Vibronic Interactions and the Jahn-Teller Effect

The multideterminental-DFT approach performed to analyze Jahn-Teller (JT) active molecules is described. Extension of this method for the analysis of the adiabatic potential energy surfaces and the multimode JT effect is presented. Conceptually a simple model, based on the analogy between the JT distortion and reaction coordinates gives further information about microscopic origin of the JT effect. Within the harmonic approximation the JT distortion can be expressed as a linear combination of all totally symmetric normal modes in the low symmetry minimum energy conformation, which allows calculating the Intrinsic Distortion Path, IDP, exactly from the high symmetry nuclear configuration to the low symmetry energy minimum. It is possible to quantify the contribution of different normal modes to the distortion, their energy contribution to the total stabilization energy and how their contribution changes along the IDP. It is noteworthy that the results obtained by both multideterminental-DFT and IDP methods for different classes of JT active molecules are consistent and in agreement with available theoretical and experimental values. As an example, detailed description of the ground state distortion of benzene cation is given.

In this article we present a symmetry-adapted approach aimed to the accurate solution of the dynamic Jahn-Teller (JT) problem. The algorithm for the solution of the eigen-problem takes full advantage of the point symmetry arguments. The system under consideration is supposed to consist of a set of electronic levels \({\Gamma }_{1},{\Gamma }_{2}\ldots {\Gamma }_{n}\) labeled by the irreducible representations (irreps) of the actual point group, mixed by the active JT and pseudo JT vibrational modes \({\Gamma }_{1},{\Gamma }_{2}\ldots {\Gamma }_{f}\) (vibrational irreps). The bosonic creation operators b ⁺(Γγ) are transformed as components γ of the vibrational irrep Γ. The first excited vibrational states are obtained by the application of the operators \({b}^{+}(\Gamma \gamma )\) to the vacuum: \({b}^{+}(\Gamma \gamma )\vert n = 0,{A}_{1}\rangle = \vert n = 1,\Gamma \gamma \rangle\) and therefore they belong to the symmetry Γγ. Then the operators b ⁺(Γγ) act on the set \(\vert n = 1,\Gamma \gamma \rangle\) with the subsequent Clebsch-Gordan coupling of the resulting irreps. In this way one obtains the basis set \(\vert n = 2,{\Gamma }^{{\prime}}{\gamma }^{{\prime}}\rangle\) with \({\Gamma }^{{\prime}}\in \Gamma \otimes \Gamma \). In general, the Gram-Schmidt orthogonalization is required at each step of the procedure. Finally, the generated vibrational bases are coupled to the electronic ones to get the symmetry adapted basis in which the full matrix of the JT Hamiltonian is blocked according to the irreps of the point group. The approach is realized as a computer program that generates the blocks and evaluates all required characteristics of the JT systems. The approach is illustrated by the simulation of the vibronic charge transfer (intervalence) optical bands in trimeric mixed valence clusters.

According to Jahn-Teller (JT) theorem any nonlinear arrangement of atomic nuclei in electron degenerate state (except an accidental and Kramers degeneracy) is unstable. Pseudo-JT systems are treated as an analogy of JT theorem for pseudo-degenerate electron states. Stable nuclear arrangements of lower energy of such systems correspond to the minima of their potential energy surfaces (PES). In large systems the analytical description of their PES is too complicated and a group-theoretical treatment must be used to describe their stable structures. This may be based either on JT active coordinates– as in the epikernel principle– or on the electron states– as in the method of step-by-step descent in symmetry. This review explains the basic terms of group theory (especially of point groups of symmetry) and potential energy surfaces (extremal points and their characteristics) as well as the principles of group-theoretical methods predicting the extremal points of JT systems– the method of epikernel principle (based on JT active coordinates) and the method of step-by-step descent in symmetry (based on a consecutive split of the degenerate electron states). Despite these methods have been elaborated for the case of electron degeneracy, they are applicable to pseudo-degenerate electron states as well. The applications of both methods to pseudo-JT systems are presented on several examples and compared with the published results.

A true dynamic Jahn-Teller effect in the solid state has proven to be quite elusive, as pure compounds suffer from cooperative effects, while doped systems are susceptible to small crystal imperfections that lock the system into static distortions. Cu(II) doped into the cubic host MgO represents a rare example of such a dynamic Jahn-Teller system and has been the subject of numerous experimental and theoretical studies. Recently we have presented high resolution low temperature Electron Paramagnetic Resonance (EPR) spectra of Cu(II)/MgO as a function of the applied field direction. These spectra indicate that at temperatures as low as 1.8 K the Cu(II) centre is in a degenerate vibronic state of E symmetry that is delocalized over the ground potential energy surface, indicating a true dynamic Jahn-Teller effect. The experiments also show us that this system has a potential energy surface with three equivalent minima, each at three equivalent tetragonally elongated geometries, separated by low barriers. Relaxation from the anisotropic E type spectrum to an isotropic spectrum occurs at temperatures above 6 K. The observation of the dynamic Jahn-Teller effect in this system is due to small barrier heights between the minima and the random crystal strain, which is small when compared to the tunneling splitting. We examine the limitations of the experiment in being able to determine these quantities separately and suggest future experiments that may shed further light on this fascinating system.

In this paper we review the concepts of dynamic and static Jahn-Teller (JT) effects and study their influence in the electron paramagnetic resonance (EPR) spectra of transition metal impurities in wide gap insulators. We show that the key quantities involved in this problem are the tunneling splitting, usually denoted 3Γ, and the splitting, δ, between∼​3z ² −r ² and∼​ x ² −y ² energy levels due to the random strain field which is present in every real crystal. It is pointed out that in the E⊗e JT problem the kinetic energy of nuclei involved in the ground state plays a key role for understanding the actual value of 3Γ and thus the existence of dynamic JT effect. The results of ab initio calculations on a variety of JT systems show that 3Γ values span a much larger range than previously suggested. In particular, we find that 3Γ=235cm⁻¹ for MgO:Cu²⁺ while 3Γ=10⁻⁴cm⁻¹ for KCl:Ag²⁺. We also show that the dynamic JT effect can only appear for such large values of 3Γ as those found in MgO:Cu²⁺ since usual strain fields lead to δ values of the order of 10cm⁻¹ that would otherwise localize the system in only one of the JT wells. The present results explain satisfactorily why the JT effect for Cu²⁺ and Ag²⁺ impurities in MgO is dynamic while static for Ag²⁺-doped CaO and alkali chlorides. The origin of such a difference is discussed in detail.

A review is presented of the worked out earlier method that uses ultrasonic experiments to evaluate Jahn-Teller effect (JTE) parameters, mostly linear vibronic coupling constants, in application to impurity centers of dopant crystals. The method employs temperature dependent ultrasound attenuation and phase velocity measurements. Distinguished from previous attempts to detect the JTE parameters by ultrasound, this method does not assume any specific mechanism of relaxation, and hence it can be applied to any JTE problem, irrelevant to its complicated dynamics of distortions. It is shown that by combining the ultrasound results with some additional information about the JT stabilization energy obtained from independent sources, the whole adiabatic potential energy surface of the JT center can be evaluated. Two examples of application of this method relevant to two JTE problems are considered, both for impurity centers in crystals with zinc-blend structure and tetrahedral coordination of the impurity ion: ZnSe:Fe²⁺ with the E⊗e problem and ZnSe:Cr²⁺ with a T term and the T⊗(e+t ₂) problem.

A theory of the Raman scattering in resonance with an electronic transition causing a strong softening of vibrations is proposed. In this case the potential surface of the excited state has a flat minimum or maximum in the configurational coordinate space. Two cases of the vibronic coupling are considered: (1) the coupling with a single coordinate and (2) the coupling with the phonon continuum. To describe the Raman scattering the Fourier-amplitude method is applied. In the first case the calculations are performed for the pseudo-Jahn-Teller effect in the excited state. In the second case, despite a strong mixing of phonons, the equations for the Raman Fourier amplitudes can be factorized and solved analytically. It is predicted that the second-order Raman scattering will be strongly enhanced. Moreover, the second-order Raman scattering is also enhanced as compared to the first-order scattering. The Raman excitation profiles show a structure caused by the Airy oscillations. The theory is applied to the triplet-triplet optical transition in Na₂ molecule confined at the surface of a ⁴He droplet.

A theory of electronic transitions from a non-degenerated to a two-fold degenerated or quasi-degenerated electronic state in an impurity centre of a crystal is proposed. The coupling of optical electron(s) with non-totally symmetrical vibration(s) and with phonons are taken into account. The latter interaction results in the replacement of vibronic lines by phonon-assisted bands. The widths of bands, the rates of the energy relaxation of vibronic states and the shapes of optical spectra are calculated for some values of parameters.

Performance and reliability in semiconductor devices are limited by electronically active defects, primarily O-atom vacancies. Synchrotron X-ray spectroscopy results, interpreted in the context of multiplet theories, have been used to analyze conduction band edge, and O-vacancy defect states in nanocrystalline transition metal oxides such as HfO₂, and the non-crystalline oxides including SiO₂, and Si₃N₄ and Si oxynitride alloys. Multiplet theory provides the theoretical foundation for an equivalentd ² model for O-vacancy transitions and negative ion states as detected by X-ray absorption spectroscopy in the O K pre-edge regime. Comparisons between theory and experiment have relied on Tanabe-Sugano energy level diagrams for identifying the symmetries and multiplicities of transition energies for an equivalent d² ground state occupancy. The equivalent d² model has been applied to nanocrystalline thin films of ZrO₂,HfO₂,TiO₂ and Lu₂O₃ and provides excellent agreement with X-ray absorption spectroscopy data. The model has also been applied to SiO₂ and other Si based dielectrics where very good agreement with multiplet theory has also been demonstrated. The spectra indicate both triplet and singlet final states indicating that the two electrons in the vacancy sites have singlet and triplet ground states that are within a few tenths of an eV of each other. For the transition metal oxides, this is explained by relatively small distortions in the vacancy geometry in which the separation between the respective transition metal atoms is 1.6 times the bond-length in an ideal tetrahedral geometry, or the same factor for two fold coordination in O-atom bonding sites in SiO₂ andGeO₂. These distortions minimize the exchange energy in triplet spin states, and reduce the radial wave function overlap in singlet spin states.

The molecular orbitals of fullerene molecules on surface substrates can be imaged experimentally using scanning tunnelling microscopy (STM). The observed images are influenced by interactions with the substrate. In addition, for fullerene ions, splitting of the orbitals by the Jahn–Teller (JT) effect also affects the observed images. In this work, we consider the effect of both static and dynamic JT interactions on the images that are expected to be obtained from the fullerene anion C₆₀ ⁻, taking into account interactions with the substrate. Our method is to use Hückel molecular orbital (HMO) theory, which is very simple and quick to implement on a desktop computer. Although our approach is not as rigorous as using density functional theory (DFT), the predicted STM images are almost indistinguishable from published DFT images. Furthermore, it readily allows us to explore different situations, such as different adsorption geometries. Our results are also compared with experimental and simulated STM images in the literature.

The fullerene trianion, C₆₀ ³⁻, and the compounds associated with it are known to have properties that differ significantly from the other fullerene ions. For example, compounds of the form A₃C₆₀ (where A is an alkali metal) which contain this ion, are known to be superconductors up to around 40K, whereas the alkali metal fullerenes containing C₆₀ ²⁻ and C₆₀ ⁴⁻ are found to be insulators, properties often attributed to the Jahn–Teller effect. In spite of this, little work has been undertaken analysing the Jahn–Teller effect in the trianion. In this work, the symmetry reduction caused by this effect is investigated by introducing quadratic terms into the Hamiltonian to model the Jahn–Teller interaction. It is found that, unlike the previously investigated ions of C₆₀, an electronic degeneracy remains if the molecular distortion were to be described by either the D _3d or D _5d point groups. Thus, a further reduction in symmetry is expected, and it is found that the distorted molecule is actually described by either the C _2h or D _2h group. A distortion of C _2h symmetry in a fullerene molecule has previously undergone little analysis, and so it is this that is then investigated by considering a set of distortional axes relating to the minimum energy wells formed under a quadratic interaction.

The vibronic coupling constants of C₆₀ ⁻ are derived experimentally and theoretically. The experimental constants are obtained by simulating the photoelectron spectrum measured by Wang et al. (J Chem Phys 123:051106, 2005). The vibronic states are calculated with the Lanczos method and second order perturbation theory. We find that the coupling constants are underestimated with the perturbation method. The couplings are calculated based on the density-functional method and state-averaged complete active space self-consistent-field methods. The vibronic coupling constants obtained from the gradient of the total electronic energy of C₆₀ ⁻ using the B3LYP functional are close to the experimental values. On the other hand, the coupling constants are overestimated with the derivative of the lowest unoccupied Kohn–Sham level of the neutral C₆₀ which is sometimes used in the literatures.

The boron fullerene B₈₀ is a spherical network of 80 boron atoms, which has a shape similar to the celebrated C₆₀. The 80 Bs span two orbits: while the first contains 60 atoms localised on the vertices of a truncated icosahedron like C₆₀, the second includes 20 extra B atoms capping the hexagons of the frame. Quantum chemical calculations showed that B₈₀ is unusually stable and has interesting physical and chemical properties. Its geometry is slightly distorted from I _h to T _h symmetry. However, the boron buckyball is only observed in silico, so far the synthesis of this molecule is only a remote possibility. Using DFT at the B3LYP/SVP level, we have analyzed the chemical bonding in B₈₀, the possibility of methyne substitution and the stability of endohedral boron buckyball complexes. A symmetry analysis revealed a perfect match between the occupied molecular orbitals in B₈₀ and C₆₀. The cap atoms transfer their electrons to the truncated icosahedral frame, and they contribute essentially to the formation of σ bonds. The frontier MOs have π character and are localised on the B₆₀ truncated icosahedral frame. The boron cap atoms can be replaced by other chemical groups, such as methyne (CH), which are also able to introduce three electrons in the cage. Symmetrical substitutions of the boron cap atoms by methyne groups in T and T _h symmetries revealed two stable endo methyne boron buckyballs, endo-\({\mathrm{B}}_{80-\mathrm{x}}\mathrm{{(CH)}_{x}}\), with x = 4, 8. The stability of these compounds seems to be due to the formation of six boron 4-centre bonding motifs in between the substituted hexagons. These localized bonding motifs are at the basis of the observed symmetry lowering, via a pseudo-Jahn-Teller effect. The methyne hydrogen atoms in the two endohedral fullerenes can be replaced by other atoms, which can lead to cubane or tetrahedral endohedral boron fullerenes. Theoretical study on encapsulated small bases molecules, tetrahedral and cubane like clusters of Group V atoms, showed that the boron buckyball is a hard acid and prefers hard bases like NH₃ or N₂H₄, to form stables off-centred complexes with B₈₀. Tetrahedral and cubane like clusters of this family are usually metastable in the encapsulated state, due to steric strain. The most favorable clusters are mixed tetrahedral and cubane clusters formed by nitrogen and phosphorus atoms such as \(\mathrm{{P}_{2}{N}_{2}@{B}_{80},\ {P}_{3}N@{B}_{80}}\) and P₄N₄@B₈₀. The boron cap atoms act as electrophilic centres, which react with nucleophilic sites rich in electrons.

The non-adiabatic coupling terms (NACTs) among the electronic states 2² E ^′ and 1² A ₁ ^′ of Na₃ system demonstrate the numerical validity of so called “Curl Condition” and thus such states closely form a sub-Hilbert space. For this subspace, we employ the NAC terms to solve the “Adiabatic–Diabatic Transformation (ADT)” equations to obtain the functional form of the transformation angles and pave the way to construct the continuous and single valued diabatic potential energy surface matrix. Nuclear dynamics has been carried out on those diabatic surfaces to reproduce the experimental spectrum for system B of Na₃ cluster and thereby, to explore the numerical validity of the theoretical development on beyond Born–Oppenheimer approach for adiabatic to diabatic transformation.

Triatomic alkali-metal clusters in their high-spin manifolds of electronically excited states provide the chance to investigate the spectroscopic consequences of the combination of Jahn–Teller effect and spin-orbit coupling with powerful methods of quantum chemistry such as open-shell coupled cluster approaches and multireference Rayleigh-Schroedinger perturbation theory. With respect to available experimental data the 2⁴E ^′ ← 1⁴A₂ ^′ transitions are selected to document the quenching of the paradigmatic E ⊗ e Jahn–Teller distortion with increasing spin-orbit coupling. The simulated spectra for potassium, rubidium and cesium trimers are provided together with all relevant parameters such as harmonic frequencies, Jahn–Teller parameters and spin-orbit splittings obtained from the ab initio approach. Beside that, the molecular geometries and formation energies of these van der Waals molecules are also listed in this chapter.

Because the orbital angular momentum L _z ^cf on the Fe(II) sites of the Fe(II)Fe(III) bimetallic oxalates is incompletely quenched by the crystal field, the spin-orbit coupling competes with the Jahn-Teller (JT) distortion energy. The value of L _z ^cf depends on the cation between the bimetallic layers. When L _z ^cf is sufficiently small, the open honeycomb lattice of each bimetallic layer is distorted at all temperatures below the JT transition temperature. But in a range of L _z ^cf , the lattice is only distorted between lower and upper JT transition temperatures, T _JT ^(l) and T _JT ^(u). For some cations, L _z ^cf may exceed the threshold required for the cancellation of the moments on the Fe(II) and Fe(III) sublattices at a temperature T _comp below the transition temperature T _c . Using elastic constants obtained from compounds that exhibit magnetic compensation, we find that T _JT ^(l) always lies between T _comp and T _c and that T _JT ^(u) always lies above T _c .

In this contribution we introduce the concept of single molecule ferroelectric based on the vibronic pseudo Jahn-Teller model of mixed valence dimeric clusters belonging to the Robin and Day class II compounds. We elucidate the main factors controlling the nonadiabatic Landau-Zener tunneling between the low lying vibronic levels induced by a pulse of the electric field. The transition probabilities are shown to be dependent on the both time of the pulse and the total spin of the cluster. A possibility to control the spin-dependent Landau-Zener tunneling by applying a static magnetic field is discussed.

A mixed valence iron dimer \([{\mathrm{L}}^{1}{\mathrm{Fe}}_{2}{(\mu -\mathrm{OAc})}_{2}]({\mathrm{ClO}}_{4})\) is investigated in the framework of the generalized vibronic model which takes into account both the local vibrations on the metal sites (Piepho-Krausz-Schatz model) and the molecular vibrations changing the intermetallic distance (suggested by Piepho). It is shown that the behaviour of the system is determined by a strong competition between three main processes: double exchange interaction and vibronic coupling with both types of vibrations. The optical and magnetic properties of the regarded compound are reported. The influence of the key parameters of the system on these properties is studied in the framework of the presented theoretical model. The degree of delocalization of the itinerant ‘extra’ electron and probability distribution in configuration (qQ) space are calculated at different values of temperature.

Only two of the first row transition metal binary oxides are either ferro- or ferri-magnetic. These are CrO₂ and Fe₃O₄. The electron spin alignment promoting electron spin alignment is associated with a double exchange mechanism requiring mixed valence as well as metallic conductivity. This chapter describes a novel way to realize these two necessary, but not sufficient conditions for double exchange magnetism. These are mixed valence and a hopping conductivity that can promote intra-plane electron spin alignment in a complex oxide host perovskite, \({\mathrm{GeSc}}_{1-\mathrm{x}}{\mathrm{Ti}}_{\mathrm{x}}{\mathrm{O}}_{3}\). This in-plane spin-correlation does necessarily produce for producing spin alignment between alternating (Sc,Ti)O₂ atomic planes, especially in distorted perovskite structures. Intra-plane alignment is obtained when the A-atom of a trivalent atom AB(D)O₃ peroskite, in this example Gd, is an ordinary metal or a rare earth atom, the B-atom, in this example Sc, is a d⁰ transition metal, and the dopant atom, D, in this example, Ti^{3 +} in d¹ state, is a dⁿ transition with n ≥ 1, as in \({\mathrm{GdSc}}_{1-\mathrm{x}}{\mathrm{Ti}}_{\mathrm{x}}{\mathrm{O}}_{3}\). This article combines X-ray absorption spectroscopy, multiplet theory, and degeneracy removal by a Jahn-Teller effect mechanisms to demonstrate intra-layer mixed valence for Sc and Ti above a percolation threshold x ∼ 0.16 at which hopping transport is associated with a metal to insulator transition. This has been observed in epitaxial films, and not in nano-grain nanocrystalline, where the number of Sc atoms in a grain with 2–5nm dimensions is orders of magnitude too small for observation of an a hopping conductivity that requires a percolation mechanism.

In this article we present a theoretical microscopic approach to the problem of cooperative spin crossover in crystals based on cyano-bridged pentanuclear metal clusters \(\{{[{\mathrm{M(CN)}}_{6}]}_{2}{[{\mathrm{Fe(tmphen)}}_{2}]}_{3}\}\) (M = Co, Os) with a trigonal bipyramidal structure. The low-spin to high-spin transition is considered as a cooperative phenomenon that is driven by the interaction of the electronic shells of the Fe ions with the full symmetric deformation of the local surrounding that is extended over the crystal lattice via the acoustic phonon field. Due to the proximity of Fe ions within the metal cluster, the short-range intracluster interactions between these ions via the optic phonon field is included as well. The interrelation between short- and long-range phonon induced interactions between metal ions was shown to determine the type and the temperature of spin transition in cluster systems. The proposed model is applied to the systems with temperature induced spin transition within the single metal ion as well as to the complexes with spin transition caused by the charge transfer between different ions in the cluster. A wide set of the experimental data on the temperature dependence of the magnetic susceptibility and Mössbauer spectra is interpreted qualitatively and quantitatively. The approach described in this paper represents a theoretical tool for the study of spin-crossover systems based on metal clusters as structural units of the crystal lattice.

Based on the model proposed by Kamimura and Suwa which bears important characteristics born from the interplay of Jahn–Teller Physics and Mott Physics, it is shown that the feature of Fermi surfaces is the Fermi pockets constructed by doped holes under the coexistence of a metallic state and of the local antiferromagnetic order. Then it is discussed that the phonon-involved mechanism based on the Kamimura–Suwa model leads to the d-wave superconductivity. Further it is shown that T _c is higher in the cuprates with CuO₅ pyramid than those with CuO₆ octahedron. Finally a new phase diagram for underdoped cuprates is proposed.

The investigation of the lowest adiabatic potential energy surface (APES) of the AO₄ structural unit in KDP (KH₂PO₄)-family materials as a function of its proton surrounding is reported. This approach results in the Ising form of the crystal pseudospin Hamiltonian. Besides the electronic structure characteristics of the AO₄ tetrahedra, the Ising model interaction parameters depend also on the vibronic constants. Numerical estimations of scale for vibronic contributions into Ising parameters (about a quarter of the total value of the maximal Ising parameter V) was obtained by using diffraction data for KH₂PO₄ ∕ KD₂PO₄ crystal structure change under ferroelectric transition together with the results of quantum-chemical calculation of PO₄ ^{3 −} force constants. The estimates show the really tangible role of the vibronic terms in the Ising parameters formation, though the physical reasons of these terms remained somewhat unclear. The situation is clarified by revealing the relationship between the above-mentioned approach and the theory of pseudospin-phonon (proton-lattice) coupling which leads to the vibronic contributions to the Ising parameters.

Mutual influence of magnetic interactions and competing two types of virtual phonon exchange interactions is considered in the framework of the cooperative Jahn-Teller effect theory. It is shown that this competition in the presence of magnetic interactions can result in a new type of structural tetragonal –tetragonal transition. The corresponding virtual phonon exchange interactions drastically affect the magnetic crystal properties. Comparison with the experimental data is presented.