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Performance of polarization-consistent vs. correlation-consistent basis sets for CCSD(T) prediction of water dimer interaction energy

  • Teobald KupkaEmail author
  • Aneta Buczek
  • Małgorzata A. Broda
  • Adrianna Mnich
  • Tapas KarEmail author
Open Access
Original Paper
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Abstract

Detailed study of Jensen’s polarization-consistent vs. Dunning’s correlation-consistent basis set families performance on the extrapolation of raw and counterpoise-corrected interaction energies of water dimer using coupled cluster with single, double, and perturbative correction for connected triple excitations (CCSD(T)) in the complete basis set (CBS) limit are reported. Both 3-parameter exponential and 2-parameter inverse-power fits vs. the cardinal number of basis set, as well as the number of basis functions were analyzed and compared with one of the most extensive CCSD(T) results reported recently. The obtained results for both Jensen- and Dunning-type basis sets underestimate raw interaction energy by less than 0.136 kcal/mol with respect to the reference value of − 4.98065 kcal/mol. The use of counterpoise correction further improves (closer to the reference value) interaction energy. Asymptotic convergence of 3-parameter fitted interaction energy with respect to both cardinal number of basis set and the number of basis functions are closer to the reference value at the CBS limit than other fitting approaches considered here. Separate fits of Hartree-Fock and correlation interaction energy with 3-parameter formula additionally improved the results, and the smallest CBS deviation from the reference value is about 0.001 kcal/mol (underestimated) for CCSD(T)/aug-cc-pVXZ calculations. However, Jensen’s basis set underestimates such value to 0.012 kcal/mol. No improvement was observed for using the number of basis functions instead of cardinal number for fitting.

Graphical Abstract

Keywords

CBS Fitting formula CCSD(T) Water dimer Basis sets Polarized-consistent Correlation-consistent 

Introduction

Rapid development in computer efficiency and progress in theoretical methods in 1980s of the previous century resulted in accurate calculation of atomic and molecular energies of small-size compounds [1, 2, 3, 4, 5]. This opened a road for thermochemical studies within chemical accuracy [6] (± 1 kcal/mol) about three decades ago. However, both electron correlation and incompleteness of one-electron wave function were recognized as the main sources of errors in theoretical calculations [2, 7, 8, 9, 10]. It was also assumed that the accuracy of the predicted results improved in calculations involving more complete and flexible basis sets. Thus, the main tool in these studies was related to simple extrapolation techniques using results of correlated calculations combined with regularly constructed correlation-consisted basis sets of Dunning and coworkers [2, 11, 12, 13].

The first mathematical formula for extrapolation of energy toward the complete basis set limit (CBS) was based on three-parameter exponential expression proposed by Dunning and coworkers [9, 12, 14, 15, 16, 17]. Thus, the results of atomic and molecular energies, calculated with a series of cc-pVXZ (or their extended aug-cc-pVXZ versions, additionally augmented with diffuse functions), where X = D, T, Q, 5, and 6 were used in single-exponential non-linear fits (1):
$$ Y(X)=Y\left(\infty \right)+B\times \exp \left(-C/X\right) $$
(1)

This crude approximation worked particularly well for Hartree-Fock (HF) energies. Later, Helgaker and coworkers [18] proposed a slower converging two-parameter formula, in particular for fitting correlation energy (2):

$$ Y(X)=Y\left(\infty \right)+B/{X}^3 $$
(2)

In these formulas Y(∞), B and C are fitting parameters and X is the cardinal number of basis set used (the largest angular momentum). Equation (2) was similar to a theoretical expression proposed by Schwarz [19] for helium atom and later elaborated by Kutzelnig and Morgan [20].

Later on, numerous molecular systems were studied and several fitting formulas proposed for isolated, as well as for weakly interacting systems [21, 22, 23, 24, 25, 26]. However, it seems that the results obtained in such extrapolations are comparable. Recently, a web-oriented application proposed 15 different formulas for such approximation [27].

Independently another hierarchy of basis sets, the so-called Jensen’s polarization consistent ones, pc-n and aug-pc-n, where n is 0, 1, 2, 3, and 4 have gained popularity [28, 29, 30, 31]. These basis sets are also tailored to calculate accurately nuclear magnetic shielding tensor [32] (aug-pcS-n), as well as indirect spin-spin coupling constants [33] (aug-pcJ-n). Jensen’s basis sets were developed for Hartree-Fock and DFT calculations. However, their use was also successful in CBS studies at MP2 and coupled cluster (CC) levels of theory [23, 25].

The use of polarization-consistent basis sets for extrapolation of Hartree-Fock and Kohn-Sham, as well as correlation energy and energy-related parameters at MP2, CCSD, and coupled cluster with single, double, and perturbative correction for connected triple excitations (CCSD(T)) levels for isolated molecules has been reported [22, 23, 24]. Elsohly and Tschumper [22] compared a performance of correlation-consistent and polarization-consistent basis sets combined with MP2 method for calculation of interaction energy of several non-covalent systems, including water dimer. They observed regular and smooth convergence of MP2-calculated interaction energy vs. the cardinal number of the studied (HF)2, (HF)3, (H2O)2, (H2O)3, and (C2H2)2 complexes toward the CBS limit [22]. The obtained raw CBS interaction energies were smaller in case of Dunning-type basis sets. However, in comparison with raw values, the counterpoise-corrected interaction energies obtained with both families of basis sets were not improved.

Moreover, it was observed in many studies on small molecules including H2, H2O, F2, F2O, and trans-N-methylacetamide that the fitting lines obtained in such extrapolations of parameters calculated with Dunning and/or Jensen basis sets converge in principle to the same CBS limit [21, 23, 34, 35].

In case of weakly interacting systems, calculations with smaller basis sets produce inaccurate results due to basis set superposition error [36, 37, 38, 39] (BSSE). Moreover, as mentioned by Klopper and coworkers [40], “non-removal of BSSE results in an unsystematic, non-monotonic convergence of the correlation contribution to the interaction energy with increasing basis set size.” In order to alleviate this problem, Boys and Bernardi [41] proposed the so-called counterpoise approach (CP). In general, raw interaction energy (without CP correction) overestimated the accurate value in calculations combined with incomplete basis sets and converged to the CBS limit from below. However, in some cases in the current studies, we noticed a reverse situation for the limiting value (the CBS-extrapolated raw energy was less attractive than the CP corrected). A similar and unrealistic raw (− 4.95 kcal/mol) and CP-corrected energy (− 5.02 kcal/mol) ordering also reported Xantheas [42].

It is also important to mention that apart from CBS fitting the results obtained with limited size basis sets in popular supermolecular (SM) studies [15], in the literature are other approaches leading to improved accuracy in calculations of interaction energy in case of weakly interacting non-bonded systems. These include the use of n-tuply augmented diffuse basis sets [43] which are responsible for long-distance attraction forces, as well as the use of so-called mid-bond functions [44, 45, 46], better describing electron density between the atoms, as well as symmetry-adapted perturbation theory [47, 48, 49, 50] (SAPT). Morokuma and Kitaura [51] also proposed energy decomposition analysis of molecular interactions. Another approach leading to higher accuracy is achieved by using directly correlated wave functions in methods applying resolution of identity [52, 53, 54] (RI), for example in MP2-RI and CCSD(T)-RI. These methods produce faster converging results which are free from BSSE. The advantage of SM approach is its simplicity and availability in user-friendly programs like Gaussian. On the other hand, the other methods mentioned above are free from BSSE and are able to produce individual, physically justified interaction components including electrostatic, repulsion, charge transfer, correlation, etc.

However, to our best knowledge, in the available literature no detailed studies on convergence of interaction energy of hydrogen bonded systems, for example in water dimer, calculated at the CCSD(T) level of theory using both Jensen’s and Dunning’s basis set families exist. In addition, little is known about the magnitude of BSSE, correlation energy and their convergence toward CBS in calculations using the polarization-consistent basis sets. It is still not clear which approach is better—CBS fitting vs. the cardinal number or vs. the number of basis functions. Finally, there is an unanswered question about the efficiency of fitting formulas in convergence studies of non-covalent systems.

The basis set requirements for efficient recovery of weak interactions and forces acting far from nuclei in non-covalent molecular systems depend heavily on the presence of diffuse functions in the basis sets applied. In fact, the more complete and flexible basis set is used the more efficiently electrostatic and dispersion forces are reproduced. On the other hand, in many cases, diffuse functions on hydrogen atoms could be omitted with only negligible loss of performance (so-called heavy atom augmented basis sets, haug-cc-pVXZ [55]). Therefore, both Dunning and Jensen basis sets are suitable for studying H-bonded systems.

In the current study, using a popular supermolecular approach, we want to address the above problems using water dimer as model. In particular, we want to assess the applicability of polarization-consistent basis sets for accurate prediction of water dimer interaction energy at frozen-core (FC) CCSD(T) level of theory. To avoid an additional impact of geometry in all calculations, we will use a well-known and very accurate water dimer CCSD(T)/cc-pVQZ structure, recently reported by Hobza and co-workers [56] in his benchmark studies on 66 selected non-covalent systems. The use of single, well-defined optimized or experimental geometry of weakly bonded molecules was often used since the geometry relaxation has a very small impact on interaction energy [15, 57]. For example, in case of water dimer, Feller et al. [58] mentioned about 0.03 kcal/mol difference.

Thus, in the current study, we want to asses two ways of determining water dimer interaction energy extrapolation toward the complete basis set limit (CBS) using Dunning- and Jensen-type basis set families and two fitting formulas (1) and (2): 3- and 2-parameter ones vs. the cardinal number and the number of basis functions at FC-CCSD(T) level of theory. Since we are not aiming at another benchmark work on water dimer interaction energy, we will concentrate on relative performance of Dunning and Jensen-type basis sets in predicting the convergence of this important parameter.

We will use a recently reported value of − 4.98065 kcal/mol, obtained from CBS extrapolation of frozen core CCSD(T)-F12 calculations with aug-cc-pVQZ and aug-cc-pV5Z basis sets as reference of interaction energy of water dimer [59]. Due to their popularity, both Eqs. (1) and (2) will be used for fitting of our results.

Methods

Interaction energy for water dimer was calculated at Hartree-Fock and coupled cluster with single, double, and non-iterative triples, CCSD(T) levels of theory, using Gaussian 09 revision E.01 program package [60]. All calculations produced raw (uncorrected) and counterpoise-corrected interaction energies of water dimer at geometry reported by Hobza and coworkers [56] in their systematic benchmark studies on 66 non-covalent molecular systems. For completeness, their dimer structure is listed in the supplementary data (Table S1). All calculations were performed using augmented polarization-consistent basis sets (aug-pc-n [28, 29, 30, 31]) with index “n” changing from 1 to 4 and the augmented correlation-consistent basis sets aug-cc-pVXZ [2, 11, 12, 13]. The cardinal numbers used with the latter basis sets were D, T, Q, 5, and 6. For fitting purposes [23], we used the following values of X = 2, 3, 4, 5, and 6.

Since Jensen’s pc-1 basis set is already of double-zeta quality, following an earlier study by Tschumper [22], for fitting purposes, we will change the corresponding index: X = n + 1. In addition, it is possible to use the number of basis function (Bf) instead of the cardinal number for studying the convergence toward the CBS limit. Therefore, we will perform fittings both with respect to cardinal number X and to Bf. Both fitting Eqs. (1) and (2) were used for fitting of CCSD(T), HF, and correlation energy. The latter was calculated as follows: Ecor = ECCSD(T) – EHF. For convenience, the following abbreviations for basis sets will be used in this paper: apcn for aug-pc-n and aXZ for aug-cc-pVX. The three-parameter CBS fits using three last data will be abbreviated as 3-par CBS(2–4) and 3-par CBS(Q-6) for Jensen and Dunning basis sets, respectively. Similarly, fits with two-parameter formula will be indicated by 2-par CBS(3,4) and 2-par CBS(5,6) notations. In case of separate fittings of HF and correlation energy with three parameter function, we will use (3 + 3) notation or (3 + 2a) and (3 + 2b) in case of correlation energy fitted with two-parameter formula using three or two last points, respectively.

The largest basis sets aug-cc-pV6Z for water dimer contained 886 basis functions (1386 primitive Gaussians and 1316 Cartesians) and the aug-pc-4 basis was significantly smaller (642 basis functions, 930 primitive Gaussians, and 856 Cartesian basis functions). In addition, in all calculations, only the spherical components of the polarization (Cartesian Gaussians) functions were used, e. g., 5D, 7F, and so on.

Finally, in agreement with some earlier studies [39], we improve our estimated interaction energies by using the average values for raw and CP-corrected energies in the complete basis set limit: (E(Raw) − E(CP))/2. Final comparison between our results and the reference energy will be expressed as deviation EcalcEref, where Eref = − 4.98065 kcal/mol, and is also marked in figures as “LANE [59]”.

Results and discussion

Convergence patterns of water dimer energy, interaction energy, and basis set superposition error calculated with polarization-consistent and correlation-consistent basis set series

The performance of different basis set hierarchies, e. g., correlation-consistent and polarization-consistent ones in total energy and interaction energy recovery could be related to their construction, including the number of primitive Gaussian and Cartesian functions, as well as the total number of basis functions [61]. Thus, we start our analysis by checking the convergence patterns of counterpoise-corrected energy of water dimer produced by Jensen’s and Dunning’s basis sets. In particular, we want to see which basis set family, correlation-consistent of polarization-consistent produce faster converging CP-corrected SCF and CCSD(T) energies.

A fairly regular convergence of both Hartree-Fock and CCSD(T) counterpoise-corrected energy of water dimer is apparent from Fig. 1a, b. Obviously, the results obtained with aug-pc-n and aug-cc-pVXZ basis sets converge to essentially the same complete basis set limit, i. a. to about − 152.1402 and − 152.7530 au for HF and CCSD(T), respectively. On the other hand, for X = 2, the total HF energy, calculated with Jensen-type basis sets (− 152.0299 au) is significantly less negative than that calculated with aug-cc-pVDZ basis set (− 152.0882 au). However, for X = 3, 4, and 5, the energies produced by Jensen-type basis sets are slightly lower than those calculated with aug-cc-pVXZ basis sets. This indicates somehow faster convergence of HF energy calculated using polarization-consistent instead of correlation-consistent basis sets for water dimer (Fig. 1a). The above picture is not so clear for CCSD(T) calculated energy of water dimer (Fig. 1b) but still the aug-pc-1 and aug-pc-2 predict somehow larger (less negative) energies than aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis sets.
Fig. 1

Convergence of counterpoise-corrected energy (au) of water dimer calculated with Jensen- and Dunning-type basis sets with a HF and b CCSD(T) methods and c and d the decrease of the corresponding basis set superposition errors (kcal/mol)

However, looking at Fig. 1c, d, it is easier to understand that the differences in performance of both basis set families are partly due to the different magnitude of basis set superposition errors calculated with HF and CCSD(T) methods. Thus, for X = 2, the HF/aug-pc-1-calculated BSSE is about three times the size of the corresponding value produced by aug-cc-pVDZ basis set (about 0.80 vs. 0.25 kcal/mol). Obviously, at the CCSD(T) level of theory, the BSSE magnitude is larger than in the case of non-correlated calculations; however, the ratio of BSSE values obtained for two studied basis set families decreases and the corresponding values are about 1.7 and 0.9 kcal/mol. For the next cardinal number (X = 3), the Jensen-type basis set performs comparable with Dunning’s one and is marginally better for X = 5.

The magnitude of water dimer interaction energy was subject of numerous accurate theoretical calculations [15, 59, 62, 63]. Since CCSD(T) method is considered a gold standard in terms of accuracy and computation cost for small-size molecular systems in Table 1, we gather such results published in recent years. It is apparent from Table 1 that all the theoretical values oscillate around − 5 kcal/mol and strongly depend on the size of basis set used for calculations. The best value, obtained at CCSD(T)-F12 level of theory and extrapolated to complete basis set limit using results calculated with aug-cc-pVQZ and aug-cc-pV5Z basis set is − 4.98065 kcal/mol [59]. A little smaller value (− 5.02 kcal/mol) was reported by Hobza as result of his composite approach [56, 64]. The most accurate result was reported by Lane [59] (− 5.017 kcal/mol). He called it the “best” one. However, he used very advanced CCSDTQ method for both geometry optimization and for energy, additionally improved by inclusion of core-valence and relativistic contributions. These accurate theoretical values of water dimer interaction energy [65, 66, 67] fall within the error bars of the experimentally determined values (− 5.44 ± 0.7 kcal/mol [68] or − 5.4 ± 0.2 [69, 70, 71]). The latter values were determined using additional terms including zero-point vibrational and temperature corrections. Turning back to our results, in Table S2, in the supplementary material, all raw and CP-corrected interaction energies of water dimer are gathered, calculated with aug-pc-n and aug-cc-pVXZ basis sets at Hartree-Fock and CCSD(T) levels of theory.
Table 1

The water dimer interaction energy (in kcal/mol) and O…O distance calculated by CCSD(T) method

Basis set

Interaction energy (kcal/mol)

  
 

Raw

CP corrected

O…O dist. (A)

Ref.

aug-c-pVTZ

− 5.0

 

2.90

[72]

CBS

 

− 4.96

 

[73]

(FC)/IO275

 

− 4.94

2.912 ± 0.05

[74]

(FULL)/IO275

 

− 4.96

 

[74]

TZ2P(f,d) + dif

 

− 5.02

2.9089

[75]

aug-cc-pVTZ

− 5.22

− 4.75

2.93

[76, 77]

aug-cc-pVDZ

− 5.29

− 4.398

2.926

[42, 78]

aug-cc-pVTZ

− 5.217

− 4.747

 

[42]

aug-cc-pVQZ

− 5.101

− 4.907

 

[42]

aug-cc-pV5Z

− 5.034

− 4.944

 

[42]

CBS

− 4.95

− 5.02 (?)

 

[42]

aug-cc-pCVDZ

− 5.07

− 4.29

2.92

[79]

aug-cc-pCVTZ

− 5.10

− 4.69

 

[79]

(F12)/aug-cc-pCVDZ

− 5.01

− 4.76

2.98

[79]

(F12)/aug-cc-pCVTZ

− 5.04

− 4.78

 

[79]

CBS

 

− 4.966

 

[80]

ha5Z

 

− 4.989

2.9127

[81]

haQZ

 

− 5.009

 

[81]

CBS (“Hobza”)

 

− 5.02

 

[56, 64]

(F12b) CBS(aug-cc-pVQZ, aug-cc-pV5Z) (“LANE”)

 

− 4.98065

2.913

[59]

CBS + CV + rel (“BEST”)

 

− 5.017

 

[59]

Exp.

 

− 5.44 ± 0.7

 

[68]

Exp.

 

−5.4 ± 0.2

 

[69, 70, 71]

It is apparent from Figs. 2a–f that both raw and CP-corrected interaction energies saturate somehow close to the value, represented by our reference (“Lane [59]”).
Fig. 2

Convergence of CCSD(T)-calculated water dimer interaction energy using Jensen’s and Dunning’s basis sets fitted with 3-parameter function and three last points with respect to a X and n + 1, b number of basis sets; fitted with 2-parameter function and three last points with respect to c X and n + 1, d number of basis sets; and fitted with 2-parameter function and two last points with respect to (e) X and n + 1 and (f) number of basis sets

The picture is very similar for energy converging along with improved basis set size and flexibility (Fig. 2a, c, e), represented by X and n + 1, as well as by increasing the total number of basis functions Bf (Fig. 2b, d, f). However, it is obvious that raw energy converges from below and the CP-corrected one from above. The results of individual fitting are shown in the figures and solid lines represent convergence of raw energies while dashed lines correspond to CP-corrected ones. Obviously, points obtained with smaller basis sets (X = 2 and 3) are less accurate and deviate from the fits.

An interesting feature of fits in Fig. 2c deserves a short comment. Raw data, corrupted by BSSE overestimate the dimer interaction energy calculated with aug-pc-1, aug-pc-2, aug-pc-3, and aug-pc-4 (the corresponding CP energies are more positive) but the lines of the corresponding fits are crossing. As result, in the complete basis set limit, the CP-corrected energy seems to be lower than the one extrapolated for raw energy. This situation is only visible in graphs plotted against the cardinal number. One should be aware that in the complete basis set limit ERaw = ECP. However, we cannot expect the magnitude of BSSE behaving exactly in the same (regular) fashion as energy (see Fig. 1c, d). As an alternative, we plotted the same values of energy vs. the number of basis functions, and this way of presentation seems to alleviate the problem of crossing both fitted lines (see Fig. 2d). Similar situation is presented in Fig. 2e, f.

Splitting of interaction energy into two components (HF and correlation energy) and separate CBS fitting is often used to obtain more precise information about non-covalent interactions [40, 73]. The limiting values of Hartree-Fock, as well as correlation interaction energy (Ecor), extrapolated toward CBS with Eqs. (1) and (2) are gathered in Table 2. As expected, the numbers are fairly similar for different combinations of fitting schemes but differences on second decimal point are apparent, in particular for results fitted with Eq. (2).
Table 2

CBS estimated valuesa of HF and Ecor interaction energies (in kcal/mol) of water dimer calculated using polarization-consistent and correlation-consistent basis set hierarchies (CSD(T)/aug-cc-pVQZ structure reported by Hobza et al. used)

CBS fit

HF

E cor

 

vs. X and n + 1

vs. no .of basis functions

vs. X and n + 1

vs. no. of basis functions

 

Raw

CP

Raw

CP

Raw

CP

Raw

CP

aug-pc-n

3-par CBS

(2–4)

− 3.58592

− 3.58448

− 3.58669

− 3.58459

− 1.387

− 1.37947

− 1.39147

− 1.35834

2-par CBS

(2–4)

− 3.54172

− 3.57359

− 3.58395

− 3.58359

− 1.30303

− 1.41869

− 1.39492

− 1.32391

2-par CBS (3,4)

− 3.57544

− 3.58251

− 3.58429

− 3.58413

− 1.35423

− 1.41686

− 1.38483

− 1.35919

aug-cc-pVXZ

3-par CBS

(Q-6)

− 3.58504

− 3.58488

− 3.58533

− 3.58488

− 1.39692

− 1.39213

− 1.41528

− 1.38582

2-par CBS

(Q-6)

− 3.56984

− 3.58466

− 3.58286

− 3.58483

− 1.40512

− 1.40539

− 1.43102

− 1.38042

2-par CBS (5,6)

− 3.58091

− 3.58513

− 3.58428

− 3.58500

− 1.39783

− 1.40360

− 1.42019

− 1.38676

aFittings are performed with 3-parameter formula using 3 last points and with 2-parameter formula using 3 and 2 last points

In Table 3 are gathered CBS-extrapolated interaction energies obtained from fits of CCSD(T) energy, as well as from separate fits of Hartree-Fock energy and correlation energy with Eqs. (1) and (2). Moreover, we calculated a simple average value of separation between raw and CP-corrected interaction energies in the basis set limit (CPcorrection). This value will be later used to improve on the CP energy. However, in this case only positive value indicating a correct behavior of fitting lines will be used to improve on the CP energy.
Table 3

CBS estimated values of CCSD(T) interaction energy and a sum of separate fits of HF and Ecor energies for water dimera

CCSD(T)

CBS fit

vs. X and n + 1

vs. no of basis functions

 

Raw

CP

CPcorrection

Raw

CP

CPcorrection

aug-pc-n

3-par CBS(2–4)

− 4.97345

− 4.97026

0.001595

− 4.97842

− 4.94574

0.01634

2-par CBS(2–4)

− 4.84475

− 4.99227

− 0.07376

− 4.97888

− 4.9075

0.03569

2-par CBS(3,4)

− 4.92966

− 4.99935

− 0.034845

− 4.96912

− 4.94331

0.012905

aug-cc-pVXZ

3-par CBS(Q-6)

− 4.99706

− 4.97759

0.009735

− 5.00626

− 4.97099

0.017635

2-par CBS(Q-6)

− 4.97496

− 4.99005

− 0.007545

− 5.01388

− 4.96525

0.024315

2-par CBS(5,6)

− 4.97874

− 4.98873

− 0.004995

− 5.00446

− 4.97176

0.01635

CCSD(T) from separate fits of HF and Ecor

aug-pc-n

3-par CBS(2–4)

− 4.97292

− 4.96395

0.004485

− 4.97816

− 4.94293

0.017615

2-par CBS(2–4)

− 4.88895

− 5.00317

− 0.05711

− 4.98161

− 4.90850

0.036555

2-par CBS(3,4)

− 4.94015

− 5.00134

−0.030595

− 4.97152

− 4.94378

0.01387

aug-cc-pVXZ

3-par CBS(Q-6)

− 4.98196

− 4.97701

0.002475

− 5.00061

− 4.97070

0.014955

2-par CBS(Q-6)

− 4.99016

− 4.99027

− 5.5E-05

− 5.01635

−4.96530

0.025525

2-par CBS(5,6)

− 4.98287

− 4.98848

− 0.002805

− 5.00552

− 4.97164

0.01694

aAll raw and CP-corrected energies calculated with polarization-consistent and correlation-consistent basis set hierarchies. Additional ad hoc corrections for CP values obtained from a simple formula [39]: E(CP) – CPcorrection, where CPcorrection = [(E(CP) – E(Raw))/2]

The performance of Jensen-type basis sets is not so obvious from data gathered in Tables 2 and 3 and Figs. 1 and 2 and therefore in Table S3, in the supplementary material, we gathered deviations of raw and CP-corrected interaction energy [39] calculated with CCSD(T) method, as well as from separate fits of HF energy and correlation energy added together. Moreover, in Table S4, are the same deviations with CP energies additionally improved by ad hoc corrections, shown in Table 3. However, in case of the confusing negative corrections (see the crossing lines in Fig. 1c), we kept the original deviations.

Performance of aug-pc-n vs. aug-cc-pVXZ basis sets in prediction of water dimer interaction energy

At the final stage of our studies, we checked the magnitudes of CBS-extrapolated CCSD(T)/aug-pc-n and CCSD(T)/aug-cc-pVXZ water dimer interaction energy deviations from the reference, reported recently by Lane [59] (− 4.98065 kcal/mol). In Fig. 3, we presented deviations from the reference for both raw and CP-corrected energies, extrapolated toward CBS limit with both Eqs. (1) and (2) vs. X, n + 1, as well as the number of basis functions (Bf). In addition, we considered both the deviations of total CCSD(T) interaction energies (Fig. 3a) and a sum of separate fits of Hartree-Fock and correlation energy (Fig. 3c). In the latter case, the HF energy was fitted with exponential formula and the correlation one with Eq. (1) using three last points or with formula (2) using the results obtained with three (2a) or two (2b) largest basis sets (Fig. 3). Each panel in Fig. 3 is clearly separated into two parts, each containing three groups of bars for aug-pc-n on the left side and for aug-cc-pVXZ basis sets on the right side.
Fig. 3

Deviations of a (top left) total CCSD(T)/aug-pc-n and CCSD(T)/aug-cc-pVXZ interaction energy (in kcal/mol) of water dimer estimated in the complete basis set limit using Eqs. (1) and (2) and (bottom left) separate HF and correlation energy fits, as well as ad hoc corrected (top right) total and (bottom right) composite CCSD(T) energy from “Lane” reference value [59]

Looking closer at the results presented in Fig. 3a, c, it is apparent that CBS extrapolations produce fairly accurate CCSD(T)/aug-pc-n interaction energies of water dimer with raw values deviating only by 0.136 kcal/mol from our reference value [59] in case of using the total energy for fitting three last points with 2-parameter function (Fig. 3a). This error decreases to about 0.092 kcal/mol in case of separate fits of HF and correlation energy (Fig. 3c). Significantly more accurate results are produced using 3-parameter fits (CCSD(T)/aug-pc-n deviations from about 0.007 to 0.008 kcal/mol are visible from the same figures). Interestingly, CBS fits of CCSD(T)/aug-pc-n counterpoise-corrected interaction energies vs. the number of basis functions are less accurate and the corresponding deviations are from 0.019 to 0.020 kcal/mol. On the overall, Eq. (2) seems to work better than the two-point fitting in case of CCSD(T) interaction energies of water dimer, calculated with polarization-consistent basis sets. Besides, the common assumption [40, 82] that CP-corrected energies are more accurate is not evident from the calculations performed with Jensen-type basis sets. However, this seems to be true in case of results obtained with correlation-consistent basis sets (see Fig. 3a and c). In addition, the deviations produced with Dunning-type basis sets seem to be 2–3 times smaller than those obtained with Jensen’s basis sets.

An additional ad hoc correction of CP interaction energies (see Table 3) seems to decrease these deviations quite significantly. Thus, the largest deviations produced with CP-corrected interaction energies calculated using aug-pc-n and aug-cc-pVXZ basis sets are now below 0.03 and 0.01 kcal/mol, respectively (see Fig. 3b, d).

The general conclusion from Fig. 3 is that for both CCSD(T)/aug-pc-n and CCSD(T)/aug-cc-pVXZ calculations extrapolations to the complete basis set limit are capable to predict interaction energy of water dimer within about 0.14 (raw data) to 0.04 kcal/mol in case of applying counterpoise correction. However, in general, the results produced with Dunning-type basis sets are closer to the reference. No gain in accuracy is observed for CBS fits vs. the number of basis functions. In addition, if possible, it is better to use 3-parameter fittings leading to deviations below 0.01 kcal/mol. In fact, deviations for CCSD(T)/aug-pc-n and CCSD(T)/aug-cc-pVXZ interaction energies fitted with 3-parameter function are the smallest ones and are nearly similar. Additional correction of CP energies is also beneficial (compare Fig. 3a, b, as well as Fig. 3c, d).

In order to better compare the performance of CBS interaction energies of water dimer using polarization-consistent and correlation-consistent in Table S5 in the supplementary material are gathered relative deviations (in %deviation with respect to Lane value). These data clearly show that in case of CBS fitting with respect to cardinal number of Dunning’s basis set (X) and n + 1 for Jensen’s ones, the uncorrected deviations are − 0.14 to 0.33% and − 0.11 to − 2.73%, respectively. The CP-corrected deviations are smaller (0.13 to − 0.18% and 0.2 to 0.4%). In the case of CBS fitting with respect to the number of basis function and applying CP correction, somehow worse results are produced (about 0.16 to − 0.37% for Dunning’s and 0.15 to − 0.75% for Jensen’s basis sets). Finally, the only advantage in using separate fits for HF and correlation energies was observed for correlation-consisted basis sets estimated toward the CBS limit using the three-parameter fitting formula (deviation of − 0.02 vs. 0.13%).

Following the reviewer suggestion, in the supplementary material, we also show fits of interaction energy calculated for all available basis sets, including results for double-zeta ones and the corresponding figures illustrating the convergence (Table S6 and Fig. S1). In three cases, fits which omitted results for the smallest basis sets resulted in closer agreement with the reference value (only 2-parameter fit of Dunning’s basis sets produced worse results).

Timing of central processor unit (cpu) is yet another important issue of very expensive CCSD(T) calculations. For a reasonable comparison, we will look here at total cpu needed for CP-calculated energies of water dimer for both series of basis sets. As expected, calculations employing the largest Dunning’s basis set (aug-cc-pV6Z) need almost five times more cpu time than the ones using Jensen aug-pc-4 basis set. Obviously, this is due to different sizes of both basis sets: The largest aug-cc-pV6Z basis sets for water dimer contains 886 basis functions and the aug-pc-4 basis is significantly smaller (only 642). Besides, only in case of the largest Jensen basis set, additionally 5 basis functions were dropped due to their linear dependencies.

Conclusions

Comparison of Jensen’s polarization-consistent basis set performance with the widely used Dunning’s correlation-consistent basis set on predicting interaction energy in water dimer at the CCSD(T), as well as at HF levels, and their convergence to CBS limit are reported in the present study. The CCSD(T)/CBS predicted non-corrected interaction energy of (H2O)2 using polarization-consistent and correlation-consistent basis sets agrees within 0.136 and 0.002 kcal/mol, respectively, with the reference value of − 4.98065 kcal/mol reported by Lane. As expected, the counterpoise procedure decreases this error by a factor of two to three. The CBS values extrapolated with 3-parameter formula produce results close to the reference value, both for Jensen- and Dunning-type basis sets. The accuracy of CCSD(T)/CBS water dimer interaction energy obtained with Dunning families of basis sets are generally 2–3 times better than obtained using polarization-consistent ones, initially designed for HF and DFT calculations.

Compared to the total CCSD(T)/CBS fitting, separate estimates of HF and correlation energies toward the CBS limit with the three-parameter fitting formula produce smaller interaction energy deviation from the reference value (deviation of − 0.007 vs. − 0.001 kcal/mol) in case of Dunning basis set series. For Jensen’s basis sets, separate 3-parameter fits of HF and correlation energy produce lesser accurate results (deviations of − 0.009 vs. − 0.012 kcal/mol.

Application of polarization-consistent basis sets for estimation of CBS interaction energy of water dimer produced somehow less accurate results in comparison to correlation-consistent basis sets but such calculations may be computationally more efficient. We hope that our detailed studies, limited to one non-covalent molecular system, could inspire future works on other H-bond complexes.

Notes

Acknowledgments

T. K., M. A. B., and A. B. were supported by the Faculty of Chemistry, Opole University. All the calculations were performed at Wrocław Supercomputing Center (WCSS). We would like to thank our two anonymous reviewers for their numerous very helpful questions and suggestions leading to significant improvement of the initial manuscript.

Supplementary material

894_2019_4200_MOESM1_ESM.doc (416 kb)
ESM 1 (DOC 416 kb)

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© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Faculty of ChemistryUniversity of OpoleOpolePoland
  2. 2.Department of Chemistry and BiochemistryUtah State UniversityLoganUSA

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