H2 1sσ bonding molecular orbitalH2 1sσ* antibonding molecular orbital

As a simple example, consider the hydrogen molecule, H2, with the two atoms labeled H' and H." The lowest-energy atomic orbitals, 1s' and 1s," do not transform according to the symmetries of the molecule. However, the following symmetry-adapted atomic orbitals do:

1s' - 1s"Antisymmetric combination: negated by reflection, unchanged by other operations1s' + 1s"Symmetric combination: unchanged by all symmetry operations

The symmetric combination (called a bonding orbital) is lower in energy than the basis orbitals, and the antisymmetric combination (called an antibonding orbital) is higher. Because the H2 molecule has two electrons, they can both go in the bonding orbital, making the system lower in energy (and hence more stable) than two free hydrogen atoms. This is called a covalent bond. The bond order is equal to the number of bonding electrons minus the number of antibonding electrons, all divided by 2. In this example, there are two electrons in the bonding orbital and none in the antibonding orbital; the bond order is 1, and there is a single bond between the two hydrogen atoms.


On the other hand, consider the hypothetical molecule of He2, with the atoms labeled He' and He. Again, the lowest-energy atomic orbitals, 1s' and 1s," do not transform according to the symmetries of the molecule, while the following symmetry-adapted atomic orbitals do:

1s' - 1s"Antisymmetric combination: negated by reflection, unchanged by other operations1s' + 1s"Symmetric combination: unchanged by all symmetry operations

Similar to the molecule H2, the symmetric combination (called a bonding orbital) is lower in energy than the basis orbitals, and the antisymmetric combination (called an antibonding orbital) is higher. However, in its neutral ground state, each helium atom contains two electrons in its 1s orbital, combining for a total of four electrons. Two electrons fill the lower energy bonding orbital, while the remaining two fill the higher energy antibonding orbital. Thus, the resulting electron density around the molecule does not support the formation of a bond (sigma bond) between the two atoms, and the molecule therefore is not formed. Another way of looking at it is that there are two bonding electrons and two antibonding electrons; therefore, the bond order is 0 and no bond exists.

Ionic bonds

Main article: ionic bond

When the energy difference between the atomic orbitals of two atoms is quite large, one atom's orbitals contribute almost entirely to the bonding orbitals, and the other's almost entirely to the antibonding orbitals. Thus, the situation is effectively that some electrons have been transferred from one atom to the other. This is called a (predominantly) ionic bond.

Molecular orbital diagrams

For more complicated molecules, the wave mechanics approach loses utility in a qualitative understanding of bonding (although is still necessary for a quantitative approach). The qualitative approach of MO uses a molecular orbital diagram. In this type of diagram, the molecular orbitals are represented by horizontal lines; the higher a line, the higher the energy of the orbital, and degenerate orbitals are placed on the same level with a space between them. Then, the electrons to be placed in the molecular orbitals are slotted in one by one, keeping in mind the Pauli exclusion principle and Hund's rule of maximum multiplicity (only two electrons per orbital (opposite spins); have as many unpaired electrons on one energy level as possible before starting to pair them).

The hardest part is to construct the MO diagram. For a simple molecule such as H2, we draw the diagram like this:

__ σ*__ σ

The σ indicates a sigma bonding orbital, while σ* indicates a sigma antibonding orbital. We know that the diagram looks like this because we know that two s orbitals will interact to form a σ bonding orbital and a σ antibonding orbital.

Now if one considers N2, one realizes that the two nitrogen atoms each have a filled 1s orbital, a filled 2s orbital, and three half-filled 2p orbitals. The 1s orbitals, being inner shell, do not interact (or, equivalently, they are not valence electrons, as explained by valence bond theory).

The two 2s orbitals do, however, interact to create a σs orbital and a σs* orbital:

__ σs*__ σs

If we assume that the interatomic axis joining the two N atoms is the z axis, we find that the two 2pz orbitals are able to overlap lobe-to-lobe to create a sigma bond. The two 2px and two 2py orbitals, lying perpendicular to the z axis, interact to create four pi orbitals (two bonding, two antibonding).

Finally, we must decide on the order of the orbitals. The 2s orbitals, since they were initially of lowest energy, interact to create the lowest-energy orbitals. The 2p sigma bonds must be stronger than the pi bonds, so we expect the σp orbital to be lower than the πp orbital. However, this is not the case, primarily because of hybridization mixing the 2s and 2p orbitals. However, we do have the expected order for the σp* and πp* orbitals:

___ σp*
___ ___ πp*___ σp
___ ___ πp___ σs*
___ σs

As promised, there are 8 orbitals, the sum of the number of atomic orbitals (4+4) which combined to create the molecular orbitals. The total number of electrons is then 10 (five valence electrons from each atom). Two go into the σs orbital; two go into the σs* orbital; four into the two πp orbitals, and two go into the σp orbital.

The sigma bond order is the total number of electrons in sigma bonding orbitals (4), minus the total number of electrons in sigma bonding orbitals (2) , all over 2 giving (4-2)/2 = 1. There is the similar pi bond order, giving (4 - 0)/2. Adding these together gives the total bond order. In this case the lowest two orbitals "cancel out"; there is one sigma bond and two pi bonds. Dinitrogen therefore has a triple bond.

Finally, we know that diatomic nitrogen is diamagnetic since there are no unpaired electrons in the diagram.

This diagram, however, is not applicable to molecules of oxygen, fluorine, and neon. Because of the higher electronegativity of these elements, the formation of hybrid orbitals is less important, and thus we get the "expected" order of energy levels:

___ σp*
___ ___ πp*
___ ___ πp___ σp___ σs*
___ σs

The observation that the formation of hybrid orbitals is much less energetically favorable for smaller, more electronegative atoms (which are found in the first row) is due to the energy difference in the atom between the 2s and 2p orbitals. This energy difference increases from left to right along a row and from top to bottom down a column of the periodic table, so is highest for fluorine, which has the lowest mixing of s and p in the MOs. Mixing is most important when the energy difference is small.

If we were working with diatomic oxygen, we would use this MO diagram. In this case, there would be 12 electrons to place into molecular orbitals; the first ten go into the five orbitals of lowest energy; the last two, however, occupy separate πp* orbitals. The bond order is reduced to 2 since this is an antibonding orbital; also, the two unpaired electrons make liquid oxygen paramagnetic, which is not explained by the localized electron model.

MO diagram for the oxygen molecule

A further observation is that molecular orbital theory explains why the dicarbon molecule, C2, does not contain a quadruple bond in its ground state although it would complete the octet - there are four bonding orbitals, but the top three cannot be occupied before one antibonding orbital is occupied.

More quantitative approach

To obtain quantitative values for the molecular energy levels, one needs to have molecular orbitals which are such that the configuration interaction (CI) expansion converges fast towards the full CI limit. The most common method to obtain such functions is the Hartree-Fock method, which expresses the molecular orbitals as eigenfunctions of the Fock operator. One usually solves this problem by expanding the molecular orbitals as linear combinations of gaussian functions centered on the atomic nuclei (see linear combination of atomic orbitals and basis set (chemistry)). The equation for the coefficients of these linear combinations is a generalized eigenvalue equation known as the Roothaan equations which are in fact a particular representation of the Hartree-Fock equation.

Simple accounts often suggest that experimental molecular orbital energies can be obtained by the methods of ultraviolet photoelectron spectroscopy for valence orbitals and X-ray photoelectron spectroscopy for core orbitals. This, however, is incorrect as these experiments measure the ionization energy, the difference in energy between the molecule and one of the ions resulting from the removal of one electron. Ionization energies are linked approximately to orbital energies by Koopmans' theorem. While the agreement between these two values can be close for some molecules, it can be poor in other cases.


HOMO and LUMO are acronyms for highest occupied molecular orbital and lowest unoccupied molecular orbital, respectively. The difference between the energies of the HOMO and LUMO, termed the band gap can sometimes serve as a measure of the excitability of the molecule: the smaller the energy, the more easily it will be excited.

The HOMO level is to organic semiconductors what the valence band is to inorganic semiconductors. The same analogy exists between the LUMO level and the conduction band. The energy difference between the HOMO and LUMO level is regarded as band gap energy.

When the molecule forms a dimer or an aggregate, the proximity of the orbitals of the different molecules induce a splitting of the HOMO and LUMO energy levels. This splitting produces vibrational sublevels, each of which has its own energy, slightly different from that of another.

There are as many vibrational sublevels as there are molecules that interact together. When there are enough molecules influencing each other (such as in an aggregate), there are so many sublevels that we no longer perceive their discrete nature: they form a continuum. We no longer consider energy levels, but energy bands.

See also

  • Atom
  • Electron
  • Hydrogen
  • Molecule
  • Quantum chemistry


  1. ↑ J. Daintith, Oxford Dictionary of Chemistry (New York: Oxford University Press, 2004).
  2. ↑ Ibid.
  3. ↑ In this case, the atomic orbitals are eigenstates of the hydrogen Hamiltonian. They can be obtained analytically.
  4. ↑ Werner Kutzelnigg, "Friedrich Hund and Chemistry" (on the occasion of Hund's 100th birthday), Angewandte Chemie 35 (1996): 573-586.
  5. ↑ "Robert S. Mulliken's Nobel Lecture," Science 157, (3785) (1967): 13-24.
  6. ↑ J. E. Lennard-Jones, Transactions of the Faraday Society 25 (1929): 668.


  • Chang, Raymond. 2006. Chemistry, 9th ed. New York: McGraw-Hill. ISBN 0073221031.
  • Daintith, J. 2004. Oxford Dictionary of Chemistry. New York: Oxford University Press. ISBN 0198609183.
  • Kutzelnigg, Werner, "Friedrich Hund and Chemistry" (on the occasion of Hund's 100th birthday), Angewandte Chemie 35 (1996): 573-586.
  • Pope, Martin, and Charles E. Swenberg. 1999. Electronic Processes in Organic Crystals and Polymers, 2nd ed. New York: Oxford University Press. ISBN 0195129636.
  • Tipler, Paul, and Ralph Llewellyn. 2003. Modern Physics, 4th ed. New York: W.H. Freeman. ISBN 0716743450.

External links

All links retrieved December 21, 2018.

  • Atomic Orbitals.
  • Covalent Bonds and Molecular Structure.
  • The Orbitron: A gallery of atomic orbitals and molecular orbitals on the WWW. from 1s to 7g.
  • Grand Orbital Table.
  • David Manthey's Atomic Orbitals.
  • Hydrogen Atom Orbital Viewer. (Java applet.)
  • Molecular Orbital Viewer. (Java applet, for the hydrogen molecular ion.)