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. As an example, consider a doublet electronic state E in cubic symmetry. The symmetric part of E ⊗ E is A1 + E. Therefore, the state E will couple to vibrational modes Q_i transforming as a1 and e. However, the a1 modes will result in the same energy shift to all states and therefore do not contribute to any JT splitting. They can therefore be neglected. The result is an E ⊗ e JT effect. This JT effect is experienced by triangular molecules X3, tetrahedral molecules ML4, and octahedral molecules ML6 when their electronic state has E symmetry. Components of a given vibrational mode are also labelled according to their transformation properties. For example, the two components of an e mode are usually labelled Q_\theta and Q_\epsilon, which in
octahedral symmetry transform as 3z^2-r^2 and x^2-y^2 respectively.
The JT Hamiltonian Eigenvalues of the
Hamiltonian of a polyatomic system define PESs as functions of normal modes Q_i of the system (i.e. linear combinations of the nuclear displacements with specific symmetry properties). At the reference point of high symmetry, where the symmetry-induced degeneracy occurs, several of the eigenvalues coincide. By a detailed and laborious analysis,
Jahn and
Teller showed that – excepting linear molecules – there are always first-order terms in an expansion of the matrix elements of the Hamiltonian in terms of symmetry-lowering (in the language of
group theory: non-totally symmetric) normal modes. These linear terms represent forces that distort the system along these coordinates and lift the degeneracy. The point of degeneracy can thus not be stationary, and the system distorts toward a stationary point of lower symmetry where stability can be attained. Proof of the JT theorem follows from the theory of
molecular symmetry (
point group theory). A less rigorous but more intuitive explanation is given in section . To arrive at a quantitative description of the JT effect, the forces appearing between the component
wave functions are described by expanding the Hamiltonian in a power series in the Q_i. Owing to the very nature of the degeneracy, the Hamiltonian takes the form of a matrix referring to the degenerate
wave function components. A
matrix element between states \Psi_a and \Psi_b generally read :H_{ab}=\langle\Psi_a|H|\Psi_b\rangle=\langle\Psi_a|H(Q_i=0)+\sum_{i}\frac{\partial V}{\partial Q_i} Q_i + ...|\Psi_b\rangle The expansion can be truncated after terms linear in the Q_i, or extended to include terms quadratic (or higher) in the Q_i. The
adiabatic potential energy surfaces (APES) are then obtained as the
eigenvalues of this matrix. In the original paper, it is proven that there are always linear terms in the expansion. It follows that the degeneracy of the
wave function cannot correspond to a stable structure.
Potential energy surfaces Mexican-hat potential In mathematical terms, the APESs characterising the JT distortion arise as the
eigenvalues of the potential energy matrix. Generally, the APESs take the characteristic appearance of a double cone, circular or elliptic, where the point of contact, i.e. degeneracy, denotes the high-symmetry configuration for which the JT theorem applies. For the above case of the linear E ⊗ e JT effect, the situation is illustrated by the APES :V=\frac{\mu\omega^2}{2}(Q_\theta^2 + Q_\epsilon^2) \pm k \sqrt{Q_\theta^2 + Q_\epsilon^2} displayed in the figure, with part cut away to reveal its shape, which is known as a Mexican Hat potential. Here, \omega is the frequency of the vibrational e mode, \mu is its mass and k is a measure of the strength of the JT coupling. The conical shape near the degeneracy at the origin makes it immediately clear that this point cannot be
stationary, that is, the system is unstable against asymmetric distortions, which leads to a symmetry lowering. In this particular case, there are infinitely many isoenergetic JT distortions. The Q_i giving these distortions are arranged in a circle, as shown by the red curve in the figure. Quadratic coupling or cubic elastic terms lead to a warping along this "minimum energy path", replacing this infinite manifold by three equivalent potential minima and three equivalent saddle points. In other JT systems, linear coupling results in discrete minima.
Conical intersections The high symmetry of the double-cone topology of the linear E ⊗ e JT system directly reflects the high underlying symmetry. It is one of the earliest (if not the earliest) examples in the literature of a
conical intersection of potential energy surfaces. Conical intersections have received wide attention in the literature starting in the 1990s and are now considered paradigms of nonadiabatic excited-state dynamics, with far-reaching consequences in molecular spectroscopy,
photochemistry and photophysics. Some of these will be commented upon further below. In general,
conical intersections are far less symmetric than depicted in the figure. They can be tilted and elliptical in shape etc., and also peaked and sloped intersections have been distinguished in the literature. Furthermore, for more than two degrees of freedom, they are not point-like structures but instead they are seams and complicated, curved hypersurfaces, also known as intersection space. The coordinate sub-space displayed in the figure is also known as a branching plane.
Implications for dynamics The characteristic shape of the JT-split APES has specific consequences for the nuclear dynamics, here considered in the fully quantum sense. For sufficiently strong JT coupling, the minimum points are sufficiently far (at least by a few vibrational energy quanta) below the JT intersection. Two different energy regimes are then to be distinguished, those of low and high energy. • In the low-energy regime the nuclear motion is confined to regions near the "minimum energy points". The distorted configurations sampled impart their geometrical parameters on, for example, the rotational fine structure in a spectrum. Due to the existence of barriers between the various minima in the APES, like those appearing due to the warping of the , motion on the low-energy regime is usually classified as either a static JTE, dynamic JTE or incoherent hopping. Each regime shows particular fingerprints on experimental measurements. :*
Static JTE: In this case, the system is trapped in one of the lowest-energy minima of the APES (usually determined by small perturbations created by the environment of the JT system) and does not have enough energy to cross the barrier towards another minimum during the typical time associated to the measurement. Quantum dynamical effects like tunnelling are negligible, and effectively the molecule or solid displays the low symmetry associated with a single minimum. :*
Dynamic JTE: In this case, the barriers are sufficiently small compared to, for example, the
zero-point energy associated to the minima, so that vibronic wavefunctions (and all observables) display the symmetry of the reference (undistorted) system. In the linear E ⊗ e problem, the motion associated to this regime would be around the circular path in the figure. When the barrier is sufficiently small, this is called (free) pseudorotation (not to be confused with the rotation of a
rigid body in space, see difference between real and pseudo rotations illustrated here for the
fullerene molecule C60). When the barrier between the minima and the saddle points on the warped path exceeds a vibrational quantum, pseudorotational motion is slowed down and occurs through tunnelling. This is called hindered pseudorotation. In both free and hindered pseudorotation, the important phenomenon of the geometric (Berry) phase alters the ordering of the levels. :*
Incoherent hopping: Another way in which the system can overcome the barrier is through thermal energy. In this case, while the system moves throughout the minima of the system, the state is not a quantum coherent one but a statistical mixture. This difference can be observed experimentally. • The dynamics is quite different for high energies, such as occur from an optical transition from a non-degenerate initial state with a high-symmetry (JT undistorted) equilibrium geometry into a JT distorted state. This leads the system to the region near the conical intersection of the JT-split APES in the centre of the figure. Here the nonadiabatic couplings become very large and the behaviour of the system cannot be described within the familiar
Born–Oppenheimer (BO) separation between the electronic and nuclear motions. The nuclear motion ceases to be confined to a single, well-defined APES and the transitions between the adiabatic surfaces occur yielding effects like Slonzcewsky resonances. In molecules, this is usually a femtosecond timescale, which amounts to ultrafast (femtosecond) internal conversion processes, accompanied by broad spectral bands also under isolated-molecule conditions and highly complex spectral features. . As already stated above, the distinction of low and high energy regimes is valid only for sufficiently strong JT couplings, that is, when several or many vibrational energy quanta fit into the energy window between the conical intersection and the minimum of the lower JT-split APES. For the many cases of small to intermediate JT couplings, this energy window and the corresponding adiabatic low-energy regime does not exist. Rather, the levels on both JT-split APES are intricately mixed for all energies and the nuclear motion always proceeds on both JT split APES simultaneously.
Ham factors In 1965, Frank Ham and allowed the results of previous
Electron Paramagnetic Resonance (EPR) experiments to be explained. In general, the result of an orbital operator acting on vibronic states can be replaced by an effective orbital operator acting on purely electronic states. In first order, the effective orbital operator equals the actual orbital operator multiplied by a constant, whose value is less than one, known as a first-order (Ham) reduction factor. For example, within a triplet T1 electronic state, the spin–orbit coupling operator \lambda \mathbf{L}.\mathbf{S} can be replaced by \gamma \lambda \mathbf{L}.\mathbf{S}, where \gamma is a function of the strength of the JT coupling which varies from 1 in zero coupling to 0 in very strong coupling. Furthermore, when second-order perturbation corrections are included, additional terms are introduced involving additional numerical factors, known as second-order (Ham) reduction factors. These factors are zero when there is no JT coupling but can dominate over first-order terms in strong coupling, when the first-order effects have been significantly reduced. Apart from
wave function-based techniques (which are sometimes considered genuinely
ab initio in the literature) the advent of
density functional theory (DFT) opened up new avenues to treat larger systems including solids. This allowed details of JT systems to be characterised and experimental findings to be reliably interpreted. It lies at the heart of most developments addressed in section . Two different strategies are conceivable and have been used in the literature. One can • take the applicability of a certain coupling scheme for granted and limit oneself to determine the parameters of the model, for example from the energy gain achieved through the JT distortion, also termed JT stabilisation energy. • map parts of the APES in whole or reduced dimensionality and thus get an insight into the applicability of the model, possibly also deriving ideas how to extend it. Naturally, the more accurate approach (2) may be limited to smaller systems, while the simpler approach (1) lends itself to studies of larger systems. ==Applications==