Discovering correlations in the interatomic distances of ... · Periodic trend: Proposed covalent...
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Periodic trend: Proposed covalent radii and threshold distances both have a rough periodic trend. Comparison with previous result: • Proposed covalent radii are almost same as those from Cordero and larger than ones from Pyykko [3] . • On average, each threshold distance is 0.63 Å shorter than the corresponding vdW radii from Alvarez [2] . Absolute error: Absolute error between sum of two proposed threshold distances and detected gap range (0 represents sum of two distances falls into the detected gap) 0 500 1000 1500 2000 2500 3000 0.725 1.475 2.225 2.975 3.725 4.475 5.225 5.975 6.725 NUMBER OF PAIRS DISTANCE (Å) Mn-O 0 1000 2000 3000 4000 5000 6000 7000 0.025 0.775 1.525 2.275 3.025 3.775 4.525 5.275 6.025 6.775 NUMBER OF PAIRS DISTANCE (Å) Na-O 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.025 0.775 1.525 2.275 3.025 3.775 4.525 5.275 6.025 6.775 NUMBER OF PAIRS DISTANCE (Å) Si-O 0 2000 4000 6000 8000 10000 12000 0.725 1.475 2.225 2.975 3.725 4.475 5.225 5.975 6.725 NUMBER OF PAIRS DISTANCE (Å) C-O Scheme of interatomic distance distribution for element pair E-X. Introduction Methods We compute the interatomic distance distribution as follows: Supercell creation: • Calculate the central point of the unit cell " # , the maximum distance $ %&' between " # , and all the atoms in the unit cell and the minimum distance $ %)* between " # and the faces of the unit cell. • Calculate the number * of repetition of the unit cell in each dimension as: • *=⎡ - . /- 012 3- 045 6- 045 ⎤∗2+1 • Where ⎡⎤ represents the ceiling function ⎡'⎤ which gives the smallest integer ≥' and $ = is the cutoff distance which is the maximum distance used in searching. Histogram Analysis Distance distribution for element pairs which contains more than 50 structures are analyzed. Covalent peak detection: The distance range for detection in each distance distribution is: $ > +$ ? +@≤B -CDEF ≤$ > +$ ? +@ 2 ⁄ Where $ > , $ ? are the covalent radii from Cordero [1] and constant @ is 0.5 Å. The peak at the largest distance would be chosen as the covalent peak due to the multiple bonds (C-O) and different oxidation states (Mn-O) as shown in Fig. A and Fig. B. Results Conclusion References 1. Beatriz Cordero et al. Dalton Transactions 21(2008), pp. 2832– 2838. 2. Santiago Alvarez. Dalton Transactions 42.24 (2013), pp. 8617– 8636. 3. Pekka Pyykko and Michiko Atsumi. Chemistry-A European Journal 15.1 (2009), pp. 186–197. Histograms of Na-O and Si-O; red line: detected threshold distance. Filtering out symmetry redundant pairs: For each symmetry-inequivalent atom in the central unit cell, we pick all the atoms located within the sphere of radii $ = centered in this atom, and compute all the distance from it. Two reasons for that: • Due to the periodicity of the crystal, other atoms which are not in the central unit cell are copies of those in the central unit cell. • Symmetry-inequivalent atoms in the central unit cell would provide unique distance information and avoid repeated pair distances from the atoms which have the same Wyckoff position. E-X distance Discovering correlations in the interatomic distances of inorganic crystals Cui Kejia, Nicolas Mounet, Giovanni Pizzi, Nicola Marzari THEOS, Materials science and engineering, École Polytechnique Fédérale de Lausanne (EPFL) Calculation of covalent radii and threshold distances: Least-square method is used to get the covalent radii and threshold distances for each element. The fitting equation is shown as following: I = J J(B >? −$ > −$ ? ) 6 D ?N> D >OP Where B >? is the detected covalent peak or threshold distance, $ > and $ ? are fitting covalent radii or threshold distance for element i and j. 0 0.5 1 1.5 2 2.5 3 3.5 4 1 7 13 19 24 29 34 40 46 51 57 63 68 73 78 83 Radius ( Å) Atomic number Proposed Covalent radii Proposed threshold distances VdW Alvarez[2] • Present a method to calculate the distance distribution in a database of crystal structures. • By using a least-square method, peak finding algorithm and detection of a minimum in the distance, two set of radii (covalent radii and threshold distances) can be obtained. • The sum of the two threshold distances can be used as a rough condition to separate bonding and non- bonding regions. We consider the crystal structures databases (ICSD and COD), and we compute and analyze the interatomic distance distribution for element pairs which have enough structures. In general, the distance distribution can be divided into three parts: covalent bond region, van der Waals region and gap between them. The covalent peak and the gap between bond and non-bond regions are detected based on a peak finding algorithm and detection of a minimum in the distribution. A set of covalent radii for 83 elements together with threshold distances for separating bond and non-bond regions are obtained from the statistical analysis using the least-square method. These two sets of radii both show periodic trends. The proposed radii are compared with values from the literature. Our results can be used to analyze crystal structures to exfoliate two-dimensional materials from theoretical and experimental structure databases. 66% 15% 10% 8% 1% 0 ≤ 0.1 Å ≤ 0.2 Å ≤ 0.4 Å > 0.4 Å Method to build a supercell large enough to contain all interatomic distances up to a value $ = . Absolute error distribution for proposed threshold distance between detected vdW gap. Van der Waals gap detection: Gap existence: histogram is normalized with the total area of bins. Below a height of 0.01, the histogram is treated as no atom pairs are found within that bin. Threshold distance: defined as the distance where the number of atom pairs is least as shown in Fig. C and Fig. D. Histograms of C-O and Mn-O; red line: detected covalent length. 0 0.5 1 1.5 2 2.5 3 1 7 13 19 24 29 34 40 45 50 56 62 67 72 77 82 94 Radius ( Å) Atomic number Proposed covalent radii Covalent radii Cordero Covalent radii Pyykko Covalent radii and threshold distances in this work, vdW radii from Alvarez. Covalent radii in this work, from Cordero and Pyykko.
Transcript of Discovering correlations in the interatomic distances of ... · Periodic trend: Proposed covalent...
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Periodic trend:
Proposed covalent radii and threshold distances both have a rough periodic trend.
Comparison with previous result:
• Proposed covalent radii are almost same as those from Cordero and larger than ones from Pyykko[3].
• On average, each threshold distance is 0.63 Å shorter than the corresponding vdW radii from Alvarez[2].
Absolute error:
Absolute error between sum of two proposed threshold distances and detected gap range (0 represents sum of two distances falls into the detected gap)
0
500
1000
1500
2000
2500
3000
0.725
1.475
2.225
2.975
3.725
4.475
5.225
5.975
6.725
NU
MBE
R O
F PA
IRS
DISTANCE (Å)
Mn-O
0
1000
2000
3000
4000
5000
6000
7000
0.025
0.775
1.525
2.275
3.025
3.775
4.525
5.275
6.025
6.775
NU
MBE
R O
F PA
IRS
DISTANCE (Å)
Na-O
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
0.025
0.775
1.525
2.275
3.025
3.775
4.525
5.275
6.025
6.775
NU
MBE
R O
F PA
IRS
DISTANCE (Å)
Si-O
0
2000
4000
6000
8000
10000
12000
0.725
1.475
2.225
2.975
3.725
4.475
5.225
5.975
6.725
NU
MBE
R O
F PA
IRS
DISTANCE (Å)
C-O
Scheme of interatomic distance distribution for element pair E-X.
Introduction
MethodsWe compute the interatomic distance distribution as follows:
Supercell creation:
• Calculate the central point of the unit cell "#, the
maximum distance $%&'
between "#, and all the atoms
in the unit cell and the minimum distance $%)*
between "#
and the faces of the unit cell.
• Calculate the number * of repetition of the unit cell in each dimension as:
• * = ⎡ -./-0123-0456-
045
⎤ ∗ 2 + 1
• Where ⎡⎤ represents the ceiling function ⎡'⎤which gives the smallest integer ≥ 'and $
=is the cutoff distance which is the
maximum distance used in searching.
Histogram Analysis
Figure A: neque dignissim, and in aliquet nisl et umis.
Distance distribution for element pairs which contains more than 50 structures are analyzed.
Covalent peak detection:
The distance range for detection in each distance distribution is:
$>+ $
?+ @ ≤ B
-CDEF≤ $
>+ $
?+ @ 2⁄
Where $>, $?
are the covalent radii from Cordero[1] and constant @ is 0.5 Å.
The peak at the largest distance would be chosen as the covalent peak due to the multiple bonds (C-O) and different oxidation states (Mn-O) as shown in Fig. A and Fig. B.
Results
Conclusion
References1. Beatriz Cordero et al. Dalton Transactions 21(2008), pp. 2832–2838.2. Santiago Alvarez. Dalton Transactions 42.24 (2013), pp. 8617–8636.3. Pekka Pyykko and Michiko Atsumi. Chemistry-A European Journal 15.1 (2009), pp. 186–197.
Histograms of Na-O and Si-O; red line: detected threshold distance.
Filtering out symmetry redundant pairs:For each symmetry-inequivalent atom in the central unit cell, we pick all the atoms located within the sphere of radii $
=
centered in this atom, and compute all the distance from it. Two reasons for that:
• Due to the periodicity of the crystal, other atoms which are not in the central unit cell are copies of those in the central unit cell.
• Symmetry-inequivalent atoms in the central unit cell would provide unique distance information and avoid repeated pair distances from the atoms which have the same Wyckoff position.
E-X distance
Discovering correlations in the interatomic distances of inorganic crystalsCui Kejia, Nicolas Mounet, Giovanni Pizzi, Nicola Marzari
THEOS, Materials science and engineering, École Polytechnique Fédérale de Lausanne (EPFL)
Calculation of covalent radii and threshold distances:
Least-square method is used to get the covalent radii and threshold distances for each element.
The fitting equation is shown as following:
I =JJ(B>?−$
>− $
?)6
D
?N>
D
>OP
Where B>?
is the detected covalent peak or threshold distance, $
>and $
?are fitting covalent radii or threshold
distance for element i and j.
0
0.5
1
1.5
2
2.5
3
3.5
4
1 7 13 19 24 29 34 40 46 51 57 63 68 73 78 83
Rad
ius
(Å)
Atomic numberProposed Covalent radii Proposed threshold distances VdW Alvarez[2]
• Present a method to calculate the distance distribution in a database of crystal structures.
• By using a least-square method, peak finding algorithm and detection of a minimum in the distance, two set of radii (covalent radii and threshold distances) can be obtained.
• The sum of the two threshold distances can be used as a rough condition to separate bonding and non-bonding regions.
We consider the crystal structures databases (ICSD and COD), and we compute and analyze the interatomic distance distribution for element pairs which have enough structures. In general, the distance distribution can be divided into three parts: covalent bond region, van der Waals region and gap between them.
The covalent peak and the gap between bond and non-bond regions are detected based on a peak finding algorithm and detection of a minimum in the distribution. A set of covalent radii for 83 elements together with threshold distances for separating bond and non-bond regions are obtained from the statistical analysis using the least-square method.
These two sets of radii both show periodic trends. The proposed radii are compared with values from the literature. Our results can be used to analyze crystal structures to exfoliate two-dimensional materials from theoretical and experimental structure databases.
66%
15%
10%8%1%
0
≤ 0.1 Å
≤ 0.2 Å
≤ 0.4 Å
> 0.4 Å
Method to build a supercell large enough to contain all interatomic distances up to a value $
=.
Absolute error distribution for proposed threshold distance between detected vdW gap.
Van der Waals gap detection:
Gap existence: histogram is normalized with the total area of bins. Below a height of 0.01, the histogram is treated as no atom pairs are found within that bin.
Threshold distance: defined as the distance where the number of atom pairs is least as shown in Fig. C and Fig. D.
Histograms of C-O and Mn-O; red line: detected covalent length.
0
0.5
1
1.5
2
2.5
3
1 7 13 19 24 29 34 40 45 50 56 62 67 72 77 82 94
Rad
ius
(Å)
Atomic numberProposed covalent radii Covalent radii Cordero Covalent radii Pyykko
Covalent radii and threshold distances in this work, vdW radii from Alvarez.
Covalent radii in this work, from Cordero and Pyykko.
