Self-consistent field theory chemical potential

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... 

Instead of the MC and MD methods using explicit particles, another method, that is, polymeric self-consistent field theory (SCFT) proposed by Edwards, is often used to study the phase separation of block copolymers. In SCFT, a polymer chain is treated as a Gaussian string, which is exposed to a set of effective chemical potentials ( ). The chemical potentials are used instead of the actual interactions between different components. Importantly, the relation between the external potentials and the concentration field ((/>) is bijective. 

Numerical self-consistent fields have been established (see, for example [9,30,31]) and we shall record below a few of the consequences of the calculations, in particular for the chemical potential p, when we have considered the relativistic generalization of the TF theory (Sect. 6.4 below). 

Our multireference M0Uer-Plesset (MRMP) perturbation method [1-4] and MC-QDPT quasi-degenerate perturbation theory (QDPT) with multiconfiguration self-consistent field reference functions (MC-QDPT) [5,6] are perturbation methods of such a type. Using these perturbation methods, we have clarified electronic stmctures of various systems and demonstrated that they are powerful tools for investigating excitation spectra and potential energy surfaces of chemical reactions [7-10]. In the present section, we review these multireference perturbation methods as well as a method for interpreting the electronic structure in terms of valence-bond resonance structure based on the CASSCF wavefunction. 

In the 1980s and 1990s, multiconfigurational self-consistent field (MCSCF) reference perturbation theories [1-6,23-30] were proposed to overcome the defects of the singlereference PT and the QDPT, and now they are established as reliable methods that can be applied to wide variety of problems potential energies surfaces of chemical reactions, excited spectra of molecules, etc. In fact, they have many advantages ... 

AMI AMBER A Program for Simulation of Biological and Organic Molecules CHARMM The Energy Function and Its Parameterization Combined Quantum Mechanics and Molecular Mechanics Approaches to Chemical and Biochemical Reactivity Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field Divide and Conquer for Semiempirical MO Methods Electrostatic Catalysis Force Fields A General Discussion Force Fields CFF GROMOS Force Field Hybrid Methods Hybrid Quantum Mechanical/Molecular Mechanical (QM/MM) Methods Mixed Quantum-Classical Methods MNDO MNDO/d Molecular Dynamics Techniques and Applications to Proteins OPLS Force Fields Parameterization of Semiempirical MO Methods PM3 Protein Force Fields Quantum Mechanical/Molecular Mechanical (QM/MM) Coupled Potentials Quantum Mecha-nics/Molecular Mechanics (QM/MM) SINDOI Parameterization and Application. 

Density Functional Applications Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field Density Functional Theory Applications to Transition Metal Problems ESR Hypeifine Calculations Gradient Theory Metal Complexes Molecular Magnetic Properties NMR Chemical Shift Computation Ab Initio NMR Chemical Shift Confutation Structural Applications NMR Data Correlation with Chemical Structure Relativistic Effective Core Potential Techniques for Molecules Containing Very Heavy Atoms Relativistic Effects of the Superheavy Elements Relativistic Theory and Applications Transition Metal Chemistry Transition Metals Applications. 

Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field Electrostatic Potentials Chemical Applications MNDO PM3 Semiempirical Methods Transition Metals SINDOl Parameterization and Application. 

Figure 7.12 Excess chemical potential of the hard-sphere fluid as a function of density. The open and filled circles correspond to the predictions of the primitive quasi-chemical theory and the self-consistent molecular field theory, respectively. The solid and dashed lines are the scaled-particle (Percus-Yevick compressibility) theory and the Carnahan-Starling equation of state, respectively (Pratt and Ashbaugh, 2003).

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