The Gaussian Approximation Potential: An Interatomic Potential Derived from First Principles Quantum Mechanics

96 Pages·2010·3.22 MB·English

by  Albert Bartόk-Pártay (auth.)

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Albert Bartók-Pártay
The Gaussian
Approximation
Potential
An Interatomic Potential
Derived from First Principles
Quantum Mechanics
Doctoral Thesis accepted
by University of Cambridge, UK
123
Author Supervisor
Albert Bartók-Pártay Prof.MikePayne
CavendishLaboratory, TCMGroup CavendishLaboratory,HeadofTCMGroup
Universityof Cambridge Universityof Cambridge
19,J J ThomsonAvenue 19,J J ThomsonAvenue
CB3 0HECambridge,UK CB3 0HECambridge,UK
[email protected] [email protected]
ISSN 2190-5053 e-ISSN2190-5061
ISBN 978-3-642-14066-2 e-ISBN978-3-642-14067-9
DOI 10.1007/978-3-642-14067-9
SpringerHeidelbergDordrechtLondonNewYork
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(cid:2)Springer-VerlagBerlinHeidelberg2010
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Supervisor’s Foreword
Atomistic simulations play an increasingly important role in many branches of
science and, in the future, they will play a crucial role in understanding complex
processes in biology and in developing new materials to confrontchallenges
concerningenergyandsustainability. Themostaccurate simulationsuse quantum
mechanics to determine the interactions between the atoms. However, such
approaches are restricted to moderate numbers of atoms, certainly less than a
million, and, perhaps more importantly, very restricted timescales, generally less
than a nanosecond and more likely of the order of picoseconds for the largest
systems. Given these restrictions, then it is clear that the most complex problems
inmaterialsandbiologywillneverbeaddressedusingquantummechanics alone.
In contrast, atomistic simulations based on empirical interatomic potentials are
computationally inexpensive. They have been used for simulations of systems
containingbillions ofatoms for timescales ofmicroseconds oreven longer. Good
empirical interatomic potentials exist for metals and for ionic materials but even
these fail to describe all atomic environments with equal accuracy. Creating
accurate interatomic potentials for covalent materials has proved to be extremely
difficult.
Thegenerationofinteratomicpotentialshas,historically,beenanartratherthan
an exact science. Furthermore, all interatomic potentials have limited regions of
applicability though it has not been possible to quantify these regions in a useful
way in order to prevent simulations performed with such potentials producing
unphysical results. The new scheme for creating interatomic potentials described
inthisthesisgeneratesaccurateforcesforanynewatomicconfigurationbasedona
database of configurations and corresponding forces and energies gathered from
any source although, realistically, the most useful source is likely to be quantum
mechanical calculations. The new scheme does notfit an explicitly parameterised
model of the interatomic potential but instead uses a Gaussian Process to deter-
mine the forces on the atoms. Indeed, one of the most interesting conclusions of
v
vi Supervisor’sForeword
thisapproachisthattheuseofparameterisedmodelsforinteratomicpotentialsis,
itself,responsibleformostofthesevereproblemsencounteredinthedevelopment
ofinteratomicpotentials.Thisnewapproachtodescribinginteratomicinteractions
is called the Gaussian Approximation Potential.
The key mathematical advance that made this approach possible was the
description of the atomic neighbourhood in terms of the elements of the bispec-
trum. The bispectrum provides a very efficient description of the local atomic
structure that takes account of the symmetries of the system, namely rotational,
permutational and translational symmetries. Hence the bispectrum is invariant
under rotations, translations and permutations of the sets atoms surrounding any
particularatom.Thebispectrumisaveryeffectivedescriptoroflocalstructureand
it will be interesting to see whether it will be applied to other problems, such as
phasetransitions,inthefuture.Importantly,theGaussianProcessalsoprovidesan
estimateoftheerrorsintheforcessosimulationsbasedonthispotentialcanavoid
the problem of generating spurious unphysical results due to errors in the inter-
atomicpotential.Thissituationwouldonlybeencounteredifthesimulationvisited
atomic configurations which were distant from the ones used to generate the
originalpotential.Oneoftheattractivefeaturesofthenewapproachisthat,atthis
point, a new potential could be generated incorporating these new atomic con-
figurations and the simulation could be continuedusing this new potential. This
procedurewouldnotreducetheaccuracyofthepotentialforatomicconfigurations
closetothoseintheoriginalfittingset.Thisisnotthecasewhenreparameterising
conventional interatomic potentials. Thus, Gaussian Approximation Potentials
maintainaccuracyastheirtransferabilitytonewatomicenvironmentsisenhanced.
In contrast, standard parameterised interatomic potentials always involve a
compromise between accuracy and transferability.
These new potentials will also solve a number of the technical problems that
presently exist in hybrid or QM/MM modelling schemes in which one or more
partsofasystemaredescribedusingquantummechanicsandtheremainderofthe
systemisdescribedusingsimpleanalyticalempiricalpotentials.Thepropertiesof
the regions described using these new potentials would accurately match the
properties of the regions described quantum mechanically thus reducing the
problems introduced by the mismatch of, say, elastic properties across the inter-
face.Furthermore,bycontinuallyaddingquantummechanicaldatatothetraining
set for the potential, the Gaussian Approximation Potential could replace explicit
quantummechanical calculations for the quantumregions in such simulations yet
maintaintheaccuracyofexplicitquantummechanicalcalculations.Thisapproach
would have a profound effect of significantly increasing the timescales that are
accessibletoQM/MMsimulations.Asaresultoftheworkdescribedinthethesis,
there is now a standard procedure to determine accurate forces to use in any
atomistic simulations. The Gaussian Approximation Potentials presented in the
thesis for gallium nitride and for iron were developed in a few days (which
Supervisor’sForeword vii
included the generation of the input quantum mechanical data), yet are more
accuratethanexistingpotentialsthatweredevelopedoveramuchlongerperiodof
time.Inthefuture,IwouldexpectthatthewholeprocessofgeneratingGAPcould
be automated to allow any user to apply the method to any system of interest.
Cambridge, May 2010 Mike Payne
Acknowledgements
I am most grateful to my supervisor, Gábor Csányi, for his advice, help and
discussions,whichoftenhappenedoutsideofficehours.Iwouldalsoliketothank
Mike Payne for giving the opportunity to carry out my research in the Theory of
CondensedMatterGroup.ThanksareduetoRisiKondor,whoseadviceonGroup
Theory was invaluable. I am especially thankful to Edward Snelson, who gave
useful advice on the details of machine learning algorithms. Further thanks are
due to my second supervisor, Mark Warner.
I am indebted to the members of Gábor Csány0s research group: Lívia Bartók-
Pártay, Noam Bernstein, James Kermode, Anthony Leung, Wojciech Szlachta,
Csilla Várnai and Steve Winfield for useful discussions and inspiration at our
groupmeetings.IamgratefultomypeersinTCM,inparticular,HatemHelaland
Mikhail Kibalchenko, from whom I received support and company.
Many thanks to Michael Rutter for his advice on computer-related matters.
Thanks also to Tracey Ingham for helping with the administration.
I would like to thank my family for their patience and my wife, Lívia Bartók-
Pártay for her love and encouragement.
Cambridge, May 2010 AlbertBartók-Pártay
ix
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Outline of the Thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Representation of Atomic Environments. . . . . . . . . . . . . . . . . . . . . 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Translational Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 Spectra of Signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.2 Bispectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.3 Bispectrum of Crystals . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Rotationally Invariant Features . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.1 Bond-Order Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.2 Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.3 Bispectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.4 4-Dimensional Bispectrum. . . . . . . . . . . . . . . . . . . . . . . 17
2.3.5 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Gaussian Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Function Inference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.1 Covariance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.2 Hyperparameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.3 Predicting Derivatives and Using Derivative Observations. 28
3.2.4 Linear Combination of Function Values. . . . . . . . . . . . . . 29
3.2.5 Sparsification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
xi