Simulation self-consistent fields

Many problems in force field investigations arise from the calculation of Coulomb interactions with fixed charges, thereby neglecting possible mutual polarization. With that obvious drawback in mind, Ulrich Sternberg developed the COSMOS (Computer Simulation of Molecular Structures) force field [30], which extends a classical molecular mechanics force field by serai-empirical charge calculation based on bond polarization theory [31, 32]. This approach has the advantage that the atomic charges depend on the three-dimensional structure of the molecule. Parts of the functional form of COSMOS were taken from the PIMM force field of Lindner et al., which combines self-consistent field theory for r-orbitals ( nr-SCF) with molecular mechanics [33, 34]. 

To overcome the limitations of the database search methods, conformational search methods were developed [95,96,109]. There are many such methods, exploiting different protein representations, objective function tenns, and optimization or enumeration algorithms. The search algorithms include the minimum perturbation method [97], molecular dynamics simulations [92,110,111], genetic algorithms [112], Monte Carlo and simulated annealing [113,114], multiple copy simultaneous search [115-117], self-consistent field optimization [118], and an enumeration based on the graph theory [119]. 

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... 

Alexander approach to spherical geometries, while making the connection between tethered chains and branched polymers. The internal structure of tethered layers was illuminated by numerical and analytical self-consistent field calculations, and by computer simulations. 

The frequency dependence is taken into accoimt through a mixed time-dependent method which introduces a dipole-moment factor (i.e. a polynomial of first degree in the electronic coordinates ) in a SCF-CI (Self Consistent Field with Configuration Interaction) method (3). The dipolar factor, ensuring the gauge invariance, partly simulates the molecular basis set effects and the influence of the continuum states. A part of these effects is explicitly taken into account in an extrapolation procedure which permits to circumvent the sequels of the truncation of the infinite sum-over- states. 

This chapter is concerned with the application of liquid state methods to the behavior of polymers at surfaces. The focus is on computer simulation and liquid state theories for the structure of continuous-space or off-lattice models of polymers near surfaces. The first computer simulations of off-lattice models of polymers at surfaces appeared in the late 1980s, and the first theory was reported in 1991. Since then there have been many theoretical and simulation studies on a number of polymer models using a variety of techniques. This chapter does not address or discuss the considerable body of literature on the adsorption of a single chain to a surface, the scaling behavior of polymers confined to narrow spaces, or self-consistent field theories and simulations of lattice models of polymers. The interested reader is instead guided to review articles [9-11] and books [12-15] that cover these topics. 

The density functional theory has the stmcture of a self-consistent field theory where the density profile is obtained from a simulation of a single chain in the... 

A more efficient way of solving the DFT equations is via a Newton-Raphson (NR) procedure as outlined here for a fluid between two surfaces. In this case one starts with an initial guess for the density profile. The self-consistent fields are then calculated and the next guess for density profile is obtained through a single-chain simulation. The difference from the Picard iteration method is that an NR procedure is used to estimate the new guess from the density profile from the old one and the one monitored in the single-chain simulation. This requires the computation of a Jacobian matrix in the course of the simulation, as described below. 

In this review, we have presented a molecular picture of the lipid bilayer system in relation to its permeation properties, as it appears from simulations and from self-consistent field calculations. Of course, it was not possible to go into all the details. In fact, the practicalities are always much more complicated, and the fine details are only of interest to the experts. 

Leermakers, F. A. M., Rabinovich, A. L. and Balabaev, N. K. (2003). Self-consistent field modeling of hydrated unsaturated lipid bilayers in the liquid-crystal phase and comparison to molecular dynamics simulations, Phys. Rev. E, 67, 011910. 

A successful theoretical description of polymer brushes has now been established, explaining the morphology and most of the brush behavior, based on scaling laws as developed by Alexander [180] and de Gennes [181]. More sophisticated theoretical models (self-consistent field methods [182], statistical mechanical models [183], numerical simulations [184] and recently developed approaches [185]) refined the view of brush-type systems and broadened the application of the theoretical models to more complex systems, although basically confirming the original predictions [186]. A comprehensive overview of theoretical models and experimental evidence of polymer bmshes was recently compiled by Zhao and Brittain [187] and a more detailed survey by Netz and Adehnann [188]. 

Various theoretical methods and approaches have been used to model properties and reactivities of metalloporphyrins. They range from the early use of qualitative molecular orbital diagrams (24,25), linear combination of atomic orbitals to yield molecular orbitals (LCAO-MO) calculations (26-30), molecular mechanics (31,32) and semi-empirical methods (33-35), and self-consistent field method (SCF) calculations (36-43) to the methods commonly used nowadays (molecular dynamic simulations (31,44,45), density functional theory (DFT) (35,46-49), Moller-Plesset perturbation theory ( ) (50-53), configuration interaction (Cl) (35,42,54-56), coupled cluster (CC) (57,58), and CASSCF/CASPT2 (59-63)). 

In addition to these experimental methods, there is also a role for computer simulation and theoretical modelling in providing understanding of structural and mechanical properties of mixed interfacial layers. The techniques of Brownian dynamics simulation and self-consistent-field calculations have, for example, been used to some advantage in this field (Wijmans and Dickinson, 1999 Pugnaloni et al., 2003a,b, 2004, 2005 Parkinson et al., 2005 Ettelaie et al., 2008). 

Continued Quantum and Molecular Mechanical Simulations, In this technique, a molecular dynamics simulation includes the treatment of some part of the system wilh a quantum mechanical technique. This approach. yMf.MM. is similar to programs that Use quantum mechanical methods to treat the n-systems of the structures in question separately from the sigma framework. The results are combined ai ihe end to render a slructure which is optimized and energy-refined in satisfy both self-consistent field (SCF) and force field energy convergence. 

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