two-state and many-state problems: the electronic state is described as either covalent or ionic, with the corresponding electronic wave functions $\phi_{cov}$ and $\phi_{ion}$ . In this diabatic representation, electronic states of the same symmetry can violate the non-crossing rule, because they are not the stationary eigenvalues which diagonalise H_e

$\displaystyle V_{ij} = < \phi_i^{ad} \vert H_e \vert \phi_j^{ad} > = \left[ \be... ...eft[ \begin{array}{cc} H_{11} & H_{12} \\ H_{21} & H_{22} \end{array} \right].$

Let us choose the wave functions $\phi_{cov}$ and $\phi_{ion}$ as the electronic basis for expanding the full molecular function $\Psi$ in the fashion of equation 0.10
$\begin{displaymath}\Psi(r,R) = \phi_{cov} \chi_{cov}(R) + \phi_{ion} \chi_{ion}(R). \end{displaymath}$ For these $\phi$ 's the Born-Oppenheimer approximation is accurate, so that the two relevant nuclear wave functions $\chi_{cov}$ and $\chi_{ion}$ satisfy the equation of motion 0.11 in the simple form

$\displaystyle [T_R + V_{cov}(R) - E ] \chi_{cov}(R)$	=	$\displaystyle - V_{cov,ion} \chi_{ion}(R)$
$\displaystyle {} [T_R + V_{ion}(R) - E] \chi_{ion}(R)$	=	$\displaystyle - V_{ion,cov} \chi_{cov}(R).{}$

The diagonal V_ion and V_cov are the potential energies for elastic motion in that particular state $\phi_{ion}$ and $\phi_{cov}$ , while the non-diagonal elements of V_ion,cov and V_cov,ion provide the coupling between the two states.

The probability of the transition from one state to the other is given by the Landau-Zener formula . The probability is only appreciable in the neighbourhood of the point where the potential energy curves V_cov and V_ion cross and for larger velocities.