Self-consistent-field calculations theory

These surfaces are all based on some combination of ab initio electronic structure calculations plus fitting. The AD and BM surfaces are based respectively in whole or in part on extended-basis-set single-configuration self-consistent-field calculations, whereas the RB and RBST calculations are based on calculations including electron correlation by Moller-Plesset fourth-order perturbation theory. For the rigid-rotator calculations R., the intramolecular internuclear distances R- and R ... 

A selection of the predictions of the equilibrium structure of DPPC bilayers as found by numerical self-consistent-field calculations is given in the following figures. In a series of articles, the SCF predictions for such membranes were published, starting in the late 1980s. As discussed above, we will update these early predictions for the theory outlined above with updated parameter sets. The calculations are very inexpensive with respect to the CPU time, and thus variations of the parameter-set will also provide deeper insight into the various subtle balances that eventually determine the bilayer structure - the mechanical properties as well as the thermodynamic properties. 

Fig. 6.22 Phase diagram for blends of PE and PEP homopolymers (A/j, - 392 and 409 respectively) with a PE-PEP diblock (iVc = 1925) (Bates et al. 1995). Open and filled circles denote experimental phase transitions between ordered and disordered states measured by SANS and rheology respectively. Phase boundaries obtained from self-consistent field calculations are shown as solid lines. The diamond indicates the Lifshitz point (LP), below which an unbinding transition (UT) separates lamellar and two-phase regions in mean field theory.

In solving Eq. (2), an iterative process is used to adjust the until the best wavefunction is found [self-consistent field (SCF) theory]. For the open shell case where incompletely filled orbitals exist, spin-restricted Hartree-Fock (RHF) methods or unrestricted Hartree-Fock (UHF) methods may be used to calculate the energies.41 The extent of calculation, approximation, or neglect of the two-electron integral terms largely defines the computation method. 

Quantum mechanical calculations are carried out using the Variational theorem and the Har-tree-Fock-Roothaan equations.t - Solution of the Hartree-Fock-Roothaan equations must be carried out in an iterative fashion. This procedure has been called self-consistent field (SCF) theory, because each electron is calculated as interacting with a general field of all the other electrons. This process underestimates the electron correlation. In nature, electronic motion is correlated such that electrons avoid one another. There are perturbation procedures whereby one may carry out post-Hartree-Fock calculations to take electron correlation effects into account. " It is generally agreed that electron correlation gives more accurate results, particularly in terms of energy. 

The quantitative evaluation of the corresponding interaction energies had to await the development of computers and ab initio systems in the 1960s. By the early 1970s it was apparent that self-consistent field (SCF) theory provides a reasonably accurate description of hydrogen-bonded complexes like (H O) while theories that explicitly account for electron correlation " must be used for systems which are predominantly bound by dispersion forces, such as He-H2 and Hcj. Rapid developments in both hardware and software have since taken place and ab initio calculations on weakly bound systems are now routinely being carried out. Useful information is gathered in this way and the potential surfaces obtained find application in simulation studies of liquids, solids and various solvation problems. ... 

A different approach to treat correlation effects which are not well described within the LSDA consists in incorporating self-interaction corrections (SIC) [111-114] in electron structure methods for solids, Svane et al. [115-120]. In the Hartree-Fock (HF) theory the electron-electron interactions are usually divided into two contributions, the Coulomb term and the exchange term although they both are Coulomb interactions. The separation though, is convenient because simplifications of self-consistent-field calculations can be obtained by including in both terms the interaction of the electron itself. In the HF theory this has no influence on the solutions because these selfinteractions in the Coulomb and exchange terms exactly cancel each other. However, when the exchange term is treated... 

In this substection we will shortly discuss the computational methods used for calculation of the spin-spin coupling constants. Two main approaches available are ab initio theory from Hartree-Fock (or self-consistent field SCF) technique to its correlated extensions, and density function theory (DFT), where the electron density, instead of the wave function, is the fundamental quantity. The discussion here is limited to the methods actually used for calculation of the intermolecular spin-spin coupling constants, i. e. multiconfigurational self consistent field (MCSCF) theory, coupled cluster (CC) theory and density functional theory (DFT). For example, the second order polarization propagator method (SOPPA) approach is not... 

This type of layer structure, with a compact inner region and a dilute outer region, was also predicted by self-consistent field theory and by computer simulations. For instance, Monte Carlo simulations show that a dense layer (1-2 nm thick) is present close to the planar interface (74). This layer contained about 70% of the segments. Further out a region of mu eh lower density was found to extent about 10 nm into the aqueous phase. Similar results were obtained by self-consistent field calculations (75), which also showed that the most hydrophilic segments reside predominantly in the outer layer. 

One feature of relativistic self-consistent field calculations to which attention should be drawn is the fact that the Breit interaction can be easily included in the self-consistent field iterations once the algebraic approximation has been invoked. This should be contrasted with the situation in atomic calculations using numerical methods in which the Breit interaction is treated by first-order perturbation theory. 

The first theories for block copolymers were introduced for the strong segregation limit (SSL) and the essential physical principles underlying phase behaviour in the SSL were established in the early 1970s [1]. Most notably, Helfand and coworkers [27-29] developed the self-consistent field (SCF) theory, this permitting the calculation of free energies, composition profiles and chain conformations. In the SCF theory, the external mean fields acting on a polymer chain are calculated self-consistently with the composition... 

To summarize, the example of homopolymer/copolymer mixtmes demonstrates nicely how field-theoretic simulations can be used to study non-trivial fluctuation effects in polymer blends within the Gaussian chain model The main advantage of these simulations is that they can be combined in a natural way with standard self-consistent field calculations. As mentioned earlier, the self-consistent field theory is one of the most powerful methods for the theoretical description of polymer blends, and it is often accurate on a quantitative level hi many regions of the parameter space, fluctuations are irrelevant for large chain lengths (large Jf) and simulations are not necessary. Field-theoretic simulations are well suited to complement self-consistent field theories in those parameter regions where fluctuation effects become important. 

I. Lindgren, A. Rosen Relativistic self-consistent field calculations with application to hyperfine interaction. Pt.I Relativistic self-consistent fields, Pt.II Relativistic theory of atomic hyperfine interaction. Case Stud. Atom. Phys. 4, 93 (1974), Pt.IlI Comparison between theoretical and experimental hyperfine structure results. Case Stud. Atom. Phys. 4, 197 (1974)... 

The computational exploration of the conformatimial landscapes of neutral hydrated monosaccharides is particularly challenging because the relative energies of the many possible conformational structures are very close and, in some cases, the associated vibrational spectra differ only very slightly. Singly hydrated pXyl-O-Ph is a typical example of these difficulties [50]. To improve the theoretical description of such systems, more sophisticated approaches than the standard DFT calculation have been applied. Vibrational anharmonicity has been accommodated ab initio, using Vibrational Self Consistent Field (VSCF) theory, and has reproduced very accurately the observed spectra [47, 50, 53]. The conformational interconversion of hydrated monosaccharides has been simulated using Ab Initio... 

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