Self-interaction error overbinds water clusters but cancels in structural energy differences

Significance

Self-interaction error has long been identified as one of the limitations of practical density functional approximations. This error originates in the inability of approximate density functionals to exactly cancel self-Coulomb and self-exchange–correlation for all one-electron densities. Self-interaction error can be subtracted from an approximate functional on an orbital-by-orbital basis, improving the description of stretched bonds. In this work, we show that, by explicitly removing self-interaction error, the hydrogen bond binding energies of water are also significantly improved. In particular, the self-interaction correction to SCAN improves binding energies and the many-body analysis without altering the correct energy ordering for small water clusters.

Abstract

We gauge the importance of self-interaction errors in density functional approximations (DFAs) for the case of water clusters. To this end, we used the Fermi–Löwdin orbital self-interaction correction method (FLOSIC) to calculate the binding energy of clusters of up to eight water molecules. Three representative DFAs of the local, generalized gradient, and metageneralized gradient families [i.e., local density approximation (LDA), Perdew–Burke–Ernzerhof (PBE), and strongly constrained and appropriately normed (SCAN)] were used. We find that the overbinding of the water clusters in these approximations is not a density-driven error. We show that, while removing self-interaction error does not alter the energetic ordering of the different water isomers with respect to the uncorrected DFAs, the resulting binding energies are corrected toward accurate reference values from higher-level calculations. In particular, self-interaction–corrected SCAN not only retains the correct energetic ordering for water hexamers but also reduces the mean error in the hexamer binding energies to less than 14 meV/${\text{H}}_2\text{O}$ from about 42 meV/${\text{H}}_2\text{O}$ for SCAN. By decomposing the total binding energy into many-body components, we find that large errors in the two-body interaction in SCAN are significantly reduced by self-interaction corrections. Higher-order many-body errors are small in both SCAN and self-interaction–corrected SCAN. These results indicate that orbital-by-orbital removal of self-interaction combined with a proper DFA can lead to improved descriptions of water complexes.

Water is a ubiquitous substance with a simple molecular structure that nonetheless exhibits complex intermolecular interactions responsible for its unique properties. These interactions have inspired intense experimental and theoretical interest for decades. Small water clusters, apart from their importance in environmental chemistry and biology, play a dual role in this history, first, by serving as simple models that give insight into the behavior of more extended systems and, second, by providing a gauge for the quality of different methodologies in modeling water. From the electronic wave function viewpoint, coupled cluster theory with single, double, and quasi-perturbative connected triple excitations (CCSD(T)), diffusion quantum Monte Carlo, and second-order Möller–Plesset perturbation theory can describe water clusters with high accuracy (1–5); however, routine simulations of extended systems are computationally prohibitive with these methods.

Kohn–Sham density functional theory (DFT) (6) is an important computational method for the electronic structure of materials, including water, from isolated monomers and clusters to extended (bulk) states (see review in ref. 7 and references therein). The accuracy of many widely used exchange–correlation (XC) functionals is not satisfactory for water systems. The dipole moment and polarizability of an isolated water monomer given by the local density approximation (LDA) are in good agreement with experiments (8), yet LDA strongly overestimates the binding energies of water clusters (9) and lattice energies of ice crystals (10). Generalized gradient approximations (GGAs) typically improve these energies, but fail in other ways. For example, the widely used Perdew–Burke–Ernzerhof (PBE) (11) functional fails to capture the correct energetic ordering of the low-lying isomers of water hexamers (12, 13) and to accurately describe the phase transition between crystalline ice polymorphs (14, 15). Other failures of GGAs include red-shifted infrared spectra (16) and overstructured pair correlation functions of liquid water (17–19), and the prediction that liquid water is less dense than ice at ambient conditions (19–22).

One of the reasons for such failures is the inability of GGAs to accurately capture the delicate energy balance between compact and extended structures for which an accurate description of intermediate-range van der Waals (vdW)/dispersion interactions is essential (7). GGA functionals lack nonlocal electron correlation and are thus unable to describe intermediate-range vdW interactions. Adding explicit, approximate dispersion interactions has improved the description of water clusters (12), ice (10, 14, 15), and liquid water (18, 23–27); however, disagreement between competing dispersion-inclusive functionals has remained, due both to different approaches for computing vdW interactions and to differences in the underlying functionals (7). On the other hand, the nonempirical strongly constrained and appropriately normed (SCAN) (28) meta-GGA functional, which satisfies all 17 exact physical constraints that a semilocal XC functional can satisfy and includes intermediate-range dispersion, reproduces the correct energetic ordering among the low-lying isomers of water hexamers (29) and the density anomaly between liquid water and ice (22). Also, SCAN significantly improves the relative lattice energies of ice polymorphs (29) and the infrared spectra of liquid water (30). The fact that the nonempirical and computationally efficient SCAN corrects many failures of its predecessors for water systems signals a leap forward for DFT simulations of water and, in particular, for condensed phase water. However, SCAN is not without its drawbacks. For example, the binding energies of individual water clusters and lattice energies of ice phases are overestimated in SCAN calculations (29).

Self-interaction error (SIE) has long been identified as one of the limitations of practical density functional approximations (DFAs) (31–35). The failure of an approximate functional to be exact for all possible one-electron densities arises due to an incomplete cancellation of self-Coulomb terms by the approximate XC energy terms in the DFA total energy expression, and causes problems even in regions with more than one electron. Evidence of SIE in the DFT description of water can be found in the results of calculations using hybrid functionals that include some fraction of exact exchange. For example, in comparison to PBE, the hybrid functional PBE0 (36) reduces the binding energies of water hexamers by $\sim$14 meV/${\mathrm{H}}_2$O (12), lowers the lattice energy of ice I$h$ by 38 meV/${\mathrm{H}}_2$O (37), and blue-shifts the infrared spectra of ambient liquid water, resulting in better agreement with experiment (16, 38). However, hybrid functionals (without dispersion) fail to capture the aforementioned balance between compact and extended water structures (7). Moreover, different choices of hybrid functionals lead to disagreements on the extent of the influence of SIE on water systems (7).

The Perdew–Zunger (PZ) (31) self-interaction correction (SIC) provides an alternative way to quantify the extent of SIE in the description of water through a fully nonlocal, orbital-by-orbital removal of electron self-interaction for LDA and semilocal functionals. Recently, the Fermi–Löwdin orbital self-interaction correction (FLOSIC) methodology (39, 40) was introduced as an efficient and unitarily invariant approach for implementing PZ SIC that can be used in conjunction with any approximate XC functional. FLOSIC has been used to study properties of chemical and physical interest for a range of systems (41–45). Typically, both total error and SIE decrease from LDA to PBE to SCAN, but this is not always so. More generally, for nonmetals (including water), any large error of SCAN is dominated by self-interaction. Improvements in the way the SIE is handled are under development (46–48), but here we will apply only a currently standard way.

For this work, we used FLOSIC to predict the binding energies of 10 small water clusters, including the global minimum configurations of the dimer through the pentamer, four low-lying isomers of the hexamer, and two isomers of the octamer, with binding energies ranging from 109 meV/H₂O to 394 meV/${\text{H}}_2\text{O}$. We used three nonempirical DFAs (LDA, PBE, and SCAN), and their self-interaction–corrected counterparts (FLOSIC–LDA, FLOSIC–PBE, FLOSIC–SCAN). To assess the influence of density-driven error (49), we also evaluated the DFA total energy using the corresponding self-consistent FLOSIC–DFA density. We refer to this approach as DFA(@FLOSIC). See SI Appendix for all of the computational details of our calculations. We find that, for all three DFAs, PZ SIC weakens the binding energies for all water clusters, bringing them into better agreement with accurate CCSD(T) values. The FLOSIC–SCAN (45) binding energies are accurate within 18 meV/${\mathrm{H}}_2$O for all clusters, which is a significant improvement over SCAN. FLOSIC preserves the ordering of isomers predicted by the corresponding uncorrected DFA calculations. Thus, both SCAN and FLOSIC–SCAN predict the correct energetic ordering of the hexamers. Decomposing the total binding energy into many-body contributions reveals that large errors in SCAN arise from the two-body interaction and that FLOSIC–SCAN reduces the errors significantly. Errors in the three-body and higher-order interactions are small in both SCAN and FLOSIC–SCAN.

Results and Discussion

The results of all methods employed herein are summarized in Table 1, including mean unsigned errors (MUEs) in the binding energies of the first four (_[H2O]n, $n$ = 2 to 5) clusters, MUE(4); the four hexamers, MUE(hexamer); and all 10 complexes, MUE(10). The performance of FLOSIC for different water clusters is discussed below.

Table 1.

CCSD(T)-F12b reference binding energies (50) for water clusters and corresponding errors in binding energies from different methods

	Signed error
Cluster	Ref.	LDA	LDA(@FLOSIC)	FLOSIC–LDA	PBE	PBE(@FLOSIC)	FLOSIC–PBE	SCAN	SCAN(@FLOSIC)	FLOSIC–SCAN
${\left({\text{H}}_2\text{O}\right)}_2$	−108.6	−65	−65	−46	−2.2	−4.1	0.8	−9.4	−11.3	−2.0
${\left({\text{H}}_2\text{O}\right)}_3$	−228.4	−148	−146	−96	−5.8	−7.2	8.7	−28.9	−30.3	−5.8
${\left({\text{H}}_2\text{O}\right)}_4$	−297.0	−196	−196	−129	−19.6	−23.9	−2.2	−36.9	−40.2	−8.7
${\left({\text{H}}_2\text{O}\right)}_5$	−311.4	−206	−207	−137	−23.4	−27.7	−4.3	−39.8	−42.5	−12.1
MUE(4)	0.0	154	154	102	12.7	15.7	4.0	28.8	31.1	7.1
${\left({\text{H}}_2\text{O}\right)}_6$ P	−332.4	−213	−211	−137	−4.9	−8.2	12.7	−42.4	−44.7	−11.7
${\left({\text{H}}_2\text{O}\right)}_6$ C	−330.5	−215	−212	−137	−9.2	−12.8	10.0	−42.7	−45.0	−12.0
${\left({\text{H}}_2\text{O}\right)}_6$ B	−327.3	−216	−214	−134	−18.7	−22.9	0.9	−41.7	−44.4	−11.3
${\left({\text{H}}_2\text{O}\right)}_6$ R	−320.1	−208	−208	−139	−23.7	−28.2	−4.0	−39.4	−42.7	−17.6
MUE(hexamer)	0.0	213	211	137	14.1	18.0	6.9	41.6	44.2	13.2
${\left({\text{H}}_2\text{O}\right)}_8$ ${\mathrm{D}}_{2d}$	−393.5	−253	−251	−162	−12.5	−16.8	9.7	−48.8	−51.0	−7.1
${\left({\text{H}}_2\text{O}\right)}_8$ ${\mathrm{S}}_4$	−393.5	−253	−251	−162	−12.5	−16.8	9.7	−48.8	−51.0	−7.1
MUE(10)	0.0	197	196	128	13.2	16.9	6.4	37.9	40.3	9.5

signed errors and mean unsigned errors (MUEs) for the computed binding energies which are calculated at the DFA, DFA(@FLOSIC), and FLOSIC–DFA evels of theory using LDA, PBE, and SCAN. The negative and positive signed errors indicate overbinding and underbinding, respectively. All energies are in millielectronvolts per H2O. DFA(@FLOSIC) is calculated from DFAs that are applied a posteriori to the corresponding FLOSIC–DFA densities.

Binding Energies of Water Dimer to Pentamer.

The binding energies of the water dimer ${\left({\text{H}}_2\text{O}\right)}_2$, trimer ${\left({\text{H}}_2\text{O}\right)}_3$, tetramer ${\left({\text{H}}_2\text{O}\right)}_4$, and pentamer ${\left({\text{H}}_2\text{O}\right)}_5$ (SI Appendix, Fig. S1 $A$ and $B$) have been investigated both experimentally and theoretically (51–54). The open form of the water dimer ($C_s$ symmetry) was found to be more favorable than the cyclic analogue, while, for the trimer, tetramer, and pentamer, the cyclic arrangements, in which a maximum number of hydrogen bonds are created from donor–acceptor arrangements of water molecules, are more stable than their open forms (55). Table 1 shows that, although FLOSIC–LDA has a smaller MUE(4) of 102 meV/${\text{H}}_2\text{O}$ compared to LDA and LDA(@FLOSIC) (154 meV/${\text{H}}_2\text{O}$), it still significantly overbinds these four clusters. FLOSIC–PBE gives the smallest MUE(4) of 4 meV/${\text{H}}_2\text{O}$, improving over PBE (13 meV/${\text{H}}_2\text{O}$) and PBE(@FLOSIC) (16 meV/${\text{H}}_2\text{O}$). Similarly, FLOSIC–SCAN reduces the SCAN and FLOSIC(@SCAN) errors by almost 20 meV/${\text{H}}_2\text{O}$, yielding an MUE(4) of 7 meV/${\text{H}}_2\text{O}$.

Binding Energies of the Water Hexamers.

The potential energy surface of the water hexamer contains many local minima, including distinct two-dimensional (2D) and three-dimensional isomers (1). Their binding energies lie within a range of a few tens of millielectronvolts (3, 50, 56). Here, we focus on the binding energies and the energy ordering of the prism (P), cage (C), book (B), and ring (R) isomers (SI Appendix, Fig. S1, $E$–$H$). The reference data (50) in Table 1 show, as do other studies (3, 4, 12, 13, 57), that the energy ordering from lowest to highest (most stable to least) is P $<$ C $<$ B $<$ R. The binding energy difference between the most and least stable hexamer ($\mathrm{\Delta }{E}_{hex}$) is 12.3 meV/${\text{H}}_2\text{O}$ from the CCSD(T)-F12b reference (Table 1).

LDA and LDA(@FLOSIC) yield hexamer energy orderings that are largely consistent with the reference (SI Appendix, Table S1). FLOSIC–LDA also reproduces the correct ordering, but with a smaller $\mathrm{\Delta }{E}_{hex}$ of 10 meV/${\text{H}}_2\text{O}$, as compared to 17 and 15 meV/${\text{H}}_2\text{O}$ for LDA and LDA(@FLOSIC), respectively. FLOSIC reduces the MUE(hexamer) from 213 meV/${\text{H}}_2\text{O}$ for LDA to 137 meV/${\text{H}}_2\text{O}$ for FLOSIC–LDA, still far from chemical accuracy. PBE provides a significant improvement, yielding a MUE(hexamer) of only 14.1 meV/${\text{H}}_2\text{O}$. However, PBE erroneously predicts the B isomer as the most stable structure and an entirely wrong order for these hexamers. This finding agrees with previous studies which showed that the majority of gradient approximations incorrectly predicted the B or R isomer to be the most stable structure among these four isomers (4, 12, 13, 57, 58). Fig. 1A shows that both PBE and PBE(@FLOSIC) predict the following order of stability: B $<$ R $<$ C $<$ P, with a $\mathrm{\Delta }{E}_{hex}$ of $\sim$9 meV/${\text{H}}_2\text{O}$. FLOSIC–PBE systematically reduces binding energies for the four hexamers by $\sim$19 meV/${\text{H}}_2\text{O}$ compared to PBE, thus retaining the same incorrect ordering of the hexamers as found with PBE. FLOSIC–PBE underbinds the P, C, and B isomers, but it gives the best overall accuracy for the four isomers, with a MUE(hexamer) of 6.9 meV/${\text{H}}_2\text{O}$ (Table 1).

Fig. 1.

The binding energies ($E_b$) per ${\mathrm{H}}_2$O molecule of the four water hexamers [prism (P), cage (C), book (B), and ring (R)] obtained from (A) PBE, PBE(@FLOSIC), and FLOSIC–PBE and (B) SCAN, SCAN(@FLOSIC), and FLOSIC–SCAN, compared with the benchmark CCSD(T)-F12b values (50).

The nonempirical SCAN meta-GGA, which captures intermediate-range vdW interactions, accurately predicts the critical energy difference between the more compact P and C structures and the less compact B and R structures (29). A few empirical meta-GGAs (M06-L, M05-2X, and M06-2X) were also successful in this regard (13). Fig. 1B shows that our SCAN calculations reproduce the correct ordering, with a slightly larger $\mathrm{\Delta }{E}_{hex}$ of 15.3 meV/${\text{H}}_2\text{O}$ compared to the reference value of 12.3 meV/${\text{H}}_2\text{O}$. SCAN and SCAN(@FLOSIC) overbind the four isomers by 13 to 14% on average, yielding MUE(hexamer) of 41.6 and 44.2 meV/${\text{H}}_2\text{O}$, respectively. The overbinding is reduced to $\sim$4% when we use FLOSIC–SCAN, which gives a MUE(hexamer) of only 13.2 meV/${\text{H}}_2\text{O}$, preserving the correct ordering (as does SCAN(@FLOSIC)).

Binding Energies of Water Octamer.

The two most stable isomers of ${\left({\text{H}}_2\text{O}\right)}_8$ are isoenergetic cubic networks of 12 hydrogen bonds (59) (SI Appendix, Fig. S1, $I$ and $J$). Table 1 shows that all tested methods predict that the ${\mathrm{D}}_{2d}$ and ${\mathrm{S}}_4$ water octamers have equal binding energies. FLOSIC–LDA overestimates $E_b$ by 162 meV/${\text{H}}_2\text{O}$, although it improves over LDA and LDA(@FLOSIC) by almost 90 meV/${\text{H}}_2\text{O}$. The ${\left({\text{H}}_2\text{O}\right)}_8$ binding energy provided by FLOSIC–PBE is underestimated by only 9.7 meV/${\text{H}}_2\text{O}$, while PBE and PBE(@FLOSIC) overbind the octamer by 12.5 and 16.8 meV/${\text{H}}_2\text{O}$, respectively. Both SCAN and SCAN(@FLOSIC) overbind ${\left({\text{H}}_2\text{O}\right)}_8$ by almost 50 meV/${\text{H}}_2\text{O}$. However, FLOSIC–SCAN significantly improves over SCAN, yielding binding energies within 7.1 meV/${\text{H}}_2\text{O}$ of the reference.

Many-Body Analysis.

Further insight into the performance of functionals for the relative stability of water clusters can be extracted from the many-body decomposition of the binding energy (60–63), which can help to predict the behavior of functionals in extended water. For example, the prediction of the local structure of ambient liquid water is largely determined by a delicate balance between two- and three-body energies (64). Here we elaborate on the relative comparison between PBE and SCAN many-body interactions in determining the energetic ordering between the P or C structures and the B or R structures of water hexamers.

The reference CCSD(T)-F12b one-body interaction is about 19 meV/${\mathrm{H}}_2$O for the four hexamers with a variation of no more than 1.5 meV/${\mathrm{H}}_2$O (50). The one-body terms are nearly the same for all four hexamers in PBE and SCAN calculations as well. The one-body interaction thus does not contribute to the ordering of the hexamers and will not be considered further in this discussion. The reference total interaction energies (excluding one-body) for the hexamers and their many-body decomposition are shown in Table 2. From the reference energies, it can be seen that the two-body interaction contributes most (71 to 81%) to the total interaction energy, and the three-body (18 to 25%) and four-body (1 to 4%) interactions contribute correspondingly less. The sum of two- and three-body interactions constitute about 96 to 99% of the stability of these hexamers. It is also evident that, from the P or C to the B or R isomers, the two-body interaction decreases, whereas the three- and four-body interactions increase. In Fig. 2A, we analyze many-body contributions to the relative stability of the C, B, and R with respect to the P structure ($\mathrm{\Delta }{E}_{rel}^{nB}$). Fig. 2A shows that the two-body interactions destabilize the B and R structures compared to the P and C structures. In contrast, both three- and four-body interactions stabilize the B and R structures compared to the P and C structures. In other words, the relative stability among these hexamers hinges upon the subtle competition between two-, three-, and four-body interactions. Thus, for DFT functionals, it becomes a critical test to reproduce each of the many-body contributions.

Table 2.

The CCSD(T)-F12b reference (50) total interaction (excluding one-body interaction), two-body ($2B$), three-body ($3B$), and four-body ($4B$) interaction energies of four isomers [prism (P), cage (C), book (B), ring (R)] of (H₂O)₆, and the corresponding errors with PBE and SCAN

	Reference				PBE error				SCAN error
Cluster	Total	$2B$	$3B$	$4B$	Total	$2B$	$3B$	$4B$	Total	$2B$	$3B$	$4B$
P	−353.2	−285.6	−63.2	−4.6	8.0	1.9	12.2	−7.9	−41.4	−40.7	−0.4	−0.8
C	−350.8	−282.8	−64.4	−3.5	4.6	−0.5	10.9	−6.5	−41.1	−39.6	−1.4	0.2
B	−347.6	−265.0	−74.7	−7.6	−4.2	−3.1	2.7	−4.0	−38.7	−33.7	−3.9	−1.1
R	−339.5	−240.3	−85.1	−12.8	−9.4	−2.6	−3.0	−3.7	−35.4	−28.1	−4.8	−3.1

The negative and positive errors indicate overbinding and underbinding, respectively. All values are in millielectronvolts per H₂O.

Fig. 2.

($A$) Two-body ($2B$), three-body ($3B$), and four-body ($4B$) interaction energies in the cage, book, and ring hexamers relative to those for the prism isomer ($\mathrm{\Delta }{E}_{rel}^{nB}$), as obtained from the CCSD(T)-F12b reference (50), PBE, and SCAN. The negative and positive errors indicate overbinding and underbinding, respectively. (B and C) Error in the $\mathrm{\Delta }{E}_{rel}^{nB}$ from ($B$) PBE and ($C$) SCAN, with respect to the reference.

As shown in Table 2, PBE overbinds the B and R isomers and underestimates the P and C isomers. The two-body energies predicted by PBE are within 3.1 meV/${\mathrm{H}}_2$O of the reference values. However, a key difference appears in the three-body interaction, which significantly underbinds the P and C isomers (11 meV/H₂O to 12 meV/${\mathrm{H}}_2$O), and marginally overbinds the R isomer (3.0 meV/${\mathrm{H}}_2$O). The four-body term overbinds by 4 meV/H₂O to 8 meV/${\mathrm{H}}_2$O in all hexamers. The finding that the three-body error for the P and C isomers is significantly larger than the two-body error is in agreement with a previous many-body analysis (65). Fig. 2A shows that the $\mathrm{\Delta }{E}_{rel}^{nB}$ obtained with PBE follows the same qualitative trend as found with the CCSD(T)-F12b reference. However, there are quantitative differences between PBE and the reference, which are summarized in Fig. 2B. PBE overstabilizes the B and R structures by 10 meV/H₂O to 15 meV/${\mathrm{H}}_2$O compared to the reference, due to the three-body interactions (Fig. 2B). The errors for PBE coming from the two- and four-body contributions are small ($\sim$5 meV/${\mathrm{H}}_2$O) and largely cancel each other. This analysis shows that the overstabilization of the B and R isomers in the three-body interaction is the key reason for the incorrect prediction of the energetic ordering of the hexamers with PBE.

In contrast, SCAN predicts similar total errors for all four hexamers. Unlike PBE, the largest contribution to the error in SCAN comes from the two-body interaction, which is about −40 meV/${\mathrm{H}}_2$O for the P and C isomers and decreases to −28.1 meV/${\mathrm{H}}_2$O for the R isomer (Table 2). Recently, relatively large two-body errors have been found with SCAN for halide–water clusters as well (66, 67). Importantly, there is a sharp decline in the magnitude of the error in three- and four-body interactions obtained from SCAN (Table 2). Fig. 2C shows that the two-body interaction in SCAN overly destabilizes the B and R isomers compared to the P isomer by 7 meV/H₂O to 13 meV/${\mathrm{H}}_2$O. On the other hand, a small overstabilization occurs in the three-body (3 meV/H₂O to 4 meV/${\mathrm{H}}_2$O) and four-body (0.2 meV/H₂O to 2.0 meV/${\mathrm{H}}_2$O) interactions for the B and R isomers compared to the P isomer. This small error compensation contributes to the correct prediction of the energetic ordering of the hexamers by SCAN. Yet, even with the success of SCAN in predicting the correct energetic ordering of the hexamers, the error in the two-body interaction remains a significant drawback.

Since FLOSIC–SCAN predicts the correct ordering of the hexamers and significantly reduces the error in the binding energies obtained from SCAN, it is interesting to determine how the many-body interactions are affected by FLOSIC–SCAN. We focus only on the two- and three-body interactions in the trimer (global minimum configuration) and the P hexamer, since SCAN significantly overestimates the two-body term and is reasonably accurate for the three-body term. Table 3 shows a comparison of the errors in SCAN and FLOSIC–SCAN. Note that the small energy differences between Tables 2 and 3 for the P hexamer using SCAN are due to the slightly different structures used (SI Appendix). Table 3 shows that, for the trimer, almost all of the error in SCAN comes from the two-body term (−29.3 meV/${\mathrm{H}}_2$O), and FLOSIC–SCAN reduces the error by 12.2 meV/${\mathrm{H}}_2$O. Similarly, for the hexamer P, FLOSIC–SCAN reduces the error in the two-body interaction by 14.5 meV/${\mathrm{H}}_2$O. The three-body interactions are only slightly ($<$1 meV/${\mathrm{H}}_2$O) altered when FLOSIC is applied on top of SCAN. The higher-order terms in FLOSIC–SCAN are also expected to be similar to those in SCAN. Overall, FLOSIC–SCAN improves the overestimation of the two-body interactions in SCAN while simultaneously retaining accurate three- and higher-body interactions in water clusters.

Table 3.

Error in the total interaction energy (excluding one-body), $2B$ energy, and $3B$ energy of (H₂O)₃ and the hexamer prism (P) calculated with SCAN and FLOSIC–SCAN with respect to CCSD(T)-F12b (50)

	SCAN error			FLOSIC–SCAN error
Cluster	Total	$2B$	$3B$	Total	$2B$	$3B$
(H2O)3	−29.7	−29.3	−0.4	−16.8	−17.1	0.3
P	−36.1	−39.5	3.5	−21.6	−25.0	3.4

The _(H2O)3 and P are chosen as representative of the test set. The negative and positive errors indicate overbinding and underbinding, respectively. All values are in millielectronvolts per H₂O.

Conclusions

We have assessed the effect of removing SIE in common DFAs for water clusters. We employed an orbital-by-orbital removal of SIE using the FLOSIC methodology to show that the resulting binding energies with self-interaction–corrected DFAs are closer to accurate reference values for clusters containing two to eight water molecules. The use of DFA(@FLOSIC), calculated from semilocal density functionals that are applied a posteriori to the corresponding FLOSIC densities, does not lead to better estimates than those obtained with the semilocal DFA. This indicates that the semilocal DFA overbinding of water clusters is not a density-driven error. Remarkably, we found that removal of SIE does not alter the ordering of the binding energies of the P, C, B, and R hexamers with respect to the underlying functional, but, instead, shifts the binding energies toward more accurate estimates. In particular, self-interaction–corrected SCAN not only yields the correct energetic ordering for water hexamers but also reduces the MUE from about 42 meV/${\text{H}}_2\text{O}$ for SCAN to less than 14 meV/${\text{H}}_2\text{O}$ for FLOSIC–SCAN. This indicates that orbital-by-orbital removal of the SIE combined with appropriate DFAs can lead to accurate energetics for water clusters. Moreover, many-body decomposition of the interaction energy shows that SCAN overestimates the two-body interactions but predicts three- and four-body interactions accurately within 5 meV/${\text{H}}_2\text{O}$ for the hexamers. FLOSIC–SCAN improves the two-body interactions over SCAN and predicts higher-order many-body interactions with about the same accuracy as SCAN. This suggests that FLOSIC–SCAN may describe liquid water and ice even better than SCAN. It will be interesting to explore this further in the future.

In additional future work, we plan to apply FLOSIC to the interaction between a small water cluster and a halide anion (66, 67). We expect a significant improvement, since a free neutral atom cannot bind a full extra electron within a semilocal approximation like LDA, PBE, or SCAN, except via basis set limitations. In the complete basis set limit, a fraction of the extra electron is excessively delocalized (31). PZ SIC is expected to correct this and other delocalization errors (68). Further improvement is expected from restoring the correct limit of slowly varying density (48).

Can we understand how the PZ SIC leaves the energetic order of the structures of a water cluster unchanged, while significantly improving the binding energy of the cluster? We know that the energy difference between the uncorrected and self-interaction–corrected electron densities is small. The additive PZ SIC depends only on the localized Fermi–Löwdin orbitals, which must change very little from one structure to another, more significantly from cluster to dissociated molecules, and disruptively if the molecules were atomized.

This article helps to explain how the SCAN functional can significantly overbind the hydrogen bond between two water molecules and still accurately predict the small energy differences between competing hydrogen bond networks in water hexamers (29), bulk liquid water (22, 30, 69, 70), and liquid water at an interface (71–73). The correct features of SCAN that improve structural energy differences are almost independent of its SIE that produces the overbinding. SCAN, by itself, is also accurate for structural energy differences in diverse other materials (29, 74).

Materials and Methods

All calculations were performed with a development version of the FLOSIC code (75). DFT-optimized atom-centered Gaussian basis sets (76) with additional polarization/diffuse functions were used. The Fermi–Löwdin orbitals and their descriptors were converged following ref. 40. More details are given in SI Appendix.

Supporting Information.

SI Appendix contains details of the computational methodology, binding energies, relaxed monomer structural comparisons, and nuclear and FOD coordinates.

Data Availability.

Additional data are available in SI Appendix.