Data on generation of Kekulé structures for graphenes, graphynes, nanotubes and fullerenes and their aza-analogs.

Two new features are added to existing algorithms for kekulization of chemical structures, i.e., handling of triple and cumulene bonds in cycles and use of random atom sorting to remove unmatched atoms. Handling of triple and cumulene bonds enables kekulization of graphynes and graphdiynes. Random sorting speeds up the calculation time, i.e., kekulization of large chemical structures containing about 107 atoms takes ≤1 min on a typical PC. Source codes (Pascal, GNU GPL license) are included as a compiled application (Windows 64). Calculation times and unmatched atom statistics are provided for graphenes, graphynes, nanotubes, graphyne nanotubes and fullerenes. Benchmark comparisons are made for some data.

Specifications table

Value of the data

This data proves the viability of a fast algorithm for the modeling of state-of-the art materials such as graphenes in scientific and commercial applications

The data is provided for very large chemical structures (10-10 atoms) and is obtained using ordinary hardware with short calculation times

This data has been benchmarked against existing algorithms and can be used for benchmarking for the future improvements to the algorithm

Data

Detailed description of an algorithm for fast generation of Kekulé structures from a list of atomic valences and connectivity matrices is given in [1]. The source code that was based on this algorithm and used for calculations is provided in this publication.

Model calculations were performed for graphenes, nanotubes, fullerenes and their aza-analogs, polycyclopentadienes and porphine, using a Windows 2012 server with 2.8 GHz i7 processor and 16 GB RAM. The procedures were not run in parallel but rather in a single thread. The Win API method GetTickCount() was used for precise time measurements. This method allows the measurement of the elapsed time with a millisecond precision. For the compounds containing less than 1,000,000 atoms, the computing time was determined as an average of 1000 calculations, less than 10,000,000 atoms – as an average of 100 calculations and for more than 10,000,000 atoms – as the time of one calculation.

For repeated calculations, the sorting order of the sequence of chemical bonds was random. For an algorithm of aromatic bond alternation, this is equivalent to the random selection of a double bond in the node. For a large number of calculations, this procedure allows estimation of the statistical distribution of unmatched atoms after alternation of aromatic bonds.

Compiled application (Windows 64) used for calculations is in the Attachment1.zip. Computer-readable chemical structures are in the Attachment2.zip.

Experimental design, materials and methods

Nanotubes

Carbon nanotubes of various lengths were generated using the pattern shown in Fig. 1. Generated nanotubes contain two acyclic methylene groups at the beginning and at the end of a nanotube.

Fig. 1

The pattern for generation of C[12,12] nanotubes.

In Fig. 1, the arrows show the points of attachment for generation of a polymer molecule. The points of attachment for the outer rings connect with the points of attachment for the inner rings. The order of an aromatic bond resulting from connecting a pair of attachment points is unknown. For the end moieties of the polymer, the points of attachment were replaced with single bonds to hydrogen atoms. The results of calculation are given in Table 1 and discussed below.

Table 1

Time required for generation of Kekulé structures and unmatched atom statistics for nanotubes and graphenes.

Chain length	Formula	Time (ms)	Unmatched atoms statistics
			2	4	6	8	10	12	14	16	18	20	22	24	26	28	30	32	34	36	38	40	42
Nanotubes
1	C144 H48	0.33		32	351	510	107
10	C1440 H48	2.42						1	28	93	111	146	232	215	113	9	3
100	C14400 H48	28.1									5	12	15	36	119	270
1000	C144000 H48	407.7										1	1	1	12	91	281	332	227	51	3
10000	C1440000 H48	8123.3															6	30	34	23	7
100000	C14400000 H48	108437																1
Graphenes
1	C144H38	0.16	31	340	442	182	5
10	C1440 H146	2.09						9	42	97	140	166	199	167	107	58	15
100	C14400 H1226	28.5									6	14	12	20	24	77	165	255	248	138	37	4
1000	C144000 H12026	451.8										2	2	2	1	5	7	36	168	421	288	65	3
10000	C1440000 H120026	8497.8																1	11	37	42	9
100000	C14400000 H1200026	118297																		1
Data for some compounds from [2] obtained with our algorithm
tube-980	C980H26	1.36	17	75	164	312	246	143	39	4
sheet-1800	C1800H178	2.19	0	0	0	0	0	0	0	0
sheet-19602	C19602H592	28.9	0	0	0	0	0	0	0	0

Where:

Chain length – number of monomeric units in a polymer,

Formula – molecular formula,

Time (ms) – time required for kekulization of a single chemical structure in milliseconds,

Unmatched atoms statistic - number of calculations for a given number of unmatched atoms (2-42).

Graphene

The pattern for generation of graphene is shown in Fig. 2.

Fig. 2

The pattern for generation of graphenes.

To generate polymers, the points of attachment on the left were connected with the points of attachment on the right of another graphene block in the way similar to that used for nanotubes. The points of attachment of the first and the last blocks were capped with hydrogen atoms via single bonds. For acyclic carbon atoms, required for the generation of repeating aromatic cycles, two hydrogen atoms were added to the point of attachment, resulting in a methylene group. The results of calculations are given in Table 1.

Kekulization of various compounds, including nanotubes and graphenes, is described [2], [3] and computing times are provided. For comparison, we performed calculations for some of the same compounds. The results are in Table 1. Computing times were dramatically shorter than those reported in [2].

Detailed discussion of this data can be found in [1].

Fullerenes and porphine

Alternation of aromatic bonds was studied for fullerenes C20, C60, C70, C80, C82, their random aza-analogs, as well as for porphine (Fig. 3).

Fig. 3

Fullerenes and porphine used for model calculations.

Polymeric analogs of these compounds do not exist. Consequently, all the calculations were performed for monomers. To validate the efficiency of the algorithm for five-member cycles, model calculations were performed for azafullerenes by randomly replacing 2, 4 or 8 carbon atoms with nitrogen. Nitrogen has a valence of 3 and three single converging bonds in each node. This substitution can be done for the even number of atoms only. Otherwise, bonds cannot be alternated, and the number of unmatched atoms is odd. The results of calculations are given in Table 2.

Table 2

Time required for generation of Kekulé structures and unmatched atoms statistics before their removal for fullerenes, their aza-analogs and porphine. The data is for 1000 transactions.

Compound	Formula	Time (ms)	No. non-existent	Avg. no. iterations	Max. no. iterations	Unmatched atoms statistic
						0	2	4	6	8	>8
С20	С20	0.078	0	1.00	1	699	301
Diaza-C20	C18N2	0.062	0	1.03	3	409	583	8
Tetraaza-C20	C16N4	0.438	33 + 59	14.2	2	499	493	8
Octaaza-C20	C12N8	0.953	541 + 218	70.6	1	769	222	9
C60	C60	0.156	0	1.00	1	329	600	68	3
Diaza-C60	C58N2	0.171	0	1.01	2	124	681	182	11	1	1
Tetraaza-C60	C56N4	0.234	4 + 6	3.72	3	78	556	337	28	1
Octaaza-C60	C52N8	1.250	63 + 83	24.4	3	146	492	306	54	1	1
C70	C70	0.187	0	1.003	2	264	656	76	4
Diaza-C70	C68N2	0.188	0	1.01	3	102	269	24
Tetraaza-C70	C66N4	0.407	2 + 3	2.54	3	61	498	382	57	2
Octaaza-C70	C62N8	0.890	49 + 52	14.86	3	79	419	390	103	9
C80	C80	0.187	0	1.093	3	291	615	94
Diaza-C80	C78N2	0.203	0	1.065	3	90	577	312	21
Tetraaza-C80	C76N4	0.344	1 + 5	1.69	4	50	421	440	84	5
Octaaza-C80	C72N8	0.860	31 + 42	12.8	4	55	303	461	162	18	1
C82	C82	0.203	0	1.014	2	223	622	149	5	1
Diaza-C82	C80N2	0.203	0	1.020	2	74	547	347	30	2
Tetraaza-C82	C78N4	0.328	2 + 4	2.84	2	42	450	425	78	5
Octaaza-C82	C74N8	0.984	28 + 37	13.97	3	46	288	468	175	21	2
Porphine	C20H14N4	0.093	0	1.00	1	441	559

Where:

Compound – chemical structure in Fig. 2,

Formula – molecular formula,

Time (ms) – time required for kekulization of a single chemical structure in milliseconds,

No. non-existent – number of compounds from a 1000 set for which Kekulé structure was not generated,

Avg.no. iterations – average number of shuffles. This number includes the maximum number of shuffles (300) after which a decision is made that a Kekulé structure does not exist,

Max. no. iterations – maximum number of shuffles required for successful kekulization,

Unmatched atom statistic - number of calculations for a given number of unmatched atoms (0->8).

Detailed discussion of this data can be found in [1].

In addition, model calculations were performed for 2488 fullerenes from a library by Yoshida [4]. This data is provided in Table S3.

The legend for the column headers in this table is the same as for Table 2, except the column No. non-existent is not provided because every compound from the set of 1000 had a Kekulé structure.

Graphynes and graphyne nanotubes

We studied graphynes GY1 and GY7 (Fig. 4) and graphyne nanotubes of various degrees of polymerization. Graphyne nanotubes were generated by replacing hydrogen atoms in GY7 with carbon atoms and adding a bond between these atoms in a vertical position.

Fig. 4

Monomer units of graphynes GY1 (A) and G7 (B). Upper half of GY7 monomer unit is shown only. The entire monomer could be visualized by connecting the upper part with its reflection relative to the zig-zag line, using two aromatic bonds.

The times required for kekulization of graphynes and graphyne nanotubes of various degrees of polymerization are shown in Table 3.

Table 3

Kekulization of graphynes and graphyne nanotubes of various degrees of polymerization.

Chain length	1		10		100		1000		10,000		100,000
	Formula	Time (ms)	Formula	Time (ms)	Formula	Time (ms)	Formula	Time (ms)	Formula	Time (ms)	Formula	Time (ms)
Tube	C192 H72	0.5	C1920 H72	2.06	C19200 H72	29.8	C192000H72	332	C1920000 H72	7201	C19200000 H72	59,515
GY1	C136 H74	0.45	C1360 H146	1.36	C13600 H866	13.7	C136000 H8066	176	C1360000 H80066	4811	C13600000 H800066	75,938
GY7	C188 H86	0.39	C1880 H122	2.44	C18800 H482	25.3	C188000 H4082	358	C1880000 H40082	6414	C18800000 H400082	82,032

Polycyclopentadienes

Polycyclopentadienes (Fig. 5) are remarkable because they contain odd-sized cycles and can be easily generated as long-chain polymers.

Fig. 5

Structural repeating unit of polycyclopentadiene.

We studied three types of polycyclopentadienes. In the first type, after generation of the structure, free valences of the end-group carbon atoms with arrows were replaced with hydrogens. That resulted in a methylene end group. In the second type, free valences in the left-hand end group were combined with these in the right-hand end group to form a cycle. In the third type, end groups were combined in a crisscross fashion for form a Moebius loop. The results of calculations are shown in Table 4.

Table 4

Results of generation of Kekulé structures for polycyclopentadienes.

Chain length	Formula	Time (ms)	No. non-existent	Avg. no. iterations	Max. no. iterations	Unmatched atoms statistic
						0	2	4	6
Linear
1	C150 H54	0.53	0 + 0	1.93	13	372	607	21	0
10	C1500H504	4.08	0 + 0	4.63	25	747	250	3	0
10	C1498H504N2	35.2	0 + 10(1)	22.9	264	477	431	81	1
10	C1496H504N4	65.75	4 + 91(71)	68.9	300	262	393	210	31
10	C1492H504N8	189	87 + 348(261)	163	299	50	183	269	247
100	C15000H5004	67.7	0 + 0	5.60	36	852	146	2	0
1000	C150000H50004	1192	0 + 0	5.61	47	850	150	0	0
10000	C1500000 H500004	47354	0 + 0	7.57	33	89	11	0	0
100000	C15000000 H5000004	505401	0 + 0	6.5	22	7	3	0	0
Cyclic
1	C150H50	0.34	0 + 0	1,23	5	289	681	30	0
10	C1500H500	2.77	0 + 0	2.89	17	624	332	44	0
100	C15000H5000	33.6	0 + 0	2.90	28	622	331	47	0
1000	C15000H50000	585	0 + 0	2.80	14	634	335	31	0
10000	C1500000 H500000	16368	0 + 0	3.09	12	69	29	2	0
100000	C15000000 H5000000	193859	0 + 0	2.2	6	7	2	1	0
Möbius
1	C150H50	0.36	0 + 0	1.26	4	189	687	121	3
10	C1500H500	3.67	0 + 0	3.66	29	544	382	73	1
100	C15000H5000	52.2	0 + 0	4.41	27	616	332	51	1
1000	C150000H50000	1381	0 + 0	4.42	29	602	350	48	0
10000	C1500000 H500000	24147	0 + 0	3.75	26	52	42	6	0
100000	C15000000 H5000000	604359	0 + 0	6.9	28	8	0	2	0

The legend for the column headers is the same as for Table 2. Parenthetical values in the column No. non-existent are the counts of structures for which Kekulé representations were found using a backtrack algorithm [5].

The calculation statistics for polycyclopentadienes differ from those for the rest of studied compounds. Specifically, 100 calculations were performed for the number of atoms in the 106-107 range and 10 calculations – for the number of atoms >107. The increase in the number of calculations was due to the probabilistic nature of the algorithm for polycyclopentadienes, requiring multiple initial approximations for the generation of Kekulé structures.

Subject area	Chemistry (General)
More specific subject area	Kekulization of large chemical structures
Type of data	Tables, text file, figures, computer-readable chemical structures
How data was acquired	Calculation using Notebook MSI GE 60 2PG Apache
Data format	*.mol, *.xyz and *.cc1 files for chemical structures
Experimental factors	2.8MHz i7 processor, 16G RAM, Windows 2012 server
Experimental features	Single-thread application; Win API method GetTickCount() used for precise time measurements
Data source location	Article text for source code and calculation statistics, Attachment1.zipfor compiled application, Attachment2.zipfor chemical structures
Data accessibility	Downloadable .zip
Related research article	Trepalin S., Gurke S., Akhukov M., Knizhnik A., Potapkin B., A fast approximate algorithm for determining bond orders in large polycyclic structures, Journal of Molecular Graphics and Modelling, vol. 86 (2019), pp. 52-65