Self-consistent procedure

Since surface charges depend on the electrostatic potential (Eq. 4.20), Eqs. 4.20-4.22 are solved in an iterative way leading to self-consistent surface charges. At the end of this procedure, surface charges and the electrostatic potential satisfy the boundary condition specified in Eq. 4.21. In practical applications, this self-consistent procedure for calculating reaction field potential is coupled to self-consistent procedure which governs solving the Kohn-Sham equations. A special case for infinite dielectric constant outside the cavity... 

Four years later, Christian Bartels and Martin Karplus [55] used the WHAM equations as the core of their adaptive US approach, in which the efficiency of free energy calculations was improved through refinement of the biasing potentials as the simulation progressed. Efforts to develop adaptive US techniques had, however, started even before WHAM was developed. They were pioneered by Mihaly Mezei [56], who used a self-consistent procedure to refine non-Boltzmann biases. 

The observations were performed at ESO using the 1.52m telescope and FEROS. The obtained spectra have high nominal resolving power (R 48000), and S/N 500 at maximum and a coverage from 4000 A to 9200 A. Many spectra were acquired for all sample stars. The atmospheric parameters (Teff, log g, [Fe/H] and microturbulence velocities) have been obtained through an iterative and totally self-consistent procedure from Fe lines of the observed spectrum. The initial values of Teg were obtained from a (B-V) vs Teg calibration and log were determined from Hipparcos parallaxes and evolutionary tracks. The [O/Fe] abundances were derived by fitting synthetic spectra to the observed one. 

Within this approach (called (SC) because it is size-consistent if localized orbitals are used and the coefficients are, in practice, yielded by a recursive self-consistent procedure) the diagonal elements An of the dressing operator become ... 

As we have satisfied the plateau condition (24) and the condition (25) for reaching the optimal value of M, we have defined a self-consistent procedure leading to the optimal averaged density matrix D°. ... 

Conceptually, the self-consistent procedure outlined below was first proposed in [21]. 

The BW form of PT is formally very simple. However, the operators in it depend on the exact energy of the state studied. This requires a self-consistency procedure and limits its application to one energy level at a time. The Rayleigh-Schrodinger (RS) PT does not have these shortcomings, and is, therefore, a more suitable basis for many-body calculations of many-electron systems than the BW form of the theory, it is applicable to a group of levels simultaneously. 

Apart from the demands of the Pauli principle, the motion of electrons described by the wavefunction P° attached to the Hamiltonian H° is independent. This situation is called the independent particle or single-particle picture. Examples of single-particle wavefunctions are the hydrogenic functions (pfr,ms) introduced above, and also wavefunctions from a Hartree-Fock (HF) approach (see Section 7.3). HF wavefunctions follow from a self-consistent procedure, i.e., they are derived from an ab initio calculation without any adjustable parameters. Therefore, they represent the best wavefunctions within the independent particle model. As mentioned above, the description of the Z-electron system by independent particle functions then leads to the shell model. However, if the Coulomb interaction between the electrons is taken more accurately into account (not by a mean-field approach), this simplified picture changes and the electrons are subject to a correlated motion which is not described by the shell model. This correlated motion will be explained for the simplest correlated system, the ground state of helium. 

Another method from the same PCM family of solvation methods, namely the IEF-PCM [24] (see also the contribution by Cances), has recently been used to develop an ab initio VB solvation method [25], According to this approach, in order to incorporate solvent effect into the VB scheme, the state wavefunction is expressed in the usual terms as a linear combination of VB structures, but now these VB structures are optimized and interact with one another in the presence of a polarizing field of the solvent. The Schrodinger equation for the VB structures is then solved directly by a self-consistent procedure. 

The nematic mean-field U, the molecule-field interaction potential, WE, and the induced dipole moment, ju d, are evaluated at different orientations using Equation (2.263), and then the coefficients of their expansion on a basis of Wigner rotation matrices can be calculated, according to Equation (2.268). The permittivity is obtained by a self-consistency procedure, because the energy WE and the induced dipole moment / md, as well as the reaction field contribution to the nematic distribution function p( l), themselves depend on the dielectric permittivity. 

With the dependence (21) of 8v on 8P p the evaluation of the density response through (17) becomes a self-consistent procedure. 

In this contribution, we will show that the spectra of He-like ions can be modeled with high accuracy, using physically relevant parameters only, such as ion and electron temperatures, plasma motion and relative ion abundances. It is organized as follows in Sect. 8.2 there is a brief discussion on X-ray spectrometers used on fusion experiments, Sect. 8.3 contains the detailed theoretical description of the He-like system, in Sect. 8.4 some results are shown as obtained by a self-consistent procedure based on the detailed modeling of the spectra. 

Converging the self-consistent procedure in periodic calculations can be a difficult task, particularly if there are free states within any band. Movement of electrons between almost identical states has little effect on the energy of the system and so the search for the optimal distribution of electrons is hampered by many trivial alterations to the occupation numbers. To make the process more efficient, partial occupancies can be used for states near the highest filled level by introducing a smoothing function which defines the occupancy as a function of state energy, Tj,. The smoothing function used in the MgO calculation was ... 

It is also possible to include in addition to the ionic part an additional external potential Vg t in the Hamiltonian in Eq. (11) in the self-consistent procedure. This could be the long-range static potential from a surrounding crystalline environment, as used in the embedded cluster method. Fig. 7. Clusters are here used to model a small section of an infinite solid or surface. 

In order to consider the periodic crystal potential in cluster calculations, we developed the "chemically complete cluster model" (MODEL II) [10], which is similar to that proposed by Goodman et al. [11]. In our cluster model, atoms in the cluster are classified into three types. Type I atoms are "seed atoms" of which basis functions are obtained by the self-consistent procedure. The seed atoms are chemically complete. Namely, they are put in a potential environment similar to that in the bulk. Type II atoms are "passive atoms" of which basis functions are solved in the same potential field as for the type I atoms of the same species. Type in atoms have atomic potentials which are the same as in the type I atoms, but their wavefunctions are not included in molecular orbital calculations. The validity of MODEL II is tested for TiC in comparison with the results obtained using MODEL I. 

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