# Molecular orbital

In quantum chemistry, a **molecular orbital** (MO) is a one-electron function that is "smeared out" over a whole molecule. Usually an MO is a linear combination of atomic orbitals (an LCAO), which is a weighted sum of (almost) all atomic orbitals in the molecule. Molecular symmetry plays an important role in molecular orbital theory. In general MOs are adapted to symmetry, which has the consequence that atomic orbitals of a symmetry different than that of an MO do not contribute to the MO (have weight factor zero).

The expansion coefficients (weight factors) of an MO can be determined by semi-empirical molecular orbital theory, which may or may not be of the self-consistent field type, or by *ab initio* methods. The most common *ab initio* method for determining MO coefficients is the self-consistent field method of Hartree and Fock. Lately, density functional theory methods for computing MOs have found increasing application.

### Definition of molecular orbital

A molecular orbital (MO) depends on several position vectors of one and the same electron, labeled 1: **r**_{A1}, **r**_{B1}, **r**_{C1}, ... where *A*, *B*, *C*, ... are different points in space (usually nuclear positions). The oldest example of an MO (without use of the name MO yet) is in the work of Burrau (1927) on the single-electron ion H_{2}^{+}. Burrau applied spheroidal coordinates (a bipolar coordinate system) to describe the wave functions of the electron of H_{2}^{+}.

Lennard-Jones^{[1]} introduced the following *linear combination of atomic orbitals* (LCAO) way of writing an MO φ:

where *A* runs over *N*_{nuc} different points in space (usually *A* runs over all the nuclei of a molecule, hence the name molecular orbital), and *i* runs over the *n*_{A} different AOs centered at *A*. The complex coefficients *c*_{ Ai} can be calculated by any of the existing effective-one-electron quantum chemical methods. Examples of such methods are the Hückel method—a non-self-consistent field method— and the Hartree-Fock method—a self-consistent field method.

## Computation of molecular orbitals

*See also Heitler-London theory for an account of the chemical bond in*H_{2}.

In the hands of Friedrich Hund, Robert S. Mulliken, John Lennard-Jones, and others, molecular orbital theory was established firmly in the 1930s as a (mostly qualitative) theory explaining much of chemical bonding, especially the bonding in diatomic molecules.

To give the flavor of the theory we look at the simplest molecule: H_{2}.
The two hydrogen atoms, labeled *A* and *B*, each have one electron (a red arrow in the figure) in an 1*s* atomic orbital (AO). These AOs are labeled in the figure *1s*_{A} and *1s*_{B}. When the atoms start to interact the AOs combine linearly to two molecular orbitals:

In equation (1) above a general expression for an LCAO-MO is given. In the present example of the σ_{g} MO of H_{2}, we have *N*_{nuc} = 2, *n*_{A} = *n*_{B} = 1, χ_{A1} = 1*s*_{A}, χ_{B1} = 1*s*_{B}, *c*_{A1} = *c*_{B1} = *N*_{g}.
With regard to the form of the MOs the following: Because both hydrogen atoms are identical, the molecule has reflection symmetry with respect to a mirror plane halfway the H—H bond and perpendicular to it. It is one of the basic assumptions in quantum mechanics that wave functions show the symmetry of the system. [Technically: solutions of the Schrödinger equation belong to a subspace of Hilbert (function) space that is irreducible under the symmetry group of the system].
Clearly σ_{g} is symmetric (is even, stays the same) under

while σ_{u} changes sign (is antisymmetric, is odd). The Greek letter σ indicates invariance under rotation around the bond axis. The subscripts *g* and *u* stand for the German words *gerade* (even) and *ungerade* (odd).
So, because of the high symmetry of the molecule we can immediately write down two molecular orbitals, which evidently are linear combinations of atomic orbitals.

The normalization factors *N*_{g} and *N*_{u} and the orbital energies are still to be computed. The normalization constants follow from requiring the MOs to be normalized to unity. In the computation we will use the bra-ket notation for the integral over *x*_{A1}, *y*_{A1}, and *z*_{A1} (or over *x*_{B1}, *y*_{B1}, and *z*_{B1} when that is easier).

We used here that the AOs are normalized to unity and that the overlap integral is real

Applying the same procedure for *N*_{u}, we find for the two normalization factors

At this point one often assumes that *S* ≈ 0, so that both normalization factors are equal to ½√2.

In order to calculate the energy of an orbital we introduce an effective-one-electron energy operator (Hamiltonian) *h*,

Since *A* and *B* are identical atoms equipped with identical AOs, we were allowed to use

Likewise

The energy term β is negative (causes the attraction between the atoms). One may tempted to assume that *q* = −½ hartree (the energy of the 1*s* orbital in the free atom). This is not the case, however, because *h* contains the attraction with both nuclei, so that *q* is distance dependent.

If we now look at the figure in which energies are increasing in vertical direction, we see that the two AOs are at the same energy level *q*, and that the energy of σ_{g} is a distance |β| below this level, while the energy of σ_{u} is a distance |β| above this level. In the present simple-minded effective-one-electron model we can simply add the orbital energies and find that the bonding energy in H_{2} is 2β (two electrons in the *bonding MO* σ_{g}, ignoring the distance dependence of *q*). This model predicts that the bonding energy in the one-electron ion H_{2}^{+} is half that of H_{2}, which is correct within a 20% margin.

If we apply the model to He_{2}, which has four electrons, we find that we must place two electrons in the bonding orbital and two electrons in the *antibonding MO* σ_{u}, with the total energy being 2β − 2β = 0. So, this simple application of molecular orbital theory predicts that H_{2} is bound and that He_{2} is not, which is in agreement with the observed facts.

## References

- ↑ J. E.Lennard-Jones,
*The Electronic Structure of some Diatomic Molecules*. Trans. Faraday Soc., vol**25**, p. 668 (1929).