8
Chapter 1 Atomic Orbitals, Electron Spin, Linear Combinations
numbers (for example, 1
and 2
involve two electrons that doubly-occupy
bonding and antibonding molecular orbitals, respectively), then lower-energy
(and higher-energy) linear combinations of these two configurations can be
constructed. This procedure is referred to as configuration interaction. Because
the molecular orbitals are orthogonal, a lower-energy linear-combination may
only be constructed if the configurations 1
ψ and 2
ψ do not differ in the orbital
and spin designations of more than two electrons
v
. (This limitation does not
necessarily apply to the bond-eigenfunctions of (i), because at least some of
the atomic orbitals for these valence-bond configurations must overlap.)
If for any of (a) and (b), the functions 1
ψ and 2
ψ can also interact with a third
wave-function 3
, then the linear combination
1 1
2 2
3 3
c
c
c
(with c 1 ,
2
c and 3
c chosen so that the energy of ψ is minimized) will have a lower energy
than has either
1 1
2 2
c
c
or 3
alone. If ψ is a molecular orbital formed
from a linear combination of three overlapping atomic orbitals χ 1 , χ 2 and χ 3
centred on three atomic nuclei, this molecular orbital is referred to as a delocalized
or 3-centre molecular orbital. We shall often encounter 3-centre molecular
orbitals, and usually use the symbols y, a and b to designate the atomic orbitals χ 1 ,
χ 2 and χ 3 .
Diatomic molecular orbitals which are either symmetric or antisymmetric with
respect to rotation around the bond-axis are designated as σ and π. Alternatively,
these orbitals have, respectively, 0 and 1 nodal planes (i.e. planes on which the
orbital wave-function is zero at all points) that pass through the atomic nuclei and
include the bond axis. There are two sets of degenerate π-type molecular orbitals.
With the z-axis as the bond axis, these orbitals are labelled here as either x
and
y
or π and , and have, respectively, the xz and yz planes as nodal planes.
Bonding and antibonding diatomic molecular orbitals have 0 and 1 nodal planes
passing through the bond axis parallel to the xy planes. Their energies are
respectively less than, and greater than the atomic orbitals from which they are
constructed. The same theory is appropriate for the delocalized molecular orbitals
of linear triatomic and linear polyatomic molecules. For non-linear planar
molecules, the delocalized molecular orbitals may be either of σ or π or
type, with π non-degenerate. The molecular orbitals have σ-symmetry with
respect to at least one pair of adjacent atoms, and symmetry with respect to at
least another pair of adjacent atoms. In Figure 1-5, atomic orbitals that may be
used to construct molecular orbitals for
2 2
O
, FNO and N 2 O 3 are
displayed.
v With self-consistent field molecular orbital theory, no direct interaction can occur between
the lowest-energy configuration with doubly-occupied molecular orbitals and singly-excited
S = 0 spin configurations with the same symmetry as that of the lowest-energy configuration,
if the orbitals used to construct all configurations are the “best” orbitals for the lowest-energy
configuration.
Chapter 1 Atomic Orbitals, Electron Spin, Linear Combinations
numbers (for example, 1
and 2
involve two electrons that doubly-occupy
bonding and antibonding molecular orbitals, respectively), then lower-energy
(and higher-energy) linear combinations of these two configurations can be
constructed. This procedure is referred to as configuration interaction. Because
the molecular orbitals are orthogonal, a lower-energy linear-combination may
only be constructed if the configurations 1
ψ and 2
ψ do not differ in the orbital
and spin designations of more than two electrons
v
. (This limitation does not
necessarily apply to the bond-eigenfunctions of (i), because at least some of
the atomic orbitals for these valence-bond configurations must overlap.)
If for any of (a) and (b), the functions 1
ψ and 2
ψ can also interact with a third
wave-function 3
, then the linear combination
1 1
2 2
3 3
c
c
c
(with c 1 ,
2
c and 3
c chosen so that the energy of ψ is minimized) will have a lower energy
than has either
1 1
2 2
c
c
or 3
alone. If ψ is a molecular orbital formed
from a linear combination of three overlapping atomic orbitals χ 1 , χ 2 and χ 3
centred on three atomic nuclei, this molecular orbital is referred to as a delocalized
or 3-centre molecular orbital. We shall often encounter 3-centre molecular
orbitals, and usually use the symbols y, a and b to designate the atomic orbitals χ 1 ,
χ 2 and χ 3 .
Diatomic molecular orbitals which are either symmetric or antisymmetric with
respect to rotation around the bond-axis are designated as σ and π. Alternatively,
these orbitals have, respectively, 0 and 1 nodal planes (i.e. planes on which the
orbital wave-function is zero at all points) that pass through the atomic nuclei and
include the bond axis. There are two sets of degenerate π-type molecular orbitals.
With the z-axis as the bond axis, these orbitals are labelled here as either x
and
y
or π and , and have, respectively, the xz and yz planes as nodal planes.
Bonding and antibonding diatomic molecular orbitals have 0 and 1 nodal planes
passing through the bond axis parallel to the xy planes. Their energies are
respectively less than, and greater than the atomic orbitals from which they are
constructed. The same theory is appropriate for the delocalized molecular orbitals
of linear triatomic and linear polyatomic molecules. For non-linear planar
molecules, the delocalized molecular orbitals may be either of σ or π or
type, with π non-degenerate. The molecular orbitals have σ-symmetry with
respect to at least one pair of adjacent atoms, and symmetry with respect to at
least another pair of adjacent atoms. In Figure 1-5, atomic orbitals that may be
used to construct molecular orbitals for
2 2
O
, FNO and N 2 O 3 are
displayed.
v With self-consistent field molecular orbital theory, no direct interaction can occur between
the lowest-energy configuration with doubly-occupied molecular orbitals and singly-excited
S = 0 spin configurations with the same symmetry as that of the lowest-energy configuration,
if the orbitals used to construct all configurations are the “best” orbitals for the lowest-energy
configuration.