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Systematic Reduction of Sign Errors in Many-Body Calculations of Atoms and Molecules
Michal Bajdich,1 Murilo L. Tiago,1 Randolph Q. Hood,2 Paul R. C. Kent,3 and Fernando A. Reboredo1
1Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA2Lawrence Livermore National Laboratory, Livermore, California 94550, USA
3Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA(Received 18 December 2009; revised manuscript received 27 January 2010; published 11 May 2010)
The self-healing diffusion Monte Carlo algorithm (SHDMC) is shown to be an accurate and robust
method for calculating the ground state of atoms and molecules. By direct comparison with accurate
configuration interaction results for the oxygen atom, we show that SHDMC converges systematically
towards the ground-state wave function. We present results for the challenging N2 molecule, where the
binding energies obtained via both energy minimization and SHDMC are near chemical accuracy
(1 kcal=mol). Moreover, we demonstrate that SHDMC is robust enough to find the nodal surface for
systems at least as large as C20 starting from random coefficients. SHDMC is a linear-scaling method, in
the degrees of freedom of the nodes, that systematically reduces the fermion sign problem.
DOI: 10.1103/PhysRevLett.104.193001 PACS numbers: 31.15.�p, 02.70.Ss, 02.70.Tt
Since electrons are fermions, their many-body wavefunctions must change sign when the coordinates of anypair are interchanged. In contrast, the sign of bosonic wavefunctions is unchanged for any coordinate interchange.Because of this misleadingly small difference, theground-state energy of bosons can be determined by quan-tum Monte Carlo (QMC) methods [1,2] with an accuracylimited only by computing time, while QMC calculationsof fermions are either exponentially difficult or are stabi-lized by imposing a systematic error, a direct consequenceof our lack of knowledge of the fermionic nodal surface.Therefore, one of the most important problems in many-body electronic structure theory is to accurately find rep-resentations of the fermion nodes [3,4], the locations wherethe fermionic wave function changes sign, the so-called‘‘fermion sign problem.’’
The sign problem limits (i) the number of physicalsystems where ab initio QMC calculations can be appliedand (ii) our ability to improve approximations of densityfunctional theory (DFT) using QMC results [5]. Moreimportantly, it limits our overall understanding of theeffects of interactions in fermionic systems. Therefore, amethod to circumvent the sign problem with reduced com-putational cost could transform condensed matter theory,quantum chemistry, and nuclear physics, among otherfields.
Arguably the most accurate technique for calculating theground state of a many-body system with more than 20fermions is diffusion Monte Carlo (DMC) calculation. Thestandard DMC algorithm [5] finds the lowest energy of allwave functions that share the nodal surface STðRÞ imposedby a trial wave function �TðRÞ. This is the fixed-nodeapproximation where the resultant energy EDMC is a rig-orous upper bound of the exact ground-state energy [6,7].The exact ground-state energy is obtained only when�TðRÞ has the same nodal surface as the exact ground-state wave function.
If the exact nodes are not provided, the implicitfixed-node ground-state wave function �FNðRÞ will ex-hibit discontinuities in its gradient [7,8] (i.e., kinks) onsome parts of STðRÞ. We recently proved [8] that by locallysmoothing these discontinuities in �FNðRÞ, a new trialwave function can be obtained with improved nodes.This proof enables an algorithm that systematically movesthe nodal surface STðRÞ towards the one of an eigenstate. Ifthe form of trial wave function is sufficiently flexible, andgiven sufficient statistics, this process leads to an exacteigenstate wave function [8,9]. We named the method self-healing DMC (SHDMC), since the trial wave function isself-corrected in DMC and can recover even from a poorstarting point.In this Letter, we report the first applications of SHDMC
to real atoms and molecules (O, N2, C20). SHDMC ener-gies are within error bars of DMC calculations using thecurrent state-of-the-art approach [10,11]. Tests of SHDMCfor C20 demonstrate that our method can be applied at thenanoscale. Its cost scales linearly with the number ofindependent degrees of freedom of the nodes with anaccuracy limited only by the achievable statistics andchoice of representation of the nodes.SHDMC is fundamentally different from optimization
methods used in variational Monte Carlo (VMC): [1,2](i) the wave function is directly optimized based on aproperty of the nodal surface and not on the local energyor its variance, and (ii) the nodes are optimized at the DMClevel (as opposed to a VMC based algorithm).Using a short-time many-body propagator, SHDMC
samples the coefficients of an improved wave functionremoving the artificial derivative discontinuities of�FNðRÞ arising from the inexact nodes. Repeated applica-tion of this method results in the best nodal surface for agiven basis. For wave functions expanded in a completebasis it can be shown that the final accuracy is limited onlyby the statistics [8,9].
PRL 104, 193001 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending14 MAY 2010
0031-9007=10=104(19)=193001(4) 193001-1 � 2010 The American Physical Society
In SHDMC (see Refs. [8,9] for details), the weightedwalker distribution is [5]
fðR; �0 þ �Þ ¼ ��TðR; �0Þ½e��ðH FN�ET Þ�TðR; �0Þ�
¼ limNc!1
1
Nc
XNc
i¼1
Wji ðkÞ�ðR�Rj
i Þ; (1)
where
�TðR; �0Þ ¼ eJðRÞ X�
n
�nð�0Þ�nðRÞ (2)
is a trial function whereP�
n represents a truncated sum,f�nðRÞg forms a complete orthonormal basis of the anti-
symmetric Hilbert space, and eJðRÞ is a symmetric Jastrow
factor. In Eq. (1), H FN is the fixed-node Hamiltonian
[H FN is the many-body Hamiltonian with an infinitepotential at the nodes of �TðR; �0Þ] and ET is an energy
reference. Next, Rji corresponds to the position of the
walker i at step j of Nc equilibrated configurations. The
weights Wji ðkÞ are given by
Wji ðkÞ¼e�½Ej
i ðkÞ�ET �� with Eji ðkÞ¼
1
k
Xk�1
‘¼0
ELðRj�‘i Þ; (3)
where ET in Eq. (3) is periodically adjusted so thatPiW
ji ðkÞ � Nc and � is k�� (with k being a number of
steps and �� a standard DMC time step).From Eq. (1), one can formally obtain
~� TðR; �0 þ �Þ ¼ fðR; �0 þ �Þ=��TðR; �0Þ: (4)
We now define the local smoothing function to be
~�ðR0;RÞ ¼ X�
n
eJðR0Þ�nðR0Þ��nðRÞe�JðRÞ: (5)
Applying Eq. (5) to both sides of Eq. (4), using Eq. (1),and integrating over R, we obtain Eq. (2) with
�nð�0 þ �Þ ¼ limNc!1
1
N
XNc
i
Wji ðkÞe�JðRj
i Þ ��nðRj
i ��
TðRji ; �
0Þ ; (6)
where N ¼ PNc
i¼1 e�2JðRj
i Þ normalizes the Jastrow factor.
These new �nð�0 þ �Þ [Eq. (6)] are used to construct a newtrial wave function [Eq. (2)] recursively within DMC(therefore the name self-healing DMC). The weights inEq. (3) can be evaluated within (i) a branching algorithm[8] for �0 ! 1 or (ii) a fixed population scheme for small�0 [9,12]. The former method is more robust, but the latterimproves final convergence.
Since SHDMC is targeted for large systems, we reportvalidations using pseudopotentials.
Validation of SHDMC with configuration interaction(CI) calculations for the O atom.—We chose to study the3P ground state of the O atom because it has at least twovalence electrons in both spin channels [13]. The single-particle orbitals were expanded in valence triple zeta(VTZ) and valence quintuple zeta (V5Z) Gaussian basissets [13] using the GAMESS [14] code. To facilitate a direct
comparison between SHDMC and CI, no Jastrow factorwas employed.Figure 1 shows a direct comparison of the first 250
converged coefficients �n obtained using SHDMC withthose from the largest CI calculation (see Table I). Theinitial SHDMC trial wave function was the Hartree-Fock(HF) solution, and the final SHDMC coefficients resultedfrom sampling the 1481 most significant excitations in theCI.We used �� ¼ 0:01 a:u:, � ¼ 0:5 a:u:, and 16 iterationsof trial wave function projection (Nc � 6� 107).Figure 1 shows the excellent agreement between the
coefficients �n obtained independently by SHDMC andCI. A perfect agreement is guaranteed only in the limit of acomplete basis and Nc ! 1. The small differences inFig. 1 are due to the truncation of the expansion and thestochastic error in �n. The inset shows the residual projec-tion as a function of the total number Nb of configurationstate functions (CSFs) included in the expansion, normal-ized either using the entire CI expansion or using a�CI thatincluded only the �n sampled in SHDMC. The residualprojection is much smaller for the truncated norm than thefull norm illustrating that most of the error in �SHDMC isfrom truncation and not limited statistics. Similar resultswere obtained for the C atom (not shown).Validation with energy minimization for N2.—We also
compared the VMC and DMC energies of wave functionsoptimized with energy minimization in VMC (EMVMC)[10,11] and SHDMC using the QWALK [15] code. Severalbases were obtained from series of complete active space(CAS) and restricted active space (RAS) [16] multiconfi-guration self-consistent field (MCSCF) calculations [dis-tributing 10 electrons into m active orbitals: CASð10; mÞ].
-0.06
-0.04
-0.02
0
0.02
0.04
2 50 100 150 200 250
λ n
n (excitation)
SHDMCCISDTQ
10-4
10-3
10-2
2 1 10 100 1000
RP
Number of CSFs (Nb)
FIG. 1 (color online). Comparison of the values of the coef-ficients �n corresponding to the first 250 excitations of a con-verged SHDMC trial wave function (large black circles) with alarge configuration interaction calculation with single, double,triple, and quadruple excitations (CISDTQ) [16] wave function(small red circles) for the oxygen atom. �0 ¼ 0:9769 is notshown. Inset: Residual projection (RP ¼ 1� jh�SHDMCj�CIi=h�CIj�CIij) as a function of the number of CSFs included:circles, RP obtained with the full CISDTQ norm; squares, RP
obtained with the truncated CISDTQ norm.
PRL 104, 193001 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending14 MAY 2010
193001-2
We retained the Nb basis functions with �n of absolutevalue larger than a given cutoff. Subsequently, for eachbasis, we performed energy minimization of the JðRÞ and�n of trial wave function using a mixture of 95% of energyand 5% of variance. We also sampled these Nb coefficientsin SHDMC recursively starting from HF solution. For aclear comparison we used the same Jastrow in EMVMCand SHDMC.
We performed these calculations for the ground state(1�þ
g ) of N2 at the experimental geometry [17]. Figure 2
shows the resulting VMC and DMC energies obtained forwave functions optimized independently with EMVMCand SHDMC methods for the largest RASð10; 43Þ(2 629 447 CSFs yielding E ¼ �19:921 717) Slater-Jastrow wave function (see also Table II). In EMVMC,as previously observed for C2 and Si2 [11], we found asystematic reduction in the fixed-node errors, even whenstarting from the smallest CAS wave function (seeTable II). When we compare with SHDMC optimizedwave functions we find an excellent agreement in bothVMC and DMC energies. Therefore, SHDMC improvesthe nodes systematically starting from the HF ground state.
Since retaining all the determinants in the wave functionwould be costly, we performed calculations with differentNb to extrapolate (quadratically) the final energies asP
nð�MCSCFn Þ2 ! 1 (see Fig. 2). The extrapolated DMC
energies reached chemical accuracy (see also Table II).Proof of principle in larger systems.—Figures 1 and 2
show that SHDMC produces reliable and accurate resultsfor small systems starting form the HF nodes. It is alsoimportant to demonstrate that SHDMC is a robust ap-proach that can find the correct nodal surface topology ofmuch larger systems even when starting from randomnodal surfaces.
Figure 3 shows proof of principle results obtained for aC20 fullerene. These calculations used the branchingSHDMC algorithm [8] implemented by us in CASINO
[18]. Two electrons were removed from the system toobtain a noninteracting DFT ground-state wave functioninvariant under any transformation belonging to the icosa-hedral group (Ih) symmetry. The orbitals were obtaineddirectly with the real space code PARSEC [19] and classifiedaccording to their irreducible representations for Ih and itssubgroup D2h. For this calculation 694 excitations (deter-minants) were sampled. No CI prefiltering of determinants
is required; we only use the selection rules of both Ih andD2h symmetries.The Cþ2
20 system has a large DFT gap (5.53 eV) which is
often associated with a dominant role of the noninteractingsolution in the many-body wave function. The �0 coeffi-cient is expected to dominate the final optimized trial wavefunction. All initial coefficients �n of �TðRÞ were set torandom values, except for �0 which was set to zero. New�n values were sampled with �5094 walkers every 100DMC steps. We found that when the quality of the wavefunction is poor, it is better (i) to update �n frequently(after only 4 samplings) and (ii) to use the T-moves ap-proximation [20] which limits persistent configurations. Asthe quality of the wave function improved, we graduallyincreased the accumulation time (up to 96 samplings) andremoved the T-moves approximation (which, in practice,hinders the final SHDMC convergence). Figure 3 showsthat SHDMC can correct nodal errors as large as 0.5 har-
TABLE I. Total energies (and correlation shown in f g) of aground state oxygen atom in CI (full CI in VTZ and CISDTQ inV5Z), coupled-cluster [CCSD(T) from Ref. [13]], and SHDMC(no Jastrow) methods.
VTZ V5Z
Method Nb E ½hartree�f½%�g Nb E ½hartree�f½%�gCI 775 182 �15:882 58f89:0g 1762377 �15:895 57f95:7gCCSD(T) � � � �15:882 04f88:8g � � � �15:901 66f98:8gSHDMC 539 �15:9003ð2Þf98g 1481 �15:9040ð4Þf100g
-19.97
-19.96
-19.95
-19.94
-19.93
-19.92
-19.91
-19.90
-19.89
0.875 0.9 0.925 0.95 0.975 1
1 9 29 82 153 809
Ene
rgy
[har
tee]
Sum of squares of CSF coefficients
Number of CSFs (Nb)
VMC
DMC
EMVMCSHDMC
FIG. 2 (color online). Total energies obtained for N2 withVMC and DMC methods for wave functions optimized viaEMVMC [10] (squares) and SHDMC (circles) as a function ofP
nð�MCSCFn Þ2. The lines are parabolic extrapolations to 1. The
dot-dashed line represents the scalar relativistic core-correctedestimate of the exact energy (see Table II). The shaded area is theregion of chemical accuracy.
TABLE II. Comparison of total and binding DMC energies ofN2 for wave functions optimized with EMVMC and SHDMC forincreasingly larger basis (see text). Binding energies were ob-tained using an atomic energyb of �9:802 13ð5Þ hartree and zeropoint energy of 5.4 mhartree [17]. Estimated exact energiesinclude core-correlations correction of 1.4 mhartree [17].
Total energy [hartree] Binding energy [eV]
Wave function EMVMC SHDMC EMVMC SHDMC
1 determinant �19:9362ð5Þ 9.07(1)
CAS(10,14) �19:9536ð6Þ �19:9536ð6Þ 9.54(2) 9.54(2)
RAS(10,35) �19:9639ð4Þ �19:9627ð4Þ 9.83(1) 9.79(1)
RAS(10,43) �19:9654ð4Þ �19:9647ð4Þ 9.87(1) 9.85(1)
Estimated exact �19:9668ð2Þ �9:900ð1ÞaaUsing the experimental value from Ref. [17].bBased on a large multideterminant DMC calculation.
PRL 104, 193001 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending14 MAY 2010
193001-3
tree. The calculation was stopped when we obtained anenergy of �112:487ð2Þ hartree compared with the singledeterminant energy of �112:473ð1Þ hartree. We have con-fidence that SHDMC can be applied to cases where thenodal structure of the ground state is completely unknownsince it is successful and converges to the expected resultstarting from random.
The SHDMC recursive runs required 220 h on 1024processors (Cray XT4). This can be reduced to �100 hstarting from the ground-state determinant. ComparableEMVMC calculations with the same basis were unsuccess-ful, presumably due to the statistical errors in the Hessianand overlap matrices. The energy was not improved withEMVMC [�112:488ð3Þ hartree] even selecting a basis withthe largest 104 coefficients of the 694 sampled in SHDMC.The estimated running time for EMVMC with CASINO 2.5using Nb ¼ 694 and just 400 configurations [21] on 1024processors is already �100 h, suggesting that for Cþ
20
SHDMC is faster than EMVMC. However, both methodscan be improved for large Nb (e.g., as in Ref. [22]), byremoving redundant input and output.
Summary.—We have shown that the SHDMC wavefunction converges to the ground state of our best CI calcu-lations and is systematically improved as the number of co-efficients sampled increases and the statistics are im-proved. SHDMC presents equivalent accuracy to theEMVMC approach [10,11] starting from random coeffi-cients. SHDMC is numerically robust and can beautomated.
The number of independent degrees of freedom of thenodes increases exponentially with the number of electrons[9]. Since EMVMC is based on VMC, the prefactor for itscomputational cost is much smaller than SHDMC.However, the number of quantities sampled in EMVMCis quadratic with respect to the number of degrees of free-dom. In addition, EMVMC requires inverting a noisy
matrix. These requirements cause EMVMC to scale at leastquadratically. In contrast, SHDMC only requires one tosample a number of quantities linear in the number ofoptimized degrees of freedom. Therefore, a crossover be-tween the methods is expected for systems of sufficientsize or complexity. Tests on the large Cþ2
20 fullerene system
demonstrate that SHDMC is robust and that the nodes aresystematically improved even starting from random coef-ficients in the trial wave function. This shows that SHDMCcan be used to find the nodes of unknown complex systemsof unprecedented size.We thank D. Ceperley, R.M. Martin, and C. J. Umrigar
for critically reading the manuscript and useful comments.This research used computer resources at NERSC andNCCS. Research sponsored by U.S. DOE BES Divisionof Materials Sciences & Engineering (F. A. R., M. L. T.)and ORNL LDRD program (M.B.). The Center forNanophase Materials Sciences research was sponsored bythe U.S. DOE Division of Scientific User Facilities(P. R. C. K.). Research at LLNL was performed underU.S. DOE Contract No. DE-AC52-07NA27344 (R.Q.H.).
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204105 (2009).
-112.50
-112.40
-112.30
-112.20
-112.10
-112.00
0 1 2 3 4 5 6 7 8 9
10 20 30 40 50 60 70 80 90 100 110 120 126 130E
nerg
y [h
artr
ee]
Number of steps [× 104]
Number of wave function updates
Single determinant DMC energy
T-moves off
FIG. 3 (color online). Proof of principle of SHDMC for largersystems. Initial evolution of the average local energy for aSHDMC run with branching [8] generated for Cþ2
20 , with random
initial coefficients (see text). Inset: Calculated icosahedral clus-ter Cþ2
20 .
PRL 104, 193001 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending14 MAY 2010
193001-4