Symmetry in Chemistry

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Electron atomic and molecular orbitals

In quantum chemistry, the molecular electronic states, i.e. the eigenstates of the electronic molecular Hamiltonian, are expanded (see configuration interaction expansion and basis) into linear combinations of anti-symmetrized products (Slater determinants) of one-electron functions. The spatial components of these one-electron functions are called molecular orbitals (MO). When one considers also their spin component, they are called molecular spin orbitals.

Most methods in computational chemistry today start by calculating the MOs of the system. A molecular orbital describes the behavior of one electron in the electric field generated by the nucleii and some average distribution of the other electrons. In the case of two electrons occupying the same orbital, the Pauli principle demands that they have opposite spin.

Hand-waving discussion

For hand-waving (imprecise, but qualitatively useful) discussion of the molecular structure, the molecular orbitals can be obtained from the "Linear combination of atomic orbitals molecular orbital method" ansatz (using eventually the concept of hybridized orbitals).

In this approach, the molecular orbitals are expressed as linear combinations of atomic orbitals, as if each atom were on its own.

The linear combination of atomic orbitals approximation for molecular orbitals was introduced in 1929 by Sir John Lennard-Jones. His ground-breaking paper showed how to derive the electronic structure of the fluorine and oxygen molecules from quantum principles. This qualitative approach to molecular orbital theory represents the dawn of modern quantum chemistry.

Some properties:

The number of molecular orbitals is equal to the number the atomic orbitals included in the linear expansion,

If the molecule has some symmetry, the degenerate atomic orbitals (with the same atomic energy) are grouped in linear combinations (called symmetry adapted atomic orbitals (SO)) which belong to the representation of the symmetry group,

The number of molecular orbitals belonging to one group representation is equal to the number of symmetry adapted atomic orbitals belonging to this representation,

Within a particular representation, the symmetry adapted atomic orbitals mix more if their atomic energy level are closer.

Examples

H₂

As a simple example consider the hydrogen molecule, H₂, with the two atoms labelled 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:

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.

H₃

On the other hand, consider the hypothetical linear molecule of H₃ with the atoms labelled H, H' (the central atom), H" and equal H-H' and H'-H" distances. Then we would expect three linear combinations:

1s - 1s' + 1s" Symmetric Anti Bonding (2 nodal surfaces perpendicular to the bonds)

1s - 1s" Antisymmetric Non bonding (1 nodal surface along the axis of symmetry)

1s + 1s' + 1s" Symmetric Bonding (0 nodal surface)

Two electrons occupy the symmetric bonding bonding orbital and the third one occupy the non bonding orbital.

Rare gases

Now let's move to larger atoms. Considering a hypothetical molecule of He₂, since the basis set of atomic orbitals is the same as in the case of H₂, we find that both the bonding and antibonding orbitals are filled, so there is no energy advantage to the pair. HeH would have a slight energy advantage, but not as much as H₂ + 2 He, so the molecule exists only a short while. In general, we find that atoms such as He that have completely full energy shells rarely bond with other atoms. (In fact there is not a single stable molecule containing He, Ne or Ar except short-lived Van der Waals complexes.)

Inner shells

Inner shell orbitals should not be included in the LCAO expansion. Molecular structure relies on the outermost (valence) electrons of the atoms, which are usually of comparable energy.

Ionic bonds

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 (mostly) ionic bond.

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.