Basis Sets in Hartree-Fock (or Other
Ab Initio) Computations
In theory, an
MO is exactly expressible (within a Hartree-Fock computation, or in general
within any ab initio one) as a linear combination of an infinite number of
constituent basis functions such as the atomic orbitals. However in practice
one must be content with representation of any MO as a linear combination of
only a finite small number of such basis functions, as per the concept of
LCAO-MO approximation. Various types of basis-function sets (called basis
sets) have been proposed to economically represent an MO in terms of a
finite number of basis functions. Basis sets are divided into two broad
classes, namely the minimal basis sets and the extended ones. A minimal
basis set uses one orbital function each to represent every inner-shell and
valence-shell AO (atomic orbital) of each constituent atom, whereas an extended
basis set uses more number of orbital functions compared to that number of
inner-shell and valence-shell AOs.
The widely
used types of basis sets are represented by definite basis-set symbols such as
STO-3G, 3-21G, 3-21G*, 6-31G, 6-31G** and so on. Thus, STO-3G stands for a
minimal basis set where three Gaussian-type functions (GTF-s) are used,
in the form of a linear combination called a contracted GTF (CGTF), to
reproduce every Slater type orbital (STO) representing an orbital
function. Here an STO means a function of spherical polar coordinates (r, q, f) with an atomic
nucleus as the origin, and is of the atomic-unit form f(r,q,f) = Nf r(n–1) e– z r Ylm(q,f), with z as a parameter, Ylm (q,f) the spherical harmonic function, and Nf the normalisation constant.
On the other hand, a GTF is expressed as a function of the Cartesian coordinates (x, y, z), is centred on the same origin, and is of the atomic-unit form g(x,y,z) = Ng xi yj zk exp {–a(x2 + y2 + z2)}, where i, j, k are non-negative integer parameters, a another parameter and Ng the normalisation constant. For the 3-21G and 6-31G types of basis sets, there are introduced more than one STO-s in linear combination to reproduce a basis function representing the valence shell AOs. The (*) or (**) notations in the above basis-set examples indicate presence of additional d-type (or, respectively, d-type plus p-type) polarisation functions within the basis set, so as to take care of the spherically non-symmetric environment near an atomic nucleus within a molecule. This means that the basis sets that include such polarisation functions result in significantly more accurate computations even though they are computationally more expensive. In particular, d-type polarisation functions must be present in the basis set for a molecular system that includes 3rd or higher row element atoms containing accessible d-orbitals.