Lecture: ab initio Methods Basis Functions

Review of the approximations for ab initio calculations,

  1. Neglect of Relativistic Effects
  2. Born-Oppenheimer
  3. Neglect of electron correlation
  4. Finite Basis Set
In this lecture we wll discuss the finite basis set approximation.

The Roothan equations.

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;

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where tex2html_wrap_inline46 are the basis functions and tex2html_wrap_inline48 are the expansion coefficients.

This set of algebraic equations can be expressed in matrix form,

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where F is called the Fock operator, S is the overlap matrix, C is the expansion coefficient vector tex2html_wrap_inline48 and tex2html_wrap_inline54 is the energy. To solve this equation, transform to a standard eigenvalue problem, solve, and then transform back.

Basis Sets.

Basis set functions commonly used are, G aussian Type Functions or
Gaussian Type Orbitals (GTF or GTO). These are written as,

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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 --

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where

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tex2html_wrap_inline76 is referred to as a primitive, and tex2html_wrap_inline78 is called the contracted orbital or basis function. The angular momentum of the shell is the sum of the exponents tex2html_wrap_inline80. The tex2html_wrap_inline82 are the contraction coefficients.

Basis functions are made not born!

Method I - Fitting

Drawback - There can be some loss of information, based on the use of too few parameters.

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

STO-nG
Slater Type Orbitals expanded as n gaussians (principally cartesian type gaussians).
Split-Valence
For example: 4-31G 4 functions for 1s, 3 gaussians for 2p shell and 1 gaussian for the tex2html_wrap_inline100 shell.
Better descriptions 6-31G.
More degrees of freedom for valence region than, for example, the 4-31G basis sets.
Add polarization functions (6-31G*).
Add d and f functions to account for polarization.
Add diffuse functions (6-31+G*).
Add d and f functions to H as well for diffusivity.



Author: Ken Flurchick