Water Hexamer Research Articles

Correcting density functionals for dispersion interactions using pseudopotentials

We present a two-channel dispersion-corrected atom-centered potential (DCACP) method for correcting BLYP and PBE density functionals for long-range dispersion. The approach, designated DCACP2, is tested on the S22X5 test set and on isomers of the water hexamer. The DCACP2 method provides a significantly improved description of the interaction energies at distances beyond Req than does the single-channel DCACP procedure.

Extreme density-driven delocalization error for a model solvated-electron system

Delocalization (or charge-transfer) error is one of the scarce but spectacular failures of density-functional theory. It is particularly apparent in extensively delocalized molecules, and manifests in the calculation of bandgaps, reaction barriers, and dissociation limits. Even though delocalization error is always present in the self-consistent electron density, the differences from reference densities are often quite subtle and the error tends to be driven by the exchange-correlation energy expression. In this article, we propose a model system (the Kevan model) where approximate density functionals predict dramatically different charge distributions because of delocalization error. The model system consists of an electron trapped in a water hexamer and is a finite representation of an experimentally observed class of solids: electrides. The Kevan model is of fundamental interest because it allows the estimation of charge transfer error without recourse to fractional charge calculations, but our results are also relevant in the context of the modeling of confined electrons in density-functional theory.

Method for Visualizing and Quantifying the Nonvalence Character of Excess Electrons.

States formed by attachment of excess electrons to molecules or clusters can be broadly classified into valence and nonvalence states, but many of these states do have both, some valence and some nonvalence character. Here a new analysis scheme for this type of state is presented. It is based on considering the density of the excess electron as a function of the distance to the nearest atom, and this vantage point yields, in the first place, an intuitive picture akin to the well-known atomic radial distribution function, and, in the second place, a distance-from-the-atoms measure that is directly related to the nonvalence character of the excess electron. As a first test the analysis scheme is applied to the occupied orbitals of the water monomer, the water hexamer, and benzene, and its properties are contrasted to those of other frequently employed measures, such as the radius of gyration. Then its utility is demonstrated for three anions: CH3NO2–, which has both a valence and a nonvalence state, NaCl–, whose ground state has been classified as valence or nonvalence by different authors, and two conformations of the water hexamer anion, (H2O)6–, the so-called AA isomer, which supports a surface state, and a Kevan-like structure, which has served as a model for a cavity state.

Hydrogen Bond Cooperativity in Water Hexamers: Atomic Energy Perspective of Local Stabilities

Atomic energies are used to describe local stability in eight low-lying water hexamers: prism, cage, boat 1, boat 2, bag, chair, book 1, and book 2. The energies are evaluated using the quantum theory of atoms in molecules (QTAIM) at MP2/aug-cc-pVTZ geometries. It is found that the simple, stabilizing cooperativity observed in linear hydrogen-bonded water systems is diminished as clusters move from nearly planar to three-dimensional structures. The prism, cage, and bag clusters can have local water stabilities differing up to 5 kcal mol(-1) as a result of mixed cooperative and anticooperative interactions. At the atomic level, in many cases a water may have a largely stabilized oxygen atom but the net water stability will be diminished due to strong destabilization of the water's hydrogen atoms. Analysis of bond critical point (BCP) electron densities shows that the reduced cooperativity results in a decrease in hydrogen bond strength and an increase in covalent bond strength, most evident in the prism. The chair, with the greatest cooperativity, has the largest average electron density at the BCP per hydrogen bond, whereas the cage has the largest total value for BCP density at all hydrogen bonds. The cage also has the second largest value (after the prism) for covalent bond critical point densities and an oxygen-oxygen BCP which may factor into the experimentally observed stability of the structure.

Self-Assembly of Water-Mediated Supramolecular Cationic Archimedean Solids

Understanding the self-assembly of small structural units into large supramolecular assemblies remains one of the great challenges in structural chemistry. We have discovered that tetrahedral supramolecular cages, exhibiting the shapes of Archimedean solids, can be self-assembled by hydrogen bonding interactions using tricationic N-donors (1 or 2) in cooperation with water (W). Single crystal X-ray analysis shows that cage (2)4(W)6, assembled in an aqueous solution of cation 2 and KPF6, consists of four tripodal trications linked by six water monomers and resembles the shape of a truncated tetrahedron. Similarly, cage (1)4(W6)4 spontaneously self-assembles in an aqueous solution of cation 1 and NH4PF6 and consists of four tripodal cations and four water hexamers. Here, each of the four (H2O)6 units act as tritopic nodes between three distinct tripodal cations forming a polyhedron similar to the cantellated tetrahedron. These two well-defined cages are assembled via total of 12 and 36 hydrogen bonds, respectively. Both cages possess interior solvent-accessible volumes exceeding 1000 Å3. Furthermore, each one of the (H2O)6 clusters in face-centered cubic structure 1b acts as a node between two distinct (1)4(W6)4 units, and thus a solvent-filled tubular three-dimensional network (tube diameter of ∼6.5 Å) is generated that mimics the structure of diamond at the nanometer scale. To our knowledge, this is the first example of such species being formed entirely via hydrogen bonding interactions.

Optimal geometries and harmonic vibrational frequencies of the global minima of water clusters (H2O)n, n = 2–6, and several hexamer local minima at the CCSD(T) level of theory

We report the first optimum geometries and harmonic vibrational frequencies for the ring pentamer and several water hexamer (prism, cage, cyclic and two book) at the coupled-cluster including single, double, and full perturbative triple excitations (CCSD(T))/aug-cc-pVDZ level of theory. All five examined hexamer isomer minima previously reported by Møller-Plesset perturbation theory (MP2) are also minima on the CCSD(T) potential energy surface (PES). In addition, all CCSD(T) minimum energy structures for the n = 2-6 cluster isomers are quite close to the ones previously obtained by MP2 on the respective PESs, as confirmed by a modified Procrustes analysis that quantifies the difference between any two cluster geometries. The CCSD(T) results confirm the cooperative effect of the homodromic ring networks (systematic contraction of the nearest-neighbor (nn) intermolecular separations with cluster size) previously reported by MP2, albeit with O-O distances shorter by ~0.02 Å, indicating that MP2 overcorrects this effect. The harmonic frequencies at the minimum geometries were obtained by the double differentiation of the CCSD(T) energy using an efficient scheme based on internal coordinates that reduces the number of required single point energy evaluations by ~15% when compared to the corresponding double differentiation using Cartesian coordinates. Negligible differences between MP2 and CCSD(T) frequencies are found for the librational modes, while uniform increases of ~15 and ~25 cm(-1) are observed for the bending and "free" OH harmonic frequencies. The largest differences between CCSD(T) and MP2 are observed for the harmonic hydrogen bonded frequencies, for which the former produces larger absolute values than the latter. Their CCSD(T) redshifts from the monomer values (Δω) are smaller than the MP2 ones, due to the fact that CCSD(T) produces shorter elongations (ΔR) of the respective hydrogen bonded OH lengths from the monomer value with respect to MP2. Both the MP2 and CCSD(T) results for the hydrogen bonded frequencies were found to closely follow the relation -Δω = s · ΔR, with a rate of s = 20.2 cm(-1)/0.001 Å for hydrogen bonded frequencies with IR intensities >400 km/mol. The CCSD(T) harmonic frequencies, when corrected using the MP2 anharmonicities obtained from second order vibrational perturbation theory, produce anharmonic CCSD(T) estimates that are within <60 cm(-1) from the measured infrared (IR) active bands of the n = 2-6 clusters. Furthermore, the CCSD(T) harmonic redshifts (with respect to the monomer) trace the measured ones quite accurately. The energetic order between the various hexamer isomers on the PES (prism has the lowest energy) previously reported at MP2 was found to be preserved at the CCSD(T) level, whereas the inclusion of anharmonic corrections further stabilizes the cage among the hexamer isomers.

Corrigendum: Inter‐ and intramolecular dispersion interactions

The authors have notified us of errors in the article. The corrected table is shown below. We apologize for any inconvenience this may have caused. For the water hexamer’s boat, book, cage, and ring isomers, M05-2X/MG3S was used and not the intended and specified MP2/aug’-cc-pVTZ structures from Reference 10. This mistake is corrected in Table 2 below, which shows relative energies that are all computed by consistently using the MP2/aug’-cc-pVTZ structures from Reference 10. For completeness, the correct Cartesian coordinates are provided on page 2.

Hexamers and witchamers: Which hex do you choose?

Water and argon hexamers are examined using correlated wavefunction methods, the effective fragment potential (EFP) method, and Hartree–Fock (HF) theory and several density functional theory (DFT) functionals. The HF and DFT methods have been employed both with and without dispersion corrections. The computationally inexpensive EFP method captures the high-level coupled cluster binding energy and relative isomer energy predictions very well for both types of hexamer, much better than the DFT methods (DFT-D), even those that include dispersion corrections. Interestingly, the dispersion corrected HF method (HF-D) does very well. When the DFT-D methods perform reasonably, they do so because of a fortuitous off-setting cancellation of errors two-body and many-body contributions to the binding energy. The HF-D method does not rely on such error cancellation, while the EFP method captures both two-body and many-body contributions to the water hexamer very well. The many body contribution to the argon cluster are small and are most likely due to dispersion.

Cage versus Prism: Electronic Energies of the Water Hexamer

Recent experiments show that the cage isomer of the water hexamer is lower in energy than the prism isomer near 0 K, and yet state-of-the-art electronic structure calculations predict the prism to be lower in energy than the cage at 0 K. Here, we study the relative energies of the water hexamers from the parametric two-electron reduced density matrix (2-RDM) method in which the 2-RDM rather than the wave function is the basic variable of the calculations. In agreement with experiment and in contrast with traditional wave function methods, the 2-RDM calculations predict the cage to be more stable than the prism after vibrational zero-point correction. Multiple configurations from the hydrogen bonding are captured by the method. More generally, the results are consistent with our previous 2-RDM applications in that they reveal how multireference correlation in molecular systems is important for resolving small energy differences from hydrogen bonding as well as other types of intermolecular forces, even in systems that are not conventionally considered strongly correlated.

The curious case of the water hexamer: Cage vs. Prism

Small water clusters, such as the hexamer, provide a unique opportunity to advance the molecular-level understanding of water in all its phases. In particular, the water hexamer is the smallest cluster that possesses several nearly iso-energetic non-planar isomeric forms whose relative stability at low temperatures can be probed experimentally and investigated theoretically. Here, we report on the equilibrium populations of the isomers in the temperature range from 30K to 150K for both H2O and D2O as predicted by four different water potentials. The simulations, performed using path-integral molecular dynamics combined with the replica exchange method, highlight some deficiencies of empirical water models while providing support for the accuracy of more recent ab initio-based potentials. The theoretical predictions for the cage/prism isomeric equilibrium upon isotopic substitution suggest that rotational spectra measured for the deuterated cluster could deliver further insights on the ground-state properties of the water hexamer.

Robustness of Frequency, Transition Dipole, and Coupling Maps for Water Vibrational Spectroscopy.

Infrared spectroscopy of the water OH stretch provides a sensitive probe of the local hydrogen-bonding structure and dynamics of water molecules. Previously, we have utilized a mixed quantum/classical model to calculate vibrational spectroscopic observables for bulk water, ice, the liquid/vapor interface, and small water clusters, as well as water interacting with ions and biological molecules. These studies rely on spectroscopic maps that relate the OH stretching frequency and transition dipole to the local environment around a water molecule. Our spectroscopic maps were parametrized based on water clusters taken from bulk water simulations; in this article, we test the robustness of these maps for water in nonbulk-liquid environments. We find that the frequency, transition dipole, and coupling maps work as well for the water surface, ice Ih, and the water hexamer as they do for liquid water. This suggests that these maps may be generally applied to study the vibrational spectroscopy of water in diverse, potentially heterogeneous environments.

Efficient Methods for the Quantum Chemical Treatment of Protein Structures: The Effects of London-Dispersion and Basis-Set Incompleteness on Peptide and Water-Cluster Geometries.

We demonstrate how quantum chemical Hartree-Fock (HF) or density functional theory (DFT) optimizations with small basis sets of peptide and water cluster structures are decisively improved if London-dispersion effects, the basis-set-superposition error (BSSE), and other basis-set incompleteness errors are addressed. We concentrate on three empirical corrections to these problems advanced by Grimme and co-workers that lead to computational strategies that are both accurate and efficient. Our analysis encompasses a reoptimized version of Hobza's P26 set of tripeptide structures, a new test set of conformers of cysteine dimers, and isomers of the water hexamer. These systems reflect features commonly found in protein crystal structures. In all cases, we recommend Grimme's DFT-D3 correction for London-dispersion. We recommend usage of large basis sets such as cc-pVTZ whenever possible to reduce any BSSE effects and, if this is not possible, to use Grimme's gCP correction to account for BSSE when small basis sets are used. We demonstrate that S-S and C-S bond lengths are very prone to basis-set incompleteness and that polarization functions should always be used on S atoms. At the double-ζ level, the PW6B95-D3-gCP DFT method combined with the SVP and 6-31G* basis sets yields accurate results. Alternatively, the HF-D3-gCP/SV method is recommended, with inclusion of polarization functions for S atoms only. Minimal basis sets offer an intriguing route to highly efficient calculations, but due to significant basis-set incompleteness effects, calculated bond lengths are seriously overestimated, making applications to large proteins very difficult, but we show that Grimme's newest HF-3c correction overcomes this problem and so makes this computational strategy very attractive. Our results provide a useful guideline for future applications to the optimization, quantum refinement, and dynamics of large proteins.

Isomer-Selective Detection of Hydrogen-Bond Vibrations in the Protonated Water Hexamer

The properties of hydrogen ions in aqueous solution are governed by the ability of water to incorporate ions in a dynamical hydrogen bond network, characterized by a structural variability that has complicated the development of a consistent molecular level description of H(+)(aq). Isolated protonated water clusters, H(+)(H2O)n, serve as finite model systems for H(+)(aq), which are amenable to highly sensitive and selective gas phase spectroscopic techniques. Here, we isolate and assign the infrared (IR) signatures of the Zundel-type and Eigen-type isomers of H(+)(H2O)6, the smallest protonated water cluster for which both of these characteristic binding motifs coexist, down into the terahertz spectral region. We use isomer-selective double-resonance population labeling spectroscopy on messenger-tagged H(+)(H2O)6·H2 complexes from 260 to 3900 cm(-1). Ab initio molecular dynamics calculations qualitatively recover the IR spectra of the two isomers and allow attributing the increased width of IR bands associated with H-bonded moieties to anharmonicities rather than excited state lifetime broadening. Characteristic hydrogen-bond stretching bands are observed below 400 cm(-1).

Water 16-mers and Hexamers: Assessment of the Three-Body and Electrostatically Embedded Many-Body Approximations of the Correlation Energy or the Nonlocal Energy As Ways to Include Cooperative Effects

This work presents a new fragment method, the electrostatically embedded many-body expansion of the nonlocal energy (EE-MB-NE), and shows that it, along with the previously proposed electrostatically embedded many-body expansion of the correlation energy (EE-MB-CE), produces accurate results for large systems at the level of CCSD(T) coupled cluster theory. We primarily study water 16-mers, but we also test the EE-MB-CE method on water hexamers. We analyze the distributions of two-body and three-body terms to show why the many-body expansion of the electrostatically embedded correlation energy converges faster than the many-body expansion of the entire electrostatically embedded interaction potential. The average magnitude of the dimer contributions to the pairwise additive (PA) term of the correlation energy (which neglects cooperative effects) is only one-half of that of the average dimer contribution to the PA term of the expansion of the total energy; this explains why the mean unsigned error (MUE) of the EE-PA-CE approximation is only one-half of that of the EE-PA approximation. Similarly, the average magnitude of the trimer contributions to the three-body (3B) term of the EE-3B-CE approximation is only one-fourth of that of the EE-3B approximation, and the MUE of the EE-3B-CE approximation is one-fourth that of the EE-3B approximation. Finally, we test the efficacy of two- and three-body density functional corrections. One such density functional correction method, the new EE-PA-NE method, with the OLYP or the OHLYP density functional (where the OHLYP functional is the OptX exchange functional combined with the LYP correlation functional multiplied by 0.5), has the best performance-to-price ratio of any method whose computational cost scales as the third power of the number of monomers and is competitive in accuracy in the tests presented here with even the electrostatically embedded three-body approximation.

Dynamics and structural changes of small water clusters on ionization

Despite utmost importance in understanding water ionization process, reliable theoretical results of structural changes and molecular dynamics (MD) of water clusters on ionization have hardly been reported yet. Here, we investigate the water cations [(H2O)(n = 2-6)(+)] with density functional theory (DFT), Möller-Plesset second-order perturbation theory (MP2), and coupled cluster theory with single, double, and perturbative triple excitations [CCSD(T)]. The complete basis set limits of interaction energies at the CCSD(T) level are reported, and the geometrical structures, electronic properties, and infrared spectra are investigated. The characteristics of structures and spectra of the water cluster cations reflect the formation of the hydronium cation moiety (H3O(+)) and the hydroxyl radical. Although most density functionals fail to predict reasonable energetics of the water cations, some functionals are found to be reliable, in reasonable agreement with high-level ab initio results. To understand the ionization process of water clusters, DFT- and MP2-based Born-Oppenheimer MD (BOMD) simulations are performed on ionization. On ionization, the water clusters tend to have an Eigen-like form with the hydronium cation instead of a Zundel-like form, based on reliable BOMD simulations. For the vertically ionized water hexamer, the relatively stable (H2O)5(+) (5sL4A) cluster tends to form with a detached water molecule (H2O).

IR Spectra of the Water Hexamer: Theory, with Inclusion of the Monomer Bend Overtone, and Experiment Are in Agreement.

Signature IR spectra of isomers of the water hexamer in the spectral range 3000-3800 cm(-1) have been reported by experimentalists, but crucial theoretical interpretation has still not been definitive. Using ab initio potential and dipole moment surfaces and a fully coupled quantum treatment of the intramolecular modes, the ring and book are assigned to spectra obtained in the He nanodroplet and Ar tagging experiments, respectively. The overtone of the intramolecular bend at ca. 3200 cm(-1) is a new calculated feature that completes an important missing piece in previous experimental and theoretical comparisons and leads to a consistent assignment of these two experimental spectra. Calculated IR spectra for the lowest energy forms of the water heptamer and octomer are also presented and compared to experiment. In all the calculated spectra, the bend overtone is demonstrated to be a noticeable feature, and this is one important conclusion from the work. Also, the danger in using scaled double-harmonic spectra to assign spectra is demonstrated.

A one-dimensional coordination polymer with a three-dimensional supramolecular framework:catena-poly[[[(3,4,7,8-tetramethyl-1,10-phenanthroline)cadmium(II)]-μ3-4,4′-sulfonyldibenzoato] trihydrate

In the title mixed-ligand metal-organic polymeric compound, {[Cd(C14H8O6S)(C16H16N2)]·3H2O}n, the asymmetric unit contains a crystallographically unique Cd(II) atom, one doubly deprotonated 4,4'-sulfonyldibenzoic acid (H2SDBA) ligand, one 3,4,7,8-tetramethyl-1,10-phenanthroline (TMPHEN) molecule and three solvent water molecules. Each Cd(II) centre is six-coordinated by two O atoms from a chelating carboxylate group of a SDBA(2-) ligand, two O atoms from monodentate carboxylate groups of two different SDBA(2-) ligands and two N atoms from a chelating TMPHEN ligand. There are two coordination patterns for the carboxylate groups of the SDBA(2-) ligand, with one in a μ1-η(1):η(1) chelating mode and the other in a μ2-η(1):η(1) bis-monodentate mode. Single-crystal X-ray diffraction analysis revealed that the title compound is a one-dimensional double-chain polymer containing 28-membered rings based on the [Cd2(CO2)2] rhomboid subunit. More interestingly, a chair-shaped water hexamer cluster is observed in the compound.

Self-consistent phonons revisited. II. A general and efficient method for computing free energies and vibrational spectra of molecules and clusters

The self-consistent phonons (SCP) method provides a consistent way to include anharmonic effects when treating a many-body quantum system at thermal equilibrium. The system is then described by an effective temperature-dependent harmonic Hamiltonian, which can be used to estimate the system's properties, such as its free energy or its vibrational spectrum. The numerical bottleneck of the method is the evaluation of Gaussian averages of the potential energy and its derivatives. Several algorithmic ideas/tricks are introduced to reduce the cost of such integration by orders of magnitude, e.g., relative to that of the previous implementation of the SCP approach by Calvo et al. [J. Chem. Phys. 133, 074303 (2010)]. One such algorithmic improvement is the replacement of standard Monte Carlo integration by quasi-Monte Carlo integration utilizing low-discrepancy sequences. The performance of the method is demonstrated on the calculation of vibrational frequencies of pyrene. It is then applied to compute the free energies of five isomers of water hexamer using the WHBB potential of Bowman and co-workers [J. Chem. Phys. 134, 094509 (2011)]. The present results predict the hexamer prism being thermodynamically most stable, with the free energy of the hexamer cage being about 0.2 kcal mol(-1) higher at all temperatures below T = 200 K.

An unprecedented mixed valence cobalt(III)/cobalt(II) ion-pair complex ([CoCO3(bipy)2]2[Co(DCA)2]⋅16H2O) with two novel water tapes

An unprecedented mixed valence cobalt(III)/cobalt(II) ion-pair complex, [CoCO3(bipy)2]2[Co(DCA)2]⋅16H2O (1) [bipy = 2,2′-bipyridine, DCA = demethylcantharidate, exo-1,4-epoxy-cyclohexyl-2,3-dicarboxylate group, (C8H8O5)2 −], has been synthesized and characterized by elemental analysis, IR, UV–vis, Thermogravimetric analysis (TGA) and single crystal X-ray diffraction. The crystal structure of 1 consists of three independent molecular moieties: [CoCO3(bipy)2]+, [Co(DCA)2]2–, and water molecules of crystallization. Interestingly, two novel water tapes, one comprises of alternate cyclic water tetramer and octamer, the other is constructed by alternate cyclic water hexamer and carboxylate–water hybrid octamer [(H2O)2(COO)2]2 −, have been detected for the first time in the host framework of 1. The two novel water tapes are linked together by the DCA and carbonate ligands via hydrogen bonds into a 2D water layer, and then the water layers assemble complex ions ([CoCO3(bipy)2]+ and [Co(DCA)2]2 −) through hydrogen bonds, π–π interactions and electrostatic interactions to form the overall 3D metal–organic framework structure.

Assessing the Accuracy of Density Functional and Semiempirical Wave Function Methods for Water Nanoparticles: Comparing Binding and Relative Energies of (H2O)16 and (H2O)17 to CCSD(T) Results.

The binding energies and relative conformational energies of five configurations of the water 16-mer are computed using 61 levels of density functional (DF) theory, 12 methods combining DF theory with molecular mechanics damped dispersion (DF-MM), seven semiempirical-wave function (SWF) methods, and five methods combining SWF theory with molecular mechanics damped dispersion (SWF-MM). The accuracies of the computed energies are assessed by comparing them to recent high-level ab initio results; this assessment is more relevant to bulk water than previous tests on small clusters because a 16-mer is large enough to have water molecules that participate in more than three hydrogen bonds. We find that for water 16-mer binding energies the best DF, DF-MM, SWF, and SWF-MM methods (and their mean unsigned errors in kcal/mol) are respectively M06-2X (1.6), ωB97X-D (2.3), SCC-DFTB-γ(h) (35.2), and PM3-D (3.2). We also mention the good performance of CAM-B3LYP (1.8), M05-2X (1.9), and TPSSLYP (3.0). In contrast, for relative energies of various water nanoparticle 16-mer structures, the best methods (and mean unsigned errors in kcal/mol), in the same order of classes of methods, are SOGGA11-X (0.3), ωB97X-D (0.2), PM6 (0.4), and PMOv1 (0.6). We also mention the good performance of LC-ωPBE-D3 (0.3) and ωB97X (0.4). When both relative and binding energies are taken into consideration, the best methods overall (out of the 85 tested) are M05-2X without molecular mechanics and ωB97X-D when molecular mechanics corrections are included; with considerably higher average errors and considerably lower cost, the best SWF or SWF-MM method is PMOv1. We use six of the best methods for binding energies of the water 16-mers to calculate the binding energies of water hexamers and water 17-mers to test whether these methods are also reliable for binding energy calculations on other types of water clusters.