Basis Sets

This chapter on basis set covers in detail all important minimal basis sets and extended basis sets such as GTOs, STOs, double zeta, triple zeta, quadruple zeta, split valence, polarized, and diffuse. A basis set is, in fact, a mathematical description of

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Basis Sets

6.1 Introduction A basis set is a mathematical description of orbitals of a system, which is used for approximate theoretical calculation or modeling. It is a set of basic functional building blocks that can be stacked or added to have the features that we need. By “stacking” in mathematics, we mean adding things, possibly after multiplying each of them by its own constant:

ψ = a1 φ1 + a2 φ2 + . . . + ak φk

(6.1)

where k is the size of the basis set, φ1 , φ2 , . . . , φk are the basis functions and a1 , a2 , . . . , ak are the normalization constants. It was John C. Slater (Fig. 6.1) who first turned to orbital computation using basis sets, known as Slater Type Orbitals (STOs). The solution of the Schrödinger equation for the hydrogen atom and other one-electron ions gives atomic orbitals which are a product of a radial function that depend on the distance of the electron from the nucleus and a spherical harmonic, as is illustrated in Table 6.1.

Fig. 6.1 John C. Slater (1900–1976). Slater is recognized for calculating algorithms which describe atomic orbitals. The algorithms became known as Slater Type Orbitals (STOs). Courtesy of “Wikipedia – The Free Encyclopedia – http://en.wikipedia.org/wiki/John_C._Slater”

K. I. Ramachandran et al., Computational Chemistry and Molecular Modeling DOI: 10.1007/978-3-540-77304-7, ©Springer 2008

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6 Basis Sets

Table 6.1 Radial and angular wavefunctions of orbitals. Here, z is the effective nuclear charge for that orbital of the atom, r is the radius in atomic units (1 Bohr radius = 52.9 p.m.), e = 2.71828 (approximately), π = 3.14159 (approximately), n = the principal quantum number, and ρ = (2Zr)/n No.

Orbital

Radial wavefunction

Angular wavefunction

1 2 3 4 5

1s 2s 2p 3s 3p

2 × z3/2 × e−ρ /2 √ ( 2/2) × (2 − ρ ) × z3/2 × e−ρ /2 √ ( 6/2) × ρ × z3/2 × e−ρ /2 √    3/9 × 6 − 6ρ + ρ 2 × z3/2 × e−ρ /2 √ ( 6/9 × ρ (4 − ρ ) × z3/2 × e−ρ /2 )

1 × (π /4)1/2 1 × (π /4)1/2 √ 3 × (x/r) × (π /4)1/2 1 × (π /4)1/2 √ 3 × (x/r) × (π /4)1/2

He pointed out that we could use functions that consisted only of the spherical harmonics and the exponential term. Slater-type orbitals represent the real situation for the electron density in the valence region and beyond, but are not so good nearer to the nucleus. Strictly speaking, atomic orbitals (AOs) are the real solutions of the Hartree-Fock (HF) equations for the atom, i.e., wavefunctions for a single electron in the atom. Anything else is not really an atomic orbital function. Hence these functions are named as “basis functions” or “contractions,” which are more appropriate. Earlier, the STOs were used as basis functions due to their similarity to atomic orbitals of the hydrogen atom. Many calculations over the years have been carried out with STOs, particularly for diatomic molecules. Slater fits linear least-squares to data that could be easily calculated. The general expression for a basis function is given in Eq. 6.2: Basis function, BF = N × e(−α r)

(6.2)

where N is the normalization constant, α is the orb