Hamiltonian geometry expansion

Non-totally symmetric vibrations lower the symmetry of a molecule and previously forbidden bands may become allowed. The Hamiltonians considered up to now were all given for a fixed nuclear equilibrium geometry. A Taylor series expansion in the normal coordinates Q around this nuclear equilibrium geometry... 

There is a general statement [17] that spin-orbit interaction in ID systems with Aharonov-Bohm geometry produces additional reduction factors in the Fourier expansion of thermodynamic or transport quantities. This statement holds for spin-orbit Hamiltonians for which the transfer matrix is factorized into spin-orbit and spatial parts. In a pure ID case the spin-orbit interaction is represented by the Hamiltonian //= a so)pxaz, which is the product of spin-dependent and spatial operators, and thus it satisfies the above described requirements. However, as was shown by direct calculation in Ref. [4], spin-orbit interaction of electrons in ID quantum wires formed in 2DEG by an in-plane confinement potential can not be reduced to the Hamiltonian H s. Instead, a violation of left-right symmetry of ID electron transport, characterized by a dispersion asymmetry parameter Aa, appears. We show now that in quantum wires with broken chiral symmetry the spin-orbit interaction enhances persistent current. 

We have previously considered the Hamiltonian as a function of the molecular geometry and described the expansion of the Hamiltonian in geometrical displacements n- We now consider the Hamiltonian as a function... 

Herzberg and Teller (1947) described the dependence of the electronic transition dipole moment on the nuclear geometry g by a McLaurin series expansion of the Hamiltonian... 

Model Hamiltonians constructed according to Elqs. (13)-(15) are particularly well suited for the calculation of low-resolution absorption spectra, photoelectron spectra and resonance-Raman spectra of polyatomic molecules. As is well known, these spectra are largely determined by the short-time dynamics in the excited state, which in turn is governed by the shape of the PE functions within the so-called Franck-Condon zone of the optical transition. In this limited range of nuclear geometries, the multidimensional PE functions are generally well approximated by the Taylor expansions of Eqs. (8) and (9). 

The aj therefore define the geometrical shape of the minimum in the usual way. If the minimum figure is a plane then the potential well is diffeomorphic to SO(3) while if it is non-planar then it is diffeomorphic to 0(3) and so the well is actually a symmetric double well. In either case, Klein et al show that the eigenvalues and eigenfunctions of the full problem can be obtained as WKB-type expansions to all orders of the expansion parameter, the square root of the ratio of the electronic to a typical nuclear mass. Because of the way the Hamiltonian is formulated the invariance of the Hamiltonian under permutations of the electronic variables is readily considered and the electronic wavefunction can easily be chosen in per-mutationally allowed form, no matter what the nuclear geometry happens to be. However it is much less easy to consider permutational invariance when some of... 

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