Optimized Potential Method OPM

The most important question is how to calculate the multiphcative potential which corresponds to xc-functionals of the type (2.19). There are three distinct ways for the derivation of the basic equation which yields this potential. As all three are rather instructive, all of them will be gone through in the following, assmning the xc-functional to be of the general form Exc l k, e/tj- 

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While the concentration dependence of the experimental fields are reproduced rather well by the theoretical fields (a phase transition to the BCC structure occurs around 65% Fe), the later ones are obviously too small. This finding has been ascribed in the past to a shortcoming of plain spin density functional theory in dealing with the core polarization mechanism (Ebert et al. 1988a). Recent work done on the basis of the optimized potential method (OPM) gave results for the pure elements Fe, Co and Ni in very good agreement with experiment (Akai and Kotani 1999). 

If one works out this expression one obtains equations that are identical to equations (316) and (317). These equations were first derived by Talman and Shadwick [45]. Since in our procedure we optimized the energy of a Slater determinant wavefunction under the constraint that the orbitals in the Slater determinant come from a local potential, the method is also known as the optimized potential method (OPM). We have therefore obtained the result that the OPM and the expansion to order e2 are equivalent procedures. The OPM has many similarities to the Hartree-Fock approach. Within the Hartree-Fock approximation one minimizes the energy of a Slater determinant wavefunction under the constraint that the orbitals are orthonormal. One then obtains one-particle equations for the orbitals that contain a nonlocal potential. Within the OPM, on the other hand, one adds the additional requirement that the orbitals must satisfy single-particle equations with a local potential. Due to this constraint the OPM total energy ) will in general be higher than the Hartree-Fock energy Fhf, i.e., Ex > E. We refer to Refs. [46,47] for an application of the OPM method for molecules. 

Given some orbital-dependent xc-functional l xc the first question to be addressed is the evaluation of the corresponding multiplicative xc-potential Wxc- This is possible via the optimized potential method (OPM) [2,3], which is described in Sect. 2.2. After an outline of three different strategies for the derivation of the crucial OPM integral equation, a few exact relations for the OPM xc-potential are summarized. In addition, the Krieger-Li-Iafrate (KLI) approximation [4] to the OPM integral equation is presented. 

Prom such an action functional, one seeks to determine the local Kohn-Sham potential through a series of chain rules for functional derivatives. The procedure is called the optimized effective potential (OEP) or the optimized potential method (OPM) for historical reasons [15,16]. The derivation of the time-dependent version of the OEP equations is very similar to the ground-state case. Due to space limitations we will not present the derivation in this chapter. The interested reader is advised to consult the original paper [13], one of the more recent publications [17,18], or the chapter by E. Engel contained in this volume. The final form of the OEP equation that determines the EXX potential is... 

If one wishes to proceed in this fashion in the relativistic case, one has to provide accurate atomic data. For this purpose, OPM, the optimized potential method [16] (in the present context the relativistic extension, the ROPM) is a valuable tool. The (R)OPM relies on the fact that the functional derivative with respect to the density (or the four-current) can be evaluated with the chain rule for functional derivatives if the dependence on the density is implicit via Kohn-Sham orbitals, E n = E[(fk =... 

Table 1. Absolute atomic total energies (Ry) calculated by Hartree-Fock (HF), optimized potential (OPM), Harbola-Sahni (HS) and using the self-consistent Xa (SC-Xa) method by Cortona [15] (table constructed from data in Table II of Cortona s paper [17])...

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