Self-consistent problem

Thereby the solution of the electronic-structure problem for an N-atomic system is decomposed into N locally self-consistent problems including only the M atoms in the LIZ associated with each atom in the system, and the computational effort now scales linearly with N, i.e. exhibits 0 N) scaling. 

Surface area A deviations from 30% to 70% were calculated for identical runs of methane and ethane, suggesting a self-consistency problem. 

This property is no longer true for thin films, where ( ) and hence the shape of the profile also depend on the boundary condition. Obviously, Eqs. (14), (19) then define a non-trivial self-consistency problem, which requires an iterative numerical treatment [58]. We note, however, that Eqs. (14), (18) yield a simple relation between ( > and ( )0=< )(z=+D/2),... 

Equation 4.7 has the form of a self-consistency problem. The solution to the equation is rffj(fi), but the exact form of the equation is determined by (/ y(/r) itself. It must be solved by an iterative procedure (called the self-consistent held, or SCF approach), in which convergence is taken to occur at the step where both sj and (/ y(/r) do not differ appreciably from the prior step. 

A great improvement can be achieved by using Newton-Raphson methods (Bendt and Zunger, 1982 Srivastava, 1984) for solving our self-consistency problem, which consists of finding the value of input potential u(r) for which the output potential u(o)C)> a functional of u(r), satisfies ... 

Strong coupling is implied when the following occurs. If, in the process of solving the self-consistent problem, one successively shifts states of the ion and positions of the wall sources to the point that... 

There is, however, a price for this versatility. The LMTO method is one of several linear methods, and like all the other linear techniques it is accurate only in a certain energy range. The present technique in particular should not be used for states too far above the Fermi level. If such states are required one may still solve the self-consistency problem by the LMTO technique and then turn to the Linear Augmented Plane Wave (LAPW) method for accurate calculation of the unoccupied high-lying levels. Furthermore, in... 

Among the assumptions of effective medium theory, for an imperfect crystal the average embedding electron density is constructed, and the total electron density in turn depends on these local densities through the Ap(r) s. Fience a self-consistent problem is at hand. Because the Ap(r) s need be calculated only... 

Needless to say, this type of self-consistent problem is very well known in quantum-chemistry, as the set of equations (17) is closely related (but by no means equivalent) to the HF equations. 

It is essential to note that the self-consistent potential U isthe same for all n (just as it is in the semi-dilute case of eq. (IX.63)). This expresses the chemical identity of all monomers and reduces our self-consistent problem considerably. There is only one unknown function U, and we can obtain it explicitly ftom the condition of constant total concentrations eq. (IX.64). Inserting eq. (IX.64) into eq. (IX.66) we get ... 

However, all relations unfortunately involve the Green s function g i this poses an additional (self-consistency) problem, in that g involves eigenstates which can only be calculated when v is already known. 

Equations (13), (16), and (17) define a self-consistent problem in which each is a solution of a single-particle Schrddinger equation in which the potential 9 is a function of all the occupied wavefunctions if j(r). The solutions of these equations have features in common with of all solutions of self-consistent equations, and we can identify six essential steps ... 

Obtaining the solution for these coupled functions represents a self-consistency problem, the subject of research in an emerging scientific field offirst-principles electrochemistry. The main objective is to employ ab initio methods for the modeling of... 

FIG U RE 3.17 The self-consistency problem in Pt electrocatalysis. The metal phase potential determines oxidation state and charging properties at the catalyst surface. These properties in turn determine the local reaction conditions at the Hehnholtz or reaction plane. At this point, structural design and transport properties of the catalyst layer come into play (as illustrated for conventional and ultrathin catalyst layers). Newly developed methods in the emerging field of first-principles electrochemistry attempt to find self-consistent solutions for this conpled problem. 

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