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J Phys Chem A
J Phys Chem A
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jpcafh
The Journal of Physical Chemistry. a
1089-5639
1520-5215
American Chemical Society

39167776
10.1021/acs.jpca.4c03012
Article
Basis Set Extrapolation from the Vanishing Counterpoise Correction Condition
https://orcid.org/0009-0004-7570-136X
Fishman Vladimir †
https://orcid.org/0000-0002-4464-4057
Semidalas Emmanouil †
https://orcid.org/0000-0002-0005-5074
Martin Jan M. L. *†‡
† Department of Molecular Chemistry and Materials Science, Weizmann Institute of Science, 7610001 Reḥovot, Israel
‡ On sabbatical at Quantum Theory Project, University of Florida, Gainesville, Florida 32611, United States
* Email: gershom@weizmann.ac.il. Phone: +972 8 9342533. Fax: +972 8 9343029.
21 08 2024
05 09 2024
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07 05 2024
13 08 2024
09 08 2024
© 2024 The Authors. Published by American Chemical Society
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https://creativecommons.org/licenses/by/4.0/ Permits the broadest form of re-use including for commercial purposes, provided that author attribution and integrity are maintained (https://creativecommons.org/licenses/by/4.0/).

Basis set extrapolations are typically rationalized either from analytical arguments involving the partial-wave or principal expansions of the correlation energy in helium-like systems or from fitting extrapolation parameters to reference energetics for a small(ish) training set. Seeking to avoid both, we explore a third alternative: extracting extrapolation parameters from the requirement that the BSSE (basis set superposition error) should vanish at the complete basis set limit. We find this to be a viable approach provided that the underlying basis sets are not too small and reasonably well balanced. For basis sets not augmented by diffuse functions, BSSE minimization and energy fitting yield quite similar parameters.

Feinberg Graduate School, Weizmann Institute of Science 10.13039/501100001811 NA Alexander S. Onassis Public Benefit Foundation 10.13039/501100005302 FZP 052-2/2021-2022 Israel Science Foundation 10.13039/501100003977 1969/20 document-id-old-9jp4c03012
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Special Issue

Published as part of The Journal of Physical Chemistry Aspecial issue “Gustavo Scuseria Festschrift”.
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pmcIntroduction

Despite great recent progress in density functional theory, wave function ab initio methods such as coupled cluster theory can still routinely exceed the accuracy of the best DFT calculations by an order of magnitude, provided they are close enough to the one-particle basis set limit.

For atom-centered orbital basis sets, basis set convergence of the correlation energy is excruciatingly slow. Schwartz1,2 showed in the early 1960s that for the second-order correlation energy of helium-like atoms, the contributions of successive angular momenta (the “partial waves”) converge as1

Then if the basis set is truncated at angular momentum L, the total residual error is2

3

where ψ is the polygamma function. For large L, this function can be approximated by the asymptotic series4

and5

Hill3 generalized this result to configuration interaction, while Kutzelnigg and Morgan4 derived a general leading L–3 dependence for singlet-coupled, and L–5 for triplet-coupled, pair correlation energies. The latter authors also showed that in the presence of explicit R12 terms in the basis set, convergence will asymptotically be accelerated to L–7.

A similar leading ∝ L–3 dependence is obtained from two different sets of considerations. Carroll, Silverstone, and Metzger in 1979 showed5 that the basis set convergence in the principal expansion asymptotically converges as . For a given principal quantum number n, however, the angular quantum number l runs from 0 to n – 1, and the magnetic quantum number m from – l to + l. This leads to ∑l = 0n–1(2l + 1) = n2 approximately equal contributions, and hence an overall ∝ n–4 leading dependence. Summing over all missing shells, from nmax+1 to infinity, again leads us to a leading inverse-cubic ∝ n–3 dependence of eq 4.

Later, Petersson and co-workers6−9 considered the convergence of the correlation energy in a natural orbital expansion, and found it to converge as ∝ N–1 (with N the number of natural orbitals retained) for opposite-spin correlation, and ∝ N5/3 for same-spin correlation. As the number of natural orbitals in a basis set series such as the correlation consistent10 cc-pVnZ or atomic natural orbital11 ANO-n will converge with the cardinal number n as6

we once again recover an inverse-cubic dependence. (See also Klopper.12 For an illustration with natural orbitals in neon atom, see Figure 1 in the present work. The natural orbitals there were obtained from the spdfghi primitives in the cc-pV10Z basis set of Feller et al.13)

Figure 1 Convergence of CCSD(T) correlation energy of neon atom as a function of the number of natural orbitals included. Natural orbitals obtained from the spdfghi part of the cc-pV10Z basis set.

Applying such extrapolation formulas to the basis set convergence in molecules entails a major leap of faith. In the mid-nineties, Helgaker and co-workers14,15 and Martin16 found that this works well enough in practice; Klopper17 introduced the additional refinement that the correlation energy is partitioned between same-spin (strictly: “triplet-coupled pair”) and opposite-spin (strictly speaking: “singlet-coupled pair”) contributions, and that these contributions are extrapolated separately assuming L–5 and L–3 behavior, respectively. (The partitioning is not unique for open-shell systems: see ref (18))

Several variants have been introduced, such as those with variable exponents α of the form E(L) = E∞ + A/Lα (e.g., ref (19)), variable L-shift E∞ + A/(L + a)3 (Petersson20,21), variable cardinal numbers X(L) for the basis sets (Varandas22), etc. As explained in ref (23), all of them can be related to the same linear two-point extrapolation of Schwenke247

where we will refer to AL as a “Schwenke coefficient”, which is specific to the level of theory and the basis set pair.

Further work by Schwenke25 going up to L = 12 appears to indicate that after initial rapid convergence, a “diminishing returns” regime quickly sets in.

Basis Set Superposition Error

BSSE (basis set superposition error) results when an interaction energy between monomers A and B is evaluated in a finite basis as E(AB) – E(A) – E(B), where A only carries the basis functions of monomer A, and likewise for B. If the basis set on A is far from the CBS (complete basis set) limit, the availability in the dimer of the additional basis functions from the other monomer leads to an artifactual stabilization of the dimer known as BSSE.

Particularly in calculations on noncovalent interactions, BSSE can rival the interaction energy itself unless well-saturated and balanced basis sets are used.

The classic remedy is the counterpoise method,26 in which the monomer energies are effectively evaluated in the whole dimer basis set. BSSE can then be defined operationally as the difference between “raw” and corrected interaction energies.8

At the complete basis set limit, BSSE should be zero — and hence if an extrapolation works correctly, then the “raw” and counterpoise answers should be the same. Discrepancies thus indicate either a flaw in the extrapolation formula, or inadequate basis sets, or both.

We now propose to invert this observation — by using the requirement that BSSE should be zero, or minimized, as a means of obtaining basis set extrapolations.

This has the advantage that it relies neither on the theoretical behavior for an idealized system, nor on fitting (possibly themselves flawed) reference interaction energies for some training dataset.

To the best of our knowledge, the concept of deriving a basis set extrapolation from the BSSE limiting condition has never been explored. However, the NASA Ames team, in the late 1980s, did advocate using a negative multiple of the calculated BSSE as a correction for basis set incompleteness (e.g., refs (27,28)). Quoting Taylor28

Since BSSE is in some sense a measure of basis set incompleteness, one can contemplate increasing the bond energy by some fraction of the counterpoise correction to correct for this residual incompleteness, rather than decreasing it to correct the computed result for BSSE. This is a completely empirical approach, but we have found (for strong interactions) that in large basis sets (up to g functions, say) increasing the computed values by 150% of the calculated BSSE gives a good approximation to the best extrapolations to the basis set limit that we can perform from very large basis set studies.

Computational Details

All quantum chemical calculations were performed using either MOLPRO 2024.129 or Gaussian 16 rev. C.0130 running on the CHEMFARM cluster of the Faculty of Chemistry at Weizmann.

Three basis set sequences were considered:1. the nZaPa sequence (n = 2–7) of Ranasinghe and Petersson (RP)21

2. the augmented correlation consistent sequence of Dunning:aug-cc-pVnZ for first row: ref (31)

aug-cc-pV(n+d)Z for second-row elements32,33 (concerning why second-row elements in high oxidation states need tight 3d functions added, see ref (34) and references therein)

cc-pV7Z hydrogen, aug-cc-pV7Z carbon through fluorine: refs (35,36)

sulfur aug-cc-pV(7+d)Z from ESI of ref (37) (see also ref (38))

3. the core–valence correlation versions39,40 of the above, but used for valence correlation only. It has previously been shown41 that this practice considerably reduces BSSE.

The CCSD(T)42,43 electronic structure method was used throughout. For open-shell systems, we adopted the Watts-Gauss-Bartlett definition43 of restricted open-shell CCSD(T).

The molecules considered in the present work were all taken from the W4-17 thermochemical benchmark.44 Reference geometries given in its Supporting Information, each optimized at the CCSD(T)/cc-pV(Q+d)Z level, were used as-is without reoptimization.

Throughout the paper, notation like cc-pV{T,Q}Z refers to extrapolation, in the given example from cc-pVTZ and cc-pVQZ basis sets. The shorthands pVTZ+d, haVTZ+d, CVTZ, and haCVTZ refer, respectively, to cc-pV(T+d)Z, heavy-aug-cc-pV(T+d)Z, cc-pCVTZ, and heavy-aug-cc-pCVTZ. (The common practice of omitting diffuse functions on hydrogen, while placing them on more electronegative elements, goes by several names in the literature: aug′-cc-pVnZ by Del Bene,45 heavy-aug-cc-pVnZ by Hobza,46 and jul-cc-pVnZ in “calendar sets” notation.47)

For BSSE evaluation in polyatomics, we exclusively use the SSFC (site–site function counterpoise) of Wells and Wilson,48 as implemented in MOLPRO’s scripting language by one of us. Operationally, SSFC entails evaluating all monomer energies in the full oligomer basis set: the unmodified procedure may be inefficient for large clusters (where some sort of screening is called for ref (49)) but this is not an issue in small-molecule systems of the W4-17 type.

In principle, one could for each pair of basis sets and for each molecule i evaluate the AL,i that would make the extrapolated BSSE vanish9

and then take the average over all molecules in the test set . However, a more solid approach would seem to be least-squares minimization with respect to AL of the aggregate BSSE over the test set.10

the solution for which is easily found to be11

For those who prefer extrapolations in the familiar L–α form or the Petersson “shift” form (L + β)−3, the exponents and shifts are easily obtained from AL as follows (e.g., ref (23))12

13

Results and Discussion

Initial Exploration with 24 Heavy-Atom Diatomics

At first, we started out with a sample consisting of the 24 nonhydrogen diatomics in the W4-17 dataset. For these, we were able to carry out calculations through cc-pV(7+d)Z, heavy-aug-cc-pV(7+d)Z, and 7ZaPa with relative ease; we did so using Gaussian 16, as the largest basis sets entail k functions and MOLPRO can handle i functions at most. (Data for subsequent tables were generated using MOLPRO, which (see the Appendix to ref (50)) carries out semicanonicalization after integral transformation rather than before like most other codes; hence, fitted parameters for the 24-system dataset may differ subtly, on the order of 1–2 units in the third decimal place.)

For energetic comparisons near the one-particle basis set limit, we employed explicitly correlated data obtained through the rigorous CCSD(F12*) method52 with the aug-cc-pwCV5Z basis set and, in the F12 geminal, an exponent of 1.4. These data were extracted from the Supporting Information of ref (53); in previous work on a smaller sample,54 aug-cc-pwCV5Z was found to agree to 0.013 kcal·mol–1 RMS with the effectively saturated “Reference-h” basis set of Hill et al.55 We believe that a conservative error bar on our reference data would be about twice that, rounded upward, or 0.03 kcal·mol–1. Hence any two extrapolations whose error statistics differ by less than that need to be regarded as of equivalent quality.

As one can see in Table 1, while the Schwenke extrapolation parameters obtained through BSSE minimization are slightly different from literature values obtained through other approaches, they largely follow the same trend. Moreover, the difference between the RMSDs of BSSE-minimizing and TAE-error-minimizing extrapolations is within the uncertainty of the reference values.

Table 1 Schwenke Extrapolation Coefficients AL and RMS Deviations (kcal·mol–1) from 24 Diatomics Dataset and from the Literature

 	Schwenke parameters AL	RMS(BSSE) or RMSD(TAE)	TAE with BSSE parameter and vice versa	
 	 	{T,Q}	{Q,5}	{5,6}	{6,7}	 	{T,Q}	{Q,5}	{5,6}	{6,7}	{T,Q}	{Q,5}	{5,6}	{6,7}	
CCSD raw	VnZ+d	0.759	0.924	1.162	1.391	TAE RAW	0.205	0.080	0.046	0.040	0.232	0.103	0.069	0.089	
CCSD CP	VnZ+d	0.751	0.922	1.140	1.472	TAE CP	0.233	0.058	0.041	0.044	0.259	0.099	0.067	0.087	
CCSD BSSE	VnZ+d	0.722	0.869	1.060	1.719	BSSE	0.076	0.056	0.029	0.033	0.085	0.060	0.034	0.040	
 	Schwenke24	0.700	0.900	1.238	 	 	 	 	 	 	 	 	 	 	
 	Varandas51	0.635	0.849	1.142	 	 	 	 	 	 	 	 	 	 	
CCSD raw	nZaPa	0.705	0.887	1.120	1.452	TAE RAW	0.183	0.080	0.053	0.037	0.183	0.119	0.054	0.058	
CCSD CP	nZaPa	0.710	0.869	1.130	1.499	TAE CP	0.235	0.094	0.043	0.035	0.236	0.130	0.044	0.057	
CCSD BSSE	nZaPa	0.706	0.805	1.148	1.669	BSSE	0.116	0.048	0.020	0.008	0.116	0.057	0.020	0.016	
CCSD raw	haVnZ+d	0.647	0.892	1.228	1.453	TAE RAW	0.128	0.095	0.041	0.041	0.356	0.095	0.070	0.075	
CCSD CP	haVnZ+d	0.677	0.906	1.189	1.541	TAE CP	0.153	0.056	0.043	0.031	0.361	0.059	0.073	0.067	
CCSD BSSE	haVnZ+d	0.774	0.892	1.071	1.799	BSSE	0.106	0.059	0.014	0.025	0.148	0.059	0.027	0.032	
 	Varandas51	0.665	0.912	1.295	[1.592]	 	 	 	 	 	 	 	 	 	
 	Schwenke24	0.700	0.930	1.266	[1.621]	 	 	 	 	 	 	 	 	 	
 	ref (23)	N/A	0.932	1.283	1.602	 	 	 	 	 	 	 	 	 	
(T) RAW	VnZ+d	0.755	0.834	1.090	1.469	TAE RAW	0.032	0.013	0.005	0.004	0.038	0.015	0.009	0.009	
(T) CP	VnZ+d	0.764	0.833	1.110	1.411	TAE CP	0.038	0.010	0.004	0.003	0.043	0.012	0.008	0.009	
BSSE (T)	VnZ+d	0.715	0.792	1.001	1.692	BSSE	0.015	0.007	0.002	0.003	0.016	0.007	0.003	0.003	
 	Schwenke24	0.695	0.741	1.102	 	 	 	 	 	 	 	 	 	 	
(T) RAW	nZaPa	0.678	0.841	1.097	1.562	TAE RAW	0.020	0.007	0.003	0.001	0.055	0.013	0.005	0.003	
(T) CP	nZaPa	0.703	0.829	1.109	1.583	TAE CP	0.022	0.007	0.003	0.001	0.055	0.013	0.005	0.003	
BSSE (T)	nZaPa	0.562	0.908	1.043	1.466	BSSE	0.004	0.003	0.001	0.001	0.011	0.003	0.002	0.001	
RP,21 eq 12	nZaPa	0.604	0.891	1.199	1.517	 	 	 	 	 	 	 	 	 	
RP,21 optimized	nZaPa	0.600	0.849	1.164	1.580	 	 	 	 	 	 	 	 	 	
(T) RAW	haVnZ+d	0.758	0.823	1.166	1.558	(T) RAW	0.030	0.009	0.003	0.003	0.049	0.019	0.014	0.006	
(T) CP	haVnZ+d	0.727	0.805	1.209	1.526	(T) CP	0.032	0.007	0.003	0.003	0.055	0.018	0.013	0.006	
BSSE (T)	haVnZ+d	0.864	0.932	0.941	1.763	BSSE (T)	0.014	0.005	0.001	0.001	0.015	0.005	0.003	0.002	
 	Schwenke	0.700	0.810	1.248	 	 	 	 	 	 	 	 	 	 	

The remaining BSSE upon extrapolation is still somewhat significant (0.12 kcal·mol–1) for {3,4}ZaPa, but dwindles to 0.05 kcal·mol–1 for {4,5}ZaPa and to essentially nil beyond that (0.02 and 0.01 kcal·mol–1, respectively, for {5,6}ZaPa and {6,7}ZaPa).

We also obtained a different set of parameters by minimizing the RMSD with respect to CCSD(F12*)/awCV5Z for this sample of 24 molecules. Unsurprisingly, this yields the lowest RMSDs of the three parameter sets — but the differences with BSSE-minimizing extrapolation, except possibly for the {4,5}ZaPa basis set pair, are again within the uncertainty of the reference values.

Using the counterpoise, rather than raw, TAEs leads to marginally different Schwenke coefficients, except for the haV{T,Q}Z+d pair where also TAE(BSSE) differs significantly.

For the connected triple excitations, BSSE minimization in the nZaPa series yields parameters fairly similar to those published by Ranasinghe and Petersson.21 In ref (56), their {6,7}ZaPa extrapolation was found to essentially represent basis set limits: our BSSE minimization has an RMSD of 0.05 kcal·mol–1 for the smallest basis set pair considered ({3,4}ZaPa), but for {4,5}ZaPa this already drops down to 0.01 kcal·mol–1, and for {5,6}ZaPa to 0.004. The RMSD(TAE) based minimizations lead to 0.02, 0.01, and 0.003 kcal·mol–1, hence only for the smallest basis set pair could the difference be considered even remotely significant. For the cc-pV(n+d)Z sequence, {T,Q}, {Q,5}, and {5,6} pairs all have essentially the same errors for the two sets of parameters: only for the {6,7} pair where the two procedures yield Schwenke parameters that differ by 0.3 (!) is there even a 0.01 kcal·mol–1 error difference. Its practical relevance is dubious, given that the {5,6} and even {Q,5} basis set pairs yields similar-quality (T) contributions at much lower cost.

Finally, we considered if the old “NASA recipe”28 of using a coefficient times the negative BSSE as a basis set incompleteness correction has any practical merit. We thus obtained coefficients more similar to 5/2 than to 3/2 — but more importantly, the RMSD are 3–5 times larger than what can be obtained by two-point extrapolation.

Further Exploration with (Most of) W4-17 for the CCSD Correlation Energy

The reader might object that two dozen diatomics is hardly a representative sample, chemically speaking. Here we repeated our analysis to nearly all of the W4-17 thermochemical benchmark, which is eight times larger.

For the CCSD correlation component, Table 2 presents extrapolation parameters, RMS(BSSE) (root-mean-square BSSE), and RMSD(TAE) (root-mean-square deviations in the total atomization energy) for several basis set sequences. (For the largest basis sets, a handful of species had to be omitted for reasons of resource constraints or, in the case of benzene, near-linear dependence of the basis set.) Once again CCSD(F12*)/awCV5Z correlation energies extracted from the ESI of ref (53) were used as the reference.

Table 2 Schwenke Extrapolation Parameters and RMS Deviations (kcal·mol–1) in the CCSD Correlation Component of the TAE for Various Basis Set Sequencesa

a “LR” refers to linear regression with slope (dimensionless) and intercept (kcal·mol–1).

For the cc-pV(n+d)Z family, the agreement between BSSE-minimizing and RMSD(TAE)-minimizing Schwenke parameters can only be described as remarkable: fitted to the W4-08 subset, we have 0.717 vs 0.690 for V{T,Q}Z+d, 0.879 vs 0.883 for V{Q,5}Z+d, and 1.063 vs 1.094 for V{5,6}Z. These differences are well within overlapping 2σ uncertainties on the fitted linear regression parameters. The RMSDs in both BSSEs and TAEs are statistically equivalent between the two basis set sequences. Only for {6,7} (Table 1) do we find a significant discrepancy of 1.794 vs 1.400: the RMSD(TAE) if we substitute the former Schwenke parameter for the latter rises from 0.04 to 0.1 kcal·mol–1 — still, not much larger than the estimated uncertainty in the reference values. RMS(BSSE) values obtained with the two extrapolation parameters are not appreciably different.

It is well-known (and standard practice in high-accuracy thermochemistry protocols like W4 theory50,57 and HEAT58−61) that adding diffuse functions speeds up basis set convergence especially if highly electronegative elements like O and F are involved. For the haVnZ+d sequence, the Schwenke parameters obtained by BSSE(CBS) minimization and by RMSD(TAE) minimization are again quite similar for the haV{Q,5}Z+d and haV{5,6}Z+d basis set pairs, the resulting RMS(BSSE) and RMSD(TAE) values being statistically equivalent. There is, however, a more pronounced difference for haV{T,Q}Z+d, AL = 0.782 vs 0.603 when fitted to the W4-08 subset. The BSSE-minimizing AL = 0.782 yields a quite poor RMSD(TAE) = 0.83 kcal·mol–1, almost three times the value obtained with AL = 0.603.

The similarity between the AL values obtained from (most of) W4-17 and of its smaller subsets W4-08 and W4-11 is indicative of the stability of the fits, especially for the smaller basis sets where we were able to include all W4-17 species.

As a further sanity check: instead of adjusting a single scaling factor, we carried out linear regression including an intercept that corresponds to correcting for a putative constant bias in the atomization energies. Ideally, said intercept should be as close to zero as possible. For this check, we used the W4-08 subset throughout as we were able to run all its species for all basis sets through n = 6, and hence we can make a fair comparison between the basis set families. In the BSSE fit, the intercept amounts to 0.10 kcal·mol–1 for the haV{T,Q}Z+d pair, but drops to insignificant values of 0.04 and 0.013 kcal·mol–1 for {Q,5} and {5,6}, respectively.

When fitted to RMSD(TAE) instead, both {T,Q} and {Q,5} have significant intercepts at −0.13 at 0.09 kcal·mol–1, respectively. For nZaPa there is a significant intercept for {T,Q} but not for the larger basis set pairs, and concomitantly with that, the Pearson coefficients of determination R2 for the BSSE fits increase sharply from 0.86 to 0.99 and 0.98, respectively, while the corresponding R2 values for the TAE-fits are 0.93, 0.96, and 0.99, respectively.

Figure 2 presents the median BSSE for the CCSD correlation component of TAE across the W4-11 subset for various basis set sequences. For each of these, the BSSE approximately halves with each step in n. (The same is seen for the triples, Figure 3.)

Figure 2 Median BSSE (cm–1) for TAE[CCSDcorr] over the W4-11 dataset for different basis set sequences.

Figure 3 Median BSSE (cm–1) for TAE[(T)] over the W4-11 dataset for different basis set sequences.

The nZaPa series, for smaller n, actually seems slightly more prone to BSSE than heavy-aug-pV(n+d)Z. Replacing haV(T+d)Z by the spdf part of haV(Q+d)Z; haV(Q+d)Z by the spdfg part of haV(5+d)Z; and so forth — i.e., the next basis set up with the top angular momentum deleted — drives down the BSSE to the same range as haV(n+1)Z+d.

If (for additional radial flexibility) we apply the cc-pCVnZ core–valence basis set sequence to valence correlation, we find that it behaves essentially like the underlying cc-pVnZ(+d) series. (There is no need to add tight d functions on second-row elements to a core–valence basis set, as the latter already will include tight d functions to describe especially 2p core–valence correlation.) Only for the cc-pCV{5,6}Z basis set pair is there a semisignificant discrepancy between BSSE- and TAE-based extrapolation coefficients — which in fact goes away when doing linear regression with an intercept. In contrast, for the diffuse function-augmented haCVnZ sequence, there is a significant difference (also in RMSD) for the {T,Q} basis set pair: interestingly, here too it disappears when an intercept is allowed into the fit. The said intercept, at +0.1 kcal·mol–1 for the BSSE fit and −0.1 kcal·mol–1 for the TAE fit, is however a bit large for the authors’ comfort. By comparison with the VnZ+d and haVnZ+d findings, we infer that the “destabilizing” factor here are the diffuse functions.

As shown earlier in ref (41), using the haCVnZ core–valence basis sets (Figure 2) for the valence correlation energy does drive down BSSE considerably. Interestingly, combining the dfg... functions from haVnZ+d with the sp set from haV(n+1)Z+d — which combination we denote haVnZ+spd — seems to be about equally effective in that regard.

For the {5,6} pair and energy-optimized extrapolations, the differences between the various basis set families are too small to make meaningful distinctions.

Further Consideration of (T) for a Larger Sample

It has been shown in great detail (see ref (56) and references therein) that basis set convergence of (T) is considerably faster than for the correlation energy overall; specifically, it was found there that for the W4-08 subset of W4-17, {4,5}ZaPa extrapolation of (T) with the Ranasinghe-Petersson formula21 causes an RMSD error in TAE[(T)] of just 0.01 kcal·mol–1 compared to (T){6,7}ZaPa. Even for the {T,Q} pair this only rose to 0.05 kcal·mol–1. Hence, we shall eschew overanalysis of results with {Q,5}, let alone {5,6} basis set pairs.

Optimized parameters and performance statistics for the connected triples contribution to TAE can be found in Table 3. Here we used (T)/{5,6}ZaPa or, for the species where available, (T){6,7}ZaPa from the ESI of ref (56) as the reference.

Table 3 Schwenke Extrapolation Coefficients AL and RMSD Deviations (kcal·mol–1) for the Connected Triples Contribution (T) to the Total Atomization Energy

For the pVnZ+d and CVnZ basis set sequences, the BSSE-fitted and TAE-fitted Schwenke parameters are again quite similar, as are their statistics. For nZaPa, haVnZ+d, and haCVnZ there is a more pronounced difference for the smaller basis set pairs; comparison for the {5,6} pair is a somewhat inane exercise, as all sets of parameters except haVnZ+d have RMSDs of 0.01 kcal·mol–1 or below for the (T) component.

Conclusions

In response to our research question — whether basis set extrapolation can viably be obtained from the condition that the extrapolated basis set superposition error should approach zero — we can conclude the following:1. For cc-pV(n+d)Z basis sets, fitting to reference TAEs or fitting to minimize extrapolated BSSE yields similar extrapolation parameters for {T,Q}, {Q,5} and {5,6} basis set pairs.

2. For other basis set sequences, this is consistently the case for the {Q,5} pair.

3. For the haV{T,Q}Z+d and haCV{T,Q}Z+d pairs there appears to be a basis set imbalance in terms of BSSE. This is much less the case for {3,4}ZaPa.

4. For 5Z and 6Z basis sets, the two approaches may still lead to different extrapolation parameters. However, owing to the smaller basis set incompleteness, the predicted basis set limits are of comparable quality considering the uncertainty in the reference values.

5. This recipe becomes less workable for angular momenta beyond i functions, as the BSSEs become too small to form a reliable foundation for fitting.

Thus, basis set extrapolation can be rationalized through BSSE minimization, which eliminates the need to rely on either analytical archetypes about the partial-wave or principal expansions, or on fitting against any sort of external reference energetics. Moreover, since no explicit connection with either the partial-wave or principal expansions exists, it may be applicable to basis set sequences that are not tied to increasing L.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpca.4c03012.Microsoft Excel workbook containing the raw data for Table 1 as well as its processing; Microsoft Excel workbook containing the raw data for Tables 2–3 and Figures 2–3, as well as its processing (ZIP)

Supplementary Material

jp4c03012_si_001.zip

The authors declare no competing financial interest.

Acknowledgments

This work was supported by the Israel Science Foundation (grant 1969/20) and by the Uriel Arnon Memorial Center for AI research into smart materials. J.M.L.M. thanks the Quantum Theory Project at the University of Florida and its head, Prof. John F. Stanton, for their hospitality, and Dr. Nisha Mehta for discussions on BSSE in chalcogen bonding that provided some inspiration for the present work. E.S. acknowledges doctoral scholarships from the Feinberg Graduate School (Weizmann Institute of Science) and the Onassis Foundation (Scholarship ID: FZP 052-2/2021-2022). All calculations were carried out on the ChemFarm HPC cluster of the Weizmann Institute Faculty of Chemistry.
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