A Variational Calculation Based on a Two-Electron Atomic Model Luis J. A. Martins lnstituto Superior de Engenharia de Coimbra, 3000 Coimbra, Portugal The oualitv electronic wave functions has . " of annroximate .. recently ( 1 ) been considrred inconnection withthevariation ~rincinle.An rxactls sol\,able heliumlike model ( 2 )was used that &owed comparison of selected average values with expectation values computed from trial Gaussian wave functions using the variation principle and a maximum overlap criterion (I).I t is the purpose of this work to present a twoelectron model from which a plausible approximate wave function may be built for the heliumlike atom. Within the framework of this model a one-parameter trial wave function is set up in the form of a product of two ground-state wave functions for one electron in a spherical box. While providing the basis for a variational calculation involving simple mathematics, the use of this model may reveal trends in the atomic behavior for the heliumlike series, namely, the effect of increasing nuclear charge on (1) the size of the atom and (2) the ionization potential of the ground-state atom. Results and Discussion The proposed model may he regarded as an extension of a modified treatment (3) of a hydrogenlike atomic model given previously (4). Accordingly, the kinetic energy of the two electrons can be obtained assuming that they are in a spherical box, with radius a, inside which the potential energy is zero, while a t the walls and outside the box i t is infinite. For spherically symmetric stationary states of a single electron, with mass m, inside a spherical box, the allowed eigenvalues and eigenfunctions are (5)
Dependenceof the klnetic (7). lhe potential (Y, and the total electmnic energy (0of the ground-state He atom on the radius of Me containing sphere
according to this model.
and finally
.
,
.....
"
.
. . . ,h being the Planck constant. he potential energy, l;y,
.
ofa sinele electron in the field ofan infinitelv heavv nucleus, with n;clear charge +Ze a t the center of the containing sphere is calculated as (3)
where the integral has been previously estimated (3) by making a Taylor series expansion. For the heliumlike atom, the approximation is then made that the electron-nuclear attractions are independent of each other for the two electrons, thus giving two such terms for the configuration ls2. Likewise, the electron-repulsion energy, V.,, is
where - is the interelectronicseparation distance. This enerev term mav he calculated using- a well-established procedure ( 6 ) ,which yields
with t = rrlla, the latter integral being estimated to five significant figures as 0.16802, giving a maximum uncertainty less than lo-=. This was achieved by a Taylor series expansion of the integrand (around t = 0) and term-hy-term integration (7).The stahility condition (4) for this two-electron atom yields the radius of the sphere a t which the total nonrelativistic electronic energy
is minimum. Substitution of VeNand V,. by their calculated values in eq 5 and minimization with respect to a, leads to a = 1.31152 A and E = -T = VI2 = -1.6068 hartree (see figure). This value enahlesus toget an estimate of theionization potential of the ground-state helium atom as IP = E H ~ + - E H=~0.4027 hartree, which is lower than the experimental one (8) (0.9035 hartree). On the other hand, the total electronic energy is overestimated in comparison with the exact nonrelativistic energy (8)(-2.9037 hartree). As noted before (9)the stability condition, which requires minimization of the total electronic energy (eq 5) in order to find the radius atwhich thevirial theorem is fulfilled, is equivalent to Volume 65 Number 10 October 1986
861
scaling the spatial wave function J.h, id = J.l&)J/~oo($, taken as an approximation to the spatial portion of the ground-state wave function of the He atom. Within this approach the expectation values of the kinetic and potential energies (TI')) and (V(l)),respectively, do not conform to the virial theorem. For this to happen a scale factor, q, is introduced (lo), which yields
with
(E(n))being the expectation value of the total electronjc energy with respect to the scaled spatial wave function J/(qrl, &.This result is now related to the more general procedure of obtaining an upper hound for the ground-state energy using the variation method. Indeed, if use is made of the correct nonrelativistic electronic Hamiltonian and the singlet two-electron wave function
Comparison of the Calculated Values ot E. IP, and Radius. ~ccordlngto Thls Model, wlth the ~orrespondlni~xperimental andlor Accurate Theoretical Values' tor the Heliumllke Atom
Miiumiike atom E(accurate)/hamee This model IP(exp.)/hartree This model Most accurate theoretical radius (takenas (
