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, and is the result of coupling between electrons and atomic nuclei. It was shown by Hermann Jahn and Edward Teller in 1937 that the orbital degeneracy of the electronic state in a polyatomic molecule cannot be stable unless the molecule is linear.

Jahn and Teller applied group theory to the perturbation calculation and examined the elements of the perturbation matrix. They showed that all the linear matrix elements vanish only if the molecule has complete axial symmetry (meaning the molecule is linear), and deduced that "all non-linear nuclear configurations are therefore unstable for an orbitally degenerate electronic state" [1]. By this reasoning it is possible to conclude that a polyatomic molecule that is not linear does not possess orbital degeneracy of the electronic ground state. Even molecules that do not have a strict degeneracy, but merely very closely spaced energy levels exhibit Jahn-Teller distortion. This effect in nearly degenerate molecules has been named the pseudo Jahn-Teller effect (PJTE).

The Jahn-Teller effect is of interest in systems with a large number of interacting electrons and nucleui, which constitues a many body problem. It cannot be solved analytically and is beyond the practicality of using a computer. Therefore, as with most many body problems, approximations must be made. The application of the adiabatic approximation, which allows separation of variables based on the difference in masses of electrons and nuclei, led to a more comprehensive understanding of the Jahn-Teller effect. In the formulation of the adiabatic approximation, the Jahn-Teller effect can be written as follows: "if the adiabatic potential of the system, which is the formal solution of the electronic part of the Schrodinger equation, has several crossing sheets, then at least one of these sheets has no extremum at the crossing point" [2]. In the image below, a characteristic adiabatic potential surface is shown (left), where the point of degeneracy does not constitute an extremum. On the right, a characteristic potential for a linear molecule is shown, where the point of degeneracy is also a minimum for each sheet. (In both plots, QFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle _{\epsilon} } and QFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle _{\theta} } are the normal coordinates of the system.)

Further application of the solution of the nuclear part specifies the expected behavior of the nuceli. It is the combined solutions of the electronic and nuclear parts of the Schrodinger equation that allow one to predict the behavior of the nuclei. Before this was well understood, many scientists were unable to detect Jahn-Teller distortion in systems they suspected should exhibit the effect.

Magnetic resonance techniques can be used to study the Jahn-Teller effect as a result of the highly sensitive magnetic nuclei present in these systems. The method for observing Jahn-Teller distortion that will be elaborated upon in this wiki is electron paramagnetic resonance (EPR), aka electron spin resonance (ESR). Other magnetic resonance methods that can be used to observe the JTE include nuclear magnetic resonance, acoustic paramagnetic resonance, paraelectic resonance, and Mossbaauer spectroscopy (nuclear gamma resonance) [3].

Orbitals and Energy Levels

Jahn-Teller distortion is most commonly observed in transition metal complexes. Transition metals occupy the middle of the periodic table, groups 4-11, and posses incomplete d-subshells. There are five d-orbitals: dFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle _{xy} } , dFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle _{xz} } , dFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle _{yz} } , dFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle _{z^2} } , dFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle _{x^2-y^2} } . In the presence of crystal fields, these orbitals split into three orbitals, denoted tFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle _{2g} } , which are lower in energy for octahedral symmetry, and two orbitals, denoted eFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle _g } , which are higher in energy for octahedral symmetry.

Jahn-Teller distortion occurs when there is an orbital degeneracy. This means that there are multiple, and energetically equivalent, ways of filling the orbitals. The CuFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{II}} ion is well known for inducing Jahn-Teller distortion in CuFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{II}} complexes. Copper has electronic structure given by: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[Ar\right] 4s^1 3d^{10}} . Therefore, we expect CuFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{II}} to have nine d-shell electrons. Since there are two energetically equivalent ways to distribute these electrons in the d-orbitals, we expect CuFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{II}} to exhibit Jahn-Teller distortion.

If we take an ion with eight d-shell electrons, we may not expect such a species to exhibit Jahn-Teller distortion. This is because there can be only a single way to distribute the electrons. (However, if the ion is low-spin both eFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle _g } electrons are in the same orbital, and therefore the ion does possess a degeneracy.) There is no degeneracy, so there is no Jahn-Teller distortion. An example of this type of ion is high spin NiFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{II}} .

Returning to the example of CuFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{II}} . A CuFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{II}} complex often consists of a CuFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{II}} surrounded by 6 coordinate atoms, forming a complex with octahedral symmetry.

This splitting occurs because the orbitals aligned with the atoms that are now farther away from the central ion drop in energy because they experience less repulsion. The orbitals closer to the in-plane atoms raise in energy.

Following are additional examples of ions that are associated with Jahn-Teller distortion. Jahn-Teller distortion is not limited to complexes with ions having the number of d-electrons listed in the following image. Different geometries can affect the number of electrons in the d-orbital that will facilitate Jahn-Teller distortion.

Approximations and the Vibronic Interaction

There are two approximations that can be used to treat the interactions between the electrons and the nuclei in a many-body system. These two methods are the adiabatic approximation, and the Born-Oppenheimer Approximation (sometimes referred to as the crude adiabatic approximation).

The adiabatic approximation is based on the significantly different electron masses and nuclear masses, which allows us to assume that the nuclei move much more slowly than the electrons. As a result, the distribution of the electrons can be determined from an instantaneous configuration of the nuclei. We assume the nuclei are fixed at some instantaneous (not necessarily equilibrium) position and perform a separation of translational and rotational variables. In effect, we are solving the equations for the electronic motion in order to determine the potential in which the nuclei move. These potentials are termed adiabatic potential energy sheets.

The Born-Oppenheimer approximation begins with the assumption that the nuclei are fixed at their equilibrium position (i.e. ideal lattice points). A static interaction between the electrons and nuclei must be added to compensate for the actual instantaneous position of the nuclei [5]. The nuclear motion is then solved for, yielding an effective potential which is used in the determination of the electronic motion.

The Born-Oppenheimer and adiabatic approximations both rely on the assumption that the mass of the electron (m) is much less than the mass of the nucleus (M). It is interesting to derive some relations between this quantity, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{m}{M} } , and the velocity of the nucleus, the displacement of the nucleus from equilibrium, and the average energy. These relations can be used to justify the validity of certain steps in the upcoming approximations.

The uncertainty principle along with the expression for the average momentum allows us to write the average velocity of the electron in terms of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar } , d, and m. In what follows, p is the average momentum of the electrons, v and V are the average velocity of the electrons and nuclei, respectively, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_0 } is the equilibrium point of the nuclei, Q is the displacement from the equilibrium point, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega } is the frequency of vibration of the nuclei.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{p} \sim \frac{\hbar}{\vec{d}} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{p} = m\vec{v} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v} \sim \frac{\hbar}{m\vec{d}} }

Now examine the nucleus. The kinetic energy and the potential energy of the nucleus can be related through the virial theorem.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2} M V^2 \sim \frac{1}{2}M\omega^2 Q^2 \sim \hbar\omega }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V \sim \left(\frac{\hbar \omega}{M}\right)^2 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q \sim \left(\frac{\hbar}{M \omega}\right)^2 }

We can find an expression for the frequency of vibration, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega } , by relating the force constant resulting from the displacement of the nucleus from its equilibrium positon to the Coulomb's interaction resulting from a given electronic distribution.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\omega^2 \sim \frac{e^2}{d^3} }

The virial theorem relates the potential energy of the electron to the kinetic energy of the electron, which can be substituted into the above equation to yield an expression for the frequency of vibration.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{e^2}{d} \sim m v^2 \sim m\left(\frac{\hbar}{m\vec{d}}\right)^2}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\omega^2 \sim \frac{\hbar^2}{md^4}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega^2 \sim \frac{\hbar^2}{m^2d^4}\frac{m}{M} \rightarrow \omega \sim \frac{\hbar}{md^2}\left(\frac{m}{M}\right) }

These results can be put back into the expressions for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V } .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V \sim v\left(\frac{m}{M}\right)^{3/4} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q \sim d\left(\frac{m}{M}\right)^{1/4} }

Thus we have found the dependence of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V } on the small quantity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{m}{M} } .

Similarly, we can use the uncertainty relation between energy and time to estimate characteristic energy gap of the electron energy spectrum, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{\Delta E} } :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{\Delta E} \sim \frac{\hbar}{\overline{\Delta t}} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{\Delta t} \sim \frac{d}{\vec{v}} \sim \frac{m d^2}{\hbar} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{\Delta E} \sim \frac{\hbar^2}{md^2} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\hbar}{md^2} \sim \left(\frac{M}{m}\right)^{1/2}\omega}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\hbar\omega}{\overline{\Delta E}} \sim \left(\frac{m}{M}\right)^{1/2}}

This relation will justify the validity of the adiabatic approximation.

The Adiabatic Approximation

We begin an overview of the adiabatic approximation by writing the Hamiltonian for a system of electrons and nuclei:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(r,Q) = T_e(r) + T_n(Q) + V(r) + V\left(r,Q\right) }

The first term is the kinetic energy of the electron, the second term is the kinetic energy of the nuclei, the third term contains the electron-electron interactions, and the fourth term contains the electron-nuclei and nuclei-nuclei interactions.

The kinetic energy of the nuclei is considered a small perturbation, and Schrodinger's equation can be solved for the electron motion assuming fixed positions of the nuclei. The exact eigenfunctions of the exact Hamiltonian can be written as (the sum takes into account the degeneracy):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi_{e,n} = \sum_e \phi_e\left(r,Q\right)\chi_{e,n}(Q) }

where the functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_e\left(r,Q\right)} and $\chi _{e,n}\left(Q\right)$ come from the equations:

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

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

In the first equation we have assumed that nuclei to be stationary and solved for the electronic wave functions $\phi _{e}\left(r,Q\right)$, which parametrically contains the variable Q. In the second equations we assume the solution to the first equation remains valid, and then solve for the vibrational wave functions, $\chi _{e,n}\left(Q\right)$.

To find a system of differential equations we apply the Hamiltonian to the eigenfunction:

$H\Psi _{e,n}=\left[T_{e}(r)+T_{n}(Q)+V(r)+V\left(r,Q\right)\right]\sum _{e}\phi _{e}\left(r,Q\right)\chi _{e,n}(Q)$

And then we multiply from the left by $\phi _{e'}^{*}\left(r,Q\right)$, and integrate with respect to r:

$\int dr\left(\phi _{e'}^{*}\left(r,Q\right)H\Psi _{e,n}\right)=\int dr\left(\phi _{e'}^{*}\left(r,Q\right)\left[T_{e}(r)+T_{n}(Q)+V(r)+V\left(r,Q\right)\right]\sum _{e}\phi _{e}\left(r,Q\right)\chi _{e,n}(Q)\right)$

Recall that:

$T(Q)=\hbar ^{2}\sum _{i}{\frac {1}{2M_{i}}}{\frac {\partial ^{2}}{\partial Q_{i}^{2}}}$

where i labels nuclear degrees of freedom and M$_{i}$ is the reduced mass related to the coordinate Q$_{i}$. We get as a result:

$\sum _{e}\left[T(Q)\delta _{e,e'}-\hbar ^{2}\sum _{i}{\frac {1}{M_{i}}}\left[\int \phi _{e'}^{*}{\frac {\partial \phi _{e}}{\partial Q_{i}}}dr{\frac {\partial }{\partial Q_{i}}}+{\frac {1}{2}}\int \phi _{e'}^{*}{\frac {\partial ^{2}\phi _{e}}{\partial ^{2}Q_{i}}}dr\right]+\epsilon _{e}(Q)\delta _{e,e'}\right]\chi _{e,n}(Q)=\epsilon _{e,n}\chi _{e',n}(Q)$

This can be written in a more compact form by defining some new variables [2]:

$\Lambda _{e,e'}=-\hbar ^{2}\sum _{i}{\frac {1}{M_{i}}}\left[A_{e,e'}^{(i)}(Q)+{\frac {1}{2}}B_{e,e'}^{(i)}(Q)\right]$

$A_{e,e'}^{(i)}(Q)=\int \phi _{e'}*{\frac {\partial \phi _{e}}{\partial Q_{i}}}dr$

$B_{e,e'}^{(i)}(Q)=\int \phi _{e'}*{\frac {\partial ^{2}\phi _{e}}{\partial ^{2}Q_{i}}}dr$

$\Rightarrow \sum _{e}\left[T(Q)\delta _{e,e'}+\Lambda _{e,e'}+\epsilon _{e}(Q)\delta _{e,e'}\right]\chi _{e,n}(Q)=\epsilon _{e,n}\chi _{e',n}(Q)$

This is a system of coupled differential equations, and is effectively the eigenvalue problem for our Hamiltonian. If we assume we have non-degenerate states, we recover the classic adiabatic approximation. Take non-degenerate state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_e\left(r,Q\right) } , and applying perturbation theory we find:

$H=T(Q)+\Lambda _{e,e}+\epsilon _{e}\left(Q\right)$

This Hamiltonian for the $\chi _{e,n}\left(Q\right)$ states is effectively the same as taking the exact eigenfunction to be the state:

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

Since we have only diagonal terms, this Hamiltonian prevents the mixing of electronic states. "The motion of the nuclei leads only to the deformation of the electron distribution and not to transitions between different electronic states. In other words, a stationary distribution of electrons is obtained for each instantaneous position of the nuclei, i.e. the electrons follow the nuclear motion adiabatically"[2]. Transitions between states would occur if the operators in $\Lambda _{e,e'}$ (where e $\neq$ e') appear in the Hamiltonian. For this reason the operators $\Lambda _{e,e'}\left(Q\right)$ are called the operators of nonadiabaticity, and the energies $\epsilon _{e}\left(Q\right)$ are called adiabatic potentials.

When treating the degenerate case, we would begin with the eigenfunction

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

and attempt to diagonalize the matrix Hamiltonian. However, when the matrix is diagonalized we are left with matrix elements that diverge near the point of degeneracy $\left(Q\rightarrow Q^{0}\right)$. Our matrix elements would contain the term [5]:

$\langle e_{2}|{\frac {\partial }{\partial Q_{i}}}|e_{1}\rangle =-\left[{\frac {1}{\epsilon _{e_{2}}-\epsilon _{e_{1}}}}\right]\langle e_{2}|{\frac {\partial V(r,Q)}{\partial Q_{i}}}|e_{1}\rangle$

The energies $\epsilon _{e_{2}}$ and $\epsilon _{e_{1}}$ are not generally the same. However, when $\left(Q\rightarrow Q^{0}\right)$, these values are the same and the matrix element diverges. The term $\langle e_{2}|{\frac {\partial V(r,Q)}{\partial Q_{i}}}|e_{1}\rangle$ is, in general, finite.

For this reason, the degenerate case is often treated by adding the term $\epsilon$ H$_{1}$ to the Hamiltonian of the non degenerate problem. Then the degenerate case can be treated as the non-degenerate case in the limit of $\epsilon \rightarrow 0$. This can be justified physically, because we expect there to be a small external perturbation which breaks the degeneracy [5].

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 [4]. 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 $\left({\frac {m}{M}}\right)^{1/4}$ [4]. (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 then 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e} denotes an electronic quantity. A subscript Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} denotes a nuclear quantity.

The total potential energy of the system is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} and the sum on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} represent the permutation of x,y,z.

The electronic kinetic energy is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_e = -\frac{\hbar^2}{2m}\sum_q\sum_k \frac{\partial^2}{\partial q_k^2}}

The nuclear kinetic energy is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_n = -\kappa^4 \frac{ \hbar^2}{2m} \sum_r\sum_l \mu_l \frac{\partial^2}{\partial r_l^2}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_l} is a dimensionless number on order 1 [4].

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = T_e + T_n + V\left(Q\right) + V(R,Q)}

where the potential has been broken into a part that depends 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[T_e + V(R,Q) + V(Q)\right]\phi_e(R,Q) = W_e(Q)\phi_e(R,Q) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[T_n + W_e(Q) \right]\chi_n(Q) = E_e\chi_n(Q)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_e(R,Q)} is the electronic wave function (Q appears only as a constant in this particular equation), Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_n(Q)} is the nuclear wave function, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_e + V} is the effective potential as seen by each electronic state [5]. In the first equation we have assumed that nuclei to be stationary and solved for the electronic wave functions. In the second equations we assume the solution to the first equation remains valid, and then solve for the vibrational wave functions.

When the electronic states are non-degenerate, the Born-Oppenheimer wave-function are the solutions to the following Hamiltonian, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta H} (the second term in parenthesis) is treated as a perturbation [5]:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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) }

Note that expanding this Hamiltonian out, the new terms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(R, Q^0)} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_e(Q)} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_e(Q^0)} cancel and we are left with the original Hamiltonian given above. These additional terms are used for mathematical convenience.

with zero order solutions[5]:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{e,n} = \phi_e\left(R,Q\right)\chi_n(Q)}

where $\phi _{e}\left(R,Q\right)$ and $\chi _{n}\left(Q\right)$ are solutions to the pair of equations given above.

When there are degenerate electronic states, the Born-Oppenheimer wave-function are the solutions to the following Hamiltonian [5]:

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

where $[\Gamma ]$ represents the dimensionality of the degeneracy, and the sum is over all degenerate states. Note that expanding this Hamiltonian out, the extra terms cancel, similar to above, and we are left with the original Hamiltonian. These additional terms are used for mathematical convenience.

The zero order solutions are now superpositions of the degenerate wavefunctions:

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

where $\phi _{e}\left(R,Q\right)$ is the solution to:

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

and $\chi _{e,n}(Q)$ is the solution to:

$\left[T_{N}+{\frac {\sum W_{e}(Q)}{\Gamma }}+V(Q)+\langle e|\Delta H|e\rangle \right]\chi _{e,n}(Q)+\sum _{e\neq e'}\langle e|\Delta H|e'\rangle \chi _{e,n}(Q)=E_{n}\chi _{e,n}(Q)$

Solution To The Jahn-Teller Problem for an Idealized System

Electronic-vibrational coupling is symbolized by $\otimes$. E, T, and G$_{3/2}$ represent 2,3, and 4-fold degenerate electronic states. The letters $\beta$, $\epsilon$, and $\tau _{2}$ representations the number of vibrational modes to which the electronic state is coupling.

E x b system

The most basic case is $E\otimes \beta$, where a doubly degenerate electronic state is coupled by a single vibrational mode. This type of coupling occurs in the cyclooctatetraene molecular ion [5], but is generally considered to have little experimental significance. However, the math is very simple so it will be presented regardless of it's lack of practical application.

The Hamiltonian for an E $\otimes \beta$ system is given by [5]:

$H(Q)=-{\frac {1}{2}}\left({\frac {\partial ^{2}}{\partial Q^{2}}}-Q^{2}\right)-kQ{\begin{pmatrix}-1&0\\0&1\end{pmatrix}}$

In this equation, Q is the normal mode coordinate and k gives the strength of the Jahn-Teller effect. This Hamiltonian can be written as a pair of Hamiltonians:

$H(Q)=-{\frac {1}{2}}{\frac {\partial ^{2}}{\partial Q^{2}}}+{\frac {1}{2}}Q^{2}\pm kQ$

These Hamiltonians can easily be recognized as a pair of harmonic oscillators, where the origin has been displaced $\pm$k from Q=0, with potential minima $E={\frac {k^{2}}{2}}$.

From this we can see that if k is large, the two minima are deep and well separated. This gives us a system where there are two equally probably ground states, corresponding to two different distortions. These states are lower in symmetry than the ground state of the uncoupled system - so the Jahn Teller effect has still broken some symmetry - but the twofold degeneracy of the ground state has not been broken.

E x e system

The next most basic, and more interesting case, is $E\otimes \epsilon$, keeping only linear vibronic coupling terms. This is a system in which a doublet of electronic states interacts with two vibrational modes. In the adiabatic approximation, Hamiltonian for the $E\otimes \epsilon$ system is given by [2]:

$H={\frac {1}{2}}\left({\frac {\partial ^{2}}{\partial Q_{\theta }^{2}}}+{\frac {\partial ^{2}}{\partial Q_{\epsilon }^{2}}}+Q_{\theta }^{2}+Q_{\epsilon }^{2}\right){\hat {\sigma }}_{0}+k\left(Q_{\theta }\sigma _{x}+Q_{\epsilon }\sigma _{y}\right)$

where the Pauli matrices are given by:

${\hat {\sigma _{0}}}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}$; ${\hat {\sigma _{x}}}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}$; ${\hat {\sigma _{y}}}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}$; ${\hat {\sigma _{z}}}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$

The first term of the Hamiltonian is the kinetic and potential energies of two simple harmonic oscillators with coordinates $Q_{\theta }$ and $Q_{\epsilon }$. These normal coordinates represent different octahedral arrangements of atoms atoms, and the subscripts give their symmetries, as shown below. The second term represents the coupling between the vibrational modes and the pair of electronic states. The electronic states are represented as $|\psi _{\theta }\rangle$ and $|\psi _{\epsilon }\rangle$

It can be convenient to write the Hamiltonian in polar coordinates, $\{R,\phi \}$

$Q_{\theta }=R\cos \left(\phi \right)={\frac {R}{2}}\left(e^{i\phi }+e^{-i\phi }\right)$

$Q_{\epsilon }=R\sin \left(\phi \right)={\frac {R}{2i}}\left(e^{i\phi }-e^{-i\phi }\right)$

$H=\left(-{\frac {1}{2}}{\frac {1}{R}}{\frac {\partial }{\partial R}}R{\frac {\partial }{\partial R}}-{\frac {1}{2}}{\frac {1}{R^{2}}}{\frac {\partial ^{2}}{\partial \theta ^{2}}}+{\frac {1}{2}}R^{2}\right)\sigma _{0}+kR\left(\cos(\theta )\sigma _{x}+\sin(\theta )\sigma _{y}\right)$

Now we can omit the kinetic energy and attempt to diagonalize the potential and interaction terms:

${\frac {1}{2}}R^{2}\sigma _{0}+kR\left(\cos(\theta )\sigma _{x}+\sin(\theta )\sigma _{y}\right)$

The first term in the potential energy is the elastic energy, and it is diagonal (a multiple of $\sigma _{0}$). The second term is the operator of the vibronic interaction. This matrix can be diagonalized via a unitary transformation (which does not affect $\sigma _{0}$).

$k\left(Q_{\theta }{\hat {\sigma }}_{x}+Q_{\epsilon }{\hat {\sigma }}_{y}\right)=k{\frac {R}{2}}\left[\left(e^{i\phi }+e^{-i\phi }\right){\begin{pmatrix}0&1\\1&0\end{pmatrix}}+{\frac {1}{i}}\left(e^{i\phi }-e^{-i\phi }\right){\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\right]$

$=k{\frac {R}{2}}{\begin{pmatrix}0&e^{i\phi }+e^{-i\phi }-e^{i\phi }+e^{-i\phi }\\e^{i\phi }+e^{-i\phi }+e^{i\phi }-e^{-i\phi }&0\end{pmatrix}}=kR{\begin{pmatrix}0&e^{-i\phi }\\e^{i\phi }&0\end{pmatrix}}$

The unitary transformation ${\hat {S}}$ diagonalizes the above matrix:

${\hat {S}}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}e^{-i\phi /2}&e^{-i\phi /2}\\e^{i\phi /2}&-e^{i\phi /2}\end{pmatrix}}$

${\hat {S}}^{\dagger }\left(Q_{\theta }{\hat {\sigma }}_{x}+Q_{\epsilon }{\hat {\sigma }}_{y}\right){\hat {S}}={\frac {R}{2}}{\begin{pmatrix}e^{i\phi /2}&e^{-i\phi /2}\\e^{i\phi /2}&-e^{-i\phi /2}\end{pmatrix}}{\begin{pmatrix}0&e^{-i\phi }\\e^{i\phi }&0\end{pmatrix}}{\begin{pmatrix}e^{-i\phi /2}&e^{-i\phi /2}\\e^{i\phi /2}&-e^{i\phi /2}\end{pmatrix}}$

${\frac {R}{2}}{\begin{pmatrix}e^{i\phi /2}&e^{-i\phi /2}\\-e^{i\phi /2}&e^{-i\phi /2}\end{pmatrix}}{\begin{pmatrix}e^{-i\phi /2}&e^{-i\phi /2}\\e^{i\phi /2}&-e^{i\phi /2}\end{pmatrix}}={\frac {R}{2}}{\begin{pmatrix}2&0\\0&-2\end{pmatrix}}=R{\begin{pmatrix}1&0\\0&-1\end{pmatrix}}=R{\hat {\sigma }}_{z}$

And so we can write the matrix ${\hat {U}}$ as the diagonal matrix ${\hat {\tilde {U}}}$:

${\hat {\tilde {U}}}={\hat {S}}^{\dagger }U{\hat {S}}={\frac {1}{2}}R^{2}{\hat {\sigma }}_{0}+kR{\hat {\sigma }}_{z}={\begin{pmatrix}{\frac {1}{2}}\mathbb {R} ^{2}+kR&0\\0&{\frac {1}{2}}\mathbb {R} ^{2}-kR\end{pmatrix}}$

It is trivial to write down the eigenvalues of this matrix of the potential energy, which are simply the diagonal terms:

$\epsilon _{\pm }={\frac {1}{2}}\mathbb {R} ^{2}\pm kR$

These energies can be represented by the "Mexican hat" adiabatic potential energy surface.

The Jahn-Teller Order Parameter

Determination of the order parameter for Jahn-Teller systems allows for a better understanding of the thermodynamic description of Jahn-Teller distortion. The order parameter for Jahn-Teller phase transitions has been a largely disagreed upon quantity. Some papers, such as [8] use the symmetrized combinations of pairwise products of electron wave functions as the order parameter for Jahn-Teller transitions. Another way to describe Jahn-Teller transitions is to analyze the distortion of nearest neighbor Jahn-Teller ions - which can be related to changes in symmetry of the electron density distribution [9]. A third way to determine an order parameter is to use components of the density matrix [9].

References

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

[2] I.B. Bersuker and V.Z. Polinger, Vibronic Interactions in Molecules and Crystals Springer-Verlag, 1989.

[3] I.B. Bersuker, The Jahn-Teller Effect A Bibliographic Review, IFI/Plenum Data Company, 1984.

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

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

[6] R. Renner, Z. Physik, 92, 172 (1934)

[7] Phil. Trans. Roy. Soc. London, 251A, 553 (1959)

[8] K.N. Kugel and D.I. Khomskii, Usp. Fix. Nauk, 136, No. 4, 621-664 (1982)

[9] O.V. Gurin and V.N. Syromyatnikov, Russian Physics Journal, 32, No. 12, 989-993 (1990)