Lobachevsky State University of Nizhni Novgorod
Polyhedron
Quick Start Guide
Nizhni Novgorod 2016
Contents
Specification of Polyhedron software ................................................................................................................ 3
Theoretical background ..................................................................................................................................... 4
1. Interface of Polyhedron.................................................................................................................................. 6
1.1. The calculation options ........................................................................................................................... 7
1.2. The polyhedron building options ............................................................................................................ 8
1.3. Main menu .............................................................................................................................................. 9
1.4. Settings of Polyhedron .......................................................................................................................... 10
1.5. Selection of coordination polyhedrons ................................................................................................. 11
1.6. Results ................................................................................................................................................... 12
2. Analysis of coordination polyhedrons ......................................................................................................... 15
3. Comparison of polyhedrons in the current structure.................................................................................... 16
4. Comparison of arbitrary polyhedrons .......................................................................................................... 17
2
Specification of Polyhedron software
The Polyhedron software is a part of а software package called PseudoSymmetry. General features
of Polyhedron are:
•
•
•
The analysis of coordination polyhedrons in crystal structures;
The quantitative estimation of the degree of similarity of polyhedrons with the point model
usage
The quantitative estimation of the degree of similarity of polyhedrons via the electron density
function
System requirements: Windows 7, 8, 10 (64 bits); PC with multicore processor; RAM 4 Gb or more;
recommended to have CUDA-device with RAM 2 Gb or more.
For more information contact Dr. Nikolay Somov ([email protected])
3
Theoretical background
The degree of similarity of two polyhedrons
Every polyhedron can be represented as a set of vectors; every vector connects the center of the
polyhedron and vertex. Let the vector set T = {t i } be the representation of the original polyhedron, and the
S = {s i } be the representation of the standard polyhedron. Sets T and S are isomorphic. Let the
of T and S coincide. The rotation of T is described by the β matrix. The scale factor and the
vector set
centers
deformation of the standard polyhedron are described by α matrix. The quantitative estimation of the
degree of similarity is determined as maximum of the following function:
n
Φ(T , S ) = max ∏ pi exp(− λ ⋅ ϑi ) ,
α,β
 i =1

where n – is a number of vectors in sets T and S ,
 x, x > 0
(α ⋅ si , β ⋅ t i ) .
pi = 
, где x =
2
max( α ⋅ s i , β ⋅ t i )
0, x ≤ 0
Parameter ϑi in (1) is an angle between vectors s i and t i
 (α ⋅ s i , β ⋅ t i ) 
.

⋅
⋅
α
s
,
β
t
i
i 

ϑi = arccos
The value of
(1)
(2)
(3)
Φ(T , S ) is the degree of similarity of polyhedrons T and S . The calculation of
Φ(T , S ) requires the optimization of (1) with α and β matrices.
The electron density method
Let some atomic cluster be corresponded to the coordination polyhedron T . Also specify the
standard polyhedron S . Matrices α and β should be obtained from (1). Electron density functions can be
calculated for these polyhedrons. Each vector s i and
ti
corresponds to i-atom from the atomic cluster. The
electron density function ρT (r ) is built on { βt i } vectors, and ρ S (r ) – on { αs i }.
The quantitative estimation of the comparison for electron densities ρ S (r ) and ρT (r ) uses the
functional


η[ρT (r ), ρ S (r )] = max K -1 ∫ ρT (r ) ⋅ ρ S (r )dV  ,
α,β
 V

(4)
where
K = ∫ ρ T2 (r )dV = ∫ ρS2 (r )dV ,
V
(5)
V
4
where V – is the volume of the unit cell. As α and β are obtained by (1), they may not correspond to the
maximum of (4), so the additional refinement of (4) is needed.
The expression (5) can be calculated in the reciprocal space. In this case (5) has the next view
η[ρT (r ), ρ S (r )] = max ( KV ) −1 ∑ FT (H ) ⋅ FS (− H ) .
α,β


H
(6)
The coefficient К in the reciprocal space can be calculated as
K=
1
2
FT (H ) .
∑
V H
(7)
5
1. Interface of Polyhedron
The main window of the Polyhedron software is presented on figure 1.1. This window contains next
parts:
Main menu – contains different options of the software;
Toolbar – contains short call of some options and filters for the polyhedron building;
Structure navigator – contains the list of atoms for the target structure;
Result viewer – contains representation of results and detailed information;
Result navigator – contains the tree of results;
Output – the log of the calculation.
Main menu
Structure navigator
Toolbar
Result viewer
Result navigator
Output
Fig. 1.1. Main window of Polyhedron
6
1.1. The calculation options
This option allows selecting the type of the calculation device:
CPU – The calculation carries out on CPU only (OpenMP).
CUDA – The calculation carries out on CPU and GPU (NVIDIA CUDA). This option is available only for
PC with CUDA 2.0 compatible devices. You may use video cards with the support of CUDA (NVIDIA
GTX 4xx and more) technology or specific HPC modules (NVIDIA Tesla and etc.)
To use this feature it is necessary to select CUDA in “Target device” menu and to select CUDA-device (if
you have more than one).
Note: GPU usage allows to accelerate the calculation substantially.
Fig. 1.1.1. The options of the target calculation device
7
1.2. The polyhedron building options
These options allow setting different filters for determination of polyhedrons:
1. Determine bonds by (fig. 1.2.1.a):
1.1
Max. distance – All atoms which are located closer than the set value belong to the
polyhedron. The value of the maximal interatomic distance is entered in the field Max. distance.
1.2
Ion radii – All atoms which are located closer than the sum of the ion radii belong to the
polyhedron (Data | Radii…).
1.3
Wan-der-Waals radii – All atoms which are located closer than the sum of the Van-der-Waals
radii belong to the polyhedron (Data | Radii…).
1.4
Bonds – All atoms which are located closer than specific distances belong to the polyhedron.
The editing of the bond length collection carries out with the bond length editor (Data | Bonds…).
2. Polyhedron type (fig. 1.2.1.b):
2.1
Face – In this case the polyhedron described by the set of faces. Caution: This method hasn’t
been tested enough yet!
2.2
Vertices – The polyhedron is described by the set of vertices (default mode).
3. Center in (fig. 1.2.1.c):
3.1
Atom –The polyhedron center is set in the central atom.
3.2
Center of mass – In this case the center of the polyhedron is set in the center of mass of
polyhedron.
4. Min. items and Max. items – These controls set the maximum and minimum number of polyhedron
items (for example, the number of vertices).
5. Target atoms – The list of selected atoms. You can enter labels or types of atom (the separator is the
space symbol).
a)
b)
c)
Fig. 1.2.1. Options for the polyhedron determination
8
1.3. Main menu
1. File (see fig. 1.3.1.a)
1.1. New project – Start a new project.
1.2. Open project… – Open saved project.
1.3. Import structure… – Import structure information from CIF-file.
1.4. Save project as… – Save the project file.
1.5. Global settings… – The global settings of PseudoSymmetry package.
1.6. PS:> – Command line of PseudoSymmetry.
1.7. Exit – Exit program.
2. Structure (see fig. 1.3.1.b)
2.1. Rebuild geometry – This option starts the process of building new geometry of coordination
polyhedrons with selected options (See section 1.2).
2.2. Out graph – Print the structure graph in the output window.
3. Data (see fig. 1.31.c)
3.1. Bonds… – Run the bond collection editor.
3.2. Radii… – Run the atomic radii editor.
3.3. Atom Information… – Run atom information tool.
3.4. Polyhedra… – Run the standard polyhedron collection editor.
4. Analysis (see fig. 1.3.1.d)
4.1. Analysis of Polyhedra – Start the calculation of the degree of similarity of polyhedrons for
selected atoms.
4.2.Comparison of polyhedra in the current structure… – This option allows to compare polyhedrons
of the current structure. You should select one polyhedron as standard, others polyhedrons will
be original (target).
4.3.Comparison of optional polyhedra… – This option allows to compare some polyhedrons.
Original polyhedrons may be imported to polyhedrons collection from the current structure or
load from the external file. One polyhedron from this collection should be selected as a standard.
All polyhedrons from polyhedron collection compare with the selected standard polyhedron.
4.4. Manual comparative analysis of polyhedra… – This option allows to compare the original
polyhedrons with the standard collection using additional options. For example, you may exclude
some standard polyhedrons from analysis.
4.5.Settings – Settings window of Polyhedron (See section 1.4).
a)
b)
c)
d)
Fig. 1.3.1. Main menu of Polyhedron
9
1.4. Settings of Polyhedron
The settings of Polyhedron contain following options (fig. 1.4.1):
Refine cycles number – The maximum number of cycles in the refinement procedure.
Maximal bond length – The default value of the maximal interatomic distance.
Refine precision – The precision of the refinement.
Map size – This value sets the number of nodes of the 3D net. 3D net is used for searching of the optimal
polyhedron orientation. With the increase of this value the accuracy of calculations also increases (default is
60).
Delta for derivative – A value of delta in numerical derivatives.
Search method – The selection of the method of the polyhedron orientation search.
Direct method of permutation – The method based on permutation of vertices, and useful for small
coordination only (less than 10).
Fast method – The method based on the optimization of the special target function (default). It
requires high values of Map size.
Calculate the electron density for all data – This option allows to run the electron density calculation for all
studied polyhedrons. The electron density calculation runs automatically after optimization of the function in
(1).
Include central atom – This option includes the central atom in coordination polyhedron for the electron
density calculation.
Resolution – This option sets the maximal resolution of data in reciprocal space (angstroms).
Fig. 1.4.1. Settings window of Polyhedron
10
1.5. Selection of coordination polyhedrons
The structure navigator contains the list of atoms. Each atom has coordination information:
interatomic distances and valence angles (fig. 1.5.1). Select target coordination polyhedrons by checking
central atoms. All atoms are checked by default, you may exclude some atoms (uncheck).
Fig. 1.5.1. The structure navigator window
11
1.6. Results
Result navigator window has the next structure (fig. 1.6.1):
-
Project – Shows the short report table (fig. 1.6.2);
o Atom – Shows the full report for the target atom.
 Standard polyhedrons – Shows the report of the refinement for the selected
standard polyhedron.
Fig. 1.6.1. The typical view of the results tree
The short report table contains a detailed information for all polyhedrons (fig. 1.6.2): the central atom
label, the list of standard polyhedrons and
Φ(T , S ) values for them.
Fig. 1.6.2. The short report table
Result reports contain different tables of data. In this part only the most important data is described.
The refinement information table (fig. 1.6.3) contains next data:
Name – The label of polyhedron;
Symmetry – Symmetry group of the standard polyhedron;
Type – The type of polyhedron (see)
Status – The refinement flag, which describes the list of refined parameters;
α, β and γ – Euler angles for the original polyhedron orientation. These angles are refined and
corresponded to the optimal orientation of original polyhedron with respect to the standard polyhedron;
12
Scale X, Y and Z – Scale factors of the standard polyhedron;
Number of items – The number of items in the standard polyhedron;
Φ – The value of Φ (T , S ) .
If the electron density method was used, the table would present additional information.
α, β and γ – Euler angles fitted by the electron density method.
The average distance for the standard polyhedron is a refined parameter.
The resolution of the electron density is the minimum of the interplanar distance.
Also the result of calculation η[ρ T (r ), ρ S (r )] via (6) and other useful data are presented.
Fig. 1.6.3. The refinement information table
The comparison of items of the standard polyhedron and the original polyhedron in “Comparison of
items” table is presented (fig. 1.6.4). In this table the minimal angles between standard and original
polyhedrons are presented.
13
Fig. 1.6.4. The comparison table
Atomic clusters of original and standard polyhedron in the table “Atomic clusters for electron density
analysis” are presented (fig. 1.6.5). Both clusters are in the same lattice. The center of polyhedrons are the
origin (0, 0, 0). Distances between the polyhedron center and dedicated atoms are also presented in the table.
Fig. 1.6.5. Atomic clusters
14
2. Analysis of coordination polyhedrons
The step by step instruction for the analysis of coordination polyhedrons is presented.
I.
II.
III.
IV.
V.
VI.
VII.
Import atomic structure information. Load CIF-file via File | Import structure… (See section
1.3).
Implement filters. Set next options:
- Max. distance = 3.2
- Polyhedron type = Vertices
- Center in = Atom
- Min. Items = 4
- Max. Items = 5 (or more)
- Target device = CPU
- Target atoms = (Empty)
Rebuild structure. This operation updates information of the atomic coordination. Use Structure |
Rebuild geometry menu option or the toolbar button
.
Selection of necessary coordination polyhedrons. See section 1.5. Check the target atoms in the
structure navigator.
Run the calculation. Use Analysis | Analysis polyhedra menu option or the toolbar button
.
View results. See section 1.6.
The electron density method. Some polyhedrons could be analyzed by the electron density
method. Use Convolution of Electron Density option of the context menu at the result navigator.
15
3. Comparison of polyhedrons in the current structure
The Polyhedron allows to compare polyhedrons with the same coordination in the current structure.
One polyhedron is set to the standard polyhedron and others are set to target. In this case we compare target
polyhedrons with the standard polyhedron which belongs to current structure.
I. – IV same section 2. Select the list of polyhedrons with the same coordination.
V. Select the standard polyhedron. You may select the target atom in the atom list and press the button
Set standard atom or click on it twice(fig. 3.1).
VI. Select target polyhedrons. Select target atoms in the atom list and press the button Add target atoms.
VII. Run the calculation by Analysis button.
Fig. 3.1. The comparison window
16
4. Comparison of arbitrary polyhedrons
The Polyhedron allows to compare arbitrary polyhedrons. The list of polyhedrons can be formed in
different ways. You may import selected polyhedrons by the context menu option (fig. 4.1). Also you may
load polyhedrons from external files.
Form the polyhedron list by context menu options.
Select one polyhedron as standard.
Run the calculation by Analysis button.
Note: In this method the standard polyhedron Is compared with all polyhedrons from the list (including
standard).
Fig. 4.1. The comparison window
17