The pseudo Jahn–Teller effect (PJTE), occasionally also known as second-order JTE, is a direct extension of the Jahn–Teller effect (JTE) where spontaneous symmetry breaking in polyatomic systems (molecules and solids) occurs even when the relevant electronic states are not degenerate. The PJTE can occur under the influence of sufficiently low-lying electronic excited states of appropriate symmetry. "The pseudo Jahn–Teller effect is the only source of instability and distortions of high-symmetry configurations of polyatomic systems in nondegenerate states, and it contributes significantly to the instability in degenerate states".
History
In their early 1957 paper on what is now called pseudo Jahn–Teller effect (PJTE), Uuno Öpik and Pryce showed that a small splitting of the degenerate electronic term does not necessarily remove the instability and distortion of a polyatomic system induced by the Jahn–Teller effect (JTE), provided that the splitting is sufficiently small (the two split states remain "pseudo degenerate"), and the vibronic coupling between them is strong enough. From another perspective, the idea of a "mix" of different electronic states induced by low-symmetry vibrations was introduced in 1933 by Herzberg and Teller to explore forbidden electronic transitions, and extended in the late 1950s by Murrell and Pople and by Liehr. The role of excited states in softening the ground state with respect to distortions in benzene was demonstrated qualitatively by Longuet-Higgins and Salem by analyzing the π electron levels in the Hückel approximation, while a general second-order perturbation formula for such vibronic softening was derived by Bader in 1960. In 1961 Fulton and Gouterman presented a symmetry analysis of the two-level case in dimers and introduced the term "pseudo Jahn–Teller effect". The first application of the PJTE to solving a major solid-state structural problem with regard to the origin of ferroelectricity was published in 1966 by Isaac Bersuker, and the first book on the JTE covering the PJTE was published in 1972 by Englman.R. Englman "The Jahn–Teller Effect in Molecules and Crystals", Wiley-Interscience, (1972). The second-order perturbation approach was employed by Pearson in 1975 to predict instabilities and distortions in molecular systems;G. Pearson, "Symmetry Rules for Chemical Reactions. Orbital Topology for Elementary Processes", Wiley: New York, 1976. he called it "second-order JTE" (SOJTE). The first explanation of PJT origin of puckering distortion as due to the vibronic coupling to the excited state, was given for the N3H32+ radical by Borden, Davidson, and Feller in 1980 (they called it "pyramidalization"). Methods of numerical calculation of the PJT vibronic coupling effect with applications to spectroscopic problems were developed in the early 1980s A significant step forward in this field was achieved in 1984 when it was shown by numerical calculations that the energy gap to the active excited state may not be the ultimate limiting factor in the PJTE, as there are two other compensating parameters in the condition of instability. It was also shown that, in extension of the initial definition, the PJT interacting electronic states are not necessarily components emerging from the same symmetry type (as in the split degenerate term). As a result, the applicability of the PJTE became a priory unlimited. Moreover, it was shown by Bersuker that the PJTE is the only source of instability of high-symmetry configurations of polyatomic systems in nondegenerate states (works cited in Refs.I. B. Bersuker, "The Jahn–Teller effect", Cambridge University Press, Cambridge UK, 2006.), and degeneracy and pseudo degeneracy are the only source of spontaneous symmetry breaking in matter in all its forms. The many applications of the PJTE to the study of a variety of properties of molecular systems and solids are reflected in a number of reviews and books J. K. Burdett, "Molecular Shapes. Theoretical Models of Inorganic Stereochemistry", Wiley: New York, 1980.R. Hoffmann, "Solids and Surfaces: A Chemist's View of Bonding in Extended Structures", VCH: New York, 1988.), as well as in proceedings of conferences on the JTE.
Theoretical background
General theory
The equilibrium geometry of any polyatomic system in nondegenerate states is defined as corresponding to the point of the minimum of the adiabatic potential energy surface (APES), where its first derivatives are zero and the second derivatives are positive. If we denote the energy of the system as a function of normal displacements as , at the minimum point of the APES (), the curvature of in direction ,
(1)
is positive, i.e., . Very often the geometry of the system at this point of equilibrium on the APES does not coincide with the highest possible (or even with any high) symmetry expected from general symmetry considerations. For instance, linear molecules are bent at equilibrium, planar molecules are puckered, octahedral complexes are elongated, or compressed, or tilted, cubic crystals are tetragonally polarized (or have several structural phases), etc. The PJTE is the general driving force of all these distortions if they occur in the nondegenerate electronic states of the high-symmetry (reference) geometry. If at the reference configuration the system is structurally unstable with respect to some nuclear displacements , then in this direction. The general formula for the energy is , where is the Hamiltonian and \psi_0 is the wavefunction of the nondegenerate ground state. Substituting in Eq. (1), we get (omitting the index for simplicity)
(2)
(3)
(4)
where are the wavefunctions of the excited states, and the expression, obtained as a second order perturbation correction, is always negative, . Therefore, if , the contribution is the only source of instability. The matrix elements in Eq. (4) are off-diagonal vibronic coupling constants,
(5)
These measure the mixing of the ground and excited states under the nuclear displacements , and therefore is termed the vibronic contribution. Together with the value and the energy gap between the mixing states, are the main parameters of the PJTE (see below). In a series of papers beginning in 1980 (see references in ) it was proved that for any polyatomic system in the high-symmetry configuration
(6)
and hence the vibronic contribution is the only source of instability of any polyatomic system in nondegenerate states. If for the high-symmetry configuration of any polyatomic system, then a negative curvature, , can be achieved only due to the negative vibronic coupling component , and only if . It follows that any distortion of the high-symmetry configuration is due to, and only to the mixing of its ground state with excited electronic states by the distortive nuclear displacements realized via the vibronic coupling in Eq. (5). The latter softens the system with respect to certain nuclear displacements (), and if this softening is larger than the original (nonvibronic) hardness in this direction, the system becomes unstable with respect to the distortions under consideration, leading to its equilibrium geometry of lower symmetry, or to dissociation. There are many cases when neither the ground state is degenerate, nor is there a significant vibronic coupling to the lowest excited states to realize PJTE instability of the high-symmetry configuration of the system, and still there is a ground state equilibrium configuration with lower symmetry. In such cases the symmetry breaking is produced by a hidden PJTE (similar to a hidden JTE); it takes place due to a strong PJTE mixing of two excited states, one of which crosses the ground state to create a new (lower) minimum of the APES with a distorted configuration.
The two-level problem
The use of the second order perturbation correction, Eq. (4), for the calculation of the value in the case of PJTE instability is incorrect because in this case , meaning the first perturbation correction is larger than the main term, and hence the criterion of applicability of the perturbation theory in its simplest form does not hold. In this case, we should consider the contribution of the lowest excited states (that make the total curvature negative) in a pseudo degenerate problem of perturbation theory. For the simplest case when only one excited state creates the main instability of the ground state, we can treat the problem via a pseudo degenerate two-level problem, including the contribution of the higher, weaker-influencing states as a second order correction. In the PJTE two-level problem we have two electronic states of the high-symmetry configuration, ground and excited , separated by an energy interval of , that become mixed under nuclear displacements of certain symmetry ; the denotations , , and indicate, respectively, the irreducible representations to which the symmetry coordinate and the two states belong. In essence, this is the original formulation of the PJTE. Assuming that the excited state is sufficiently close to the ground one, the vibronic coupling between them should be treated as a perturbation problem for two near-degenerate states. With both interacting states non-degenerate the vibronic coupling constant in Eq. (5) (omitting indices) is non-zero for only one coordinate with . This gives us directly the symmetry of the direction of softening and possible distortion of the ground state. Assuming that the primary force constants in the two states are the same (for different see ), we get a 2×2 secular equation with the following solution for the energies of the two states interacting under the linear vibronic coupling (energy is referred to the middle of the gap between the levels at the undistorted geometry):
(7)
It is seen from these expressions that, on taking into account the vibronic coupling, , the two APES curves change in different ways: in the upper sheet the curvature (the coefficient at in the expansion on ) increases, whereas in the lower one it decreases. But until the minima of both states correspond to the point , as in the absence of vibronic mixing. However, if
(8)
the curvature of the lower curve of the APES becomes negative, and the system is unstable with respect to the displacements (Fig. 1). Under condition (8), the minima points on the APES are given by
(9)
From these expressions and Fig. 1 it is seen that while the ground state is softened (destabilized) by the PJTE, the excited state is hardened (stabilized), and this effect is the larger, the smaller and the larger F. It takes place in any polyatomic system and influences many molecular properties, including the existence of stable excited states of molecular systems that are unstable in the ground state (e.g., excited states of intermediates of chemical reactions); in general, even in the absence of instability the PJTE softens the ground state and increases the vibrational frequencies in the excited state.
Comparison with the Jahn–Teller effect