Mean-field Hamiltonian

T. Dauxois, V. Latora, A. Rapisarda, S. Ruffo, and A. Torcini, The Hamiltonian Mean Field Model From Dynamics to Statistical Mechanics and Back, in Ref. [13] also cond-mat/0208456. 

Hamiltonian = t+ The additivity of implies that the mean-field energies il/are additive and the wavefunctions [Pg.2162]

The one-electron additivity of the mean-field Hamiltonian gives rise to the concept of spin orbitals for any additive bi fact, there is no single mean-field potential different scientists have put forth different suggestions for over the years. Each gives rise to spin orbitals and configurations that are specific to the particular However, if the difference between any particular mean-field model and the fiill electronic... 

Hamiltonian is Hilly treated, corrections to all mean-field results should converge to the same set of exact states. In practice, one is never able to treat all corrections to any mean-field model. Thus, it is important to seek particular mean-field potentials for which the corrections are as small and straightforward to treat as possible. 

In the mean field considerations above, we have assumed a perfectly flat interface such that the first tenn in the Hamiltonian (B3.6.21) is ineffective. In fact, however, fluctuations of the local interface position are important, and its consequences have been studied extensively [, 58]. 

Here we review the properties of the model in the mean field theory [328] of the system with the quantum APR Hamiltonian (41). This consists of considering a single quantum rotator in the mean field of its six nearest neighbors and finding a self-consistent condition for the order parameter. Solving the latter condition, the phase boundary and also the order of the transition can be obtained. The mean-field approximation is similar in spirit to that used in Refs. 340,341 for the case of 3D rotators. 

As an example of a multilayer system we reproduce, in Fig. 3, experimental TPD spectra of Cs/Ru(0001) [34,35] and theoretical spectra [36] calculated from Eq. (4) with 6, T) calculated by the transfer matrix method with M = 6 on a hexagonal lattice. In the lattice gas Hamiltonian we have short-ranged repulsions in the first layer to reproduce the (V X a/3) and p 2 x 2) structures in addition to a long-ranged mean field repulsion. Second and third layers have attractive interactions to account for condensation in layer-by-layer growth. The calculations not only successfully account for the gross features of the TPD spectra but also explain a subtle feature of delayed desorption between third and second layers. As well, the lattice gas parameters obtained by this fit reproduce the bulk sublimation energy of cesium in the third layer. 

For a given Hamiltonian the calculation of the partition function can be done exactly in only few cases (some of them will be presented below). In general the calculation requires a scheme of approximations. Mean-field approximation (MFA) is a very popular approximation based on the steepest descent method [17,22]. In this case it is assumed that the main contribution to Z is due to fields which are localized in a small region of the functional space. More crudely, for each kind of particle only one field is... 

A chemical interconversion requiring an intermediate stationary Hamiltonian means that the direct passage from states of a Hamiltonian Hc(i) to quantum states related to Hc(j) has zero probability. The intermediate stationary Hamiltonian Hc(ij) has no ground electronic state. All its quantum states have a finite lifetime in presence of an electromagnetic field. These levels can be accessed from particular molecular species referred to as active precursor and successor complexes (APC and ASC). All these states are accessible since they all belong to the spectra of the total Hamiltonian, so that as soon as those quantum states in the active precursor (successor) complex that have a non zero electric transition moment matrix element with a quantum state of Hc(ij) these latter states will necessarily be populated. The rate at which they are populated is another problem (see below). 

The above observation suggests an intriguing relationship between a bulk property of infinite nuclear matter and a surface property of finite systems. Here we want to point out that this correlation can be understood naturally in terms of the Landau-Migdal approach. To this end we consider a simple mean-field model (see, e.g., ref.[16]) with the Hamiltonian consisting of the single-particle mean field part Hq and the residual particle-hole interaction Hph-... 

Table 7.2 Comparison between the vibrational frequencies of linear triatomic molecules obtained by exact diagonalization of the Hamiltonian and the l/N (mean field) result."...

The mean-field approximation has been extensively applied in many-body physics. Its application to molecular algebraic Hamiltonians and the connection with the coherent-states expectation method was begun by van Roosmalen (1982). See also, van Roosmalen and Dieperink (1982), and van Roosmalen, Levine, and Dieperink (1983). For applications in the geometrical context see Bowman (1986) and Gerber and Ratner (1988). 

The problem of time evolution for a Hamiltonian bilinear in the generators (Levine, 1982) has been extensively discussed. The proposed solutions include the use of variational principles (Tishby and Levine, 1984), mean-field self-consistent methods (Meyer, Kucar, and Cederbaum, 1988), time-dependent constants of the motion (Levine, 1982), and numerous others, which we hope to discuss in detail in a sequel to this volume. 

The model of a reacting molecular crystal proposed by Luty and Eckhardt [315] is centered on the description of the collective response of the crystal to a local strain expressed by means of an elastic stress tensor. The local strain of mechanical origin is, for our purposes, produced by the pressure or by the chemical transformation of a molecule at site n. The mechanical perturbation field couples to the internal and external (translational and rotational) coordinates Q n) generating a non local response. The dynamical variable Q can include any set of coordinates of interest for the process under consideration. In the model the system Hamiltonian includes a single molecule term, the coupling between the molecular variables at different sites through a force constants matrix W, and a third term that takes into account the coupling to the dynamical variables of the operator of the local stress. In the linear approximation, the response of the system is expressed by a response function X to a local field that can be approximated by a mean field V ... 

In this section, we introduce the model Hamiltonian pertaining to the molecular systems under consideration. As is well known, a curve-crossing problem can be formulated in the adiabatic as well as in a diabatic electronic representation. Depending on the system under consideration and on the specific method used, both representations have been employed in mixed quantum-classical approaches. While the diabatic representation is advantageous to model potential-energy surfaces in the vicinity of an intersection and has been used in mean-field type approaches, other mixed quantum-classical approaches such as the surfacehopping method usually employ the adiabatic representation. 

Let us first consider the relation to the mean-field trajectory method discussed in Section III. To make contact to the classical limit of the mapping formalism, we express the complex electronic variables imaginary parts, that is, [Pg.308]

To alleviate this problem, Bonella and Coker [118, 119] have proposed a different variant of the semiclasscial mapping approach, where the trajectories are calculated according to the mean-field Hamiltonian... 

The terms etc. represent the one-body mean-field potential, which approximates the two-electron interaction in the Hamiltonian, as is the practice in SCF schemes. In the DFB equations this interaction includes the Breit term (3) in addition to the electron... 

The different techniques utilized in the non-relativistic case were applied to this problem, becoming more involved (the presence of negative energy states is one of the reasons). The most popular procedures employed are the Kirznits operator conmutator expansion [16,17], or the h expansion of the Wigner-Kirkwood density matrix [18], which is performed starting from the Dirac hamiltonian for a mean field and does not include exchange. By means of these procedures the relativistic kinetic energy density results ... 

The determination of the ground state energy and the ground state electron density distribution of a many-electron system in a fixed external potential is a problem of major importance in chemistry and physics. For a given Hamiltonian and for specified boundary conditions, it is possible in principle to obtain directly numerical solutions of the Schrodinger equation. Even with current generations of computers, this is not feasible in practice for systems of large total number of electrons. Of course, a variety of alternative methods, such as self-consistent mean field theories, also exist. However, these are approximate. 

Many methods in chemistry for the correlation energy are based on a form of perturbation theory, but the positivity conditions are quite different. Traditional perturbation theory performs accurately for all kinds of two-particle reduced Hamiltonians, which are close enough to a mean-field (Hartree-Fock) reference. There are a myriad of chemical systems, however, where the correlated wave-function (or 2-RDM) is not sufficiently close to a statistical mean field. Different from perturbation theory, the positivity conditions function by increasing the number of extreme two-particle Hamiltonians in which are employed as constraints upon the 2-RDM in Eq. (50) and, hence, they exactly treat a certain convex set of reduced Hamiltonians to all orders of perturbation theory. For the... 

In this paper we will not pursue such formal developments any further, and instead use mean field ideas and heuristic arguments to motivate the choice of the appropriate free energy functional. We represent the intrinsic free energy functional in the form of an effective 2D step Hamiltonian H and imagine on physical grounds that it has the... 

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