Self-consistent integral equations scheme

We now consider the PPP, CNDO, INDO, and MINDO two-electron semiempirical methods. These are all SCF methods which iteratively solve the Hartree-Fock-Roothaan equations (1.296) and (1.298) until self-consistent MOs are obtained. However, instead of the true Hartree-Fock operator (1.291), they use a Hartree-Fock operator in which the sum in (1.291) goes over only the valence MOs. Thus, besides the terms in (1.292), f/corc(l) m these methods also includes the potential energy of interaction of valence electron 1 with the field of the inner-shell electrons rather than attempting a direct calculation of this interaction, the integrals of //corc(/) are given by various semiempirical schemes that make use of experimental data furthermore, many of the electron repulsion integrals are neglected, so as to simplify the calculation. 

Now, the OEP scheme is complete The integral equation (17), determining the local xc potential vxc(r) corresponding to an orbital-dependent approximation of Exc, has to be solved self-consistently with the KS equation (1) and the differential equation for Gsia(r, r), Eq. (20). 

The constraint forces depend linearly on the multipliers which have to be determined in accordance with the numerical integration scheme. This usually leads to nonlinear equations which can in special cases be solved directly. However, the most common algorithm, called SHAKE [15], solves the equations iteratively, until self consistency between input and output multipliers is achieved. 

At this point a side remark seems appropriate. All potentials shown in Fig. 2.3 originate from self-consistent calculations within the corresponding schemes. One might then ask how these curves change if the same density (and thus the same orbitals) are used for the evaluation of the different functionals This issue is addressed in Fig. 2.4 in which the solution of the OPM integral equation on the basis of three different sets of orbitals is plotted. 

The solution of Equations 47, 48, and 49 requires numerical techniques. For such nonlinear equations, it is usually wise to employ a simple numerical integration scheme which is easily understood and pay the price of increased computational time for execution rather than using a complex, efficient, numerical integration scheme where unstable behavior is a distinct possibility. A variety of simple methods are available for integrating a set of ordinary first order differential equations. In particular, the method of Huen, described in Ref. 65, is effective and stable. It is self-starting and consists of a predictor and a corrector step. Let y = f(t,y) be the vector differential equation and let h be the step size. 

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