Molecular mean field approach

J. Sikkema, L. Visscher, T. Saue, M. Ilias. The molecular mean-field approach for correlated relativistic calculations. /. Chem. Phys., 131 (2009) 124116. 

Tike all effective one-electron approaches, the mean-field approximation considerably quickens the calculation of spin-orbit coupling matrix elements. Nevertheless, the fact that the construction of the molecular mean-field necessitates the evaluation of two-electron spin-orbit integrals in the complete AO basis represents a serious bottleneck in large applications. An enormous speedup can be achieved if a further approximation is introduced and the molecular mean field is replaced by a sum of atomic mean fields. In this case, only two-electron integrals for basis functions located at the same center have to be evaluated. This idea is based on two observations first, the spin-orbit Hamiltonian exhibits a 1/r3 radial dependence and falls off much faster... 

An additional factor driving the segregation exists whenever the anchoring moieties change interactions at the interface. The related reduction A in brush chain free energy may depend on the molecular architecture of diblocks (i.e.,the size of blocks and of copolymer) as was recognized by our recent studies [254] described in Sect. 4.2.3. The results of these studies are analyzed with mean field and self consistent mean field approaches, both yielding identical adsorption parameters. 

A. Stochastic Difference in Time Definition A Stochastic Model for a Trajectory Weights of Trajectories and Sampling Procedures Mean Field Approach, Fast Equilibration, and Molecular Labeling Stochastic Difference Equation in Length Fractal Refinement of Trajectories Parameterized by Length... 

D. Mean Field Approach, Fast Equilibration, and Molecular Labeling... 

These expressions however are essentially, mean-field approaches as little account was taken of the molecular diversity that exists in living cell membranes. This is in part due to the evident complexity of such membranes but some modelling simulations have now begun to incorporate such diversity (eg. lipid-protein interactions are emphasised within the context of computational modelling). These are important tasks as there is clearly much local variation in the magnitude of the membrane surface potentials. We have engaged this problem for some years and are able to visualise the spatial variation of the potential about the membrane and cell surface with high molecular and spatial resolution (see below). 

In addition, using molecular dynamics (MD) simulations, it was demonstrated that the second term is directly related to the result expected from a mean-field approach, that is, in this term the fluctuations have been averaged out. This could also be argued from the fact that the second term presents the cross term of two amplitudes. The third term, on the other hand, is the scattering intensity of the cotmterions and presents the... 

Within the standard quantum-chemistry approach we begin our description with solving the Hartree-Fock (HF) equations. This is a mean-field approach and the solution, Le. an electronic state, is represented by the set of occupied molecular orbitals. The expectation value of the molecular Hamiltonian with the Hartree-Fock wave function is an approximation to the ground state energy of the system [33, 34]... 

This simple approach also does not take into account any potencial explicit dependence of the exchange field on temp>erature. While this dependence is possible, it is generally not considered in the molecular mean-field framework. On the examples we have shown earlier, no such A(T) dependence was considered. 

Polymer+low-molecular-liquid system. Mean field approaches. Liquid-liquid phase separation... 

When two incompatible polymeric melts A and B are contacted, the equilibrium interface width a is dependent on the Flory-Huggins interaction parameter / and the molecular weights of each poljmier. A mean field approach has been used (23,24) to predict that a is given by... 

Within the mean-field approach to the many-molecule problem, the free energy is represented by considering an ensemble of independent molecules in an effeaive external field density profile as in the real system of interest. The problem is thus reduced to that of a single ideal (maao)molecule in a complex external mean field, which characterizes the many-body effects of the other (macro)molecules. Physically, this self-consistent calculation means that the molecular conformations are assumed to be in local equilibrium under the given density profiles (r). 

Within a mean-field approach the R-T exchange interaction is described by molecular fields that are aeting on the magnetic moments. In a two-sublattice model, the suitable effective molecular fields acting on the R and T moments can be written as... 

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