This appendix contains several techniques for analyzing data generated from AIMD simulations.
Configurations in an AIMD trajectory are sampled in the phase space, and physical properties such as RDF and diffusion coefficients are obtained by averaging along the a whole simulated trajectory; therefore, the longer trajectory we have, the more accurate our analyzed results become. However, since the AIMD is computationally costly, we wish to find an
appropriate length for simulated trajectories where the computed properties are reliable enough, and the computational cost is reasonable enough. Appx B.1 discusses techniques regarding such concern, and introduces several ways to guarantee converged properties for our analyses.
Sometimes, we wish to get physical properties from simulations that are experimentally measurable, for example dipole moments or diffusion coefficients, so that we could compare our results with the experimental values. In other times, however, we wish to compute order
parameters that, although may not be measured experimentally, could capture the particular structural or dynamical properties of concern. It also happens that the property that we want to analyze has multiple definitions existing in the literature. Therefore, for the clarity of this thesis, it is important to list and discuss the definitions, usages, and limitations of such order parameters, which is the content for Appx. B.2-4.
143 B.1 Checking for converged physical properties
To produce a trajectory for the MD simulation of a system that is ready for further analyses, one needs to check whether the system has reached the state of equilibrium. To reach equilibrium means that the observable quantities of the system no longer change with time. For example, if one has a trajectory of 100 ps, and the RDF đ""(đ) for the timespan of [50:100] ps is identical to đ""(đ) for the timespan of [0:50], then we could claim that the system is equilibrium for this 100-ps trajectory. Sometimes, however, after checking đ""(đ) for periods of [0:20], [20:40], âĻ [80:100], it may be discovered that for this 100-ps trajectory the first 20 ps has not reached equilibration. In this case, the first 20 ps needs to be abandoned in the future analyses.
It also happens that some observable quantities reach equilibrium far slower than the others. For example, compared to properties such as RDF, the diffusion coefficient of ions in a system is very hard to converge. In fact, diffusion coefficients of H3O+(aq) and OH-(aq) ions in liquid water, as discussed in Chapter 5, Appx D, and Appx E, is exceptionally hard to converge.
Fig. D.3 shows how the different selection of dT, which determines the number of
sub-trajectories for averaged MSD, influence the value of MSD. Besides, it was argued that instead of directly comparing diffusion coefficient values of H3O+(aq) and OH-(aq) with experiments, the ratio between the two could be a robust way to scrutinize the simulated scenarios
qualitatively (Tuckerman et al., 2006).
In order to guarantee that a AIMD trajectory is meaningful, some quantities of the system, for example, forces acting on the ions, fluctuation of the potential energy surface, etc., also need to be constantly checked. The reader is referred to Ref. (Z. Li, 2012) for a detailed description on how to monitor and perform meaningful and correct CPMD simulations.
144 B.2 H-bonds
The notion of H-bonds is not uniquely defined. Many definitions of the H-bonds, which use various geometrical, topological, electronic structure-based, or potential-based information, exist in the literature. As H-bonds are weak inter-molecular bonds, any quantitative description of the statistics for intact H-bonds yields certain degree of ambiguity. Nevertheless, a general agreement among many proposed definitions for intact H-bonds still exist (Prada-Gracia et al., 2013); in liquid water at ambient conditions, the average number of H-bonds per molecule nHB
are typically between 3 and 4 (Kumar, Schmidt, & Skinner, 2007). In our works, we follow the H-bond definition proposed by Luzar and Chandler (Luzar & Chandler, 1996a) and define an intact H-bond when O-O distance ROO < 3.5 à and b < 30°, where b ⥠â OAâĻOD â HD denotes the bond angle formed by an oxygen atom OA that accepts a H-bond (HD) from a nearby oxygen atom OD.
B.3 Identifying the ion and the proton-transfer moment
For the liquid water system with a H3O+ or OH- ion, in order to find the ion we first need to define the covalent bond. In this thesis, the covalent bond between the O atom and H atom is defined with a upper distance cutoff of d(OH) < 1.24 Ã .
Using this definition in the pure liquid water system, for almost all cases, for each of the O atom in the system, two H atoms would be found as covalently bonded to the oxygen. In the system containing a H3O+ (or OH-) ion, in most snapshots there would be one O atom (Oi) identified as covalently bonded with three (or one) H atom, which compose the H3O+ (or OH-) ion. However, in some snapshots not one, but zero or two ions would be identified. This is due to the fact that when PT is taking place, the H atom in transition is between two adjacent O atoms
145
(Oi1 and Oi2), and would be identified as covalently bonded to both or neither of Oi1 and Oi2. As the PT process is very fast, this ambiguous period would only last for a few snapshots. After this ambiguous period, the index of the ion would be exclusive.
The PT moment is defined as (1) the snapshot in which Oi changes, or (2) the snapshot when the aforementioned ambiguous period finishes, and Oi after the period is different from Oi
before the period.
B.4 Ring Statistics
In Chapter 3 and 4, the statistics of closed rings are used for analyzing the H-bond network structure in liquid water. Ring statistics is a useful set of analyses for the study of
topological networks in systems like the silicate (King, 1967; Yuan & Cormack, 2002) and liquid water (Belch & Rice, 1987; Hassanali et al., 2013; Matsumoto et al., 2007). In liquid water, for example, where rings are threaded by O atoms that are connected with H-bonds, properties like size of rings and number of rings threading a molecule provide detailed information on the H-bond network.
There are various ways to define a closed ring in a system. In this thesis, rings were defined adopting the shortest-path definition from Ref. (Belch & Rice, 1987), in which only the shortest circuit with two continuous H-bonds from one water molecule is recorded. The
algorithm for finding the shortest-path rings is described as follows (see Ref. (Yuan & Cormack, 2002) for definitions of terminologies). First, a pair of consecutive H-bonds are found and recorded as the first two edges of a potential ring. Secondly, for each pair of H-bonds, search for the smallest closed elementary paths starting from 3-edge path onward. For example, if a 3-edge closed path is not found, the algorithm will proceed to look for possible 4-edge closed paths. If a
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4-edge closed path is found for this pair of H-bonds, then the algorithm will not search whether a 5-edge closed elementary path exists; in the meantime, however, all the 4-edge closed paths for this pair of H-bonds are recorded.
147 APPENDIX C