Optimised Projections for the Ab Initio Simulation of Large and Strongly Correlated Systems

223 Pages·2012·4.25 MB·English

by  David D. O'Regan (auth.)

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David D. O’Regan
Optimised Projections for
the Ab Initio Simulation
of Large and Strongly
Correlated Systems
Doctoral Thesis accepted by
The University of Cambridge, UK
123
Author Supervisor
Dr. DavidD.O’Regan Prof.MikeC. Payne
CavendishLaboratory CavendishLaboratory
TCMGroup TCMGroup
Universityof Cambridge Universityof Cambridge
JJ ThomsonAvenue JJ ThomsonAvenue
Cambridge,CB3 0HE Cambridge,CB3 0HE
UK UK
e-mail: [email protected] e-mail: [email protected]
ISSN 2190-5053 e-ISSN 2190-5061
ISBN 978-3-642-23237-4 e-ISBN978-3-642-23238-1
DOI 10.1007/978-3-642-23238-1
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2011936135
(cid:2)Springer-VerlagBerlinHeidelberg2012
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To my beloved parents
Supervisor’s Foreword
Density functional theory is remarkable. By searching over the single particle
electron density alone, in principle, it provides the exact quantum mechanical
ground state energy of a given system and the corresponding exact ground state
electron density. It achieves this incredible feat via the exact density functional,
which, for any input electron density, outputs the sum of the kinetic, Hartree and
exchange-correlation energies for the ground state many-body wavefunction
which generates this input electron density. As we add the final electrons to a
semiconductor when filling the valence bands, the density functional tells us that
the change in energy on adding each successive electron is almost constant.
However, when we add an extra electron, which has to enter the conduction
band, the exact density functional tells us that the change in energy must differ,
discontinuously, from its previous value. The difference in these energies is, in
fact, the band gap of the semiconductor and exact density functional theory
reproduces its value precisely even though the single particle electronic density
changes by an infinitesimally small amount each time an electron is added to the
system. Additionally, although the contribution to the single particle electron
density from a given set of orbitals might equate to four and a half electrons, for
example, the exact density functional captures the knowledge that electrons
cannot be divided and that this occupancy can only occur because the true many-
body wavefunction is a superposition of configurations, perhaps one which puts
four electrons in this set of orbitals while another represents five electrons. In
situations like this, the exact density functional is capable of precisely deter-
mining the energies associated with this many-body wavefunction.
Unfortunately, we are not as clever as the true density functional. Our crude
approximations of the density functional break some of the physical constraints
on the true many-body wavefunction, with a consequent detrimental effect on the
predicted energies and densities. For example, all of the available density func-
tionals get band gaps wrong, often badly underestimating them, and numerous
approximate functionals quite happily place non-integer numbers of electrons on
particular sites. It is my view that no density functional that we may create will
ever overcome all of these problems simultaneously, although it is often possible
vii
viii Supervisor’sForeword
to overcome one shortcoming using an approximate functional designed for that
purpose. However, while such a functional will, by construction, give better
results for the targeted property, it may then give worse predictions of other
properties than standard functionals and it is thus in no real way closer to the
exact density functional.
Given the complexity of many-body wavefunctions it is, perhaps, remarkable
that available density functionals work as well as they do—often predicting
physical properties to within an accuracy of a few percent. Furthermore, density
functional theory allows us to perform predictive calculations on systems con-
taining many thousands of atoms, while we can only compute many-body wave-
functionsforahandfulofelectrons.Oneapproachtoalleviatetheshortcomingsof
the available density functionals is simply to insert the physics that is missing in
these approximate functionals. For instance, it has now become very common to
add an explicit van der Waals interaction between the atoms as an additional
contributiontothetotalenergyinadensityfunctionaltheorycalculation.Another
widely used approach is the so called DFT ? U method whereby a Hubbard U
interaction is added to reproduce the physics of strongly correlated localised
electronic orbitals. The weakness of previous implementations of DFT ? U, in
which the occupancy of the orbitals is constrained to be an integer number of
electrons, was that the results depended on the choice of the projectors used to
determinetheoccupancyofthelocalisedorbitals.Thisthesispresentsamethodin
which these projectors may be determined self-consistently during the
DFT ? U calculation, thus providing an approach to overcome this weakness in
previous implementations. This approach has been implemented in the linear
scalingdensityfunctionalcode ONETEP andisshowntoretainthelinearscaling
of computational cost with system size. This thesis contains applications of this
technique to bulk nickel oxide, ligated iron porphyrins of biological interest and
the copper phthalocyanine dimer, as well as scaling tests on nickel oxide nano-
clusters containing over 7,000 atoms.
InordertodeveloptheprojectorselfconsistentDFT ? Umethodology,itwas
necessarytomasterthefullmathematicalcomplexitiesoftensorialcalculusinthe
context of electronic structure calculations. This thesis contains a detailed
exposition on the use of nonorthogonal orbitals, the construction of contracted
tensorial invariants, energy minimisation algorithms on curved spaces and the
Christoffel symbol corrections needed to ensure that the density matrix retains its
idempotency, to first order, as the functions in which it is expanded are updated.
This thesis provides a very detailed, yet readable, account of these issues and
could become the standard reference on this topic for the electronic structure
community.
Many technological materials rely on strongly correlated electronic systems
for their functional properties and atoms that host strongly correlated electronic
orbitalsarefound intheactivesites ofmanyproteins.DFT methods have usually
struggledtodescribe such systemsaccurately andthe resultsof DFT ? Ustudies
have fundamentally depended on the set of projectors used in such calculations.
As a result of the work presented in this thesis, we are moved a step closer to the
Supervisor’sForeword ix
accurate and routine description of such systems using first principles quantum
mechanical approaches.
Cambridge, June 2011 Prof. Mike C. Payne
Acknowledgments
Thisdissertationcomes,ostensibly,asaculminationofoneman’slaboursoverthe
past three years. That it is but, as Donne wrote, ‘‘No man is an island, entire of
itself’’ and it is a pleasure take some time here to thank those organisations and
individuals who have contributed to this work and to my life over this period.
My research has been generously supported by the UK Engineering and
Physical Sciences Research Council and the National University of Ireland. The
Cambridge HPCS and, via the UK Car-Parrinello Consortium, the UK National
SupercomputingServiceHECToRhaveprovidedmuchoftherequiredcomputing
resources. Pembroke College has provided travel grants, much pastoral support
and a welcoming home for a good part of my time in Cambridge. My sincere
thanks extends to these organisations for their assistance.
The design of this dissertation is derived from a style package due to Thomas
Fink and Robert Farr, though any inconsistencies in the layout are purely of my
own making.
The Thomas Young Centre at Imperial College London has allowed me to
make frequent visits to the Mostofi group, which has been my academic home
away from Cambridge for the past three years. I would like to warmly thank my
friends at Imperial for their excellent welcome and all that they have taught me.
Onecouldnotaskforafriendlierandmorestimulatingworkenvironmentthan
the TCM group at Cavendish Laboratory; I have very much landed on my feet in
thatsense.ItwasagreatprivilegetoshareanofficewithJamieBlundellandJohn
‘‘Maestro’’Bigginsforthreeyears;Ithinkthatthesupportsharedtheremorethan
outweighed the ample distractions! All members of TCM have enriched my
experience in some way, but I would like to particularly acknowledge Andrew
Morris, Hatem Helal, Alex Silver, Robert Lee, Jonathan Edge, Gareth Griffiths,
Priyanka Seth, Danny Cole, Patricia Silas, Mark Robinson, Sˆian Joyce, Professor
MarkWarnerandProfessorDavidKhmelnitskiifortheirsupportandadvice.Iam
also grateful to the regulars at the TCM DFT meetings for their help and enthu-
siasm.Last,butbynomeansleast,IextendmysincerestthankstoTraceyIngham
and Michael Rutter for unstinting generosity with their time and expertise.
xi
xii Acknowledgments
Much of my efforts have centred around the ONETEP code and it is my
pleasuretoacknowledgeallofthedevelopersandfellowcontributorstothisgreat
work for their patience and professionalism. In particular, I thank Simon
Dubois, Peter Haynes and Chris-Kriton Skylaris, who also kindly proof-read
Chaps. 5 and 6, for stimulating discussions and suggestions. Nicholas Hine
deserves a very special mention and thanks for a great deal of time spent guiding
me; I have learned a huge amount from him in many matters and it is doubtful
whether linear-scaling could be achieved for DFT ? U without his help.
ProfessorCharlesFalco,ProfessorStephenFahyandDr.MichelVandyckhave
been my academic mentors prior to postgraduate study, and without their
invaluable encouragement I might not have commenced this work at all.
Dr. Jonathan Yates and Professor Matteo Cococcioni have helped to direct my
researchviastimulatingdiscussions.Myexaminers,ProfessorNicolaMarzariand
Professor EmilioArtacho, offered someveryhelpfuladvice andcommentsonthe
manuscript. I am very grateful.
The friends I have made in Cambridge have got me through this process and
madeitpossible,we’ve shared manyupsanddowns.IparticularlymentionKatia
Shutova, Michelle Rigozzi, Kelsey Edwardsen, Matt Smith, Taylor Hathaway-
Zepeda,ElizabethDearnley,EmmaFirestone,MatthiasWivel,RobinPayne,Peter
EvanandKrishnaaMahbubanifortakingcareofme,Icannotthankthemenough.
I fear that my exile might be made permanent if I neglected to thank Linda
Mason, Jennifer Lavin, Niall Johansson, Aoife FitzGibbon O’Riordan, David
O’Farrell, Sinéad Rose and David Sheehan for their loyal friendship. I promise I
will try harder to stay in touch. Thank you too to all at Munster Vintage Motor
Cycle and Car Club.
TheproximityofmydearfriendShaneMansfieldhasbeenaverygreatcomfort
to me. Thánamair abhus anso le chéile sa bhád agus, le cúnamh Dé, is sa chaoi
chéanna go bhfillfimíd thar n-ais aríst lá éigin.
My teacher, advisor, critic, counsellor and friend; the game would be lost
completelyifitwerenotfortheunwaveringguidanceandgenerosityofDr.Arash
Mostofi. My obligation to Arash is great, I thank him a thousand times.
I thank my supervisor, Professor Mike Payne, for his excellent advice over
these years, long hours spent proof-reading and straight answers when I needed
themmost.MikehasbeenanenthusiasticadvocateatimportantmomentsandI’m
very grateful indeed.
The love and kindness shown by Florence Paul over these years has truly kept
me going. I hope to be repaying it for a very long time.
Finally,Iwouldliketothankallofmyfamilyfortheirloveandmindingsince
dayone.My newgoddaughter,Isabel, has brightenedupa verycold winterspent
in writing. I would be lost without my wonderful sister, Aoife, and my beloved
Mother and Father, Bernice and John, to whom all of this is dedicated.