Phy5670/JahnTellerEffect

Introduction

The Jahn-Teller effect (JTE), also called Jahn Teller distortion, is the spontaneous geometrical distortion of a nonlinear molecule to lower the overall energy of the system by breaking the degeneracy of a ground or excited state.

The Born-Oppenheimer Approximation

The Born-Oppenheimer approximation is often the preferred perturbative method in which to describe the JTE. The Born-Oppenheimer approximation, which is similar in some respects to the adiabatic approximation, is based on the largest contribution to a molecular spectra arising from the electronic motion, followed by the contribution of nuclear vibration, followed by nuclear rotation [1]. These three terms appear in increasing order with respect to $\left({\frac {m}{M}}\right)^{1/4}$, where $m$ is the electronic mass and $M$ is the average nuclear mass. Nuclear vibrations corresponds to second order, while nuclear rotational motion corresponds to fourth order in $\left({\frac {m}{M}}\right)^{1/4}$ [1]. (The electronic motion is zeroth order in $\left({\frac {m}{M}}\right)^{1/4}$.)

In the BO approximation, the nuclei of atoms are assumed to be fixed in space, and therefore their kinetic energy can be neglected. Schrodinger's equation for the electrons is solved - while still including the Coulomb potential resulting from the interaction between the electrons and nuclei. This problem is solved repeatedly for very small changes in position, resulting in an expression for the electronic energy as a function of nuclei position. This is called the potential energy surface. The nuclear kinetic energy is the reintroduced to the Hamiltonian, and Schrodinger's equation is solved again, this time yielding the total energy of the molecule.

For the remained of this section I will define $\kappa =\left({\frac {m}{M}}\right)^{1/4}$, $r_{l}$ is the local nuclear coordinate (where r can be x,y,z), $q_{k}$ is the electron coordinate (where q can be x,y,z). A subscript $e$ denotes an electronic quantity. A subscript $n$ denotes a nuclear quantity.

The total potential energy of the system is

$\sum _{q}\sum _{r}U\left(q_{1},q_{2},q_{3}...;r_{1},r_{2},r_{3},...\right)=U(Q,R)$

where Q encompasses all electronic coordinates, and R encompasses all nuclear coordinates, and the sum on $q$ and the sum on $r$ represent the permutation of x,y,z.

The electronic kinetic energy is

$T_{e}=-{\frac {\hbar ^{2}}{2m}}\sum _{q}\sum _{k}{\frac {\partial ^{2}}{\partial q_{k}^{2}}}$

The nuclear kinetic energy is

$T_{n}=-\kappa ^{4}{\frac {\hbar ^{2}}{2m}}\sum _{r}\sum _{l}\mu _{l}{\frac {\partial ^{2}}{\partial r_{l}^{2}}}$

where $\mu _{l}$ is a dimensionless number on order 1 [1].

The Hamiltonian for a system of electrons and nuclei is given by:

$H=T_{e}+T_{n}+V\left(Q\right)+V(R,Q)$

where the potential has been broken into a part that relies only on the electron coordinates, and a part that includes the electron-nucleus interaction. Using the Born-Oppenheimer approximation we can write a pair of equations:

$\left[T_{e}+V(R,Q)\right]\phi _{e}(R,Q)=W_{e}(Q)\phi _{e}(R,Q)$

$\left[T_{n}+W_{e}(Q)+V(Q)\right]\chi _{n}(Q)=E_{e}\chi _{n}(Q)$

where $\phi _{e}(R,Q)$ is the electronic wave function (Q appears only as a constant in this particular equation), $\chi _{n}(Q)$ is the nuclear wave function, and $W_{e}+V$ is the effective potential as seen by each electronic state [2].

When the electronic states are non-degenerate, the Born-Oppenheimer wave-function are the solutions to the following Hamiltonian, where $\Delta H$ (the second line) is treated as a perturbation [2]:

$H=H_{0}+\Delta H=\left(T_{e}+T_{n}+V(R,Q^{0})+V(Q)+W_{e}(Q)-W_{e}(Q^{0})\right)+\left(V(R,Q)-V(R,Q^{0})-W_{e}(Q)+W_{e}(Q^{0})\right)$

with zero order solutions[2]:

$\psi _{e,n}=\phi _{e}\left(R,Q\right)\chi _{n}(Q)$

When there are degenerate electronic states, the Born-Oppenheimer wave-function are the solutions to the following Hamiltonian [2]:

$H=H_{0}+\Delta H=\left(T_{e}+T_{n}+V(R,Q^{0})+V(Q^{0})+\sum _{e}\left({\frac {W_{e}(Q)-W_{e}(Q^{0})}{[\Gamma ]}}\right)\right)+\left(V(R,Q)-V(R,Q^{0})-\sum _{e}\left({\frac {W_{e}(Q)-W_{e}(Q^{0})}{[\Gamma ]}}\right)\right)$

where $[\Gamma ]$ represents the dimensionality of the degeneracy, and the sum is over all degenerate states. The zero order solutions are given by[2]:

$\psi _{\{e\},n}=\sum _{e}\phi _{e}\left(R,Q\right)\chi _{e,n}(Q)$

References

[1] M. Born and R. J. Oppenheimer, Ann. Physik (Leipzig) 89, 457 (1927)

[2] R. Englman, The Jahn-Teller Effect in Molecules and Crystals John Wiley and Sons, 1972

[3] H.A. Jahn and E. Teller. Proc. Roy. Soc. A, 161, 220 (1937)