Thursday, September 25, 2014

When is water quantum? II

A previous post focused on quantum effects largely associated with hydrogen bonding associated with the O-H stretch vibration. Here, I consider effects largely associated with angular motion, known as librational modes.

Feynman Path Integral computer simulations performed by Peter Rossky, Bruce Berne, Greg Voth, and Peter Kusalik have led to the following key ideas.

1. Quantum water is less structured than classical water.

This is seen in the figure below taken from a 2004 paper by Hernandez de la Pena and Kusalik.

2. This is largely do to quantisation of orientational rather than translational degrees of freedom.

The clear evidence for this is from Kusalik's path integral simulations. There, the water molecules are rigid and only the orientational motion of the molecules is quantised. Similar results for the structure factor, (translational) diffusion constant, and orientational relaxation times (and their isotope effects) are obtained from simulations with flexible molecules and quantised translational modes.

3. Many of these quantum effects are similar to those of classical water with a higher temperature by 50 K. 

The figure below taken from a 2005 paper by Hernandez de la Pena and Kusalik. The structure functions are virtually identical for quantum water at 0 degrees C and classical water at 50 degrees C.

The temperature dependence of the potential energy is shown below. The quantum energy is larger than the classical by about 350 cm^-1.

N.B. one should not use this result to justify the claim that ALL quantum effects are similar to raising the temperature by 50 K. For example, none of the quantum effects discussed in the earlier post can be understood in these terms.

What is going on?

Here is one idea about the essential physics. I think it is similar to earlier ideas going back to Feynman and Hibbs (page 281) about effective "classical potentials", and discussed in detail by Greg Voth.

Consider a simple harmonic (angular/torsional) oscillator of frequency Omega and moment of inertia I at temperature T. The RMS fluctuation in the angle Phi are given below for the quantum and classical cases.

The two temperature dependences of the quantum case [purple curve] and classical curve [straight red line] are plotted below. The temperature [horizontal scale] is in units of the zero point energy.  The vertical scale is in units of the zero-point motion.

The RMS is about the same [horizontal line] when the quantum temperature (273 K) is such that
Omega/k_B T ~ 3 and the classical temperature is about 15 per cent larger.

This suggests that Omega ~ 3 k_B T ~ 600 cm^-1.

Is this reasonable?

The figure below shows the spectrum of the librational modes.

This frequency scale is also consistent with the differences in potential energy.

How do the quantum fluctuations lead to softening of the structure?

Quantum fluctuations lead to "swelling" of the polymer beads in the discrete path integral or the new effective classical potential, a la Feynman-Hibbs/Voth...
This swelling reduces the effective interaction between molecules and reduces structure

Some of the papers mention the role of tunnelling.
It is not clear to me what this is about.
Can anyone clarify?

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