Assessing Many-Body Effects of Water Self-Ions. I: OH (H2O)n Clusters

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Assessing Many-Body Effects of Water Self-Ions. I: OH(HO) Clusters Colin K. Egan, and Francesco Paesani J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b01273 • Publication Date (Web): 15 Mar 2018 Downloaded from http://pubs.acs.org on March 16, 2018

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Journal of Chemical Theory and Computation

Assessing Many-Body Effects of Water Self-Ions. I: OH−(H2O)n Clusters Colin K. Egan† and Francesco Paesani∗,†,‡,¶ †Department of Chemistry and Biochemistry, University of California San Diego, La Jolla, California 92093, United States ‡Materials Science and Engineering, University of California San Diego, La Jolla, California 92093, United States ¶San Diego Supercomputer Center, University of California San Diego, La Jolla, California 92093, United States E-mail: [email protected] Abstract The importance of many-body effects in the hydration of the hydroxide ion (OH− ) is investigated through a systematic analysis of the many-body expansion of the interaction energy carried out at the CCSD(T) level of theory, extrapolated to the complete basis set limit, for the low-lying isomers of OH− (H2 O)n clusters, with n = 1 − 5. This is accomplished by partitioning individual fragments extracted from the whole clusters into “groups” that are classified by both the number of OH− and water molecules and the hydrogen bonding connectivity within each fragment. With the aid of the absolutely localized molecular orbital energy decomposition analysis (ALMO-EDA) method, this structure-based partitioning is found to largely correlate with the character of different many-body interactions, such as cooperative and anticooperative hydrogen bonding, within each fragment. This analysis emphasizes the importance of a many-body representation of inductive electrostatics and charge transfer in modeling OH− hydration.

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Furthermore, the rapid convergence of the many-body expansion of the interaction energy also suggests a rigorous path for the development of analytical potential energy functions capable of describing individual OH− −water many-body terms, with chemical accuracy. Finally, a comparison between the reference CCSD(T) many-body interaction terms with the corresponding values obtained with various exchange-correlation functionals demonstrates that range-separated, dispersion corrected, hybrid functionals exhibit the highest accuracy, while GGA functionals, with or without dispersion corrections, are inadequate to describe OH− −water interactions.

1

Introduction

Water is essential to life as we know it. This is the case because of its role as the primary solvent in biological systems, due to its polarity, the vastness of the conditions in which it exists in the liquid phase, and its ability to form hydrogen bonds. 1 After decades of research, there is now little question that the unique behavior of water results not from a simple three-nucleus, 10-electron molecule, but rather from the delicate interplay of many-body interactions between individual water molecules from clusters to the bulk. 2–9 Many applications of water as a solvent rely on its amphoterism. The self-ions of water, H3 O+ and OH− , function as the mediators of aqueous acid/base chemistry, and therefore play key roles in many fields within physical and natural sciences, including chemistry, 10–12 biology, 10,13–19 and environmental sciences. 20,21 Arguably the most fundamental consequence of the amphoteric character of water is the tendency of pairs of molecules to undergo the so-called autoionization reaction (Eq. 1), whose equilibrium constant corresponds to the widely-known KW = 10−14 for liquid water at room temperature and atmospheric pressure, 2H2 O OH − + H3 O+

(1)

The autoionization (and subsequent recombination) of water continues to be the subject

2


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of ongoing research. 22–36 However, practical issues, such as the short timescale and infrequency of reactive events, 22,23,27 along with the complexity of the associated (free-)energy landscape, 25,27,29,31,34,35 lead to great difficulty in probing the dissociation of neutral water on both the experimental and theoretical fronts. Consequently, the exact molecular mechanisms through which water autoionization occurs remain poorly understood, and related questions, such as whether the air/liquid interface of neutral water is basic, 24,36–47 or acidic 32,33,47–53 are the subject of intense debate. However, there is some evidence that the autoionization mechanism involves concerted motions of four water molecules on the sub-picosecond timescale, 22,23,27,34,35 suggesting that the reaction might, in fact, be better represented by

4H2 O 2H2 O + OH − + H3 O+

(2)

Molecular dynamics (MD) is particularly well-suited to study the energetics and dynamics of proton transport in aqueous systems, as it allows for an atomic-scale investigation of structural, thermodynamic, and dynamical properties of molecular systems. In this context, the MB-pol many-body potential energy function (PEF) 9,54–56 has recently emerged as one of the most successful molecular models for MD simulations of neutral water. The success of MB-pol in accurately predicting the properties of water across different phases, 9,57–59 is due, in part, to the fact that it was systematically derived from high-level electronic structure data for individual terms of the many-body expansion (MBE) of water−water interaction energies 8,60 E int (x1 , x2 , x3 , . . . , xN ) =

X

E 1B (xa ) +

a

X

E 2B (xa , xb ) +

a>b

X

E

3B

(xa , xb , xc ) + . . . + E

(3) NB

(x1 , x2 , x3 , . . . , xN )

a>b>c

Here, each xi collectively represents the coordinates of all atoms within the ith molecule. Building on the accuracy of MB-pol, many-body PEFs (called MB-nrg for many-body en-

3



ergy) describing monoatomic ion-water interactions have recently been developed. 61,62 In order to extend the MB-nrg formalism to OH− −water and H3 O+ −water interactions, it is necessary to establish a cohesive picture of many-body effects inherent to these systems. It is known that for systems whose MBEs converge quickly, knowledge about molecular clusters can be systematically extended to bulk systems. Many studies have been reported in the literature investigating OH− (H2 O)n and H3 O+ (H2 O)n clusters using both theoretical 63–101 and experimental 73,91,102–116 approaches. However, a rather small fraction of the theoretical work investigated many-body effects in these clusters. 64,65,69,72,74,89,95–98 As a first step toward a quantitative assessment of many-body effects of water self-ions in aqueous systems, we report here an analysis of OH− −water interactions in low-energy isomers of OH− (H2 O)n clusters, with n ≤ 5. The article is organized as follows: In section 2 we describe the theoretical and computational methods used in our analysis. In section 3.1 we demonstrate the convergence of the MBE of the interaction energies of the four lowest-lying isomers of the hexamer (6-mer) cluster OH− (H2 O)5 , and discuss in detail the 2-mer OH− (H2 O), and 3-mer OH− (H2 O)2 clusters to establish some context regarding the nature of OH− −water hydrogen bonding, as well as to introduce the most common type of 3-body interactions. In section 3.2 we analyze the geometries and relative energies of the clusters studied. In section 3.3 we present a detailed discussion of many-body interactions in OH− (H2 O)n clusters, using hydrogen bond connectivities (represented by directed graphs 90,93,117 ) to aid our analysis of many-body effects and hydrogen bond cooperativity/anticooperativity. In section 3.4, we evaluate the accuracy of various density functional theory (DFT) models in reproducing many-body effects in OH− (H2 O)n clusters calculated at the coupled cluster level of theory. The conclusions and outlook are discussed in Section 4.

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2

Theoretical and Computational Methodology

2.1

Cluster Structures

The clusters investigated in this study were selected to represent various coordination numbers and hydrogen bonding connectivities. Most clusters were optimized from coordinates reported in the literature, 84,85 or adapted from previous studies. 84 Each cluster was optimized at the RI-MP2 118,119 level of theory with the aug-cc-pVTZ 120 basis set using a gradient optimization-threshold of 1.0 × 10−6 a.u., an optimization step-size of 1.0 × 10−6 a.u., and a gradient precision of 1.0 × 10−8 a.u.

2.2

Electronic Structure Calculations

In order to guarantee chemical accuracy at all orders, references energies for the individual MBE terms used in the analysis reported in Section 3 were calculated using the “stratified approximation” many-body approach (SAMBA), 60 through which successively higher-body interaction energy terms are calculated with successively lower levels of theory. This approach effectively allows one to calculate the each contribution to the interaction energy (2B, 3B, etc.) with the highest possible accuracy, given finite computational resources. The reference 2B interaction energies were computed at the coupled cluster with single, double, and perturbative triple excitations, i.e., CCSD(T), level of theory 121,122 extrapolated to the complete basis set limit (CBS). The energies obtained with the augmented, correlation-consistent triple (aug-cc-pVTZ) and quadruple (aug-cc-pVQZ) basis sets 120 were extrapolated via the two-point extrapolation recommended in Ref. 60 for CCSD(T) calculations, CBS EHF

=

X EHF

CBS Ecorr

−

X Y EHF − EHF √ √ e−α X − e−α Y

Y X Y β Ecorr − X β Ecorr = Y β − Xβ

5


√

e−α

X

(4)

(5)


Here, the Hartree Fock (Eq. 4) and correlation (Eq. 5) contributions to the total CCSD(T) energy are extrapolated separately. In both equations, X denotes the lower-basis set cardinal number (e.g., X = 3 for aug-cc-pVTZ), and Y denotes the upper-basis set cardinal, while α and β are parameters that must be optimized for the given basis sets used in the extrapolation. Following Ref. 123, the two extrapolations were carried out with α = 5.790 and β = 3.050, respectively. Additionally, it was found necessary to correct for the basis set superposition error (BSSE) in computing all CCSD(T) reference energies. Within the SAMBA framework, multiple BSSE correction schemes can be used for different terms in the MBE. The two BSSE correction schemes used in this study are the “site-site function counterpoise” method (SSCP) proposed, 124 and the “K-centered CP” method originally proposed in Ref. 125 and generalized in Ref. 126. The SSCP correction requires the calculation of each (sub-cluster) fragment energy in the basis of the entire cluster. 124 In the K-centered CP correction, each K-mer fragment’s contribution to the total K-body term in the MBE is calculated in the basis of the K-mer fragment. 126 By construction, the SSCP-corrected calculations become prohibitively expensive for large clusters and large basis sets, while the number of K-centered CP corrected calculations grows rapidly with K. Based on this, the reference 2B interaction energies were computed using K=2-centered CP corrections, and the 3B interaction energies were computed at the CCSD(T)/aug-cc-pVTZ level of theory using K=3-centered CP corrections. All reference energies for 4B and higher-order terms were computed within the SSCP scheme at the explicitly correlated, CCSD(T)-f12b/aug-cc-pVTZ level of theory. 127,128 To investigate the accuracy of different DFT exchange-correlation models in describing many-body effects in OH− −water interactions, calculations with the aug-pc-3 129 basis set were performed with: BLYP, 130 PBE, 131 revPBE, 132 SCAN, 133 TPSS, 134 B3LYP, 135 PBE0, 136 revPBE0, M06-2X, 137 ωB97X-D, 138 and ωB97M-V. 139 Excluding ωB97X-D and ωB97M-V, all DFT energies were calculated with and without Grimme’s D3(0) empirical dispersion correction (GD3), 140 so that the effect of the GD3 corrections could be evaluated

6


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along with the corresponding functionals. In the analysis of 3B interactions, we made use of the (1st generation) absolutely-localized molecular orbital energy decomposition analysis (ALMO EDA) method, 141 including the associated charge transfer analysis (CTA), 142 to dissect 3B interactions into individual, physical contributions. All ALMO EDA calculations were performed with ωB97M-V in combination with the Karlsruhe def2-TZVPPD 143,144 basis set. ωB97M-V was used because it yields, on average, energies closer to the SAMBA reference data than any other functional tested (see section 3.4 and the Supporting Information), and the def2-TZVPPD basis set because it is large enough to fully capture charge transfer (CT) effects, but still small to avoid intermonomer-orbital linear-dependence, which can lead to unwanted muddling between polarALM O O ) ALMO EDA energy terms. The ALMO EDA method was ization (EALM P OL ) and CT (ECT O O ), (∆EALM devised in such a way that pairwise, directional (monomer → monomer) EALM CT CT

and CT “electron reorganization” (∆QCT ), are well-defined, even in clusters with more than two fragments. 141,142 All CCSD(T) calculations and RI-MP2 optimizations were carried out using MOLPRO, version 2015.1. 145 DFT calculations were carried out with Gaussian 09 146 for all functionals except revPBE, revPBE0, SCAN, and ωB97M-V, for which Q-Chem 4.4 147 was used. Additionally, Q-Chem 4.4 was used for all ALMO EDA/CTA calculations. 141,142

3 3.1

Results Building Blocks: OH− (H2 O) and OH− (H2 O)2

The convergence of the MBE for the hydration of OH− is presented in Fig. 1 for the OH− (H2 O)6 clusters shown in Fig. 5. This analysis indicates that 2B and 3B contributions dominate the total interaction energies, which suggests that 2- and 3-mer clusters can effectively be considered as the primary ‘building blocks’ of larger deprotonated water cluster. 7



25 0 Energy (kcal/mol)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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2

-25

1 0

-50

-1 4B

5B

6B

-75 −

OH (H2O)5 isomer (a) −

OH (H2O)5 isomer (b)

-100

−

OH (H2O)5 isomer (c) −

OH (H2O)5 isomer (d)

-125 2B

3B

4B

5B

6B

Many-body term

Figure 1: Convergence of the many-body expansion of the low-lying isomers of OH− (H2 O)5 shown in Fig. 5. The most prevalent and, arguably, most important building block that constitutes all OH− (H2 O)n clusters is the OH− (H2 O) 2-mer in which the water molecule donates a hydrogen bond to OH− , as is the case in the OH− (H2 O) global minimum-energy (GME) structure (Fig. 2). In general, hydrogen bonds, especially when involving charged species, are highly electrostatic in nature and can effectively be seen as “incipient chemical bonds.” 148,149 This is very much the case in OH− /H2 O systems, where efficient proton transfer (i.e., incipient chemical bond → chemical bond) leads to seemingly “anomalously” high ion mobility. 150,151 In addition to a strongly attractive interaction, a lengthening of the donor-hydrogen (covalent) bond length is characteristic of the hydrogen bond. This structural effect is a result of the electrostatic attraction between the donor-hydrogen and the acceptor atom, as well as charge transfer. 149 These effects are fully realized in the OH− (H2 O) GME structure for which the interaction energy is −37.092 kcal/mol and the length of the water donor-OH bond is

8


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a)

b)

c)

d)

e)

Figure 2: OH− (H2 O)n building blocks. a) OH− (H2 O) global minimum-energy structure, b) OH− (H2 O)2 global minimum-energy structure along with its constituent 2-mer fragments (c-e). 1.11 ˚ A. For reference, the equilibrium OH bond length in an isolated water is 0.96 ˚ A, optimized at the same level of theory (RI-MP2/aug-cc-pVTZ). The relatively large strength of the interaction is emphasized when compared to that of the F− (H2 O)1 GME structure, which has an interaction energy of −32.704 kcal/mol and a water donor-OH bond length of 1.06 ˚ A, also obtained at the RI-MP2/aug-cc-pVTZ level of theory. It is worth noting that the GME structure of OH− (H2 O) reported here, wherein the donor-proton is localized on one oxygen—allowing one to unequivocally identify individual OH− and H2 O molecular identities within the 2-mer—corresponds to the Born-Oppenheimer equilibrium geometry. However, previous experimental and theoretical studies 67,75,109 have shown that the actual, quantum-mechanical ground state of OH− (H2 O) is delocalized along the proton transfer coordinate, lying above the associated potential energy barrier. This implies that, in a fully quantum-mechanical picture of the GME structure, the donor-proton is actually shared equally between the two oxygen atoms. The presence of a second water molecule in OH− (H2 O)2 introduces additional 2B contributions from a second OH− −water interaction and a single water−water interaction, as well as a 3B contribution associated with the interaction of OH− with the two water molecules. The total 2B interaction energy for the OH− (H2 O)2 GME structure is −58.333 kcal/mol, with each OH− −water interaction contributing, on average, −29.576 kcal/mol, which is ∼7 kcal/mol higher than in the OH− (H2 O) GME structure. The weakening of the average OH− −water interaction compared the 2-mer is reflected in a lower degree of water distor-

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tions, with both donor-OH bond lengths being equal to ∼1.03 ˚ A, as well as in the lengthening of the OH− −water O-O distance from 2.48 ˚ A in the 2-mer to 2.57 ˚ A in the 3-mer. The water−water 2B contribution is relatively small and repulsive, with a magnitude of 0.820 kcal/mol and a water−water O-O distance of 4.59 ˚ A. 3B effects contribute 3.402 kcal/mol to the total interaction energy of OH− (H2 O)2 in the GME structure. An early study based on symmetry-adapted perturbation theory (SAPT) determined that the dominant component of the 3B term is the induction energy (including CT effects) arising from the deformation of the isolated-monomer electron orbitals. 69

3.2

Structures and Relative Energies of OH− (H2 O)n , with n = 3−5

The interaction energies of the low-lying isomers of OH− (H2 O)n are reported in Table 1. To facilitate the analysis, we first introduce the notation used to describe hydrogen bonding connectivities. The label i WX is used to indicate each water molecule X ∈ {A, . . . , E} within the cluster as shown in Figs. 3-5, with i defining the solvation shell that the water molecule resides in. The solvation shell of a water molecule corresponds to the minimum number of hydrogen bonds that separate the molecule from OH− . All water−water and OH− −water hydrogen bonds are classified according to 152

2 ROacc −Odon ≤ −0.00044 θO + 3.3 acc −Odon −Hdon

(6)

Here, a pair of molecules are classified as hydrogen bonded if their O-O distance, ROacc −Odon , falls within the inequality depending on the acceptor-donor-donor O-O-H angle, θOacc −Odon −Hdon , where all lengths have units of ˚ A, and angles are in degrees. The different types of water molecules within a cluster can be further differentiated by adding the subscript Y ∈ {A, . . . , E} that identifies the hydrogen bond partner of X, with a left or right subscript indicating that Y is a hydrogen-bond acceptor or donor, respectively. Within this notation, for example, if 1 WB accepts a hydrogen bond from 2 WD , then the complete notation would

10


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Table 1: Reference interaction energies (in kcal/mol) for the low-lying isomers of OH− (H2 O)n clusters shown in Figs. 2−5. Cluster size

(a)

(b)

(c)

(d)

(e)

(f)

−72.080

−71.762

2-mer

−37.092

3-mer

−54.931

4-mer

−74.543

−72.829

−72.720

−72.428

5-mer

−90.717

−89.552

−87.655

−86.896

6-mer

−106.151

−104.728

−104.450

−99.516

be 1 WB D , and

D 2 BW

for the two molecules, respectively. The first six low-lying isomers of

OH− (H2 O)3 are shown in Fig. 3. Five of the isomers have 3-coordinated OH− , while the remaining isomer (isomer a) has a 2-coordinated OH− within a 4-membered ring structure in which one water molecule occupies the second solvation shell of OH− and donates hydrogen bonds to each of the remaining molecules residing in the inner solvation shell (i.e., C 2 AB W ).

Interestingly, this isomer has the lowest interaction energy of the 4-mer cluster set,

despite having the fewest OH− −water hydrogen bonds. The physical effects that cause the stabilization of this ring structure can be traced back to the subtle balance of many-body interactions in deprotonated water clusters, and will be analyzed in detail in the fragment analysis in Section 3.3. The 3-coordinated 4-mer with the lowest interaction energy (isomer b) is non-symmetric (C1 symmetry) although it resembles a distorted C3V structure. Two of the remaining 3coordinated structures have C3V symmetry (isomers c and e) while the other two isomers have CS symmetry (isomer d and f ) and only differ by the orientation of the water molecule lying in the plane of symmetry (1 WC ). The CS isomer whose 1 WC free-OH bond points antiparallel to the OH− OH bond has the lower energy of the two. Among the four low-lying OH− (H2 O)4 isomers shown in Figs. 4, the two isomers with the lowest interaction energies have structures containing 4-membered rings (isomers a and b).

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Each of these 4-membered ring fragments has the same connectivity as the 2-coordinated, GME 4-mer. The lowest-energy 5-mer (isomer a) is characterized by two additional 4membered rings, which are nonplanar, with hydrogen bond connectivities corresponding to {OH− ,

1 B 2 D 1 C D W , C WB , WD },

and {OH− ,

1 B 2 D 1 A D W , A WB , WD },

respectively. The presence

of the two nonplanar rings endows the GME 5-mer with a necessarily three-dimensional structure, whereas all three of the other isomers can be deformed into two dimensions without breaking any hydrogen bonds or distorting any water molecules. The single isomer with a 2-coordinated OH− (isomer d) has the highest interaction energy of the 5-mers considered in this study. Its structure is linear such that no water molecule donates nor accepts more than one hydrogen bond. The third-lowest energy 5-mer (isomer c) has a branched-chain structure with the branching occurring at the OH− location (i.e., OH− is 3-coordinated). The first solvation shell of this isomer is reminiscent of the second-lowest-energy 4-mer (isomer b in Fig. 3) in that both appear to have distorted C3V symmetry (although they are strictly non-symmetric, with C1 symmetry). However, the presence of the 2C WD molecule leads to a ˚ contraction of the OH− −1 WC D O-O distance to 2.52 A, compared to an O-O distance of 2.61 ˚ A in the corresponding OH− −1 W 2-mer fragment in the C1 4-mer. The two 6-mers with the lowest interaction energies (isomers a and b in Fig. 5) have structures which resemble two low-energy isomers of the neutral (H2 O)6 cluster, the “glove” and “book” isomers, respectively. 6 However, the structures of the deprotonated glove and book isomers are distinct from those of their neutral-water counterparts in their hydrogen bonding topologies, due to both the strength of the OH− −water 2B and 3B interactions, and

a)

b)

B

c)

C

d) C

C

A

B A

e)

C

f)

C

C

A A

B

B

A

A B

B

Figure 3: Low-lying isomers of the OH− (H2 O)3 cluster shown in order of increasing interaction energy from isomer a to isomer f .

12


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a)

b)

c)

B

d)

D

C

D

C A D

D

C

C

A B

B

B A

A

Figure 4: Low-lying isomers of the OH− (H2 O)4 cluster shown in order of increasing interaction energy from isomer a to isomer d. the fact that OH− accepts numerous hydrogen bonds, but donates none. These properties restrict the possible hydrogen bonding connectivities of the surrounding five water molecules, which results in the deprotonated isomers having different numbers of double-donor and double-acceptor water molecules than the neutral isomers. One notable consequence of these structural effects is the difference in symmetry of the deprotonated book compared to the neutral book, which are CS and C1 , respectively. While there are several neutral book isomers whose energies are within 1 kcal/mol of each other, 7 the additional symmetry effectively imposed on the deprotonated book makes it unique, since nontrivial rearrangements of any hydrogen bonds in OH− (H2 O)5 will exchange attractive 3B interactions with repulsive 3B interactions (see Table 3). The third-lowest-energy 6-mer (isomer c) has a 4-coordinated OH− , whereas all three other low-lying 6-mers examined in this study have 3-coordinated OH− . The single water molecule in the 2nd solvation shell of isomer c, 1 C 1 D that the fragment {OH− , 1 WB E , E W , WE ,

2 E DB WC ,

2 E BD WC }

closes three 4-membered rings so

has the same connectivity as the 5-mer

GME isomer (Fig. 4a). The 6-mer with the highest interaction energy has a branched-chain structure that resembles the branched-chain 5-mer (Fig. 4c), and the C1 4-mer (Fig. 3b). In the same way that the OH− −1 W O-O distance contracts when 1 W accepts a hydrogen bond from a 2 W molecule in the branched chain 5-mer (isomer c), both O-O distances between the hydrogen bond-accepting 1 W molecules and OH− in the branched chain 6-mer are contracted compared to those in the C1 isomer (isomer b) of OH− (H2 O)3 . 13



a)

b)

C

C

E

B

c)

E

D

d)

D

D

B

A

C

E

A

A

C

A

Page 14 of 53

E

B

B D

Figure 5: Low-lying isomers of the OH− (H2 O)5 clusters shown in order of increasing interaction energy from isomer a to isomer d.

3.3

Fragment Analysis and Many-Body Effects in Deprotonated Water Clusters

In this section, we analyze the individual many-body contributions to the interaction energies associated with the different clusters analyzed in Sec. 3.2. Since there are over 400 fragments within the 16 OH− (H2 O)n clusters considered in this study, which precludes the possibility of analyzing each fragment separately, all fragments are partitioned into a relatively small number of “fragment groups,” which are defined in terms of configurational geometry. Three properties are used to classify the different fragment groups: 1) presence of OH− in the fragment, 2) number of molecules in the fragment, and 3) (directional) hydrogen bonding connectivity, not distinguishing between permutations of water molecules. In addition, the fragment notation introduced in Section 3.2 is extended to describe fragment groups by retaining the solvation shell and connectivity rules when relevant, but only assigning rightsuperscript letters to distinguish water molecules, rather than to identify particular molecules in a cluster. Finally, a subscript “G” is appended outside the curly braces to specify that the notation refers to a group of fragments. For example, keeping in mind that OH− does not donate a hydrogen bond, there are four possible 2-mer fragment groups within OH− (H2 O)n clusters: 1. water donating a hydrogen bond to OH− : {OH− , 1 W}G B 2. water donating a hydrogen bond to water: {WA B , A W }G

14


Page 15 of 53 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3. OH− (H2 O) with no hydrogen bond: {OH− , 2 W}G 4. (H2 O)2 with no hydrogen bond: {WA , WB }G Table 2 reports the statistics for the four 2-mer fragment groups enumerated above. The “freq.” column reports the number of fragments within each group from all clusters. The “max.,” “min.,” and “mean” columns report the signed-maximum, signed-minimum (i.e., the most repulsive and attractive fragment energy contributions, respectively), and signed-mean of the 2B interaction energy contributions within each fragment group. The “RSD” (relative standard deviation) column reports the standard deviation of the 2B interaction energy contributions divided by the absolute value of their signed-mean. Values in the “sig. factor” column quantify the ‘significance’ of each fragment group, and are calculated by multiplying the unsigned-mean energy (not shown) by the occurrence frequency of that fragment group, and dividing the resulting product by the sum of all frequencies of the fragment groups with the same fragment size. The fragment groups are listed in descending significance order. Tables 2, 3, 4, and 5 show that most fragment groups contain exclusively attractive or repulsive fragments, indicating that the classification adopted in this analysis reveals information about specific many-body interactions within OH− (H2 O)n clusters. The groups that contain both attractive and repulsive fragments turn out to be structurally ambiguous, since the non-hydrogen bonded region of a given configurational space is much larger than the complementary hydrogen bonded region. The most prevalent example of these ‘structurally Table 2: Statistics for the 2B energies (in kcal/mol) of 2-mer fragments. See main text for details. Frag. Group

freq.

max.

min.

mean

RSD

sig. factor

{OH− , 1 W}G

44

−23.949

−37.092

−27.779

0.077

8.731

{OH− , 2 W}G

13

−5.208

−12.006

−9.582

0.200

0.890

B {WA B , A W }G

24

−2.125

−4.184

−3.307

0.147

0.567

{WA , WB }G

59

+1.720

−0.525

+0.462

1.119

0.242

15



Page 16 of 53

Table 3: Statistics for the 3B energies (in kcal/mol) of 3-mer fragments. See main text for details. Frag. Group

freq.

max.

min.

mean

RSD

sig. factor

{OH− , 1 WA , 1 WB }G

42

+4.315

+1.948

+2.862

0.159

0.829

2 B {OH− , 1 WA B , A W }G

20

−0.817

−5.278

−2.246

0.642

0.310

{OH− , 1 W, 2 W}G

14

+1.244

+0.265

+0.757

0.356

0.073

8

+1.022

+0.737

+0.876

0.110

0.048

{OH− , 1B WA , 2 WB A }G

3

+2.215

+2.004

+2.105

0.041

0.044

B C {WA B , A WC , B W } G

6

−0.331

−0.889

−0.589

0.309

0.024

{WA , WB , WC }G

19

−0.005

−0.467

−0.172

0.617

0.023

B C {WA B , A W , W }G

27

+0.255

−0.307

+0.059

2.127

0.022

B C { C WA B , A WC , B WA } G

1

−1.503

−1.503

−1.503

N/A

0.010

{OH− , 2 WA , 2 WB }G

3

+0.349

+0.126

+0.256

0.370

0.005

B C {WA BC , A W , A W }G

1

+0.571

+0.571

+0.571

N/A

0.004

B 2 {OH− , 2 WA B , A W }G

1

+0.163

+0.163

+0.163

N/A

0.001

B {WA C , WC ,

AB W

C

}G

ambiguous’ fragment groups (of all sizes) is the non-hydrogen bonded (H2 O)2 group, {WA B, AW

B

}G , whose most attractive and most repulsive fragments contribute −0.525 and 1.720

B kcal/mol, respectively, with a RSD of 1.119. However, the majority of the {WA B , A W }G

fragments (50 out of 59 fragments) tend to be repulsive, which results in a mean contribution of 0.462 kcal/mol. Although the {OH− , 2 W}G fragment group also appears to be structurally ambiguous, due to the lack of direct hydrogen bonding, each fragment is associated with a negative (attractive) 2B energy contribution, despite a RSD of 0.200, and a difference of almost 7 kcal/mol between minimum and maximum values. This results from the fact that each fragment considered in this analysis was generated as a ‘byproduct’ of a geometry optimization performed on one of the low-lying OH− (H2 O)n clusters. This implies that 2B interaction energies associated with particular fragment groups can have a greater influence than others on determining the minimum-energy structures of the parent clus-

16


Page 17 of 53

ters on the Born-Oppenheimer potential energy surface, thus biasing the type of fragments that can be generated. The remaining two 2-mer fragment groups are hydrogen-bonded OH− (H2 O) and (H2 O)2 which contain exclusively attractive fragments since hydrogen bonds are attractive, by definition. 148 These two fragment groups have been extensively investigated in previous studies and we refer the reader to the original references for specific details. 54,66,67,75,78–80,80,82,83,105,107–109,149,153–167 It should only be noted that the “min.” value for {OH− , 1 W}G in Table 2, corresponding to the GME OH− (H2 O) structure, is significantly lower than the 2B energy (−31.421 kcal/mol) of the next most attractive fragment. This difference is due to repulsive 3B interactions present in clusters with two or more 1 W molecules, which limit how close the 1 W molecules can get to OH− in a stable geometry. Among the 14 possible 3-mer fragment groups (not counting bifurcated hydrogen bonds, or pairs of water molecules donating hydrogen bonds to each other) that can be generated from the low-lying OH− (H2 O)n=3−5 isomers, 12 are represented in the fragment set that is 4.5 3B energy contribution (kcal/mol)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


SAMBA 3B interaction energy ALMO EDA 3B POL energy ALMO EDA 3B CT energy

4.0 3.5 3.0 2.5 2.0 1.5 1.0 2.50

2.55

2.60

2.65

2.70

2.75

− 1

Average OH − W O-O distance (Å)

Figure 6: Analysis of SAMBA 3B interaction energy contributions and ALMO energy decompositions as a function of the average OH− −water O-O distance in {OH− , 1 WA , 1 WB }G fragments. 17



Page 18 of 53

analyzed in Table 3. The key concept that shapes the discussion of the 3-mer fragments is that of cooperative vs. anticooperative hydrogen bonding, an idea that has received substantial attention in (neutral) water systems. 6,155,168–172 In general terms, cooperative effects occur when the formation of one hydrogen bond between two molecules promotes further hydrogen bonding with the surrounding molecules due to the redistribution of the electron density, which results in negative 3B (and higher-body) contributions to the total interaction energy. On the contrary, anticooperativity occurs when the formation of one hydrogen bond between two molecules weakens the capability of these two molecules to engage in further hydrogen bonding, manifesting in positive 3B (and higher-body) energies. 155,168,171,172 In the case of neutral water trimers, cooperativity occurs when at least one water molecule acts as a hydrogen bond donor and acceptor simultaneously, while anticooperativity occurs when a water molecule acts as a double hydrogen bond donor, or as a double acceptor. 6,155,168–172 The most ‘significant’ 3-mer fragment group among the low-lying isomers of OH− (H2 O)n=3−5 is {OH− , 1 WA , 1 WB }G . As briefly mentioned in Section 3.1, fragments of this group have OH− accepting two hydrogen bonds, which is a common motif in large OH− (H2 O)n clusters where OH− tend to be at least 3-coordinated. All fragments in this group are associated with positive 3B contributions to the interaction energy, with a mean value of 2.862 kcal/mol; the analysis of Fig. 6 shows that a nearly linear relationship exists between the 3B energies and the corresponding average OH− −water O-O distances. Given the inherently repulsive interactions and specific hydrogen bond connectivity, it is possible to conclude that OH− −water hydrogen bonds in this fragment group follow the same anticooperativity trends as water−water hydrogen bonds in the corresponding neutral water analog {WA BC , AW

B

, A WC }G . 6,168–171

To gain further insights, 1st generation ALMO EDA calculations 141,142 were performed on the {OH− , 1 WA , 1 WB }G fragments using ωB97M-V/def2-TZVPPD (refer to Section 2.2 for specific details). The dependence of the different 3B terms on the average OH− −1 W O,3B O,3B O-O distance is shown in Fig. 6. EALM and EALM calculated for {OH− , 1 WA , P OL CT

18


Page 19 of 53

1

WB }G fragments tend to be roughly equal and together dominate the 3B interaction energy,

effectively showing the same dependence on the OH− −1 W O-O distances. Fig. 7 shows the correlation between the sum of the 3B CT “electron reorganization,” ∆QCT (in units of millielectrons, m¯ e), from OH− to 1 W (OH− → 1 W CT) and the average OH− −1 W O-O distance calculated for the {OH− , 1 WA , 1 WB }G fragments. Each monomer → monomer ∆Q3B CT term was calculated by subtracting the value calculated for the relevant 2-mer from the value of the corresponding term calculated for the full 3-mer fragment. This leads to a negative value when less charge is transferred from the donor-monomer to the acceptor-monomer in the presence of the third monomer than when isolated as a 2ALM O,3B contribution since any CT mer. Therefore, a negative ∆Q3B CT implies a positive ECT

necessarily results in a lower total electronic energy within the ALMO EDA framework, wherein CT interactions correspond to the relaxation of the monomer-specific absolutelylocalized molecular orbitals into cluster-wide delocalized molecular orbitals. 141 Summed and

-1.5 -2.0 -2.5 -3.0 -3.5

−

1

OH → W 3B CT reorganization sum (më)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-4.0 2.50

2.55

2.60

2.65

2.70

2.75

− 1

Average OH − W O-O distance (Å)

Figure 7: Correlation plot between the sum of 3B OH− → 1 W CT “electron reorganization” (in millielectrons, m¯ e) and average OH− −water O-O distance in {OH− , 1 WA ,1 WB }G fragments. 19



Page 20 of 53

averaged quantities are used in this analysis because of the equivalence of the 1 W molecules in the definition of the fragment group. Fig. 7 shows that all OH− →

1

W ∆Q3B CT contributions associated with fragments in

{OH− , 1 WA , 1 WB }G are negative. This is explained by considering the finite extent of electron charge available for hydrogen bond donation from the OH− lone pairs (see Section 4.2.1 of Ref. 149) which must be shared between both of the 1 W molecules in the {OH− , 1 WA , 1

WB }G fragments. This picture is completely consistent with the description of anticoopera-

tive hydrogen bonding and, as expected, parallels the occurrence of anticooperativity seen in B C 6,168–171 {WA The correlation seen in Fig. 7 is also similar to the BC , A W , A W }G fragments.

correlation between the total-fragment 3B energy contributions and the average OH− −1 W O-O distance found in Fig. 6 (keeping in mind the inverse relationship between ∆QCT and O mentioned above). These results are consistent with the SAPT analysis reported EALM CT

in Ref. 69 showing that the interactions in {OH− , 1 WA , 1 WB }G fragments are primarily inductive in nature, since SAPT induction energy includes both POL and CT contributions that are defined separately within the ALMO EDA framework. 142 As shown in Table 3, the second-most ‘significant’ 3-mer fragment group within OH− (H2 O)n=3−5 B 1 2 clusters is {OH− , 1 WA B , A W }G . By construction, given the simultaneous presence of W and 2

W molecules, these fragments occur with frequency p×(n−p) in p-coordinated OH− (H2 O)n

clusters with two solvation shells. 3B energy contributions for these fragments are found to correlate with the OH− −1 W−2 W O-O-O angle, with the 3B contributions becoming more 2 B attractive as the angle becomes more linear (Fig. 8). Furthermore, the {OH− , 1 WA B , A W }G

fragments analyzed here have exclusively attractive 3B energy contributions and, therefore, are characterized by cooperative hydrogen bonds. This parallels the connectivity trends of cooperative hydrogen bonding in neutral water clusters, since the analogous {WA B, BW

C

B A WC ,

}G 3-mer fragments exhibit similar cooperativity. 6,168–171

2 B Further insights into the nature of hydrogen bonding in {OH− , 1 WA B , A W }G are gained − 1 2 from the analysis of ∆Q3B CT shown in Fig. 9 as a function of the OH − W− W O-O-O angle.

20


Page 21 of 53

3B energy contribution (kcal/mol)

0

SAMBA 3B interaction energy ALMO EDA 3B POL energy ALMO EDA 3B CT energy

-1 -2 -3 -4 -5 70

80

90 − 1

100

110

120

130

140

2

OH − W− W O-O-O angle (degrees)

Figure 8: Analysis of 3B SAMBA interaction energy contributions and ALMO energy decom2 B positions as a function of OH− −1 W−2 W O-O-O angle in {OH− , 1 WA B , A W }G fragments. 2.5 Pairwise 3B CT reorganization (më)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


−

1

OH → W CT − 2 OH → W CT 1 2 W→ W CT

2.0 1.5 1.0 0.5 0.0 -0.5 -1.0

80

90 − 1

100

110

120

130

140

2

OH − W− W O-O-O angle (degrees)

Figure 9: 3B ALMO monomer → monomer CT reorganization vs OH− −1 W−2 W O-O-O 2 B angle for {OH− , 1 WA B , A W }G fragments. 21



1 A First, ∆Q3B CT calculated for WB →

2 B AW

Page 22 of 53

(green symbols) are exclusively positive. This

result demonstrates that the presence of OH− enhances the strength of hydrogen bonding between the two water molecules, which is precisely the definition of cooperative hydrogen bonding. Second, ∆QCT calculated for OH− →

2 B AW

(blue symbols) are enhanced by the

− presence of 1 WA B , which acts as an electrical wire, shuttling electron charge from OH straight − 1 A to 2A WB . Finally, ∆Q3B CT from OH to WB (orange symbols) are negative, indicating that

cooperative effects result in 1 WA B transferring an excess of the additional charge it receives from OH− in the presence of 2A WB , clearly demonstrating the inherent many-body nature of the underlying interactions. B C As expected, the neutral water analog of this fragment group, {WA B , A WC , B W }G , also

has cooperative hydrogen bonding. 6,168–171 All 3B interaction energies in this set are attractive, with a mean contribution of −0.589 kcal/mol (Table 3). However, these fragments have B 2 qualitatively different CT behavior than the analogous deprotonated {OH− , 1 WA B , A W }G

fragments. The ALMO EDA analysis indicates that the average ∆QCT for WA B → WA B →

BW

C

, and A WB C →

BW

C

B A WC ,

are 0.149, −0.055, and 0.192 m¯ e, respectively (see Sup-

porting Information), whereas the corresponding averages for the deprotonated fragments are −0.375, 0.772, and 0.923 m¯ e. This implies that, while both hydrogen bonds in {WA B, C B A WC , B W } G B A WC ,

fragments are enhanced at the 3B level, the donor/acceptor water molecule,

does not shuttle excess charge in the presence of OH− , as the analogous 1 WA B molecule

does in the corresponding deprotonated fragments. As shown in Table 4, all 4B contributions to the total interaction energies of 4-mer fragments extracted from the low-lying OH− (H2 O)3−5 clusters tend to be small (statistics for all 4-mer fragment groups can be found in the Supporting Information). Nevertheless, it is still important to comment on the most ‘significant’ 4-mer fragment group, {OH− , 1

1 B WA C , WC ,

2 C AB W }G .

Fragments within this group are cyclic with the single 2 W molecule

donating a hydrogen bond to each 1 W molecule, and contains the two most ‘significant’ types 2 B of 3-mer fragments, {OH− , 1 WA , 1 WB }G and {OH− , 1 WA B , A W }G , as sub-fragments. The

22


Page 23 of 53 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1 B Figure 10: The {OH− , 1 WA C , WC , structures.

2 C AB W }G

fragments within the 4-, 5-, and 6-mer GME

most notable feature of this group is its presence in each of the GME 4-mer, 5-mer, and 6-mer clusters (Fig. 10), despite contributing, on average, a relatively high 4B energy (0.692 kcal/mol). The repulsive nature of the 4-body interactions inherent to these fragments is consistent with the above analysis of cooperativity and CT in the 3-mer fragments, given the finite amounts of charge that OH− can donate, and that 2 W can accept. Finally, it is not surprising that all 5-mer fragment groups are essentially not significant (Table 5) given the convergence rate of the MBE for OH− (H2 O)n clusters shown in Fig. 1. However, it is important to point out that the single instance of the {OH− , 1 WA , 1 WB , 1

WC , 1 WD }G 5-mer fragment group, corresponding to a 4-coordinated OH− , stands out,

being associated with a relatively large 5B contribution (−0.175 kcal/mol). Table 4: Statistics for ‘significant’ 4B energies (in kcal/mol) of 4-mer fragments. See main text for details. See Supporting Information for a complete analysis of all 4B energies. Frag. Group

freq.

max.

min.

mean

RSD

sig. factor

7

+0.861

+0.542

+0.692

0.142

0.056

{OH− , 1 WA , 1 WB , 1 WC }G

15

+0.297

+0.013

+0.144

0.598

0.025

2 C {OH− , 1 WA , 1 WB C , B W }G

19

+0.234

−0.069

+0.024

3.234

0.011

1 B 2 C {OH− , 1 WA C , C W , A WB } G

5

+0.181

−0.065 −0.005

17.883

0.005

1 B {OH− , 1 WA C , WC ,

2 C AB W }G

23



Page 24 of 53

Table 5: Statistics for ‘significant’ 5B energies (in kcal/mol) of 5-mer fragments. See main text for details. See Supporting Information for a complete analysis of all 5B energies. Frag. Group

freq.

max.

min.

mean

RSD

sig. factor

{OH− , 1 WA , 1 WB , 1 WC , 1 WD }G

1

−0.175 −0.175 −0.175

N/A

0.006

1 B 1 C {OH− , 1 WA D , WD , W ,

5

−0.019 −0.039 −0.027

0.264

0.005

3

−0.033 −0.051 −0.043

0.178

0.005

2 D AB W }G

2 D {OH− , 1 WA , 1D WB , 1 WC D , C WB } G

3.4

Assessment of DFT in reproducing many-body interactions

The fragment analysis introduced in section 3.3 is used to evaluate the accuracy of various DFT models (see Section 2.2) in reproducing the interaction energies of OH− (H2 O)n=1−5 clusters and constituting fragments. As discussed above, 2B and 3B terms dominate the MBE, which implies that the errors for the 2-mer and 3-mer fragments are indicative of the overall accuracy of different exchange-correlation functionals in correctly reproducing the energetics of deprotonated water systems. Figure 11 shows the errors of the tested exchange-correlation functionals in reproducing the 2B SAMBA energy contributions for the hydrogen bonded 2-mers {OH− , 1 W}G (panel B a) and {WA B , A W }G (panel b). These comparisons show that the net negative charge on

{OH− , 1 W}G fragments tends to further emphasize the errors present in the neutral water fragments. This appears to be the case for all functionals, except for B3LYP(-D3), which is slightly more accurate for {OH− , 1 W}G fragments, as well as for the range-separated hybrids (ωB97X-D and ωB97M-V) which essentially reproduce the reference energies for both fragment groups. These errors may be attributed to well-known deficiencies of several functionals in describing electron density tails (i.e., the valence electrons) of anionic species. 173 This problem is thought to be due to the overestimation of the HOMO energy of molecular species, resulting from incorrect features of the exchange-correlation functional, particularly in the asymptotic region. 174 While the error increase is fairly drastic for some functionals, like M06-2X, and quite 24


Page 25 of 53

small for other functionals, such as PBE, the average errors of nearly all functionals for the B deprotonated fragments show qualitative correlation with those for the {WA B , A W }G frag-

ments, illustrating the similarity of the interactions present in the two fragment groups. The observed correlation suggests that the magnitude of the error increase may reflect precisely the deficiencies of a given functional in the asymptotic region mentioned above. For example, the tendency of the BLYP and revPBE GGA functionals to underbind the neutral water dimer (Fig. 11b) is a result of the strongly-repulsive behavior of their exchange-enhancement factors in the high-density-gradient limit, 130,175,176 particularly in regions of inter-monomer orbital overlap. Since the electron density of OH− is far more diffuse than that of water, this deficiency is thus emphasized in the charged fragments (Fig. 11a). 4.0

a)

Mean Signed Error Mean Unsigned Error * Max Unsigned Error *

2.0 1.0 0.0 -1.0 -2.0 -3.0

V

ωB

97

M-

XD

X

97

Mean Signed Error

M0

rev

ωB

D3

97

X6-2

ωB

3

X

-D

6-2

M0

PB E0

3

E0

-D

PB

rev

E0

PB

E0

PD3

B3

LY

PB

3

YP

-D

B3 L

SS TP

3

SS

-D

TP

SC

AN

E

PB

ED3 SC AN

PB

D3 E-

rev

rev

3.0

PB

3

E

-D

YP

PB

YP

b)

BL

4.0

mean reference energy: -27.779 kcal/mol BL

-4.0

Mean Unsigned Error * Max Unsigned Error *

2.0 1.0 0.0 -1.0 -2.0 -3.0

M-

V

D X-

97 ωB

ωB 97

X

ωB

97

D3

M0

6-2

X-

X 6-2

3

M0

0-D PB E

rev

3

E0 PB

rev

E0

E0 -D

PB

D3 PLY

B3

B3

LY

P

3 -D TP SS

S TP S

3 -D SC

AN

3

AN

SC

ED

E PB

rev

PB rev

3 -D PB E

BL

YP

-D 3 PB E

mean reference energy: -3.307 kcal/mol BL YP

-4.0

PB

EDFT - Ereference (kcal / mol)

3.0


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 11: DFT errors for the hydrogen-bonded 2-mer fragment groups: (a) {OH− , 1 W}G ; B (b) {WA B , A W }G . *Mean Unsigned Errors are arbitrarily given the same sign as the Mean Signed Errors to ease comparison between the two, and the Maximum Unsigned Error is arbitrarily given the sign of the corresponding error value.

25



2.0


a)


1.0

0.0

-1.0

V M-

XD

ωB 97

X 97

Mean Signed Error

M0

rev

ωB 97

XD3

ωB

6-2

M0

PB E0

-D 3 6-2 X

E0 PB

3 PB

rev

E0

E0 -D

-D 3

B3

LY P

PB

3

YP

B3 L

SS -D

3

SS

-D

TP

AN

TP

rev

SC

PB

ED3 SC AN

PB E

D3

rev

BL

b)

PB E-

3

E PB

YP -D

BL

2.0

YP

mean reference energy: -9.582 kcal/mol -2.0



1.0

0.0

-1.0

M-

V

D X-

97 ωB

97

X

ωB

ωB

97

D3 X-

X M0

6-2

6-2

3

M0

-D E0 PB

rev

3

E0

-D

rev

PB

E0

E0 PB

PB

D3 PLY

B3

B3

LY

P

3 -D SS TP

SS TP

3 -D

AN

SC

AN

D3 E-

SC

PB

E

PB rev

rev

ED3 PB

3

E PB

-D BL

YP

YP

mean reference energy: 0.462 kcal/mol -2.0

BL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 53

Figure 12: DFT errors for the non-hydrogen-bonded 2-mer fragment groups {OH− , 2 W}G (panel a) and {WA , WB }G (panel b). Several functionals tend to have smaller average errors for the non-hydrogen bonded 2mer fragments, {OH− , 2 W}G , and {WA , WB }G , than for the hydrogen bonded fragments as seen by comparing Fig. 12 with Fig. 11. In this context, it is important to note the differences between the mean signed errors (MSEs) and mean unsigned errors (MUEs) for the non-hydrogen bonded deprotonated fragments (Fig. 12a). These differences quantify the amount of error cancellation present in the 2B estimates, averaged over all clusters (the MUEs, which are technically positive by definition, are arbitrarily given the same sign as the MSEs to ease comparisons between the two sets of results). Error cancellation for the {OH− , 2 W}G fragments is likely related to the ‘structural ambiguity’ associated with the non-hydrogen bonded fragment groups mentioned in Section 3.3 and is significant for BLYP, revPBE, and TPSS, whose MSEs are deceptively small.

26


Page 27 of 53

Interestingly, there does not appear to be significant error cancellation for the {WA , WB }G fragments (Fig. 12b). However, several functionals are associated with maximum unsigned errors that are substantially larger in magnitude than the corresponding MUEs (note that the signs of the maximum errors shown in the figure reflect the signs of the particular errors with the largest magnitude for each functional). The cause of these large maximum errors is related to the ‘structural ambiguity’ inherent to this fragment group. Specifically, several functionals are unable to reproduce 2B energy contributions of {WA , WB }G fragments at small O-O separations. This can be seen in Figure 13, which shows relative errors associated with selected functionals for {WA , WB }G fragments plotted as a function of the WA −WB O-O distance from 3 to 4 ˚ A. It should be noted that the reference CCSD(T)/CBS energies and associated errors remain close to zero beyond 4 ˚ A. The nonuniformity of the relative errors shown in Figure 13 indicates deficiencies in the overall shape 20 10

Relative 2B energy contribution error

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


BLYP revPBE

BLYP-D3 revPBE-D3

SCAN TPSS

SCAN-D3 TPSS-D3

B3LYP revPBE0

B3LYP-D3 revPBE0-D3

0 -10 -20 20 10 0 -10 -20 20 10 0 -10 -20 3.0

3.2

3.4

3.6

3.8

4.0 3.0

3.2

3.4

3.6

3.8

4.0

Water−water O-O distance (Å)

Figure 13: Relative 2B errors, for the {WA , WB }G fragments.

EDF T − Eref , Eref

as a function of the water−water O-O distance

27



of the potential energy surfaces predicted by these functionals, which may thus result in an incorrect description of the local hydration structure of OH− in aqueous solutions. For some functionals (e.g., BLYP, revPBE, and revPBE0) the largest maximum errors predicted for these fragments are reduced upon inclusion of the GD3 dispersion correction. It should be noted that relatively large maximum errors are also predicted by several functionals for the {OH− , 2 W}G fragments, although they tend to be closer to the corresponding MUEs (Fig. 12a), thus indicating, overall, a smaller degree of error cancellation. Overall, ωB97M-V and ωB97X-D exhibit the highest accuracy for all 2-mer fragments considered in this study, followed by the dispersion-corrected, hybrid functionals, B3LYP-D3 and revPBE0-D3. Finally, PBE is the best performer among the GGA functionals. It should be noted, however, that the dispersion-corrected PBE-D3 tends to perform worse than the original PBE, suggesting that the apparent success of the latter is largely incidental. This is also manifested by the rather poor accuracy of PBE in reproducing the 2B energies of the {OH− , 2 W}G fragments as well as other higher-body terms (see Supporting Information). DFT errors for the {OH− , 1 WA , 1 WB }G 3-mer fragments are shown in Fig. 14. As discussed above, OH− acts as a double-hydrogen bond acceptor in these fragments, leading to anticooperativity which results in exclusively repulsive 3B energies. Among all functionals 2.0 Mean Signed Error Mean Unsigned Error *


Max Unsigned Error *

1.0

0.0

-1.0

M-

V

D X97

97 ωB

X

ωB

D3

97

ωB

X-

X 6-2

6-2 M0

-D 3 E0

M0

rev PB

E0

E0 -D 3 rev PB E0

PB

D3 PLY

B3

PB

P

3 -D

LY B3

SS TP

3

SS

-D

TP

AN SC

ED3 SC AN

PB

rev

E

3

D3 rev PB E

EPB

PB

-D YP

BL

YP

mean reference energy: 2.862 kcal/mol -2.0

BL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 53

Figure 14: DFT errors for the anticooperative 3-mer fragment group {OH− , 1 WA , 1 WB }G .

28


Page 29 of 53

considered in this study, BLYP and revPBE underestimate, while SCAN, PBE0, and M06-2X overestimate these 3B contributions. This can be related to the tendency of BLYP and revPBE to underbind {OH− , 1 WA }G 2-mer fragments, and the tendency of SCAN, PBE0, and M06-2X to overbind the OH− −water hydrogen bond (Fig. 11). 2 B A B C Errors for the {OH− , 1 WA B , A W }G , and {WB , A WC , B W }G fragment groups are shown

in Fig. 15. Both groups are characterized by cooperative hydrogen bonds, resulting in exclusively attractive 3B energies. On average, the errors for the deprotonated water fragments (Fig. 15a) tend to be larger than those for the neutral water fragments, particularly for the GGA and meta-GGA functionals which overestimate the 3B contributions of {OH− , 1 WA B, 2.0


a)


1.0

0.0

-1.0

-V

D

97 M

97 X-

ωB

rev

ωB

-D 3 97 X ωB

Mean Signed Error

M0

PB

6-2 X

-D 3 M0 6-2 X

E0

E0

-D 3

PB

E0 PB

rev

E0

D3 P-

B3 LY

PB

P

3

LY

-D

B3

SS TP

SC

AN -D 3 TP SS

D3

AN

E-

SC

E PB

rev PB

D3

E

BL

b)

rev

E-

PB

PB

-D 3 YP

2.0

BL YP




1.0

0.0

-1.0

ωB

97

M-

V

D X97

X

ωB

ωB

97

D3 X-

X 6-2

6-2 M0

3 -D

M0

E0 PB

rev

E0 -D 3 rev PB E0

PB

E0

D3 P-

PB

LY B3

B3

LY

P

3

SS

-D SS TP

TP

3 -D SC

AN

AN

D3 E-

SC

PB

PB

E rev

D3

rev

E PB

EPB

3 -D YP

BL

YP


BL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 15: DFT errors for the cooperative 3-mer fragment groups {OH− , 1 WA B, B C (panel a) and {WA , W , W } (panel b). B A C B G

29


2 B A W }G


2 B A W }G

fragments, artificially counteracting cooperative effects that characterize the refer-

ence data (Fig. 15b). The remaining (hybrid) functionals show a high degree of accuracy for 2 B the OH− , 1 WA B , A W }G group, suggesting that the inclusion of exact exchange is needed to

correctly reproduce 3B interactions in these fragments. On the other hand, 3B energy conB C tributions in the {WA B , A WC , B W }G fragments are reproduced by all functionals considered

in this study. This result agrees with previous studies demonstrating that many functionals are able to semi-quantitatively reproduce cooperative effects in water clusters. 176

4

Conclusions

In this study, we have conducted a detailed analysis of many-body interactions in OH− (H2 O)n clusters, with n ≤ 5. The reference energies calculated at the CCSD(T)/CBS level of theory indicate that the many-body expansion for OH− −water interactions converges rapidly, suggesting that the analysis presented here extends to larger clusters and to bulk systems as well. In order to approach the analysis from the top-down, without sacrificing the specificity of studying each fragment individually, we devised a scheme to classify the fragments into groups that correlate with the type of interactions present in each fragment. In this analysis, all fragments extracted from the low-lying OH− (H2 O)n clusters were classified according to three criteria: presence of OH− in the fragment, number of water molecules in the fragment, and hydrogen bond connectivity. This classification allowed for the identification of hydrogen bonding connectivities of K-mer fragments with exclusively repulsive or attractive K-body energies. Cooperativity and anticooperativity effects were further investigated using the ALMO EDA/CTA method, which supported conclusions drawn from the fragment analysis. The CCSD(T)/CBS reference data for the individual many-body contributions to the total interaction energy for OH− (H2 O)n clusters were then used to assess the accuracy of various exchange-correlation functionals in reproducing OH− −water interactions. From the

30


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analysis of the energetics of different fragment groups, it was possibly to identify specific strengths and weaknesses of the functionals with respect to the type of interactions inherent to each group. This analysis demonstrated that several functionals (e.g., BLYP, revPBE, SCAN, PBE0, and M06-2X) exhibit serious difficulties in describing OH− −water interactions, and in fact that the presence of OH− emphasize the errors seen in analogous neutral water fragments. Overall, ωB97M-V and ωB97X-D provide the highest accuracy. Additionally, hybrid functionals, like B3LYP and revPBE0, exhibit satisfactory accuracy after the addition of the empirical GD3 dispersion correction.

Supporting Information Available XYZ-coordinates for all OH− (H2 O)n=1−5 isomers studied here. Scatter plots of the manybody energies for all OH− (H2 O)3 and OH− (H2 O)4 isomers, demonstrating convergence of their respective MBEs. Tables containing statistics for all fragment groups studied here. B C CTA data for the {WA B , A WC , B W }G fragment group. Bar graphs of DFT errors for each

fragment group studied here.

This material is available free of charge via the Internet at

http://pubs.acs.org/.

Acknowledgement We would like to thank Yuezhi Mao for advice regarding ALMO EDA calculations, Zhengting Gan and Zhi-Qiang You for their help with Q-Chem, and Mahidhar Tatineni for his help with SDSC Comet. We also thank Marc Riera and Pushp Bajaj for stimulating discussions. This research was supported by the National Science Foundation through Grant CHE1453204. and used resources of the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by the National Science Foundation (Grant ACI-1053575).

31



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TOC Figure

4.0


3.0


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2.0 1.0 0.0 -1.0 -2.0 -3.0

mean reference energy: -27.779 kcal/mol

-4.0

BL

Y

3 P-D

3 3 3 3 3 V -D3 -D3 P-D E0-D X-D 97ME-D AN-D E0 SS LY PB 6-2 PB PB TP SC ωB B3 rev M0 rev

PB

3 E-D

53


Assessing Many-Body Effects of Water Self-Ions. I: OH (H2O)n Clusters

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