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[Kousuke Nakano](https://orcid.org/0000-0001-7756-4355), Benjamin X. Shi, Dario Alfè, Andrea Zen

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[Basis Set Incompleteness Errors in Fixed-Node Diffusion Monte Carlo Calculations on Noncovalent Interactions](https://mdr.nims.go.jp/datasets/bef1f052-e8da-4c08-8a04-84353cebb58a)

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Basis Set Incompleteness Errors in Fixed-Node Diffusion Monte Carlo Calculations on Noncovalent InteractionsBasis Set Incompleteness Errors in Fixed-Node Diffusion MonteCarlo Calculations on Noncovalent InteractionsKousuke Nakano,* Benjamin X. Shi, Dario Alfe,̀ and Andrea Zen*Cite This: J. Chem. Theory Comput. 2025, 21, 4426−4434 Read OnlineACCESS Metrics & More Article Recommendations *sı Supporting InformationABSTRACT: Basis set incompleteness error (BSIE) is a commonsource of error in quantum chemistry calculations, but it has notbeen comprehensively studied in fixed-node Diffusion Monte Carlo(FN-DMC) calculations. FN-DMC, being a projection method, isoften considered minimally affected by basis set biases. Here, weshow that this assumption is not always valid. While the relativeerror introduced by a small basis set in the total FN-DMC energy isminor, it can become significant in binding energy (Eb) evaluationsof weakly interacting systems. We systematically investigated BSIEsin FN-DMC-based Eb evaluations using the A24 data set, a well-known benchmark set of 24 noncovalently bound dimers. We foundthat BSIEs in FN-DMC evaluations of Eb are indeed significantwhen small localized basis sets, such as cc-pVDZ and cc-pVTZ, areemployed. Our study shows that the aug-cc-pVTZ basis set family strikes a good balance between computational cost and BSIEs inthe Eb calculations. We also found that augmenting the basis sets with diffuse orbitals, using counterpoise correction, or both,effectively mitigates BSIEs, allowing smaller basis sets such as aug-cc-pVDZ to be used.1. INTRODUCTIONDiffusion Monte Carlo (DMC)1,2 is a state-of-the-artelectronic structure method used for predicting and under-standing phenomena in materials science, chemistry, andphysics. In particular, DMC can achieve highly accuratequantitative predictions, typically surpassing those of mean-field approaches like density functional theory (DFT). Thislevel of accuracy has proven essential for studying systemschallenging for DFT, such as high-pressure hydrogen,3−9layered materials,10−14 molecular crystals,15,16 and molecularadsorption on surfaces.17−21In theory, DMC is an exact technique to project the groundstate (GS) of a Hamiltonian. However, in practical applicationsto Fermionic systems (e.g., atoms, molecules, and materials), itrelies on the fixed-node (FN) approximation to maintain theantisymmetry of the wave function. The FN approximationconstrains the nodal surface of the projected state to that of atrial wave function, which can be generated by methods suchas DFT, Hartree−Fock (HF), or correlated quantumchemistry (QC) methods, including the complete activespace self-consistent field (CASSCF) method.The approaches used to generate the trial wave function arenot exact, so its nodal surface is not exact either, yielding anerror on the FN-DMC evaluations called the FN error. Thecloser the nodal surface of the trial wave function to the nodalsurface of the exact GS, the smaller the FN error. There areother approximations in FN-DMC, but typically the majorsource of error is the FN error. The FN error depends on theaccuracy of the trial wave function, which in turn depends onthe level of theory employed to generate it (e.g., we expect aCASSCF wave function to have a better nodal surface than aDFT or a HF wave function) and on the completeness of theemployed basis set representation. In general, a larger basis setgives a better wave function (i.e., nodal surface), although afew exceptions are reported.22 One typically chooses a basis setby considering the trade-off between costs (i.e., CPU time +memory requirement) and accuracy. On the one hand, using atoo large basis set increases the computational cost and thememory requirement with little benefit. This issue becomesparticularly prominent when constructing a multideterminanttrial wave function23 or generating a single-determinant trialwave function with methods beyond mean-field theory.24Specifically, several of the authors have recently proposed away to generate a trial wave function based on the naturalorbitals constructed from a second-order Møller−Plesset(MP2) calculation,24 allowing one to go beyond the single-reference fixed-node approximation. However, the workflowbecomes impractical for large molecules in terms of memoryReceived: November 30, 2024Revised: April 20, 2025Accepted: April 22, 2025Published: April 30, 2025Articlepubs.acs.org/JCTC© 2025 The Authors. Published byAmerican Chemical Society4426https://doi.org/10.1021/acs.jctc.4c01631J. Chem. Theory Comput. 2025, 21, 4426−4434This article is licensed under CC-BY 4.0Downloaded via NATL INST FOR MATLS SCIENCE (NIMS) on December 1, 2025 at 07:15:53 (UTC).See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.https://pubs.acs.org/action/doSearch?field1=Contrib&text1="Kousuke+Nakano"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Benjamin+X.+Shi"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Dario+Alfe%CC%80"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Andrea+Zen"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/showCitFormats?doi=10.1021/acs.jctc.4c01631&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c01631?ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c01631?goto=articleMetrics&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c01631?goto=recommendations&?ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c01631?goto=supporting-info&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c01631?fig=tgr1&ref=pdfhttps://pubs.acs.org/toc/jctcce/21/9?ref=pdfhttps://pubs.acs.org/toc/jctcce/21/9?ref=pdfhttps://pubs.acs.org/toc/jctcce/21/9?ref=pdfhttps://pubs.acs.org/toc/jctcce/21/9?ref=pdfpubs.acs.org/JCTC?ref=pdfhttps://pubs.acs.org?ref=pdfhttps://pubs.acs.org?ref=pdfhttps://doi.org/10.1021/acs.jctc.4c01631?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://pubs.acs.org/JCTC?ref=pdfhttps://pubs.acs.org/JCTC?ref=pdfhttps://acsopenscience.org/researchers/open-access/https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/and storage when the basis set size is increased, due to thesteep scaling of MP2. On the other hand, using a too smallbasis set risks introducing bias into FN-DMC results. Thechoice of a basis set family balancing the accuracy and thecomputational cost is also particularly pertinent for acalculation spanning chemical space, such as developingmachine-learning models.25 While previous works haveexplored the influence of trial wave function accuracy,26−32the impact of the basis set incompleteness errors (BSIEs) inFN-DMC has yet to be comprehensively and systematicallyexplored in the context of noncovalent interaction evaluations,which is one of the most prominent applications of FN-DMC.10−1415,21,27,33−40In QC and DFT methods, BSIEs are a dominant errorsource that requires careful control, yet it has been oftenassumed that FN-DMC is relatively immune to BSIEs from thetrial wave function41 because it depends only on the nodalsurface, not on the full wave function amplitude. In this work,we systematically investigate how these assumptions hold upby analyzing BSIEs in FN-DMC calculations.BSIEs are especially pronounced in QC and DFT methodswhen describing noncovalent interactions. In this context, thequantity of interest is typically the binding energy of a dimercomplex (AB), defined as=E E E EbAB A B (1)where EA, EB, and EAB are the total energies of monomer A,monomer B, and the AB dimer complex, respectively. Thisstudy focuses on the propagation of BSIEs from the trial wavefunction in FN-DMC calculations of Eb, a particularly relevantarea of investigation given the high sensitivity of noncovalentinteractions to basis set quality.24,39,42A basis set consists of a number of basis functions that areused to represent the electronic wave function, with thecomplete basis set (CBS) limit achieved when expandedtoward an (infinite) set of functions. The BSIE is the deviationfrom the CBS limit43,44 and for a binding energy Eb, it isdefined as43,44=E M M M E M M M E( , , ) ( , , )bBSIE A B ABbA B ABbCBS (2)where MA, MB and MAB denote the number and type of basisfunctions employed in the calculation of EA, EB and EABrespectively within eq 1, and EbCBS denotes the binding energyin the CBS limit. Two common choices of basis function typesare plane waves (PWs) and atom-centered Gaussian TypeOrbitals (GTOs). On the one hand, BSIEs are well-controlledwith PWs because systematic convergence toward the CBSlimit can be achieved by monotonically increasing the kinetic(i.e., cutoff) energy of the included PWs. On the other hand,errors in GTOs are less well-behaved, with users selecting from‘families’ of available basis sets consisting of increasing sizes,often denoted by the number of ‘zeta’ basis functions peroccupied valence orbital. A popular example is the correlationconsistent basis-set family, developed by Dunning and co-workers,45 for instance the correlation-consistent polarizedvalence n-zeta (cc-pVnZ), where n, the cardinal number, cantake on double (D), triple (T), quadruple (Q), quintuple (5)and sextuple (6) zeta functions on each atom. It is alsocommon to augment these with additional diffuse functions,which are denoted by an ‘aug-’ prefix in front.When using GTOs, or any other set of atom-centered basisfunctions, to compute binding energies, it is crucial todistinguish BSIEs from basis-set superposition errors(BSSEs),43,44 a related source of error. BSSE occurs whenbasis functions of interacting molecular systems A and B in theAB dimer overlap, increasing the variational space for the ABdimer with respect to the A and B monomers, thus leading toan overestimation of Eb.1 This error is defined by Boys andBernardi46 as= [ ] + [ ]E M M ME M E M E M E M( , , )( ) ( ) ( ) ( )bBSSE A B ABA AB A A B AB B B(3)involving two separate calculations on each monomer. Formonomer A, alongside the original basis set EA(MA), acalculation including additional empty ‘ghost’ functions frommonomer B is also performed to get EA(MAB), as proposed byBoys and Bernardi.46 The difference between the twoquantities, appearing in eq 3, then provides an estimate onthe effect of the basis set superposition on the energy of eachmonomer. Thus, the BSSE error EbBSSE can be used to correctthe original Eb evaluation to obtain a counterpoise (CP)corrected estimate of the binding energy: EbCP = Eb − EbBSSE. Itmust be emphasized that the CP corrected estimates still sufferfrom BSIE, although they are typically closer to the CBSlimit,44 and typically underbind Eb.2 In the CBS limit, bothBSIE and BSSE will vanish.To date, only a few studies have reported BSIEs in FN-DMC for Eb calculations of noncovalent interactions and toour knowledge, none have studied the effect of CP corrections.Korth et al.26 reported the difference between noncovalentinteraction energies of the Li-thiophene complex obtained withcc-pVTZ and cc-pVQZ basis sets. The results from the cc-pVQZ basis were close to the CCSD(T)/CBS reference value.Dubecky ́ et al.34 studied the effect of the cardinal number nand augmentation functions in ammonia dimer. On the onehand, they revealed that the higher cardinality number n (fromcc-pVTZ to cc-pVQZ) has a smaller effect on the overallaccuracy than the augmentation does. On the other hand, theadditional diffuse functions (aug-) were found to be crucial toreach the reference CCSD(T)/CBS interaction energy valuebecause the augmentation functions likely improve the tails oftrial wave functions that are crucial for describing van derWaals complexes correctly. They recommended the aug-cc-pVTZ basis set as the most reasonable choice with respect tothe price/performance ratio. Very recently, Zhou et al.47evaluated barrier heights and complexation energies in smallwater, ammonia, and hydrogen fluoride clusters using FN-DMC with basis sets of increasing completeness, andrecommend basis sets containing diffuse basis functions.In this paper, we present a detailed analysis of the basis seteffects, BSIEs and BSSEs, in DMC binding energy calculations,specifically focusing on noncovalent interactions. Our findingsindicate that while BSIEs and BSSEs in FN-DMC aresubstantially reduced compared to those in the trial wavefunction, they are not negligible. The key conclusions to getCBS-limit binding energies (i.e., negligible BSIEs and BSSEs)from our work are (1) aug-cc-pVDZ is sufficient when CPcorrection is applied and (2) the aug-cc-pVTZ basis setperforms well without the need for CP correction.2. COMPUTATIONAL DETAILSTo investigate BSIEs in DMC calculations systematically, wecomputed binding energies (Eb) of the complex systemsJournal of Chemical Theory and Computation pubs.acs.org/JCTC Articlehttps://doi.org/10.1021/acs.jctc.4c01631J. Chem. Theory Comput. 2025, 21, 4426−44344427pubs.acs.org/JCTC?ref=pdfhttps://doi.org/10.1021/acs.jctc.4c01631?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asincluded in the A24 data set.48 The A24 data set is a set ofnoncovalently bound dimers, consisting of systems dominatedby H-bonding, dispersion and a mixture of both.48 The data setwas intended to test the accuracy of computational methodsthat are used as benchmarks in larger model systems. Weemployed the correlation consistent (cc) GTOs accompaniedby the correlation consistent effective core potentials49,50(ccECP) in this study. The majority of the QMC resultsreported in this work are obtained using the TurboRVB51 abinitio QMC packages. TurboRVB performs QMC calcu-lations using trial wave functions expressed in terms oflocalized atomic orbitals, such as GTOs. TurboRVB supportsthe CP correction for QMC calculations using trial wavefunctions with GTOs, allowing one to study both BSIEs andBSSEs. More specifically, TurboRVB can assign GTOs to theso-called ghost atoms (i.e., with zero nuclear charges), as inQC calculations.TurboRVB implements the lattice discretized version ofthe FN-DMC calculations (LRDMC).51,52 Notice that theinfinitesimal mesh limit of LRDMC evaluations is equivalent tothe infinitesimal time step limit in standard DMC evaluations,provided that the computational setup (i.e., trial wave function,pseudopotential, localization approximation of the nonlocalpseudopotential terms) is the same. The LRDMC calculationswith TurboRVB were performed by the single-grid scheme52with lattice spaces a = 0.30, 0.25, 0.20, and 0.10 Bohr. BSSEswere computed at each lattice space according to eq 3, andthen the obtained values were extrapolated to a → 0 usingEbBSSE(a2) = k2 · a2 + EbBSSE, where EbBSSE is the extrapolatedBSSE. In computing BSIEs, the binding energies computedwith the aug-cc-pV6Z were used as the reference values, i.e.,EbCBS in eq 2, for each complex system because, as shown in thefollowing section, the aug-cc-pV6Z basis has reached the CBSlimit. The binding energy obtained with each basis set wasextrapolated to a → 0 using Eb(a2) = k4 · a4 + k2 · a2 + Eb,where Eb is the extrapolated binding energy, and then BSIEswere computed according to eq 2. While the wave functionvariances differ across the various basis sets, this does not affectthe LRDMC extrapolations because we reduced the error barsfor each lattice space to the same value in all basis sets (SeeTable SII and Figure S2 of the SI).The ccECP pseudopotentials are semilocal effective corepotentials, as with most available pseudopotentials, so theDMC results depend on how the sign problem from itsnonlocal term is addressed. In this study, we used thedeterminant locality T-move (DTM)53 scheme in the majorityof the calculations shown here, which are performed withTurboRVB.For the DFT calculations that generate trial wave functionswith GTOs for subsequent QMC calculations via TREX-IO54files, we used the PySCF55,56 package, with the PZ-LDA57exchange-correlation functional. For LRDMC calculations withTurboRVB the obtained trial wave functions are combinedwith the two-body and the three-body Jastrow factors.51 Thethree-body Jastrow factors are not attached to the ghost atomsin CP calculations. The parameters in the Jastrow factors wereoptimized using the Stochastic Reconfiguration method.58 Wenotice that the optimization of the Jastrow factor does notaffect the extrapolated LRDMC total and binding energiessince the DTM is employed in this study. In this sense, theobtained conclusions in this study are deterministic.We also compare QMC evaluations obtained using PW basissets in comparison with localized GTO basis sets (Table SI ofthe Supporting Information (SI)). The comparison uses resultsobtained with the QMCPACK package,59,60 which implementswave functions using either PW or GTO basis sets. Detailsabout the QMCPACK calculations are provided in Section 1 ofthe SI.Figure 1. (a) The BSSEs in the binding energies of the A24 set computed by LRDMC with cc-pVDZ, cc-pVTZ, aug-cc-pVTZ, and aug-cc-pV6Zbasis sets. The plotted BSSEs are the values extrapolated to the infinitesimal lattice space. The error bars represent 1σ. (b) The violin plots for theobtained BSSEs, with median of the distribution indicated with a gray line inside the violin plot.Journal of Chemical Theory and Computation pubs.acs.org/JCTC Articlehttps://doi.org/10.1021/acs.jctc.4c01631J. Chem. Theory Comput. 2025, 21, 4426−44344428https://pubs.acs.org/doi/suppl/10.1021/acs.jctc.4c01631/suppl_file/ct4c01631_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jctc.4c01631/suppl_file/ct4c01631_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jctc.4c01631/suppl_file/ct4c01631_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jctc.4c01631/suppl_file/ct4c01631_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c01631?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c01631?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c01631?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c01631?fig=fig1&ref=pdfpubs.acs.org/JCTC?ref=pdfhttps://doi.org/10.1021/acs.jctc.4c01631?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-as3. BASIS-SET CONVERGENCE CHECKS TO ESTIMATETHE BINDING ENERGIES IN THE CBS LIMITTo estimate BSIEs, the binding energies in the CBS limit areneeded, as described in eq 2. Since zero BSSE implies zeroBSIE in binding energy calculation, computing BSSEs ishelpful to decide which basis set should be used to computeEbCBS in eq 2.Figure 1(a) shows BSSEs in the binding energies of the A24set computed by LRDMC implemented in TurboRVB. Theywere obtained using cc-pVDZ, cc-pVTZ, aug-cc-pVTZ, andaug-cc-pV6Z basis sets. Figure 1 (b) shows the violin plots ofthe BSSEs. The figures reveal that the binding energiesobtained with the cc-pVDZ and cc-pVTZ basis sets havesignificant BSSEs, indicating that the small basis sets are farfrom the CBS limit. BSSEs vanish for all molecules with theaug-cc-pV6Z basis set within an interval of three standarddeviations (±3σ, corresponding to a confidence of 99.7%),indicating that the aug-cc-pV6Z basis set has reached the CBSlimit.In addition, to double-check that the aug-cc-pV6Z basis setgives the CBS-limit binding energies, we computed the DMCbinding energies on all A24 dimers using aug-cc-pV6Z, as wellas smaller GTO basis sets, and PW basis sets with a very largecutoff. We made this calculations using QMCPACK, whichallows to use both localized and PW basis sets. The bindingenergies values are reported in Table SI of the SI, and acomparison between the evaluations with different basis sets isshown in Figure S1 of the SI. The results indicate that aug-cc-pV6Z and large-cutoff-PW basis sets give consistent bindingenergies within an interval of ±3σ, supporting the aboveargument that the aug-cc-pV6Z basis set gives convergedbinding energies.Therefore, both the BSSEs evaluation and the comparisonwith large-cutoff-PW indicate that the aug-cc-pV6Z hasreached the CBS limit. Thus, we can use the binding energiesobtained with the aug-cc-pV6Z basis sets (without the CPcorrection) as reference values (i.e., EbCBS in eq 2) in thefollowing BSIE analysis. The reference DMC values arereported in Table 1.Additionally, for the ammonia dimer, we also tested trialwave functions with the following XC functionals: PBE,61PBE0,62 B3LYP,63 ωB97M-V64 and Hartree−Fock, todetermine whether the convergence behavior depends on thechoice of XC, as shown in Table SV and SVI of the SI. Theresults demonstrate that neither the binding energy nor theconvergence behavior depends on the XC functionalemployed.4. BIAS AGAINST THE BINDING ENERGIES IN THECBS LIMITBSIEs in binding energies obtained from LRDMC calculationswith the cc-pVDZ and aug-cc-pVDZ basis sets (for theccECPs49,50) with and without the CP corrections are shownin Figure 2 for each of the 24 dimers of the A24 data set. Thedistribution of the BSIEs across the data set for the same basisset and CP correction combinations is shown in Figure 3(a)via a violin plot. By comparison, BSIEs in MP2 calculations areshown in Figure 3(b).In Figure 2, the comparison between the BSIEs with cc-pVDZ (without CP) and with aug-cc-pVDZ (without CP)reveals that the augmentation of the basis set drasticallydecreases BSIEs, specifically for the complex systems withhydrogen-bond interactions. The most significant discrepancyis seen for the ammonia dimer, for which Dubecky ́ et al.34 alsoreported that the additional diffuse functions (i.e., augmenta-tion) were crucial to reach the reference CBS interactionenergy value. They interpreted the outcome such thataugmentation functions likely improve the tails of trial wavefunctions that are crucial for describing the weak interactionscorrectly.34 The wider set of results reported in this worksupports the above interpretation. The interaction amongmolecules included in the A24 data set are categorized intothree groups:48 Hydrogen bonds (index 1 to 5), mixedinteractions (index 6 to 15), and dispersion-dominatedinteractions (index 16 to 24). Dimers in the hydrogen-bondgroup show the most significant BSIEs, while the dispersion-dominated dimers are less affected by BSIEs. The hydrogenbond, which originates from the Coulomb interactions, has thelong-tail effect (e.g., 1/r) compared with the dispersion-dominated ones, which are typically shorter-range interactions(e.g., 1/r6). It appears that the long-tail of the interaction hasan effect on the nodal surface (affecting the FN-DMCevaluations), which can be improved if diffuse functions areavailable in the basis set.In Figure 2, the comparison between BSIEs with cc-pVDZwith and without the CP correction of the basis set shows thatthis correction alleviates the BSIEs. It implies that the basis setsTable 1. Binding Energies Eb, in kcal/mol, of the 24Molecular Dimers Contained in the A24 Dataset48,alabel EbDMC EbCCSD(T) Δwater--ammonia −6.75(7) −6.493 0.26(7)water dimer −5.10(8) −5.006 0.09(8)HCN dimer −5.09(7) −4.745 0.34(7)HF dimer −4.74(7) −4.581 0.16(7)ammonia dimer −3.10(6) −3.137 −0.04(6)HF--methane −1.64(7) −1.654 −0.01(7)ammonia--methane −0.80(7) −0.765 0.04(7)water--methane −0.58(6) −0.663 −0.08(6)formaldehyde dimer −4.42(9) −4.554 −0.13(9)water--ethene −2.50(10) −2.557 −0.06(10)formaldehyde--ethene −1.71(10) −1.621 0.09(10)ethyne dimer −1.44(7) −1.524 −0.08(7)ammonia--ethene −1.38(6) −1.374 0.01(6)ethene dimer −0.97(9) −1.090 −0.12(9)methane--ethene −0.56(6) −0.502 0.06(6)borane--methane −1.46(7) −1.485 −0.03(7)methane--ethane −0.65(9) −0.827 −0.18(9)methane--ethane −0.57(8) −0.607 −0.04(8)methane dimer −0.58(6) −0.533 0.05(6)Ar--methane −0.36(8) −0.405 −0.05(8)Ar--ethene −0.24(7) −0.364 −0.12(7)ethene--ethyne 1.04(9) 0.821 −0.22(9)ethene dimer 1.04(8) 0.934 −0.11(8)ethyne dimer 1.32(8) 1.115 −0.21(8)RMSD � � 0.135aEbDMC column shows results obtained in this work, from LRDMCcalculations employing the ccECP pseudopotentials49,50 with theDTM approximation,53 and a trial wavefunction with the determinantfrom a LDA-PZ DFT calculation, constructed with ccecp-aug-cc-pV6Z basis sets. EbCCSD(T) column shows the evaluations from Řezać ̌and Hobza,48 computed by CCSD(T) with extrapolations to the CBSlimits. The last column shows the differences Δ = EbCCSD(T) − EbDMCbetween the LRDMC and CCSD(T) values, with the root meansquare deviation RMSD at the end.Journal of Chemical Theory and Computation pubs.acs.org/JCTC Articlehttps://doi.org/10.1021/acs.jctc.4c01631J. Chem. Theory Comput. 2025, 21, 4426−44344429https://pubs.acs.org/doi/suppl/10.1021/acs.jctc.4c01631/suppl_file/ct4c01631_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jctc.4c01631/suppl_file/ct4c01631_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jctc.4c01631/suppl_file/ct4c01631_si_001.pdfpubs.acs.org/JCTC?ref=pdfhttps://doi.org/10.1021/acs.jctc.4c01631?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asassigned to the ghost atoms can compensate missing diffusefunctions in the cc-pVDZ basis set, thus improving the nodalsurface of the monomers and decreasing the FN error on thebinding energy evaluations. This suggests that the CPcorrection is an alternative way to reduce BSIEs in DMCcalculations. The simultaneous use of augmentation and CPleads to a synergistic effect, as can be appreciated in Figure 2observing the evaluations obtained using aug-cc-pVDZ withCP.Figure 3(a) summarizes the BSIEs obtained with all thefamily members of the cc basis sets and PW used in this study.The left panel of Figure 3(a) plots the BSIEs with thenonaugmented cc basis sets (cc-pVnZ: n = D,T,Q,5,6),revealing that, to get binding energies in the CBS limit withintheir statistical errors (3 σ ∼ 0.25 kcal/mol), one needs the cc-pVQZ without the CP corrections or the cc-pVTZ with the CPcorrection. The cc-pVTZ without the CP correction introducesnon-negligible BSIEs in binding energies of several moleculesin the A24 set (see. Table SIII of the SI), such as water−ammonia, ammonia dimer, and formaldehyde−ethene. Thecentral panel of Figure 3(a) plots BSIEs with the augmented ccbasis sets (aug-cc-pVnZ: n = D,T,Q,5), indicating that theaugmentations of the basis sets improve the situation. To getbinding energies in the CBS limit within their statistical errors,one needs the aug-cc-pVTZ without the CP correction or theaug-cc-pVDZ basis with the CP correction. The right panel ofFigure 3(a) plots BSIEs with PW basis set, confirming thataug-cc-pV6Z basis set gives binding energies in the CBS limit(i.e., zero BSIEs within the statistical errors).It is informative to compare the BSIEs obtained by DMCwith those obtained using a quantum chemistry method, suchas MP2, to understand the impact of basis sets. Thecomparison between Figure 3(a),(b) reveals that BSIEs inthe DMC calculations are not as significant as in the MP2calculations, as believed in the QMC community. This is truenot only for the binding energies, but also the total energies ofFigure 2. BSIEs in the binding energies of the A24 set, estimated from LRDMC calculations in the limit of infinitesimal lattice spaces. The errorbars represent 1σ.Figure 3. Violin plots of BSIEs in the binding energy calculations of the A24 data set with and without the CP corrections. (a) LRDMC with cc-pVnZ (n = D,T,Q,5,6) and aug-cc-pVnZ (n = D,T,Q,5) and DMC with PW. (b) MP2 with cc-pVnZ (n = D,T,Q,5,6) and with aug-cc-pVnZ (n =D,T,Q,5). The reference binding energies are those obtained with aug-cc-pV6Z basis without CP correction.Journal of Chemical Theory and Computation pubs.acs.org/JCTC Articlehttps://doi.org/10.1021/acs.jctc.4c01631J. Chem. Theory Comput. 2025, 21, 4426−44344430https://pubs.acs.org/doi/suppl/10.1021/acs.jctc.4c01631/suppl_file/ct4c01631_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c01631?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c01631?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c01631?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c01631?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c01631?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c01631?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c01631?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c01631?fig=fig3&ref=pdfpubs.acs.org/JCTC?ref=pdfhttps://doi.org/10.1021/acs.jctc.4c01631?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asfragments and complexes, as shown in Figure S3. In panel(b),the asymptotic behaviors with n are seen in the bindingenergies computed by MP2. For QC calculations, the mostcommon and established procedure to reach the CBS limit isthe extrapolation of the binding energies with consecutivecardinal numbers.65 The asymptotic behaviors allow theextrapolation, and, in fact, the CP correction in MP2calculations is necessary for smoother extrapolations to theCBS limit (c.f. Figures S8 and S9 of the SI plot Eb for all 24systems), as already mentioned in ref 44. Instead, in DMCcalculations, the extrapolation is no longer needed, as shown inpanel (a) (c.f. Figures S10 and S11 plot Eb for all 24 systems).The observed differences between MP2 and DMC likely arisebecause the energy from the latter is affected only by the nodalsurface, while the former depends on the entire wave function.While BSIEs are correlated among these methods, thecorrelation is inadequate for discussions at the subchemicalaccuracy level (see Figure S16 of the SI). In other words, wefound that the BSIEs in DMC calculations cannot be estimatedaccurately from those in other QC calculations.Our study reveals that, in DMC calculations, one can get thebinding energy in the CBS limit only with a single medium-sizebasis set (such as cc-pVQZ and aug-cc-pVTZ). It helpsdecrease a DMC computational cost to reach the CBS limitbecause we can avoid using atomic orbitals with higher angularmomenta (e.g., h and i orbitals).3 For instance, the LRDMCcomputational cost of the water dimer with respect to basis setfamilies is plotted in Figure S17. It indicates that the aug-cc-pVTZ basis set strikes a good balance between the computa-tional cost and BSIE in the binding energy calculation.Furthermore, it should be emphasized that the use of amedium-size basis set is also important from the perspective ofreducing memory requirements because memory limitationcan be critical for large systems rather than the computationalcost.In summary, we revealed that both BSSE and BSIE are notnegligible in DMC binding energy calculations if one targets tocompute binding energies of complex systems within thesubchemical accuracy (i.e., ∼0.1 kcal/mol). The augmentation(i.e., more diffuse functions) of a basis set and the CPcorrection for a basis set are both helpful to reduce BSIEs, i.e.,to get binding energies in the CBS limit.5. A24 BENCHMARK TEST REVISITEDBenchmarks for the A24 set were done by Dubecky ́ et al.35 andby Nakano et al.67 with the aug-TZV basis sets associated withthe ECPs developed by Burkatzki et al.68 and the cc-pVTZbasis sets associated with the ECPs developed by Bennett etal.,49,50 respectively. Root mean square deviation (RMSD) ofthe binding energies from CCSD(T) reported by Dubecky ́ etal.35 and Nakano et al.67 are 0.15 and 0.315 kcal/mol,respectively. Table 1 shows the binding energies obtained inthis study by DMC calculations with the aug-ccpV6Z basis sets(without CP) associated with the ECPs developed by Bennettet al.,49,50 and those obtained by CCSD(T) in the CBS limittaken from Benchmark Energy and Geometry DataBase(BEGDB).69 In this work, we obtained a RMSD of 0.135kcal/mol, which is very close to the value obtained byDubecky,́ while ∼0.2 kcal/mol off from the value reported byNakano et al. As mentioned in the previous section, Figure3(a) indicates that the cc-pVTZ basis set without the CPcorrection shows non-negligible BSIEs and the augmentation(aug-ccpVTZ) reduces the BSIEs significantly. In fact, we got0.247(14) and 0.131(14) kcal/mol for RMSD with cc-pVTZand aug-ccpVTZ basis sets, respectively. The obtained cc-pVTZ value (0.247(14) kcal/mol) is very close to thosepreviously reported by Nakano et al.67 (0.315 kcal/mol),although the treatments of the nonlocal terms are different(DLA was employed in the previous study, while DTM isemployed in the present study). As such, the RMSD obtainedby Nakano et al.67 should be a little affected by BSIEs, whilethe values obtained by Dubecky ́ et al.35 with the augmentedbasis sets should already reach the CBS limit. Thus, as thebenchmark values for the A24 data set, one should refer to thebinding energies obtained by Dubecky ́ et al.35 or thoseobtained in this work.6. CONCLUSIONSIn this study, we investigated two basis-set related errors,BSIEs and BSSEs, in binding energy calculations by ab initioFN-DMC calculations using the A24 benchmark set. Werevealed that BSIE and BSSE are not negligible in DMCcalculations when a small basis set, such as cc-pVDZ and cc-pVTZ, is used without the CP correction. Our study impliesthat, to get binding energies in the CBS limit with GTOs, oneshould use, at least, a medium-size basis set, such as cc-pVQZor aug-cc-pVTZ basis set. We found that the CP correction isalso helpful in DMC calculations to reduce BSIEs, as in QCcalculations. With the CP correction, one can use a smallerbasis, such as cc-pVTZ or aug-cc-pVDZ basis sets. This workraises awareness of BSSEs and BSIEs in binding energycalculations by DMC, which have not been extensively studiedpreviously. In the future, it would be interesting to perform amore comprehensive study investigating BSIEs in DMC forlarger molecules or periodic systems.■ ASSOCIATED CONTENTData Availability StatementThe QMC kernels used in this work, TurboRVB andQMCPACK, are available from their GitHub repositories,[https://github.com/sissaschool/turborvb] and [https://github.com/QMCPACK/qmcpack], respectively.*sı Supporting InformationThe Supporting Information is available free of charge athttps://pubs.acs.org/doi/10.1021/acs.jctc.4c01631.Figures plotting binding energies obtained by themethods employed in this study (HF, MP2, andDMC) with various basis sets (PDF)■ AUTHOR INFORMATIONCorresponding AuthorsKousuke Nakano − Center for Basic Research on Materials,National Institute for Materials Science (NIMS), Tsukuba,Ibaraki 305-0047, Japan; orcid.org/0000-0001-7756-4355; Email: kousuke_1123@icloud.comAndrea Zen − Dipartimento di Fisica Ettore Pancini,Universita ̀ di Napoli Federico II, I-80126 Napoli, Italy;Department of Earth Sciences, University College London,London WC1E 6BT, United Kingdom; orcid.org/0000-0002-7648-4078; Email: andrea.zen@unina.itAuthorsBenjamin X. Shi − Yusuf Hamied Department of Chemistry,University of Cambridge, Cambridge CB2 1EW, UnitedKingdom; orcid.org/0000-0003-3272-0996Journal of Chemical Theory and Computation pubs.acs.org/JCTC Articlehttps://doi.org/10.1021/acs.jctc.4c01631J. Chem. Theory Comput. 2025, 21, 4426−44344431https://pubs.acs.org/doi/suppl/10.1021/acs.jctc.4c01631/suppl_file/ct4c01631_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jctc.4c01631/suppl_file/ct4c01631_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jctc.4c01631/suppl_file/ct4c01631_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jctc.4c01631/suppl_file/ct4c01631_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jctc.4c01631/suppl_file/ct4c01631_si_001.pdfhttps://github.com/sissaschool/turborvbhttps://github.com/QMCPACK/qmcpackhttps://github.com/QMCPACK/qmcpackhttps://pubs.acs.org/doi/10.1021/acs.jctc.4c01631?goto=supporting-infohttps://pubs.acs.org/doi/suppl/10.1021/acs.jctc.4c01631/suppl_file/ct4c01631_si_001.pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Kousuke+Nakano"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0001-7756-4355https://orcid.org/0000-0001-7756-4355mailto:kousuke_1123@icloud.comhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Andrea+Zen"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0002-7648-4078https://orcid.org/0000-0002-7648-4078mailto:andrea.zen@unina.ithttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Benjamin+X.+Shi"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0003-3272-0996https://pubs.acs.org/action/doSearch?field1=Contrib&text1="Dario+Alfe%CC%80"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfpubs.acs.org/JCTC?ref=pdfhttps://doi.org/10.1021/acs.jctc.4c01631?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asDario Alfe ̀ − Dipartimento di Fisica Ettore Pancini, Universita ̀di Napoli Federico II, I-80126 Napoli, Italy; Department ofEarth Sciences, University College London, London WC1E6BT, United Kingdom; Thomas Young Centre and LondonCentre for Nanotechnology, London WC1H 0AH, UnitedKingdom; orcid.org/0000-0002-9741-8678Complete contact information is available at:https://pubs.acs.org/10.1021/acs.jctc.4c01631NotesThe authors declare no competing financial interest.■ ACKNOWLEDGMENTSK.N. is grateful for computational resources from theNumerical Materials Simulator at National Institute forMaterials Science (NIMS). K.N. is grateful for computationalresources of the supercomputer Fugaku provided by RIKENthrough the HPCI System Research Projects (Project ID:hp240033). K.N. acknowledges financial support from MEXTLeading Initiative for Excellent Young Researchers (Grant No.JPMXS0320220025) and from Iketani Science and Technol-ogy Foundation (Grant No. 0361248-A). B.X.S. acknowledgessupport from the EPSRC Doctoral Training Partnership (EP/T517847/1) and from the European Union under the “n-AQUA” European Research Council project (Grant No.101071937). D.A. and A.Z. acknowledge support fromLeverhulme grant no. RPG-2020-038. D.A. and A.Z. alsoacknowledge support from the European Union under theNext generation EU (projects 20222FXZ33 andP2022MC742). The authors acknowledge the use of theUCL Kathleen High Performance Computing Facility(Kathleen@UCL), and associated support services, in thecompletion of this work. This research used resources of theOak Ridge Leadership Computing Facility at the Oak RidgeNational Laboratory, which is supported by the Office ofScience of the U.S. Department of Energy under Contract(No. DE-AC05-00OR22725). Calculations were also per-formed using the Cambridge Service for Data DrivenDiscovery (CSD3) operated by the University of CambridgeResearch Computing Service (www.csd3.cam.ac.uk), providedby Dell EMC and Intel using Tier-2 funding from theEngineering and Physical Sciences Research Council (capitalgrant EP/T022159/1 and EP/P020259/1). This work alsoused the ARCHER U.K. National Supercomputing Service(https://www.archer2.ac.uk), the United Kingdom Car−Parrinello (UKCP) consortium (EP/F036884/1).■ ADDITIONAL NOTES1Note that PW basis sets are not affected by any BSSE, whilethey can be affected by a BSIE when the PW cutoff is toosmall.2Note that EbCP ≥ Eb, because EbBSSE ≤ 0 as MAB > MA and MAB> MB.3A DMC computation with the spline basis is independent of achosen grid-size,66 while its memory requirement scales.■ REFERENCES(1) Ceperley, D. M. The statistical error of green’s function MonteCarlo. J. Stat. Phys. 1986, 43, 815−826.(2) Foulkes, W. M. C.; Mitas, L.; Needs, R. J.; Rajagopal, G.Quantum Monte Carlo simulations of solids. Rev. Mod. Phys. 2001,73, No. 33.(3) Drummond, N. D.; Monserrat, B.; Lloyd-Williams, J. H.; Ríos, P.L.; Pickard, C. J.; Needs, R. J. Quantum Monte Carlo study of thephase diagram of solid molecular hydrogen at extreme pressures. 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