Two- and three-body, and relaxation energy terms in water clusters: Application of the hierarchical BSSE corrected decomposition scheme
Introduction
There has been a general consensus that the unique physical, chemical and structural properties of liquid water are originated from the existence of a complex and dynamic three-dimensional hydrogen bonded network [[1], [2], [3], [4], [5], [6]]. There is a series of hypotheses which suggest that the cooperativity of hydrogen bonds has an important role in determining the anomalous properties of water [[7], [8], [9], [10], [11], [12]]. The collective interactions between the water molecules strongly influence a variety of important chemical and physical processes in liquid and solid states, as well. Water clusters [[13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]] can serve as model systems for the precise investigation of H-bonding in condensed phase, thus providing a quantitative estimation of the cooperative effects that determine the structure of aqueous environments such as in liquid water and ice. There are several investigations in the literature in which the authors assessed the influence of cooperativity in water clusters [[29], [30], [31], [32], [33], [34], [35]].
A typical manifestation of the cooperativity is the large proportion of many-body interactions in the clusters. Higher order (three-body and above) non-additive energy terms have been shown to contribute significantly to the energetics of water clusters, up to about 30% of the total interaction energy [[36], [37], [38], [39], [40], [41], [42]]. However, a major problem emerges from many-body energy decompositions schemes when basis set superposition error (BSSE) [[43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57]] corrections are also to be computed. BSSE arises as a consequence of using incomplete basis sets for the individual subsystems (“monomers”). There are at least two different approaches for taking into account the BSSE in supramolecular (water cluster) calculations. The “hierarchical” BSSE-corrected energy decomposition scheme named as VMFC(m) (Valiron and Mayer) [[44], [45], [46], [47]] is based on the assumption that for every k-body subcluster [[50], [51], [52], [53], [54], [55]] one can perform an energy decomposition by calculating each 1, 2, …, k-body energy component by using its own k-body basis set (more detailed discussions are given later). The second scheme can be regarded as a direct generalization of the most conventional interpretation of the Boys−Bernardi counterpoise (CP) [43] method, in which every interaction energy term is calculated within the full basis set of the system.
Here we applied the VMFC(m) method for water clusters in the size range of 3 to 30. We systematically assessed the accuracy of our calculations by comparing to high level LNO-CCSD(T)/CBS results [58,59]. We also investigated the contributions of various terms (relaxation energy, two-, three- and many-body terms) in these clusters. In addition, we investigated the relationship between the relaxation energy term and the structure of the H-bonded environment around a central molecule.
Section snippets
Many body energy decomposition scheme
It is widely accepted [46] that one of the main problems when calculating weak intermolecular interactions is the proper consideration of the basis set superposition error (BSSE). The BSSE is an artificial “mathematical effect” which is related to the incomplete and unbalanced quantum chemical description of the monomers in clusters. Valiron and Mayer introduced a generalized method (VMFC(m), where “m” is the last term in the hierarchical scheme) for a consistent treatment of BSSE [44,45] as
Computational details
The geometries of water clusters consisting of 3 to 30, 35, 42, 50(a, b, c, d), 54, 55, 80 and 81 monomers were optimized at the M05-2X/6-311G** level of theory, which method is suitable for such calculations [64]. The initial cluster geometries were taken from the literature [[13], [14], [15],19,20]. These geometries are one of the typical low energy conformers of these water clusters.
In order to assess the accuracy of the correction scheme applied, two different reference calculations were
Hydrogen bond properties of water clusters at their global minima
The structures investigated consist of 3 to 30, 35, 42, 50(a, b, c, d), 54, 55, 80 and 81 water molecules and are denoted as “wan” throughout this article. Cartesian coordinates of the optimized geometries are collected in the Supplementary material. The structures of some representative clusters investigated are shown in Fig. 1. The properties of hydrogen-bonded network topologies have already been studied by several authors [[13], [14], [15],19,20]. It is known that hydrogen bonding
Conclusions
In this study, detailed analyses are presented for the two- and three-body water-water interactions in water clusters of 3 to 30 water molecules, as evaluated at the M05-2X/6-311G** level of theory, by using the hierarchical BSSE-corrected energy decomposition scheme VMFC(3) of Valiron and Mayer [[41], [42], [43]]. At this level of theory, the classical CP, two-body and three-body correction terms are 39%, 5% and 2% of total BSSE-corrected interaction energy, respectively. The accuracy of
Acknowledgements
The authors are grateful to the National Research Development and Innovation Office (NRDIO, NKFIH) for financial support K 124885 K128136. We would like to express appreciation to NIIF for the provided computational resources.
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