Hamiltonian with Fermi coupling

Freund, Herbst, Mariella and Klemperer [112] expressed their magnetic hyperfine constants in the form originally given by Frosch and Foley [117]. As discussed elsewhere in this book, particularly in chapters 9, 10 and 11, we prefer to separate the different physical interactions, particularly the Fermi contact and dipolar interactions, in our effective Hamiltonian. This separation is usually made by other authors even when the effective Hamiltonian is expressed in terms of Frosch and Foley constants, because it is the natural route if the molecular physics of a problem is to be understood. Nevertheless since so many authors, particularly of the earlier papers, use the magnetic hyperfine theory presented by Frosch and Foley, we present in appendix 8.5 a detailed comparison of their effective Hamiltonian with that adopted in this book. The merit of the Frosch and Foley parameters is that they form the linear combination of parameters which is best determined (i.e. with least correlation) for a molecule which conforms to Hund s case (a) coupling. The values of the constants determined experimentally from the 7 LiO spectrum were therefore, in our notation (in MHz) ... 

The model described here is based on a series of approximations, but it has nonetheless been found to be quantitatively successful in fitting the observed vibrational levels of a considerable number of small molecules [6,7,9,15,16,17,18]. The most obvious approximations concern the use of perturbation theory and the implicit use of harmonic basis functions in deriving the vibrational hamiltonian matrices, and the neglect of matrix elements coupling different values of the total vibrational excitation quantum number V. However when the same approximations are applied to a diatomic Morse oscillator they yield the exact solution, as observed by Mills and Roblette [7], and some similar cancellation of approximations holds for the polyatomic examples. Another more serious problem concerns the mixing with other vibrations this has been discussed here only in connection with Fermi resonance, but its importance in other ways is only just being investigated. 

Any temperature dependence of the CEF halfwidth can be attributed to either s-f interaction or the interaction of f states with phonons, s-f line broadening for simple metals has been attributed to the effects of the carrier spin dynamics on the CEF-state lifetimes and has been described by the Fermi liquid theory, introduced by Becker et al. (1977). The s-f interaction Hamiltonian describing the coupling between the 4f electrons and the conduction electrons can be written as... 

In the weak-coupling limit unit cell a (, 0 7a for fra/u-polyacetylene) and the Peierls gap has a strong effect only on the electron states close to the Fermi energy eF-0, i.e., stales with wave vectors close to . The interaction of these electronic states with the lattice may then be described by a continuum, model [5, 6]. In this description, the electron Hamiltonian (Eq. (3.3)) takes the form ... 

The complete Hamiltonian of the molecular system can be wrihen as H +H or H =H +H for the commutator being linear, where is the Hamiltonian corresponding to the spin contribution(s) such as, Fermi contact term, dipolar term, spin-orbit coupling, etc. (5). As a result, H ° would correspond to the spin free part of the Hamiltonian, which is usually employed in the electron propagator implementation. Accordingly, the k -th pole associated with the complete Hamiltonian H is , so that El is the A -th pole of the electron propagator for the spin free Hamiltonian H . 

Several studies of Fermi resonances in the absence of H bond have been made [76-80]. We shall account for this situation by simply ignoring the anharmonic coupling between the fast and slow modes (a = 0). The theory then describes the coupling between the fast mode and a bending mode through the potential Htf, with both of these modes being damped in the same way. Because aG = 0, the slow mode does not play any role, so that the total Hamiltonian does not refer to it ... 

Consequently, the Hamiltonian of the dimer that involves Davydov coupling, Fermi resonances between the g excited state of the fast mode and the g first harmonics of the bending mode, together with the damping of is... 

The Fermi resonance Hamiltonian consists of two terms. The first one, Ho, is the Dunham expansion, which characterizes the uncoupled system, while the second term, Hp, is the Fermi resonance coupling, which describes the energy flow between the reactive mode and one perpendicular mode. For the three systems, HCP CPH, HOCl HO - - Cl and HOBr HO + Br, the reactive degree of freedom is the slow component of the Fermi pair and will therefore be labeled s, while the fast component will be labeled /. Thus, the resonance condition writes co/ w 2c0s. More explicitly, for HCP the slow reactive mode is the bend (mode 2) and the fast one is the CP stretch (mode 3), while for HOCl and HOBr the slow mode is the OX stretch (X = Cl,Br) (mode 3) and the fast one is the bend (mode 2). The third, uncoupled mode— that is, the CH stretch (mode 1) for HCP and the OH stretch (mode 1) for HOCl and HOBr—will be labeled u. With these notations, the Dunham expansion writes in the form... 

Let us now consider the dynamics of the coupled system with Hamiltonian H = Hp, + Hp. Ju and Jp remain good quantum numbers for this Hamiltonian and are quantized according to Eq. (31). It is known that the dynamics of the coupled system is governed by the shape of its stable periodic orbits (POs) in the subspace of the normal coordinates involved in the Fermi... 

In this Hamiltonian (5) corresponds to the orbital angular momentum interacting with the external magnetic field, (6) represents the diamagnetic (second-order) response of the electrons to the magnetic field, (7) represents the interaction of the nuclear dipole with the electronic orbital motion, (8) is the electronic-nuclear Zeeman correction, the two terms in (9) represent direct nuclear dipole-dipole and electron coupled nuclear spin-spin interactions. The terms in (10) are responsible for spin-orbit and spin-other-orbit interactions and the terms in (11) are spin-orbit Zeeman gauge corrections. Finally, the terms in (12) correspond to Fermi contact and dipole-dipole interactions between the spin magnetic moments of nucleus N and an electron. Since... 

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