The polarisable ion tight binding model for water and ice

I am actually quite proud of our tight binding parameterisation for water. In many ways our "TB water" is better than "DFT water". Our diffusivity is closer to experiment by a factor of two and, unlike in DFT, our ice floats!

While we initially developed the PITB theory to study transition metal oxides (particularly stoichiometric and non stoichiometric oxides of zirconium and titanium) [1], it occurred to me that we could also study "hydrogen oxide". In fact the oxygen–oxygen bond length in water and ice is about the same as in zirconia, and to me the hydrogen bond is really a covalent bond with a proton interposed (off centre and slightly non linear). A number of clever undergraduate students helped me in this, especially Catherine Walsh and Jeff Armstrong, and we published our first findings in 2011 [2]. Then a very clever and dedicated PhD student named Terence Sheppard embarked on a massive undertaking to find a transferable parameter set to describe carbon, oxygen and hydrogen in small organic molecules [3] and in water [4]. He succeeded after much trial and tribulation (and sadly he left physics). We were working on a large project in catalysis and our collaborators in physical chemistry were working on benzyl acetone. So Terence succeeded in simulating the "enolisation" of that molecule. His movie is next:

Later, Sasha Lozovoi, extended the model to include parameters for titanium dioxide [5]. The essential two points are:

1. Adding parameters for new elements musn't muck up the model for those elements for which it is already parameterised.
2. The parameters must be transferable, which means that, say, oxygen in water is the same element as oxygen in an organic molecule and in the transition metal oxide. This may seem obvious, but it is far from the case in classical force field models such as amber or gromacs. Moreover these models do not permit bond breaking and making, so they cannot be used in studies like this.

Tigany Zarouk has now extended the TB model to including metallic titanium, and Sasha is currently studying the behaviour of water on Ti as part of our "quantum corrosion" project.

Meanwhile we were very lucky to have Sebastian (Seb) Budd researching his MSci project in the group in 2020. Since the enolisation shows the transfer of a proton between water molecules, which is equivalent to the diffusion of the hydronium, Seb chose to study the Grotthuss process using our TB water. Here are a couple of his movies.

We are currently continuing from where Seb left off to try and calculate the proton diffusivity quantitatively and to assess the role of quantum tunelling on the rate coefficient. Watch this space…!

Right. Well I wrote that on 6 Jan 2021, and we have done some investigation of the proton diffusivity, but we have not got so far as to apply our TB implementation in Michele Ceriotti's i-pi code which is what I had planned in order to look into the role of quantum mechanical tunnelling. Now that I'm retired probably this will never get done which is a shame. On the other hand retiring with still lots of research plans is no bad sign.

What we have done is two-fold.

1. Tigany Zarouk has made a really effective and reliable TB model for pure titanium. When we wrote the trilogy of papers [3-5] I thought at the time that it would be trivial to extend to a pure-Ti transferable model, but that was wrong. I tried it myself a few years ago and got bogged down in soft phonon modes which arise because Ti takes the hexagonal close packed crystal structure. Tigany fixed all this in his thesis [6]. He may or may not publish his work on Ti and on steel; but all my group have moved on so that and the following work may go unpublished.
2. Sasha Lozovoi used Tigany's pure-Ti model in conjunction with our water to look at the Ti-water interface and to investigate at the crudest level the effect of raising or lowering the electric potential of the metal in mimcry of its connexion to a battery (this should have been done using the hairy probes but we ran out of time).

The first video shows an equilibrated excerpt from an MD simulation of a periodic supercell of water and a slab of Ti. You see that since Ti is a reactive metal it seeks to form an oxide; but it can't do this on the timescale of the molecular dynamics. But two observations are important.

1. Some water molecules dissociate into hydrogen and hydroxyl (OH) ions. The hydrogen ions (protons) penetrate into the first sub layer of Ti atoms, but they do not at this timescale diffuse further and dissolve. Actually hydrogen is not very soluble in hcp Ti (but is in bcc Ti).
2. The remaining hydroxyl ions bind to the surface through the formation of Ti-oxygen bonds. You can see quite energetic vibrations of the O-H bonds of the hydroxyl ions absorbed on the surface.

In the second video, we wanted to observe a hydronium ion in the water of the interphase. We neutralised it with Sasha's and Seb's invention, the "Ab-ion" which is an atom with no off diagonal hamiltonian matrix elements (hopping integrals) to the other atoms, but a repulsive pair potential and diagonal hamiltonian matrix element (on-site energy) large and negative so it is guaranteed to maintain a charge of one electron. We made two very significant observations.

1. The bonding of the Ab ion to the hydrogen ions of the water molecules that are screening it is not exactly the hydrogen bonding that we'd expect in a fully quantum chemistry scheme but over time the arrangement fluctuates between hydrogen bonding-like and a sort of "kite arrangement" that you'd expect from simple electrostatics. This argues that the hydrogen bonding to, say, a chloride ion must be regarded as a consequence of the covalent bonding that is missing for our simple Ab ion.
2. Unlike in the simulation of hydronium in pure water that I show above in this page, the third proton of the hydronium is not strongly bound to its water molecule. In fact we found, not only at the potential of zero charge (PZC), that the proton runs away and hangs about in the spaces between the water molecules. This means that unlike in pure water in which the excess hydrogen ions that give rise to lowered pH are always associated with hydronium or Zundel ions, in an electrolyte near a metal electrode (that is, in the double layer) protons are to found freely moving about. At first we thought this an unlikely conclusion and that maybe this is an artefact of our simple level of quantum theory in the TB model; but now we believe this result to be correct. The explanation as we find in our analysis of the Mullikan charges is that the run-away hydrogen ion carries a smaller positive charge than its counterparts in the other water molecules; we think this is because its charge is compensated by charge transfer from the metal. The situation becomes exaggerated as we raise the electric potential above the PZC as it becomes more willing to accept a fractional electron charge and partly neutralise the run-away hydrogen ion. I expect this is a general feature of the double layer of electrochemistry and that as quantum chemistry and DFT simulations advance this effect will be reproduced by other researchers.

In this simulation we applied an electric potential to the titanium atoms (through a shift in on-site energies) to bring its potential close to the potential of zero charge.

1. M. W. Finnis, A. T. Paxton, M. Methfessel and M. van Schilfgaarde Crystal structures of zirconia from first principles and self-consistent tight binding Physical Review Letters, 81, 5149 (1998)
2. A. T. Paxton and J. J. Kohanoff, A tight binding model for water J. Chemical Physics, 134 art. 044130 (2011)
3. T. J. Sheppard, A. Y. Lozovoi, D. L. Pashov, J. J. Kohanoff and A. T. Paxton, Universal tight binding model for chemical reactions in solution and at surfaces. I Organic molecules J. Chemical Physics, 141 art. 044503 (2014)
4. A. Y. Lozovoi, T. J. Sheppard, D. L. Pashov, J. J. Kohanoff and A. T. Paxton, Universal tight binding model for chemical reactions in solution and at surfaces. II Water J. Chemical Physics, 141 art. 044504 (2014)
5. A. Y. Lozovoi, T. J. Sheppard, D. L. Pashov, J. J. Kohanoff and A. T. Paxton, Universal tight binding model for chemical reactions in solution and at surfaces. III Stoichiometric and reduced surfaces of titania and the adsorption of water J. Chemical Physics, 141 art. 044505 (2014)
6. Tigany Zarouk, Simple quantum-mechanical models for defects in titanium and iron—how far can they take us?, PhD Thesis, Department of Physics, King's College London (2022)