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Searching for all three-periodic uninodal subnets of diamond (dia) and lonsdaleite (lon) topological types. Searching for the shortest transformation paths between dia and lon
Algorithm:
1. Open the database dia_lon. Be sure that the adjacency matrix is computed for the first record (dia). Open Representation Parameters window (Compound/Generate Representations). Specify the options as shown below.
Press Ok and specify the database name dia. You will store in the database all uninodal nets (Max. NumAt = 1) with the diamond topology having an extended unit cell (MaxVolRatio = 1000, i.e. the unit cell can be 1000 times larger than the initial unit cell) and a lower symmetry of one of the subgroups of the space group Fd3m of the highest-symmetry embedding of diamond net.
2. Open the dia database; there are 47 records: the initial diamond net and 46 nets with a lower symmetry. The symmetry transformations are shown in the net names.
Detailed information on the symmetry transformations is given in Comment tab of the Crystal Data window for every net. The first line shows the successive transformation path, the second one contains the cumulative transformation codes in relation to the initial symmetry. If the transformation is done in one step nothing is written in the comment tab, because the transformation itself is written in the name.
3. Select all records in the dia database and open Representation Parameters window again. Change the parameters as shown below.
Click Ok and specify the database name dia_subnets. You will store in the database all subnets of the selected nets keeping the net symmetry. Open the dia_subnets database with 215 records. All the subnets are obtained by breaking some sets of equivalent bonds in a dia net. For example, the net
obtained from Fd3m diamond by decreasing symmetry down to P4132 and shifting the origin by the (5/8,5/8,5/8) vector, has two bond sets; therefore the initial net is stored under the name
It produces two subnets where either the second or the first set is retained; they have the names
and
respectively.
4. Remove all the dia nets except the most symmetrical embedding. All the initial nets have a label All bond sets in the formula name, therefore they can be selected using the Formula filter. Click Filter/Fragment/Formula and type the label. Run the filter.
5. Select all the 47 records found except the first one (select all of them and then unselect the first record). Remove selected records (Compound/(Un)Delete). Return to the main list (right-click and choose Show Main).
6. Determine the dimensionality of the subnets. For this purpose, select all records and open an IsoTest window (Program/IsoTest). Specify the IsoTest options as shown below.
7. Run IsoTest. You may stop the screen output to speed up the calculation by clicking Data/Show Data. After finishing the session close the IsoTest window.
8. Search for all non-three-periodic structures. Go to Filter/Topology/Dimensionality and specify the options as shown below. Remove current (previously used) filter by clicking Remove or Clear button.
Run the filter. You will get the following message that reminds you about the records found by the previous filter. Reply Yes.
9. Remove all the 156 found 0-, 1-, or 2-periodic structures. Return to the main list that now contains 13 records (the initial dia net and 12 its 3-periodic subnets).
10. Obtain the information about the topology of the subnets and store it as a net relation graph. For this purpose, select all the records and open an ADS window. Specify the ADS options as shown below. Run ADS and specify the name dia_lon for the net relation graph file.
11. Close the ADS window after finishing the calculation. Be sure that you have four selected records in the database list. They correspond to topologically different dia subnets. Other 9 subnets are different embeddings of the four distinct topologies; the four selected embeddings has maximal found symmetry. However, the topologies can have more symmetrical embeddings (that are not pcu subnets). To check it, use Systre program (http://www.gavrog.org). Copy the four records into a new dia_subnets_diff database.
12. Open the net relation graph window.
You see that there is the initial dia net and four its three-periodic subnets (bto, srs, ths and utp) in the net relation graph. Right-click on the dia record and choose Grow Branch. You see the hierarchical relations between the nets.
13. Repeat the whole procedure for the lon net. Store the net relations for lon in the same net relation graph. Do not forget to clear the current filter on step (4)! You have to get five lon three-periodic subnets (copy them into a new lon_subnets_diff database) and the extended net relation graph. To reload it you need to clear current net relation graph (Database/TTD Collection/Clear) and load dia_lon.nrl again (Database/TTD Collection/Net Relations). The resulting graph with 8 nets is shown in the last of the four pictures below.
14. Look at the dia and lon subnets using Grow Branch command. Find the shortest transformation paths from dia to lon through their common subnets. For this purpose, select the dia and lon records (Ctrl-click), right-click and choose Pair Relations command. You will get three possible paths through bto, ths and utp subnets.
15. Check the symmetry of the common subnets saved in the dia_subnets_diff and lon_subnets_diff databases (use ADS to identify the subnet topologies). You will find that the ths (dia/F d -3 m->C 2/c (-5/2a-3b-5/2c,1/2a-1/2c,2a+2b+2c; 1/2,1/2,1/2);Bond sets: 1,3,4 and lon/P 63/m m c->C 2/c (-a-b,a-b,c);Bond sets: 1,2,4) and utp (dia/F d -3 m->P n n a (a-b,c,-1/2a-1/2b; 1/2,1/4,1/4);Bond sets: 1,2,4 and lon/P 63/m m c->P n n a (a-b,c,-a-b; 0,1/2,0);Bond sets: 1,2,4) subnets have the same symmetry both for dia and lon (C2/c and Pnna, respectively), whereas the bto subnet has P3221 and P6522 space groups for dia (dia/F d -3 m->P 32 2 1 (-1/2a+1/2b,-1/2b+1/2c,a+b+c; 1/6,1/2,5/6);Bond sets: 1,2) and lon (lon/P 63/m m c->P 65 2 2 (a,b,3c);Bond sets: 1,3), respectively.
16. Try to decrease the symmetry for the bto subnet of lon from P6522 down to P3221. For this purpose, open a Crystal Data window and click Symmetry button. Select the P3221 translation-equivalent subgroup and click Change Group button.
You will find that at this symmetry the subnet becomes binodal (see tab Atoms). So the transformation from dia to lon through uninodal nets is possible only with ths and utp subnets. Close the Crystal Data window without saving the changes.
Exercise: find all uninodal three periodic subnets for quartz (qtz database). Can qtz be transformed to dia or lon through uninodal subnets?
Answer: There is only one uninodal subnet for qtz, it is bto in P3221 and P6522 space groups (totally 6 embeddings), so qtz can be transformed to dia through uninodal subnet bto in symmetry P3221, and to lon in P6522 space group.
Algorithm:
1. Open the database dia_lon. Be sure that the adjacency matrix is computed for the first record (dia). Open Representation Parameters window (Compound/Generate Representations). Specify the options as shown below.
Press Ok and specify the database name dia. You will store in the database all uninodal nets (Max. NumAt = 1) with the diamond topology having an extended unit cell (MaxVolRatio = 1000, i.e. the unit cell can be 1000 times larger than the initial unit cell) and a lower symmetry of one of the subgroups of the space group Fd3m of the highest-symmetry embedding of diamond net.
2. Open the dia database; there are 47 records: the initial diamond net and 46 nets with a lower symmetry. The symmetry transformations are shown in the net names.
Detailed information on the symmetry transformations is given in Comment tab of the Crystal Data window for every net. The first line shows the successive transformation path, the second one contains the cumulative transformation codes in relation to the initial symmetry. If the transformation is done in one step nothing is written in the comment tab, because the transformation itself is written in the name.
3. Select all records in the dia database and open Representation Parameters window again. Change the parameters as shown below.
Click Ok and specify the database name dia_subnets. You will store in the database all subnets of the selected nets keeping the net symmetry. Open the dia_subnets database with 215 records. All the subnets are obtained by breaking some sets of equivalent bonds in a dia net. For example, the net
obtained from Fd3m diamond by decreasing symmetry down to P4132 and shifting the origin by the (5/8,5/8,5/8) vector, has two bond sets; therefore the initial net is stored under the name
It produces two subnets where either the second or the first set is retained; they have the names
and
respectively.
4. Remove all the dia nets except the most symmetrical embedding. All the initial nets have a label All bond sets in the formula name, therefore they can be selected using the Formula filter. Click Filter/Fragment/Formula and type the label. Run the filter.
5. Select all the 47 records found except the first one (select all of them and then unselect the first record). Remove selected records (Compound/(Un)Delete). Return to the main list (right-click and choose Show Main).
6. Determine the dimensionality of the subnets. For this purpose, select all records and open an IsoTest window (Program/IsoTest). Specify the IsoTest options as shown below.
7. Run IsoTest. You may stop the screen output to speed up the calculation by clicking Data/Show Data. After finishing the session close the IsoTest window.
8. Search for all non-three-periodic structures. Go to Filter/Topology/Dimensionality and specify the options as shown below. Remove current (previously used) filter by clicking Remove or Clear button.
Run the filter. You will get the following message that reminds you about the records found by the previous filter. Reply Yes.
9. Remove all the 156 found 0-, 1-, or 2-periodic structures. Return to the main list that now contains 13 records (the initial dia net and 12 its 3-periodic subnets).
10. Obtain the information about the topology of the subnets and store it as a net relation graph. For this purpose, select all the records and open an ADS window. Specify the ADS options as shown below. Run ADS and specify the name dia_lon for the net relation graph file.
11. Close the ADS window after finishing the calculation. Be sure that you have four selected records in the database list. They correspond to topologically different dia subnets. Other 9 subnets are different embeddings of the four distinct topologies; the four selected embeddings has maximal found symmetry. However, the topologies can have more symmetrical embeddings (that are not pcu subnets). To check it, use Systre program (http://www.gavrog.org). Copy the four records into a new dia_subnets_diff database.
12. Open the net relation graph window.
You see that there is the initial dia net and four its three-periodic subnets (bto, srs, ths and utp) in the net relation graph. Right-click on the dia record and choose Grow Branch. You see the hierarchical relations between the nets.
13. Repeat the whole procedure for the lon net. Store the net relations for lon in the same net relation graph. Do not forget to clear the current filter on step (4)! You have to get five lon three-periodic subnets (copy them into a new lon_subnets_diff database) and the extended net relation graph. To reload it you need to clear current net relation graph (Database/TTD Collection/Clear) and load dia_lon.nrl again (Database/TTD Collection/Net Relations). The resulting graph with 8 nets is shown in the last of the four pictures below.
14. Look at the dia and lon subnets using Grow Branch command. Find the shortest transformation paths from dia to lon through their common subnets. For this purpose, select the dia and lon records (Ctrl-click), right-click and choose Pair Relations command. You will get three possible paths through bto, ths and utp subnets.
15. Check the symmetry of the common subnets saved in the dia_subnets_diff and lon_subnets_diff databases (use ADS to identify the subnet topologies). You will find that the ths (dia/F d -3 m->C 2/c (-5/2a-3b-5/2c,1/2a-1/2c,2a+2b+2c; 1/2,1/2,1/2);Bond sets: 1,3,4 and lon/P 63/m m c->C 2/c (-a-b,a-b,c);Bond sets: 1,2,4) and utp (dia/F d -3 m->P n n a (a-b,c,-1/2a-1/2b; 1/2,1/4,1/4);Bond sets: 1,2,4 and lon/P 63/m m c->P n n a (a-b,c,-a-b; 0,1/2,0);Bond sets: 1,2,4) subnets have the same symmetry both for dia and lon (C2/c and Pnna, respectively), whereas the bto subnet has P3221 and P6522 space groups for dia (dia/F d -3 m->P 32 2 1 (-1/2a+1/2b,-1/2b+1/2c,a+b+c; 1/6,1/2,5/6);Bond sets: 1,2) and lon (lon/P 63/m m c->P 65 2 2 (a,b,3c);Bond sets: 1,3), respectively.
16. Try to decrease the symmetry for the bto subnet of lon from P6522 down to P3221. For this purpose, open a Crystal Data window and click Symmetry button. Select the P3221 translation-equivalent subgroup and click Change Group button.
You will find that at this symmetry the subnet becomes binodal (see tab Atoms). So the transformation from dia to lon through uninodal nets is possible only with ths and utp subnets. Close the Crystal Data window without saving the changes.
Exercise: find all uninodal three periodic subnets for quartz (qtz database). Can qtz be transformed to dia or lon through uninodal subnets?
Answer: There is only one uninodal subnet for qtz, it is bto in P3221 and P6522 space groups (totally 6 embeddings), so qtz can be transformed to dia through uninodal subnet bto in symmetry P3221, and to lon in P6522 space group.
