Hamiltonian characterized

In the full quantum mechanical approach [8], one uses Eq. (22) and considers both the slow and fast mode obeying quantum mechanics. Then, one obtains within the adiabatic approximation the starting equations involving effective Hamiltonians characterizing the slow mode that are at the basis of all principal quantum approaches of the spectral density of weak H bonds [7,24,25,32,33,58, 61,87,91]. It has been shown recently [57] that, for weak H bonds and within direct damping, the theoretical lineshape avoiding the adiabatic approximation, obtained directly from Hamiltonian (22), is the same as that obtained from the RR spectral density (involving adiabatic approximation). 

Starting from the standard QED Lagrangian, the Hamiltonian characterizing a system of interacting electrons in a static external potential (x) is readily obtained as the 00-component of the energy-momentum tensor (see e.g. [35]),... 

The inclusion of the PCM solvent effects in the scheme described in the previous section is neither straightforward nor unequivocal. The complexity is caused by the use of an effective solute Hamiltonian, characterized by a nonlinear potential term Va (depending on the solute charge distribution) that takes into account the polarization interaction with the solvent [27], namely ... 

The complex energy levels (198) and (199) characterizing the initial and final states may be considered as the eigenvalues of the effective non-Hermitean Hamiltonians characterizing the H-bond bridge ... 

To determine an effective dressed Hamiltonian characterizing a molecule excited by strong laser fields, we have to apply the standard construction of the free effective Hamiltonian (such as the Born-Oppenheimer approximation), taking into account the interaction with the field nonperturbatively (if resonances occur). This leads to four different time scales in general (i) for the motion of the electrons, (ii) for the vibrations of the nuclei, (iii) for the rotation of the nuclei, and (iv) for the frequency of the interacting field. It is well known that it is a good strategy to take into account the time scales from the fastest to the slowest one. 

From a quantum mechanical perspective, the starting point for any analysis of the total energy is the relevant Hamiltonian for the system of interest. In the present setting, it is cohesion in solids that is our concern and hence it is the Hamiltonian characterizing the motions and interactions of all of the nuclei and electrons in the system that must be considered. On qualitative grounds, our intuition coaches us to expect not only the kinetic energy terms for both the electrons and nuclei, but also their mutual interactions via the Coulomb potential. In particular, the Hamiltonian may be written as... 

Symplecticness is a characterization of Hamiltonian systems in terms of their solution. The solution operator t, to) defined by... 

The mathematical machinery needed to compute the rates of transitions among molecular states induced by such a time-dependent perturbation is contained in time-dependent perturbation theory (TDPT). The development of this theory proceeds as follows. One first assumes that one has in-hand all of the eigenfunctions k and eigenvalues Ek that characterize the Hamiltonian H of the molecule in the absence of the external perturbation ... 

The previous treatment relied on the assumption that the transition occurs on a single potential energy surface V(x) characterized by a barrier separating two wells. This potential is actually created from the terms of the initial and final electronic states. The separation of electron and nuclear coordinates in each of these states gives rise to the diabatic basis with nondiagonal Hamiltonian matrix... 

In the CHS model only nearest neighbors interact, and the interactions between amphiphiles in the simplest version of the model are neglected. In the case of the oil-water symmetry only two parameters characterize the interactions b is the strength of the water-water (oil-oil) interaction, and c describes the interaction between water (oil) and an amphiphile. The interaction between amphiphiles and ordinary molecules is proportional to a scalar product between the orientation of the amphiphile and the distance between the particles. In Ref. 15 the CHS model is generalized, and M orientations of amphiphiles uniformly distributed over the sphere are considered, with M oo. Every lattice site is occupied either by an oil, water, or surfactant particle in an orientation ujf, there are thus 2 + M microscopic states at every lattice site. The microscopic density of the state i is p.(r) = 1(0) if the site r is (is not) occupied by the state i. We denote the sum and the difference of microscopic oil and water densities by and 2 respectively and the density of surfactant at a point r and an orientation by p (r) = p r,U(). The microscopic densities assume the values = 1,0, = 1,0 and 2 = ill 0- In close-packing case the total density of surfactant ps(r) is related to by p = Ylf Pi = 1 - i i. The Hamiltonian of this model has the following form [15]... 

Hamiltonian Systems A Hamiltonian system is characterized by an even number of dimensions N = 2n = number of degrees of freedom), with variables conventionally labeled as (representing canonical positions) and (representing... 

The critical points of the equivalent classical Hamiltonian occur at stationary state energies of the quantum Hamiltonian H and correspond to stationary states in both the quantum and generalized classical pictures. These points are characterized by the constrained generalized eigenvalue equation obtained by setting the time variation to zero in Eq. (4.17)... 

This corresponds to a Hamiltonian system which is characterized by a weak oscillatory perturbation of the SHV streamfunction T r, ) —> Tfr, Q + HP, (r, ( ) x sin(fEt). The equations of fluid motion (4.4.4) are used to compute the inertial and viscous forces on particles placed in the flow. Newton s law of motion is then... 

The point q = p = 0 (or P = Q = 0) is a fixed point of the dynamics in the reactive mode. In the full-dimensional dynamics, it corresponds to all trajectories in which only the motion in the bath modes is excited. These trajectories are characterized by the property that they remain confined to the neighborhood of the saddle point for all time. They correspond to a bound state in the continuum, and thus to the transition state in the sense of Ref. 20. Because it is described by the two independent conditions q = 0 and p = 0, the set of all initial conditions that give rise to trajectories in the transition state forms a manifold of dimension 2/V — 2 in the full 2/V-dimensional phase space. It is called the central manifold of the saddle point. The central manifold is subdivided into level sets of the Hamiltonian in Eq. (5), each of which has dimension 2N — 1. These energy shells are normally hyperbolic invariant manifolds (NHIM) of the dynamical system [88]. Following Ref. 34, we use the term NHIM to refer to these objects. In the special case of the two-dimensional system, every NHIM has dimension one. It reduces to a periodic orbit and reproduces the well-known PODS [20-22]. 

A convenient quantitative characterization of the stable and unstable manifolds themselves as well as of reactive and nonreactive trajectories can be obtained by noting that the special form of the Hamiltonian in Eq. (5) allows one to separate the total energy into a sum of the energy of the reactive mode and the energies of the bath modes. All these partial energies are conserved. The value of the energy... 

For magnetostructural correlation, intimate knowledge of the molecular structure is a prerequisite, and hence the following discussion is mainly restricted to compounds characterized by X-ray crystallography. /-values are given with respect to the H= —J,jS,Sj Hamiltonian (reported values have been adapted for cases where the H= —2J S,Sj Hamiltonian has been used in the literature). 

Almost all problems that require knowledge of free energies are naturally formulated or can be framed in terms of (1.15) or (1.16). Systems 0 and 1 may differ in several ways. For example, they may be characterized by different values of a macroscopic parameter, such as the temperature. Alternatively, they may be defined by two different Hamiltonians, 3%o and 3%, as is the case in studies of free energy changes upon point mutation of one or several amino acids in a protein. Finally, the definitions of 0 and 1 can be naturally extended to describe two different, well-defined macroscopic states of the same system. Then, Q0 is defined as ... 

Let us start by considering an. /V-particle reference system described by the Hamiltonian. (x, p , j, which is a function of 3N Cartesian coordinates, x, and their conjugated momenta p,. We are interested in calculating the free energy difference between this system and the target system characterized by the Hamiltonian... 

The eigenvalue equation of the representation of the effective Hamiltonian operators (28) in the base of the number occupation operator of the slow mode is characterized by the equation... 

The eigenvalue equations of the two diagonal blocks of the effective Hamiltonian matrix is characterized by the equations... 

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