A self-consistent set of equations is derived for an atomic central potential such that the energy given by the orbitals for the potential is minimized. It is shown that this effective potential behaves like -e /r for large r values. The equations have been solved for carbon, neon, and aluminum, and the resulting total energies exceed the Hartree-Fock total energies by less than 0.005%. The theory leads to an effective, local, central exchange potential analogous to the Xa potential.

Local ͑multiplicative͒ effective exchange potentials obtained from the linear-combinationof-atomic-orbital ͑LCAO͒ optimized effective potential ͑OEP͒ method are frequently unrealistic in that they tend to exhibit wrong asymptotic behavior ͑although formally they should have the correct asymptotic behavior͒ and also assume unphysical rapid oscillations around the nuclei. We give an algebraic proof that, with an infinity of orbitals, the kernel of the OEP integral equation has one and only one singularity associated with a constant and hence the OEP method determines a local exchange potential uniquely, provided that we impose some appropriate boundary condition upon the exchange potential. When the number of orbitals is finite, however, the OEP integral equation is ill-posed in that it has an infinite number of solutions. We circumvent this problem by projecting the equation and the exchange potential upon the function space accessible by the kernel and thereby making the exchange potential unique. The observed numerical problems are, therefore, primarily due to the slow convergence of the projected exchange potential with respect to the size of the expansion basis set for orbitals. Nonetheless, by making a judicious choice of the basis sets, we obtain accurate exchange potentials for atoms and molecules from an LCAO OEP procedure, which are significant improvements over local or gradient-corrected exchange functionals or the Slater potential. The Krieger-Li-Iafrate scheme offers better approximations to the OEP method.

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The problem of calculating the eigenvalues of the Dirac equation by the finite-basis expansion method is studied. Bounds for the eigenvalues are obtained explaining the numerical results on the spectrum that have been observed previously. It is argued that the problem of variational collapse can be avoided by finding the minimum over the wave-function large component of the maximum over the wave-function small component of the energy functional. A numerical example is discussed.PACS numbers: 03.65. Ge, 02.70.+d, 31.15.+q There has been a great deal of recent interest in the problem of solving the Dirac equation for particle bound states by finite-basis expansion methods. This problem is becoming increasingly important since the study of relativistic effects in molecular physics is of increasing interest and finite-basis expansions are an important practical method of constructing molecular wave functions. The problem of solving the Dirac equation in a finite basis has proved to be more difficult than the corresponding problem for the Schrodinger equation because of the so-called "variational collapse." 1 The problem is that one is seeking a highly excited state above all the negative-energy states. It turns out that it is not always possible to identify which physical state, if any, corresponds to a particular state arising from a matrix diagonalization, and that if nonlinear parameters are varied, almost any values for the matrix eigenvalues may be obtained.In this article the matrix problem associated with the Dirac equation will be analyzed and bounds for the eigenvalues will be obtained. The Dirac equation and its matrix approximation can be formulated in terms of a minimax principle and a version of the Hylleraas-Undheim-MacDonald theorem 2 ' 3 (HUM theorem) applies to the matrix formulation. The advantage of this minimax principle is that, as well as providing a formulation for the matrix problem, it provides a guide for the determination of nonlinear parameters.There is no difficulty for the Schrodinger equation since it has been shown by Poincare 4 thatis the / th eigenvalue of the matrix approximation and e t is the / th exact eigenvalue. (In this article the eigenvalues of an Nx N matrix are always taken to be ordered e x ^ e 2 ^ ... ^ e N .) Thus one can, for example, minimize ^( iV) on nonlinear parameters and be assured of improving the energy and wavefunction approximations. This is not possible for the Dirac equation since a particular eigenvalue can decrease arbitrarily into the hole-state continuum. In connection with this, we note that the HUM theorem asserts that as the size of a finite basis increases, a particular eigenvalue decreases, i.e., e^N^l ) ^ e t {N \ For N -* oo, e^-* ej if the basis is complete in the first Sobolev space (i.e., for first derivatives).In this article the two-component radial Dirac equation in the form given by Drake and Goldman 5 (to be referred to as DG) will be considered:The functions g and /are traditionally called the large and small components of the wave f...

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