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. 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 quantum-mechanical 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) basis--which can either be Pseudo-Atomic Orbitals (PAOs) or B-Spline functions--and 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 eigen-value problem
It can be shown that the hermitian matrices
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)
are used to calculate the Fermi-energy 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
pzhegvxbecause 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
pzhegvxall eigenvectors are then need to be stored. By calling
pzhegvxtwice we can save significant memory by simply accumulating the eigenvectors into the density matrix. It was found2 also that the call to
pzhegvxfor 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 self-consistent 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.