Self-consistent field theory expressions

Equation (75) expresses the partition function of the many-polymer system in terms of the partition functions of single polymers subjected to external fluctuating fields. The self-consistent field theory approximates this functional integral over the fields by the value of the integrand evaluated at those values of the fields, and Wb, that minimize the functional F[h, 4, vb]. From the definition of F it follows that these functions satisfy the self-consistent equations... 

The complete treatment of solvation effects, including the solute selfpolarization contribution was developed in the frame of the DFT-KS formalism. Within this self consistent field like formulation, the fundamental expressions (96) and (97) provide an appropriate scheme for the variational treatment of solvent effects in the context of the KS theory. The effective KS potential naturally appears as a sum of three contributions the effective KS potential of the isolated solute, the electrostatic correction which is identified with the RF potential and an exchange-correlation correction. Simple formulae for these quantities have been presented within the LDA approximation. There is however, another alternative to express the solva-... 

This expression excludes self-interaction. There have been a number of attempts to include into the Hartree-Fock equations the main terms of relativistic and correlation effects, however without great success, because the appropriate equations become much more complex. For a large variety of atoms and ions both these effects are fairly small. Therefore, they can be easily accounted for as corrections in the framework of first-order perturbation theory. Having in mind the constantly growing possibilities of computers, the Hartree-Fock self-consistent field method in various... 

Exactly as In lattice models, the walks are assumed to take place In a (self-consistent) field Ulz), which depends on the concentration profile

relation between U[z) and (p[z) one may use the Floiy-Hugglns theory usually in an expanded form, but other models, such as a generalized Van der Waals equation of state ) can also be taken. The most general expression for the self-consistent mean field U z) has been given by Hong and Noolandl K It has been shown ) that this expression is the continuum analogue of the lattice version of Scheutjens and Fleer, to be discussed in sec. 5.5. 

At the HF-SCF level of theory, the wave function is determined completely by the molecular orbitals. In the vast majority of cases, these are given by linear combinations of atom-centered basis functions, and these MO coefficients are obtained by the self-consistent field procedure. The first-order change to the wave function is therefore governed by the first-order change in the MO coefficients. It is not difficult to work out expressions for the derivatives of the MO coefficients,22 and one obtains... 

D11.8 In ab initio methods an attempt is made to evaluate all integrals that appear in the secular determinant. Approximations are still employed, but these are mainly associated with the construction of the wavefunctions involved in the integrals. In semi-empirical methods, many of the integrals are expressed in terms of spectroscopic data or physical properties. Semi-empirical methods exist at several levels. At some levels, in order to simplify the calculations, many of the integrals are set equal to zero. Density functional theory (DFT) is considered an ab initio method, but it is different from the Hartree-Fock (HF) or self-consistent field (SCF) approach in that DFT focuses on the electron density while HF/SCF methods focus on the wavefunction. They are both iterative self consistent methods in that the calculations are repeated until the energy and wavefunctions (HF) or energy and electron density (DFT) are unchanged to within some acceptable tolerance. 

The MOs are themselves expressed as linear combinations of atomic orbitals (AOs) the latter are usually a basis set of known functions. With a given basis set, the problem of variationally optimizing the energy transforms into that of finding the coefficients of the orbitals. Expressed in matrix form in an AO basis, and in the independent particle approximation of Hartree-Fock theory, this leads to the well-known self-consistent field (SCF) equations. 

The Roothaan-Hall equations are nonlinear because the Fock matrix depends upon the orbital coefficients c, through the density matrix expression (6. 111). Solution therefore involves an iterative process, as we discussed previously for atomic systems, and the technique is therefore called self-consistent-field (SCF) theory. 

Quantitative predictions of surfactant phase behavior can be made by constructing a thermodynamic model. The classical expression for the free energy of a microemulsion is a function of the interfacial tension, bending moment, and micelle-micelle interactions [47]. Two quantitative models have been developed to describe supercritical microemulsions based on this concept. Here, the key challenge is to find accurate expressions for the oil-surfactant tail interactions and the tail-tail interactions. To do this, the first model uses a modified Flory-Krigbaum theory [43,44], and the second a lattice fluid self-consistent field (SCF) theory [25]. 

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