Quantifying cooperativity in water clusters: an attempt towards obtaining a generalised equation

The current study describes the development of a general equation of cooperativity energy (CE) for an n-mer system based on the difference between interaction energy of the system and the sum of interaction energy of its dimeric subunits. The counterpoise corrected generalised CE equation of n-mer system has also been developed. In order to validate these two generalised equations, we have considered four different kinds of water clusters, viz. W1D, W2D, W2D-H and W3D, ((H₂O)_n: n = 3–10), with the increasing cluster size as well as changing arrangements. The CE obtained from the two generalised equations follow the trend W3D > W2D-H > W2D > W1D. The deviation at n = 6 for W3D clusters in the plot of CE with cluster size has also been observed for O–H stretching frequencies, electron density and the Laplacian of the electron density plots. The contrastingly higher CE for cyclic pentamer compared to hexamer is in accordance with earlier observations as well as computed average hydrogen bond lengths. We have observed that the average charge value is more on oxygen atoms of pentamer than hexamer resulting in the shortening of hydrogen bond in pentamer and hence the CE values are more for pentamer than hexamer. We have shown that CE has the capability to quantify the cooperativity effect in water clusters.