
The CONQUEST implementation of matrices do not regard any system (bulk or otherwise) to have periodic boundary conditions, instead the code treats any location in real space as it is[5]. The cell from user input is regarded as the Fundamental Simulation Cell (FSC), and the FSC is repeated in all lattice directions so that all atoms taking part of interaction with that inside the FSC are taken account of (see figure 9). All quantummechanical operators are represented by matrices using support functions (which for the purpose of this report may be regarded as a set of basis functions upon which we have our matrix representation). For details on the meaning of support functions, how these are formed from the (actual) basiswhich can either be PseudoAtomic Orbitals (PAOs) or BSpline functionsand how are the quantum mechanical quantities represented by these support functions, please refer to [4,5].

The CONQUEST method of storing matrices of the form
To update the electron density CONQUEST solves the
generalised eigenvalue problem
It can be shown[3] that the hermitian matrices
such as
and
defined in equation
(16) and (17) can be calculated from the native
CONQUEST matrices using the relationship
A call to ScaLAPACK subroutine pzhegvx
is made to obtain the
set of eigenvalues (the band structure)
which
are used to calculate the Fermienergy and occupation function (this
is discussed briefly in section 3). Once this
is done another call to pzhegvx
is made to get the eigenvectors
for each point, the new electronic density can than be
calculated using formula[3]
pzhegvx
because we cannot calculate the density without knowing
the occupation function first, but on the other hand since the band
structure needs eigenvalues calculated for all if we are
going to make only one call to pzhegvx
all eigenvectors are
then need to be stored. By calling pzhegvx
twice we can save
significant memory by simply accumulating the eigenvectors into the
density matrix. It was found^{2} also that
the call to pzhegvx
for only calculating eigenvalues is only
about 10% the cost of the full call that also calculates the
eigenvectors.
The original CONQUEST implementation solves equation (15) one point at a time. And the matrices and are then mapped onto new matrices and distributed across all available processors arranged in a BLACS processor grid according to ScaLAPACK cyclic block format. The calculated eigenvectors from ScaLAPACK for each are then transfered from ScaLAPACK data format and accumulated into stored across the processors in CONQUEST format, and the selfconsistent calculation carries on from there.
We note that calculations involved for solving eigenvectors for different are independent from each other. If we could add a degree of freedom of allowing subgroups of processors working on different then it would allow us to choose better optimised parameters for the ScaLAPACK calculations. For matrices of a given size there is an optimal number of processors that should be allowed to work on it, and too many processors means inefficient communications taking over. Hence parallelising calculation in would in theory allow one to use more processors more efficiently by having groups of optimal number of processors working for each ScaLAPACK subroutine call. Since for metallic calculations, te number of points required are in the order of 1000s, this is a real degree of freedom we can exploit, especially for calculations running on HPC systems such as HECToR.