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 electronic wave functions that together span the irreducible representation E_u of this group and assume that the molecule is in the corresponding electronic state. One wave function transforms as x and will be denoted by |X⟩. Its partner transforms as y and is denoted by |Y⟩. In the case of the perfect square both have the same energy $\varepsilon _{E}$.

In the middle of Fig.1 a normal mode Q (v₄) is indicated by red arrows, explicitly it is

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

where the deviations of the atoms are of the same length q. For the record: Q transforms as B_2g of D_4h. When q is positive, the molecule is elongated along the y-axis; this is the leftmost molecule in Fig. 1. Similarly, negative q implies an elongation along the x-axis (the rightmost molecule). The case q = 0 corresponds to the perfect square.

The 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 $\varepsilon _{X}$ and $\varepsilon _{Y}$. Although group theory tells us that the energies are different, without explicit calculation it is not a priori clear which energy 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 is that the leftmost and rightmost molecules in the figure 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: $\varepsilon _{X}>\varepsilon _{Y}$.

For small q-values it is fair to assume that both $\varepsilon _{X}$ and $\varepsilon _{Y}$ are quadratic functions of q. They are shown in Fig. 2. For q = 0 the curves cross and $\varepsilon _{X}=\varepsilon _{Y}\equiv \varepsilon _{E}$. 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.

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.