Self-consistency equation

Kohn W and Sham L J 1965 Self-consistent equations including exchange and correlation effects Phys. Rev A 140 1133-8... 

Kohn W and L J Sham 1965. Self-consistent Equations Including Exchange and Correlation Effects. Physical Review A140 1133-1138. 

Self-Consistent Equations Including Exchange and Correlation Effects W. Kohn and L. J. Sham Physical Review 140 (1965) All33... 

Within the mean-field SCFT, the integral in Eq. (37) is approximated by the extremum of the integrand. Thus the equilibrium free energy, F = —kgTlnZ, is a functional of the fields (f), B, uA, ub, , at which F attains its minimum. From the definition of F follows that these fields should satisfy the set of self-consistent equations... 

Derive the following self-consistent equation for the mean field F ... 

This self-consistent equation has a simple message the relevant friction for the reaction is determined by the (Laplace) frequency component of the time dependent friction at the reactive frequency X. This frequency sets the basic time scale for the microscopic events affecting k. 

Returning to (4.76), we see that the right-hand side depends on eaa, which in turn is a function of (na-a) (4.34). We can express this dependence via the self-consistent equations... 

The surface CP as is found by putting gs in place of Ge in the CPA self-consistency equation (6.39), whence... 

Once the double self-consistency equations have been solved numerically, it is trivial to calculate the charge transfer (4.102) from the surface to the... 

Equation (7.19) is a self-consistent equation for AEn, in the form of a sum of a pair of continued fractions (CFs). Although numerical solutions to (7.19) are feasible, we are only concerned with its qualitative features. In particular, we note that an exact WSL occurs when AEn = 0, which happens only if both CFs contain the same number of terms (apart from the trivial case 0 = 0). For the infinite chain, this situation is the case for every allowed energy, so an exact WSL is indeed found. But, for a finite chain, AEn = 0 only for the center state, which thus possesses the exact WSL energy. Therefore, the set of energies for a finite chain form only an approximate WSL. 

The numerical solution of the self-consistency equation (8.35) is rather difficult, so we refer readers to Schranz and Davison (1998) for a few details. However, for illustrative purposes, we present, in Fig. 8.5, a typical graph-... 

Taking into account the lowest diagram, we can then write down the self-consistent equations for the mean-fields, V and A ... 

Gathering all these stuffs to put them in the self-consistent equations, we have the coupled gap equations for As,... 

This full set of self-consistent equations is clearly very difficult to solve, even numerically. However, good approximations of closed integral type have been proposed. These essentially ignore the s-dependence of the survival and orientation functions, which makes them a physically appeaUng approach in the case of wormlike surfactants [71,72]. For ordinary monodisperse polymers the following approximate integral constitutive equation results ... 

The occupation probability can be obtained from the self-consistency equation,... 

Careri (Ref 7b) suggested a procedure for computing exactly the thermodynamic functions of a nonequilibrium system. The state of the system was then varied, at fixed volume and temperature, so as to give a minimum Helmholz free energy, consistent with such conditions as are imposed to permit the exact computation. The condition under which this method leads to self-consistent equations is discussed in detail. The method is then applied in a way that is very close to the Lennard-Jones and Devonshire cell method, bur with cells of variable size. The distribution within a cell is assumed to be Gaussian. Mayer Careri claimed that the method is easier to apply than the cell method, but it seems to be rather complicated... 

The self consistent equation, for the ground state energy gap An (ky) is ... 

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