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 calculations of atomic enereies since the ouhlication of the Kohnand Sham paper (1);wo decades a& A renewed interest in the orohlem started with the vuhlicarion hv Parr et al. (2) of an energy function written c~mpletelyin terms of the electron densitv D. I t is known that the electron-electron repulsion and resonance terms do not allow an analytic solution to the Schrodinger equation for the many-electron atom. The mathematical treatment of such terms is not simple enough, with the~ossihleexceotion 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., i o avoid the as shown in the cited literature makesit complicated two-electron integrals. The purpose of this article is to present an energy function E(p) containing a single integral and one variational parameter a.Within the LDA all two-electron integrals in an atom are represented in a single integral. The total electron-electron interaction V , is given by a function of the total electron 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 i 4 ) ~
~
Table 1. Ground State Energy E, Klnetlc Energy T, and Potential Energy V(in atomlc unlts) where a is the Variational Parameter Atom
Table 2.
~
B C N
normalized to unity
'&for
-E
-Ed
Estimated ionization Energles E, (au) for the Slngly ionized Atoms of Second Row ol
-4
-E+
E?
E:
c
3.84 4.33 5.20 6.07
14.57 24.53 37.69 54.40
14.27 24.10 37.26 54.16
0.30 0.43 0.43 0.24
0.30 0.29 0.40 0.51
0.34 0.30 0.41 0.53
Atom Be
-V
T
rn
me emund state.
bE, = E+-E.. "Reference 6.
dobserved ionization potential (7) 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,
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 numher of the u ~ o e r m o soccuoied t orbital and reo~timizineEM. ,-, Since the dehations in ~ i b l 1 e 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. Therefore the deviations for those atoms increase with increasing u
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
Z. Volume 66
Number 1 January 1989
45
In conclusion a LDA for V,,, together with hydrogenic orbitals, can be used variationally as an elementary method for the calculation of atomic energies.
3- Gadn. S. R.; Bartolotti, L.J.: Handy, N. C. C h m Phys. 1980.72,1034-1038. 4.
Psuling,L.;Wilson,E.B.lnfroduefionroQuonrumMechoni~~;Dover:NewYorL:1985: Chapter5.
5. Fischer, C. F. ThoHnrtme-Pock Method for Atoms; Wiley: New Yorlt: 1977.
Literature Cited I. Kohn, W.;Sham,L. J.Phys.Rou.A 1965,140,1133. 2. Psrr, R. G.; Gadre, S. R.; Bartolotti, L. J. Proc. Noll. Acod. Sri. U S A 1979,76,25222526.
46
Journal of Chemical Education
6. Frsga,8.;Kawawski, J.: Ssrens. K. M. S. Handbook York, 1976.
of
Atomic Doto; Elsevier: Nelu
7. Condon,E.U.;Odabssi. H.Atomic Sfruefure: Cambridge. New Yark, 1980.
