Noncovalent Interactions by Fixed-Node Diffusion Monte Carlo

Article pubs.acs.org/JCTC

Noncovalent Interactions by Fixed-Node Diffusion Monte Carlo: Convergence of Nodes and Energy Differences vs Gaussian Basis-Set Size Matús ̌ Dubecký* Department of Physics, Faculty of Science, University of Ostrava, 30. dubna 22, 701 03 Ostrava, Czech Republic ATRI, Faculty of Materials Science and Technology, Slovak University of Technology, Paulínska 16, 917 24 Trnava, Slovakia ABSTRACT: Convergence of fixed-node (FN) shape and FN diffusion Monte Carlo (FNDMC) interaction energies is studied vs the Gaussian basis set saturation level in HF and CH4 dimers and one-determinant Slater-Jastrow trial wave functions (ΨT). The tested 25 distinct basis sets obtained by stepwise trimming of aug-VDZ and aug-VTZ bases suggest minimum basis set requirements to achieve reasonable results. A single selected trimmed basis set, about 2 times smaller in size than aug-VTZ, is extensively tested on a set of 12 noncovalent complexes including formic acid dimer, benzene-methane, or coronene-H2. The results indicate that equivalent noncovalent FNDMC energy differences are available at costs lower than assumed before. Additional insights from electron density differences and comparison of dimer vs monomer ΨT nodes explain this observation.

regime, the nodes of ΨA are merely unaffected by the presence of a weakly interacting system described by ΨB and vice versa.17,21,30−32 It turned out that, in a number of closed-shell noncovalent (NC) systems, interaction energies (ΔE) from single-point FNDMC using orbital-unoptimized SD SJ ΨT’s attain accuracy competitive to CCSD(T)/CBS17,32−39 or sometimes even CCSDT(Q).40 Some of the NC interactions therefore appear to be sufficiently weak so that the FN bias present in approximate ΨT mostly cancels out. This justifies use of FNDMC in studies of many NC systems21 including molecular crystals,41−43 materials,39,44−50 and NC complexes,51,52 some far beyond the possibilities of CCSD(T) or MP2.38,49,53−57 The FN bias cancellation approach has indeed its limits,21,58 and more research is required along this line.59 One of the FNDMC limitations is a large CPU cost prefactor that stems mainly from its stochastic nature. Millions of CPU-hours are often invested in promising large-scale projects.53,54,57,60 It would be desirable to single out ways to reduce the related CPU costs to a maximum possible extent. This can be acomplished by algorithmic and methodological improvements61−65 as well as by simplification of ΨT. Traditionally, three-center J terms and high-quality oneparticle basis sets have been used to construct ΨT for FNDMC calculations,21 keeping their CPU cost rather high. Focusing on NC systems and SJ wave functions, it recently turned out that three-center terms in J may be replaced by two-center ones without a significant effect on accuracy and local energy

1. INTRODUCTION Fixed-node diffusion Monte Carlo (FNDMC) is a projector many-body electronic structure method1−3 promising for its accurate description of electron correlations, direct treatment of extended systems, low-order polynomial CPU cost scaling with the number of electrons N, ∝ O(N3−4), and massive parallelism.4 FNDMC recovers all symmetric correlations exactly as it samples electron configurations in real-space, and its accuracy is limited by the node of the supplied trial wave function ΨT that fixes antisymmetry of the simulated state. The total FNDMC energy is an upper bound to the exact energy,5 and the total energy bias due to the approximate ΨT scales quadratically with the nodal displacement error.6 Economic ΨT’s thus often lead to reliable FNDMC results. Extensive details on the FNDMC method, its performance, and applications can be found elsewhere.4,7−17 In principle, ΨT can be expanded in a large number of manybody terms (e.g., determinants) so that the related FN bias becomes sufficiently small;18−20 however, such an approach is not feasible for large systems. The practical FNDMC computations of energy differences therefore rely on FN bias cancellation21 based on compact ΨT’s like the Slater−Jastrow (SJ) ansätze, most often expressed as a product of a single Slater determinant (SD) and explicit correlation Jastrow (J) term.22−29 Although the FN bias caused by approximate ΨT is in general hard to control, in certain circumstances, it largely cancels out in energy differences. For instance, this happens in the limit of sufficiently weak interaction when the state of the interacting system ΨAB may be represented to a good approximation as a product of its subsystem states: ΨAB ≈ ΨA ⊗ ΨB. In this © XXXX American Chemical Society

Received: May 24, 2017 Published: July 7, 2017 A

DOI: 10.1021/acs.jctc.7b00537 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

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Journal of Chemical Theory and Computation

Figure 1. SD SJ FNDMC interaction energies ΔE in HF−HF and CH4−CH4 vs the one-particle basis set size (specified for F/C) used to expand B3LYP orbitals in ΨT (using J with ee and en terms). Each segment was obtained with a fixed valence basis set (indicated at the bottom), and the diffuse functions vary within each segment (range indicated at the top). The trimmed basis sets were derived from aug-VDZ (blue triangles) and aug-VTZ (black circles) basis sets. The 1s-rVTZ results are indicated by the filled black circles. As a guide to the eye, the reference ΔE corresponding to aug-VTZ FNDMC results is indicated by the solid red line, and red dashed lines correspond to a ±0.1 kcal/mol margin from this reference.

fluctuations, leading to significant speedup.58 What should also be explored is a possibility of one-particle basis-set size reduction in the Slater part, the main motivation for the present work. It is known that total energies from FNDMC are not as critically sensitive to one-particle basis sets66,67 used to express the one-particle orbitals as the wave function methods of quantum chemistry, like, e.g., CCSD(T).21 Although the occupied shells in ΨT for FNDMC should be saturated as much as possible,68 high-angular momentum functions can often be safely removed, since correlations are recovered in a differet way.31,60,69 For some NC complexes, it was found that the cardinality of Gaussian basis sets beyond the VTZ level plays a less important role than the presence of diffuse functions.32,58,70 In larger NC complexes, dense Gaussian basis sets may lead to linear dependency problems that artificially reduce the FNDMC accuracy.31,60 Dunning-type aug-VTZ basis sets used in SD ΨT were found to provide a reasonable trade-off between the size and accuracy of FNDMC ΔE’s (0.1−0.2 kcal/mol) in a number of small NC closed-shell s/p complexes.32,58 Interest in large models challenges the FNDMC CPU cost and puts forward the following question: what is the smallest possible basis set leading to converged NC energy differences? Our recent progress report,71 where we provide an example of a single trimmed basis set (tested on seven complexes), indicates that significantly but sensibly trimmed basis sets in ΨT may lead to results comparable to those from the full aug-VTZ basis set. Limits of the basis-set saturation in NC energy differences nevertheless still remain unknown and call for more attention. Here, we present results that extend our preliminary work71 in two distinct ways. First, we study the convergence of nodal shapes and FNDMC ΔE’s in NC systems vs the saturation level of Gaussian basis sets used to express the orbitals in SD SJ ΨT’s. To this end, we stepwise trim the aug-VDZ and aug-VTZ basis sets to produce 25 different basis sets with an increasing level of flexibility. For each such basis set, we construct ΨT and use it in FNDMC to compute ΔE and to study the nodal shape

convergence, in dimers of HF and CH4. Second, we test the performance of the selected best-trade-off basis set (referred to as 1s-rVTZ) on an extended set of 12 NC complexes, and in four cases, we also consider an alternative type of J. It turns out that the results produced by the 1s-rVTZ basis set are identical with the aug-VTZ data in all considered settings. Since the local energy fluctuations are not significantly affected by the trimming, this finding indicates that equivalent NC FNDMC energy differences are available at costs lower than those assumed to date. An extensive analysis of basis set effects on electron density differences and monomer vs dimer ΨT nodes justifies the observed trends.

2. MODELS AND METHODS Below, we report on FNDMC interaction energies defined as ΔE = EAB − EA − E B

(1)

where E’s denote the total energies of the interacting complex AB and its constituent molecules A and B in dimer geometry; i.e., deformation energy is not considered. Note that accurate ΔE’s that are of our sole interest in this work represent a first and necessary step of FNDMC development toward subkcal/mol accuracy before its future applications involving temperature72,73 and/or quantum nuclei74,75 at this high-standard accuracy level. We consider 12 closed-shell s/p complexes: eight from the database A24,76 ammonia dimer (NH3−NH3), water−ammonia (H2O−NH3), water dimer (H2O−H2O), ammonia−methane (NH3−CH4), HF dimer (HF−HF), ethene dimer (C2H2− C2H2), methane dimer (CH4−CH4), and HCN dimer (HCN− HCN); three from the set S22,77 benzene−water (Bz−H2O), benzene−methane (Bz-CH4), and formic acid dimer (FA−FA); and coronene−H2 (Cor−H2) complex.51 The atomic cores in the studied complexes were replaced by the effective core potentials (ECPs).78,79 The corresponding VDZ and VTZ valence basis sets were used together with the diffuse (aug-) functions from the related all-electron Dunning bases.80 In the case of aug-VDZ, the related reduced basis sets B


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The J parameters were optimized for each system and aug-VTZ-based ΨT separately for dimer/monomer within the variation MC (VMC) by the Hessian driven method consisting of at least 10 × 10 VMC iterations (i.e., 100 iterations where the distribution of walkers was refreshed every 10 iterations), using a linear combination83 of energy (95%) and variance (5%) as a cost function. Since we are interested in differences between FNDMC results caused by the basis sets used in ΨT, in HF−HF and CH4−CH4, we minimize the effect of J by keeping it fixed across the whole considered basis set sequence, i.e., without reoptimization (that has only a negligible effect, see Table 1). The ΔE’s reported in Table 3 were obtained with the consistently optimized J as usual. The respective ΨT’s were subsequently used in the production FNDMC runs using the T-moves scheme to treat ECPs84 and a time step of 0.005 au.58 The target FNDMC populations amounted to 16 000−32 000 walkers. All VMC and DMC calculations were performed by the QWalk85 code. The ribbon-like cuts through the many-body nodal hypersurfaces32 (Figures 2, 3, and 5) were produced by singleelectron scans of ΨT’s in a squeezed 3D cuboid volume, keeping the remaining N − 1 electron positions (sampled from ΨT) fixed in both dimer and monomer alike. To produce directly comparable slices, the monomer scans used a subset of (well separable) dimer electron configurations selected by the criterion of the smallest distance from the monomer.

were produced by the aug- part trimming. In the case of bases derived from aug-VTZ, both valence as well aug- basis set parts were trimmed in a stepwise fashion, as indicated in Figure 1. In total, this procedure led to 25 individual basis sets including aug-VDZ, aug-VTZ, and the selected 1s-rVTZ. For comparison, in the case of a carbon atom, the reference aug-VTZ basis set contains 45 basis functions ([3s3p2d1f] + [1s1p1d1f]), while the 1s-rVTZ contains 23 of them ([3s3p2d] + [1s]). In the hydrogen atom, aug-VTZ contains 23 ([3s2p1d] + [1s1p1d]) and 1s-rVTZ contains nine ([2s2p] + [1s]) basis functions. All FNDMC calculations used SD SJ9 ΨT constructed of canonical unoptimized B3LYP orbitals (GAMESS81 code) and the Schmidt−Moskowitz82 J expanded in polynomial Padé functions9 containing two-center electron−electron (ee) and electron− nucleus (en) terms, or three-center een terms in addition.58 Table 1. Effect of J Optimization (Here J with ee and en Terms) on 1s-rVTZ-based FNDMC ΔE’s (kcal/mol)a complex

reused J

optimized J

HF−HF CH4−CH4

−4.75 ± 0.05 −0.62 ± 0.02

−4.79 ± 0.06 −0.65 ± 0.04b

The “reused J” denotes 1s-rVTZ ΔE’s (that correspond to the filled circles in Figure 1) produced with the J term variationally optimized in the presence of Slater determinant orbitals expressed in the aug-VTZ basis set. The “optimized J” ΔE values correspond to the calculaitons with consistently optimized J (used in Table 3). bRef 71.

a

Figure 2. Convergence of the many-body nodes of ΨT in the HF−HF complex (zoom to the right monomer) vs one-particle basis set saturation (cf. convergence of ΔE in Figure 1, left). Rows, valence basis sets; columns, diffuse functions; red, actual basis set; gray, reference aug-VTZ basis set. C


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3. CONVERGENCE OF INTERACTION ENERGIES AND NODES Convergence of FNDMC energy differences, ΔE, vs basis set quality is reported in Figure 1 for HF−HF and CH4−CH4 complexes. As expected, with an increasing basis set size, ΔE’s approach a constant limiting value that solely depends on J (via ECP term in FNDMC) and type of orbitals (that determine the nodes) used in our ΨT’s. The convergence may not be monotonous over the whole sequence of basis sets considered, because of (i) statistical uncertainty (one σ reported) and (ii) since some of the individual basis sets may be less or more suboptimal (some contain higher angular momentum in diffuse part than in valence part, or, excess of certain basis functions) after trimming since the basis functions are not consistently optimized as in the original Dunning construction.80 This may cause some bias in ΔE, although it looks that this effect is not critical since only deviations of up to 0.1 kcal/mol (the most prominent one being ΔE in CH4−CH4 from [3s3p2d] + [1s1p1d]) are observed for reasonable basis sets (above [3s3p2d]). In the hydrogen bound HF−HF complex, we observe that less flexible valence basis sets derived from aug-VTZ (Figure 1 left, black circles) perform poorly ([3s3p], [3s3p] + [1s], [3s3p] + [1s1p] and [3s3p1d]) and significantly overestimate the reference ΔE unless additional higher angular-momentum functions bring more flexibility to the respective ΨT’s

(e.g., [3s3p] + [1s1p1d], [3s3p1d] + [1s1p1d], [3s3p2d] and above). These data indicate that, indeed, the d functions are necessary to capture the polarization effects to approach the reference ΔE limit; however f functions have only little effect in FNDMC. The nodal shape convergence in the HF−HF complex vs the basis set (Figure 2) provides transparent understanding of the observed ΔE behavior. One can clearly identify that the nodal surface ΨT based on the [3s3p] + [1s1p1d] basis set is rather far from the reference, and therefore, the agreement of ΔE with the reference appears fortuious. On the other hand, the nodes obtained with the [3s3p2d] basis set, and above, but also [3s3p1d] + [1s1p1d] qualitatively reproduce the aug-VTZ nodes well, so the similar ΔE may be attributed to similar FN shapes and related similar FN biases that cancel out in energy differences. The observed variations of ΔE from the reference (Figure 1) for bases above [3s3p2d] are therefore attributed to statistical uncertainty and possible suboptimality (see above) of the Gaussian basis set exponents. Finally, for comparison, the VDZ-related basis sets (Figure 1 left, blue triangles) show convergence toward the reference if at least diffuse functions of p type are included in the basis set. But, the whole aug-VDZ-derived segment is quite disperse, and we discourage the use of VDZ-based basis sets without further tests. In the CH4 dimer, with an increasing basis set size, contrary to HF−HF, the reference ΔE limit is achieved on average from

Figure 3. Convergence of the many-body nodes of ΨT in the CH4−CH4 complex (zoom to the left monomer) vs one-particle basis set saturation (cf. convergence of ΔE in Figure 1, right). Rows, valence basis sets; columns, diffuse functions; red, actual basis set; gray, reference aug-VTZ basis set. D


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hard to accept, since, if the valence polarization effects are captured well, the dispersion interactions of s type that take place in low-density86−88 areas with negligible exchange contributions dominate the energy differences, and residual valence FN bias cancels out.21 This is confirmed by the observation that all ΔE’s in CH4, including those produced with poor basis sets, lie acceptably close to the reference. This finding also indicates that in dispersion complexes, ΔE itself is not a good indicator of the basis-set related FN convergence. Putting it together, the current data suggest that the trimmed basis sets with at least two valence polarization functions, much smaller in size than the reference aug-VTZ basis set, are capable of reaching accuracy competitive to the full aug-VTZ basis set. Since the polarization effects generally take place in the valence region, we take the [3s3p2d] valence basis set as a reasonable starting point (instead of [3s3p1d] + [1s1p1d] that could be also inferred from Figures 2 and 3 as a reasonable candidate). In addition, since London dispersion is unidirectional, i.e., of s symmetry, and reference ΔE’s are approached to within 0.1 kcal/mol from the reference even without diffuse functions, we assume that the polarization effects are well captured by the valence functions and p-type, or higher-order diffuse functions are not crucial in SD FNDMC. Our data in HF−HF and CH4−CH4 (Table 2) as well as previous experience89 show that diffuse functions improve ΔE’s, and in some complexes, sufficient accuracy cannot be achieved if they are omitted.58 For instance, in the NH3−NH3 complex (slightly different geometry than used here), a sequence of VTZ, VQZ, and aug-VTZ basis sets leads to ΔE = −3.33 ± 0.07, −3.47 ± 0.07, and −3.10 ± 0.06 kcal/mol,58 and only the value obtained with the augmented basis set approaches the CCSD(T)/CBS benchmark (−3.15 kcal/mol).77,90 In this complex sensitive to the presence of diffuse functions, the presence of a single diffuse s function (1s-rVTZ vs [3s3p2d] or rVTZ) makes a non-negligible difference in ΔE on the order of ∼0.25 kcal/mol (Table 2),

above (Figure 1, right). For smaller aug-VTZ-based bases, two isolated ΔE’s fall close to the reference ([3s3p] + [1s1p1d1f] and [3s3p1d]), but this appears to be fortuious. The convergence of ΔE is obtained for [3s3p1d] + [1s1p1d1f], [3s3p2d], and [3s3p2d] + [1s] basis sets and above. Note that in one case ([3s3p2d] + [1s1p1d]), ΔE lies quite away from the reference that is attributed to a suboptimal combination of basis set primitives (overabundant d and no f functions, cf. above). The general nodal surface convergence in CH4−CH4 (Figure 3) is akin to HF−HF and shows that only basis sets above [3s3p2d] + [1s] reach convergence due to the similar FN properties. Note that in this complex, walkers with visible differences between [3s3p2d] and [3s3p2d] + [1s] basis-set results were more abundant than in HF−HF. The reference ΔE limit is nevertheless achieved in a similar basis set range like in the HF−HF complex. The VDZ-based ΔE’s (Figure 1 right, blue triangles) show a good agreement vs the reference ΔE and a steady improvement along the considered sequence. An important and promising observation from Figures 1, 2, and 3 is that the convergence to the aug-VTZ reference is approached for bases much smaller in size than aug-VTZ, in both complexes. It looks that since dispersion effects are recovered in FNDMC very accurately, and only weakly depend on the intermolecular node location errors, the nodal bias cancellation21,30−32 starts to be operative for bases able to sufficiently describe the valence polarization effects. This is not Table 2. Effect of Diffuse Functions on ΔEs (kcal/mol)a J

complex HF−HF CH4−CH4 NH3−NH3 NH3−NH3

ee, ee, ee, ee,

en en en en, een

rVTZ −4.63 −0.58 −3.48 −3.42

± ± ± ±

1s-rVTZ

0.09 0.02 0.06 0.04

−4.79 −0.65 −3.36 −3.17

± ± ± ±

0.06 0.04 0.08 0.04

aug-VTZ −4.71 −0.63 −3.30 −3.24

± ± ± ±

0.05 0.03 0.04 0.04

a Reported are results obtained with [3s3p2d] (rVTZ), [3s3p2d] + [1s] (1s-rVTZ), and aug-VTZ basis sets and consistent J.

Table 3. Comparison of FNDMC Interaction Energies ΔE (kcal/mol), Local Energy Variances σ2, Number of aug-VTZ/ 1s-rVTZ Basis Functions (Mref/Mtrim), Relative Deviations RD (%), and Ideal and Real Speedups si and sr (See Text), in the Dimer of the Given Complex aug-VTZ complex

ΔE

J with ee, en terms NH3−NH3a −3.30 H2O−NH3a −6.71 H2O−H2Oa −5.30 HF−HF −4.71 NH3−CH4a −0.83 CH4−CH4a −0.63 C2H2−C2H2 −1.08 HCN−HCNa −5.09 Bz−H2O −3.21 Bz−CH4 −1.50 FA−FA −19.60 Cor−H2 −1.03 J with ee, en, een terms NH3−NH3 −3.24 H2O−H2O −5.12 HF−HF −4.76 CH4−CH4 −0.54 a

σ

1s-rVTZ 2

ΔE

Mref

σ2

Mtrim

RD

si

sr

± ± ± ± ± ± ± ± ± ± ± ±

0.04 0.07 0.05 0.05 0.06 0.03 0.07 0.08 0.15 0.15 0.10 0.16

0.464 0.601 0.706 0.960 0.370 0.266 0.473 0.619 1.080 0.846 1.583 2.590

228 205 182 136 251 274 364 226 499 545 362 1402

−3.36 −6.64 −5.25 −4.79 −0.77 −0.65 −1.07 −4.97 −3.17 −1.59 −19.90 −0.97

± ± ± ± ± ± ± ± ± ± ± ±

0.08 0.09 0.09 0.06 0.07 0.04 0.08 0.08 0.09 0.10 0.10 0.19

0.514 0.637 0.762 0.974 0.371 0.292 0.483 0.629 1.105 0.834 1.621 2.707

106 96 86 66 116 126 172 112 241 261 178 692

−1.82 1.04 0.94 −1.70 7.23 −3.17 0.93 2.36 1.25 −6.00 −1.53 5.83

1.94 2.01 1.96 2.03 2.16 1.99 2.07 1.99 2.02 2.12 1.99 1.94

1.30 1.29 1.27

± ± ± ±

0.04 0.04 0.03 0.04

0.275 0.411 0.585 0.171

228 182 136 274

−3.17 −5.15 −4.75 −0.58

± ± ± ±

0.04 0.06 0.05 0.05

0.285 0.434 0.588 0.180

106 86 66 126

2.16 −0.59 0.21 −7.41

2.07 2.00 2.05 2.06

1.02

1.30 1.27 1.50 1.35 1.67

1.03

Ref 71. E


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where “r” stands for “reduced,” and we explore its properties in more detail in the following section.

analogous to the case of aug-VTZ vs VTZ, again confirming the importance of diffuse functions. We thus come to the conclusion that the minimum reasonable choice of a basis set for FNDMC in NC s/p complexes should contain well saturated occupied shell contractions with flexible polarization functions that guarantee nodal shape convergence (here achieved with two polarization functions per main-group atom) and at least a single diffuse function of s type that improves sampling of intermolecular regions. These requirements are satisfied by the [3s3p2d] valence basis set augmented with a single diffuse s function per atom (i.e., [3s3p2d] + [1s]). This basis set (Figure 1, filled circles) approaches the reference ΔE’s well in both tested complexes, as well as in the sensitive NH3 dimer. We denote it as 1s-rVTZ,

4. SELECTED BASIS SET The FNDMC results obtained with ΨT’s expanded in aug-VTZ and the reduced 1s-rVTZ basis set are reported in Table 3 and Figure 4. Although the 1s-rVTZ basis set is about 2 times smaller in size than aug-VTZ, and such a reduction might indeed seem drastic (as usual in wave function theory), FNDMC ΔE’s obtained with 1s-rVTZ are, within error, clearly compatible with the aug-VTZ data in all the considered cases. Even in the strongly bound FA−FA complex, the overall relative deviation amounts to −0.3 kcal/mol or −1.5%, which lies well within the statistical uncertainty and within the benchmark accuracy bound (2%). Interestingly, this holds also for different types of J used in four complexes (Table 3, bottom). We find that, although the aug-VTZ results differ for the two types of J considered, the 1s-rVTZ data vary in the same way. It looks that the bias coming from J (via ECP term in FNDMC) alone is unaffected by the basis set trimming, and it also appears that the basis-set incompleteness compensation is similar for both bases irrespective of J. We attribute these findings to a similar degree of bias compensation in FNDMC ΔE’s and sufficient sampling of intermolecular regions secured by a single diffuse function per atom (as confirmed for NH3−NH3 above). In Figure 5, we provide a set of illustrative cuts through the monomer and dimer ΨT nodal hypersurfaces of HF−HF obtained with aug-VTZ and 1s-rVTZ basis sets. Clearly, the nodes from both basis sets visually appear very similar. For both bases, the monomer vs dimer nodes lie on top of each other in the intramolecular regions where the electron density is high (places that dominate total energies). This supports the hypothesis that similar ΔE’s result from a similar degree of

Figure 4. Identity of FNDMC interaction energies ΔE (kcal/mol) from Table 3 (FA−FA not shown) obtained with 1s-rVTZ vs aug-VTZ basis set. The error bars (not shown) are smaller than the symbol size.

Figure 5. Representative ribbon-like 3D cuts through the many-body nodal hypersurfaces of ΨT expanded in aug-VTZ and 1s-rVTZ basis sets for the HF dimer (D) and its right monomer (M). For an easy comparison, overlap of M and D nodes is depicted in a pair of panels labeled “M, D.” Note that the nodal surfaces overlap well in the selected monomer region (the same is true for the remaining monomer, not shown). They appear very similar for the two distinct basis sets considered, which rationalizes a similar degree of bias cancellation in FNDMC energy differences. F


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some complexes, such an enhancement may be crucial to achieving reasonable accuracy. The 1s-rVTZ total energies (in dimers) are typically higher by less than 0.001 au per atom vs the aug-VTZ values. For instance, in the Bz−CH4 complex, the total energy related to 1s-rVTZ is higher by only 0.003 au (−45.8096 au vs −45.8124 au) and 0.0002 au per atom. Together with the insights from the nodes (Figures 2, 3, and 5) this indicates that the proposed trimming does not significantly affect the flexibility of the basis set vs the node location. The local energy variances (σ2) also differ only slightly (Table 3, σ2), which indicates that the places of ΨT that dominate the statistical sampling efficiency are not significantly affected by such a trimming. The similarity in total energies and σ’s in FNDMC can be attributed to the use of identical occupied-shell contractions in reference vs trimmed one-particle basis sets and negligible importance of high angular-momentum functions in the considered closed-shell s/p complexes. Finally, we briefly discuss the CPU cost benefits of the 1s-rVTZ vs aug-VTZ basis set usage. Since the Slater matrix evaluation step in FNDMC scales as O(MN2),4,13,61 where M is the number of basis functions and N is the number of electrons, an ideal speedup si of a trimmed basis set containing Mtrim basis functions vs the reference basis set containing Mref independent functions is proportional to

si =

2 σref M ref 2 σtrim M trim

(2)

where σ2’s denote the respective local energy variances. The CPU cost benefits thus become interesting for Mtrim’s significantly smaller than Mref and only if the ratio σ2ref/σ2trim does not compensate the gain from Mref/Mtrim. Furthermore, the overall speedup observed in practical computations using the trimmed basis sets is smaller than the theoretical limit si, since other routines than the Slater matrix updates are executed as well. For instance, evaluation of J or ECPs. The real speedup sr thus approaches si only for large N and only if the Slater matrix updates asymptotically dominate the overall CPU cost (this is true for two-center J but not for the three-center J, see sr in Table 3). The use of the 1s-rVTZ basis set instead of aug-VTZ in combination with a two-center J containing ee and en terms asymptotically offers about a 2-fold speedup (Table 3, si). The following examples illustrate the observed speedups of SD SJ FNDMC runs with J containing ee and en terms (for more details, see ref 71). In the HCN dimer (20 electrons) where Mref/Mtrim = 2.02, an evaluation cost speedup achieves sr = 1.27, rather far from the expected value si = 1.99. This indicates that the system is “small” in a sense that routines other than the Slater matrix updates significantly contribute to the computation cost. This finding also confirms that in systems where other routines than Slater matrix evaluation dominate, the basis set trimming is not very useful. On the other hand, a larger Cor-H2 complex (110 electrons, si = 1.94) shows a speedup factor of sr = 1.67. Clearly, in larger systems (Table3, sr), the speedups grow toward the theoretical limit and become more interesting.

Figure 6. Comparison of CH4−CH4 electron density differences (in plane passing through the monomer centers of masses) derived from ΨT’s using three pairs of basis sets: 1s-rVTZ vs rVTZ (top), aug-VTZ vs VTZ (middle), and aug-VTZ vs 1s-rVTZ (bottom). Red/blue color corresponds to enhancement/depletion of electron density caused by the presence of diffuse functions.

FN bias compensation in FNDMC energy differences. A comparison of electron density differences (Figure 6) obtained from ΨT’s expressed in terms of rVTZ vs 1s-rVTZ (in CH4−CH4) demonstrates how a single diffuse s function leads to an enhancement of the electron density in the intermolecular region and thus improves its sampling by FNDMC, in a way (qualitatively) similar to diffuse functions in aug-VTZ vs VTZ. As shown above for the NH3 dimer, in

5. CONCLUSIONS In summary, the convergence of FNDMC interaction energies and nodal surface shape was studied vs the Gaussian basis set size saturation level. The results obtained for 25 distinct basis sets in dimers of HF and CH4 provide interesting insights and G


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suggest room for basis set trimming without an effect on the accuracy of FNDMC ΔE’s. We stress that visual inspection of nodal shape convergence of ΨT vs the basis set saturation level provides a low-cost tool (a few DFT calculations) that enables one to optimize the number of basis set primitives vs the quality standard desired for production FNDMC runs. A single selected basis set (that should be considered as an interesting example) with a trimmed VTZ valence basis set and a single diffuse s function per atom (called 1s-rVTZ), about 2 times smaller in size than the reference aug-VTZ basis set, reaches sufficient nodal-shape convergence and statistically indistinguishable FNDMC results, while the local energy fluctuations increase only marginally. Additional analyses of the monomer vs dimer nodes of ΨT’s in the HF dimer and electron densitity differences in CH4 dimer rationalize such behavior. The results indicate that noncovalent FNDMC energy differences based on Gaussian-expanded ΨT’s with a two-center J are available at costs lower than assumed to date. In addition to CPU cost benefits, trimmed basis sets can readily find use in stacked π-conjugated complexes where the large basis sets like aug-VTZ show convergence and/or linear dependency problems91 that may hinder FNDMC accuracy.31

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS The author is grateful to Lubos Mitas, René Derian, and Petr Jurečka for fruitful discussions and useful comments on the manuscript. This work was financially supported by the University of Ostrava (UO, IRP201558), VEGA (2/0130/15, 1/0279/16), and the Ministry of Education, Science, Research and Sport of the Slovak Republic (003STU-2-3/2016). The calculations were performed at a local facility of UO (purchased from EU funds, CZ.1.05/2.1.00/19.0388), Metacentrum CESNET (LM2015042) and CERIT (LM2015085), TACC under XSEDE allocation (provided by Lubos Mitas), and Slovak infrastructure for high-performance computing (ITMS 26230120002 and 26210120002) funded by ERDF. This research also used a Director’s Discretionary allocation at the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357. Kind assistance by Dr. Anouar Benali is gratefully acknowledged.

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