Optimization self-consistent iteration

Hartree -Fock or Self-Consistent Field (SCF) Method Spin Optimized Self-Consistent Field Method Configuration Interaction Iterative Natural Orbital Method Multi-Configuration SCF Many Body Perturbation Theory Valence-Bond Method Pair-Function or Geminal Method... 

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. 

Since we do not know the value of C in advance, the optimal C and thus the free energy difference A A can be solved in practice by iterating self-consistently (6.65) and (6.66) or (6.67). A convenient way to do so is to record all the perturbation data during the simulation, then compute C and AA in a postsimulation analysis. This method is also referred to as Bennett s method or the acceptance ratio method. 

Continuum solvation models consider the solvent as a homogeneous, isotropic, linear dielectric medium [104], The solute is considered to occupy a cavity in this medium. The ability of a bulk dielectric medium to be polarized and hence to exert an electric field back on the solute (this field is called the reaction field) is determined by the dielectric constant. The dielectric constant depends on the frequency of the applied field, and for equilibrium solvation we use the static dielectric constant that corresponds to a slowly changing field. In order to obtain accurate results, the solute charge distribution should be optimized in the presence of the field (the reaction field) exerted back on the solute by the dielectric medium. This is usually done by a quantum mechanical molecular orbital calculation called a self-consistent reaction field (SCRF) calculation, which is iterative since the reaction field depends on the distortion of the solute wave function and vice versa. While the assumption of linear homogeneous response is adequate for the solvent molecules at distant positions, it is a poor representation for the solute-solvent interaction in the first solvation shell. In this case, the solute sees the atomic-scale charge distribution of the solvent molecules and polarizes nonlinearly and system specifically on an atomic scale (see Figure 3.9). More generally, one could say that the breakdown of the linear response approximation is connected with the fact that the liquid medium is structured [105],... 

Alternatively, reaction field calculations with the IPCM (isodensity surface polarized continuum model) [73,74] can be performed to model solvent effects. In this approach, an isodensity surface defined by a value of 0.0004 a.u. of the total electron density distribution is calculated at the level of theory employed. Such an isodensity surface has been found to define rather accurately the volume of a molecule [75] and, therefore, it should also define a reasonable cavity for the soluted molecule within the polarizable continuum where the cavity can iteratively be adjusted when improving wavefunction and electron density distribution during a self consistent field (SCF) calculation at the HF or DFT level. The IPCM method has also the advantage that geometry optimization of the solute molecule is easier than for the PISA model and, apart from this, electron correlation effects can be included into the IPCM calculation. For the investigation of Si compounds (either neutral or ionic) in solution both the PISA and IPCM methods have been used. [41-47]... 

In order to determine vibrational NLO properties efficiently it is necessary to carry out finite field geometry optimizations as we have seen. In principle, Eq. (35) can be used directly for this purpose. There are, however, practical considerations related to convergence of the self-consistent field (SCF) iterations. The most obvious iterative sequence is (i) determine the zero-field solution (ii) evaluate dC/dk, (iii) substitute dC/dk from the previous step into the TDHF equation (iv) solve for C(k) and return to step (ii) etc. until convergence is achieved. In order to carry out step (iv) the normalization condition C SC = 1 may be used to write dC/dk = [(5C/5A )OS]C. Then the multiplicative form of the field-free equation is preserved and the polarization matrix will remain Hermitian for all iterations. Investigations are underway to test the convergence properties of the above iterative sequence and to determine how the convergence properties depend upon the magnitude of the field as well as the number of -points that are sampled in the band structure treatment. 

In complete active space self-consistent field (CASSCF) calculations with long configuration expansions the most expensive part is often the optimization of the Cl coefficients. It is, therefore, particularly important to minimize the number of Cl iterations. In conventional direct second-order MCSCF procedures , the Cl coefficients are updated together with the orbital parameters in each micro-iteration. Since the optimization requires typically 100-150 micro-iterations, such calculations with many configurations can be rather expensive. A possible remedy to this problem is to decouple the orbital and Cl optimizations , but this causes the loss of quadratic convergence. The following method allows one to update the Cl coefficients much fewer times than the orbital parameters. This saves considerable time without loss of the quadratic convergence behaviour. 

As indicated above, the butane, HCHO, CH3CHO and MEK systems were considered iteratively, such that the optimized chamber radical sources and the changes made to the photolysis parameters for HCHO and MEK led to a self-consistent description of all the systems. 

Quantum Mechanics (QM). The objective of QM is to describe the spatial positions of electrons and nuclei. The most commonly implemented QM method is the molecular orbital (MO) theory, in which electrons are allowed to flow around fixed nuclei (the Born-Oppenheimer approximation) until the electrons reach a self-consistent field (SCF). The nuclei are then moved, iteratively, until the energy of the system can go no lower. This energy minimization process is called geometry optimization. 

The atomic orbitals that are used constitute was is known as the basis set and a minimum basis set for compounds of second-row elements is made up of the 2s, 2p, 2py, and 2p orbitals of each atom, along with the 1 orbitals of the hydrogen atoms. In MO calculations, an initial molecular structure and a set of approximate MOs are chosen and the molecular energy is calculated. Iterative cycles of calculation of a self-consistent electrical field (SCF) and geometry optimization are then repeated until a... 

In this section we briefly outline the parametrization protocol for determining the partial atomic charges, atomic polarizabilities, and the atom-based Thole damping factors, as well as the optimization procedure of the force field parameters not dependent of the Drude oscillator positions, namely the bonded and Lennard-Jones terms. While the overall parameter optimization is described linearly in the text, it is important to bear in mind that the bonded and nonbonded parameters are strongly interdependent, such that in practice, an iterative procedure is adopted, with the electrostatic and LJ nonbonded and bonded parameters optimized in turn until a self-consistent solution is reached, offering optimal agreement with all sets of target data. 

Finally, given the above SIC-LSD total energy functional, the computational procedure is similar to the LSD case, that is minimization is accomplished by iteration until self-consistency. In the present work, the electron wavefunctions are expanded in LMTO basis functions (Andersen, 1975 Andersen et al., 1989), and the energy minimization problem becomes a non-linear optimization problem in the expansion coefficients, which is only slightly more complicated for the SIC-LSD functional than for the LDA/LSD functionals. Further technical details of the present numerical implementation can be found in Temmerman et al. (1998). 

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