# Monthly Archives: January 2010

## Basis sets

In this new post I will address some issues regarding the correct use of the terminology used about **basis sets** in *ab initio* calculations.

One of the keys to achieve good results in *ab initio* calculations is to wisely select a basis set; however this requires some previous insight about the specific model to use, the system (molecule/properties) to be calculated and the computational resources at hand. Most of the basis sets available today remain in our codes due to historical reasons more than to their real practical use. We know the Schrödinger equation is not analytically solvable for an interestingly big enough molecule, so the Hartree-Fock (HF) approach approximates its solution in terms of MOs but these MOs have to be constructed of smaller functions, ideally AOs but even these are constructed as linear combinations of simpler, linearly independent, mutually orthogonal functions which we call **Basis Sets**.

INTRODUCTION

For true beginners: Imagine the 3D vector space as you know it. The position vector corresponding to any point in this space can be deconstructed in three different vectors: R =*a*x+*b*y+*c*z In this case x, y and z would be our basis vectors which comply with the following rules: A) They are linearly independent; none of them can be expressed in terms of the others. B) They are orthogonal; their pairwise scalar product is zero. C) Their pairwise vector product yields the remaining one with its sign defined by the range three tensor *epsilon*. In a vector space with more than three dimensions we can always find a basis which has the same properties described above with which we are able to uniquely define any other vector belonging to this hypothetical space. In the case of Quantum Mechanics we are dealing with *function spaces* (since our entity of interest is the Wavefunction of a quantum system) instead of vector ones, so what we look for are basis functions that allow us to generate any other function belonging to this space.

SLATER TYPE ORBITALS (STO’s)

This is one of those examples that survive for historical reasons. Its value relies on the fact that is a good first start to obtain the properties of small systems. *EXPAND*

*minimal basis*: This term refers to the fact of using a single STO for each occupied atomic orbital present in the molecule.

*double zeta basis*: Here each STO is replaced by two STO’s wich differ in their *zeta *values. This improves the description of each orbital at some computational cost.

SPLIT VALENCE

A single STO is used to describe core orbitals (a minimal core basis set) while two or more are used to describe the valence orbitals.

DIFFUSION AND POLARIZATION FUNCTIONS

PLANE WAVES

A plane wave is a wave of constant frequency whose wavefronts are described as infinite parallel planes. When dealing with -tranlational- symmetrical systems (such as crystals) the total wavefunction can be deconstructed as a combination of plane waves. This kind of basis set is suitable for Periodic Boundary Conditions (PBC) computations if a suitable code is available for it, since plane wave solutions converge slowly. Softwares such as CRYSTAL make use of plane wave solution to find the electronic properties of crystaline solids.

As usual I hope this post is of help. Please rate or comment on this post just to know we are working on the right path!

REFERENCES