# Order Parameter for Water in an MD Trajectory: Understanding Changes in Oxygen-Oxygen-Oxygen Triplet Angular Distributions

I'm interested in understanding how changes in the oxygen-oxygen-oxygen (O-O-O) triplet angular distribution can provide insights into the order parameter.

In literature, the tetrahedral order parameter q, an index to describe the similarity of the simulated water structure with the perfect tetrahedral structure would yield a value of 1 for the perfect tetrahedral structure and 0 for the ideal gas

My question is how the changes observed in these distributions can provide information about the order parameter of water. I tried to calculate the probability plot of angles for two different frames of the trajectory (first and last frame). Attached below sequentially.

Now, I would like to seek guidance on how to further analyze these changes in the O-O-O triplet angular distribution and extract meaningful insights about the order parameter and how to calculate the q-value.

While there are a number of order parameters which can track the phase transitions and other physio-chemical changes, what you are asking here is about the tetrahedral orientational factor (OTO) which can be written as: $$q_\text{OTO} = 1-\frac{3}{8}\Sigma^3_i\Sigma^{4}_{j+1}\bigg[\text{cos}\ \theta_{jk}+\frac{1}{3}\bigg]^2$$ which gives a measure of the order of the bulk material in terms of molecular near-neighbours. Consider a simple case of a water molecule surrounded by neighbouring water molecules. which due to h-bonding effects form a tetrahedron surrounding the original water molecule.If we can extract the internal angles of this tetrahedron, for ex:- the angle between the neighbours $$j$$ and $$k$$ being $$\theta_{jk}$$ (the O-O-O angle defined in your question), we can describe the local tetrahedral order $$q_\text{OTO}$$.The $$q_\text{OTO}$$ values ranges from $$0\leq q_\text{OTO}\leq1$$, where $$0$$ implies no order (something which might be possible for water vapour) and $$1$$ which implies a maximum ordered system (close to ice $$Ih$$).

So for a simplistic implementation, you can:

1. Select a water molecule at random from your bulk system.
2. Find the closest neighbouring four water molecules.
3. Find $$q_\text{OTO}$$ value for the chose water molecule.
4. Repeat 1-3 until you have cycled through all water molecules under analysis. Save all the $$q_\text{OTO}$$ values
5. Bin and normalize (to 1) the distribution of the $$q_\text{OTO}$$ values and plot the distribution, which gives you probability of $$q_\text{OTO}$$ $$P(q_\text{OTO})$$ vs $$q_\text{OTO}$$ plot, which describes your system. Something like this: The above figure is from Tetrahedral Ordering in Water: Raman Profiles and Their Temperature Dependence. For further reading look into the following articles: