Hamiltonian phenomenological

Hamiltonian, but in practice one often begins with a phenomenological set of equations. The set of macrovariables are chosen to include the order parameter and all otlier slow variables to which it couples. Such slow variables are typically obtained from the consideration of the conservation laws and broken synnnetries of the system. The remaining degrees of freedom are assumed to vary on a much faster timescale and enter the phenomenological description as random themial noise. The resulting coupled nonlinear stochastic differential equations for such a chosen relevant set of macrovariables are collectively referred to as the Langevin field theory description. 

One can regard the Hamiltonian (B3.6.26) above as a phenomenological expansion in temis of the two invariants Aiand//of the surface. To establish the coimection to the effective interface Hamiltonian (b3.6.16) it is instnictive to consider the limit of an almost flat interface. Then, the local interface position u can be expressed as a single-valued fiinction of the two lateral parameters n(r ). In this Monge representation the interface Hamiltonian can be written as... 

The first theoretical handling of the weak R-T combined with the spin-orbit coupling was carried out by Pople [71]. It represents a generalization of the perturbative approaches by Renner and PL-H. The basis functions are assumed as products of (42) with the eigenfunctions of the spin operator conesponding to values E = 1/2. The spin-orbit contribution to the model Hamiltonian was taken in the phenomenological form (16). It was assumed that both interactions are small compared to the bending vibrational frequency and that both the... 

If it cannot be guaranteed that the adsorbate remains in local equilibrium during its time evolution, then a set of macroscopic variables is not sufficient and an approach based on nonequihbrium statistical mechanics involving time-dependent distribution functions must be invoked. The kinetic lattice gas model is an example of such a theory [56]. It is derived from a Markovian master equation, but is not totally microscopic in that it is based on a phenomenological Hamiltonian. We demonstrate this approach... 

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... 

The other class of phenomenological approaches subsumes the random surface theories (Sec. B). These reduce the system to a set of internal surfaces, supposedly filled with amphiphiles, which can be described by an effective interface Hamiltonian. The internal surfaces represent either bilayers or monolayers—bilayers in binary amphiphile—water mixtures, and monolayers in ternary mixtures, where the monolayers are assumed to separate oil domains from water domains. Random surface theories have been formulated on lattices and in the continuum. In the latter case, they are an interesting application of the membrane theories which are studied in many areas of physics, from general statistical field theory to elementary particle physics [26]. Random surface theories for amphiphilic systems have been used to calculate shapes and distributions of vesicles, and phase transitions [27-31]. 

First, as the molecule on which the chromophore sits rotates, this projection will change. Second, the magnitude of the transition dipole may depend on bath coordinates, which in analogy with gas-phase spectroscopy is called a non-Condon effect For water, as we will see, this latter dependence is very important [13, 14]. In principle there are off-diagonal terms in the Hamiltonian in this truncated two-state Hilbert space, which depend on the bath coordinates and which lead to vibrational energy relaxation [4]. In practice it is usually too difficult to treat both the spectral diffusion and vibrational relaxation problems at the same time, and so one usually adds the effects of this relaxation phenomenologically, and the lifetime 7j can either be calculated separately or determined from experiment. Within this approach the line shape can be written as [92 94]... 

These constructions are evidently phenomenological in that they rely on consistency between the stochastic forces and their correlations. A more rigorous construction of these terms is therefore desirable. This has led to the use of the Mori projections of large-dimensional Hamiltonian systems. In particular, the projection of the Hamiltonian, ... 

Remarkably, when our general ME is applied to either AN or PN in Section 4.4, the resulting dynamically controlled relaxation or decoherence rates obey analogous formulae provided the corresponding density matrix (generalized Bloch) equations are written in the appropriate basis. This underscores the universality of our treatment. It allows us to present a PN treatment that does not describe noise phenomenologically, but rather dynamically, starting from the ubiquitous spin-boson Hamiltonian. 

In this review we shall first establish the theoretical foundations of the semi-classical theory that eventually lead to the formulation of the Breit-Pauli Hamiltonian. The latter is an approximation suited to make the connection to phenomenological model Hamiltonians like the Heisenberg Hamiltonian for the description of electronic spin-spin interactions. The complete derivations have been given in detail in Ref. (21), but turn out to be very involved and are thus scattered over many pages in Ref. (21). For this reason, we aim here at a summary that is as brief and concise as possible so that all relevant connections between different levels of approximation are evident. This allows us to connect present-day quantum chemical methods to phenomenological Hamiltonians and hence to establish and review the current status of these first-principles methods applied to transition-metal clusters. 

A special focus will be on phenomenological Hamiltonians involving electronic spin interactions. For this it is necessary to define atomic surrogate spin operators—so-called local spin operators—that may be directly related to the effective spins in... 

These fields become important in magnetic resonance spectroscopies, where the interaction between electronic and nuclear spins is considered in phenomenological Spin Hamiltonians. Note... 

The phenomenological Spin Hamiltonian widely applied in quantum chemistry—often also referred to as the Dirac-Heisenberg-Van... 

Though the true electron spin operators were employed here as well as in the Breit-Pauli Hamiltonian, the phenomenological Spin Hamiltonian, in which the spin coupling is an exchange effect, is in sharp contrast to the Breit-Pauli Hamiltonian, that is including the (magnetic) spin-spin interactions. Since the exchange effect is an effect introduced by the Pauli principle imposed on the wave function, we may write the electron-electron interaction as an expectation value,... 

For over a century it has been known that two classes of variables have to be distinguished the microscopic variables, which are functions of the points of ClN and thus pertain to the detailed positions and motions of the molecules and the macroscopic variables, observable by operating on matter in bulk, exemplified by the temperature, pressure, density, hydro-dynamic velocity, thermal and viscous coefficients, etc. And it has been known for an equally long time that the latter quantities, which form the subject of phenomenological thermo- and hydrodynamics, are definable either in terms of expected values based on the probability density or as gross parameters in the Hamiltonian. But at once three difficulties of principle have been encountered. 

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