CHEM 614
HOMEWORKS 9 and 10
2002
me = 9.1095x1031 kg
mp = 1.6726x10-27 kg
h = 6.6262x10-34 Js
c = 2.998x108 ms-1
e = -1.6022x10-19 C
1:  a and  b are chosen to be a normalized set of basis functions for LCAO wavefunction
for a one-electron homonuclear diatomic system. It is found that the values for the
integrals involving these functions are
  a *Hˆ  a dv  2 au,
  b* Ĥ  b dv  2 au,
  a* Ĥ  b dv  1
au,
1
  a*  b dv  4
Find an upper bound for the exact lowest electronic energy for this system. Find the
corresponding LCAO normalized approximate wavefunction.
2:  a and  b are chosen as the normalized basis functions for an LCAO wavefunction for a
one electron, heteronuclear, diatomic molecule. It is found that the values for some integrals
involving these functions are
1
  a Ĥ  a dv  2 au,
  a Ĥ  b dv   2 au,
  b Ĥ  b dv  1
au,
1
  a  b dv  3 ,
where Ĥ is the molecular hamiltonian. Set up the secular determinantal equation and find
the lowest electronic energy that can be computed from an LCAO wavefunction
c a  a + cb  b. Find ca and cb such that E is minimized and the wavefunction is normalized.
3: In the equation
H AA  H AB
1  S AB
let H AB = kHAA. What relationship between k and SAB is necessary if the  g MO is to be
lower in energy than the  u MO? [Assume that HAA is negative, and that k and S AB are
positive.].
E 
4: Evaluate the energy of the gerade orbital of the hydrogen molecule ion at R = 0 au, R =
2.0 au, R = 2.5 au, R = 4.0 au and R = 100.0 au.
5: For a homonuclear diatomic molecule aligned with the inter-nuclear direction as the
defined z-axis, characterize each of the following MO's as  ,  ,  , and g or u, and bonding or
antibonding.
a) 2pya + 2pyb
b) 2pza + 2pzb
c) 3dz2a + 3dz2b
d) 3dxya + 3dxyb
e) 3dxza - 3dxzb
6: Indicate whether you expect each of the following homonuclear diatomic MO's to be
bonding or antibonding. Sketch the MO in each case:
a)  u
b)  u
c)  g
7: A homonuclear diatomic molecule MO of  u symmetry is to be expressed as a linear
combination of AOs centered on the nuclei, which lie on the z axis. Which of the AOs in the
following list can contribute to the MO?
1s 2s 2px 2py 2pz 3s 3px
3py 3pz 3dxy 3dxz 3dyz 3dz2 3dx2-y2
8: Use the sketches and symmetry arguments to decide which of the following integrals
vanish for diatomic molecules (the z axis is the inter-nuclear one):
(a)  2 p za1sb dv, (b)  2 p ya1sb dv, (c)  2 p za2pyb dv,
(d)  2 p ya3dyzb dv
(e)  2 p za3dyzb dv
9: Assuming the inter-nuclear axis to lie along the z coordinate, what are the possible ML
quantum numbers for an MO constructed from 3dz2a - 3dz2b?
10: a) write out the ground state configuration for O2 and O2+ using MO symmetry symbols
(1  g2, etc.)
b) What is the net number of bonding electrons in both molecules?
c) What is the bond order of both species?
d) How does the dissociation energy for O2+ compare to that for O2?
e) What spin multiplicity would you expect for the two ground states?
f) In O2+, which occupied MO's may contain contributions from 2pz AO's, assuming z is
the inter-nuclear axis?