Jahn–Teller Effects in Molecular Cations Studied by Photoelectron Spectroscopy and Group Theory

The traditional “ball‐and‐stick” concept of molecular structure fails when the motion of the electrons is coupled to that of the nuclei. Such a situation arises in the Jahn–Teller (JT) effect which is very common in open‐shell molecular systems, such as radicals or ions. The JT effect is well known to chemists as a mechanism that causes the distortion of an otherwise symmetric system. Its implications on the dynamics of molecules still represent unsolved problems in many cases. Herein we review recent progress in understanding the dynamic structure of molecular cations that have a high permutational symmetry by using rotationally resolved photoelectron spectroscopy and group theory. Specifically, we show how the pseudo‐Jahn–Teller effect in the cyclopentadienyl cation causes electronic localization and nuclear delocalization. The fundamental physical mechanisms underlying the vaguely defined concept of “antiaromaticity” are thereby elucidated. Our investigation of the methane cation represents the first experimental characterization of the JT effect in a threefold degenerate electronic state. A special kind of isomerism resulting from the JT effect has been discovered and is predicted to exist in all JT systems in which the minima on the potential‐energy surface are separated by substantial barriers.     

Jahn–Teller Effect in Circulenes: X‐ray Diffraction Study of Coronene and Corannulene Radical Anions

Single‐crystal X‐ray diffraction studies of two polyaromatic radical anions crystallized as sodium salts, namely [Na(DME)₃]⁺[C₂₀H₁₀^?] ( 1 ) and [Na(DME)₃]⁺[C₂₄H₁₂^?] ( 2 ) are reported. This allowed the first structural evaluation of Jahn–Teller (JT) effects for monoreduced circulenes and a comparison between bowl‐shaped corannulene and planar coronene. The C_s and D_2h symmetrical distortions are found to fit the experimental data for C₂₀H₁₀^.? and C₂₄H₁₂^.?, respectively. The continuous symmetry measure (CSM) analysis was carried out to provide a quantitative measure of the JT distortions in 1 and 2 . In addition, the X‐ray crystallographic results were fully supported by DFT calculations.     

Simplest proof of the Jahn–Teller theorem for molecular systems

A new simple proof of the Jahn–Teller theorem for molecular systems is presented. The proof is based on some general properties of symmetric square representation characters that simplify their explicit treatment and minimize the use of tables. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

A Genuine Jahn–Teller System with Compressed Geometry and Quantum Effects Originating from Zero‐Point Motion

First‐principle calculations together with analysis of the experimental data found for 3d⁹ and 3d⁷ ions in cubic oxides proved that the center found in irradiated CaO:Ni²⁺ corresponds to Ni⁺ under a static Jahn–Teller effect displaying a compressed equilibrium geometry. It was also shown that the anomalous positive g_∥ shift (g_∥?g₀=0.065) measured at T=20 K obeys the superposition of the |3 z²?r²? and |x²?y²? states driven by quantum effects associated with the zero‐point motion, a mechanism first put forward by O'Brien for static Jahn–Teller systems and later extended by Ham to the dynamic Jahn–Teller case. To our knowledge, this is the first genuine Jahn–Teller system (i.e. in which exact degeneracy exists at the high‐symmetry configuration) exhibiting a compressed equilibrium geometry for which large quantum effects allow experimental observation of the effect predicted by O'Brien. Analysis of the calculated energy barriers for different Jahn–Teller systems allowed us to explain the origin of the compressed geometry observed for CaO:Ni⁺.     

Geminal approach to vibronic models of superconductivity in crystals. I. The Jahn–Teller effect

Electronic geminals constructed as linear combinations of binary products of site functions are used to formulate a vibronic model of superconductivity in crystals that is based upon the approximation of independent correlated electron pairs obtained variationally from an electron‐pair Hamiltonian and the Jahn–Teller effect. The cyclic symmetry of the system is taken into account and the geminals are sorted into doubly degenerate pairs. The Herzberg–Teller expansion of the pair Hamiltonian in terms of vibrational modes leads directly to the Jahn–Teller effect. A contact transformation of the vibronic Hamiltonian containing only linear terms lowers the energy of the system by a second‐order term associated with the Jahn–Teller stabilization energy. A possible model for superconductivity in solids is proposed on the basis of the Jahn–Teller effect. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Degree of electron–nuclear entanglement in molecular states

The degree of electron–nuclear entanglement in molecular states is analyzed. This entanglement has, generally, two sources: delocalization of the electronic and nuclear wave functions and vibronic coupling. For a diatomic molecular ground‐state with a single potential energy minimum, it is demonstrated that the entanglement is a function of the product of the vibrational energy and the Born–Huang potential energy correction evaluated at the minimum. In the case of a double‐well potential energy surface, the deviation from maximal entanglement is determined by the overlap of the electronic and nuclear wave functions evaluated at and around the two minima. The adiabatic states of the E⊗ϵ Jahn–Teller model are shown to be maximally entangled and a relation between the degree of entanglement and Ham's reduction factor for this model is derived. Numerical calculations in the E⊗ϵ model demonstrate a nontrivial relation between entanglement and vibronic coupling. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 526–533, 2000     

Distortions of π‐Coordinated Arenes with Anionic Character

A qualitative analysis of the distortions that operate on the π system of bridging arenes with anionic character is presented and substantiated by computational studies at the density functional B3LYP and CASSCF levels. The observed effects of bonding to two metal atoms and of the negative charge are an expansion of the arene ring due to the partial occupation of π* orbitals, an elongation or compression distortion accompanied by a loss of the equivalence of carbon‐carbon bonds due to a Jahn–Teller distortion of the arene dianions, and a ring puckering due to a second‐order Jahn–Teller distortion that may appear independently of the existence of the first‐order effect. The workings of the orbital mixing produced by these distortions have been revealed in a straightforward way by a pseudosymmetry analysis of the HOMOs of the distorted conformations. The systems studied include Li^I and Y^III adducts of benzene, as well as trimethylsilyl‐substituted derivatives in the former case. An analysis of the structural data of a variety of purported di‐ and tetraanionic arene ligands coordinated to transition metals in several bridging modes has reproduced the main geometrical trends found in the computational study for the benzene and trimethylsilyl‐substituted benzene dianions, allowing a classification of the variety of structural motifs found in the literature.     

An investigation of the electron density of a Jahn–Teller‐distorted CrII cation: the crystal structure and charge density of hexakis(acetonitrile‐κN)chromium(II) bis(tetraphenylborate) acetonitrile disolvate

In the crystal structure of the title homoleptic Cr^II complex, [Cr(CH₃CN)₆](C₂₄H₂₀B)₂·CH₃CN, the [Cr(CH₃CN)₆]²⁺ cation is a high‐spin d⁴ complex with strong static, rather than dynamic, Jahn–Teller distortion. The electron density of the cation was determined by single‐crystal X‐ray refinements using aspherical structure factors from wavefunction calculations. The detailed picture of the electronic density allowed us to assess the extent and directionality of the Jahn–Teller distortion of the Cr^II cation away from idealized octahedral symmetry. The topological analysis of the aspherical d‐electron density about the Cr^II cation showed that there are significant valence charge concentrations along the axial Cr—N axes. Likewise, there were significant valence charge depletions about the Cr^II cation along the equatorial Cr—N bonds. These charge concentrations are in accordance with a Jahn–Teller‐distorted six‐coordinate complex.     

Jahn‐Teller‐Ordnung in Mangan(III)‐fluoridsulfaten. II. Phasenübergang und Verzwillingung bei K2[MnF3(SO4)] und 1D Magnetismus in Verbindungen A2[MnF3(SO4)] (A = K,NH4, Rb,Cs)

Jahn‐Teller Ordering in Manganese(III) Fluoride Sulfates. II. Phase Transition and Twinning of K₂[MnF₃(SO₄)] and 1D Magnetism in Compounds A₂[MnF₃(SO₄)] (A = K, NH₄, Rb, Cs) According to single‐crystal X‐ray investigations, K₂[MnF₃(SO₄)] crystallizes at low temperature, like the isostructural Rb, NH₄, and Cs analogues in space group P2₁/c, Z = 4, e.g. at 100 K with a = 7.197, b = 10.704, c = 8.427Å, β = 91.84°. Below about 300 K, the crystals are found to be [001] axis twins. Using a new integration method for area detector records, nearly complete intensity data could be gained allowing for structure refinements of similar quality as for untwinned crystals (e.g. at 100 K: wR₂ = 0.050, R = 0.020 for all reflections). With rising temperature, the monoclinic angle approaches continuously 90°. For an ordering parameter Δβ = β?90° a 2^nd‐order phase transition is observed with an exponent λ = 0.17. At the transition temperature of 280 K resulting from the fit, the monoclinic structure changes – with delay – to orthorhombic with the minimum super‐group Pnca, a = 7.243, b = 10.763, c = 8.457Å, R = 0.024, as found in an early structure determination at room temperature by Edwards 1971. In the chain‐like [MnF₃(SO₄)]^2? anions, manganese(III) is octahedrally coordinated by two trans‐terminal and two trans‐bridging fluorine ligands as well as by the O atoms of two trans‐bridging sulfate ligands. At low temperature, the octahedral elongation by the Jahn‐Teller effect alternates between a F–Mn–F and an O–Mn–O axis (antiferrodistortive ordering). All bridges are asymmetric. From about 320 K on they become symmetric. Due to 2D dynamical Jahn‐Teller effect all octahedra appear compressed. All compounds A₂[MnF₃(SO₄)] show 1D antiferromagnetism. The antiferrodistortive Jahn‐Teller order at low temperatures and the small bridge angles explain the much lower magnetic exchange energies and their inverse relation to the bridge angles as compared with other fluoromanganate(III) chain compounds with the usual ferrodistortive ordering.     

Nonadiabatic theory of triatomics: Formalism for 1Πu/1Φg interaction,for electronic spin,and for 2Λ Renner–Teller effect

A recent theory of nonadiabatic effects in triatomic molecules is specialized to the four-state Renner–Teller and Jahn–Teller ¹Π_u/¹Φ_g interactions and is then generalized by including the electronic spin and by considering the ²Λ Renner–Teller effect.     

Magnetic Anisotropy in “Scorpionate” First‐Row Transition‐Metal Complexes: A Theoretical Investigation

In this work we have analyzed in detail the magnetic anisotropy in a series of hydrotris(pyrazolyl)borate (Tp^?) metal complexes, namely [VTpCl]⁺, [CrTpCl]⁺, [MnTpCl]⁺, [FeTpCl], [CoTpCl], and [NiTpCl], and their substituted methyl and tert‐butyl analogues with the goal of observing the effect of the ligand field on the magnetic properties. In the [VTpCl]⁺, [CrTpCl]⁺, [CoTpCl], and [NiTpCl] complexes, the magnetic anisotropy arises as a consequence of out‐of‐state spin–orbit coupling, and covalent changes induced by the substitution of hydrogen atoms on the pyrazolyl rings does not lead to drastic changes in the magnetic anisotropy. On the other hand, much larger magnetic anisotropies were predicted in complexes displaying a degenerate ground state, namely [MnTpCl]⁺ and [FeTpCl], due to in‐state spin–orbit coupling. The anisotropy in these systems was shown to be very sensitive to perturbations, for example, chemical substitution and distortions due to the Jahn–Teller effect. We found that by substituting the hydrogen atoms in [MnTpCl]⁺ and [FeTpCl] by methyl and tert‐butyl groups, certain covalent contributions to the magnetic anisotropy energy (MAE) could be controlled, thereby achieving higher values. Moreover, we showed that the selection of ion has important consequences for the symmetry of the ground spin–orbit term, opening the possibility of achieving zero magnetic tunneling even in non‐Kramers ions. We have also shown that substitution may also contribute to a quenching of the Jahn–Teller effect, which could significantly reduce the magnetic anisotropy of the complexes studied.     

Einfluss von H‐Brückenbindungen auf die Jahn–Teller‐Verzerrung in den Kettenstrukturen von 2,2′‐bipyMn(H2PO4)F2·H2O und 2,2′‐bipyMn(H2PO4)2F

In the system 2,2′‐bipyridine/Mn^III/HF/H₃PO₄/H₂O two compounds with chain structures could be prepared and characterised by X‐ray structure analyses. 2,2′‐bipyMn(H₂PO₄)F₂·H₂O ( 1 ): monoclinic, twinned, space group P2₁/c, Z = 4, a = 6.7883(4), b = 10.9147(5), c = 17.8102(8) Å, β = 100.142(4)°, R = 0.0328. 2,2′‐bipyMn(H₂PO₄)₂F ( 2 ): triclinic, space group P , Z = 2, a = 6.675(1), b = 10.715(1), c = 11.013(1) Å, α = 107.595(9)°, β = 90.994(9)°, γ = 95.784(8)°, R = 0.0252. Both compounds show chain structures with trans‐bridging dihydrogenphosphate ligands and bipy and two fluorine ligands for ( 1 ), or bipy, fluorine and an additional dihydrogenphosphate, respectively, for ( 2 ) in equatorial positions. Due to the pseudo‐Jahn–Teller effect, Mn^III shows elongated octahedral coordination with ferrodistortive ordering along the chain direction. The distortion is remarkably higher in ( 1 ) than in ( 2 ). This is discussed in context with additional hydrogen bonds along the chain in ( 2 ).     

VIBPACK: A package to treat multidimensional electron‐vibrational molecular problems with application to magnetic and optical properties

We present a FORTRAN code based on a new powerful and efficient computational approach to solve multidimensional dynamic Jahn–Teller and pseudo Jahn–Teller problems. This symmetry‐assisted approach constituting a theoretical core of the program is based on the full exploration of the point symmetry of the electronic and vibrational states. We also report some selected examples of increasing complexity aimed to display the theoretical background as well as the advantages and capabilities of the program to evaluate of the energy pattern, magnetic and optical properties of large multimode vibronic systems. © 2018 Wiley Periodicals, Inc.     

Sodium and Manganese Stoichiometry of P2‐Type Na2/3MnO2

To realize a reversible solid‐state Mn^III/IV redox couple in layered oxides, co‐operative Jahn–Teller distortion (CJTD) of six‐coordinate Mn^III (t_2g³–e_g¹) is a key factor in terms of structural and physical properties. We develop a single‐phase synthesis route for two polymorphs, namely distorted and undistorted P2‐type Na_2/3MnO₂ having different Mn stoichiometry, and investigate how the structural and stoichiometric difference influences electrochemical reaction. The distorted Na_2/3MnO₂ delivers 216 mAh g^?1 as a 3 V class positive electrode, reaching 590 Wh (kg oxide)^?1 with excellent cycle stability in a non‐aqueous Na cell and demonstrates better electrochemical behavior compared to undistorted Na_2/3MnO₂. Furthermore, reversible phase transitions correlated with CJTD are found upon (de)sodiation for distorted Na_2/3MnO₂, providing a new insight into utilization of the Mn^III/IV redox couple for positive electrodes of Na‐ion batteries.     

Enhanced Pseudocapacitance in Multicomponent Transition‐Metal Oxides by Local Distortion of Oxygen Octahedra

Anomalously high pseudocapacitance of a metal oxide was observed when Ni, Co, and Mn were mixed in a solid solution. Analysis by X‐ray absorption near‐edge spectroscopy (XANES) identified a wider redox swing of Ni as the origin of the enlarged pseudocapacitance. Ab initio DFT calculations revealed that aliovalent species resulting from the copresence of multiple transition metals can generate permanent local distortions of [NiO₆] octahedra. As this type of distortion breaks the degenerate e_g level of Ni²⁺, the Jahn–Teller lattice instability necessary for the Ni^2+/3+ redox flip can be effectively diminished during charge–discharge, thus resulting in the significantly increased capacitance. Our findings highlight the importance of understanding structure–property correlation related to local structural distortions in improving the performance of pseudocapacitors.     

Structural Effects of Cu/Zn Substitution in the Malachite–Rosasite System

Synthetic zincian malachite samples (Cu_1–xZn_x)₂(OH)₂CO₃ with x = 0, 0.1, 0.2 and 0.3 were characterized by powder X‐ray diffraction and optical spectroscopy. The XRD patterns of the samples up to x = 0.2 indicate single phase materials with an approximately linear dependence of the refined lattice parameters on the zinc content. In contrast, the sample with a nominal zinc content x = 0.3 shows the formation of a small amount of aurichalcite (Zn,Cu)₅(OH)₆(CO₃)₂ as an additional phase. Based on the lattice parameter variations, the zinc content of the zincian malachite component in this sample is estimated to be x ≈? 0.27, which seems to represent the maximum possible substitution in zincian malachite under the synthesis conditions applied. The results are discussed in relation to preparation of Cu/ZnO catalysts and the crystal structures of the minerals malachite and rosasite. One striking difference between these two structurally closely related phases is the orientation of the Jahn–Teller elongated axes of the CuO₆ octahedra in the unit cell, which seems to be correlated with the placement of the monoclinic β angle. The structural and chemical relationship between these crystallographically distinct phases is discussed using a hypothetical intermediate Zn₂(OH)₂CO₃ phase of higher orthorhombic symmetry. In addition to the crystallographic analysis, optical spectroscopy proves to be a useful tool for estimation of the Cu:Zn ratio in (Cu_1–xZn_x)₂(OH)₂CO₃ samples.     

Analogies between Jahn–Teller and Rashba spin physics

In developing physical theories, analogical reasoning has been found to be very powerful, as attested by a number of important historical examples. An analogy between two apparently different phenomena, once established, allows one to transfer information and bring new concepts from one phenomenon to the other. Here, we discuss an important analogy between two widely different physical problems, namely, the Jahn–Teller distortion in molecular physics and the Rashba spin splitting in condensed matter physics. By exploring their conceptual and mathematical features and by searching for the counterparts between them, we examine the orbital texture in Jahn–Teller systems, as the counterpart of the spin texture of the Rashba physics, and put forward a possible way of experimentally detecting the orbital texture. Finally, we discuss the analogy by comparing the coexistence of linear Rashba + Dresselhaus effects and Jahn–Teller problems for specific symmetries, which allow for nontrivial spin and orbital textures, respectively.     

Exact S‐wave solution of the trigonometric pöschl‐teller potential

The trigonometric Pöschl‐Teller (PT) potential describes the diatomic molecular vibration. By using the Nikiforov‐Uvarov method, we have obtained the exact analytical s‐wave solutions of the radial Schrödinger equation (SE) for the trigonometric PT potential. The energy eigenvalues and corresponding eigenfunctions are calculated in closed forms. Some numerical results are presented too. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

P2‐Na0.67AlxMn1−xO2: Cost‐Effective,Stable and High‐Rate Sodium Electrodes by Suppressing Phase Transitions and Enhancing Sodium Cation Mobility

Sodium layered P2‐stacking Na_0.67MnO₂ materials have shown great promise for sodium‐ion batteries. However, the undesired Jahn–Teller effect of the Mn⁴⁺/Mn³⁺ redox couple and multiple biphasic structural transitions during charge/discharge of the materials lead to anisotropic structure expansion and rapid capacity decay. Herein, by introducing abundant Al into the transition‐metal layers to decrease the number of Mn³⁺, we obtain the low cost pure P2‐type Na_0.67Al_xMn_1?xO₂ (x=0.05, 0.1 and 0.2) materials with high structural stability and promising performance. The Al‐doping effect on the long/short range structural evolutions and electrochemical performances is further investigated by combining in situ synchrotron XRD and solid‐state NMR techniques. Our results reveal that Al‐doping alleviates the phase transformations thus giving rise to better cycling life, and leads to a larger spacing of Na⁺ layer thus producing a remarkable rate capability of 96 mAh g^‐1 at 1200 mA g^‐1.     

Investigations of the local distortions and EPR parameters for Cu2+ in xNa2O‐(30–x)K2O‐70B2O3 (5 ≤ x ≤ 25 mol%) glasses

The local distortions and electron paramagnetic resonance parameters for Cu²⁺ in the mixed alkali borate glasses xNa₂O‐(30–x)K₂O‐70B₂O₃ (5 ≤ x ≤ 25 mol%) are theoretically studied with distinct modifier Na₂O compositions x. Owing to the Jahn–Teller effect, the octahedral [CuO₆]¹⁰⁻ clusters show significant tetragonal elongation ratios p ~19% along the C₄ axis. With the increase of composition x, the cubic field parameter Dq and the orbital reduction factor k exhibit linearly and quasi‐linearly decreasing tendencies, respectively, whereas the relative tetragonal elongation ratio p has quasi‐linearly increasing rule with some fluctuations, leading to the minima of g factors at x = 10 mol%. The composition dependences of the optical spectra and the electron paramagnetic resonance parameters are suitably reproduced by the linear or quasi‐linear relationships of the relevant quantities (i.e., Dq, k, and p) with x. The above composition dependences are analyzed from mixed alkali effect, which brings forward the modifications of the local crystal‐fields and the electronic cloud distribution around Cu²⁺ with the variation of the composition of Na₂O.