Self-consistent field equations

Drowicz F W and W A Goddard IB 1977. The Self-Consistent Field Equations for Generalized Valence Bond and Open-Shell Hartree-Fock Wave Functions. In Schaeffer H F III (Editor). Modem Theoretical Chemistry III, New York, Plenum, pp. 79-127. 

What has been said applies to approximate as well as to ab-initio molecular orbital wavefunctions,i.e. those obtained by solving the self-consistent-field equations exactly. Hence, the localized orbital approach also offers an attractive tool for bridging the gap between rigorous quantitative calculations and qualitative chemical intuition. The experience gained so far has shown that interpretations suggested by the localized orbital picture correspond closely to intuitive chemical thinking. 

Thus, the orbitals uk and vk satisfy Hartree-Fock equations which are identical in form and differ only in the numerical values of the constants X/Jt and Ajk respectively. But since the latter are unknowns in the equation, and since 7(p) is itself invariant as shown in Eq. (21), we can say that the Hartree-Fock self-consistent-field equations are invariant under the orbital transformation given by Eqs. (5) and (6). This means in effect, that the energy integral ( H "X11/0 is minimized by the vk s as well as by the uk s — a circumstance which is in agreement with the invariance of and ( 1 under the transformation (5). 

Stewart, J. J. P. (1996) Applications of localized molecular orbitals to the solution of semiemperical self consistent field equations. Int. J. Quantum Chem. 58, 133-146. 

A mean field approach was applied to determine homopolymer distributions in the lamellar phase of a blend of AB diblock and A homopolymer by Shull and Winey (1992). In the strong segregation limit, complete segregation of the A homopolymer into the A microdomain was predicted. Furthermore, in this limit, the diblocks were treated as brushes , wetted by homopolymer in the A domain. Composition profiles showing the distribution of homopolymer and copolymer were determined by numerical solution of the self-consistent field equations. 

F. W. Bobrowicz and W. A. Goddard III, Self-Consistent Field Equations for Open-Shell Hartree-Fock and Generalized Valence Bond Wave Functions, in Modem Theoretical Chemistry, vol. 3, H. F. Schaeffer III Ed., Plenum, New York, 1977... 

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. 

Semiempirical techniques are the next level of approximation for computational simulation of molecules. Compared to molecular mechanics, this approach is slow. The formulations of the self-consistent field equations for the molecular orbitals are not rigorous, particularly the various approaches for neglect of integrals for calculation of the elements of the Fock matrix. The emphasis has been on versatility. For the larger molecular systems involved in solvation, the semiempirical implementation of molecular orbital techniques has been used with great success [56,57]. Recent reviews of the semiempirical methods are given by Stewart [58] and by Rivail [59],... 

This factorization amounts to the statement thatEq. (2.25) breaks down into two separate linear systems, one for the determination of a orbitals, and the other for n orbitals. In the Hartree-Fock scheme, symmetry-adapted functions (i.e. in the case of planar unsaturated molecules symmetric or antisymmetric functions with respect ot the molecular plane), at least for closed-shell ground states iM8,20,2i) ... 

Reiher, M. (1998) Development and implementation of numerical algorithms for the solution of multi-configuration self-consistent field equations for relativistic atomic structure calculations. PhD thesis, FakultSt fUr Chemie, University Bielefeld, Germany. 

ROKS is based on the sum method by Ziegler, Rauk, and Baerends [45], and allows for MD simulations in the first excited singlet state (Sj) [24]. The theoretical framework has been generalized to arbitrary spin states [25], and a new algorithm for solving the self-consistent field equations has been introduced recently [46]. 

Ochsenfeld C, Head-Gordon M. A reformulation of the coupled perturbed self-consistent field equations entirely within a local atomic orbital density matrix-based scheme. Chem Phys Lett 1997 270 399 405. 

There are other reasons why methods based on Dirac Hamiltonians have been unpopular with quantum chemists. Dirac theory is relatively unfamiliar, and the field is not well served with textbooks that treat the topic with the needs of quantum chemists in mind. Matrix self-consistent-field equations are usually derived from variational arguments, and as a result of the debates on variational collapse and continuum dissolution , many people believe that such derivations are invalid for relativistic problems. Most implementations of the Dirac formalism have made no attempt to exploit the rich internal structure of Dirac... 

M. Reiher, Development and Implementation of Numerical Algorithms for the Solution of Multi-Configuration Self-Consistent Field Equations for Relativistic Atomic Structure Calculations, Dissertation, Fakultat fiir Chemie, Universitat Bielefeld (1998). 

Bobrowicz F W and Goddard WAIN 1977 The self-consistent field equations for generalized valence bond and open-shell Hartree-Fock wave functions Modern Theoretical Chemistry vol 3, ed H F III Schaefer (New York Plenum) pp 97-127... 

Dolan and Edwards (1975) write the self-consistent field equation in a. slightly modified form compared with equation (11.58)... 

Equation (55) is a generalized Poisson-Boltzmann equation including the free ions and the charged polymers. The first term represents the salt contrihu-tion and the second term is due to the charged monomers and their counterions. Equation (56) is a generalization of the self-consistent field equation of neutral polymers [9]. In the bulk, the above equations are satisfied by setting i/r — 0 and f (pb. 

When the surface is taken as ideal, that is, flat and homogeneous, the physical quantities depend only on the distance a from the surface. The surface imposes boundary conditions on the polymer order parameter fix) and electrostatic potential fix). In thermodynamic equihhrium, all charge carriers in solution should exactly balance the surface charges (charge neutrality). The Poisson-Boltzmann Equation (55), the self-consistent field Equation (56), and the boundary conditions uniquely determine the polymer concentration profile and the electrostatic potential. In most cases, these two coupled nonlinear equations can only he solved numerically. 

A symmetrical orthogonalization corresponds to a transformation that has the property X SX = I, where X denotes all the x, coordinate vectors and S contains the overlap elements. One such transformation is given by the inverse square root of the overlap matrix (X = ), a procedure used in solving the self-consistent field equations in... 

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