Brian C. Hoffman: SCF Theory

Basic Principles and
Hartree-Fock Theory

Basis Sets

The previous section reduces the Hartree-Fock equations to a set of matrix equations which contain matrix elements of the one-electron Hamiltonian and $\frac{1}{{r_1}_2}$ with elements of the basis set, mathematical functions which are chosen for their utility and how well they model the system. The matrix elements with the basis set can be evaluated either analytically or numerically and are the last level of abstraction. Two general types of basis functions are in common use, Slater[17] and Gaussian [18] functions. Slater type orbitals (STOs) are of the general form

$\begin{displaymath}r^L e^{-\zeta r} \; \; \; r = {[X^2+Y^2+Z^2]}^{\frac{1}{2}} \end{displaymath}$

(71)

and are chosen because they correctly describe the electron density of a molecule, having the correct cusp behavior near the nucleus and the correct fall-off behavior far away from the nucleus. In equation (1.71) L corresponds to the angular momentum quantum number of the orbital and $\zeta$ reflects the spatial extent of the orbital. Unfortunately, STOs are difficult to integrate because it is not possible to separate out the X, Y, and Z components, and two-electron integrals may contain STOs centered on as many as four different atoms. Therefore, in the majority of ab initio methods an STO is approximated by either a single Gaussian type orbital (GTO) or a linear combination of GTOs:

$\displaystyle {\psi}_{GTO}$	=	$\displaystyle X^a Y^b Z^c e^{-\alpha r^2}$	(72)
$\displaystyle {\psi}_{STO}$	=	$\displaystyle \sum_{r=1}^{n} {\psi}_{{GTO}_r} dV.$	(73)

In the expression for the GTO, the orbital exponent $\alpha$ defines the spatial extent of the orbital; whereas, the sum of the exponents a, b, and c represents the angular momentum. Gaussian functions do not have a cusp at the nucleus and do not fall-off correctly far from the nucleus, but they have the advantage of being much easier to use. In addition to being easier than STOs to integrate straight out, Gaussian orbitals have the beneficial property that the product of two Gaussian functions is a third Gaussian function centered between the original two. Known as the Gaussian Product Theorem, this property allows two-electron integrals to be written with respect to at most two centers.

In addition to varying in the form of the functions comprising them, basis sets differ in the number of functions used to represent the atomic orbitals. In a minimum basis set each occupied set of atomic orbitals is represented by only a single Slater function or its Gaussian counterpart. For example, B, C, N, O, and Ne atoms would all be represented by 2 S-type functions and a set of 3 P-type functions. An STO-NG basis set is a minimal basis set in which STOs are represented by N GTOs. In order to provide a better description of a molecule, more expansive basis sets have been built using the STO-NG basis sets as a starting point. For example, basis sets designated by a-bcG, where each lower case letter indicates an integer, would model the core atomic orbitals using an STO-aG basis, inner valence orbitals using an STO-bG basis, and outer valence orbitals using an STO-cG basis. Another common method is to use basis sets which describe atomic orbitals by representing each with two or three distinct Slater functions or their Gaussian counterparts. The later kinds of basis sets are called double and triple $\zeta$ basis sets, respectively. Lastly, basis sets may include functions with higher angular momentum to account for distortions, such as polarization, caused by the interactions of neighboring atoms.

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