Many properties of bulk water, including its

many anomalous properties, can be described/understood in terms of the classical dynamics of interacting "molecules" that consist of localised point charges. However, there is more to the story. In particular, it turns out some of the success of classical calculations arise from a fortuitous cancellation of quantum effects. Some quantum effects can just be

mimicked by using a higher temperature or a softer potential in a classical simulation.

Properties to consider include thermodynamics, structure, and dynamics. Besides bulk homogeneous liquid water, there is ice under pressure, and water in confined spaces, at surfaces, and interacting with ions, solutes, and biomolecules.

Distinctly quantum effects that may occur in a system include

**zero-point motion**,

**tunnelling**, reflection at the top of a barrier, coherence, interference, entanglement, quantum statistics, and collective phenomena (e.g. superconductivity). As far as I am aware only the first two are relevant to water: they involve the nuclear degrees of freedom, specifically the motion of hydrogen atoms or protons. A definitive experimental signature of such quantum effects in seen by substitution of hydrogen with deuterium.

Although there have been exotic claims of entanglement, from both experimentalists and theorists, I think these are based on such dubious data and unrealistic models they are not worthy of even referencing. More positively, there may also be some collective tunnelling effects involving hexagons of water molecules, as described

here.
**When are quantum nuclear effects significant? **
**What are the key physico-chemical descriptors?**
I believe there are two:

the distance R between a proton donor and acceptor in a hydrogen bond and

epsilon, the difference between the proton affinity of the donor and the acceptor.

Quantum nuclear effects become significant when

**both** R is less than 2.6 Angstroms

**and** epsilon is less than roughly 20 kcal/mol.

In

**bulk water** at room temperature the

**average value **of R is about 2.8 Angstroms and epsilon is of the order of 20 kcal/mol and so

**quantum nuclear effects are not significant for properties that are dominated by averages**. However, there are significant thermal fluctuations than can make R as small at 2.4 Angstroms and epsilon smaller, for times less than hundreds of femtoseconds. More on that below. Furthermore, when water interacts with protons, to produce entities such as the Zundel cation, or with biomolecules, the average R can be as small at 2.4 Angstroms and epsilon can be zero. Also, in high pressure phases of ice (such as in

Ice X) average values of R of order 2.4 Angstrom occur.

This way of looking at quantum nuclear effects is discussed at length in

this paper.
**What are the key organising principles for understanding quantum nuclear effects?**

I

proposed before these two.

**1. Competing quantum effects: O-H stretch vs. bend**

Hydrogen bonding changes vibrational frequencies and thus the zero-point energy. As the H-bond strength increases (e.g. due to decreasing R) the O-H stretch frequency decreases while the O-H bend frequencies increases. These changes compete with each other in their effect on the total zero-point energy. Also, the quantum corrections associated with the two types of vibrations have the opposite sign reducing the total quantum effects.

I think this idea was first clearly stated by

Markland, Habershon, and Manolopoulos.
**2. Dynamics dominated by rare events**

For example, consider proton transfer in water. When R is at the average value of 2.8 Angstroms the energy barrier is very large. However, if there a very short fluctuation in R so that it reduces to 2.4 Angstroms the barrier disappears and the proton transfers.

**What are the implications of these two organising principles for computer simulations?**

Pessimism and caution!

1. Subtracting two numbers of about the same size can be error prone.

Suppose that we can calculate the quantum effects of each of the vibrational modes to an accuracy of about 10 per cent. Then we add the contributions (picking some representative numbers):

100 - 50 - 30 = 20. The problem is that the total error is about +/-20.

2. The probability of the rare events is related to the tails of the nuclear wave function. The problem is that this is very sensitive to the exact form of the effective potential energy surface (PES) for the nuclear motion. Tunneling is very sensitive to the height and shape of energy barriers. The problem is that is

very difficult, particularly for hydrogen bonding, to calculate these PES accurately, especially at the level of Density Functional Theory. These issues are nicely illustrated in a recent paper by

Wang, Ceriotti, and Markland.
This post is motivated by preparing to be part of a working group on quantum water at

a Nordita workshop.
My questions are:

Is the picture above valid? Is it oversimplified? Are there exceptions?

Are the two organising principles valid and important? Are there other relevant principles?