Self-consistency loop

To solve the full problem of finding an approximate ground state to Hamiltonian (13), one is faced to a self-consistent loop which can be proceeded in two steps. First one can get the occupations nia)o from a HWF, and a set of bare levels. Then one obtains a set of configuration parameters, the probabilities of double occupation, di by minimizing (18) with respect to these probabilities. Afterwards the on-site levels are renormalized according to (21) and the next loop starts again for the new effective Hamiltonian He// till convergence is achieved. 

From the density, it is now possible to construct the potential for the next iteration, and also to calculate total energies. During the self-consistency loop, we make a spherical approximation to the potential. This is because the computational cost for using the full potential is very high, and that the density converges rather badly in the corners of the unit cell. In order not to confuse matters with the ASA (Atomic Sphere Approximation) this is named the Spherical Cell Approximation (SC A) [65]. If we denote the volume of the Wigner-Seitz cell centered at R by QR, we have fiR = QWR — (4 /3), where w is the Wigner-Seitz radius. This means that the whole space is covered by spheres, just as in the ASA. We will soon see why this is practical, when we try to create the potential. 

A self-consistent loop is set up, with the weighting coefficients altered until the ground-state electronic configuration is obtained. Or, put another way, we need to minimize the change in energy with respect to the change in the basis-set coefficients, i.e. d /dc. 

The Hartree approximation is summarized in Figure 2 in schematic form. The heavy arrow denotes the basic band structure problem as stated in the previous section. The p H arrow in the self-consistent loop denotes, in the unrefined Hartree method, Poisson s equation, by which the charge density p is converted into an electrostatic potential which is to be incorporated in the Hamiltonian. With... 

Returning to the block diagram, the screening potential VSCR is added to the electron-core potential which is now called the pseudopotential, and this new total potential is used together with the new structure to start the loop over again. When input and output agree, the calculation is self-consistent. ... 

When Eq. (233) is compared with Eq. (230), the zero-frequency value of the longitudinal viscosity in first order is found to be larger than its zeroth-order value. This suggests that in every loop of the self-consistent calculation the zero-frequency longitudinal viscosity will increase, which might lead to a divergence of the zero-frequency value of rfo(z) and [Pg.134]

It is not possible to understand the nature of the divergence just by doing a self-consistent calculation since that implies an infinite loop calculation. An alternative way is to make an ansatz for [Pg.134]

In Section XI we discussed the calculational method of the dynamic structure factor in the supercooled regime. We also discussed that the memory function F// needs to be calculated self-consistently with the dynamic structure factor itself. Near the glass transition, the dynamic structure factor is expected to diverge. This leads to an infinite loop numerically formidable calculation. 

Self-consistency of postulated forward and reverse rate equations can be tested with the principles of thermodynamic consistency and so-called microscopic reversibility. The former invokes the fact that forward and reverse rates must be equal at equilibrium the latter is for loops in networks and can be stated as requiring that the products of the clockwise and counter-clockwise rate coefficients of the loop must be equal, or, for catalytic cycles, that the product of the forward coefficients must equal that of the reverse coefficients multiplied with the equilibrium constant of the catalyzed reaction. 

Abbreviations used in table MC - Monte Carlo aa - amino acid vdW - van der Waals potential Ig - immunoglobulin or antibody CDR - complementarity-determining regions in antibodies RMS -root-mean-square deviation r-dependent dielectric - distance-dependent dielectric constant e - dielectric constant MD - molecular dynamics simulation self-loops - prediction of loops performed by removing loops from template structure and predicting their conformation with template sequence bbdep - backbone-dependent rotamer library SCMF - self-consistent mean field PDB - Protein Data Bank Jones-Thirup distances - interatomic distances of 3 Ca atoms on either side of loop to be modeled. 

A network may have more than two parallel pathways. In such cases, several loops can be constructed from pairs of pathways, and the rules for self-consistency of the rate coefficient values are more convenient when reformulated ... 

Self-consistency of postulated forward and reverse rate equations and their coefficients can be tested with the principles of thermodynamic consistency and so-called microscopic reversibility. The former invokes the fact that forward and reverse rates must be equal at equilibrium. The latter is for loops of parallel pathways and for catalytic cycles. Thermodynamic consistency allows the reverse rate equation to be constructed from the forward one if at least one of its reaction orders is known, and requires the ratio of the products of the forward and reverse rate coefficients to be equal to the thermodynamic equilibrium constant. Microscopic reversibility leads to several useful conclusions The products of the clockwise and counter-clockwise rate coefficients of a loop must be equal the product of the forward rate coefficients of a catalytic cycle must be equal that of the reverse rate coefficients multiplied with the equilibrium constant of the catalyzed reaction forward and reverse reaction must occur along the same pathway and the ratio of the products of forward and reverse rate coefficients must be the same along all parallel pathways from same reactants to same products. The latter two rules apply regardless of whether or not any of the reactions are catalytic. 

In summary, the self-consistency procedure has two iteration loops. One is based upon the enerqy-soaKng principle of Sect.2.6 and implemented in SCFC, and the other is based upon band calculations and therefore requires consecutive execution of LMTO, DDNS and SCFC. At the end of the scaling iterations, and hence also at the end of a band iteration, one may compare the calculated ground-state properties with previous band iterations. If convergence is obtained one may stop at this point. If not, one may start a new band iteration using the potential parameters from the last scaling iteration. 

At this stage the loop may be closed and steps two through four iterated to self-consistency. The self-consistency criterion is that the first-order moments, i.e. q = 1, vanish. In that case, E is the centre of gravity of the occupied part of the z band, and hence that range of the band structure which is important for self-consistency is described with reasonable accuracy. 

This equation can be solved numerically. It is important here to incorporate infinite terms corresponding to the same class of diagrams so that at low temperature and high anharmonicity the self-consistent equation will not diverge. Even higher-order corrections to the propagator diagrams will consist of multiline loops and their combinations [3]. 

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