Atomic Structure Theory

52 2 Central-Field Schrödinger Equation
where N is the number of bound electrons in the atom.
In the Thomas-Fermi theory, the electronic potential is given by the classical
potential of a spherically symmetric charge distribution,
R
1
Ve(r) = 4πr ′2 ρ(r ′ )dr ′ , (2.99)
0
r>
where r> = max (r, r ′ ). The total energy of the atom in the Thomas-Fermi
theory is obtained by combining (2.97) for the kinetic energy density with the
classical expressions for the electron-nucleus potential energy and the electronelectron
potential energy to give the following semi-classical expression for the
energy of the atom:
R
3
E =
0 10 (3π2 ) 2/3 ρ 2/3 − Z
R
1 1
+ 4πr
r 2 0 r>
′2 ρ(r ′ )dr ′
4πr 2 ρ(r)dr .
(2.100)
The density is determined from a variational principle; the energy is required
to be a minimum with respect to variations of the density, with the
constraint that the number of electrons is N. Introducing a Lagrange multiplier
λ, the variational principal δ(E − λN) =0gives
R
1
0 2 (3π2 ) 2/3 ρ 2/3 − Z
r +
R
1
4πr
0 r>
′2 ρ(r ′ )dr ′
− λ 4πr 2 δρ(r)dr =0.
(2.101)
Requiring that this condition be satisfied for arbitrary variations δρ(r) leads
to the following integral equation for ρ(r):
1
2 (3π2 ) 2/3 ρ 2/3 − Z
r +
R
1
4πr
0 r>
′2 ρ(r ′ )dr ′ = λ. (2.102)
Evaluating this equation at the point r = R, whereρ(R) = 0, we obtain
λ = − Z
R
1
+ 4πr
R R 0
′2 ρ(r ′ )dr ′ Z − N
= − = V (R) , (2.103)
R
where V (r) is the sum of the nuclear and atomic potentials at r. Combining
(2.103) and (2.102) leads to the relation between the density and potential,
1
2 (3π2 ) 2/3 ρ 2/3 = V (R) − V (r) . (2.104)
Since V (r) is a spherically symmetric potential obtained from purely classical
arguments, it satisfies the radial Laplace equation,
which can be rewritten
1
r
d 2
rV (r) =−4πρ(r) , (2.105)
dr2