Low-lying electronic states of M(3)O(9)(-) and M(3)O(9)(2-) (M = Mo, W)

Multiple low-lying electronic states of M(3)O(9)(-) and M(3)O(9)(2-) (M = Mo, W) arise from the occupation of the near-degenerate low-lying virtual orbitals in the neutral clusters. We used density functional theory (DFT) and coupled cluster theory (CCSD(T)) with correlation consistent basis sets to study the structures and energetics of the electronic states of these anions. The adiabatic and vertical electron detachment energies (ADEs and VDEs) of the anionic clusters were calculated with 27 exchange-correlation functionals including one local spin density approximation functional, 13 generalized gradient approximation (GGA) functionals, and 13 hybrid GGA functionals, as well as the CCSD(T) method. For M(3)O(9)(-), CCSD(T) and nearly all of the DFT exchange-correlation functionals studied predict the (2)A(1) state arising from the Jahn-Teller distortion due to singly occupying the degenerate e' orbital to be lower in energy than the (2)A(1)' state arising from singly occupying the nondegenerate a(1)' orbital. For W(3)O(9)(-), the (2)A(1) state was predicted to have essentially the same energy as the (2)A(1)' state at the CCSD(T) level with core-valence correlation corrections included and to be higher in energy or essentially isoenergetic with most DFT methods. The calculated VDEs from the CCSD(T) method are in reasonable agreement with the experimental values for both electronic states if estimates for the corrections due to basis set incompleteness are included. For M(3)O(9)(2-), the singlet state arising from doubly occupying the nondegenerate a(1)' orbital was predicted to be the most stable state for both M = Mo and W. However, whereas M(3)O(9)(2-) was predicted to be less stable than M(3)O(9)(-), W(3)O(9)(2-) was predicted to be more stable than W(3)O(9)(-).     

Benchmark calculations on the electron detachment energies of MO3* and M2O6* (M = Cr, Mo, W)

Neutral and anionic molecules of the monomers and dimers of the group VIB transition metal oxides (MO3 and M2O6) were studied with density functional theory (DFT) and coupled cluster CCSD(T) theory. Franck-Condon simulations of the photoelectron spectra were carried out for the transition from the ground state of the anion to that of the neutral molecule. Molecular structures from the DFT and CCSD(T) methods are compared. Electron detachment energies reported in the literature were evaluated. The calculated adiabatic and vertical electron detachment energies (ADEs and VDEs) were compared with the experimental results. CCSD(T) gives results within 0.12 eV for the ADEs. CCSD(T) predicts VDEs that are in error by as much as 0.3 eV for M = Cr. DFT hybrid functionals were found to give poor results for the ADEs and VDEs for M = Cr due to the substantial amount of multireference character in the wavefunction, whereas the pure DFT functionals give superior results. For M = Mo and W, excellent agreement was found for both CCSD(T) and many DFT fucntionals. The BP86 functional yields the best overall results for the VDEs of all the metal oxide clusters considered. Heats of formation calculated at the CCSD(T) level extrapolated to the complete basis set limit are also in good agreement with available experimental data.     

Enhanced stability by three-dimensional aromaticity of endohedrally doped clusters X(10)M(0/-) with X = Ge, Sn, Pb and M = Cu, Ag, Au

The group 14 clusters encapsulated by coinage metals in neutral and anionic states X(10)M(0/-) (X = Ge, Sn, Pb and M = Cu, Ag, Au) are investigated using quantum chemical calculations with the DFT/B3LYP functional and coupled-cluster CCSD(T) theory. Addition of transition metals into the empty cages forms high symmetry endohedral structures, except for Ge(10)Ag(0/-). In agreement with experiments available for X(10)Cu, the D(4d) global minima of the anions are calculated to be magic clusters with large frontier orbital gaps, high vertical and adiabatic detachment energies, and large embedding energies and binding energies as compared to those of the empty cages X(10)(2-). The enhanced stability of these magic clusters can be rationalized by the three-dimensional aromaticity.     

Stoichiometric and oxygen-rich M2O(n)- and M2O(n) (M = Nb, Ta; n = 5-7) clusters: molecular models for oxygen radicals, diradicals, and superoxides

We investigated the structures and bonding of two series of early transition-metal oxide clusters, M(2)O(n)(-) and M(2)O(n) (M = Nb, Ta; n = 5-7) using photoelectron spectroscopy (PES) and density-functional theory (DFT). The stoichiometric M(2)O(5) clusters are found to be closed shell with large HOMO-LUMO gaps, and their electron affinities (EAs) are measured to be 3.33 and 3.71 eV for M = Nb and Ta, respectively; whereas EAs for the oxygen-rich clusters are found to be much higher: 5.35, 5.25, 5.28, and 5.15 eV for Nb(2)O(6), Nb(2)O(7), Ta(2)O(6), and Ta(2)O(7), respectively. Structural searches at the B3LYP level yield triplet and doublet ground states for the oxygen-rich neutral and anionic clusters, respectively. Spin density analyses reveal oxygen radical, diradical, and superoxide characters in the oxygen-rich clusters. The M(2)O(7)(-) and M(2)O(7) clusters, which can be viewed to be formed by M(2)O(5)(-/0) + O(2), are utilized as molecular models to understand dioxygen activation on M(2)O(5)(-) and M(2)O(5) clusters. The O(2) adsorption energies on the stoichiometric M(2)O(5) neutrals are shown to be surprisingly high (1.3-1.9 eV), suggesting strong capabilities to activate O(2) by structural defects in Nb and Ta oxides. The PES data also provides valuable benchmarks for various density functionals (B3LYP, BP86, and PW91) for the Nb and Ta oxides.     

Molecular structures and energetics of the (TiO2)n (n = 1-4) clusters and their anions

The (TiO2)n clusters and their anions for n = 1-4 have been studied with coupled cluster theory [CCSD(T)] and density functional theory (DFT). For n > 1, numerous conformations are located for both the neutral and anionic clusters, and their relative energies are calculated at both the DFT and CCSD(T) levels. The CCSD(T) energies are extrapolated to the complete basis set limit for the monomer and dimer and calculated up to the triple-zeta level for the trimer and tetramer. The adiabatic and vertical electron detachment energies of the anionic clusters to the ground and first excited states of the neutral clusters are calculated at both levels and compared with the experimental results. The comparison allows for the definitive assignment of the ground-state structures of the anionic clusters. Anions of the dimer and tetramer are found to have very closely lying conformations within 2 kcal/mol at the CCSD(T) level, whereas that of the trimer does not. In addition, accurate clustering energies and heats of formation are calculated for the neutral clusters and compared with the available experimental data. Estimates of the titanium-oxygen bond energies show that they are stronger than the group VIB transition metal-oxygen bonds except for tungsten. The atomization energies of these clusters display much stronger basis set dependence than the clustering energies. This allows the calculation of more accurate heats of formation for larger clusters on the basis of calculated clustering energies.     

Photoelectron spectroscopy of doubly and singly charged group VIB dimetalate anions: M2O72-, MM'O72-, and M2O7- (M, M' = Cr, Mo, W)

We produced both doubly and singly charged Group VIB dimetalate species-M(2)O(7)(2-), MM'O(7)(2-), and M(2)O(7)(-) (M, M' = Cr, Mo, W)-using two different experimental techniques (electrospray ionization for the doubly charged anions and laser vaporization for the singly charged anions) and investigated their electronic and geometric structures using photoelectron spectroscopy and density functional calculations. Distinct changes in the electronic and geometric structures were observed as a function of the metal and charge state. The electron binding energies of the heteronuclear dianions MM'O(7)(2-) were observed to be roughly the average of those of their homonuclear counterparts (M(2)O(7)(2-) and M'(2)O(7)(2-)). Density functional calculations indicated that W(2)O(7)(2-), W(2)O(7)(-), and W(2)O(7) possess different ground-state structures: the dianion is highly symmetric (D(3d),(1)A(1g)) with a single bridging oxo ligand, the monoanion is a doublet (C(1), (2)A) with two bridging oxo ligands and a radical terminal oxo ligand, whereas the neutral is a singlet (C(1), (1)A) with two bridging oxo ligands and a terminal peroxo ligand. The combined experimental and theoretical study provides insights into the evolution of geometric and electronic structures as a function of charge state. The clusters identified might provide insights into the possible structures of reactive species present in early transition-metal oxide catalysts that are relevant to their reactivity and catalytic function.     

On the structure and chemical bonding of tri-tungsten oxide clusters W3On- and W3On (n=7-10): W3O8 as a potential molecular model for O-deficient defect sites in tungsten oxides

Electronic and structural properties of a series of tri-tungsten oxide clusters, W3On- and W3On (n=7-10), are investigated using photoelectron spectroscopy and density functional theory (DFT) calculations. Both W 5d and O 2p detachment features are observed for n=7-9, whereas only detachment features from O 2p-type orbitals are observed for W3O10- at high electron binding energies (>7 eV). A large energy gap (approximately 3.4 eV) is observed for the stoichiometric W3O9 cluster, which already reaches the bulk value, suggesting that W3O9 can be viewed as the smallest molecular model for bulk WO3. DFT calculations are carried out to locate the most stable structures for both the anion and neutral clusters; time-dependent DFT method is used to predict the vertical detachment energies and to compare with the experimental data. It is shown that W3O9 possesses a D3h structure, in which each W atom is tetrahedrally coordinated with two bridging O atoms and two terminal O atoms. W3O8 and W3O7 can be viewed as removing one and two terminal O atoms from W3O9, respectively, whereas W3O1) can be viewed as replacing a terminal O in W3O9 by a peroxo O2 unit. We show that W3O8 contains a localized W4+ site, which can readily react with O2 to form the W3O10 clusters with a calculated O2 adsorption energy of -78 kcal/mol. It is suggested that the W3O8 cluster can be viewed as a molecular model for O-deficient site in tungsten oxides.     

Accuracy and limitations of second-order many-body perturbation theory for predicting vertical detachment energies of solvated-electron clusters

Vertical electron detachment energies (VDEs) are calculated for a variety of (H(2)O)(n)(-) and (HF)(n)(-) isomers, using different electronic structure methodologies but focusing in particular on a comparison between second-order M?ller-Plesset perturbation theory (MP2) and coupled-cluster theory with noniterative triples, CCSD(T). For the surface-bound electrons that characterize small (H(2)O)(n)(-) clusters (n< or = 7), the correlation energy associated with the unpaired electron grows linearly as a function of the VDE but is unrelated to the number of monomers, n. In every example considered here, including strongly-bound "cavity" isomers of (H(2)O)(24)(-), the correlation energy associated with the unpaired electron is significantly smaller than that associated with typical valence electrons. As a result, the error in the MP2 detachment energy, as a fraction of the CCSD(T) value, approaches a limit of about -7% for (H(2)O)(n)(-) clusters with VDEs larger than about 0.4 eV. CCSD(T) detachment energies are bounded from below by MP2 values and from above by VDEs calculated using second-order many-body perturbation theory with molecular orbitals obtained from density functional theory. For a variety of both strongly- and weakly-bound isomers of (H(2)O)(20)(-) and (H(2)O)(24)(-), including both surface states and cavity states, these bounds afford typical error bars of +/-0.1 eV. We have found only one case where the Hartree-Fock and density functional orbitals differ qualitatively; in this case the aforementioned bounds lie 0.4 eV apart, and second-order perturbation theory may not be reliable.     

Geometries and electronic structures of metastable C2N4 and its ions

We carried out the computational studies on the geometric and electronic properties of electronic states of metastable C(2)N(4) (m-C(2)N(4)) and corresponding ions using the CASSCF and DFT(B3LYP)/CCSD(T) techniques. The optimized geometries of electronic states, vibrational frequencies, Mulliken populations, bond orders, and average polarizabilities are computed at the DFT level while the relative energies of the electronic states, ionization energy, electron affinity, binding energy of m-C(2)N(4) are calculated at the CCSD(T) level. The anion photoelectron spectra of m-C(2)N(4)(-) are also predicted. It is interesting to find that the relative energies of the electronic states of m-C(2)N(4) cluster linearly correlate with the amount of charge transfer between N and C atoms and that, however, there is no charge transfer between C and N atoms upon electron ionization or electron attachment.     

Electronic and structural evolution and chemical bonding in ditungsten oxide clusters: W2O(n)- and W2O(n) (n = 1-6)

We report a systematic and comprehensive investigation of the electronic structures and chemical bonding in a series of ditungsten oxide clusters, W2O(n)- and W2O(n) (n = 1-6), using anion photoelectron spectroscopy and density functional theory (DFT) calculations. Well-resolved photoelectron spectra were obtained at several photon energies (2.331, 3.496, 4.661, 6.424, and 7.866 eV), and W 5d-based spectral features were clearly observed and distinguished from O 2p-based features. More complicated spectral features were observed for the oxygen-deficient clusters because of the W 5d electrons. With increasing oxygen content in W2O(n)-, the photoelectron spectra were observed to shift gradually to higher binding energies, accompanied by a decreasing number of W 5d-derived features. A behavior of sequential oxidation as a result of charge transfers from W to O was clearly observed. A large energy gap (2.8 eV) was observed in the spectrum of W2O6-, indicating the high electronic stability of the stoichiometric W2O6 molecule. Extensive DFT calculations were carried out to search for the most stable structures of both the anion and neutral clusters. Time-dependent DFT method was used to compute the vertical detachment energies and compare to the experimental data. Molecular orbitals were used to analyze the chemical bonding in the ditungsten oxide clusters and to elucidate their electronic and structural evolution.     

Comparative density functional theory and post-Hartree-Fock (CCSD, CASSCF) studies on the electronic structure of halogen nitrites ClONO and BrONO using quantum chemical topology

In this paper, the electronic structures of cis- and trans-ClONO and BrONO are studied at the CCSD∕aug-cc-pVTZ, CASSCF(14,12)/aug-cc-pVTZ, and B3LYP/aug-cc-pVTZ computational levels. For the Cl-O bond, topological analysis of the electron density field, ρ(r), shows the prevalence of the shared-electron type bond (?(2)ρ((3,-1)) < 0). The Br-O bond, however, represents the closed-shell interaction (?(2)ρ((3,-1)) > 0). Topological analysis of the electron localization function, η(r), and electron localizability indicator (ELI-D), (D) (σ)(r), shows that the electronic structure of the central N-O bond is very sensitive to both electron correlation improvements (coupled-cluster single double (CCSD), CASSCF, density functional theory (DFT)) and bond length alteration. Depending on the method used, the N-O bond can be characterized as a "normal" N-O bond with a disynaptic V(N,O) basin (DFT); a protocovalent N-O bond with two monosynaptic, V(N) and V(O), basins (CCSD, CASSCF); or a new type, first discovered for FONO, characterized by a single monosynaptic, V(N) basin (CCSD, DFT). The total basin population oscillates between 0.46-0.96 e (CCSD) and 0.86-1.02 e (CASSCF). The X-O bond is described by the single disynaptic basin, V(X,O), with a basin population between 0.76 and 0.81 e (CCSD) or 0.77 and 0.85 e (CASSCF). Analysis of the localized electron detector distribution for the cis-Cl-O1-N=O2 shows a manifold in the Cl···O2 region, associated with decreased electron density.     

First principles study on the solvation and structure of C2O4(2-)(H2O)n, n = 6-12

The structures and energies of hydrated oxalate clusters, C2O4(2-)(H2O)n, n = 6-12, are obtained by density functional theory (DFT) calculations and compared to SO4(2-)(H2O)n. Although the evolution of the cluster structure with size is similar to that of SO4(2-)(H2O)n, there are a number of important and distinctive futures in C2O4(2-)(H2O)n, including the separation of the two charges due to the C-C bond in C2O4(2-), the lower symmetry around C2O4(2-), and the torsion along the C-C bond, that affect both the structure and the solvation energy. The solvation dynamics for the isomers of C2O4(2-)(H2O)12 are also examined by DFT based ab initio molecular dynamics.     

Electronic states of Ga(3)Si, GaSi(3), and their ions

The equilibrium geometries and electronic states of Ga(3)Si, GaSi(3), and their ions are investigated using the complete active space self-consistent-field (CASSCF) and DFT(B3LYP)/CCSD(T) techniques. The (2)B(1), (3)B(1), and (1)A(1) states in C(2v) symmetry with a planar quadrilateral geometry are found to be the ground states of Ga(3)Si, Ga(3)Si(+), and Ga(3)Si(-), respectively. On the other hand, the ground states of GaSi(3), GaSi(3)(-) are also predicted to undergo Jahn-Teller distortion to the (2)A' and (1)A' states in C(s) with a distorted triangular pyramid geometry, respectively, whereas that of GaSi(3)(+) is found to be the (1)A(1) state in C(3v) with symmetric triangular pyramid structure. Binding energies, electron affinities, ionization energies of Ga(3)Si and GaSi(3) are computed at the CCSD(T)/QCISD(T) level and discussed.     

Silicon monohydride clusters Si(n)H (n = 4-10) and their anions: structures, thermochemistry, and electron affinities

The molecular structures, electron affinities, and dissociation energies of the Si(n)H/Si(n)H- (n = 4-10) species have been examined via five hybrid and pure density functional theory (DFT) methods. The basis set used in this work is of double-zeta plus polarization quality with additional diffuse s- and p-type functions, denoted DZP++. The geometries are fully optimized with each DFT method independently. The three different types of neutral-anion energy separations presented in this work are the adiabatic electron affinity (EA(ad)), the vertical electron affinity (EA(vert)), and the vertical detachment energy (VDE). The first Si-H dissociation energies, D(e)(Si(n)H --> Si(n) + H) for neutral Si(n)H and D(e)(Si(n)H- --> Si(n)- + H) for anionic Si(n)H- species, have also been reported. The structures of the ground states of these clusters are traditional H-Si single-bond forms. The ground-state geometries of Si5H, Si6H, Si8H, and Si9H predicted by the DFT methods are different from previous calculations, such as those obtained by Car-Parrinello molecular dynamics and nonorthogonal tight-binding molecular dynamics schemes. The most reliable EA(ad) values obtained at the B3LYP level of theory are 2.59 (Si4H), 2.84 (Si5H), 2.86 (Si6H), 3.19 (Si7H), 3.14 (Si8H), 3.36 (Si9H), and 3.56 (Si10H) eV. The first dissociation energies (Si(n)H --> Si(n) + H) predicted by all of these methods are 2.20-2.29 (Si4H), 2.30-2.83 (Si5H), 2.12-2.41 (Si6H), 1.75-2.03 (Si7H), 2.41-2.72 (Si8H), 1.86-2.11 (Si9H), and 1.92-2.27 (Si10H) eV. For the negatively charged ion clusters (Si(n)H- --> Si(n)- + H), the dissociation energies predicted are 2.56-2.69 (Si4H-), 2.80-3.01 (Si5H-), 2.86-3.06 (Si6H-), 2.80-3.03 (Si7H-), 2.69-2.92 (Si8H-), 2.92-3.18 (Si9H-), and 2.89-3.25 (Si10H-) eV.     

Exploring the rich energy landscape of sulfate-water clusters SO4(2-) (H2O)(n=3-7): an electronic structure approach

We present a reinvestigation of sulfate-water clusters SO4(2-) (H2O)(n=3-7), which involves several new aspects. Using a joint molecular mechanics/first principles approach, we perform exhaustive searches for stable cluster geometries, showing that the sulfate-water landscape is much richer than anticipated previously. We check the compatibility of the new structures with experiment by comparing vertical detachment energies (VDEs) calculated at the B3LYP/6-311++G** level of theory and determine the energetic ordering of the isomers at the RI-MP2/aug-cc-pVTZ level. Our results are bench-marked carefully against reference energies of estimated CCSD(T)/aug-cc-VTZ quality and VDEs of CCSD(T)/aug-cc-pVDZ quality. Furthermore, we calculate anharmonic vibrational corrections for up to the n = 6 clusters, which are shown to be significant for isomer energy ordering. We use energy decomposition analysis (EDA) based on the absolutely localized fragment (ALMO) expansion to gain chemical insight into the binding motifs.     

Geometries and electronic properties of carbon-nitrogen clusters C8N4 and C8N4+/-

The geometry optimization and frequency analysis for the low-lying electronic states of C(8)N(4) and its ions are performed at the DFT/6-31G(d) level. Their energies are calibrated at the CCSD(T)/6-31G(d) level of theory. Ionization energy, electron affinity, binding energy of C(8)N(4), and anion photoelectron spectra of C(8)N(4)(-) are provided at the CCSD(T)/6-31G(d) level. Mulliken populations, leading configurations, bond orders, and compositions of molecular orbitals are used to examine the bonding characteristics in the low-lying electronic states of C(8)N(4) and its ions. It is surprising to find that the ground state of C(8)N(4) is the open shell (5)A(1) state. Interestingly for the low-lying electronic states of C(8)N(4) and its ions, their structures significantly corrugated, which may be caused by their larger [N]/([N]+[C]) ratios. In addition, the similarities and differences between C(8)N(4) and C(10)N(2)(II) are analyzed and discussed.     

Electronic structure of the hydroxo and methoxo oxometalate anions MO3(OH)- and MO3(OCH3)- (M = Cr, Mo, and W)

The electronic structure of the mononuclear hydroxo MO3(OH)- and methoxo MO3(OCH3)- Group 6 oxometalate anions (M = Cr, Mo, and W) were examined by photodetachment photoelectron spectroscopy and electronic structure calculations at the density functional and CCSD(T) levels of theory. All of the anions exhibited high electron binding energies (>4.9 eV), with the lowest-energy detachment features arising from oxygen 2p-based orbitals. The combined experimental and theoretical results allowed the change in molecular orbital energy levels to be investigated as a function of metal (Cr, Mo, or W) and ligand (-OH, -OCH3). A number of fundamental thermodynamic properties of the anions and corresponding neutrals were predicted on the basis of the theoretical calculations. The calculations indicate high O-H bond dissociation energies for MO2(OR)(O-H) (R = H, CH3) and MO3(O-H), consistent with their high Br?nsted acidities (just below that of H2SO4 in the gas phase) and the high ionization energies of their conjugate base anions. This suggests that the corresponding radicals should readily abstract H atoms from organic molecules.     

Geometries and electronic structures of nitrogen-doped fullerene fragment C10N2(I) and its ions

The geometries and relative energies of the low-lying electronic states of C(10)N(2)(I), cation, and anion are investigated by the DFT/CCSD(T) method. Vibrational frequency calculation is performed to analyze the stability of optimized geometries of these states. The binding energy, ionization energy, electron affinity of C(10)N(2)(I) and the anion photoelectron spectra are estimated at the CCSD(T)/6-31G(d) level. The ground states of neutral C(10)N(2)(I), cation, and anion are the (1)A(1), (4)B(2), and (2)A(2) states, respectively. The structure of C(10)N(2)(I) can be described as resulting from the fusion of 2 five-numbered rings and 1 six-numbered ring. Results demonstrate that the 2 five-numbered rings are more active than the six-numbered ring in C(10)N(2)(I) during electron excitation and the C(1) atom site within each N(11)-C(1)-C(5)-C(10) unit exhibits more inert than other atom sites during electron ionization and electron attachment.     

Spectroscopic properties of gallium arsenide tetramers: Ga2As2, Ga2As2+ and Ga2As2-

Spectroscopic properties of the low-lying electronic states of Ga2As2 and its ions are studied using the complete active-space self-consistent field (CASSCF) and density function theory (DFT) followed by the coupled-cluster single and double substitutions (including triple excitations) (CCSD(T)) calculations. The stability of low-lying electronic states is examined by computing vibrational frequency. The energies of the ground states and a number of excited electronic states have been computed to predict the spectra of Ga2As2 and its ions. The ionization energy, electronic affinity, and atomization energy are estimated at the CCSD(T)/6-311+G(d) level and compared with the available experimental results.     

Development of a 3-body:many-body integrated fragmentation method for weakly bound clusters and application to water clusters (H2O)(n = 3-10, 16, 17)

A 3-body:many-body integrated quantum mechanical (QM) fragmentation method for non-covalent clusters is introduced within the ONIOM formalism. The technique captures all 1-, 2-, and 3-body interactions with a high-level electronic structure method, while a less demanding low-level method is employed to recover 4-body and higher-order interactions. When systematically applied to 40 low-lying (H(2)O)(n) isomers ranging in size from n = 3 to 10, the CCSD(T):MP2 3-body:many-body fragmentation scheme deviates from the full CCSD(T) interaction energy by no more than 0.07 kcal mol(-1) (or <0.01 kcal mol(-1) per water). The errors for this QM:QM method increase only slightly for various low-lying isomers of (H(2)O)(16) and (H(2)O)(17) (always within 0.13 kcal mol(-1) of the recently reported canonical CCSD(T)/aug-cc-pVTZ energies). The 3-body:many-body CCSD(T):MP2 procedure is also very efficient because the CCSD(T) computations only need to be performed on subsets of the cluster containing 1, 2, or 3 monomers, which in the current context means the largest CCSD(T) calculations are for 3 water molecules, regardless of the cluster size.