scholarly journals Graph theory and the Jahn–Teller theorem

2011 ◽  
Vol 468 (2140) ◽  
pp. 971-989 ◽  
Author(s):  
A. Ceulemans ◽  
E. Lijnen ◽  
P. W. Fowler ◽  
R. B. Mallion ◽  
T. Pisanski
Keyword(s):  
Energy Levels ◽  
Electronic States ◽  
Geometric Distortion ◽  
Hamiltonian Matrix ◽  
Weighted Graphs ◽  
General Graph ◽  
Full Resolution ◽  
Jahn Teller ◽  
Degenerate Eigenvalues ◽  
Eigenvectors And Eigenvalues

The Jahn–Teller (JT) theorem predicts spontaneous symmetry breaking and lifting of degeneracy in degenerate electronic states of (nonlinear) molecular and solid-state systems. In these cases, degeneracy is lifted by geometric distortion. Molecular problems are often modelled using spectral theory for weighted graphs, and the present paper turns this process around and reformulates the JT theorem for general vertex- and edge-weighted graphs themselves. If the eigenvectors and eigenvalues of a general graph are considered as orbitals and energy levels (respectively) to be occupied by electrons, then degeneracy of states can be resolved by a non-totally symmetric re-weighting of edges and, where necessary, vertices. This leads to the conjecture that whenever the spectrum of a graph contains a set of bonding or anti-bonding degenerate eigenvalues, the roots of the Hamiltonian matrix over this set will show a linear dependence on edge distortions, which has the effect of lifting the degeneracy. When the degenerate level is non-bonding, distortions of vertex weights have to be included to obtain a full resolution of the eigenspace of the degeneracy. Explicit treatments are given for examples of the octahedral graph, where the degeneracy to be lifted is forced by symmetry, and the phenalenyl graph, where the degeneracy is accidental in terms of the automorphism group.

ESR STUDY OF SPIN-HAMILTONIAN PARAMETERS AND CRYSTAL FIELD ENERGY LEVELS FOR THE LOW C3 SYMMETRY Fe3+ CENTER IN GREEN SAPPHIRE CRYSTALS AND POWDER

2007 ◽  
Vol 21 (04) ◽  
pp. 225-236 ◽  
Author(s):  
P. LIMSUWAN ◽  
N. UDOMKAN ◽  
P. WINOTAI
Keyword(s):  
Rotation Angle ◽  
Energy Levels ◽  
Heat Treating ◽  
Esr Spectra ◽  
Field Energy ◽  
Hamiltonian Matrix ◽  
Magnetic Field Direction ◽  
Spin Hamiltonian ◽  
Corundum Structure ◽  
Hamiltonian Parameters

In this report, Fe 3+ impurity ions present in green sapphire ( Al 2 O 3) were studied experimentally, by heating a light green sapphire in flowing oxygen atmosphere for 12 h from 1200, 1300, 1400, 1500 and 1600°C, respectively. Electron spin resonance (ESR) spectra in X-band (~9.45 GHz ) were recorded by mounting the crystal with the c-axis perpendicular (θ = 90°) to the magnetic field direction. The spectra were recorded and simulated by a numerical diagonalization of spin Hamiltonian matrix in the range from 0 to 180 degrees for every 15 degrees of rotation angle (φ). In our case, only the last two sets of peaks strongly depend on the rotation angle (φ), and each exhibits C 3 symmetry due to two magnetically inequivalent Fe 3+ sites in the corundum structure. For polycrystalline ESR spectra, seven main Fe 3+ ESR absorption peaks occur at the resonance magnetic fields of 100.20, 310.24, 486.80, 525.00, 550.60, 761.00 and 777.00 mT respectively. Specifically, ESR signals show that the number of paramagnetic Fe 3+ ions increase roughly linearly with the heat treating temperature, having the [Formula: see text] ratio ~1.41 at 1600°C.

A Jahn–Teller analysis of K3 and Rb3 in the electronic states 12E′ and 12E″

2010 ◽  
Vol 375 (1) ◽  
pp. 73-84 ◽  
Author(s):  
Andreas W. Hauser ◽  
Carlo Callegari ◽  
Pavel Soldán ◽  
Wolfgang E. Ernst
Keyword(s):  
Electronic States ◽  
Jahn Teller

Cooperative Pseudo Jahn-Teller Effect as Microscopic Mechanism of (OMT)-(LTT) Structural Phase Transition in La2−xBaxCuO4Superconductors

1992 ◽  
Vol 291 ◽  
Author(s):  
Michael D. Kaplan
Keyword(s):  
Phase Transition ◽  
Structural Phase Transition ◽  
Structural Phase ◽  
Electronic States ◽  
Teller Effect ◽  
Excited Electronic States ◽  
Microscopic Mechanism ◽  
Jahn Teller ◽  
Jahn Teller Effect ◽  
Local Distortions

ABSTRACTA microscopic mechanism for the structural phase transition from the orthorhombic mediate temperature (OMT) phase into the low temperature tetragonal (LTT) one is suggested on the basis of the cooperative pseudo Jahn-Teller effect. The local distortions mixing the ground and the first excited electronic states are ordered antiferrodistortively and are connected in part, with the oxygen octahedra rotations around the [100] axis. The results are in agreement, with the neutron scattering experiments data.

THE ROTATIONAL ENERGY LEVELS OF DIATOMIC MOLECULES IN 4Σ ELECTRONIC STATES

1962 ◽  
Vol 40 (5) ◽  
pp. 598-606 ◽  
Author(s):  
Jon T. Hougen
Keyword(s):  
Experimental Data ◽  
Diatomic Molecules ◽  
Energy Levels ◽  
Electronic States ◽  
Rotational Energy

Expressions are derived for the rotational energy levels of diatomic molecules in 4Σ states. These expressions contain two rho-type doubling parameters (γ's), and thus differ from earlier expressions which contain only one such parameter. The new expressions are in better agreement with the experimental data, though some discrepancy still exists.

Studies of the Jahn-Teller effect. I. A survey of the static problem

1957 ◽  
Vol 238 (1215) ◽  
pp. 425-447 ◽  
Keyword(s):  
Electronic States ◽  
Static Problem ◽  
Electronic Energy ◽  
Tetragonal Symmetry ◽  
Linear Molecule ◽  
Subsequent Paper ◽  
Jahn Teller ◽  
Near Degeneracy ◽  
Static Distortion ◽  
Physical Features

According to the Jahn-Teller theorem, symmetrical molecules with degenerate electronic states are unstable. Such molecules therefore take up a distorted shape. If there is near-degeneracy, the symmetrical shape may also be unstable. We have studied the distortion in some particular cases. The approach is to minimize the total electronic energy with respect to distortions of the nuclear framework, the latter being considered to be static. There are always several equivalent distortions of equal energy, so that a static distortion fails to remove the degeneracy. The discussion of dynamic effects is postponed to a subsequent paper. A linear molecule of formula BAB , for which two electronic states of opposite symmetry are sufficiently nearly degenerate, will be stable in a configuration with unequal A—B separations, and unstable when symmetrical. This example illustrates some of the main physical features of Jahn-Teller distortions in a simple manner. Although we know of no example where the symmetrical structure is actually unstable, there are examples where the tendency towards distortion noticeably reduces the force constant of the asymmetric vibration. Octahedral complexes AB 6 with degenerate electronic states occur in many situations (e. g. paramagnetic crystals, F-centres, luminescent centres and exciton states in cubic crystals). The orbital degeneracy may be threefold ( T 1 and T 2 ) or twofold. In the former case the stable distortion is found to be either of tetragonal symmetry about a [100] direction, or of trigonal symmetry about a [111] direction. The twofold degenerate situation leads to a more complicated situation. If one neglects anharmonic effects there appears to be an infinity of distortions minimizing the energy; a more detailed consideration of the anharmonic terms shows that the stable distortions are of elongated tetragonal character. This result has an important bearing on complexes involving the cupric ion.

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