6.4: Linear combination of atomic orbitals (LCAO)

The expansion of a molecular orbital in terms of atomic orbitals is an extremely important approximation, known as the linear combination of atomic orbitals (LCAO). The atomic orbitals used in this expansion constitute the basis set for the calculation. Ideally, the number of atomic orbitals used should be infinite such that we could reexpress any given wavefunction exactly in terms of a linear combination of atomic orbitals. In this case, we say that the basis set is also infinite. But computational limitations usually force the basis set to be finite in practice. Choice of the basis set is an especially important consideration in numerical simulations; for example we might consider s, p and d orbitals, but not f or higher orbitals.

In some cases, we can take good guesses at the weighting coefficients, \(c_{r}\), based on the likely nuclear arrangement. However, depending on the nuclear arrangement, it often helps to define new atomic orbitals that are linear combinations of the familiar s and p atomic orbitals. These are known as symmetry adapted linear combinations (SALCs) because they are chosen based on the nuclear symmetry. They are also known as hybrid atomic orbitals. We discuss SALCs in Appendix 4.