Self-consistent field external fields

However, another study concluded that the changes of the hydrogen-bond stability may be important in biological processes. For these, the influence of local electric fields created by Li+, Na+, and Mg2+ ions on the properties and reactivity of hydrogen bonds in HF and HC1 dimer has been carried out by means of ab initio self-consistent field (SCF) method [33]. A few years later, the effect of intensity and vector direction of the external electric field on activation barriers of unimole-cular reactions were studied using the semiempirical MINDO/3 method [34]. However, both semiempirical and ab initio calculations were performed to study the multiplicity change for carbene-like systems in external electric fields of different configurations (carbene and silylene) and the factor that determines the multiplicity and hence the reactivity of carbene-like structures is the nonuniformity of the field [35]. 

Methods are introduced for generating many-electron Sturmian basis sets using the actual external potential experienced by an N-electron system, i.e. the attractive potential of the nuclei. When such basis sets are employed, very few basis functions are needed for an accurate representation of the system the kinetic energy term disappears from the secular equation solution of the secular equation provides automatically an optimal basis set and a solution to the many-electron problem is found directly, including electron correlation, and without the self-consistent field approximation. In the case of molecules, the momentum-space hyperspherical harmonic methods of Fock, Shibuya and Wulfman are shown to be very well suited to the construction of many-electron Sturmian basis functions. 

Lin C-Y, Schick M, Andelman D (2005) Structural changes of diblock copolymer melts due to an external electric field a self-consistent-field theory study. Macromolecules 38(13) 5766—5773... 

Julg et al.24 employed an approximation based on the self consistent field molecular orbital method to evaluate the average energy per atom for various structures. They calculate that whereas the normal b.c.c. structure is more stable for clusters containing more than 106 atoms, smaller clusters prefer to take up pentagonal symmetry. However, these authors make an important point, namely, that the calculated energies for different structures are very similar. Interconversion of different structures will be facile, and external factors such as the method of deposition, level of impurities, support effects, etc., may cause the less stable structure to grow. For example, impurities on... 

Eq. (10) represents the self-consistent field equation for the local segment density of the polymer chains subject to an external electrical potential ip, a van der Waals interaction with the plates —UkT and an excluded volume interaction. Eq. (11) is a modified Poisson-Boltzmann equation in which the first term accounts for the charges of the small ions of the salt, the second term for the charges of the polyelectrolyte chains and the third one for the charges of the ions dissociated from the polyelectrolyte molecules. 

Of course, this dissymmetry can be remedied by replacing the semi-infinite chain by a finite chain and by introducing a self-consistent field without spherical symmetry however, the remedy would be only partial. In fact, in all cases, when N is large, in any self-consistent potential V(r), the chain has a tendency to follow the lines of force. Thus, in the asymptotic limit, the chain always has the shape of a needle, which may be curved. This means that the action of an external potential cannot produce a swelling of the chain in all directions. For this, it is necessary to introduce true correlations between chain links. 

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 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... 

Lifshitz (1968) has developed a now formalism of polymer theory, b/used on the mathematical analogy between the state equation of a molecular coil with Schrodinger s equation of a quantum mechanical particle placed in an external potential field. This analogy was mentioned in section 3.1 when speaking of Edwards solution of the conformational problem of a molecular coil in a good solvent with the help of the solutions of the diffusion equations for Green s functions in the self-consistent field approximation. 

Keywords Polymer blends Self-consistent field theory External potential dynamics Field-theoretic polymer simulations Polymeric microemulsion Polymer dynamics... 

External Potential (EP) Theory and Self-Consistent Field (SCF) Theory... 

This trick was called Douglas-Kroll the easy way by Jensen [725], who introduced it in a presentation at the REHE2005 conference in Miilheim, Germany, in April 2005. It was the starting point for extensive work by Kutzel-nigg and Liu [726,727,732]. If V is simply taken as the electron-nuclei potential energy operator, then no self-consistent-field problem arises and an (approximate) decoupling transformation can be obtained noniteratively. This approach is called the external-field no-pair projection. 

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