Solid Hamiltonian

Fig. 5. The left hand side figure shows a contour plot of the potential energy landscape due to V4 with equipotential lines of the energies E = 1.5, 2, 3 (solid lines) and E = 7,8,12 (dashed lines). There are minima at the four points ( 1, 1) (named A to D), a local maximum at (0, 0), and saddle-points in between the minima. The right hand figure illustrates a solution of the corresponding Hamiltonian system with total energy E = 4.5 (positions qi and qs versus time t).

Hamiltonians equivalent to (1) have been used by many authors for the consideration of a wide variety of problems which relate to the interaction of electrons or excitons with the locaJ environment in solids [22-25]. The model with a Hamiltonian containing the terms describing the interaction between excitons or electrons also allows for the use of NDCPA. For example, the Hamiltonian (1) in which the electron-electron interaction terms axe taken into account becomes equivalent to the Hamiltonians (for instance, of Holstein type) of some theories of superconductivity [26-28]. 

Fig. 5. Band gap as a function of nanotube radius calculated using empirical tight-binding Hamiltonian. Solid line gives estimate using Taylor expansion of graphene sheet results in eqn. (7).

Most microscopic theories of adsorption and desorption are based on the lattice gas model. One assumes that the surface of a sohd can be divided into two-dimensional cells, labelled i, for which one introduces microscopic variables Hi = 1 or 0, depending on whether cell i is occupied by an adsorbed gas particle or not. (The connection with magnetic systems is made by a transformation to spin variables cr, = 2n, — 1.) In its simplest form a lattice gas model is restricted to the submonolayer regime and to gas-solid systems in which the surface structure and the adsorption sites do not change as a function of coverage. To introduce the dynamics of the system one writes down a model Hamiltonian which, for the simplest system of a one-component adsorbate with one adsorption site per unit cell, is... 

Note that wq cannot be fixed by detailed balance, the reason being that it contains the information about the energy exchange with the solid which is not contained in the static lattice gas Hamiltonian. However, by comparison with the phenomenological rate equation (1) we can identify it as... 

Eckart, criteria, 264, 298 procedure, 267 Effective charge, 274, 276 Effective Hamiltonian, 226 Elastic model, excess entropy calculation from, 141 of a solid solution, 140 Electric correlation, 248 Electric field gradient, 188, 189 Electron (s), 200... 

Hamiltonian in Eq. (39). The solid curve, for m = 0, indicates stable isomers at 0 = 0 and 0 = tt and a saddle point at 0 O.dStt. Note, however, that the secondary minimum disappears as the angular momentum increases. 

Figure 10. Divisions of the Em map for the quadratic Hamiltonian in Eq. (39). Va, Vb, and Vx are the energies of the two isomers and that of the saddle point between them. The solid lines are relative equilibria, and the cusp points, a , indicate points at which the secondary minimum and the saddle point in Fig. 9 merge at a point of inflection.

Figure 11. Quantum monodromy in the spectrum of the quadratic Hamiltonian of Eq. (38). The solid lines indicate relative equilibria. Filled circles mark the eigenvalues of the most stable isomer and those above the relevant effective potential barrier in Fig. 8. Open circles indicate interpenetrating eigenvalues of the secondary isomer. The transported unit cell moves over the hlled circle lattice, around the curved fold line connecting the two spectra.

Initially we consider a simple atom with one valence electron of energy and wave function which adsorbs on a solid in which the electrons occupy a set of continuous states Tj, with energies Ej. When the adsorbate approaches the surface we need to describe the complete system by a Hamiltonian H, including both systems and their interaction. The latter comes into play through matrix elements of the form Vai = / We assume that the solutions T j to this eigen value problem... 

The Hamiltonian in Eq. (104) may describe both the process of tunnel inversion or isomerization of a molecule and the inertia effects arising from the symmetric vibrations of the reaction complex AH- B in the cage of the solvent or solid matrix (Fig. 9). In the latter case, the coordinate and the frequency of the symmetric vibration correspond to R and w0. 

The anisotropies that lead to line broadening in isotropic ESR spectra influence solid-state spectra more directly. Accordingly a more complex spin Hamiltonian is required to interpret such spectra ... 

This Hamiltonian leads to dephasing of the S -spin signal recorded as function of time (increasing number of rotor periods Nc in the REDOR experiment) as illustrated in Fig. lb. REDOR has been a key experiment in biological solid-state NMR, as for example used recently for determination of statherin binding to biomineral surfaces as illustrated in Fig. lc, with numerous REDOR determined intemuclear distances high-lighted in Fig. Id [79]. 

Fig. 6 Temperature dependence of %MT for [Fe(bpym)(NCS)2]2(bpym) at different pressures (a). The solid lines, together with estimated concentrations of [HS-LS] and [HS-HS] species, correspond to calculations using the appropriate Hamiltonian. Temperature dependence of %MT for [Fe(bpym)(NCSe)2]2(bpym) at different pressures (b). The magnetic behaviour of [Fe(bt)(NCS)2]2(bpym) at room pressure has been also included for comparison (adapted from [9])...

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