Energy Splitting of MOs
Orbital Interactions
• most molecules include more than two atoms, and the
atoms are not all of the same type
• the relative strength of the interaction between two
atomic (or fragment) orbitals depends on a number of
properties
o energy difference (ΔE) between orbitals
o overlap (Sij) between orbitals
o hamiltonian mediated interaction (Hij) between
orbitals
large
splitting
smaller
splitting
degenerate
different
electronegativity
Figure 1 Range of orbital interactions
dependent on fragment energy levels
Overlap
• the relative overlap of orbitals effects the magnitude of
the total interaction between two orbitals and is
represented by S, Sab = ψ a ψ b = ∫ ψ aψ b ∂τ where
ΔE decreases
A
B
r
A
r < r' < r'
r'
B
Energy difference
• the greater the energy difference between two interacting
atomic orbitals the smaller the interaction between them,
and the smaller the splitting between the resultant MOs,
Figure 1
• there is a sliding scale:
o degenerate orbitals have the greatest interaction
o as the energy difference between the orbitals
increases the extent of their interaction decreases
o eventually the orbitals too widely separated in
energy and they do not interact
• and don't forget the antibonding interaction is always
larger than the bonding interaction
A
r"
B
Figure 2 distance dependence in energy
splitting
•
strong overlap
S=1
weaker overlap
0<S<1
Figure 3 The effect of angular position of
orbital overlap
•
•
Sab is the "overlap" of orbitals ψ a and ψ b over all space
(and spin!)
orbitals that are close together overlap more than those
that are further apart, so the longer the distance
between two atoms the smaller the interaction between
two orbitals, Figure 2, note we have just discussed the
energy difference between orbitals, now we are talking
about the physical location of the orbitals and the atomatom distance increasing
in addition orbitals do not necessarily have to belong to
atoms that are directly "bonded" to overlap!
overlap is also dependent on the relative type and
position of the two orbitals involved, Figure 3
1
negative
•
when the net overlap between two orbitals includes inphase overlap, and an equivalent amount of out-ofphase overlap there is no net interaction, Figure 4sigma
type interactions are stronger than pi type interactions
(Figure 5) which are stronger than delta type
interactions (when two dAOs "face" each other),
Figure 5
o s − s > pσ − pσ > pπ − pπ
•
the relative electron density of the orbitals involved is
important
o 2s orbitals will interact more strongly than 3s
orbitals which are more diffuse
o eg F > Cl > Br > I the F orbitals are small but dense
while I orbitals are large but very diffuse
positive
net effect S=0
Figure 4 net zero overlap
Figure 5 Relative strength of σ and π
interactions
Hamiltonian Mediated Interaction
• the Hamiltonian represents the interaction of the
electrons with other electrons, and electrons with the
nuclei (and nuclear-nuclear interactions)
• so the Hamiltonian mediated interaction between two
orbitals really represents how these orbitals interact in
within the environment of the molecule, taking into
account the effects of other electrons and nuclei
• in Huckel theory (which you learn about in the
Theoretical methods course) only orbitals on adjacent
nuclear sites are allowed to interact this way, in
addition H ab = ψ a H ψ b = β , H ab is asumed to be
the same for all sites and is denoted β sometimes β is
also called the resonance integral.
• this approximation is not a very good one and in reality
orbitals on distant atomic sites can interact.
H2-
He2+
Figure 6 H2- and He2+ are isoelectonic
Stability
• MO diagrams can tell us qualitatively about the
stability of related compounds, if in addition they have
the same number of electrons then they are also
isoelectronic
• For example, H2- and He2+ are isoelectonic, Figure 6
• molecules isolobal (ie MO diagrams involve the same
type of orbitals) with H2 (Figure 7) include
o H2+, H2-, H22- (1s valency)
o He2, He2+,
(1s valency)
o Li2, Be2,
(2s valency)
o Na2, Mg2,
(3s valency) … and so on
2
• Li2, dilithium (star treck fans) has configuration
(1 σ g+ )2(1 σ u+ )2(2 σ g+ )2 is a weakly bound gaseous
H2
He2
Figure 7 H2 and unstable He2
H2+
E ∝ ΔEσ
BO=1/2
H2
E ∝ 2ΔEσ
BO=1
H2E ∝ 2ΔEσ − 1ΔEσ *
BO=1/2
molecule
• He2 and Be2 … do not exist … He2 is unstable (Error!
Reference source not found.) as the antibonding orbital
is filled and destabilises the molecule more than the
bonding orbitals stabilise it.
• the bond order is given by the number of electrons in
bonding MOs minus the number in antibonding MOs
all divided by two (Figure 8)
• the bond order of the species H2+, H2-, H22- is shown to
the left
Figure 8 Relative stability of H2 species
3