Mean-field method

The mean field teclmique is one of the most robust and simple methods used to handle larger molecules in gas and liquid enviromnents [M, 134. 135 and 136]. The basic premise of all mean-field methods is that the fiill wavefiinction represents N very weakly coupled modes (2 ) and can be approximated as... 

The fimdamental disadvantage of the mean-field method is that it does not allow modes to respond in a correlated maimer to each other. This problem can be somewhat alleviated by a good definition of the relevant coordinate system [134. 136]. (An extension of mean-field methods that does allow for coupling [137. 138 and 139] will be discussed later.)... 

We close these introductory remarks with a few comments on the methods which are actually used to study these models. They will for the most part be mentioned only very briefly. In the rest of this chapter, we shall focus mainly on computer simulations. Even those will not be explained in detail, for the simple reason that the models are too different and the simulation methods too many. Rather, we refer the reader to the available textbooks on simulation methods, e.g.. Ref. 32-35, and discuss only a few technical aspects here. In the case of atomistically realistic models, simulations are indeed the only possible way to approach these systems. Idealized microscopic models have usually been explored extensively by mean field methods. Even those can become quite involved for complex models, especially for chain models. One particularly popular and successful method to deal with chain molecules has been the self-consistent field theory. In a nutshell, it treats chains as random walks in a position-dependent chemical potential, which depends in turn on the conformational distributions of the chains in... 

To describe nonequilibrium phase transitions, there have been developed many methods such as the closed-time path integral by Schwinger and Keldysh (J. Schwinger et.al., 1961), the Hartree-Fock or mean field method (A. Ringwald, 1987), and the l/lV-expansion method (F. Cooper et.al., 1997 2000). In this talk, we shall employ the so-called Liouville-von Neumann (LvN) method to describe nonequilibrium phase transitions (S.P. Kim et.al., 2000 2002 2001 S.P. Kim et.al., 2003). The LvN method is a canonical method that first finds invariant operators for the quantum LvN equation and then solves exactly the... 

In the presence of magnetic exchange a treatment by the mean-field method gives an implicit expression similar to equation (57) ... 

This paper deals with one of the mean-field methods of modeling the connectivity build-up that can be applied to polymerization processes. As in the other mean-field methods of modeling, certain physical features such as concentration fluctuations or fluctuation coupled diffusion control of reaction steps, etc., are neglected. 

The asymmetric spin boson model presents a significantly more challenging non-adiabatic condensed phase test problem due to the asymmetry in forces from the different surfaces. Approximate mean field methods, for example, will fail to reliably capture the effects of these different forces on the dynamics. 

For small values of e = 4 - d, a polymer chain in solution is nearly Brownian and a mean field method might reasonable results in this limit. Thus it is possible consider that the chains feel a potential V (x) which is the sum of the (attractive and repulsive) surface potential and of a self - consistent potential produced by the other chains. 

The shortcoming of the mean field method is that it admits no correlation between the motions of the individual particles. This correlation can be introduced by means of the random phase approximation (RPA) or time-dependent Hartree (TDH) method. In order to formulate this method, we introduce excitation operators (Ep) which replace f) p by when applied to the mean field ground state of the crystal when applied to any other state, they yield zero. Then, we write the Hamiltonian as a quadratic form in the excitation operators (Ep)+ and their Hermi-tean conjugates Ep... 

Instead of using the traditional LS-representation for the spin-orbit coupling we return to more fundamental representation (Eq. 7) and note that since all basis-set oriented mean-field methods expand the singleelectron wavefunctions according to Eq. (3), it is only necessary to determine matrix elements of the form ... 

Computationally less demanding mean-field methods provide a tool to account for the out-of-plane constraints, but have the disadvantage of using phase averaged stress and stain fields. In the present work, an incremental Mori Tanaka approach is employed, which is implemented as a constitutive material law in a finite element code. Both two-dimensional and three-dimensional investigations are performed and the results are compared to the predictions of the extended unit cell approaches. 

As a complementary approach a mean-field method is used in combination with the finite element method to investigate the FGM. To compare the predictions with the periodic unit cell simulations, the FGM part is divided into nine sublayers. Each of them consists of two bi-quadratic 8-node plane elements over the thickness, and only one element is used in the horizontal direction. The parts of pure alumina and pure nickel are modeled by three and 12 elements, respectively. In each sublayer the volume fractions of the phases are constant. In addition, the center sublayer can be split into a metal-matrix and a ceramic-matrix half sublayer. The properties of the particular material on the meso-structural level within the finite element calculation are described via a con-... 

The periodic unit cell results are directly comparable to the IMT predictions, because both approaches represent the same matrix/inclusion type microstructure. However, such comparisons have to be done carefully since some assumptions regarding the finite element calculations are not equivalent for the extended unit cell approaches and the present mean-field method. The plane stress analysis of the unit cell models does not take into account the constraints in the out-of-plane direction. In contrast, within the present IMT formulation the inclusions are enclosed three-dimensionally by the matrix material. In contrast to the plane stress unit cell models, the constraint in the out-of plane direction is accounted for. Accordingly, these predictions are denoted as full internal constraint. To overcome this internal constraint in order to simulate the plane stress model assump-... 

Finally and perhaps most usefully, mean-field methods are computationally... 

Aspects of the relativistic theory of quantum electrodynamics are first reviewed in the context of the electronic structure theory of atoms and molecules. The finite basis set parametrization of this theory is then discussed, and the formulation of the Dirac-Hartree-Fock-Breit procedure presented with additional detail provided which is specific to the treatment of atoms or molecules. Issues concerned with the implementation of relativistic mean-field methods are outlined, including the computational strategies adopted in the BERTHA code. Extensions of the formalism are presented to include open-shell cases, and the accommodation of some electron correlation effects within the multi-configurational Dirac-Hartree-Fock approximation. We conclude with a survey of representative applications of the relativistic self-consistent field method to be found in the literature. 

Even more interesting are carbon clusters. Raghavachari and his co-workers [97] discovered that mean-field methods have led to dramatically different results for the low-energy Cjo isomers. The structures of these isomers are very different a ring (Djo .) symmetry that is essentially... 

As an example of an application of the mean field method we shall consider the theory of the dielectric anisotropy of the nematic phase. The low frequency dielectric anisotropy of a molecule is determined by two factors (i) the polarizability anisotropy which for the elongated molecules of nematogenic compounds always makes a positive contribution (i.e., a... 

Translating it into the quantum mechanical language, the underlying assumptions of the mean field method for the N identical particles (here electrons) are as follows ... 

Any mean field method needs to solve two jxoblmis ... 

It is worth remembering that the mean field method is known by several different names in chanistiy ... 

It will be shown how the mean field method implies that milestme of chemistry the periodic table of chranical elements. 

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