Roothan's contribution is to use a set of basis functions to expand the molecular wave function in terms of a set of basis functions to recast the integro-differential equations into a set of algebraic equations. These basis functions must span Hilbert space and be physically adequate. Thus the wave function looks like;
where
are the basis functions and
are the expansion coefficients.
This set of algebraic equations can be expressed in matrix form,
where F is called the Fock operator, S is the overlap
matrix, C is the expansion coefficient vector
and
is the energy. To solve this equation, transform to a standard
eigenvalue problem, solve, and then transform back.
Basis set functions commonly used are,
G aussian Type Functions
or
Gaussian Type Orbitals
(GTF or GTO). These are written as,
where the k's are the paramemters for the respective coordinate functions and a is the exponential parameter.
Also known as cartesian gaussians.
Notation;
Sum of gaussians to describe an orbital --
where
is referred to as a primitive, and
is called the contracted orbital or basis function. The angular
momentum of the shell is the sum of the exponents
.
The
are the contraction coefficients.
Basis functions are made not born!
Method I - Fitting
For example, this method gives the STO-nG basis set.
Method II - Minimizing the energy
generally known as
LEMAO - Least Energy Minimized
Atomic Orbitals
Construct a basis set that looks like STO-nG and optimize all exponents and contraction coefficients.
This includes the split valence type as well
Split valence - allows the valence region added degrees of freedom to respond to the molecular environment. Break the valence part into more than one region, each described by a different function. This allows for more flexibility in the valence region.
Example basis sets