Supplemental Materials II
Examples of Found Nets
Contents
About the Outputs 1
Table 1 and Outputs from Version 1/3 5
Representative CGD and Systre output for 3,4-nets 5
Representative CGD and Systre output for 3,6-nets 32
Representative CGD and Systre output for 4,4-nets 67
Representative CGD and Systre output for 4,6-nets 108
Table 2 and Outputs from Version 2/3 155
Representative CGD output for 3,6-nets 155
Representative CGD output for 4,4-nets 160
Representative CGD output for 4,8-nets 164
Some Nets Found by Version 2/3 But Not Listed in Table 2 199
TOPOS Outputs of Selected Nets 210
About the Outputs
Since embedded nets are enumerated as geometric objects, based on a naive breadth-first search, the raw outputs are merely embedded nets with no particularly desirable properties. Although an embedded net of optimal symmetries will eventually be enumerated by the program, there is as yet no firm estimate of how many sub-optimal but topologically equivalent embedded nets one would encounter first.
For example, among the 4,4-coordinated nets, the diamond (dia) net is frequently encountered. The small search space for Version 2/3 suggests how severe the problem may become. Among dia embedded nets of cubic symmetry, one finds examples like:
CRYSTAL
NAME spg3s0et3s0NET8
GROUP P1
CELL 2.00000 2.00000 2.00000 90.00000 90.00000 90.00000
ATOM 1 4 0.50000 0.00000 0.00000
ATOM 2 4 0.00000 0.00000 0.00000
ATOM 3 4 0.50000 0.50000 0.50000
ATOM 4 4 0.00000 0.00000 0.50000
ATOM 5 4 0.00000 0.50000 0.00000
ATOM 6 4 0.00000 0.50000 0.50000
ATOM 7 4 0.50000 0.00000 0.50000
ATOM 8 4 0.50000 0.50000 0.00000
EDGE 0.50000 0.00000 0.50000 0.50000 -0.50000 0.50000
EDGE 0.50000 0.00000 0.00000 1.00000 0.50000 -0.50000
EDGE 0.50000 0.00000 0.00000 0.50000 -0.50000 0.00000
EDGE 0.00000 0.00000 0.50000 0.00000 0.50000 0.50000
EDGE 0.00000 0.50000 0.00000 -0.50000 1.00000 0.50000
EDGE 1.00000 -0.50000 0.00000 0.50000 0.00000 0.50000
EDGE 0.00000 0.50000 0.50000 -0.50000 0.50000 0.50000
EDGE 0.00000 0.00000 0.00000 0.00000 0.00000 0.50000
EDGE 0.00000 0.00000 0.50000 -0.50000 0.00000 0.50000
EDGE 0.50000 0.00000 0.50000 1.00000 0.00000 0.50000
EDGE 0.00000 0.50000 0.00000 0.50000 0.50000 0.00000
EDGE 0.00000 0.00000 0.00000 0.00000 0.50000 0.00000
EDGE 0.00000 0.50000 -0.50000 0.00000 0.50000 0.00000
EDGE -0.50000 0.00000 1.00000 0.00000 0.50000 0.50000
EDGE 0.00000 0.00000 0.00000 0.50000 0.50000 0.50000
EDGE 0.50000 0.00000 0.00000 0.00000 0.00000 0.00000
EDGE 0.50000 1.00000 0.00000 0.50000 0.50000 0.00000
EDGE 0.50000 -0.50000 1.00000 0.00000 0.00000 0.50000
EDGE 0.50000 0.50000 -0.50000 0.50000 0.50000 0.00000
EDGE 0.00000 0.50000 0.50000 0.00000 0.50000 1.00000
EDGE 0.50000 0.00000 0.00000 0.50000 0.00000 0.50000
EDGE 0.50000 0.50000 0.50000 0.50000 1.00000 0.50000
EDGE 0.50000 0.50000 0.50000 0.50000 0.50000 1.00000
EDGE 0.50000 0.50000 0.50000 1.00000 0.50000 0.50000
EDGE 0.00000 1.00000 -0.50000 0.50000 0.50000 0.00000
END
Maple drew this one as:
This one looks a little better:
CRYSTAL
NAME spgdmm2s0etdmm2s0NET25873
GROUP P1
CELL 1.41421 1.41421 1.41421 120.00000 60.00000 90.00000
ATOM 1 4 0.00000 0.00000 0.00000
ATOM 2 4 0.50000 0.50000 0.00000
EDGE 0.00000 0.00000 0.00000 -0.50000 0.50000 1.00000
EDGE 0.00000 0.00000 0.00000 0.50000 -0.50000 0.00000
EDGE 0.00000 1.00000 0.00000 0.50000 0.50000 0.00000
EDGE 1.00000 0.00000 0.00000 0.50000 0.50000 0.00000
EDGE 0.50000 0.50000 0.00000 1.00000 1.00000 0.00000
EDGE 0.50000 0.50000 0.00000 1.00000 0.00000 -1.00000
EDGE 0.00000 0.00000 0.00000 -0.50000 0.50000 0.00000
EDGE 0.00000 0.00000 0.00000 -0.50000 -0.50000 0.00000
END
But this one (which occurred frequently) is still sub-optimal:
Looking at dia embedded nets of hexagonal symmetry is not much better. For example:
CRYSTAL
NAME spg2s5et2s5NET14855
GROUP P1
CELL 1.00000 1.73205 2.00000 90.00000 90.00000 90.00000
ATOM 1 4 0.00000 0.00000 0.00000
ATOM 2 4 0.50000 0.50000 0.00000
ATOM 3 4 0.50000 0.50000 0.50000
ATOM 4 4 0.00000 0.00000 0.50000
EDGE 0.00000 0.00000 0.00000 0.00000 0.00000 0.50000
EDGE 0.00000 0.00000 0.00000 -0.50000 0.50000 0.00000
EDGE 0.50000 0.50000 0.00000 0.50000 0.50000 0.50000
EDGE 0.50000 0.50000 0.50000 1.00000 1.00000 0.50000
EDGE 0.00000 0.00000 0.00000 0.00000 0.00000 -0.50000
EDGE 0.00000 0.00000 0.00000 0.50000 0.50000 0.00000
EDGE 0.50000 -0.50000 0.50000 0.00000 0.00000 0.50000
EDGE 0.50000 0.50000 0.50000 0.00000 1.00000 0.50000
EDGE 0.50000 0.50000 0.00000 0.50000 0.50000 -0.50000
EDGE 0.50000 0.50000 0.50000 0.50000 0.50000 1.00000
EDGE 1.00000 0.00000 0.00000 0.50000 0.50000 0.00000
EDGE 0.00000 0.00000 1.00000 0.00000 0.00000 0.50000
EDGE -0.50000 -0.50000 0.50000 0.00000 0.00000 0.50000
END
Maple’s picture of this one is:
Since dia is uninodal and edge transitive, while Version 2/3 starts with two edges and two vertices, we should not be surprised to find that a small search space produces suboptimal examples of dia. In fact, Version 1/3 had a larger search space, and as can be seen by comparing dmin and Dmin on Table 1, even Version 1/3 did not find an optimal embedding. Either a larger point group assignment or a larger search space is required.
Versions 1/3 and 2/3 both generate a lot of chaff, which had to be searched through manually. One could address it in the standard way by massaging the enumerated nets to optimize the symmetry, or by expanding the search space to look for more nets and then automating the chaff-elimination process. Version 1 will take the latter approach.
Below, the examples from Version 1/3 are typical outputs. They may be of interest geometrically, but they are primarily of interest as an indication of what randomly selected embedded nets look like. The industrious reader who looks them up will find pyramidal vertices and edge crossings. The examples from Version 2/3 have been screened for at least esthetics, and for some of the nets exhibited in table 2, we have presented several embeddings of the same net for readers to compare.
Table 1 and output from Version 1/3
Representative CGD and Systre output for 3,4-nets
These are binodal edge transitive (embedded) nets.
Nets generated within m-3m (cubic)
Note: It is impossible for a 3,4-coordinated edge-transitive embedded net to be hexagonal: the 3-coordinated nodes would need to have vertical rotational axes, and the resulting embedded net would be at most a layer.
The following is a selection of embedded nets generated by Version 1/3 of the Crystal Turtlebug, one per isomorphism (topology) equivalence class. This file list a representative of every 3,4-coordinated (binodal, edge transitive) topology found by Version 1/3. Given are the CGD file (position of the second node is given in the bold CGD file line) and the Systre output.
The names are also generated by the program: “spg”, followed by the (conjugate of) the point group applied to the first node at the origin, followed by “et”, followed by the (conjugate of) the point group applied to the second node, followed by “NET” and the number of the embedded net in the given topological density (td10) of all nets generated in that “run”.
CGD File, second vertex at [-1, 2, 0]:
CRYSTAL
NAME spg222s1et3s1NET1
GROUP P1
CELL 6.92820 6.92820 6.92820 109.47122 70.52878 70.52878
ATOM 1 4 0.50000 0.25000 0.75000
ATOM 2 4 0.12500 0.25000 0.87500
ATOM 3 4 0.12500 0.62500 0.00000
ATOM 4 4 0.00000 0.00000 0.00000
ATOM 5 3 0.75000 0.87500 0.12500
ATOM 6 4 0.87500 0.12500 0.75000
ATOM 7 3 0.75000 0.37500 0.12500
ATOM 8 3 0.25000 0.37500 0.12500
ATOM 9 3 0.75000 0.87500 0.62500
ATOM 10 3 0.25000 0.87500 0.62500
ATOM 11 3 0.75000 0.37500 0.62500
ATOM 12 4 0.87500 0.00000 0.37500
ATOM 13 3 0.25000 0.87500 0.12500
ATOM 14 3 0.25000 0.37500 0.62500
EDGE 0.00000 1.00000 0.00000 0.25000 0.87500 0.12500
EDGE 1.12500 0.25000 -0.12500 0.75000 0.37500 0.12500
EDGE 0.12500 0.25000 0.87500 0.25000 -0.12500 0.62500
EDGE 0.00000 0.00000 0.00000 0.25000 -0.12500 -0.37500
EDGE 0.87500 1.00000 0.37500 0.75000 0.87500 0.62500
EDGE 0.00000 0.00000 0.00000 -0.25000 -0.12500 0.12500
EDGE 0.12500 0.62500 0.00000 0.25000 0.87500 0.12500
EDGE 0.12500 0.25000 0.87500 0.25000 0.37500 0.62500
EDGE 0.12500 1.25000 0.87500 0.25000 0.87500 0.62500
EDGE 0.50000 0.25000 0.75000 0.25000 0.37500 1.12500
EDGE 0.25000 0.87500 0.62500 -0.12500 1.12500 0.75000
EDGE 0.12500 0.62500 0.00000 0.25000 0.37500 -0.37500
EDGE 0.00000 1.00000 1.00000 0.25000 0.87500 0.62500
EDGE 0.75000 -0.12500 0.62500 0.87500 0.12500 0.75000
EDGE 0.50000 0.25000 0.75000 0.75000 0.37500 0.62500
EDGE 0.75000 0.37500 1.12500 0.87500 0.12500 0.75000
EDGE 0.75000 0.87500 0.12500 1.00000 1.00000 0.00000
EDGE 1.25000 -0.12500 0.12500 0.87500 0.00000 0.37500
EDGE 0.75000 -0.12500 0.62500 0.87500 0.00000 0.37500
EDGE 0.75000 -0.12500 0.12500 0.87500 0.00000 0.37500
EDGE 0.12500 0.62500 0.00000 -0.25000 0.87500 0.12500
EDGE 0.50000 0.25000 0.75000 0.75000 -0.12500 0.62500
EDGE 1.00000 0.00000 0.00000 0.75000 0.37500 0.12500
EDGE 0.87500 1.12500 0.75000 0.75000 0.87500 0.62500
EDGE 0.75000 0.87500 0.12500 1.12500 0.62500 0.00000
EDGE 0.75000 0.37500 0.62500 0.87500 0.00000 0.37500
EDGE 0.50000 0.25000 0.75000 0.25000 0.37500 0.62500
EDGE 0.12500 0.25000 0.87500 0.25000 0.37500 1.12500
EDGE 0.12500 0.62500 0.00000 0.25000 0.37500 0.12500
EDGE 0.25000 0.37500 0.12500 0.12500 0.25000 -0.12500
EDGE 0.87500 1.00000 0.37500 0.75000 0.87500 0.12500
EDGE 0.12500 0.62500 1.00000 0.25000 0.37500 0.62500
EDGE 0.50000 1.25000 0.75000 0.75000 0.87500 0.62500
EDGE 0.00000 0.00000 0.00000 0.25000 -0.12500 0.12500
EDGE 0.87500 0.12500 0.75000 1.25000 -0.12500 0.62500
EDGE 0.12500 0.25000 0.87500 -0.25000 0.37500 1.12500
EDGE 0.50000 0.25000 -0.25000 0.25000 0.37500 0.12500
EDGE 0.00000 0.00000 0.00000 -0.25000 0.37500 0.12500
EDGE -0.12500 1.00000 0.37500 0.25000 0.87500 0.12500
EDGE 0.75000 0.37500 0.12500 0.87500 0.12500 -0.25000
EDGE 0.87500 0.12500 0.75000 0.75000 0.37500 0.62500
END
Systre Output (not locally stable):
Structure #1 - "spg222s1et3s1NET1".
Structure of dimension 3.
Given space group is P1.
14 nodes and 24 edges in repeat unit as given.
======
!!! ERROR (STRUCTURE) - Structure is not locally stable.
!!! In structure #1 - "spg222s1et3s1NET1".
!!! Last status: Computing barycentric placement...
======
Finished structure #1 - "spg222s1et3s1NET1".
CGD File, second vertex at [-1, 1, 3]:
CRYSTAL
NAME spg222et3s1NET1
GROUP P1
CELL 4.00000 4.00000 4.00000 90.00000 90.00000 90.00000
ATOM 1 4 0.00000 0.00000 0.00000
ATOM 2 3 0.25000 0.25000 0.25000
ATOM 3 3 0.75000 0.25000 0.75000
ATOM 4 3 0.75000 0.75000 0.25000
ATOM 5 4 0.50000 0.50000 0.00000
ATOM 6 4 0.50000 0.00000 0.50000
ATOM 7 3 0.25000 0.75000 0.75000
EDGE 0.75000 0.25000 0.75000 0.50000 -0.50000 1.00000
EDGE 0.25000 0.25000 1.25000 0.50000 0.00000 0.50000
EDGE 0.75000 -0.25000 0.25000 0.50000 0.50000 0.00000
EDGE 0.75000 0.25000 0.75000 0.50000 0.00000 1.50000
EDGE 0.25000 -0.25000 -0.25000 0.50000 0.50000 0.00000
EDGE 0.75000 0.75000 0.25000 0.50000 1.00000 -0.50000
EDGE 0.50000 1.50000 1.00000 0.25000 0.75000 0.75000
EDGE 0.50000 1.00000 1.50000 0.25000 0.75000 0.75000
EDGE 0.00000 0.00000 0.00000 -0.75000 -0.25000 -0.25000
EDGE 0.75000 0.75000 0.25000 0.50000 1.50000 0.00000
EDGE 1.00000 1.00000 1.00000 0.25000 0.75000 0.75000
EDGE 0.25000 1.25000 0.25000 0.50000 0.50000 0.00000
EDGE 0.00000 0.00000 1.00000 0.75000 0.25000 0.75000
EDGE 0.25000 0.25000 0.25000 0.50000 -0.50000 0.00000
EDGE 0.00000 1.00000 0.00000 0.75000 0.75000 0.25000
EDGE 0.75000 1.25000 -0.25000 0.50000 0.50000 0.00000
EDGE 0.75000 0.25000 -0.25000 0.00000 0.00000 0.00000
EDGE 0.00000 0.00000 0.00000 0.75000 -0.25000 0.25000
EDGE 0.25000 -0.25000 -0.25000 0.50000 0.00000 0.50000
EDGE 1.00000 0.00000 0.00000 0.25000 0.25000 0.25000
EDGE -0.75000 0.25000 0.25000 0.00000 0.00000 0.00000
EDGE 0.75000 0.25000 -0.25000 0.50000 0.00000 0.50000
EDGE 0.25000 0.25000 0.25000 0.50000 0.00000 -0.50000
EDGE 0.75000 -0.25000 1.25000 0.50000 0.00000 0.50000
END
Systre Output (bor):
Structure #1 - "spg222et3s1NET1".
Structure of dimension 3.
Given space group is P1.
7 nodes and 12 edges in repeat unit as given.
Given repeat unit is accurate.
Point group has 24 elements.
2 kinds of node.
Equivalences for non-unique nodes:
3 --> 2
4 --> 2
5 --> 1
6 --> 1
7 --> 2
Coordination sequences:
Node 1: 4 8 20 30 60 68 120 126 200 180
Node 2: 3 9 15 33 45 84 90 152 150 240
TD10 = 819.8571
Ideal space group is P-43m.
Ideal group differs from given (P-43m vs P1).
Structure was found in builtin archive:
Name: bor
Relaxed cell parameters:
a = 2.44949, b = 2.44949, c = 2.44949
alpha = 90.0000, beta = 90.0000, gamma = 90.0000
Cell volume: 14.69694
Relaxed positions:
Node 1: 0.00000 0.50000 0.50000
Node 2: 0.33333 0.33333 0.66667
Edges:
0.00000 0.50000 0.50000 <-> 0.33333 0.33333 0.66667
Edge centers:
0.16667 0.41667 0.58333
Edge statistics: minimum = 1.00000, maximum = 1.00000, average = 1.00000
Angle statistics: minimum = 70.52878, maximum = 131.81031, average = 114.82988
Shortest non-bonded distance = 1.15470
Degrees of freedom: 2
Finished structure #1 - "spg222et3s1NET1".
CGD File, second vertex at [-3, 1, 1]:
CRYSTAL
NAME spg222s1et3s1NET3
GROUP P1
CELL 5.65685 5.65685 5.65685 60.00000 120.00000 120.00000
ATOM 1 4 0.50000 0.50000 0.50000
ATOM 2 4 0.00000 0.00000 0.00000
ATOM 3 3 0.87500 0.62500 0.62500
ATOM 4 4 0.00000 0.00000 0.50000
ATOM 5 3 0.12500 0.87500 0.37500
ATOM 6 3 0.87500 0.62500 0.12500
ATOM 7 4 0.00000 0.50000 0.00000
ATOM 8 3 0.87500 0.12500 0.62500
ATOM 9 3 0.62500 0.37500 0.37500
ATOM 10 3 0.12500 0.37500 0.87500
ATOM 11 3 0.12500 0.37500 0.37500
ATOM 12 3 0.37500 0.62500 0.62500
ATOM 13 4 0.50000 0.50000 0.00000
ATOM 14 4 0.50000 0.00000 0.50000
EDGE 0.12500 0.87500 0.37500 0.50000 1.50000 0.00000
EDGE 0.00000 0.00000 0.50000 0.12500 0.37500 -0.12500
EDGE 0.87500 0.62500 0.12500 1.00000 1.00000 -0.50000
EDGE 0.50000 1.50000 0.50000 0.12500 0.87500 0.37500
EDGE 0.87500 0.62500 -0.37500 0.50000 0.50000 0.00000
EDGE 0.12500 0.37500 0.37500 0.50000 0.50000 0.00000
EDGE -0.12500 -0.37500 1.12500 0.00000 0.00000 0.50000
EDGE 0.50000 0.50000 0.50000 0.87500 0.62500 1.12500
EDGE 0.87500 1.12500 -0.37500 0.50000 0.50000 0.00000
EDGE 0.00000 0.00000 0.00000 -0.12500 -0.37500 -0.37500
EDGE 0.87500 0.12500 0.62500 0.50000 -0.50000 0.50000
EDGE 1.00000 -0.50000 1.00000 0.87500 0.12500 0.62500
EDGE 0.50000 0.50000 0.50000 0.87500 1.12500 0.62500
EDGE 0.50000 0.50000 -0.50000 0.87500 0.62500 0.12500
EDGE 0.87500 -0.37500 0.62500 0.50000 0.00000 0.50000
EDGE -0.62500 -0.37500 -0.37500 0.00000 0.00000 0.00000
EDGE 0.87500 -0.37500 1.12500 0.50000 0.00000 0.50000
EDGE 0.50000 0.50000 0.50000 0.12500 -0.12500 0.37500
EDGE 0.00000 0.00000 0.00000 0.12500 0.37500 0.37500
EDGE 0.87500 0.12500 0.62500 0.50000 -0.50000 1.00000
EDGE 0.50000 0.50000 0.50000 0.12500 0.37500 -0.12500
EDGE 0.87500 0.62500 0.12500 0.50000 1.00000 -0.50000