The Jahn–Teller effect (JT effect or JTE) is an important mechanism of spontaneous symmetry breaking in molecular and solid-state systems which has far-reaching consequences in different fields, and is responsible for a variety of phenomena in spectroscopy, stereochemistry, crystal chemistry, molecular and solid-state physics, and materials science. The effect is named for Hermann Arthur Jahn and Edward Teller, who first reported studies about it in 1937. Simplified overview The Jahn–Teller effect, sometimes also referred to as Jahn–Teller distortion, describes the geometrical distortion of molecules and ions that results from certain electron configurations. The Jahn–Teller theorem essentially states that any non-linear molecule with a spatially degenerate electronic ground state will undergo a geometrical distortion that removes that degeneracy, because the distortion lowers the overall energy of the species.
Transition metal chemistry
The Jahn–Teller effect is most often encountered in octahedral complexes of the transition metals. The phenomenon is very common in six-coordinate copper(II) complexes. The d9 electronic configuration of this ion gives three electrons in the two degenerate eg orbitals, leading to a doubly degenerate electronic ground state. Such complexes distort along one of the molecular fourfold axes (always labelled the z axis), which has the effect of removing the orbital and electronic degeneracies and lowering the overall energy. The distortion normally takes the form of elongating the bonds to the ligands lying along the z axis, but occasionally occurs as a shortening of these bonds instead (the Jahn–Teller theorem does not predict the direction of the distortion, only the presence of an unstable geometry). When such an elongation occurs, the effect is to lower the electrostatic repulsion between the electron-pair on the Lewis basic ligand and any electrons in orbitals with a z component, thus lowering the energy of the complex. The inversion centre is preserved after the distortion.
In octahedral complexes, the Jahn–Teller effect is most pronounced when an odd number of electrons occupy the eg orbitals. This situation arises in complexes with the configurations d9, low-spin d7 or high-spin d4 complexes, all of which have doubly degenerate ground states. In such compounds, the eg orbitals involved in the degeneracy point directly at the ligands, so distortion can result in a large energetic stabilisation. Strictly speaking, the effect also occurs when there is a degeneracy due to the electrons in the t2g orbitals (i.e. configurations such as d1 or d2, both of which are triply degenerate). In such cases, however, the effect is much less noticeable, because there is a much smaller lowering of repulsion on taking ligands further away from the t2g orbitals, which do not point directly at the ligands (see the table below). The same is true in tetrahedral complexes, e.g. the distortion of manganate is very subtle because there is less stabilisation to be gained because the ligands are not pointing directly at the orbitals.
The expected effects for octahedral coordination are given in the following table: + Jahn–Teller effect Number of d electrons 1 2 3 4 5 6 7 8 9 10 High/Low Spin HS LS HS LS HS LS HS LS Strength of J–T Effect w w s w w w w s s
w: weak Jahn–Teller effect (t2g orbitals unevenly occupied)
s: strong Jahn–Teller effect expected (eg orbitals unevenly occupied)
blank: no Jahn–Teller effect expected.
The Jahn–Teller effect is manifested in the UV-VIS absorbance spectra of some compounds, where it often causes splitting of bands. It is readily apparent in the structures of many copper(II) complexes. Additional, detailed information about the anisotropy of such complexes and the nature of the ligand binding can be however obtained from the fine structure of the low-temperature electron spin resonance spectra.
Related effects The underlying cause of the Jahn–Teller effect is the presence of molecular orbitals that are both degenerate and open shell (i.e., incompletely occupied). This situation is not unique to coordination complexes and can be encountered in other areas of chemistry. In organic chemistry, the phenomenon of antiaromaticity has the same cause and also often sees molecules distorting; as in the case of cyclobutadiene and cyclooctatetraene (COT).
Advanced treatment
The Jahn–Teller theorem The JT theorem can be stated in different forms, two of which are given here:
A nonlinear polyatomic system in a spatially degenerate electronic state distorts spontaneously in such a way that the degeneracy is lifted and a new equilibrium structure of lower symmetry is attained.
Alternatively and considerably shorter: ... stability and degeneracy are not possible simultaneously unless the molecule is a linear one ....
Spin-degeneracy was an exception in the original treatment and was later treated separately.
The formal mathematical proof of the Jahn–Teller theorem rests heavily on symmetry arguments, more specifically the theory of molecular point groups. The argument of Jahn and Teller assumes no details about the electronic structure of the system. Jahn and Teller made no statement about the strength of the effect, which may be so small that it is immeasurable. Indeed, for electrons in non-bonding or weakly bonding molecular orbitals, the effect is expected to be weak. However, in many situations the JT effect is important.
Historic developments Interest in the JTE increased after its first experimental verification. Various model systems were developed probing the degree of degeneracy and the type of symmetry. These were solved partly analytically and partly numerically to obtain the shape of the pertinent potential energy surfaces (PES) and the energy levels for the nuclear motion on the JT-split PES. These energy levels are not vibrational energy levels in the traditional sense because of the intricate coupling to the electronic motion that occurs, and are better termed vibronic energy levels. The new field of 'vibronic coupling' or 'vibronic coupling theory' was born.
A further breakthrough occurred upon the advent of modern ("ab initio") electronic structure calculations whereby the relevant parameters characterising JT systems can be reliably determined from first principles. Thus one could go beyond studies of model systems that explore the effect of parameter variations on the PES and vibronic energy levels; one could also go on beyond fitting these parameters to experimental data without clear knowledge about the significance of the fit. Instead, well-founded theoretical investigations became possible which greatly improved the insight into the phenomena at hand and into the details of the underlying mechanisms.
While recognizing the JTE distortion as a concrete example of the general spontaneous symmetry breaking mechanism, the exact degeneracy of the involved electronic state was identified as a non-essential ingredient for this symmetry breaking in polyatomic systems. Even systems that in the undistorted symmetric configuration present electronic states which are near in energy but not precisely degenerate, can show a similar tendency to distort. The distortions of these systems can be treated within the related theory of the pseudo Jahn–Teller effect (in the literature often referred to as "second-order JTE"). This mechanism is associated to the vibronic couplings between adiabatic PES separated by nonzero energy gaps across the configuration space: its inclusion extends the applicability of JT-related models to symmetry breaking in a far broader range of molecular and solid-state systems.
Chronology:
1934: Lev Landau, in discussion with Edward Teller, suggested that electronic states of certain degenerate nuclear configurations are unstable with respect to nuclear displacements that lower the symmetry (see 'An historical note' by Englman). 1937: Hermann Arthur Jahn and Edward Teller formulated what is now known as the Jahn–Teller theorem. 1939: John Hasbrouck Van Vleck extended the Jahn–Teller theorem to ions in crystals. As attempts to observe the Jahn–Teller effect experimentally had been unconvincing, he noted that 'it is of great merit of the Jahn–Teller effect that it disappears when not needed'. 1950–2: Brebis Bleaney and co-workers first obtained unambiguous experimental evidence of the Jahn–Teller effect, by carrying out electron paramagnetic resonance studies on paramagnetic ions in crystals 1957–8: Öpik and Pryce showed that spin–orbit coupling can stabilise symmetric configurations against distortions from a weak JTE. Moffitt et al. and Longuet-Higgins et al. argued that the states of JT systems have inextricably mixed electronic and vibrational components, which they called vibronic states, with energies very different to the electronic states. 1962–4: Isaac Bersuker and Mary O'Brien investigated tunnelling in the lowest-energy vibronic states, the so-called tunnelling splitting, and the dynamic nature of the JT effect. The article by O'Brien shows the influence of the geometric phase factor (later called Berry phase) on the ordering of the vibronic states. 1965: Frank Ham realised the effect of coherent dynamics on the measurement of observables. This influence can be described in terms of reduction factors multiplying orbital operators and specific formulae were proposed for the magnetism of JT ions. 1984: Generalization of the concept of geometric phase by Berry (or Berry phase as it is also known) provided a general background to aid understanding of the rotation-dependent phase associated with the electronic and vibrational wavefunction of JT systems, as discovered by Longuet-Higgins, and further discussed by Herzberg and Longuet-Higgins, Longuet-Higgins, O'Brien, and Mead and Truhlar. 1990s: Advances in computing power meant that ab initio methods including those based on the Density Functional Theory started to be used to solve JT problems.
Relation to important discoveries
In 1985, Harry Kroto and co-workers discovered a class of closed-cage carbon molecules known as fullerenes. Buckminsterfullerene (C60), which has icosahedral symmetry, becomes JT-active upon addition or removal of one electron. The ordering of energy levels may not be the same as that predicted by Hund's rule. Discovery in 1986 by Bednorz and Müller of superconductivity in cuprates with a transition temperature of 35 K, which was higher than the upper limit allowed according to standard BCS theory, was motivated by earlier work by Müller on JT ions in crystals. Colossal magnetoresistance, a property of manganese-based perovskites and other materials, has been explained in terms of competition between dynamic Jahn–Teller and double-exchange effects. Peierls theorem, which states that a one-dimensional equally spaced chain of ions with one electron per ion is unstable, has common roots with the JT effect.
Theory
Symmetry of JT systems and categorisation using group theory A given JT problem will have a particular point group symmetry, such as Td symmetry for magnetic impurity ions in semiconductors or Ih symmetry for the fullerene C60. JT problems are conventionally classified using labels for the irreducible representations (irreps) that apply to the symmetry of the electronic and vibrational states. For example, E ⊗ e would refer to an electronic doublet state transforming as E coupled to a vibrational doublet state transforming as e.
In general, a vibrational mode transforming as Λ will couple to an electronic state transforming as Γ if the symmetric part of the Kronecker product [Γ ⊗ Γ]S contains Λ, unless Γ is a double group representation when the antisymmetric part {Γ ⊗ Γ}A is considered instead. Modes which do couple are said to be JT-active.