Basis Sets and the Road to Computational Chemistry

Introduction

Every idea in the variation principle — trial functions, adjustable parameters, linear combinations and secular determinants — comes together in modern computational chemistry. Programs that predict molecular structures, reaction energies and spectra all express molecular orbitals as linear combinations of atom-centred functions, just as we did for H₂⁺. The set of functions chosen is called the basis set, and choosing it wisely is one of the central practical decisions in any calculation.

Core explanation

Molecular orbitals from a basis. Each molecular orbital is written as ψᵢ = Σ μ c μi φ μ, where the φ μ are basis functions centred on the atoms. The coefficients are found variationally from a matrix equation of the form FC = SCε, which is the secular problem HC = ESC generalised so that each electron feels the average field of all the others (the Hartree–Fock method). Because the field depends on the orbitals, the equations are solved repeatedly until they stop changing: a self-consistent field.

Two kinds of function.

- Slater-type orbitals (STOs) , r^(n−1) e^(−ζr) times an angular part, have the correct cusp at the nucleus and the correct exponential tail, but their multi-centre integrals are slow to compute. - Gaussian-type orbitals (GTOs) , e^(−αr²) times an angular part, have the wrong shape but give integrals that are fast and analytical, thanks to the Gaussian product theorem.

The compromise is the contracted Gaussian : a fixed combination of several primitive Gaussians fitted to resemble an STO.

The hierarchy of basis sets.

- Minimal (STO-3G): one function per occupied atomic orbital, each made of three Gaussians. For carbon: 1s, 2s, 2px, 2py, 2pz — five functions. - Split-valence (3-21G, 6-31G): the core orbitals get one function, but each valence orbital gets two of different sizes, an inner and an outer one. The variational mixing of the two lets orbitals expand or contract as bonding demands. In 6-31G, the core function uses six Gaussians and the two valence functions use three and one. - Polarisation (6-31G , also written 6-31G(d)): adds d functions on heavy atoms (and p functions on hydrogen in 6-31G ). These let the electron density shift off-centre, essential for describing bond angles and hypervalent bonding. - Diffuse functions (6-31+G ): very spread-out functions, needed for anions, weak interactions and excited states. - Correlation-consistent (cc-pVDZ, cc-pVTZ, cc-pVQZ): designed by Dunning to converge systematically, so results can be extrapolated towards the complete basis set limit.

What a basis can and cannot fix. Enlarging the basis lowers the Hartree–Fock energy towards the Hartree–Fock limit — every HF energy is a variational upper bound. But even at that limit, HF misses the correlation energy , because each electron sees only the average of the others. Methods such as configuration interaction, coupled cluster, Møller–Plesset perturbation theory and density functional theory address correlation, and they all still need a good basis.

Cost. The number of two-electron integrals grows roughly as N⁴ for N basis functions, so doubling the basis can multiply the cost by about sixteen. Choosing a basis balances accuracy against computing time.

Step-by-step reasoning

To count basis functions for a molecule:

1. List the atoms and their occupied atomic orbitals. 2. For a minimal basis, count one function per orbital (three for a p set). 3. For split-valence, count core orbitals once and valence orbitals twice. 4. Add polarisation functions (six Cartesian d functions per heavy atom in 6-31G ). 5. Sum over all atoms.

Visual explanation

Picture the carbon 2p orbital built in a split-valence basis as two nested dumb-bells, a tight inner one and a larger outer one. A variational calculation mixes them in different proportions: more of the inner function for a compact orbital, more of the outer one for an expanded orbital. Adding a d function lets the dumb-bell bend towards a neighbouring atom.

Real-world analogy

A basis set is like a photographer's set of lenses. A single fixed lens can take a usable picture of anything, but a zoom lens and a macro lens let the photographer frame each subject properly. More lenses cost more to carry, so the kit is chosen to suit the job.

Real-world example

Pharmaceutical researchers routinely optimise the geometries of candidate drug molecules with density functional theory in a split-valence polarised basis such as 6-31G or def2-SVP, then refine key energies with a larger triple-zeta basis. The approach predicts bond lengths to within about 1 to 2 pm.

Why?

Why do polarisation functions matter so much? An isolated hydrogen atom needs only a spherical 1s function, but in a molecule its density is pulled towards bonded neighbours. A spherical function cannot describe that distortion; mixing in a p function shifts density to one side, just as mixing 1s and 2p produces a lopsided hybrid.

Common misconception

"A bigger basis set always gives answers closer to experiment." A bigger basis brings a given method closer to its own limit. If the method neglects correlation, as Hartree–Fock does, the limit itself can still disagree with experiment, and a small basis sometimes agrees better only through lucky cancellation of errors.

Worked example

Question: How many basis functions does water have in STO-3G and in 6-31G ?

Reasoning: STO-3G: oxygen has 1s, 2s and three 2p orbitals, giving 5; each hydrogen has 1s, giving 1 each. Total 5 + 2 = 7. In 6-31G : oxygen has 1 core function, 2 × 4 = 8 valence functions and 6 d functions, giving 15; each hydrogen has 2 functions. Total 15 + 4 = 19.

Answer: 7 basis functions in STO-3G and 19 in 6-31G .

Quick check

1. What does the "31" in the name 6-31G tell you about how the valence orbitals are described? Answer: Each valence orbital is split into two functions, one built from three Gaussians and one from a single Gaussian.

Exam focus

Know the difference between STOs and GTOs, what "minimal", "split-valence", "polarisation" and "diffuse" mean, and be able to count basis functions. Distinguish clearly between basis-set error and the correlation error of the method.

Advanced insight

Because atom-centred basis functions move with the atoms, a molecule in a complex can borrow its partner's functions to improve its own description. This artificially stabilises the complex and is called basis set superposition error; the counterpoise correction of Boys and Bernardi estimates and removes it.

Summary

Modern calculations build molecular orbitals as linear combinations of basis functions and solve a secular-type problem self-consistently. Gaussian functions are used for speed, contracted to resemble Slater functions. Basis sets range from minimal (STO-3G) through split-valence (6-31G), polarised (6-31G ) and diffuse sets to systematic correlation-consistent sets. Enlarging the basis approaches the Hartree–Fock limit, but correlation requires better methods as well.

Practice questions

1. Explain the difference between a minimal basis and a split-valence basis. Answer: A minimal basis has one function per occupied atomic orbital; a split-valence basis gives each valence orbital two or more functions of different size so its shape can adjust variationally. 2. How many basis functions does methane have in STO-3G? Answer: Carbon has 5 and each of the four hydrogens has 1, giving 9. 3. Why are diffuse functions important for anions? Answer: The extra electron in an anion is weakly bound and spread far from the nuclei, so functions that extend to large distances are needed to describe it. 4. Why can no basis set, however large, make a Hartree–Fock energy equal the exact energy? Answer: Hartree–Fock treats each electron in the average field of the others and so omits the correlation energy; enlarging the basis only approaches the Hartree–Fock limit.