1. Molecular orbitals (MOs) are made of
fractions of atomic orbitals. All atoms in the molecule provide their atomic orbitals for construction
of MOs, but not all atomic orbitals must participate in all MOs. The number of
MOs is equal to the number of atomic orbitals used to generate them.
Instead of making bonds one at a time by overlapping pairs
of atomic or hybridized orbitals, in the the MO procedure all available atomic orbitals are
mixed into multiple combinations (MOs). This mixing procedure is called the Linear Combination of Atomic Orbitals (LCAO), and it
simply means that we "add" and "subtract" fractions of atomic orbitals (wavefunctions) to make new
molcular orbitals. We have to use each atomic orbital completely, we have to generate
normalized molecular orbitals (MOs), and the number of MOs must be equal to the number of
atomic orbitals that we have started with. Again, it is a bit
like making mixed drinks (except for the "subtracting" part), but now we mix orbitals
of different atoms all at once, instead of just premixing individual atom's
orbital as we din in hybridization.
2. The MO are delocalized over many atoms. In general, they do
not directly correspond to specific bonds (the exceptions include simple diatomic
molecules or some isolated π bonds, such as one in
Since many atomic orbitals participate in the "mixture"
to form the molecular orbitals, the volumes of these new orbitals encompass many atoms,
sometimes even the whole molecule. The so formed MOs do not any longer correspond to specific
bonds (as they were in VB theory). In fact, they can be bonding between some pairs
of atoms (where "addition" of wavefunctions took place) and antibonding between other pairs of atoms
(where "subtraction" of wavefunctions happend), and sometimes they will have a node
at a given atom (like in the allyl system). On some occasions the MO and VB orbitals will "look" exactly the same. For example, we may find such cases
in isolated π bonds, or orbitals containing lone pairs that are
not adjacent to π systems.
σ (or σ*) type MOs are usually separated from π
(or π*) type MOs. (One exception that will
be discussed by us is called hyperconjugation).
This separation is the consequence of symmetry.
π bonds are usually perpendicular to σ bonds, i.e. they cannot mix (because the overlap is zero). In many cases
this arrangement simply means that π and σ
networks do not interact and can be treated separately. This situation simplifies the
analysis. For example, look at benzene: its σ and π networks are essentially independent; when we analyze
benzene (and aromaticity later on) we talk exclusively about π electrons. Of course, there are exceptions. There are many situations where
σ bonds are in geometry that allows for overlap with a π system. One such exception is called hyperconjugation. In
fact, the concept is (again) derived from the VB theory to account for the delocalization
of electrons from the σ bond to the π system. And like resonance (= conjugation) it is a
fix of our model, and not the problem of molecular structure. In MO theory, the hyperconjugation shows naturally: the appropriate,
mostly π-type MO show contributions from some atomic orbitals (s
or p) of adjacent atoms that are properly aligned with the π
system (but would not be, in the VB language, considered a part of it).
4. The MO are filled by all available electrons
(no more than two per orbital), starting from the lowest energy MO orbital.
Each MO has energy associated with it (see above). All available electrons (from all
participating atoms) are placed (two per orbital) in the molecular orbitals, starting at
the bottom of the energy scale and moving up, until no more electrons are left. What
counts is not whether the orbital is bonding or antibonding between
specific atoms within the molecule, but what is the energy of the
5. All electrons in all MOs
determine the structure of the molecule, but the Highest (in energy) Occupied MO (HOMO) and the Lowest (in energy) Unoccupied MO (LUMO) are the most important from the point of view of
|The energy of the specific molecular structure depends on energy
of its electrons in occupied molecular orbitals. Different structures (i.e. molecular
geometries) will have different energies of their molecular orbitals. Thus, all electrons
will influence the structure (remember the compromises
discussed above). But from the point of view of reactivity some electrons and some
orbitals are more important than others. The electrons of the highest energy are the ones
that the molecule would like to "dump", and empty orbitals of the lowest energy
(in the reaction partner) are the best "dumping grounds". In some chemical
reactions (for example electron-transfer reactions or Lewis acid-base
coplex formations) this is exactly what takes place, in
others the interactions between the HOMO (occupied) and the LUMO (unoccupied)
reorganization of bonding of both reacting partners.
6. The usual energy ordering of
MOs is as follows: σ-type orbitals (the lowest energy), π-type orbitals,
nonbonding orbitals (atomic orbitals, lone
pair orbitals, or non-bonding π-type orbitals), π*-type orbitals and σ*-type orbitals (the highest in energy). The exceptions are known (for example, CO molecule). This ordering may be used
to rapidly identify the HOMO and the LUMO in organic molecules.
Identifying the HOMO and the LUMO can be
easily accomplished, if all MOs (and their energy are known. But, that
usually requires extensive computer-based quantum calculations. The MOs
of various molecules presented on this website were, in fact, obtained
in that way.
However, the HOMO and the LUMO may be also recognized
correctly in most cases based on the generic energy ordering of orbitals. This ordering is qualitative, but
it yields useful information from analysis of Lewis structures for types
of electrons present.
σ-types orbitals are occupied in
all molecules (at least one). π orbitals are
occupied in compounds with multiple bonds. Orbitals at non-bonding level
(n) may be occupied (lone pairs or non-bonding
π) or remain empty (atomic p or non-bonding π).
π* are very rarely occupied in the ground
state (see O2),
and σ* are essentially never occupied in
If the orbital type for the HOMO or the LUMO is
identified, its shape and "location" within the molecule can be
approximated using the modified VB theory orbitals. The
approximation is not perfect, but gives good understanding of molecular
interactions controlling their reactivity to a large degree. The only
complication is encountered if resonance is present which requires the
analysis of the whole
In general for each type (σ or π) the energy of the MO increases with the increasing
number of nodes (in the bonding sense).
|Predicting the ordering of the energy levels of the orbitals that
are farther removed from the line dividing the occupied and unoccupied orbitals is more
difficult, short of performing calculations. But we rarely need it anyway. On the
other hand, it is useful to know that the energy of each type of orbital (σ
and especially π) increases with the number of
nodes. Here, we want to count the nodes that result in antibonding interaction between
atoms that are bonded in the molecule. The more nodes of this type,
the higher the energy of the orbital. So, although we cannot order the σ– and
π–type orbitals relative to each other, we can
arrange the π orbitals according to their energy and decide easily (for example)
which is the highest occupied.
8. The electronegativity of atoms is reflected
in the size of their lobes within the MO. Usually, in bonding
orbitals there is more participation (the lobes are larger) by the more electronegative
atoms. In the corresponding antibonding orbitals the lobe sizes are reversed.
|The situation here is slightly complicated. The size of the lobe
of the atomic orbital participating in a given molecular orbital depends on the energy of
that orbital and its size (these two are of course related, see above). The bonding molecular orbital will
have larger contributions from the lower energy (i.e. more electronegative) atoms, and the
antibonding orbitals will have larger contributions from the higher energy (i.e. less
electronegative) atoms. This usually works well for atoms from the
same row of periodic table, but deviations from this pattern can be expected
if the orbital lobes are contributed by atoms
belonging to different rows (compare for example the size of the p orbitals of oxygen and sulfur in the
table in Part I).
9. Most often, the HOMO corresponds to a filled
π-type orbital or a lone pair (nonbonding electrons), and the LUMO corresponds to an empty
π*-type orbital or (if there is no π system) to an empty σ* orbital.
This is just the consequence of orbital ordering discussed in p.
A minor complication (not really) is when a lone pair orbital is part of
the π system
(as happens when resonance is present). Well... then the whole
π system: needs to be analyzed. Typically, the HOMO in such situations is
nonbonding π type orbital.
10. The chemical reactions between molecules are
largely governed by HOMO-LUMO interactions (the highest-energy electrons (HOMO) of one
molecule "looking for" the lowest-energy unfilled space (LUMO) in the other
molecule). The electron-rich component's (see: Brønsted base, Lewis base,
nucleophile, electron donor) HOMO will interact strongly with the electron-deficient
component's (see: Brønsted acid, Lewis acid, electrophile, electron acceptor) LUMO.
The difference in energy between these orbitals and the overlap between them (orbital lobe
size) will largely determine the facility of the reaction and the site of attack (bond
Since interaction between filled (occupied) orbitals does not
result in net bonding (electron-electron repulsion), and interaction between empty
(unoccupied) orbitals cannot contribute to bonding (no electrons to be shared) only the
interaction between occupied and unoccupied orbitals may provide the initial impetus for the
reorganization of existing bonding. Of course the highest energy electrons (HOMO) and the
lowest energy empty orbitals (LUMO) will interact the strongest (they are closest in
energy, see above). Essentially all
chemical reactions are dependent on these HOMO-LUMO interactions.
The strength of interaction between orbitals is proportional to
their overlap (here assumed the same for both pairs of interaction)
and inversely proportional to their energy separation. HOMOB–LUMOA
interaction will control the reactivity between A (Lewis acid or
electrophile) and B (Lewis base or nucleophile).
back to Part II ...