Jahn-Teller Hamiltonian

We have followed a phenomenological approach and used the cluster model [18]. In this model the eg-type distortion interacts more strongly with the electronic state of an octahedral coordinated Cr3+ ion than the distortions of t2g symmetry. According to Ham [19], we assume that the continuum of vibrational modes with eg character can be approximated by a single mode with an effective frequency o>, mass /r and coupling constant V. The collective coordinates of the eg mode are conventionally known as Qe x2 — y2) and Qs ( 3z2 — r2). The linear Jahn-Teller Hamiltonian in equation (1) for the X state is [18] ... 

Vibronic Coupling Constant and Jahn-Teller Hamiltonian... 

E e Jahn-Teller and (E A) e Pseudo Jahn-Teller Hamiltonian for the Benzene Cation... 

It is possible to evaluate Eq. (3.3) for arbitrary pulse shapes using the well known numerical diagonalization of the Jahn-Teller Hamiltonian [10]. However, geometric phase development will be most clearly manifest under simplifying experimental conditions which make possible an... 

There has been some work on semi-classical quantization of the linear Jahn-Teller Hamiltonian (2.2) [41,42]. The quantization scheme which bears the closest relation to the present wave packet treatment involved the calculation of classical trajectories while slowly turning on the nonadiabatic interaction [41(a)]. The main emphasis of that work was the development of a method for obtaining energy levels in molecules with nonadiabatic dynamics which might then be applied to larger multimode systems. 

We begin by describing the origin of the geometric phase effect using the quadratic Jahn-Teller Hamiltonian of Longuet-Higgins [4] ... 

The Ex E Jahn-Teller Hamiltonian then takes the well-known form ... 

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

The quadratic Jahn-Teller effect is switched on by including the ijiiadratic tenns in Hq. (7) thus, with the inclusion of the additional diagonal Hamiltonian iij. 

The simplest way to write down the 2 x 2 Hamiltonian for two states such that its eigenvalues coincide at trigonally symmetric points in (x,y) or (q, ( )), plane is to consider the matrices of vibrational-electronic coupling of the e Jahn-Teller problem in a diabatic electronic state representation. These have been constructed by Haiperin, and listed in Appendix TV of [157], up to the third... 

If the states are degenerate rather than of different symmetry, the model Hamiltonian becomes the Jahn-Teller model Hamiltonian. For example, in many point groups D and so a doubly degenerate electronic state can interact with a doubly degenerate vibrational mode. In this, the x e Jahn-Teller effect the first-order Hamiltonian is then [65]... 

There is no analytic proof of the Jahn-Teller theorem. It was shown to be valid by considering all possible point groups one by one. The theorem is traditionally treated within perturbation theory The Hamiltonian is divided into three parts... 

II electronic states, 638-640 vibronic coupling, 628-631 triatomic molecules, 594-598 Hamiltonian equations, 612-615 pragmatic models, 620-621 Kramers doublets, geometric phase theory linear Jahn-Teller effect, 20-22 spin-orbit coupling, 20-22 Kramers-Kronig reciprocity, wave function analycity, 201 -205 Kramers theorem ... 

The above results mainly apply to the Longuet-Higgins E x e problem, but this historical survey would be incomplete without reference to early work on the much more challenging problems posed by threefold or higher electronic degeneracies in molecules with tetrahedral or octahedral symmetry [3]. For example, tetrahedral species, with electronic symmetry T or T2, have at least five Jahn-Teller active vibrations belonging to the representations E and T with individual coordinates (Qa,Qb) and (Qx. Qx. Q ) say. The linear terms in the nine Hamiltonian matrix elements were shown in 1957 [3] to be... 

Kambara (1979) has proposed a microscopic model in which the coupling between the d-electrons and the lattice has been given a definitive meaning. Assuming that there is Jahn-Teller coupling between the d-electrons and the local intramolecular distortion, the Hamiltonian of the system is written as... 

A quantitative treatment of the Jahn-Teller effect is more challenging (46). A major issue is that many theoretical models explicitly or implicitly assume the Bom—Oppenheimer approximation which, for octahedral Cu(II) systems in the vibronic coupling regime, cannot be correct (46,51). Hitchman and co-workers solved the vibronic Hamiltonian in order to model the temperature dependence of the molecular structure and the attendant spectroscopic properties, notably EPR spectra (52). Others, including us, take a more simphstic approach (53,54) but, in either case, a similar Mexican hat potential energy description of the principal features of the Jahn-Teller effect in homoleptic Cu(II) complexes emerges (Fig. 13). 

Fig. 5. The pseudo-Jahn-Teller effect in ammonia (NH3). (a) CCSD(T) ground state potential energy curve breakdown of energy into expectation value of electronic Hamiltonian (He), and nuclear-nuclear repulsion VNN. (b) CASSCF frequency analysis of pseudo-Jahn-Teller effect showing the effect of including CSFs of B2 symmetry is to couple the ground and 1(ncr ) states to give a negative curvature to the adiabatic ground state potential energy surface for the inversion mode.

To prove the Jahn-Teller theorem and to discuss related phenomena we consider the change in the electronic Hamiltonian of a molecule by making small distortions of the nuclei from some chosen origin. For convenience these distortions are represented by vectors in the space of the normal coordinates of the original structure. The change in the Hamiltonian can therefore be written as the Taylor expansion... 

A Jahn-Teller system modeled through generalized spin Hamiltonian the... 

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