Numerical-parameterized optimized effective potential ...

Numerical-parameterized optimized effective potential approximation for atoms Pablo Maldonado Jim´ enez Departamento de F´ısica At´ omica, Molecular y Nuclear, Facultad de Ciencias, Universidad de Granada, E-18071 Granada, Spain Abstract. A numerical-parameterized solution of both the non relativistic and the relativistic optimized effective potential equations for atoms is proposed. The analytic continuation method is used to solve the single particle Schr¨ odinger and Dirac equations. This method provides an accurate solution and allows for a straightforward use of the logarithmic transformation. The equations are solved within both a single and a multi configurational framework. The single configuration results for the ground state of a number of closed and open shells are compared with those obtained from a fully numerical solution of the optimized effective potential equations in both the non relativistic and relativistic frameworks, as well as with the Hartree–Fock and Dirac–Hartree–Fock results. We have also calculated the ionization potential and the electronic affinity of a number of atoms. Finally, the performance of the Multi Configuration version of the method is illustrated by studying a number of excited states of the iron atom.

PACS numbers: 31.10.+z, 31.15.-p, 31.15.Pf, 31.15.Ne, 31.30.Jv, 02.70.-c

1. Introduction The main approximation in the study of the electronic structure of atoms is the so called single particle approximation, in either a non–relativistic or in a relativistic framework. Within this approximation, the non relativistic Hartree–Fock (HF) and the relativistic Dirac–Hartree–Fock (DHF) methods are the best ones in determining the single particle wave functions to built the total A–particle wave function. Both of them are variational approaches which minimize the expectation value of the Hamiltonian (relativistic or non-relativistic) with respect to the single particle orbitals. Therefore, HF and DHF methodologies provide the best non correlated solution to the many-electron non-relativistic or relativistic Hamiltonian, respectively. A different approach for this problem is provided by the Optimized Effective Potential (OEP) method. This is a variational approach with a trial wave function written in terms of Slater determinants as in the HF method. The difference arises from an extra constraint imposed in the OEP method, in which the orbitals are the eigenfunctions of a single particle Hamiltonian with a local potential, the same for all of the orbitals. This potential is fixed by minimizing the expectation value of the atomic Hamiltonian with respect to the effective potential. This method was proposed by Sharp and Horton [1], and implemented by Talman and Shadwick[2] in a non relativistic and single configuration framework. The Relativistic Optimized Effective Potential (ROEP) method [3] is a generalization of the OEP method to the relativistic atomic Hamiltonian. The single configuration OEP and ROEP results

Numerical-parameterized optimized effective potential approximation for atoms

2

compare very well with the corresponding HF and DHF ones for all atoms considered [4]. The OEP and ROEP equations, giving the total energy of the atom and the single particle wave functions, contain an effective potential which is determined through an integral equation coupled to either the Schr¨odinger or the Dirac equation, respectively, which must be solved numerically. To simplify the application of the OEP method, an approximation in which both the effective potential and the single particle wave functions are parameterized was introduced. With this approximations all the calculations can be performed analytically. Contrary to what one should expect, the energies thus obtained are in a number of cases better than those obtained by solving numerically the OEP equations. To study more deeper this approximation we have proposed to solve the OEP approximation by parameterizing the effective potential and by solving numerically (and then exactly within our numeric window) the single-particle Schr¨odinger or Dirac equations by using the very accurate analytic continuation method. The method thus built is called NPOEP (RNPOEP) in the non relativistic (relativistic) case. 2. Numerical-parameterized approximation The non relativistic Hamiltonian of a N –electron atom with nuclear charge Z is X 1 X ∇ ~2 Z (1) − ]+ [− HN R = 2 ri r i