Lecture 25
Molecular orbital theory I
(c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has
been developed and made available online by work supported jointly by University of Illinois, the
National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus
Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and
conclusions or recommendations expressed in this material are those of the author(s) and do not
necessarily reflect the views of the sponsoring agencies.
Molecular orbital theory




Molecular orbital (MO) theory provides a
description of molecular wave functions and
chemical bonds complementary to VB.
It is more widely used computationally.
It is based on linear-combination-ofatomic-orbitals (LCAO) MO’s.
It mathematically explains the bonding in H2+
in terms of the bonding and antibonding
orbitals.
MO versus VB

Unlike VB theory, MO theory first combine
atomic orbitals and form molecular orbitals in
which to fill electrons.
MO theory
VB theory
MO theory for H2

First form molecular orbitals (MO’s) by
taking linear combinations of atomic
orbitals (LCAO):
X  A  B and Y  A  B
MO theory for H2

Construct an antisymmetric wave function by
filling electrons into MO’s
X(1)X(2)[a (1)b (2) - b (1)a (2)]
Singlet and triplet H2
(X)2 singlet
éë X (1) X (2) ùû éëa (1)b (2) - b (1)a (2) ùû far more stable
ì a (1)a (2)
ï
(X)1(Y)1 triplet
éë X (1)Y (2) - Y (1) X (2) ùû í b (1)b (2)
ï a (1)b (2) + b (1)a (2)
î
(X)1(Y)1 singlet
éë X (1)Y(2) + Y(1) X (2) ùû a (1)b (2) - b (1)a (2)
{
} least stable
Singlet and triplet He (review)

In the increasing order of energy, the five
states of He are
{
}
Y ( r1 ,r2 ) » j1s ( r1 )j1s ( r2 ) {a (1)b (2) - b (1)a (2)}
(1s)2 singlet
by far most stable
{
}
(1s) (2s) triplet
Y ( r ,r ) » {j ( r )j ( r ) - j ( r )j ( r )} b (1)b (2)
Y ( r ,r ) » {j ( r )j ( r ) - j ( r )j ( r )} {a (1)b (2) + b (1)a (2)}
Y ( r1 ,r2 ) » j1s ( r1 )j 2s ( r2 ) - j1s ( r2 )j 2s ( r1 ) a (1)a (2)
1
2
1s
1
2s
2
1s
2
2s
1
1
2
1s
1
2s
2
1s
2
2s
1
{
}
1
1
Y ( r1 ,r2 ) » j1s ( r1 )j 2s ( r2 ) + j1s ( r2 )j 2s ( r1 ) {a (1)b (2) - b (1)a (2)}
(1s)1(2s)1 singlet
least stable
MO versus VB in H2
VB
MO
[ A(1)B(2) + B(1)A(2)][a (1)b (2) - b (1)a (2)]
éë X (1) X (2) ùû éëa (1)b (2) - b (1)a (2) ùû
X(1)X(2) = { A(1) + B(1)} { A(2) + B(2)}
= A(1)B(2) + B(1)A(2) + A(1)A(2) + B(1)B(2)
MO versus VB in H2
VB [ A(1)B(2) + B(1)A(2)][ spin ]
covalent
covalent
MO
[ X(1)X(2)][spin] = [ A(1)B(2) + B(1)A(2) + A(1)A(2) + B(1)B(2)][spin]
ionic
H −H +
covalent
=
covalent
ionic
H +H −
MO theory for H2+



The simplest, one-electron molecule.
LCAO MO is by itself an approximate wave
function (because there is only one electron).
Energy expectation value as an approximate
energy as a function of R.
e
2
æ 1 1 1ö
e
2
Hˆ = Ñ + - ÷
ç
2me
4pe 0 è rA rB R ø
2
Parameter
rA
A
rB
R
B
LCAO MO

MO’s are completely determined
by symmetry:
y ± = N ± ( A ± B)
Normalization
coefficient
LCAO-MO
A
B
Normalization

Normalize the MO’s:
N± =
=
(
)
d t ± ( ò A B dt + ò B Ad t )}
*
(
A
±
B)
( A ± B) d t
ò
{ ò A dt + ò B
2
= ( 2 ± 2S )
- 12
2
- 12
*
*
2S
- 12
Bonding and anti-bonding MO’s
φ+ = N+(A+B)
φ– = N–(A–B)
bonding orbital – σ
anti-bonding orbital – σ*
Energy
2
æ 1 1 1ö
e
2
Hˆ = Ñ + - ÷
ç
2me
4pe 0 è rA rB R ø
2


Neither φ+ nor φ– is an eigenfunction of the
Hamiltonian.
Let us approximate the energy by its
respective expectation value.
Energy
2
2
æ
e
1ö
2
ç - 2m Ñ - 4pe r ÷ A = E1s A
è
e
0 Aø
2
æ
e2 1 ö
2
ç - 2m Ñ - 4pe r ÷ B = E1s B
è
e
0 Bø
e2
2
1
e
1
*
*
j=òA
Adt = ò B
Bdt
4pe 0 rB
4pe 0 rA
2
1
e
1
*
*
k=òB
Adt = ò A
Bdt
4pe 0 rB
4pe 0 rA
ˆ ( A ± B )dt
A
±
B
H
(
)
ò
=
e2
*
E±
2 ± 2S
2
2
æ 1 1 1 ö üï
*ì
e
ï
2
ò ( A ± B ) íï- 2me Ñ - 4pe 0 çè rA + rB - R ÷ø ýï( A ± B)dt
î
þ
=
2 ± 2S
e2
j±k
= ... = E1s +
4pe 0 R 1± S
S, j, and k
A=
e- rA /a0
p a03
rB
rA
A
, B=
e- rB /a0
p a03
B
R
rA
rB
1.0
S = ò A* Bdt
0.8
A
R
j = ò A*
e2
B
0.6
e2
1
k = ò B*
Adt
0.4
4pe 0 rB
1
Adt
4pe 0 rB
0.2
0
2
4
6
8
10
R
Energy
j±k
E± = E1s +
4pe 0 R 1± S
e2
e2
1.0
4pe 0 R
3.0
2.5
S = ò A* Bdt
0.8
0.6
j+k
2.0
1.5
0.4
1.0
0.2
0.5
0
2
e2
4
1
*
k=òB
Adt
4pe 0 rB
6
8
10
e2
R
0
1
*
j=òA
Adt
4pe 0 rB
2
4
6
j-k
8
10
R
Energy
j±k
E± = E1s +
4pe 0 R 1± S
e2
e2
4pe 0 R
3.0
2.5
e2
4pe 0 R
j+k
2.0
1.5
- ( j - k)
φ– = N–(A–B)
anti-bonding
1.0
0.5
0
2
4
6
j-k
8
10
R
R
e2
(
)
φ+ = N+(A+B)
- j+k
bonding
4pe 0 R
Energy
j±k
E± = E1s +
4pe 0 R 1± S
e2
e2
4pe 0 R
- ( j - k)
j-k
E1s +
4pe 0 R 1- S
e2
φ– = N–(A–B)
anti-bonding
φ– is more anti-bonding
than φ+ is bonding
R
e2
(
)
φ = N+(A+B)
- j + k +bonding
4pe 0 R
E1s
E1s +
e2
4pe 0 R
-
j+k
1+ S
Summary



MO theory is another orbital approximation
but it uses LCAO MO’s rather than AO’s.
MO theory explains bonding in terms of
bonding and anti-bonding MO’s. Each MO
can be filled by two singlet-coupled electrons
– α and β spins.
This explains the bonding in H2+, the simplest
paradigm of chemical bond: bound and
repulsive PES’s, respectively, of bonding
and anti-bonding orbitals.