Main page QUANTUM  CHEMISTRY  PRIMER:  PART  III
 
Part I:   What are orbitals? Part II: Valence Bond Theory   Part III: Molecular Orbital Theory 
Atomic orbitals Hybridized orbitals Molecular orbitals 
Representations of orbitals Electronic structure  HOMO & LUMO
How are bonds made? σ and π bonds Energy ordering of MOs
Why are bonds made? Resonance   Lobe size
 

MOLECULAR ORBITAL THEORY

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 ethylene).

 

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.

 

3. The 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 orbital.

 

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 reactivity.

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) "starts" the 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 ground-state molecules.

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 π system.

 

  

7. 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 atomic p,  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. 6.  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 an occupied 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 formation).

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.  HOMOBLUMOA interaction will control the reactivity between A (Lewis acid or electrophile) and B (Lewis base or nucleophile).

  back to Part II ...

 
Quantum Primer Last updated 07/29/10 Copyright 1997-2013
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