The approximate charge density is expanded in a set of basis functions [38,39],

(19) |

We can now write an approximation to the Hartree energy in
Equation 2.4.15. This is always lower than *E*_{H}, but tends to
*E*_{H} as tends to and so provides a
way of estimating the quality of the basis function fit,

(20) |

Then we can select *c*_{k} in Eq. 3.2.7 to minimise the error

(21) |

If this is now differentiated with respect to *c*_{k} in order to
determine the minimum we get an important expression,

Evaluation of these `*t* matrices' is often the slowest step in these
cluster calculations, so obviously a careful choice of fitting
function, *g*_{k}, is required. Simple Gaussian functions have the
advantage of being analytically soluble, but if there are roughly as
many *g*_{k} as basis functions (commonly the case), this means
the *t* matrices are still *O*(*N ^{3}*) integrals. Nonetheless, this is
the way all the O in Si work was performed. However it is possible to
split the

The first of these gives a Gaussian potential,

(22) |

making *t*_{ijk} a simple product of three Gaussians which
is quick to calculate. However the form of these expressions means
that their integral vanishes, and hence they do not contribute to the
total charge density. Therefore the second set of expressions for
*g*_{k} are introduced (simple Gaussians) to ensure the integrated
charge density gives the correct number of electrons.

This formalism was adopted for the III-V work in Chapter 4,
however when testing different fitting functions for small Si-O based
molecules such as disiloxane, (SiH_{3})_{2}O, we found that a much
better fit was obtained using simple Gaussian functions. Therefore
these have been used for all the Si work despite the loss in computing
speed.

The exchange-correlation is similarly approximated as

(23) |

However the expression used to minimise the error in for
*E*_{H} (Equation 3.2.9) was integrated over all and does not necessarily minimise the error at any specific value of
. Therefore it is necessary to choose a new approximation
for and a good choice is a sum of Gaussians,

(24) |

with *d*_{k,s} obtained by minimising

(25) |

If this is differentiated with respect to *d*_{k,s} we obtain

The integrals are the same for both *s*, and if *g*_{k} is set to the
first form in Equation 3.2.13 then *u*_{ijk} are just
proportional to *t*_{ijk} which makes calculation extremely quick.

- Spin averaged and improvements to the approximations
- Spin polarised exchange correlation
- Summary so far
- Matrix Formalism