CHEM 333
QUANTUM THEORY AND SPECTROSCOPY
PROBLEM SET IV
1. Molecular configuration of the OF molecule
a. Consider the 8O and 9F atoms. Write the atomic configurations of the atoms and
schematically represent the occupation of shells and subshells.
b. Draw the molecular orbital diagram of the OF molecule, knowing that F is more
electronegative than O, and that π orbitals are less bonding σ orbitals.
c. Write down the molecular electronic configuration.
d. Calculate the bond order of the OF molecule.
e. Write down the molecular electronic configuration of OF- and calculate the bond order.
f. Write down the molecular electronic configuration of OF+ and calculate the bond order.
g. Sort the OF, OF-, OF+ species in order of increasing stability and explain.
2. Shape of molecules predicted by quantum theory and IR spectroscopy
a. Write down the electronic configuration of 1H and 8O.
b. Using valence-bond theory and orbital hybridization arguments, predict the shape of the
H2O molecule. Is it linear or bent?
c. Consider the vibrational modes of H2O. Knowing that 3 distinct vibrational bands appear
in the IR spectrum of H2O, can you confirm your earlier prediction about the shape of the
water molecule?
3. A spectral line of
48
Ti8+ (mass 47.95 g.mol-1) in a distant star was found to be shifted from
654.2 nm to 706.5 nm and to be broadened by 61.8 pm. What is the speed of recession and
the temperature of the star?
CONCORDIA UNIVERSITY – CHEM 333 – Problem Set IV
4. Rotational-vibrational spectroscopy of HCl
a. Explain why HCl is IR-active.
b. What are the allowed values for the changes in vibrational and rotational quantum
numbers for the P, Q and R branches of the IR spectrum?
c. What is the frequency of the fundamental mode of vibration of HCl?
d. Calculate the HCl vibration zero-point energy.
e. Calculate the force constant of the HCl bond.
f. What is the rotational constant of HCl?
g. Calculate the HCl equilibrium distance.
5. It is customary to form bonding and antibonding molecular orbitals using the Linear
Combination of Atomic Orbital method. Show that the combinations
ψ = c Aψ A + c Bψ B
and
ψ * = c Bψ A − c Aψ B ,
where ψ A and ψ B are orthonormal atomic orbitals, are orthogonal molecular orbitals. What
additional condition should be imposed on the coefficients c A and c B for the molecular
orbitals to be normalized?
-2-
CONCORDIA UNIVERSITY – CHEM 333 – Problem Set IV
Constants and Units
hv =
1
mv 2 + I .E.
2
= h / 2π
h = 6.62618 10
-34
E = −hcR H / n 2
J.s
ψ nlm (r , θ , φ ) = Rnl (r )Yl m (θ , φ )
c = 2.997925 108 m.s-1
k B = 1.38066 10-23 J.K-1
l = l (l + 1)
N A = 6.02 1023
l z = m
β = 9.274 10-24 J.T-1
∆l = ±1
me = 9.10953 10-31 kg
J = L + S ,..., L − S
∆E = E m +1 − E m = β ⋅ B
m H = 1.0 g/mol
mCl = 35.4 g/mol
1s 2 s 2 p 3s 3 p 4 s 3d 4 p 5s 4d ...
1 eV = 1.602 10-19 J
b = 1 / 2(n − n*)
1 Å = 10-10 m
µ=
Elements of Mathematics
∫
∞
∫
∞
0
0
e
− ax 2
2n
x e
1⋅  π 
dx =  
2 a
− ax 2
E J = hc BJ ( J + 1)
v receding =
c
λ
v=
v=
1
2π
k
µ
1/ 2
Equations
v=
I = µre2
1

E v =  v + hv
2

1/ 2
1 ⋅ 3 ⋅ 5 ⋅ ⋅ ⋅ (2n − 1)  π 
dx =
 
2 n +1 a n
a
m1 m2
m1 + m2
1
λ
v
1+ v /c
2v  2k T ln 2 
δv =  B

c 
m

-3-
B=
1/ 2
h
8π 2 Ic