Self-consistent optimization

KK Koretke, Z Luthey-Schulten, PG Wolynes. Self-consistently optimized statistical mechanical energy functions for sequence structure alignment. Protein Sci 5 1043-1059, 1996. 

In M0ller-Plesset theory, first-order perturbation theory does not improve on the HF energy because the zeroth-order Hamiltonian is not itself the HF Hamiltonian. However, first-order perturbation theory can be useful for estimating energetic effects associated with operators that extend the HF Hamiltonian. Typical examples of such terms include the mass-velocity and one-electron Darwin corrections that arise in relativistic quantum mechanics. It is fairly difficult to self-consistently optimize wavefunctions for systems where these tenns are explicitly included in the Hamiltonian, but an estimate of their energetic contributions may be had from simple first-order perturbation theory, since that energy is computed simply by taking the expectation values of the operators over the much more easily obtained HF wave functions. 

The GMS wave function [1,2] combines the advantages of the MO and VB models, preserving the classical chemical structures, but dealing with self-consistently optimized orbitals. From a formal point of view, it is able to reproduce all VB or MO based variational electronic wave functions in its framework. Besides that, it can deal in a straightforward way with the nonadiabatic effects of degenerate or quasi-degenerate states, calculating their interaction and properties. 

For a given chain, the strain ratios 5 (q) fully determine the mean-square dimensions of the chain. In the following, they will be derived from free-energy self-consistent optimization. 

AIMD data were used to improve the classical description of C mim Cl] by applying the force matching approach [72], A self-consistent optimization method for the generation of classical potentials of general functional form was presented and applied. A force field that better reproduces the observed first-principles forces was obtained [72],... 

This is the zero-order (reference) multiconfigurational description, with configurations consisting of the self-consistently optimized "Fermi-Sea" (FS) orbitals [40-42,131]. In other words. 

Here, D is the self-consistent, optimized density matrix. Using a time-reversible Verlet scheme we get an explicit integration of the form... 

Fig. 5.8 Top direct (through space) hopping between two magnetic centers by tat with strongly locaUzed atomic orbittils Bottom effective (through Ugand) hopping by t"fj with self-consistently optimized magnetic orbitals, which have delocalization tails on the (bridging) ligands...

As usual, the CAS wavefunction incorporates complete Cl over a chosen number of electrons (N) and orbitals (M) that constitute the active space [CAS(N,M)]> with iterative self-consistent optimization of all occupied and partially occupied orbitals. Compared with conventional CAS/MO, the following advantages of CAS/NBO may be noted ... 

The basic self-consistent field (SCF) procedure, i.e., repeated diagonalization of the Fock matrix [26], can be viewed, if sufficiently converged, as local optimization with a fixed, approximate Hessian, i.e., as simple relaxation. To show this, let us consider the closed-shell case and restrict ourselves to real orbitals. The SCF orbital coefficients are not the... 

A configuration interaction calculation uses molecular orbitals that have been optimized typically with a Hartree-Fock (FIF) calculation. Generalized valence bond (GVB) and multi-configuration self-consistent field (MCSCF) calculations can also be used as a starting point for a configuration interaction calculation. 

To overcome the limitations of the database search methods, conformational search methods were developed [95,96,109]. There are many such methods, exploiting different protein representations, objective function tenns, and optimization or enumeration algorithms. The search algorithms include the minimum perturbation method [97], molecular dynamics simulations [92,110,111], genetic algorithms [112], Monte Carlo and simulated annealing [113,114], multiple copy simultaneous search [115-117], self-consistent field optimization [118], and an enumeration based on the graph theory [119]. 

The self-consistent field function for atoms with 2 to 36 electrons are computed with a minimum basis set of Slater-type orbitals. The orbital exponents of the atomic orbitals are optimized so as to ensure the energy minimum. The analysis of the optimized orbital exponents allows us to obtain simple and accurate rules for the 1 s, 2s, 3s, 4s, 2p, 3p, 4p and 3d electronic screening constants. These rules are compared with those proposed by Slater and reveal the need for the screening due to the outside electrons. The analysis of the screening constants (and orbital exponents) is extended to the excited states of the ground state configuration and the positive ions. 

Multiconfiguration self-consistent field (MCSCF) theory aims to optimize simultaneously the LCAO coefficients and the Cl expansion coefficients in a wavefunction such as... 

The Multi-configuration Self-consistent Field (MCSCF) method can be considered as a Cl where not only the coefficients in front of the determinants are optimized by the variational principle, but also the MOs used for constructing the determinants are made optimum. The MCSCF optimization is iterative just like the SCF procedure (if the multi-configuration is only one, it is simply HF). Since the number of MCSCF iterations required for achieving convergence tends to increase with the number of configurations included, the size of MCSCF wave function that can be treated is somewhat smaller than for Cl methods. 

Fan, L., Ziegler, T., 1991, Optimization of Molecular Structures by Self-Consistent and Nonlocal Density-Functional Theory , J. Chem. Phys., 95, 7401. 

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