Hamiltonian inhomogeneous

Numerical solution of Eq. (51) was carried out for a nonlocal effective Hamiltonian as well as for the approximated local Hamiltonian obtained by applying a gradient expansion. It was demonstrated that the nonlocal effective Hamiltonian represents quite well the lateral variation of the film density distribution. The results obtained showed also that the film behavior on the inhomogeneous substrate depends crucially on the temperature regime. Note that the film exhibits different wetting temperatures on both parts of the surface. For chemical potential below the bulk coexistence value the film thickness on both parts of the surface tends to appropriate assymptotic values at x cx) and obeys the power law x. Such a behavior of the film thickness is a consequence of van der Waals tails. The above result is valid when both parts of the surface exhibit either continuous (critical) or first-order wetting. 

Ab-initio studies of surface segregation in alloys are based on the Ising-type Hamiltonian, whose parameters are the effective cluster interactions (ECI). The ECIs for alloy surfaces can be determined by various methods, e.g., by the Connolly-Williams inversion scheme , or by the generalized perturbation method (GPM) . The GPM relies on the force theorem , according to which only the band term is mapped onto the Ising Hamiltonian in the bulk case. The case of macroscopically inhomogeneous systems, like disordered surfaces is more complex. The ECIs can be determined on two levels of sophistication ... 

Similar considerations lead to the transformation properties of the one-photon states and of the photon in -operators which create photons of definite momentum and helicity. We shall, however, omit them here. Suffice it to remark that the above transformation properties imply that the interaction hamiltonian density Jf mAz) = transforms like a scalar under restricted inhomogeneous Lorentz transformation... 

This chapter considers the distribution of spin Hamiltonian parameters and their relation to conformational distribution of biomolecular structure. Distribution of a g-value or g-strain leads to an inhomogeneous broadening of the resonance line. Just like the g-value, also the linewidth, W, in general, turns out to be anisotropic, and this has important consequences for powder patterns, that is, for the shape of EPR spectra from randomly oriented molecules. A statistical theory of g-strain is developed, and it is subsequently found that a special case of this theory (the case of full correlation between strain parameters) turns out to properly describe broadening in bioEPR. The possible cause and nature of strain in paramagnetic proteins is discussed. 

Multiple-pulse measurements were performed on both the LP and HP samples at 20° and — 80° C, and when no differences were noted, lower temperature measurements were performed only on the LP sample. Multiple-pulse spectra for the LP sample are illustrated in Figure 4 together with the eight-pulse spectrum of the reference used for the low-temperature measurements, Ca(OH)2. The lineshapes observed are quite broad, and the line center is a function of temperature. The line width was separated into three contributions by performing three related multiple-pulse measurements (I). These indicated that the main contributions to the linewidth came from both relaxation and second-order dipolar effects. The maximum possible field inhomogeneity Hamiltonian is estimated to be less than 16 ppm by this means, which indicates that the com-... 

Therefore the scaling transformation of the quantum-mechanical force field is an empirical way to account for the electronic correlation effects. As far as the conditions listed above are not always satisfied (e.g. in the presence of delocalized 7r-electron wavefunctions) the real transformation is not exactly homogeneous but rather of Puley s type, involving n different scale constants. The need of inhomogeneous Puley s scaling also arises due to the fact that the quantum-mechanical calculations are never performed in the perfect Hartree-Fock level. The realistic calculations employ incomplete basis sets and often are based on different calculation schemes, e.g. semiempirical hamiltonians or methods which account for the electronic correlations like Cl and density-functional techniques. In this context we want to stress that the set of scale factors for the molecule under consideration is specific for a given set of internal coordinates and a given quantum-mechanical method. 

Let us, for a moment, consider a single particle in one dimension with a Hamiltonian of the type// = p2/2m + V(x). This is a second-order differential operator, and this means that the general solution to the inhomogeneous Eq. (3.51)—considered as a second-order differential equation—will consist of a linear superposition of two special solutions, where the coefficients will depend on the boundary conditions introduced. As a specific example, one could think of the two solutions to the JWKB problem, their connection formulas, and the Stoke s phenomenon for the coefficients. 

Here we provide some details on inhomogeneous systems from the viewpoint of a weakly perturbed electron gas, i.e. we consider the external potential in the Hamiltonian (2.20) as a small perturbation. In this case one can expand the four current 6j r) induced by a static perturbation in a power series with respect to... 

Statistical thermodynamics of the electric double layer starts with modelling the electrolyte and the Interface. This can be done by specifying all inter-molecular and external Interactions in the phase space as a Hamiltonian. The notion of phase space was defined in sec. 1.3.9a and the Hamiltonian H was introduced in [1.3.9.11. As the kinetic part of the Hamiltonian does not contribute to the configuration Integrals, we sum only over the potential energies of the ions. In the Inhomogeneous system it Is customary to separate the interactions with the charged wall (the external" field) from the interlonlc ones. 

This approach for the development of multiple-pulse sequences is only practical if a large number of sequences can be assessed in a short period of time. The final assessment of the quality of a multiple-pulse sequence must always be based on experiments. However, for the optimization of multiple-pulse sequences, experimental approaches are, in general, too slow and too expensive (instrument time ). An attractive alternative to experiments at the spectrometer is formed by numerical simulations, that is, experiments in the computer. In simulations it is also possible to take relaxation and experimental imperfections such as phase errors or rf inhomogeneity into account. In addition to the direct translation of a laboratory experiment into a computer experiment, it is possible to numerically assess the properties of a multiple-pulse sequence on several abstract levels, for example, based on the created effective Hamiltonian. If simple necessary conditions can be defined for a multiple-pulse sequence with the... 

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