Topological network criteria

This page is the definitions. The runnable checks that the implementations still return the documented counts live under Tutorials.

Bond correlation

CHILL and CHILL+ look at \(l = 3\). With \(N_b(i)\) neighbours,

\[q_{3m}(i) = \frac{1}{N_b(i)} \sum_{j \in N(i)} Y_{3m}(\hat r_{ij})\]

\[c_{ij} = \mathrm{Re} \frac{\sum_m q_{3m}(i)\, q_{3m}(j)^{*}} {\sqrt{\sum_m |q_{3m}(i)|^2 \sum_m |q_{3m}(j)|^2}}\]

A bond is staggered when \(c_{ij} \le -0.8\). The eclipsed window is the part that differs:

rule	eclipsed	staggered	neighbours
CHILL	\(-0.2 \le c_{ij} \le -0.05\)	\(c_{ij} \le -0.8\)	4
CHILL+	\(-0.35 \le c_{ij} \le 0.25\)	\(c_{ij} \le -0.8\)	4

The code stores those edges as chill::chillRule (-0.8, -0.2, -0.05, 4) and chill::chillPlusRule (-0.8, -0.35, 0.25, 4). Boundaries are inclusive.

The four-bond star is the phase. Cubic is four staggered bonds. Hexagonal is three staggered and one eclipsed. CHILL+ adds clathrate (four eclipsed) and interfacial clathrate (three eclipsed). Water is what remains. The full star table is in bop.hpp.

Primitive rings

A cycle is primitive, in the Franzblau sense, when for every pair of its vertices a shortest path in the bond graph lies along the cycle. primitive::ringNetwork enumerates cycles up to max_depth and drops the ones that fail that test. The bond graph is an argument. Mutual four-nearest, the union of those nominations, and a distance cutoff are three different graphs. Quote the graph with the census.

Cages

A hexagonal cage is two basal six-rings joined by prismatic six-rings. A double-diamond cage is an equatorial six-ring and its peripheral rings. ring::cageAffiliation evaluates those conditions for every six-ring, without the claim order of the older findHC / findDDC walk. The seeded variant takes a strict graph for the seeds and a permissive graph for the completions. An atom on the permissive graph is kept only when its component contains a seed. --complete then fills a six-ring that is missing one labelled vertex. That walk is the least fixed point of the all-but-one rule; see the ring-completion page.

Confined ice

A prism block is two rings of the same size that satisfy the basal conditions in ring::basalPrismConditions (or the relaxed form). ring::findPrisms returns which rings are blocks, split into perfect and imperfect. A monolayer analysis keeps primitive rings of each size and, when you ask for the files, writes the coverage of the sheet. The in-plane \(g(r)\) normalises the pair histogram by the slab factor

\[ f(r) =

begin{cases} 1 - r / (2h) & r le h \ h / (2r) & r > h end{cases}

\]

where \(h\) is the confined height. That is rdf2::normalizeRDF.

Order parameters

\[q_l(i) = \sqrt{\frac{4\pi}{2l+1} \sum_{m=-l}^{l} |q_{lm}(i)|^2}\]

\(\bar q_l(i)\) replaces \(q_{lm}(i)\) with the average of \(q_{lm}\) over \(i\) and its neighbours before the same sum. That is the Lechner-Dellago form. The implementation is chill::steinhardtQl.