18.6 The Lithium Atom 781
PROBLEMS
Section 18.5: Angular Momentum in the Helium Atom
18.9 Find the possible term symbols for the excited
configuration (2p)(3d) of the helium atom.18.10Find the possible term symbols for the excited
configuration (1s)(3d) for the helium atom. Which will
probably have the lowest energy?
18.11Pretend that the direction of angular momentum vectors
can be specified exactly. If the spin angular momentum
vector for one electron with spin up lies in the yz plane,
and the spin angular momentum vector for another
electron with spin down also lies in the yz plane, find the
possible magnitudes and directions of their vector sum.
To what values ofSandMS(if any) do these values
correspond?
18.12Pretend that the direction of angular momentum vectors
can be specified exactly. Assume an electron in a
hydrogen atom is in aψ 322 orbital with spin up and
pretend that both the orbital angular momentum vector
and the spin angular momentum vector lie in the xz plane.
Find the possible vector sums and the possiblez
projection of the sum. To what values ofJandMJ(if
any) do the sums correspond?18.6 The Lithium Atom
A lithium atom has three electrons and a nucleus withZ3. The Hamiltonian operator
for a lithium atom with a stationary nucleus isĤ−h ̄2
2 m(∇ 12 +∇ 22 +∇^23 )
+
1
4 πε 0(
−
3 e^2
r 1−
3 e^2
r 2−
3 e^2
r 3+
e^2
r 12+
e^2
r 13+
e^2
r 23)
(18.6-1)
The zero-order Hamiltonian omits the electron–electron repulsion terms and consists
of three hydrogen-like Hamiltonians withZ3:
̂H(0)ĤHL(1)+ĤHL(2)+̂HHL(3) (18.6-2)where we as usual abbreviate the coordinates by giving only the subscript specifying
which electron is meant.
Solution of the Schrödinger equation in the zero-order approximation leads to a
wave function that is a product of hydrogen-like orbitals withZ3:Ψ(0)ψn 1 l 1 m 1 ,ms 1 (1)ψn 2 l 2 m 2 ,ms 2 (2)ψn 1 l 1 m 1 ,ms 1 (3)
ψ 1 (1)ψ 2 (2)ψ 3 (3) (18.6-3)where we abbreviate the subscripts in the second version of this equation. The zero-
order electronic energy of the atom is the sum of three hydrogen-like energy eigenvalues
withZ3. From Eq. (17.3-19),E(0)E(0)n 1 n 2 n 3 En 1 (HL)+En 2 (HL)+En 3 (HL)−(13.60 eV)(
32
n^21+
32
n^22+
32
n^23)
(18.6-4)
Exercise 18.5
Carry out the steps to obtain Eq. (18.6-4).