Thesis
On Linear-Scaling Methods for Quantum Mechanical
First-Principles Calculations
Arash A. Mostofi
CHRIST'S COLLEGE
CAMBRIDGE
A dissertation
submitted for the degree
of Doctor of Philosophy at the
University of Cambridge
2003
Abstract
Quantum mechanical modelling is invaluable for
understanding the structure of matter, permitting investigations that
it is impossible to perform experimentally.
First-principles quantum mechanical modelling
is especially useful as it requires no prior assumptions regarding the
properties of the system under investigation, only specification
of the atoms present through their atomic numbers.
Consequently, these calculations are particularly reliable as
they are parameter-free and preconceptions about the final result
cannot bias the outcome.
A rigorous theoretical framework to solve quantum mechanical
equations is provided by density-functional theory,
which, in conjunction with the pseudopotential approximation and
physical insight into the exchange and correlation effects of the
interacting electrons, has become the paradigm for modern
first-principles calculations.
Even with this method, though, the computational effort required
increases asymptotically with the third power of the system size and,
despite the relentless progress of computer technology, this scaling
places a limit on the scientific problems that can be tackled with
this approach.
As a result, there has been much recent interest in the subject of
this dissertation: linear-scaling methods, whose
computational cost scales only linearly with the size of the
system. Such schemes are essential if calculations on systems such as
large biological molecules and other nanostructures, which are beyond
the reach of current methods, are to become routine.
In this work, a new set of localised basis functions for representing
the density matrix in real space is introduced. These
functions have a number of desirable properties: they are
closely related to the plane wave basis; basis set completeness is
controlled by a single parameter; they naturally obey periodic
boundary conditions; and they are orthogonal.
The implementation of this basis set in a minimisation method for
calculating total ground-state energies of arbitrary systems is
described.
Linear-scaling is achieved using a novel "fast Fourier transform
box" approach, which exploits localisation of the orbitals in real
space.
The issue of length-scale ill-conditioning, which causes unacceptably
slow convergence in methods that use large systematic basis sets, is
addressed. A general preconditioning scheme to overcome this
ill-conditioning is presented and its implementation is described.
With this approach, the rate of convergence to the ground-state
solution is improved by an order of magnitude.
Results of the linear-scaling method are presented. Convergence of the
total energy is investigated with respect to the size of the fast
Fourier transform box, the range of the density matrix, and the energy
cut-off of the basis. Physical properties such as charge densities,
equilibrium bond lengths and vibrational frequencies are compared
against results from conventional plane wave calculations.
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