Understanding and Improving the Efficiency of Full Configuration Interaction Quantum Monte Carlo

0.25 MB·

by  W. A. Vigor

Understanding and Improving the Efficiency of Full Configuration Interaction
Quantum Monte Carlo
W. A. Vigor,1 J. S. Spencer,2,3 M. J. Bearpark,1 and A. J. W. Thom1,4
1)Department of Chemistry, Imperial College London, Exhibition Road, London, SW7 2AZ,
United Kingdom
2)Department of Physics, Imperial College London, Exhibition Road, London, SW7 2AZ,
United Kingdom
3)Department of Materials, Imperial College London, Exhibition Road, London, SW7 2AZ,
6 United Kingdom
1 4)University Chemical Laboratory, Lensfield Road, Cambridge, CB2 1EW, United Kingdom
0
2 (Dated: 15 March 2016)

r WithinFullConfigurationInteractionQuantumMonteCarlo,weinvestigatehowthestatisticalerrorbehaves
a
asafunctionoftheparameterswhichcontrolthestochasticsampling. Wedefinetheinefficiencyasameasure
M
ofthestatisticalerrorperparticlesamplingthespaceandpertimestepandshowthereisasizeableparameter
regime where this is minimised. We find that this inefficiency increases sublinearly with Hilbert space size
4
1 and can be reduced by localising the canonical Hartree–Fock molecular orbitals, suggesting that the choice
of basis impacts the method beyond that of the sign problem.

]
h
p
I. INTRODUCTION suredby computationalcost(CPUtime andmemoryre-
-
m quirements)necessaryforconvergencetoagivenstochas-
tic error. The FCIQMC algorithm is not a ‘black box’
e TheFCIQMCmethod1,2 isastochasticalgorithmthat
h hasenabledcalculationofthe groundstateenergyofthe method and the manner in which sampling is performed
c largest molecules to date3 to FCI accuracy, i.e. exact canimpactthe computationalresourcesused whilst per-
. forming a calculation and the resultant statistical error
s givenafixedbasisset. Ithasalsofoundsuccesscalculat-
c bar. We believe that understanding the efficiency of a
ing the ground state energy of model Hamiltonians such
i FCIQMC calculation is important for two main reasons.
s astheuniformelectrongas4andthethreebandHubbard
y Firstlysothatonecanuseacomputationalbudgetasef-
model5, as well as solid state systems.6 Recent develop-
h fectively aspossible,ideally leadingto the ability to per-
p ments meanthat propertiesbeyondthe groundstate en-
form FCIQMC calculations automatically and efficiently
[ ergy, such as forces7 and excited state energies8–10 can
without careful tuning of input parameters. Secondly
also be calculated. Further, efficient stochastic imple-
2 it helps compare the effectiveness of different algorithms
mentationsofothermethodssuchassecondorderMøller
v based upon and additional approximationsto FCIQMC.
Plesset Perturbation theory (MP2)11–13, Density Func-
5
6 tional Theory (DFT)14 and Coupled Cluster theory15
8 have been developed, forming a new field of stochastic
0 computational chemistry. These Monte Carlo methods
Previous studies of the efficiency of FCIQMC investi-
0 differ from the more traditional Diffusion Monte Carlo
1. method16, which requires the use of a fixed node ap- gaantdedfotchuesesdtaonndacordmeprarroinrgasvearyfudnicffteiorennotfaclgoomrpituhtmers.t2i1m,2e2
0 proximationto preventcollapse of the wavefunctionto a
Here we take a different tack, running many empirical
6 bosonic state. The eventual hope is that these methods
1 calculations to understand how the stochastic sampling
willallowthepropertiesandreactivityoflargemolecules
: of the wavefunctionaffects the errorbar for FCIQMC in
v to be investigated to unprecedented accuracy.
general. Once we understand this effect, we form a met-
i
X Quantum Monte Carlo simulations are plaguedby the ricwhichisindependentoftheparameterchoicesanduse
fermion sign problem: the quantity being sampled can
r it to measure how the error bar scales with system size.
a beeitherpositiveornegativewhichmaycausethecalcu-
lationtoconvergetothewronganswer.17,18 InFCIQMC
the sign problem appears as the critical population of
particles required to correctly sample the ground state We begin with an overview of the FCIQMC method.
wavefunction.18 Whilst the scaling of this with system We then explore the dependence of the statistical error
size has been investigated in a number of systems18–20, bar on the total population and the timestep in Item 4
the effect of it on the stochastic error has not been and on the system size, both basis set and number of
well studied. The (systematically improvable) initiator electrons,in Fig. 2. Finally, we discuss in Fig.3 how the
approximation2 can reduce the critical population by total number of particles (relative to the plateau height)
many orders of magnitude. However, the impact of this affects the stochastic error bar and draw some conclu-
keyadvanceonthestochasticerrorhassimilarlynotbeen sions about how to choose parameters to maximise the
thoroughly investigated. efficiency of the FCIQMC algorithm ‘a priori’ from the
The efficiency of a stochastic algorithm can be mea- plateau height.
2
II. A RECAP OF THE FCIQMC METHOD where n (τ) is the (signed) number of psips on determi-
i
nant i in ψ(τ) and ... represents the time average.
τ
TheFCIQMCalgorithm1,2 canbeviewedasastochas- The ini|tiatoriapprohximiation2 can dramatically reduce
tic power method. On every iteration the representation the sign problem in FCIQMC.6,7,9,20,22,26–28 In initiator
of the wavefunction, ψ(τ), at imaginary time τ is up- FCIQMC (iFCIQMC), previously unoccupied determi-
dated by sampling the action of the operator: nants can only be spawned onto from determinants with
a population above a certain threshold; a threshold of 3
ψ(τ +δτ)= 1 (Hˆ S)δτ ψ(τ) (1) is typical and used throughout this paper. The initiator
(cid:16) (cid:17)
− − approximationintroduces a systematic error in the sam-
pling of Eq. (1), which is evident in both estimates of
whereHˆ istheHamiltonian,S isanoffsetusedtocontrol the energy, with unbiased sampling restored in the limit
normalisation, and δτ is the timestep. As long as ψ(0) of an infinite population of psips. A monotonic conver-
hasanon-zerooverlapwithit,thegroundstatewillbeall gence to the FCI energy as a function of N (τ) has
p
that remains once τ is large enough, assuming δτ is suf- been observed for many simple systems,2 thhough tihis is
ficiently small18 and S is carefully controlled. Di ψ(τ) not universally the case.
h | i
is represented by a number of (signed) unit weights lo-
catedon D ;wecallasingleunitapsi-particleorpsip.23
i
Othercho|iceisofdiscretisationarepossible16;forthepur- III. SYSTEMS STUDIED
poses of this work we consider the simplest case.
Eq. (1) is sampled in three stages to make up a single
In this paper we study the following systems:
timestep of δτ1:
1. AnisolatedneonatominthefollowingDunningba-
Spawning: each psip (with weight wi) attempts to sis sets29: cc-pVDZ, aug-cc-pVDZ, cc-pVTZ, aug-
spawn a child psip on a randomly selected cc-pVTZ, cc-pVQZ, aug-cc-pVQZ.
D withprobability D Hˆ D δτ/p(j i),where
j i j
|p(jii) is the probab|ihlity| th|ati|D is| selected 2. The hydrogenfluoride,HF,molecule ina cc-pVDZ
j
give|n the parent psip is on Di| aind with sign basis at bond lengths of R = R0, R = 1.5R0 and
sgn( D Hˆ D w ). | i R=2R0,whereR0 =0.91622˚A,the Hartree–Fock
i j i
−h | | i equilibrium bond length in this basis.
Death: each psip dies (is removed from the simulation)
3. Chainsofbetween2and7heliumatomsatintervals
withprobability K δτ,whereK = D Hˆ D S,
| | h i| | ii− of 3˚A in a 6-31G basis set.30
if K < 0; for K > 0, instead, a copy of the psip is
made with probability Kδτ. 4. Chains of between 5 and 7 helium atoms at inter-
valsof3˚Aina6-31Gbasissetwithlocalisedmolec-
Annihilation: psips on the same determinant with op- ular orbitals. We used Pipek–Mezey localisation31
posite signs cancel.
tolocalisetheoccupiedHartree–Fockmolecularor-
bitals.
Initially S is set to the Hartree–Fock energy, which is
larger(lessnegative)thantheFCIenergyandthiscauses Hartree–Fock calculations were performed using Q-
the populationNp(τ) (totalnumber ofpsips)tobeginto Chem32, with local modifications to obtain the re-
grow exponentially. If there is a sign problem24 Np(τ) quired integrals. FCIQMC calculations were performed
will plateau as the sign structure is projected out. It and analysed using HANDE33 and figures plotted using
is only after this point that ψ(τ) becomes a stochastic matplotlib.34 Unless otherwise stated, atomic units are
representation of the wavefunction and Np(τ) grows ex- used throughout. Raw and analysed data and analysis
ponentially again.18 To counter this exponential growth scripts are availale at Ref. 35.
S is periodically adjusted every A timesteps according
to:
IV. POPULATION DYNAMICS AND THE
N (τ +Aδτ)
S(τ +Aδτ)=S(0) ξlog p , (2) STOCHASTIC ERROR
− N
s
If the Hamiltonian doesn’t have a ‘sign problem’ in
whereS(0)is the initialvalueofthe shift, N is the pop-
s
FCIQMC, it is impossible for psips of opposite signs to
ulation at the end of the equilibration phase and ξ, the
damping factor, is usually fixed during a simulation.25 be generated on the same determinant and so annihila-
tion cannot occur. In this case each psip samples the
In addition to the shift, the energy of the system can
wavefunction independently from each other and hence
beestimatedviatheprojectedestimator,whichtypically
the error in the estimate of the energy, σ , must behave
has a smaller statistical noise: E
as:
EProj = hDhD0|0H|ˆee−−HˆHˆττ|D|D00ii = hPi6=h0nH0i0τiniiτ, (3) σE = phNpaiNδτ, (4)
3
where a is a constant of proportionality and N is the Thebehaviourofawith N inFCIQMCsimulations
p
h i
numberoftimestepsfromwhichσ isestimated. Eq.(4) is shown in Fig. 1 for hydrogen fluoride. This behaviour
E
arisesbecausethecontributiontotheestimateofE from has been seen in a wide range of systems, which are
eachpsipcanbecombinedinasimilarwaytothecontri- included in the Supplemental Information.40 Owing to
bution for running from multiple timesteps. Increasing the sign problem, only after the population reaches the
δτ simply increases the probability of spawning/death plateaudoesthe vectorofpsipsbecomeastochasticrep-
and so decreases σ in the same way as increasing N or resentation of the eigenvector.18 Populations below the
E
N . plateau could either have a divergent σ or an incor-
p E
h i
a can be estimated from a FCIQMC calculation using rect average correlation energy with a finite σ ; we see
E
σ estimatedbyblockinganalysisoftheprojectedenergy the former behaviour. For populations greater than the
E
and N . Forconsistencythroughoutthispaper,weuse plateau, a decays as a function of N and tends to a
h pi h pi
an automatic approach to calculate the optimal block finite constant in the large population limit.
length36, which has previously been applied to DMC37 Iftheinitiatorapproximationisused,thenwefindthat
and is implemented in pyblock.38 aisaconstantasafunctionof N forpopulationsmuch
p
h i
For systems with a ‘sign problem’, which is the usual smaller than the plateauand, for a fixedδτ, the same as
case, a will depend on hNpi and δτ. We will call a in FCIQMC in the large hNpi limit. We call this limit
a( N ,δτ) the inefficiency and note from Eq. (4) that on a the iFCIQMC limit. For example in Fig. 1 ineffi-
p
h i
a smaller a implies a more efficient calculation. In Secs. ciency in FCIQMC doesn’t hit the iFCIQMC limit until
IVAandIVBweinvestigatethedependenceofaonhNpi abouthNpi=2×106,approximately3timesthe plateau
and δτ respectively. height. Thisisanimportantpoint: ifthelargestpopula-
tion affordable in FCIQMC is not sufficient to reach the
iFCIQMC limit (but is sufficient to exceed the plateau),
then the initiator approximation is still useful as it pro-
A. Effect of population on inefficiency of FCIQMC
simulations videsasignificantreductioninthestochasticerrorforthe
same computational cost. It may, however, be difficult
to quantify if the introduction of an initiator error is a
price worthpayingfor a potentially significantreduction
in statistical error bar.
1.4 HF cc-pVDZ R0 FCIQMC This behaviour has implications about how best to
HF cc-pVDZ R0 iFCIQMC run parallel FCIQMC and iFCIQMC simulations given
1.2 a fixed amount of computational resources.
In the canonical parallel FCIQMC implementation41
1.0
the Hilbert space is partitioned over the processors, re-
sulting in efficient distribution of the memory demands
u. 0.8
a. across the processors. Psips spawned from a parent de-
a / terminant in one part of the Hilbert space onto a child
0.6
determinant in another part of the space are communi-
catedbetweenprocessors. Itisthis communicationover-
0.4
head that limits the parallel scalability of FCIQMC.
0.2 The simplest way to parallelise a Monte-Carlo al-
gorithm is to run independent calculations and com-
0.0 bine statistics gathered in each calculation. The overall
stochasticerrorscalesas1/√N ,whereN isthenumber
106 107 I I
of independent calculations. Therefore once the popula-
Np tionissuchthatareachestheiFCIQMClimit, itismore
› fi
efficient to use multiple independent simulations to min-
imise interprocess communication.
FIG.1. Theinefficiencyaasafunctionoftheaveragenumber
of psips for HF in the cc-pVDZ basis, R = 0.91622˚A, and
0
δτ =0.00175 for both FCIQMC and iFCIQMC. The plateau
B. Effect of timestep on inefficiency of FCIQMC
heightis∼6.5×105 (measuredusingthemethoddescribedin
simulations
39) and is indicated by the vertical line. The horizontal line
showsafittotheiFCIQMCdataforconstanta. Theinitiator
erroris converged belowthestochastic error inall iFCIQMC a is approximately constant for a sufficiently small
calculations shown; a may not remain constant where this is timestep,δτ,andotherwiseaincreaseswithδτ;thevalue
nottrue. ForapopulationofhNpi=50000iniFCIQMC(not of δτ after which a is non-constant is system dependent.
shown), we find a = 0.318(18) compared to a = 0.2682(34) We have seen this behaviour for all systems investigated
from the fit, showing that sufficiently small populations do in this paper (see Supplemental Information40) and for
havean effect on theinefficiency in iFCIQMC calculations. FCIQMCandiFCIQMC.Fig.2showsthisforanisolated
4
1.2 600000
Ne cc-pVDZ FCIQMC
Ne aug-cc-pVDZ FCIQMC
1.0 Ne aug-cc-pVDZ iFCIQMC 500000
0.8 400000
u.
a / a. 0.6 Np 300000
0.4
200000
δτ=0.0006 δτ=0.005
0.2 δτ=0.001 δτ=0.007
100000
δτ=0.002 δτ=0.009
δτ=0.003 δτ=0.01
0.0
0
10-3 10-2
0 5 10 15 20 25 30 35 40
δτ
τ
FIG.2. Left: Theinefficiencyaasafunctionofδτ forNeincc-pVDZ(hNpi≈46100)andaug-cc-pVDZbases(hNpi≈590000).
Once δτ is large enough to raise the plateau height above hNpi, a diverges in a similar way as in Eq. (4). Right: The plateau
height as a function of the timestep for FCIQMC on Ne in aug-cc-pVDZ; note the region in the plateau is roughly minimum
matches thesimilar region for a.
neon atom in cc-pVDZ and aug-cc-pVDZ bases. define a metric that lets us investigate the scaling of the
A previous investigation42 has shownthat the plateau error bar as we change system. The simplest such met-
height exhibits a similar behaviour to a as a function ric is to find the minimum value of a, a , i.e. when
min
of δτ in FCIQMC calculations. As shown in Fig. 2 for it is a constant as a function of N and δτ. This re-
p
the Ne atom(andin the SupplementalInformation40 for quirescalculations with δτ <δτ handieither the initiator
0
other systems) the plateau height appears to be a good approximation (with the initiator error converged) or a
metric for when a remains constant as a function of δτ. large N . Thebehaviourofa asafunctionofHilbert
p min
h i
This is useful as the plateau height is easier and cheaper spacesizeisshowninFig.3forallsystemsstudiedinthis
to measure than statistical accumulation of the energy work: theneonatom,hydrogenfluorideatdifferentbond
andhenceevaluationofa. Wedenotethelargestδτ such lengths and chains of helium atoms in the canonicaland
that a is constant as δτ . localised basis sets. We see a sublinear relationship be-
0
In the Supplemental Information40 we investigate tween the scaling of a with size of Hilbert space for
min
whether N has an impact on δτ and find this is not the same chemicalspecies and a significantimpact when
p 0
h i
the case. We do however find that a increases faster as the degree of strong correlation is increased.
a function of δτ when δτ > δτ0 for a small Np than a FCIQMCisveryefficientatfindingthegroundstateof
h i
large hNpi and hence the most efficient timestep has an the neon atom: fc, the ratio of plateau height to Hilbert
implicit dependency on N . spacesize,is 10−4.1ThesizeoftheHilbertspacescales
p
h i ∼
The population dynamics can become unstable if the factoriallywiththebasissetyetthereisasublinearscal-
δτ is set large enough such that the exponential growth ing of the stochastic error with the Hilbert space size.
cannotbecounteredbypopulationcontrol. Wefindthat In contrastf scales superlinearly for chains of helium
c
thepointatwhichthepopulationexplodesisbeyondδτ0 atoms separated by 3 ˚A in the canonical Hartree–Fock
forthesystemsstudiedherewiththe exceptionofNe cc- basis: for He f =0.3 whereas for He f =0.97. How-
4 c 7 c
pVDZ (see Fig. 2 left) which doesn’t have a noticeable ever,a againseemstoscalesublinearlywiththesizeof
min
plateau.43 theHilbertspace. Thesimilarbehaviour,withinstatisti-
calerrorbars,ofa forNeatomandheliumchainssug-
min
gestssublinearscalingofthestochasticerrorinFCIQMC
V. THE SCALING OF INEFFICIENCY AS A FUNCTION withHilbertspacesizeforweaklycorrelatedsystemswith
OF SYSTEM SIZE no strong dependence on the FCIQMC sign problem. In
a similar style study in DMC44, Nemec et al. investi-
In the previous sections we investigatedboth the scal- gatedcomputationalscaling as a function of the number
ing of the inefficiency, a, as a function of (mean) popu- ofhydrogenatomsatalargeseparation,findingthatthe
lation, N , and timestep, δτ. Following this, we can computertimerequiredtoachieveafixederrorbarscales
p
h i
5
VI. CONCLUSIONS
100 2R0
He Canonical We have defined a metric, inefficiency, for measuring
He Local
the stochastic efficiency ofFCIQMCcalculationsandin-
Ne
HF 1.5R0 vestigated its behaviour as a function of the parameters
controllingthe populationdynamicsandasafunctionof
system size. We have found that a sizeable reduction in
R0
the stochastic error is possible by increasing the popula-
u.
a. tionofpsipsandusingthe largestpossibletimestepsuch
a / that the plateauheight remains constant.46 The optimal
timestepistransferablebetweenFCIQMCandiFCIQMC
10-1 simulations. Theefficiencydecreases(andinefficiencyin-
creases) sublinearly with the size of the Hilbert space in
weakly correlated systems. The population required to
exceed the plateau height or converge the initiator ap-
proximation becomes a limiting factor much faster than
the rise in the stochastic error bar for a given computa-
1011021031041051061071081091010101110121013
tional effort. This suggests that if improved approxima-
Hilbert Space Size
tions which reduce the sign problem can be made, then
the stochastic noise will not be insurmountable when
treating yet larger systems.
FIG. 3. The scaling of a as a function of system size.
Eq. (3) describes the systemmins studied in more detail. a This analysis ignores the impact of population and
min
was calculated by fitting a for iFCIQMC calculations with timestep on the computational cost of the calculation,
differingpopulationswiththeexceptionofNecc-pVDZ,He2, whichisdetailedinAppendixii. Ideallywewouldliketo
He3and He4,which havenonoticeable plateau,where many set the population and timestep such that the stochas-
FCIQMCcalculations wereused. Thetimestep wasset tobe tic error decays as fast as possible as a function of the
sufficiently small such that a is constant. The Supplemental computational effort. This analysis doesn’t change the
Information40 shows the studies used to find amin for each guidelinesgivenabove: i)the populationshouldbe large
system. enough such that the inefficiency metric, a, is converged
to the iFCIQMC limit; ii) the largesttimestep for which
the plateau remains constant represents a lower bound
on the most efficient timestep and is difficult to improve
exponentially as the square root of the system size. As
upon without running lots of expensive calculations at
the Hilbert space size scales loosely exponentially with
different timesteps.
the systemsize, we see this trend is similar,though with
Orbitallocalisationwasfoundtobeeffectiveforreduc-
such few systems studied it seems premature to general-
ing the plateau height, and thus the cost of a FCIQMC
ize this trend.
calculation for chains of helium atoms, as well improv-
Localisation of the molecular oribitals breaks symme- ing the stochastic efficiency. Consequently, it may be
try and so increases the size of the accessible Hilbert worthwhile investigating FCIQMC algorithms tuned to
space. However we find in this case that the plateau localised orbitals.
height decreases45; for example localisation causes the More broadly, we have provided a framework for as-
plateau for He7 to decrease from 5.7 106 (fc = 0.97) sessing the statistical efficiency of FCIQMC and related
to 4.6 106 (fc = 0.39). Importantly×localisation also methods. Given the recent activity in such methodolog-
×
appears to decrease amin compared to using the canon- ical development15,21,47–50, this approach offers a useful
ical orbitals, especially for He7, though it is difficult to meanstofairlycomparealternativeimplementationsand
draw definitive conclusions from three data points with algorithms.
non-negligible standard errors.
The behaviour of a is not solely governed by the
min
size of the Hilbert space, however. Stretching hydrogen ACKNOWLEDGMENTS
fluoride in a fixed basis set (and hence Hilbert space)
causes the correlation energy to increase. We find that We are indebted to Dr. George Booth, Prof. Ali
a increases significantly with bond length but have Alavi and Prof. Matthew Foulkes for enlightening dis-
min
notfoundasimple linearscalingwiththe correlationen- cussions. WAV is grateful to EPSRC for a studentship
ergy or plateau height. Given this behaviour doesn’t fit and for a knowledge transfer secondment under Grant
with the weaklycorrelatedsystems studied above,inves- No. EP/K503733/1. JSS acknowledges the research en-
tigationofthescalingoftheerrorbarinFCIQMCinthe vironment provided by the Thomas Young Centre un-
stronglycorrelatedlimitwouldbeaninterestingtopicfor der Grant No. TYC-101. AJWT thanks Imperial Col-
future investigation. lege for a Junior Research Fellowship and the Royal So-
6
ciety for a University Research Fellowship under Grant that introduces a logarithmic dependency on the num-
No. UF110161. Calculations were performed using the ber of occupied determinants. For the purposes of this
ImperialCollegeHighPerformanceComputingService.51 analysis,weshallneglectthelogarithmicdependencyand
instead focus on the optimal case.
Under these assumptions the average total amount of
computer time to perform a single timestep t is:
Appendix A: Computational cost of an FCIQMC calculation
h i
t =C N +C δτ N . (A1)
1 p 2 p
h i h i h i
C andC dependonboththecomputerarchitectureand
1 2
700 the chemical system of interest. All timing calculations
He5 2000000 He4 500000 were run on one processor (core) of the same machine.52
650 He5 50000 He3 50000 In Fig. 4 we show the computational time, t, of a sin-
He5 500000 He3 500000 gle timestep per psip for a range of different populations
600 He4 50000 He3 2000000 and systems. If Eq. (A1) was obeyed exactly then there
He4 2000000 to be no dependence between the number of psips and
ns 550 the intercept or gradient of Fig. 4 for a given system;
N / p fi the approximations made are good at large hNpi. The
/ ›500 approximation is less suitable for a small population as
t
it assumes that the population is such that the average
number of occupied determinants is approximately con-
450
stant.
From Eq. (A1) and Eq. (4), it follows that
400
a( N ,δτ) C /δτ +C
350 σ = h pi p 1 2. (A2)
E
N t
0.0000.0020.0040.0060.0080.0100.0120.0140.016 p h i
δτ
As N t is the total computer time for running for N
h i
steps, minimising a′ =a C /δτ +C gives the most ef-
1 2
p
ficient population dynamics. Whilst the exact form of
FIG. 4. The computer time of a single iteration per psip
a( N ,δτ) is not known, we can draw qualitative con-
as a function of the timestep. There is a linear relationship p
h i
(Eq. (A1)) with the timestep and a near-linear relationship clusions from the behaviour observed in IVA and IVB:
with the number of psips for a large population. All cal- i)a′andasharethesamedependenceon Np ;FCIQMC
h i
culations were run on a single core of an Intel Core i7-2600 calculations should therefore be performed with suffi-
processor. cientlylargepopulationtoplaceitintheiFCIQMClimit;
ii) if δτ <δτ then a′ will decrease as δτ−1/2, given that
0
An efficient FCIQMC implementation uses a sparse aisconstantwithδτ inthisregion. Ifδτ >δτ0 a willin-
storageschemetostorethelistofpsips: arepresentation creaseswith δτ which will compete with (and dominate)
ofa determinantalongwiththe numberofpsipsoccupy- the C1/δτ +C2 term, making it difficult to quantify
p
ingthedeterminantarestoredtogether.41 Theexpensive what will happen to a′. Thus δτ0 forms a lower bound
steps are evolving the psips (spawning and death) and on the most efficient timestep.
annihilation(combiningpsipsonthe samedeterminant).
1G. H. Booth, A. J. W. Thom, and A. Alavi, J. Chem. Phys.
The cost of spawning and death is linear in the popu- 131,054106(2009).
lation; the death step can be efficiently performed on a 2D. Cleland, G. H. Booth, and A. Alavi, J. Chem. Phys. 132,
per-determinantratherthanper-psipbasisbuttheeffect 41103(2010).
of this can absorbed into a system-dependent prefactor. 3C. Daday, S. Smart, G. H. Booth, A. Alavi, and C. Filippi, J.
Chem.TheoryComput.8,4441–4451 (2012).
Theprefactoralsoincludesthecostofselectingarandom
4J.J.Shepherd,G.H.Booth, andA.Alavi,J.Chem.Phys.136,
excitationandcalculatingthenecessaryHamiltonianma-
244101(2012).
trix elements. 5L. R. Schwarz, G. H. Booth, and A. Alavi, Phys. Rev. B 91,
The cost of annihilation depends on the number of 045139(2015).
6G.H.Booth,A.Gru¨neis,G.Kresse, andA.Alavi,Nature493,
psipscreatedinasingletimestep. Assumingthe calcula-
365–370(2013).
tionhasequilibratedatapopulationinexcessofthecriti- 7R. E. Thomas, D. Opalka, C. Overy, P. J. Knowles, A. Alavi,
calpopulation,thenumberofpsipscreatedscaleslinearly andG.H.Booth,J.Chem.Phys.143,054108(2015).
with the populationandthe timestep with a ratedepen- 8G. H. Booth and G. K.-L. Chan, J. Chem. Phys. 137, 191102
dent on the average absolute value of the off-diagonal (2012).
9N.S. Blunt, A.Alavi, and G. H.Booth, Phys. Rev. Lett. 115,
elementsintheHamiltonianmatrix. Whilstannihilation
050603(2015).
canbeperformedwithalinearscaling41,theimplementa- 10N. S. Blunt, S. D. Smart, G. H. Booth, and A. Alavi, ArXiv
tion in HANDE currently uses a binary searchapproach e-prints (2015), arXiv:1508.04680[physics.comp-ph].
7
11A.J.W.ThomandA.Alavi,Phys.Rev.Lett.99,143001(2007). FullConfigurationInteractionQuantumMonteCarlo”,” (2016),
12S. Y. Willow, K. S. Kim, and S. Hirata, J. Chem. Phys. 137, http://dx.doi.org/10.5281/zenodo.45831.
204122(2012). 36U.Wolff,Comput.Phys.Commun.156,143–153(2004).
13S.Y.Willow,M.R.Hermes,K.S.Kim, andS.Hirata,J.Chem. 37R.M.Lee,G.J.Conduit, N.Nemec, P.LopezRios, andN.D.
TheoryComput.9,4396–4402 (2013). Drummond,Phys.Rev.E83,066706(2011).
14R. Baer, D. Neuhauser, and E. Rabani, Phys. Rev. Lett. 111, 38Seehttps://github.com/jsspencer/pyblock forcode.
106402(2013). 39J. S. Spencer and A. J. W. Thom, ArXiv e-prints (2015),
15A.J.W.Thom,Phys.Rev.Lett. 105,263004(2010). arXiv:1511.05752[physics.chem-ph].
16C. J. Umrigar, M. P. Nightingale, and K. J. Runge, J. Chem. 40W.A.Vigor,J.S.Spencer,M.J.Bearpark, andA.J.W.Thom,
Phys.99,2865–2890 (1993). “Supplemental material for“understanding the efficiency of full
17D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566–569 configurationinteractionquantum montecarlo”,”.
(1980). 41G.H.Booth,S.D.Smart, andA.Alavi,Mol.Phys.112,1855–
18J. S. Spencer, N. S. Blunt, and W. M. C. Foulkes, J. Chem. 1869(2014).
Phys.136,054110(2012). 42G. Booth, A novel quantum Monte Carlo method for molecular
19D.M.Cleland,G.H.Booth, andA.Alavi,J.Chem.Phys.134, systems,Ph.D.thesis,UniversityofCambridge(2010).
024112(2011). 43WhenrunningthecalculationsforFig.2toprightwefoundthat
20J.J.Shepherd,G.E.Scuseria, andJ.S.Spencer,Phys.Rev.B thepopulationexplodeswhenδτ isslightlygreaterthan0.05for
90,155130(2014). Necc-pVDZ.
21F.R.Petruzielo,A.A.Holmes,H.J.Changlani,M.P.Nightin- 44N.Nemec,Phys.Rev.B81,035119(2010).
gale, andC.J.Umrigar,Phys.Rev.Lett.109,230201(2012). 45Note that this behaviour is system specific. Ref. 42 found that
22N.S.Blunt, S.D.Smart,J.A.F.Kersten,J.S.Spencer, G.H. systems where the plateau height was unaffected by orbital lo-
Booth, andA.Alavi,J.Chem.Phys.142,184107(2015). calisation.
23J.B.Anderson,J.Chem.Phys.63,1499–1503 (1975). 46Thiscanbeeasilyassessedpriortoproductioncalculationsgiven
24ThereisasignprobleminFCIQMCifitispossiblethatpsipsof that propagation to the plateau is computationally cheap com-
oppositesignscanbegenerated onthesamedeterminant.18. pared to an entire FCIQMC calculation and the plateau height
25As discussed in Ref. 53 this is equivalent to the original formu- caneasilybefoundinanautomated fashion.20,39.
lationoftheshiftinRef.1. 47N.S.Blunt,T.W.Rogers,J.S.Spencer, andW.M.C.Foulkes,
26G. H. Booth, D. Cleland, A. Alavi, and D. P. Tew, J. Chem. Phys.Rev.B89,245124(2014).
Phys.137,1–11(2012). 48F. D. Malone, N. S. Blunt, J. J. Shepherd, D. K. K. Lee, J. S.
27C.Overy,G.H.Booth, N.S.Blunt,J.J.Shepherd, D.Cleland, Spencer, and W. M. C. Foulkes, J. Chem. Phys. 143, 044116
andA.Alavi,J.Chem.Phys.141,1–11(2014). (2015).
28J.J.Shepherd,G.Booth,A.Gru¨neis, andA.Alavi,Phys.Rev. 49S.Ten-no,J.Chem.Phys.138,164126 (2013).
B85,081103(2012). 50M.KolodrubetzandB.K.Clark,Phys.Rev.B86,075109(2012).
29T.H.Dunning,J.Chem.Phys.90,1007–1023 (1989). 51http://www.imperial.ac.uk/ict/services/
30R.Ditchfield,W.J.Hehre, andJ.A.Pople,J.Chem.Phys.54, teachingandresearchservices/highperformancecomputing.
724–728 (1971). 52By running on a single core we avoid complications resulting
31J.PipekandP.G.Mezey,J.Chem.Phys.90,4916–4926(1989). from Np affecting load balancing, which will in turn affect t.
32Y.e.a.Shao,Mol.Phys.113,184–215 (2015). Loadbalancinghasbeenextensivelyinvestigated inRef.41.
33http://www.hande.org.uk/. 53W.A.Vigor,J.S.Spencer,M.J.Bearpark, andA.J.W.Thom,
34J.D.Hunter,Comput.Sci.Eng.9,90–95(2007). J.Chem.Phys.142,104101(2015).
35W.A.Vigor,J.S.Spencer,M.J.Bearpark, andA.J.W.Thom,
“Data from ”Understanding and Improving the Efficiency of