MO Theory
H2+ and H2 solns
Solutions to Hydrogen Molecule
Ion
Y2, E2 = -10.16 eV (for H2 )
Y1, E1 = 1.37 eV
(for H2)
Solutions to Hydrogen Molecule
MOs created from combinations of p-orbitals
pxA + pxB, pyA + pyB
pxA - pxB, pyA - pyB
pzA - pzB
pzA + pzB
Solutions to Hydrogen Molecule
px+ px OR py + py
px - px OR py - py
pz - pz
pz + pz
represents
center of
inversion
Parity
inversion
inversion
Gerade =
symmetric
with
inversion
Ungerade =
antisymmetric
with inversion
MO Energy Level Diagram for
Homonuclear Diatomics
lone atom
s*
lone atom
p*
2p
p
2p
s
s*
2s
1s
s
s*
s
2s
1s
Molecular Term Symbols
• ML = S (over all e-) l
• l identifies “z-component” of angular momentum
of an e• Symbols used to id l
|l|
0
1
2
3
4
s
p
d
f
g
Molecular Term Symbols
• Angular momentum about “z-axis” for all
electrons is M L 
L = |ML|
Symbol used to id L
L
0
1
2
3
4
S
P
D
F
G
Molecular Term Symbols
• Symbol is 2S + 1 L g/u
• 2S + 1 is multiplicity as already used for atomic
term symbols
• g or u identifies overall parity
– To determine overall parity, make use of multiplication
of symmetric and antisymmetric functions
• If the term is a S term, a right superscript of + or
– is added to indicate whether the wavefunction
is symmetric or antisymmetric with respect to
reflection through a plane containing the two
nuclei
Molecular Term Symbols
Remember sigma orbs:
From s orbs
From pz orbs
Remember pi orbs:
From px orbs
From py orbs
Molecular Term Symbols
Remember sigma-star orbs:
From s orbitals
From p orbitals
Remember pi-star orbs:
From px orbitals
From py orbitals
Spectroscopy – Selection Rules
DL = 0, +1, -1
DS = 0
note S = Ms
DS = 0
note W refers to spin-orbit coupling and
W=L+S
DW = 0, +1, -1
Molecular Term Symbols
• Molecular Orbitals not always so “clearcut”
• Remember how orbitals change energy as
go across PT
– Can affect MO energy pattern too
MO Energy Level Diagram for
Homonuclear Diatomics
As you move
to the right on
PT, 2s and 2p
energy gap
increases.
Early, in the
period, then,
this permits
mixing of 2s
and 2pz
orbitals.
Atkins, Fig 14.30
Essentially LCAOs
involving four
orbitals are made.
The sigma orbitals
that we thought of
as being made by
the 2s orbitals are
lowered in E while
the sigma orbitals
that we thought of
as being made by
the 2pz orbitals are
raised in E.
MO Energy Level Diagram for Homonuclear
Diatomics (N2 and “before”)
lone atom
s*
lone atom
p*
2p
Use this
diagram for N2
and earlier in
PT
2s
1s
s
p
2p
s*
s
s*
s
2s
1s
Taking a look
at
heteronuclear
diatomic
molecules
Taking a look
at
heteronuclear
diatomic
molecules
MOs of HF
Unoccupied, E = -0.124 eV
Occupied, E = -0.3523 au
E = -0.491 au
E = -1.086 au
MOs of HF
H atom
H – F molecule
F atom
s
1s
p
2p
s
s
2s
s
1s
Computational Chemistry
• Considering complexity of the calculations we’ve been
doing, certainly, using computers to do these calcs
should be useful  Computational Chemistry
• For polyatomic molecules can make LCAOs
yMO = S ciyi
- Yi constitute basis set (computational forms of atomic
orbitals)
– Use variation theory to find ci
– To find structure of molecule, must move nuclei and
find MOs  find structure with lowest overall energy
Computational Chemistry
• May “solve” for MOs using ab initio or semiempirical methods
– Semi-empirical methods: empirical parameters
substituted for some “integrals” to save time in
calculations
– Ab initio methods: supposedly make no assumptions
• NOTE: computational chemistry may determine Energy
and some other properties without using quantum
chemistry
– Such calculations are referred to as molecular
mechanics calculations
Valence Bond Theory
• H2
• Initial approx is y = y1sA(1) y1sB(2)
– But, is this a symm or antisymm wavefxn?
• So, make LCs
– y+ = y1sA(1) y1sB(2) + y1sB(1) y1sA(2)
– y- = y1sA(1) y1sB(2) - y1sB(1) y1sA(2)
• In this case, turns out that y+ is lower E
Valence Bond Theory
• Ground state wavefunction would be
ybond = [y1sA(1) y1sB(2) + y1sB(1) y1sA(2)][a(1)b(2) – a(2)b(1)]
• 2 electrons in overlapping orbitals – with
spins paired
Remember CH4
• If try to make combinations of the valence s of C
with s of H, will be different type of wavefxn,
hence diff’t kind of bond than when make
combination of a p of C with an s of H
• DON’T see any diff in bonding of 4 H’s
–
–
–
–
Make LCs of valence orbitals on central atom
Call these LCs hybrid orbitals
Use these hybrid orbitals to make sigma bonds with H
Atomic orbitals NOT used to make sigma bonds used
to make pi bonds (Huckel method for conjugated)
Hybrid Orbitals
• Valence s and p orbitals on C  hybrids
y1 = a12s + a22px + a32py + a42pz
y2 = b12s + b22px + b32py + b42pz
y3 = c12s + c22px + c32py + c42pz
y4 = d12s + d22px + d32py + d42pz
• Consider ethyne
– Only two hybrids
y1 = s + pz and y2 = s – pz
– Leftover px and py on one C overlap with px and py on
other C
Simplification to MO
Approach
Huckel Approach
Symmetry of Molecules
Determining Point Groups
Special Group?
No
Yes
C∞v , D∞h , Td , Oh , Ih , Th
Cn
No
i
C1
No
sh
No
Yes
Yes
Yes
Cs
Ci
No
n sv
No
Yes
Yes
No
Cn
sh
Cnv
S2n or S2n and i only, collinear
with highest order Cn
No
Sn
Yes
nC2 perpendicular to Cn
Yes
No
Cnh
Dn No
n sd
Yes
sh
Yes
Dnh
Dnd
C2v Character Table
C2v
E
C2
sv(xz)
sv’(yz)
A1
1
1
1
1
A2
1
1
-1
-1
B1
1
-1
1
-1
B2
1
-1
-1
1
Now go practice!!!
Applying Symmetry to MOs
Water
MOs of Water
HOMO-4
a1
Looks like s orbital on O, nbo
E = -18.6035 au
MOs of Water
HOMO-3 from two viewpoints
a1
Looks like s orbital
on O with
constructive
interference with
c1 - bo
E = -0.9127 au
MOs of Water
HOMO-1
HOMO-2
b2
a1
Looks like combination of p on
O along C2 with constructive
interference with c1, bo (close
to nbo)
E = -0.3356 au
Looks like combination of p on
O (perp to C2, but in plane of
molecule) with constructive
interference with c2, bo
E = -0.4778 au
MOs of Water
HOMO from two viewpoints
b1
Looks like p orbital
on O, perpendicular
to plane of
molecule - nbo
E = -0.2603 au
MOs of Water
LUMO
LUMO +1
b2
a1
Looks like combination of p on
O along C2 with destructive
interference with c1, abo
E = -0.0059 au
Looks like combination of p on
O (perp to C2, but in plane of
molecule) with destructive
interference with c2, abo
E = 0.0828 au
Filling Pattern for Water
2b2 (abo)
4a1 (abo)
1b1 (nbo)
3a1 (bo/nbo)
1b2 (bo)
2a1 (bo)
1a1 (nbo)
Molecular Spectroscopy
• Molecule has a number of motions
– Translational, vibrational, rotational, electronic
• Sum them to get total energy of molecule
• Changes may occur in any of these modes
through absorption or emission of energy
– Vibrational: IR
– Rotational: Microwave
– Electronic: UV-Vis
CHP 16, 17, 18 of text
Statistical Mechanics
• Quantum gives you possible energy levels
(states)
– In a real sample, not all molecules in the same energy
level
• With statistics and total energy, can predict (on
average) how many molecules in each state
– Dynamic Equilibrium
– Role of Temperature
• Can predict macroscopic properties/behavior
– Heat capacity, pressure, etc.
CHP 19, 20 of text