There is a nice PNAS paper Nuclear quantum effects and hydrogen bond fluctuations in water

Michele Ceriotti, Jerome Cuny, Michele Parrinello, and David Manolopoulos

The authors perform path integral molecular dynamics simulations where the nuclei are treated quantum mechanically, moving on potential energy surfaces that are calculated "on the fly" from density functional theory based methods using the Generalised Gradient Approximation. A key technical advance is using an approximation for the path integrals (PI) based on a mapping to a Generalised Langevin Equation [GLE] [PI+GLE=PIGLET!].

In the figure below the horizontal axis co-ordinate nu is the difference in the O-H distance between the donor and acceptor atom. Thus, nu=0 corresponds to the proton being equidistant between the donor and acceptor.

In contrast, for the Zundel cation, H5O2+ the proton is most likely to be equidistant due to the strong H-bond involved [the donor-acceptor distance is about 2.4 A]. In that case quantum fluctuations play a significant role.

The most important feature of the probability densities shown in the figure is the difference between the solid red and blue curves in the upper panel. This is the difference between quantum and classical. In particular, the probability of a proton being located equidistant between the donor and acceptor becomes orders of magnitude larger due to quantum fluctuations. It is still small (one in a thousand) but

**relevant to the rare events that dominate many dynamical processes**(e.g. auto-ionisation).

On average the H-bonds in water are weak, as defined by a donor-acceptor distance [d(O-O), the vertical axis in the lower 3 panels] of about 2.8 Angstroms. However, there are rare [but non-negligible] fluctuations where this distance can become shorter [~2.5-2.7 A] characteristic of much stronger bonds. This further facilitates the quantum effects in the proton transfer (nu) co-ordinate.

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