1-3 Linear Combinations of Wave Functions
7
requirement that
1
2
/
/
0
E c
E c
as a necessary condition that E is a
minimum gives the secular equations
11
1
12
12
2
(
)
(
)
0
Η
E c
H
ES c
21
21 1
22
2
(
)
(
)
0
H
ES c
H
E c
for normalized 1
and 2
, with
ψ ψ dτ
ij
i
j
ˆ
H
and
d
ij
i
j
S . In this
book, the wave-functions 1
and 2
are of two main types:
a) Atomic and molecular orbitals: If 1
and 2
are a pair of (real) atomic
orbitals centred on two atomic nuclei, the interaction integral (or resonance
integral)
d
ˆ 2
1 H
is non-zero if these atomic orbitals overlap, i.e. the
overlap integral S 12 =
1 2 d 0
v
. The lower and higher energy linear
combinations of 1
and 2
are designated as bonding and antibonding
molecular orbitals, respectively. If the atomic orbitals are located on the same
atomic centre, then usually they are orthogonal (i.e.
1 2 d 0
v
). When this
is the case, no lower-energy linear combination may be constructed. Hybrid
atomic orbitals, examples of which were provided in Section 1-1, are linear
combinations of atomic orbitals located on the same atomic centre.
b) Two-electron and many-electron configurations of electrons: An electron
configuration designates the orbital occupancies and spins for the electrons.
The following two-electron or many-electron configurations need to be
considered here:
i) Valence-bond structure functions (bond-eigenfunctions): These wave-functions
describe the configurations of electrons that are associated with valence-bond
structures. If a pair of valence-bond structures (for example, Li — and
Li : H
for LiH, or
and
for 6 6
C H ) have configuration wavefunctions (or structure wavefunctions or bond-eigenfunctions) designated as
1
and 2
for their electrons, then the construction of linear combinations of
these wave functions is equivalent to invoking resonance between the valencebond structures. The valence-bond structures are said to be stabilized by
resonance if one of the linear combinations has a lower energy than has either
1
and 2
alone. Resonance stabilization can only occur if
1
2
ψ ψ dτ
ˆ
H
(the
exchange integral) is non-zero. A necessary (but not necessarily sufficient)
condition for this to occur is that the bond-eigenfunctions 1
ψ and 2
ψ must
have the same sets of values for their S and z
S spin quantum numbers.
ii) Molecular orbital configurations: If 1
and 2
are two different molecular
orbital configurations with the same spatial symmetry and sets of spin quantum
7
requirement that
1
2
/
/
0
E c
E c
as a necessary condition that E is a
minimum gives the secular equations
11
1
12
12
2
(
)
(
)
0
Η
E c
H
ES c
21
21 1
22
2
(
)
(
)
0
H
ES c
H
E c
for normalized 1
and 2
, with
ψ ψ dτ
ij
i
j
ˆ
H
and
d
ij
i
j
S . In this
book, the wave-functions 1
and 2
are of two main types:
a) Atomic and molecular orbitals: If 1
and 2
are a pair of (real) atomic
orbitals centred on two atomic nuclei, the interaction integral (or resonance
integral)
d
ˆ 2
1 H
is non-zero if these atomic orbitals overlap, i.e. the
overlap integral S 12 =
1 2 d 0
v
. The lower and higher energy linear
combinations of 1
and 2
are designated as bonding and antibonding
molecular orbitals, respectively. If the atomic orbitals are located on the same
atomic centre, then usually they are orthogonal (i.e.
1 2 d 0
v
). When this
is the case, no lower-energy linear combination may be constructed. Hybrid
atomic orbitals, examples of which were provided in Section 1-1, are linear
combinations of atomic orbitals located on the same atomic centre.
b) Two-electron and many-electron configurations of electrons: An electron
configuration designates the orbital occupancies and spins for the electrons.
The following two-electron or many-electron configurations need to be
considered here:
i) Valence-bond structure functions (bond-eigenfunctions): These wave-functions
describe the configurations of electrons that are associated with valence-bond
structures. If a pair of valence-bond structures (for example, Li — and
Li : H
for LiH, or
and
for 6 6
C H ) have configuration wavefunctions (or structure wavefunctions or bond-eigenfunctions) designated as
1
and 2
for their electrons, then the construction of linear combinations of
these wave functions is equivalent to invoking resonance between the valencebond structures. The valence-bond structures are said to be stabilized by
resonance if one of the linear combinations has a lower energy than has either
1
and 2
alone. Resonance stabilization can only occur if
1
2
ψ ψ dτ
ˆ
H
(the
exchange integral) is non-zero. A necessary (but not necessarily sufficient)
condition for this to occur is that the bond-eigenfunctions 1
ψ and 2
ψ must
have the same sets of values for their S and z
S spin quantum numbers.
ii) Molecular orbital configurations: If 1
and 2
are two different molecular
orbital configurations with the same spatial symmetry and sets of spin quantum