Atomic Calculations with a One-Parameter, Single Integral Method

Reinaldo Baretty Departamento de Fisica y Electronica, Colegio Universitario de Humacao. Humacao, PR

Carmelo Garcia Departamento de Quimica, Colegio Universitario de Humacao, Humacao, PR 00661

Various energy functions have heen proposed for the cal- culations of atomic enereies since the ouhlication of the Kohnand Sham paper (1);wo decades a& A renewed inter- est in the orohlem started with the vuhlicarion hv Parr et al.

Table 1. Ground State Energy E, Klnetlc Energy T, and Potential Energy V(in atomlc unlts) where a is the Variational Parameter

Atom rn T - V -E - E d

(2) of an energy function written c~mpletely in terms of the electron densitv D.

I t is known that the electron-electron repulsion and reso- nance terms do not allow an analytic solution to the Schro- dinger equation for the many-electron atom. The mathe- matical treatment of such terms is not simple enough, with the~ossihle exceotion of the helium atom. to he resented in an introductory course in quantum mechanics. i ow ever, the use of a local densitv ao~roximarion (LDA) to reorcsent V., as shown in the cited literature makesit io avoid the complicated two-electron integrals.

The purpose of this article is to present an energy function E(p) containing a single integral and one variational parame- ter a. Within the LDA all two-electron integrals in an atom are represented in a single integral. The total electron-elec- tron interaction V, is given by a function of the total elec- tron density. We chose here a representation proposed by Gadre et al. (3) given by

where En = 0.8727. E, = -0.6349. and N is the numher of ~ ~ ~" ~ . .

electrons. An enormous simplification in the integration of eo 1 results if the densitv is exoressed in terms of the hvdro- &dike radial wave func"tions i4 )

normalized to unity

where f,, are the occupation numbers of the filled orhitals. The total energy function (in atomic units) is given by

E = X f, (J.,hb,) + v,, (4)

When the atomic numher (Z) is a in all orhitals, the problem becomes a variational one. The total energy is a function of the variational parameter a,

where the sum is over all occupied orhitals and n is the principal quantum numher. The integral in the Ve part reduces to a one-dimensional quadrature over the radial part.

The optimized ground state energy obtained with this

Table 2. Estimated ionization Energles E, (au) for the Slngly ionized Atoms of Second Row

Atom ol -4 -E+ E? E: c Be 3.84 14.57 14.27 0.30 0.30 0.34 B 4.33 24.53 24.10 0.43 0.29 0.30 C 5.20 37.69 37.26 0.43 0.40 0.41 N 6.07 54.40 54.16 0.24 0.51 0.53

'&for me emund state. bE, = E+-E.. "Reference 6. dobserved ionization potential (7)

single variational parameter is summarized in Table 1. The average deviation with respect to the Hartree-Fock (HF) values is just 1.4%. This is consistent with the plot of V.. and its root mean square deviation given in ref 3. I t is clear that in the atomic range explored here (3 5 Z 5 18) the energies are in general overestimated.

Table 2 shows the estimated energies for the singly ionized atoms. This is done by reducing by one the occupation num- her of the u~oermost occuoied orbital and reo~timizine EM.

u ,-, Since the dehations in ~ i b l e 1 are 1% we cannot expect the ionization ootentials to coincide with the H F values to more than an orier of magnitude. Furthermore, the normalization to unity in eq 3 does not allow the explicit consideration of the Hund's rule for atoms having p and d electrons. There- fore the deviations for those atoms increase with increasing Z.

Volume 66 Number 1 January 1989 45

In conclusion a LDA for V,,, together with hydrogenic 3- Gadn. S. R.; Bartolotti, L. J.: Handy, N. C. C h m Phys. 1980.72,1034-1038.

orbitals, can be used variationally as an elementary method 4. Psuling,L.;Wilson,E.B.lnfroduefionroQuonrumMechoni~~;Dover:NewYorL:1985:

for the calculation of atomic energies. Chapter5.

5. Fischer, C. F. ThoHnrtme-Pock Method for Atoms; Wiley: New Yorlt: 1977. Literature Cited 6. Frsga, 8.; Kawawski, J.: Ssrens. K. M. S. Handbook of Atomic Doto; Elsevier: Nelu I. Kohn, W.;Sham,L. J.Phys.Rou.A 1965,140,1133. York, 1976. 2. Psrr, R. G.; Gadre, S. R.; Bartolotti, L. J. Proc. Noll. Acod. Sri. U S A 1979, 76,2522-

2526. 7. Condon, E. U.; Odabssi. H. Atomic Sfruefure: Cambridge. New Yark, 1980.

46 Journal of Chemical Education