User talk:Paul Wormer/scratchbook1

The Jahn-Teller effect has a quantum mechanical origin and no classical physics description of it exists. Some knowledge of quantum mechanics is prerequisite to the reading of this article. Further some chemically oriented group theory (Schönflies notation for point groups and Mulliken notation for their irreducible representations) is used.

The Jahn-Teller effect is the distortion of a highly symmetric—but non-linear—molecule to lower symmetry and lower energy. The effect occurs if the molecule is in a degenerate state of definite energy, that is, if more than one wave function is eigenfunction of the molecular Hamiltonian with the same energy. In other words, energy degeneracy of a state implies that there are two or more orthogonal wave functions describing the state. Due to Jahn-Teller distortion, the molecule is lowered in symmetry and the energy degeneracy is lifted. Some of the wave functions obtain lower energy, while others obtain higher energy by the distortion.

The effect is named after H. A. Jahn and E. Teller who predicted it in 1937. It took some time before the effect was experimentally observed, because it was masked by other molecular interactions. However, there are now numerous unambiguous observations that agree well with theoretical predictions. These range from the excited states of the simplest non-linear molecule H₃, through moderate sized organic molecules, like ions of substituted benzene, to complex crystals and localized impurity centers in solids.

Explanation

The Jahn-Teller effect is best explained by an example. Consider the square homonuclear molecule in the middle of Fig. 1. Its symmetry group is D_4h. Consider two—in principle exact—electronic wave functions that together span the irreducible representation E_u of this group and assume that the molecule is in the corresponding E_u state. One wave function transforms as an x-coordinate and is denoted by |X ⟩. Its partner transforms as a y-coordinate and is denoted by |Y ⟩. In the case of the perfect square both wave functions have the same energy $\varepsilon _{E}$.

A simple approximate model for the wave functions is obtained by considering two degenerate molecular orbitals |x⟩ and |y⟩ (carrying E_u) that are outside a closed-shell and share a single unpaired electron. A closed-shell (a number of doubly occupied molecular orbitals) is invariant under the group operations. The wave function |X ⟩ is modeled by the closed-shell times the single unpaired electron in orbital |x⟩, while |Y ⟩ is modeled by the closed-shell times the electron in |y⟩. The orbital |x⟩ has the yz-plane—a mirror plane—as a nodal plane, that is, the orbital vanishes in this plane and has opposite sign on either side of the plane. Similarly the orbital |y⟩ has the xz-plane as a nodal plane.

In the middle of Fig.1 a vibrational normal mode Q of the molecule is indicated by red arrows. Explicitly the mode is

$Q=-\Delta x_{1}+\Delta x_{3}+\Delta y_{2}-\Delta y_{4},\,$

where the deviations of the atoms are all of the same length |q|. When q is positive, the molecule is elongated along the y-axis; this is the leftmost molecule in Fig. 1. Similarly, negative q implies a compression along the y-axis and an elongation along the x-axis (the rightmost molecule). The case q = 0 corresponds to the perfect square.

Both distorted molecules are of D_2h symmetry; D_2h is an Abelian group that has—as any Abelian group—only one-dimensional irreducible representations. Hence all electronic states of the distorted molecules are non-degenerate. The function |X ⟩ transforms as B_3u and |Y ⟩ transforms as B_2u of D_2h. Let the respective energies be written as $\varepsilon _{X}$ and $\varepsilon _{Y}$. Group theory tells us that these energies are different, but without explicit calculation it is not a priori clear which of the two energies is higher. Let us assume that for positive q : $\varepsilon _{X}<\varepsilon _{Y}$. This is shown in the energy level scheme in the bottom part of Fig. 1. An important observation now is that the leftmost and rightmost distorted molecules are essentially the same, they follow from each other by rotation over ±90° around the z-axis (a rotation of the molecules in the xy-plane). The x and y direction are interchanged between the left- and rightmost molecules by this rotation. Hence the molecule distorted with negative q-value has the energies satisfying: $\varepsilon _{X}>\varepsilon _{Y}$. Summarizing,

$\varepsilon _{X}(q)=\varepsilon _{Y}(-q)\quad {\hbox{and}}\quad \varepsilon _{X}(-q)=\varepsilon _{Y}(q)\quad {\hbox{with}}\quad \varepsilon _{X}(q)<\varepsilon _{Y}(q)\quad {\hbox{for}}\quad q>0.$

For small q-values it is reasonable to assume that both $\varepsilon _{X}(q)$ and $\varepsilon _{Y}(q)$ are quadratic functions of q. The parabolic energy curves are shown in Fig. 2. For q = 0 the energy curves cross and $\varepsilon _{X}(0)=\varepsilon _{Y}(0)\equiv \varepsilon _{E}$. The crossing point, corresponding to the perfect square, is clearly not an absolute minimum; therefore, the totally square symmetric configuration of the molecule will not be a stable equilibrium for the degenerate electronic state. At equilibrium, the molecule will be distorted from a square, and its energy will be lowered. This is the Jahn-Teller effect.

The above arguments are not restricted to square molecules. With the exception of linear molecules, which show Renner-Teller effects, all polyatomic molecules of sufficiently high symmetry to possess spatially degenerate electronic states will be subject to the Jahn-Teller instability. The proof, as given by Jahn and Teller, proceeds by application of point group symmetry principles.