The Self-Consistency Loop

Before discussing the usage of the individual programmes indicated by an asterisk in Table 9.1, I shall briefly describe a typical application of these programme in a self-consistent calculation of the ground-state properties for a monoatomic metal. The structure of such a calculation is indicated in Fig.9.1. 

DDNS Calculates -projected and total state densities plus -projected and total number of states. 

FSAR Calculates extremal areas and effective masses on the Fermi surface. 

READB Reads and prints energy-band data sets. 

RHFS Generates atomic and frozen-core charge densities. 

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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. 

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... 

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. 

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]

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]. 

Figure Al. The lowest level hierarchical structure for most quantum mechanical computational algorithms. The inner loop is used to converge the self-consistent field in order to establish the electronic structure to within a user-defined tolerance. The outer loop is used to optimize the structure to within a defined geometric tolerance.

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. 

Figure 11.11 Flow chart for the self-consistent construction of a CE Hamiltonian, (a) Initial input data from DFT and information about all possible clusters on a lattice L form the initial setup, (b) Some clusters C are chosen and the CE sum is fitted to the energies of the input structures

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. ... 

The architecture of the self-tuning regulator is shown in Fig. 7.99. It is similar to that of the Model Reference Adaptive Controller in that it also consists basically of two loops. The inner loop contains the process and a normal linear feedback controller. The outer loop is used to adjust the parameters of the feedback controller and comprises a recursive parameter estimator and an adjustment mechanism. 

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