CHEM 442 Lecture 21 Problems
21-1. Suggest the normalized, antisymmetrized wave functions of the helium atom in the
singlet (1s)2 and triplet (1s)1(2s)1 configurations in the orbital approximation.
21-2. Show that the (1s)2 wave function of 21-1 is indeed normalized, provided that
individual orbitals and spin functions are orthonormal (“orthonormal” means orthogonal
and normalized).
21-3. Show that the expectation value of the Hamiltonian in the (1s)2 wave function of
21-1 is given by,
ì 2 2
ì 2 2
2e2 ü
2e2 ü
*
*
E = ò j1s ( r1 ) íÑ1 Ñ ýj ( r ) dr + j ( r ) íýj ( r ) dr
4pe 0 r1 þ 1s 1 1 ò 1s 2 î 2m 2 4pe 0 r2 þ 1s 2 2
î 2m
+ ò j1s ( r1 )
2
e2
4pe 0 r12
j1s ( r2 ) dr1dr2
2
21-4. Show that the expectation value of the Hamiltonian in the (1s)1(2s)1 wave
function of 21-1 is given by,
ì 2 2
ì 2 2
2e2 ü
2e2 ü
*
E = ò j1s* ( r1 ) íÑ1 j
r
dr
+
j
r
Ñ
(
)
(
)
ý
í
ýj ( r ) dr
4pe 0 r1 þ 1s 1 1 ò 2s 2 î 2m 2 4pe 0 r2 þ 2s 2 2
î 2m
+ ò j1s ( r1 )
2
e2
4pe 0 r12
j 2s ( r2 ) dr1dr2 - ò j ( r1 )j
2
*
1s
*
2s
e2
( r2 ) 4pe r
j 2s ( r1 )j1s ( r2 ) dr1dr2
0 12
21-5. Which terms in the energy expressions in 21-3 and 21-4 explain the effect of
shielding and Hund’s rule (namely, the energy lowering due to spin correlation or Pauli
exclusion)?
21-6. Show that the last term in the energy expression in 21-3 is positive.
21-7. Show that the last term in the energy expression in 21-4 is real.
21-8. Show that the (1s)2 wave function of 21-1 is an eigenfunction of Ŝz = ŝz (1) + ŝz ( 2 )
operator. What is the eigenvalue? On this basis, suggest the total spin angular momentum
quantum number and total spin angular momentum of the (1s)2 helium atom.
21-9. Show that the triplet (1s)1(2s)1 wave functions of 21-1 are eigenfunctions of
Ŝz = ŝz (1) + ŝz ( 2 ) operator. What are the eigenvalues? On this basis, suggest the total spin
angular momentum quantum number and total spin angular momentum of the triplet
(1s)1(2s)1 helium atom.
21-10. What is the spin multiplicity of the hydrogen atom? What spin multiplicities the
lithium atom can take?