MASSACHUSETT 'rITI ITF
OF TECHNOLULG'Y
JUN 0 8 2015
2015 Chen
All rights reserved

Signature of A uthor!...... ......,.. ................ .............................................
by....................................
A ccepted by .....................................
... , ..... ... ..... ............. ...
Beac
'58
1
by
Local coordination and topology of ions determine several important properties of materials, including electronic structure, migration barrier, and diffusivity. In this thesis, we employ the Materials
Project Database to investigate the coordination preferences of cations and topology of coordination polyhedra in oxides. We calculate the coordination environment preferences of several common cations in oxides, identifying lithium, sodium, calcium, and magnesium ion's preferred coordination numbers are 4fold/6fold, 6fold, 6fold, and 6/4fold coordination respectively. We also develop a method to quantify the connectivity between two polyhedra and determine whether they are pointsharing, edgesharing, or facesharing. We find that 4fold coordinated lithium polyhedra mainly pointshare while the 6fold coordinated lithium polyhedra connectivites are facesharing. We then build a tool to identify and insert
"empty polyhedra" (i.e. coordination polyhedra which are bounded by ions but contain no central ion) which can help to provide a better descriptor of the structure topology. We also find that most connections with lithium polyhedra are with empty polyhedra and that in a connected set of two lithium polyhedra and one empty polyhedron, the coordination polyhedra tend to be either 646 or 464 with the empty polyhedron in the center. Finally, we utilize the database to evaluate Pauling's first and second rules, which are guidelines for current understanding of coordination and topology, and observe that the rules are generally accurate only within a 30% error margin.
Thesis advisor: Gerbrand Ceder
Title: R.P. Simmons Professor Materials Science and Engineering
2
A bstract ......................................................................................................................................................
Table of Contents
1
.......................................................................................................................................3
List of Figures............................................................................................................................................4
List of Tables.............................................................................................................................................5
1. Introduction ...........................................................................................................................................
1.1 Coordination Preferences..........................................................................................................................................6
1.2 Topology in M aterials .................................................................................................................................................
1.3 Datam ining and Databased Analyses................................................................................................................9
6
8
2. Coordination Preference A nalysis .............................................................................................. 11
2.1 Coordination Num ber Definitions ......................................................................................................................
2.2 Coordination Num ber Frequency Distributions.....................................................................................
11
13
2.3 Coordination Num ber as a Function of Energy Above Convex Hull ................................................ 17
2.4 Form ation Energy......................................................................................................................................................19
2.5 Discussion ..................................................................................................................................................................... 21
3. Topology A nalysis and Tool Developm ent.................................................................................
3.1 Topology Tool M ethodology .................................................................................................................................
3.2 Lithium Topology as a Case Study ......................................................................................................................
3.3 Discussion .....................................................................................................................................................................
4. Em pty Polyhedra A nalysis and Tool Developm ent................................................................... 30
4.1 Em pty Polyhedra Tool M ethodology.................................................................................................................
4.2 Em pty Polyhedra in Li Oxides as a Case study........................................................................................
4.3 Discussion .....................................................................................................................................................................
30
32
35
24
24
26
28
5. Pauling Rules......................................................................................................................................39
5.1 Pauling's First Rule....................................................................................................................................................
5.2 Pauling's Second Rule ..............................................................................................................................................
39
42
6. Conclusion..........................................................................................................................................45
7. Bibliography.......................................................................................................................................47
8. Appendix.............................................................................................................................................47
8.1 Appendix A: Uncom m on Coordination Num ber and Outliers........................................................... 49
3
Figure 1: Coordination numbers of univalent, divalent, and trivalent cations.......................... 6
Figure 2: LiO bond length frequency distribution.....................................................................7
Figure 3: Pauling 's third rule......................................................................................................
Figure 5: Frequency distributions of Li, Na, Ca, and Mg coordination numbers....................15
1
Figure 6: Comparison of O'Keeffe and effective coordination numbers...................................16
Figure 7: Energy above the convex hull...................................................................................
Figure 8: Coordination number and energy above hull plots....................................................18
Figure 9: Formation energy vs. coordination number plots.................................................... 20
Figure 10: Coordination numbers and formation energy differences between lithiated and
delithiated states. ..........................................................................................................................
Figure 11: Polyhedra connectivity types...................................................................................25
Figure 12: Counting of shared anions.......................................................................................25
Figure 13: Overall Li to Li Polyhedra Connectivity Distribution.............................................27
Figure 14: Distributions of Li to Li polyhedra connectivity by coordination number.............
Figure 15: Processing steps for adding empty polyhedra to a structure.................................
Figure 16: Frequency distributions of Li to all polyhedra connectivities.................................32
Figure 17: Frequency distributions of the connectivities in Liempty polyhedraLi paths...........34
Figure 18: Frequency distribution of the coordination numbers in Liempty polyhedraLi
p a th s...............................................................................................................................................3 4
Figure 19: Structures with common Liempty polyhedraLi coordination numbers.................36
Figure 20: Pauling's second rule error distribution................................................................ 44
Figure 21: Exampled of 5fold coordinated Li......................................................................... 49
Figure 23: Examples of unstable 12coordinated Li................................................................
Figure 24: Examples of unstable 8coordinated Li..................................................................
4
Table 1: Rules for
topology method...................................................................................................................24
Table 2: Pauling's Radius Ratio rule for coordination numbers. ....................................................
Table 3: Evaluation of Pauling's First Rule's Prediction Ability ..................................................... 41
5
Local coordination and topology are two properties of atoms in materials that determine many other properties, such as electronic conduction and response to magnetic fields. Another property that local coordination and topology may help to predict is ion diffusion. Specifically, an ion's preference in coordination and topological environment may indicate the stability of that ion in different sites of a material. In this thesis, we aim to elucidate the local coordination and topological preferences of common cations with the end goal of developing tools for materials design.
1.1 Coordination Preferences
The current understanding of coordination in ionic compounds is primarily guided by the wellknown Pauling's rules.' Pauling's first and second rules present a framework for understanding how ions coordinate. Pauling's first rule provides a guideline based on the ratio between the radii of the cation (re) and the anion (ra). In general, larger cations will have higher coordination numbers. Pauling's second rule ensures local electronegativity by considering the charge of coordinating ions.
Later studies looked more closely at coordination numbers of specific cations.
Brown analyzed the relative frequencies of coordination numbers for different cations, showing that cations in inorganic structures have "preferences" toward certain coordination numbers.
2
Using ICSD data
2345678911213 2345678912 23456789Ii12
Li 891  
K oooeeeev? cc,
eeeo 0 Ga OD(
Rb 6ee'ee
Cs ooooeeeeeo
Be<D
Mg
Sr e0eo
.eeeo
1
00 in
B OD
Q o
* ee.4eO TBa 0 o
It wobeee e
Cu Q
"
Sn
0911wo* AS
Pb *0oeese 'o
Sb
Bi
O
Fe &a
CO Odd
*
N
Cu Qeo
Zn .Ye
x
Mn 0 x
Fe 0 x y
LO
OO
9 o available at the time, Brown conducted an analysis of over
envionmntsof 00 iffeentcatonsand environments of 100 different cations and compared coordination numbers to those
O96s
Figure 1: Coordination numbers of univalent, divalent, and trivalent cations.
The area of the circle corresponds to the number of samples observed (normalized to the same total for each cation). The coordination number predicted by ionic radius ratio is represented by an x. Figure from Brown, 1988.2
6
predicted by ionic radius ratios. One result of Brown's study was that some cations tended to have strong preferences for certain coordination environments. For example, Mg ions strongly prefer the 6 coordination environment, while Ca ions prefer 6 8 coordinations as seen in Figure 1. There were also, however, ions which did not show preference for a single coordination environment but instead tended to appear in a range of coordination environments. He also discovered that the observed coordination numbers tended to be smaller than or equal to the coordination numbers predicted by Pauling's first rule.
Smaller coordination numbers were associated with stronger bonds, and the Lewisbase strength of the anion strongly affected the coordination number.
While some studies attempted to find trends in coordination number, a few looked more closely at specific cations. In particular, Wenger studied Li's coordination and bonding to oxygen, comparing bond lengths and strengths for different coordinations of Li using a data set of 134 Li polyhedrons from 94 compounds, as shown in Figure 2.3 Wenger found that Li's coordinations ranged from 3fold to 8fold, and 4fold coordination was the most preferred. Wenger also discovered that octahedral (6fold) coordination sometimes has uneven bonds, instead bonding in a 4+2 substructure rather than a symmetric
26
1IS
U coordination:
E 4
0 5
0t 6
10
6
1.8
2
1.9
2.0 2.1 2.2
2.3 2.38
Figure 2: LiO bond length frequency distribution.
The frequency distribution of bond lengths between Li and 0, also grouped by Li coordination, shows a peak around 2.0
angstroms for 4coordinated Li. 6coordinated Li has bond lengths ranging from 2.1 angstroms to 2.3 angstroms. Based on this, we expect to find most 0 coordinated to Li within a 2.4 angstrom radius. Figure from Wenger, 1991.3
7
1.0 ~~,
0.38
Figure 3: Pauling's third rule.
An illustration of Pauling's Third Rule shows that two polyhedra with higher connectivity between them have closer cations.
This results in higher connectivity being less stable due to the electrostatic repulsive forces between central cations. Figure from Nelson, 2014.4
octahedral substructure. Brown and Wenger demonstrate that cations often have preferred coordination numbers, and we look to verify these coordination number preferences using largerscale data.
While Pauling's first and second rules have provided a framework for understanding how ions coordinate, his third and fourth rules describe features of topology that are generally preferred in ionic crystals. Pauling's third rule, demonstrated in Figure 3, uses electrostatic interactions to explain why pointsharing connections are more stable than edge or facesharing connections. Pauling's fourth rule states, again using an argument based on electrostatic interactions, that cations with higher charge and lower coordination number tend to form fewer connections with other polyhedra. Pauling's third and fourth rules thus provide an explanation for the trends observed in the topology of structures.
Other work in topology of structures has centered around understanding networks in structures. In one of the earlier works describing topology in terms of nets, Wells investigates the properties of periodic graphs using planar tessellations. Blatov et al analyzes nets through the enumeration of periodic tiling with application to zeolite nets.
6
Carlucci et al analyzes the types of entanglement that appear in materials such as organic and metalorganic networks.
7 More recently, Blatov et al have proposed the use of a computational software package that helps to visualize and analyze topology and networks. While these studies have focused on the overall connectivity and network of structures, here we focus not only on the network but also on the local topology and how polyhedra in the network are connected.
8
1.3 Datamining and Databased Analyses
In the studies of Brown and Wenger, only 100200 Li samples were considered. Recent efforts to expand current materials databases, such as the Materials Project, 9 have increased the potential to use data mining and data based analysis to make predictions about materials. One major application of data mining has been the prediction of crystal structures. Fischer combined data mining and quantum mechanics to predict binary crystal structures as well as the probabilities they will form.1 Hautier expanded on
Fischer's work by predicting ternary crystal structures" as well as structures that could be formed from ionic substitutions.
Based on Hautier's work predicting ionic substitutions, Yang looked to expand on the ionic substitution similarity function Hautier et al developed by proposing methods for assessing the similarity of compositions, substructures, and crystal structures based on datamined probabilities.
3
'
14
One specific application of her substructural similarity function was the prediction of Li intercalation sites in oxides.
To predict Li intercalation sites, Li ions were removed from structures, and all possible Li sites were considered with respect to the Voronoi polyhedrons in the delithiated oxides. From the possible Li sites, the most likely Li sites were found by comparing the substructure similarity between the site and all known substructures in a training set. In our study, we develop tools that allow topology to be a feature added to Yang's structural similarity function.
This thesis aims to obtain a better understanding of coordination and topology with the goal of helping to develop tools for material design of oxides. A better understanding of coordination preferences could help us design a better descriptor for diffusion. Because diffusion involves the migration of a species with a changing local coordination environment, the local environment should affect the migration barrier.'
5
If we consider that how strongly an ion prefers its coordination environments is related to its energy in that site, then it is possible to trace out possible diffusion paths knowing the coordination environments of all potential cation sites within a structure.
9
A better understanding of topology would help current efforts to design a function for measuring the similarity between structures. Specifically, a tool that could describe topology of a structure could be used as a descriptor by which structures are compared. By adding topology as a criterion for a structural similarity function, we allow the function to identify materials with similar topologydependent properties.
Preferred coordination environments for several common cations will be found using data analysis of the Materials Project Database. We then follow with the development of tools to obtain the connectivity of coordination polyhedra and the positions of empty polyhedra, which are subsequently used to find trends in the connectivity between cation polyhedra. Finally, we investigate the utility of
Pauling's first and second rules using data from the Materials Project Database.
10
The environment of a site has a strong relationship with the stability of any cation in that site.
Specifically, the interactions between the cation and its nearby neighbors determine how strongly the cation is tied to the site. We define coordination environment of a cation as the set of nearby anions bonded, or coordinated, directly to the cation. The coordination number of that cation is then the number of nearby anions coordinated to the cation. These anions form a polyhedron around the central site, with the shape of the polyhedra depending on the number of anions coordinated to the central site.
We employ the Materials Project Database to sample the coordination preferences of several cations in order to identify their preferred coordination. To do this, we first require a mathematical definition of coordination number to process the structutal data from the database. We then measure coordination preferences by the frequency of the appearance of coordination numbers based on our definition of coordination number. We also explore the use of other measures of coordination preference, including the energy above the convex hull and the formation energy. Finally, we discuss the potential materials design applications, specifically for diffusion, of being able to calculate coordination preferences as well as evaluate ours measures of coordination preference.
While some structures have clear coordination polyhedra formed around cation sites, in others it is often unclear how atoms are coordinated to each other. We use two definitions of coordination number to measure the coordination environment of any structure.
(b e
(a)
2.1. O'Keeffe Coordination Number
In his 1979 paper, O'Keeffe offers a coordination number
Figure 4: Voronoi polyhedra.
a) The Voronoi polyhedron for the bodycentered cubic array. b) The Voronoi polyhedron for the simple cubic array after a trigonal distortion. c) The Voronoi polyhedron for the facecentered cubic array after a small tetragonal distortion.
Figure from O'Keeffe 1979.1s
11
based on weighting the faces of the Voronoi polyhedron, examples shown in Figure 4, around the central ion.' 6 The planes that bisect lines drawn from the central ion to neighboring ions create the faces of the
Voronoi polyhedron. Thus each coordinating anion contributes a face to the Voronoi polyhedron around the central cation. This method weighs the coordination number in proportion to the solid angle Qy subtended by that faces at the center. The neighbor subtending the largest angle is given a weight of 1, and other coordinating anions are given weights of Qx/Qmax. The coordination number is then the sum of these weights:
CNO'Keeffe = ,(Equation
DmaxI
1)
2.1.2 Effective Coordination Number
Hoppe proposes the use of the effective coordination number (CNeffective).'
7 The effective coordination number is calculated by weighting the nearby anions based on the bond lengths (li) as compared to the bond length (min) of the closest anion. This method results in higher coordination numbers when anions are equidistant from the central ion and lower coordination numbers when anions have very different distances from the central ion.'
8
Effective coordination number is calculated by summing the weights:
CNEffective = i exp

6 , (Equation 2) where lav is the weighted average bond length is equal to
= i l
1 exp (1(
lav =mi
YEj exp (1
(min
(Equation 3)
However, to use this method, a radius must be given to determine the number of ions included in the calculation of the coordination number. In doing this, we make assumptions about the cation to anion bond lengths. To mitigate this difficulty, several radiuses are tested to approximate which radius gives the most accurate coordination number. Another possible way to address this difficulty is by varying the radius used based on a property like the distance to the nearest anion.
In order to understand the preferences of cations, we look at the stability of the cations in different coordination environments. The main metric we use to measure stability is the frequency distribution of coordination numbers over all calculated coordination numbers for a single cation.
Through this distribution, we find the coordination numbers that occur most frequently.
2.2.1 Data Collection and Analysis Methods
Data sets of 1,000 structures and their energy above the convex hull data were obtained from the
Matgen version of the Materials Project Database. The Materials Project Database includes crystals from the ICSD database and from the Materials Project. This data was filtered using several criteria similar to the filtering methods described by Yang.' 4
In Yang's thesis, only compounds satisfying the following criteria were included the data set:
(1) Compounds must be oxides; consider a compound as an oxide if it contains more than 20% oxygen.
(2) Compounds must not be peroxides or superoxides; consider a compound nonperoxide and nonsuperoxide if 00 bond lengths are less than 1.5 angstroms.
(3) Compounds must not be marked high temperature or high pressure.
(4) Compounds must not have a mismatch between the reported composition and the ions given in the structure.
(5) Compounds must not contain hydrogen.
13
(6) Compounds must be charge balanced; consider a compound chargebalanced if the sum of the charges of all species is less than 0.001 e.
From the data within the Materials Project Database, we filtered using several criteria based on those given in Yang's thesis.'
4 Specifically, we used criteria (1), (2), (5), and (6). We exclude criteria (3) and (4) from our search because the Materials Project Database has already been cleaned of these data.
The coordination numbers were then calculated using both the O'Keeffe coordination number and effective coordination number formulae. Frequency distributions of O'Keeffe coordination number and effective coordination number were graphed as separate histograms with bin size of 0.2 coordination number units. The data set was further filtered by the energy above the convex hull value of the structure to obtain data sets containing more stable materials. We use 0.1 eV/atom as the cutoff for a subset of stable compounds.
2.2.2 Coordination Number Distributions of Several Common Cations
We look at the frequency distributions of several common cations, namely Li, Na, Ca, and Mg.
Based on the frequency distributions in Figure 5, we observe relative preferences of coordination numbers for each cation. For Li, the frequency distribution peaks at 4fold coordination and is also relatively high in the 4fold to 6fold range using the O'Keeffe coordination number. With the effective coordination number, a peak at 6fold coordination rather than a plateau from 4 to 6 coordination appears. This is in agreement with Wenger's results, which show that 4fold coordination is the most preferred.
3
For the Na distribution, we observe peaks at 4fold and 6fold coordination when using effective coordination number but a much broader peak at 6fold coordination that decreases in gradually intensity farther from
6fold coordination when using the O'Keeffe coordination number. The Ca distribution shows a greater preference for 6fold coordination, with a strong peak in the effective coordination number distribution and a broad peak in the O'Keeffe coordination number distribution. The broad peak in the O'Keeffe
14
e
10 o
Keefe Coordination Number Frequency
Energy Above Hull Cutoff 0.1 I
E above hull <
M E above hull infinite
0. 1
Effective Coordination Number Frequency
Energy Cutoff 0.1
M E above hull < infinite
M
E Above hull <
0.1
a.
400
200
100
1200
1000
800
600
400
200
00
45s0
40D
33O
2 4 6 10
0 Keefe Coordination Number
12
0 Keefe Coordination Number Frequency
Enermw Above Hull CutOff 0.1
M E above hull < Infinite
M E above hull
<
0.1
14
0
2
1200
4 6 a 10 12
Effective Coordination Number
Effective Coordination Number Frequency
Eneray Cutoff 
0.1
M E above hull < infinite
M
E Above hull < 0.1
14
C.
1000
800
1250
300
250
400
200 so
500
400 f
00
200
1
6
8 o Keefe Coordination Number
10
0 Ke tfe Coordination Number Frequency
Energy Above Hull CutOff 0.1
12
M
E above hull <
Infinite
I E above hull<01
14 f.
8go
400
300
200
100
0
700
600
500
2 4 6
8 10
Effective Coordination Number
12
Effective Coordination Number Frequency ne'ov
Cutoff  0.1
M
E above hull
< infinite
M E Above hull < 0.1
14
1200
1000
I
2 4 6 8 10
Effective Coordination Number
12
Effective Coordination Number Frequency
Enerov Cutoff 
0.1
1
E above hull < Infinite
E
Above hull < 0.1
14 g.
400
350
300
250
0 2
4
0 6eefe
6
8
Coarditsion Number
10 12 o
Keefe Coordination Number Frequency
Energy Above Hull CutOff 0.1
M
E above hull < infinite
E above hu
<
0. 1
14
800
Mg
600
150
100 so
0
400
200
2 4 8
0 Cedfe Coordination Number
2 4
6 8
10
Effective Coordination Number
12 14
Figure 5: Frequency distributions of Li, Na, Ca, and Mg coordination numbers.
Histograms showing frequency of appearance of coordination numbers, binned with bin size 0.2. Blue data includes all data in the data set. Green data includes only data having energy above hull less than 0.1. Graphs on the left (a, c, e, g) use O'Keeffe coordination number to calculate coordination number while graphs on the right (b, d, f, h) use effective coordination number. The first row (a, b) are frequency distributions for Li coordination numbers, the second row (c, d) are frequency distributions for Na coordination numbers, the third row (e, f) for Ca coordination numbers, and the fourth row (g, h) for Mg coordination numbers. Each set consists of 1,000 oxides containing the specified cation.
15
distribution gradually decreasing in intensity away from 6fold coordination with the ends of the peak ranging from 4fold to 6fold coordination, similar to the Na O'Keeffe distribution. The Mg distribution prefers both 4fold and 6fold coordination, showing strong peaks for those coordination numbers when using effective coordination number. It also shows these peaks when O'Keeffe coordination number is used, but these peaks decrease in intensity asymmetrically farther from top of the peaks at 4 and 6fold coordination. O'Keeffe coordination numbers below 4fold and above 6fold coordination have very little intensity while the region between 4 and 6fold coordination has a valley at 5fold coordination. These results match those of Brown as well as common knowledge, which expects Li to have 4 and 6fold coordination, Na to have 6fold coordination, Ca to have 6 and 8fold coordination, and Mg to have 6 fold coordination.
2
A brief exploration of some of the substructures with uncommon coordination numbers is given in Appendix A.
In the histograms in Figure 5, we note the preference of more plateaus in O'Keeffe coordination numbersbased frequency
14
12
.
Comparison of O'Keefe and Effective
Coordination Number
..
distributions and more distinct peaks in effective coordination numberbased frequency distributions. Distributions using effective z
C
8
E
561
0 6 u
O coordination number also tend to extend to very low coordination numbers (below 3fold
2
0.
coordination). This is likely because distances
0
S 2 4 6 8 10
O'Keefe Coordination Number
12 between the cation and coordinating anions are not
Figure 6: Comparison of O'Keeffe and effective evenly weighted, causing coordinating anions that are coordination numbers.
For each Li oxide in the sample of 1000 Li oxides, the farther away to be very lightly weighted. Based on this, the peaks around coordination 2 and 3 may be from distorted coordination 4 and coordination 6 coordination numbers of each Li site were averaged separately for both methods to obtain a single O'Keeffe coordination number and a single effective coordination number for each oxide. The y = x line is given as a reference for the case where O'Keeffe and effective coordination number calculate the same coordination number.
14
16
polyhedron (for example, in a coordination 4 polyhedron, 2 anions are closer to the central ion than the other 2 anions, resulting in a effective coordination number close to 2). This is supported by Figure 6, which plots the O'Keeffe coordination number versus the effective coordination number of each substructure. Most points fall below the y = x line, implying that the O'Keeffe coordination number of a substructure is almost always greater than or equal to the effective coordination number, and the effective coordination number is much more likely to underestimate the actual coordination number. Based on
Figure 5, it appears that the O'Keeffe method is better at classifying coordination number outside of the
4fold to 6fold coordination number range. However, it does not differentiate between 4fold and 6fold coordination as well as the effective coordination number does.
2.3 Coordination Number as a Function of Energy Above Convex Hull
In addition to frequency, we also explore the use of energy above the convex hull as a measure of site stability. Specifically, when a cation is in its preferred site, it should be in a low energy state. This should be reflected in the energy above the convex hull, which is a measure of the stability of a structure relative to the most stable phase of the material.
2.3.1 Data Collection and Analysis Methods formation energy
Energy above convex hull is calculated as the formation energy difference between a compound and the convex hull trace.'
9
In Figure 7, phase y is an unstable phase with the convex hull traced out by the green line.
ao
A2B AB AB2 B fraction
Phases A, B, a,, and P2 are all stable phases. The energy above convex hull for phase y is calculated by the difference between the formation energy of y and the
Figure 7: Energy above the convex hull.
is from its most stable form. The green line shows the stable phases of the material. The arrow from the unstable phase, y, to the convex hull is the eneir above hull. Figure from
Hautier, 2004.' formation energy of stable phase on the convex hull with the same composition. Materials with zero energy above hull are in their most stable form. On the other hand, materials with positive energy above
17
hull are unstable because a more stable phase exists on the convex hull. Thus, energy above hull is a metric that measures the stability of compounds.
Again, we obtained data sets consisting of 1,000 structures and their energy above the convex hull data from the Materials Project Database as in Section 2.2.1. The coordination numbers were then calculated using both the O'Keeffe coordination number and effective coordination number formulae. We then bin with a bin size of 0.5 coordination number units and average the lowest 1 0'hpercentile of data.
This attempts to remove data from structures that are unstable due to factors other than the specific cation site.
2.3.2 Coordination Number and Energy Above Hull Results
We first look at the relationship between coordination number and energy above hull for
Licontaining oxides. In Figure 8, the trends between coordination number and energy above hull are much less clear than those from the frequency distributions. Li is expected to have preferred coordination numbers of 4 and 6fold coordination, and these specific coordination numbers indeed appear to have lower energy above convex hull. However, the data outside the 4 and 6
0.6
Evaluation of OKeefe Coordination Number
Using 10 Percentile Values
* 10th percentile
.
median
0.6
Evaluation of Effective Coordination Number
Using 10 Percentile Values

*
10th percentile median
IV
0.4
0.3
00.4
0.3
< 0.2
< 0.2
0.1 0.
4.0
2 4 6
8 10
OKeefe Coordination Number
2
14 10.
2 4 6
10
Effective Coordination Number
2
Figure 8: Coordination number and energy above hull plots.
O'Keeffe coordination number (left) and effective coordination number (right) vs. the 10th percentile energy above hull value in eV/atom. The 10th percentile was chosen to remove unstable oxides since that instability likely came from sources other than Li sites. The 50th percentile (median) is given in blue as a reference.
Data was taken from a sample of 1,000 Li oxides.
14
18
fold coordination number region are statistically unreliable as the bins outside this area contain very few data points. In particular, bins above 8fold coordination generally had less than 10 data points in their bins. Because of this and reasons mentioned further in the discussion, we believe that energy above hull is not an adequate measure of site stability for identifying coordination preferences.
2.4 Formation Energy
We also explored formation energy as another metric of site stability. Similar to the energy above convex hull, formation energy should be low when a cation is in a site with its preferred coordination number. One important distinction of the formation energy is that it is calculated relative to an unchanging reference, which is the energy of the elements in their elemental forms.
2.4.1 Data Collection and Analysis Methods
Formation energy, also called heat or enthalpy of formation, is also used to measure stability of a system. Formation energy is calculated as the change in energy when one mole of a substance is formed from its pure elements under standard state (standard temperature and pressure). The more negative formation energies indicate more stable compound because a system with lower formation energy requires more energy to be reverted to its pure elements. We then expect materials with more negative formation energy to be more stable and to be more likely to have preferred coordination numbers.
We obtained data sets consisting of 1,000 structures and their formation energy per atom data from the Materials Project Database as in Section 2.2.1. The coordination numbers were then calculated using both the O'Keeffe coordination number and effective coordination number formulae. We then bin with a bin size of 0.2 coordination number units and average the lowest
1 0 th percentile of data. This attempts to remove data from structures that are unstable due to factors other than the specific cation site.
Additionally, we perform an analysis in which we isolate the formation energy of the cation by by subtracting the formation energy of the delithiated structure from that of the lithiated structure of a
19
material. For this, we obtained 146 stable materials and matched the space groups of the lithiated and delithiated states of the material. This decreases the sample size we are able to work with but allows us to isolate the energy from the site from energy due to other factors in the structure.
We start by looking at the plots for Licontaining oxides of coordination number and formation energy per atom. In Figures 9, showing the plots for Licontaining oxides, a trend is less evident than in the plots of energy above the convex hull. In particular, there appears to be no dip in formation energy correctly identifying Li's preferences as 4fold coordination. In addition, the preferred coordination number (the coordination number with the lowest formation energy per atom) seems to be 2fold coordination, which is incorrect One explanation is that many of the 4fold coordinated materials are actually misclassified as 2fold coordinated based on the method used to calculate effective coordination number. Another is that these 2fold coordinated materials exist in 2fold formation but are unstable.
However, overall, the plots indicate that formation energy by itself is also not an adequate indicator of the coordination preferences of cations in oxides.
Evaluation of OKeefe Coordination Number Using
10 Formation Energy Percentile Values
10hpe l
S10th percentile
E
*median
Evaluation of Effective Coordination Number Using
1
120h
10 Formation Energy Percentile Values
*10th
.ceti
percentile median
E2.0
3.0
0I
00
OC.,
_2.
0
2.0
0 2
4 6 a 10
OKeefe Coordination Number
12 2 4 6 8 10
Effective Coordination Number
12
Figure 9: Formation energy vs. coordination number plots.
O'Keeffe coordination number (left) and the effective coordination number (right) vs. the average of the 10th percentile formation energy per atom value (in kJ/atom) for a sample of 1000 Li oxides. The 10th percentile was chosen to remove unstable oxides since that instability likely came from sources other than Li sites. The 50t percentile (median) is given in blue as a reference.
14
20
We then isolate formation energy contributions from Li by taking the difference in formation energies between the lithiated and delithiated states of a material and view these differences in relation to the effective coordination numbers of Li in the material. In
Figure 10, we observe the lowest differences in formation energy around 4fold and 6fold coordination. This agrees with the results from the Li frequency distributions. However due to our data set, which is all stable materials and only those with both lithiated and delithiated states available, we are only able to obtain data
E
0
~ o 4
.
5
E!
_
C
0
E 7
0
2
Evaluation of Effective Coordination Number Using
Formation Energy
0 2 4 6 8 10
Effective Coordination Number
12 14
Figure 10: Coordination numbers andformation energy differences between lithiated and delithiated states.
The effective coordination number vs. the difference between formation energies of lithiated and delithiated states of 146 stable materials (in kJ/Li atom). For this graph, we used bins of size 1 coordination number unit. Due to the nature of the materials in our dataset, only coordination numbers between 3 and 6 were obtained.
for coordination numbers between 3 and 8. Therefore even though it may be the most accurate approach, the method of using lithiated and delithiated materials to measure site stability is limited.
In our study of coordination preferences, we identified the coordination preferences of cations using the frequency distribution. For Li we identified 4fold coordination as the most preferred coordination number environment. 6fold coordination was also a preferred coordination environment but was less preferred than 4fold coordination. Originally we predicted that cations would have low energy when they were in their most preferred sites and higher energy when they were not in their preferred sites.
The energy of a cation in a site is related to its migration energy barrier to other sites." We can then
describe a cation's likelihood of migrating from a site in terms of the cation's preference (or lack therefore) for the site.
Movement of a cation between two sites requires the cation to overcome the migration barrier between the sites. Part of this migration barrier is the energy difference between the target site and the current site of the cation. This means that when a cation is in its most preferred site, it is the least able to migrate to another site. Conversely, when a cation attempts to move to a target site that does not have preferred coordination, it is unable to enter the target site due to the high energy state of the target site.
15
Based on this reasoning, the best sites for cation migration are those that are moderately preferred that is, they should be preferred enough that the cation is willing to enter but also not so preferred that the cation is unwilling to leave. In the case of Li, if we wanted to look for materials with better diffusion, based on Figure 5a and 5b, we might start by looking for materials with a large number of 5 or 6fold coordinated Li sites because those coordination numbers seem relatively stable or low energy but still not as stable as 4fold coordinated sites. In this way, coordination preferences can potentially help to identify materials with higher diffusion of different cations.
Our main metrics of stability in this thesis were frequency, energy above hull, and formation energy. While frequency is not a direct measure of stability (like energy above hull and formation energy are), frequency is a strong metric to use because it is able to isolate the effect of the coordination number.
That is, a cation in a specific coordination environment will only appear frequently if it is relatively stable in that environment.
Energy above hull is a very good measure for the general stability of a compound but not for identifying the preference of coordination numbers. In particular, we are able to determine whether a compound is stable or not because materials with energy above hull of 0 are in their most stable form by definition. However, energy above hull is a characteristic of the entire material. It therefore takes into account instabilities from sources other than cation sites. This means that when we use energy above hull
22
as a metric for site stability, rather than compound stability, a site may be stable even though the energy above hull value for the material is nonzero.
Formation energy is similar to energy above hull in that it is a good measure of the stability of the compound but lacks as a measure of site stability for similar reasons. One advantage of formation energy is that it has consistent energy references. While energy above hull has a different ground or lower boundary for each different compound, formation energy is always measured against the compound in its pure elemental form (that is when the compound is decomposed into its pure elements in standard state).
However this also means that formation energy can only be used to measure relative stability between compounds since there is no clear binary way of determining whether a compound is more or less stable.
Further, a more accurate floor to compare the lithiated and delithiated states to would be oxides of the same valence rather than the compound in pure elemental form. However, the formation energies with respect to oxides of the same valence are not readily available.
Part of our inability to use formation energy comes from a lack of data. Ideally, if we wanted to measure the stability of a site, say for Li in a compound, we could delithiate the compound, and then put a single Li ion in a specific site. Alternatively we could fully lithiate the compound and take out a single
Li ion. In these cases, if we were able to calculate formation energy for the states before and after we add or remove a single Li ion, then this would be a much more direct measure of the stability of the cation in that specific site with that specific coordination number as most of the structure is unchanged.
23
The topology of a structure describes the next level of environment around a cation site. While coordination polyhedra describe how atoms are connected, topology describes how coordination polyhedra are connected. Understanding how polyhedra are connected has several applications, one of which is as a descriptor for structure. Specifically, if we want to be able to compare two structures, topology could be one descriptor with which we evaluate how similar those structures are. Another potential application is diffusion. Having a descriptor that measures how two polyhedra are connected can help us predict the migration paths between those polyhedra. We will focus mainly on Li oxides as they are abundant in the Materials Project Database due to their application as battery electrodes. In this section, we develop a tool to calculate how any two polyhedra are interconnected. We then use this tool to examine trends in topology by using data from the Materials Project Database.
3.1 Topology Tool Methodology
In order to understand topology, we first need a way to measure and calculate how two polyhedral are connected. Specifically, we want to identify whether two polyhedra share a point, an edge, or a face. The types of connectivity are shown in Figure 11. We do this by counting the number of anions shared between the polyhedra, with the rules shown in Table 1.
Number of Shared Anions
1
0
2
> 3
Connection Type
Unconnected
Pointsharing
Edgesharing
Facesharing
Table 1: Rules for topology method.
The rules used in the topology tool to define how polyhedra are connected. We count the number of shared anions between any two polyhedra to classify how those two polyhedra are connected.
24
Figure 11: Polyhedra connectivity types.
Structures demonstrating the types of connections between different polyhedra. The structure on the left shows a set of two pointsharing tetrahedra; the structure in the center shows a set of two edgesharing tetrahedra; the structure on the right shows two sets of facesharing polyhedra, one tetrahedron and one octahedron, on the top and bottom of the structure. The faces shared between the pairs of polyhedra are triangular faces.
If only one anion is shared between the two polyhedra, then the two polyhedra are considered to be pointsharing. This is because the corners of the polyhedra are defined by the anions. If only one corner is shared between the polyhedra then by convexity that corner is the only point that is shared between the polyhedra. If the polyhedra share two anions, then the polyhedra must be edgesharing, as in Figure 12. If the polyhedra share three or more anions, then they are most likely facesharing.
Figure 12: Counting of shared anions.
Schematic demonstrating how the topology method counts the number of shared anions to determine the connectivity between two polyhedra. The shared anions are circled in red and the polyhedra are circled in blue.
Specifically, two polyhedra sharing three anions would be facesharing via a triangular face, and two polyhedra sharing four anions would be facesharing via a square face. We note that it is possible for three anions in a single line to result in edgesharing between the polyhedra, but the occurrence of a polyhedra with this condition is very uncommon. If the polyhedra do not share any anions, then the polyhedra must be nonsharing and have no points of contact with each other.
25
The tool then uses this method of calculate connectivity to identify, for any cationcontaining oxide, how polyhedra in the structure are connected. More specifically, it looks through the occurrences of one type of cation in the structure and checks how it is connected to every other cationcentered polyhedron in the structure. It then counts the number of each type of connection (pointsharing, edgesharing, or facesharing). This tool then provides a description of the topology of the structure by quantifying how the polyhedra are connected within a single structure. In Figure 12, the structure on the left would give one pointsharing connection; the structure in the center would give one edgesharing connection; and the structure on the right would give two facesharing connection and two unconnected polyhedra.
3.2 Lithium Topology as a Case Study
We now investigate Li oxides using this tool in order to discern some trends in cation polyhedra topology. These trends give us a better general understanding of how Li polyhedral are connected in Li oxides.
3.2.1 Data Collection and Analysis Methods
We obtained a data set from the Materials Project Database with 10,000 Licontaining oxide using the method and filters described in Section 2.2.1. We then used the topology tool on each of these
10,000 structures and summed up the counts of each type of connection to see the overall trend of which connectivities occurred most frequently between Li polyhedra. We then grouped the connections based on the coordination environment of the polyhedra involved in the connection and plotted the counts of each type of connection for common Li site coordination numbers.
For this second analysis, we remove the structures with energy above convex hull above 10eV/atom so that we only account for the coordination numbers of stable structures. We also define coordination number using the effective coordination number, as described in Equation 2. While effective coordination number may sometimes underestimate coordination number, the effective coordination
26
Connectivity Frequency Distribution of
Li Polyhedra in Li Oxides
25000F
U
20000
I
W
0*
S15000
I
LL
1000%
100
800
U
.
600
400
5000[ ft
Non Point sharing Sharing
Edge Face Connectivity
Sharing Sharing
Figure 13: Overall Li to Li polyhedra connectivity distribution.
Distribution of count of each type of Li polyhedra to Li polyhedra connection based on a data set of 10,000
Licontaining oxides.
120C
Connectivity Frequency Distribution of
3.0 Coordination Number 1200
Connectivity Frequency Distribution of
4.0 Coordination Number
100
200
100
U
Cr 600
U
400
Non Point sharing Sharing
Edge Face Connectivity
Sharing Sharing
1200
Connectivity Frequency Distribution of
5.0 Coordination Number
1000 srNon
Point Edge Face Connectivity sharing Sharing Sharing Sharing
1200
Connectivity Frequency Distribution of
6.0 CoordInation Number
2000
600
U
Cr
600
U
400
800
U
600
U.
400
6 0 Corinto Nube
200
200 
. mE
Non Point sharing Sharing
Edge Face Connectivity
Sharing Sharing
Non Point sharing Sharing
Edge Face Connectivity
Sharing Sharing
Figure 14: Distributions of Li to Lipolyhedra connectivity by coordination number.
The top left figure shows the distribution for Li polyhedra that are (rounded) 3fold coordinated; the top right figure shows the distribution with (rounded) coordination number 4; the bottom left figure shows the distribution with (rounded) coordination number 5; and the bottom right figure shows the distribution with (rounded) coordination number 6.
number can be calculated much more quickly, and resulted in more precise peaks in the coordination number frequency distributions presented in Section 2.2.2.
3.2.2 Li Polyhedra Connection Distributions
We first look at the counts of each type of connection for all 10,000 Licontaining oxides. In
Figure 12, we observe that the most common type of connection between Li polyhedral is pointsharing.
Facesharing connections between Li polyhedral is the least common, while edgesharing is moderately common. There are also more nonsharing polyhedral than there are edge or faceconnections.
Then we look at the distributions of connections grouped by the coordination number of the Li polyhedra involved in the connection. These distributions account only for stable materials (defined as below 10 eV/atom). In Figure 13, we see that the Li polyhedra with coordination numbers 3 and 4 tend to connect to other Li polyhedra mainly through pointsharing. Li polyhedra with coordination numbers 5 and 6, on the other hand, tend to connect to other Li polyhedra mainly through edgesharing. Although we only investigated Li in this analysis, other cations (Na, K, Ca, Mg) on which we used the topology tool showed similar trends with lowercoordinated polyhedra (particularly tetrahedra) more commonly connected to other polyhedra by pointsharing and highercoordinated polyhedra (namely octahedra) more commonly connected to other polyhedra by edgesharing.
3.3 Discussion
Topology helps us to understand how polyhedra are connected throughout a structure. From
Figure 12, Li polyhedra seem to have relatively low connectivity to each other, with most connections between Li polyhedra being pointsharing and very few being facesharing. In addition we observe more nonsharing polyhedra than there are edge or facesharing connections. Nonsharing and pointsharing connectivity are the dominant connectivity types in Li polyhedra. This agrees with Pauling's third rule, which states that pointsharing connections are more stable than edgesharing or facesharing connections. Because edgefacing and facesharing connections place the polyhedra closer to each other, it also brings the cations of the polyhedra closer together, increasing the electrostatic repulsion between
28
the two cations. The effect becomes stronger for lower coordination numbers because coordination polyhedral with lower coordination numbers are smaller. Thus Pauling's third rule also provides an argument for the higher proportion of edgesharing in for 6coordinated Li polyhedra compared to 4coordinated polyhedra.
A possible application of the topology tool is as a descriptor for comparing structures. One objective of comparing structures is to streamline the search for better materials. Because of the dependence between structure and many different properties, materials with similar structures should have similar properties. Thus by searching for materials with structures similar to materials with desirable properties, we are essentially searching for materials with those desirable properties. However, the ability to search for similar structures depends on the ability to compare those structures. Because some properties depend on the topology of the structure, this information should also be a criterion to evaluate structural similarity.
Another application of the topology tool is toward understanding how cations migrate between cation sites. In the Li polyhedra case study, we discovered trends in how polyhedra connect to other polyhedra using the topology tool developed. In particular, we noted that there were in general fewer facesharing connections between polyhedra than either pointsharing or edgesharing connections. In terms of migration, facesharing polyhedra can potentially correspond to the easiest path from one polyhedron to another. A cation in a polyhedron that is faceconnected to another polyhedron can simply pass through the face to the other cation site. Edgesharing and pointsharing polyhedra on the other hand lack such a direct path between the polyhedra because cations very rarely pass through an edge and cannot pass through a point. Instead, they must pass through one or more empty sites that could be occupied by a cation but are vacant, which we hereafter refer to as "empty polyhedra". Nonsharing polyhedra are required to pass through more than one empty polyhedron to migrate from one cation site to the other. Further investigation of empty polyhedra will be presented in the following section.
29
Structures can be thought of as networks of coordination polyhedra. Cations coordinate to surrounding anions or anions coordinate to surrounding cations, creating occupied polyhedra throughout the structure. In this thesis we consider polyhedra with cations in the center and anions as coordinating ions because we focus on the coordination of cations. Between occupied polyhedra there is often empty space where there are no cation polyhedra. However, oftentimes this space still contains anions which together form polyhedron with no central cation. These polyhedra are empty but could be occupied by cations and therefore represent space through which cations could occupy or pass through during migration. They also represent another way of measuring the topology of a structure and can be used as a criterion in evaluating similarity between structures. However, as mentioned earlier, cations prefer to exist in only certain coordination polyhedra. These cations can only occupy empty space in which they would have these preferred coordination numbers. Thus, identifying empty polyhedra can help us predict which empty areas could feasibly be occupied by cations.
In order to better under connectivity between cation polyhedra, we build a tool to identify empty polyhedra in a structure. In this section, we build a tool to identify the empty polyhedra in oxides. We then investigate these empty polyhedra in Li oxides as a case study.
The empty polyhedra tool developed in this thesis is able to section empty space within a structure into empty polyhedra. It first identifies all the potential empty polyhedron centers of the structure, following the methodology developed by Yang
1 4
, and then removes those that could not be possible cation sites.
Yang developed a methodology to identify possible Li sites in a structure based on a similarity criterion between the substructures of occupied Li sites and potential Li sites. The set of potential Li sites she uses is composed of all the corners of Voronoi polyhedra and the center of the faces of Voronoi polyhedra; this set is referred to as the Voronoi points of the structure.
30
This tool uses Yang's code to identify all the Voronoi points in the structure as the set of all possible empty polyhedron center sites. The tool then removes any repeated Voronoi points as well as any
Voronoi points in any occupied cation polyhedra. Of those remaining points, the tool keeps only those
Voronoi polyhedra that have the minimum number of anion neighbors in a given radius, with both the minimum neighbors and radius as designated by the user. This number should be based on the coordination number preferences and the cation to anion bond lengths of the cation whose migration paths we are interested in. Additionally, for any Voronoi points that share four or more anion neighbors, we remove all Voronoi points in that set except for the one with the most neighbors. While this may remove square facesharing polyhedra, it also avoids the scenario in which multiple Voronoi points represent the same empty polyhedra. Because the number of actual instances where polyhedra are facesharing with square faces is very low, this criterion cleans data while likely not removing many actual empty polyhedron sites. The remaining Voronoi points are the set of empty polyhedron center sites. The process
Figure 15: Processing steps for adding empty polyhedra to a structure.
The original structure (top left) is given Voronoi points in yellow, resulting in a structure with all Voronoi points (top right).
Voronoi points that are not empty polyhedra structures are then taken out, resulting in a structure with empty polyhedra (bottom left without empty polyhedra visualized, bottom right with empty polyhedra visualized).
31
20000
Connectivity Frequency Distribution of
4 Coordination Number
M LiL
M UOther Cation
15000
0I
10000
LaL
5000
5000
4000
Connectivity Frequency Distribution of
6 Coordination Number

UU
111 UOther Cation
LIEmpty
U
3000
2000
1000
0
Sharing
Point
Sharing
Edge
Sharing
Face
Sharing
Connectivity ?I0
Sharing
Point
Sharing
Edge
Sharing
Face
Sharing
Connectivity
Figure 16: Frequency distributions of Li to all polyhedra connectivities.
Left: 4coordinated Lipolyhedra, right: 6coordinated Lipolyhedra. Plots are grouped by the type of polyhedra the Li polyhedra is connected to as well as by the type of connection between the polyhedra. Note that the scale in the two plots is different. Data was taken from a set of 500 Lioxide structures.
for adding and removing Voronoi points is shown in Figure 14.
We now look more closely at Li oxides as a case study for the empty polyhedra tool. Through this case study, we show how we can further describe connectivity not just between a pair of cation polyhedra but also between a cation polyhedron and an empty polyhedron.
4.2.1 Data Collection and Analysis Methods
The structural data from a sample of 500 Lioxides was collected from the Materials Project
Database as in Section 2.2.1. Using the topology tool, we counted the connectivity between our cation of interest, Li, and all other polyhedra in the structure (other Li polyhedra, other types of cation polyhedra, and empty polyhedra). In order to investigate how empty polyhedra connect to cation polyhedra, we take a closer look at all connections between two Li polyhedra that share an empty polyhedron.
4.2.2 Li Polyhedra Connectivity to Other Polyhedra
We first look at how Li polyhedra are connected to all types of polyhedra. In Figure 15, we can observe the distributions of each type of connection for each type of polyhedra. Among the different
32
types of polyhedra, Li polyhedra appear to connect the most often to empty polyhedra and, in general, the least often to other Li polyhedra, except for 6coordinated edgesharing Li polyhedra, which coordinated
by edgesharing to Li polyhedra slightly more than other types of polyhedra. We also notice that, while very few Li or other cation polyhedra faceshare with Li polyhedra, significantly more empty polyhedra faceshare with Li polyhedra. It is noticeable that there is a relatively high proportion of facesharing 6 coordinated Li polyhedra to empty polyhedra. We also note that there are no nonsharing Li polyhedra, which supports the idea that the empty polyhedra fill up the empty space in the structure. Any polyhedra should be connected to occupied polyhedra or empty polyhedra, with the total number of connected polyhedra defined by the shape of the polyhedra. For example, a tetrahedron should share faces with 4 polyhedra because it has 4 faces, edges with 6 polyhedra because it has 6 edges, and points with 4 polyhedra because it has 4 corners.
4.2.3 Li Polyhedra to Empty Polyhedra to Li Polyhedra Paths
We then look more closely at the connections between two Li polyhedra that also share an empty polyhedra to investigate how an empty polyhedron interacts with connected cation polyhedra. From
Figure 16, we observe relatively high numbers of pointsharing and facesharing between Li polyhedral and empty polyhedra for both pointsharing connected and edgesharing connected polyhedra. This indicates that, when two Li polyhedra are connected by either points or edges and they also share an empty polyhedra, that empty polyhedra is usually pointconnected or edgeconnected to the two Li polyhedra. In particular, the relatively high number of Li polyhedra to empty polyhedra facesharing suggests that the empty polyhedron is a polyhedron that fills the empty space between adjacent occupied polyhedra. We also note that a pointsharing and edgesharing Li site seem to have around equal probability of sharing a face with an empty polyhedron. This implies that a pair of pointshared Li polyhedra and a pair of edgeshared Li polyhedra are equally likely to have a faceshared empty polyhedron between the pair.
33
25000
1
M PointSharing to Empty Polyhedra
EdgeSharing to Empty Polyhedra
FaceSharing to Empty Polyhedra
20000
U
M 15000
0r
4)
U
10000
5000
Point
Sharing
Edge
Sharing
Face LiLi
Sharing Connectivity
Figure 17: Frequency distributions of the connectivities in Liempty polyhedraLi paths.
Data is based on all connected set of polyhedra in which two Li polyhedra are connected and both are connected to an empty polyhedra in a set of 500 Li oxides. The plot groups frequency by both the connectivity between the two Li polyhedra and the connectivities between the Li polyhedra and the empty polyhedra.
14000
16000
5
12000
UL
10000
8000
6000
4000
2000
0 0 2 4 6
Li Coordination Number
8
Figure 18: Frequency distribution of the coordination numbers in Liempty polyhedraLi paths.
Data is based on all connected set of polyhedra in which two Li polyhedra are connected and both are connected to an empty polyhedra in a set of 500 Li oxides. The plot groups frequency by the coordination numbers of the Li sites and the empty polyhedra.
34
We next look at the frequency distribution of coordination numbers. In Figure 17, we note the relatively high proportion of 4coordinated Li polyhedra connected to 6coordinated empty polyhedra and 6 coordinated Li polyhedra connected to 4coordinated empty polyhedra. This suggests that, in a connected set of two Li polyhedra and one empty polyhedra, lowercoordinated Li polyhedra are more likely to coordinate to highlycoordinated empty polyhedra and highercoordinated Li polyhedra are more likely to coordinate to lowercoordinated empty polyhedra. This is likely due to geometrical constraints where lowercoordinated Li polyhedra have wider angles in which highercoordinated empty polyhedra fit while highercoordinated Li polyhedra have narrower angles in which lowercoordinated empty polyhedra fit.
We notice a relatively high number of 5coordinated Li polyhedra. These are likely polyhedra with underestimate coordination numbers due to the use of effective coordination number, as many of the Li polyhedral reported as 5coordinated were actually 6coordinated upon inspection of the actual structure.
4.3.1 Understanding of Topological Environment
The empty polyhedra tool provides a way to further understand the topological environment in oxides. Using the tool we investigated not only the general topology of Li oxides but also more specifically the local topology between Li polyhedra and empty polyhedra.
In the study on Li oxides, several trends were found that give us a sense for how coordination polyhedra are generally connected and separated by empty space within a structure. One trend we note from Figure 15 is that Li polyhedra tend to connect more often to empty polyhedra than to either Li or any other cation polyhedra. This gives us a sense of the compactness of the cations and cation polyhedra in Li oxide structures.
Another trend we observed was that, for 4coordinated polyhedra, the number of connection types to each polyhedra type followed the trend of decreasing frequency for increasing connectivity. For Li to
Li polyhedra connections, Li to other cation polyhedra connections, and Li to empty polyhedra connections, point sharing was most frequency, followed by edgesharing, and facesharing was least
35
frequent. However, this trend does not hold for 6coordinated polyhedra, in which there is much more Li to Li polyhedra edgesharing and much less Li to empty polyhedra edgesharing. This raises the question of how much the polyhedra center site species (or lack thereof) affects its connections to other polyhedra or if connections are mostly dictated by geometry. Pauling's fourth rule indicates that more charged central cations and cations with smaller coordination number tend to be less connected in order to reduce cationcation electrostatic repulsion. One further study to verify this would be to include information on the coordination number and the charge of the central ion of the polyhedra. This would allow us to observe the effects of different polyhedra center site species and varying coordination polyhedra.
We next explore more local topology, specifically between two connected Li polyhedra sharing an empty polyhedron in Figures 16 and 17. In this specific case, we were able to see that the common connections among these three polyhedra were Li to empty polyhedra pointsharing and facesharing with
Li to Li polyhedra pointsharing and edgesharing. The most common coordination number combinations among the connected polyhedra were 4coordinated Li polyhedra to 6coordinated empty polyhedra and
6coordinated Li polyhedra to 4coordinated empty polyhedra, with examples shown in Figure 18.
Figure 19: Structures with common Liempty polyhedraLi coordination numbers.
Structures with empty polyhedra that demonstrate the main types of coordination polyhedra involved in a pair of Li polyhedra sharing an empty polyhedra. Li polyhedra are colored light green while empty polyhedra are colored yellow. The figure on the left shows pairs of 4coordinated Li polyhedra connected by an edge that are connected via facesharing to a 6coordinated empty polyhedra. The structure on the right shows pairs of 6coordinated Li polyhedra connected by an edge that are connected via facesharing to a 4coordianted empty polyhedra.
36
4.3.2 Similarities to Li SiteFinding Code in Yang 2014
The empty polyhedra tool developed here has a similar goal to the Li sitefinding code in Yang's thesis.' 4 In her code, Yang utilizes her algorithm for substructural similarity to identify Li sites in delithiated materials. These potential Li sites are similar to the empty polyhedra that the tool here identifies in that both are sites where Li could potentially be or migrate to. Both codes also use Voronoi points as the initial set of possible Li sites. The main differences arises from the method of identifying either Li sites or empty polyhedra. Yang uses substructural similarity to identify similarities in structure between current Li sites and Voronoi points that are potentially Li sites. The empty polyhedra tool described here only need to consider distances from potential sites to nearby anions. Yang's Li sitefinding code looks specifically for stable Li sites, while the empty polyhedra here should identify stable
Li sites in addition to other possible intermediary (less stable) sites.
4.3.3 Further Applications for the Empty Polyhedra Tool
In addition to providing insight into the topological environment of structures, the empty polyhedral tool has other possible applications. One possible application is the prediction of possible diffusion paths. Specifically, the empty polyhedra that our tool identifies are also those that Li could potentially migrate through when diffusing between Li sites. If Li polyhedra are not directly connected through facesharing, then Li ions diffusing from one polyhedron to the other must go through some other path. This path is likely to consist of a series of one or more empty polyhedra connected to each other as well as at the ends to the Li polyhedra by facesharing. The empty polyhedra tool, which identifies empty polyhedra, and the topology tool, which identifies connections between polyhedra, could then be used in conjunction to identify possible diffusion paths between the two Li polyhedra.
Another potential application for the empty polyhedra tool is to evaluate similarity between structures. The coordination of the empty polyhedra could be another characteristic to be accounted for in the calculation of structural similarity. Similar structures should not only have similar substructures, but they should also have similar empty spaces. Two structures that are similar should have matching
37
substructures that are similarly oriented in space. The addition of empty polyhedra supplies this topology information on how substructures are spaced in the structure.
38
5.
First proposed by Linus Pauling in 1929, Pauling's rules are rules based on geometry and electrostatic interactions that help describe how and why ions tend to coordinate in certain ways and form specific topologies.' Pauling's first and second rules in particular can be used to predict coordination number. In this section, we use data from the Materials Project Database to evaluate the accuracy of
Pauling's first and second rules.
5.1 Pauling's First Rule
Pauling's first rule, known as the Radius Ratio Rule, allows for prediction of a cation's coordination number based on its radius and its peripheral ion's radii. Specifically, it notes that at large ratios of the cation radius to the anion radius, the cation fits stably in a hexagonal closestpacking structure. However, if the radius of the cation decreases enough, it hits the "no rattle limit", at which the cation is no longer stable in its position, as seen in Figure 19.1 As a result, the structure must become cubic (8fold) coordinated to compensate for the decreased radius. As the radius continues to decrease, the structure must also change for the cation to remain stably coordinated to its anions. Table 2 identifies
Rg/Ra Lower Limit
I
0.645
0.414
0.225
0.155
0
RI/Ra Upper Limit Predicted Coordination
12fold (hexagonal close packing)
1
0.645
0.414
0.225
0.155
8fold (cubic coordination)
6fold (octahedral coordination)
4fold (tetrahedral coordination)
3fold (triangular coordination)
2fold (linear coordination)
Table 2: Pauling's Radius Ratio rule for coordination numbers.
Using the ratio between the radius of the cation and the radius of its surrounding anions, it is possible to predict the coordination number of the cation based on geometrical arguments. As the radius ratio between the cation and the anion decreases, fewer anions can be coordinated to the cation (Pauling 1929).1
39
subsequent "no rattle limits" and their accompanying structures.
However, a recent study by Gibbs challenges Pauling's first rule by presenting an argument against the spherical model of the atom. This argument centers around electron density distributions, which demonstrate that the electron density of an atom can be highly distorted depending on the atoms it bonds to.
2
0
Based on Gibbs' findings, Pauling's first rule, dependent on the assumption of atoms as spheres, will not hold. In this section, we aim to evaluate whether Pauling's first rule holds for structures in the Materials Project Database.
5.1.1 Testing Methodology
To evaluate the validity of Pauling's first rule, we observe the coordination environments of several common cations, namely Li, Na, K, Mg, Ca, Ba, Al, Cr, Mn, Fe, Co, Ni, Cu, Ag, and Hg. For each of these common cations, we obtained data sets of all available oxides with these cations and an energy above hull value below 106 eV/atom to ensure the materials are stable. The data was retrieved from the Materials Project Database as specified in Section 2.2.1.
For each cation, we investigated the cation surrounded by oxygen atoms since we only looked for oxides in this data set. The expected coordination number of each cation is then described by the cation's ionic radius divided by oxygen's ionic radius. When available, we used Pauling's radiil' 2 2
, otherwise we use the effective ionic radii described by Shannon 2 3 for calculating the expected coordination numbers.
From the database, we obtained the O'Keeffe and effective coordination numbers for a variety of cations.
To evaluate these coordination numbers, we looked at the most frequent coordination number to compare to the expected coordination number given by the radius rule.
5.1.2 Results and Discussion
In Table 3 we present a comparison between the most frequent coordination numbers calculated using the O'Keeffe and effective coordination number definitions and the expected coordination number according to Pauling's first rule. Pauling's first rule seems to predict coordination number relatively well.
40
Cation
Co
Mn
Fe
Cr
Mg
Al
Ag
Ni
Li
Na
K
Ca
Ba
Hg
Cu
Sn
Radius (pm)2
0.574*
0.873
1.173*
1.020
1.400**
0.750**
0.390
1.260
0.600
0.610
0.645
0.645
0.615
1.02
0.57
0.69
O'Keeffe
CN
4.4
5.6
6.4
5.6
6.8
5.2
4.0
5.4
5.4
5.8
5.4
5.8
3.8
4.6
4.8
5.4
Most Frequent
Effective CN
3.8
5.8
3.8
5.8
5.8
5.8
5.8
5.8
11.8
5.8
3.8
1.8
5.8
2.0
3.8
5.8
Table 3: Evaluation of Pauling's First Rule's Prediction Ability
Pauling's first rule was used to predict coordination numbers for the ions in the leftmost column. The radii used or each cation are given in the second column. The predictions from Pauling's first rule are compared to the most frequent
O'Keeffe and effective coordination numbers.
In particular, it appears to predict the coordination number of most transition metals and some Group
and Group 2 cations within an error margin of 1 of the most frequent O'Keeffe or effective coordination number. There are some outliers, such as K, Ca, Ag, and Hg, which Pauling's first rule was unable to correctly predict the coordination number within the
error margin. For these outliers, the coordination numbers predicted using Pauling's first rule tend to be larger than the most frequent O'Keeffe or effective coordination numbers, which agrees with Brown's results, who also found that his actual coordination numbers were less than the expected coordination number calculated using the radius rule. This may result from Pauling's first rule's assumption that ions act as solid spheres. Because ions have electron
Expected CN
4
6
6
8
4
8
6
6
6
8
8
12
6
6
4
6
*Based on Pauling's radii (1927)"
41
clouds rather than actual physical boundaries, this assumption of ions as solid spheres may indicate that
Pauling's first rule is actually an upper limit rather than a hard rule.
The predictions made by Pauling's first rule depend significantly on the cation radius. The radius used to calculate Pauling's first rule is given in the second column of Table 3, but in actuality not only is it difficult to define an ion's radius but the ion's radius also depends on the coordination number (which we are attempting to predict) and the charge of the ion (which we do not consider for ions with multiple charge states). To add to that point, Shannon asserts that crystal radii better represents the physical size of ions than Pauling's radii or ionic radii. Pauling's first rule is a purely physical/geometrical argument, so we would expect that the crystal radii should yield more accurate predictions for Pauling's first rule; however, using crystal radius resulted in coordination numbers that consistently overestimated the coordination number.
Another issue to consider is that a cation can have several different species in its set of peripheral anions. Currently Pauling's first rule appears to be best suited to materials in which cations are surrounded by a single species of ion because with the ratios between a single species of cation and a single species of anion. However, in actuality, many cations are surrounded by multiple species of ions
(for example, a 6fold coordinated Li ion may be surrounded by 3 F ions and 3 0 ions). We do not consider multiple anions here but subsequent studies of Pauling's first rule should factor in the presence of different anion species.
5.2
Pauling's second rule investigates the local charge balance of cations. Specifically, it states that the sum of the electrostatic valencies (e.v.) of any ion should equal its charge. Electrostatic valency, which describes bond strength to an extent, is calculated by dividing the ion's charge by its coordination number. In his original paper, Pauling' described bond strengths between a central ion and one of its peripheral ions as the charge of the peripheral ion divided by the peripheral ion's coordination number.
Numerically,
S z
(Equation 4) where z is the peripheral ion charge and v is the peripheral ion coordination number. While Pauling's second rule was used extensively at the time to determine structures, Baur demonstrated that bond strength sums to the oxide ions in numerous silicates, borates, phosphates, and sulfates could deviate as much as 40% from the charge of the ion. The method of calculating bond strength was also improved to factor in the influence of bond length on bond strength. In this section, we will measure the strength of the electrostatic valencies of anions to determine the accuracy of Pauling's second rule.
5.2.1 Testing Methodology
To evaluate the validity of Pauling's second rule, we compute the sum of electrostatic valences for anions in materials from the Materials Project Database and compare these to their nominal charges.
The dataset we obtain contains 3,400 materials containing 0, F, Cl, I, Br, S, or N anions and with energy above hull below 102 eV/atom (to ensure the materials are relatively stable) from the Materials Project
Database as described in Section 2.2.1, excluding the criterion that requires the materials to contain 20%
0. To calculate the electrostatic valences of an anion, we divide the charge of each neighboring ion by its effective coordination number.
In order to evaluate the performance of Pauling's second rule, we investigate the frequency distribution of the errors between charge expected by Pauling's second rule and the nominal charge. For cations with several common positive charged states, the average of the common charged states were used since it was not feasible to measure the charged state of each individual cation in a material, which presents a potential limitation to this analysis method.
43
Ann
500 o400 m 300
0
S00 u 200
Frequency of Error in Charge
0 50 100 150
Error in Charge, binned by 5% (in %)
200
Figure 20: Pauling's second rule error distribution.
The frequency distribution of errors in the charges predicted by Pauling's second rule shows a very broad range of errors.
Charges are based on the average of the most common positive charges of cations, and coordination numbers calculated using the effective coordination number.
In Figure 20, the frequency distribution of the errors in Pauling's second rule's predicted charge demonstrates that the rule is relatively imprecise. The average error in predicted charge is 22% while the median error is 17%. The range of the distribution is from around 0% up to 200% error while most of the estimates have an error between 0% and 50%. Based on the fundamental nature of the logic in Pauling's second rule, the errors in charge are higher than expected.
There are some issues that may have affected these results, including the use of effective coordination number, the use of Pauling's original method of calculation bond strength, and the use of average nominal charge of an ion rather than its true nominal charge. Specifically, if the effective coordination number method underestimates or overestimates the coordination number of neighboring cations, the electrovalence of the cations are also underestimated or overestimated. In addition, Pauling's method of measuring electrostatic valence does not account for bond length, even though longer bonds imply less charge is distributed to that bond. Further studies should be done to see if the error is lower or higher depending on the species of anion, its cation, or the average bond length of the anion to its cations.
44
In this thesis, we utilized the Materials Project Database to conduct data analysis on coordination and topology in oxides. Our goal was to understand how atoms coordinate in structures by investigating coordination and topological environments. We first quantified the connectivity between ions by measuring the frequency of different polyhedra. Then, we quantified the connectivity between polyhedra
by measuring the frequency of different types of connectivities. Subsequently, we created a tool to find empty polyhedra in oxide structures. Finally we evaluated the accuracy of Pauling's first and second rules.
We first observed coordination preferences of several common cations using the frequency distributions of their coordination numbers to find what types of polyhedra these cations tend to form. We found that Li prefers 4fold coordination and, to a lesser extent, 6fold coordination, Na prefers 6fold coordination, Ca prefers 6fold coordination, and Mg prefers 6fold coordination and, to a lesser extent, 4fold coordination. We focused on Li when investigating topology and empty polyhedra. Li polyhedra were most often connected to other Li polyhedra through pointsharing but also frequently edgeshared in the case of 6coordinated Li polyhedra. After adding empty polyhedra to the topology network we consider, we observed that Li polyhedra are highly connected to empty polyhedra, usually via point and facesharing, implying that the Li oxides we observed were relatively uncompact. We also looked more closely at local topology among Li polyhedra and empty polyhedra, noting that common connections involved either 4coordinated Li polyhedra and 6coordinated empty polyhedra or 6coordinated Li polyhedra and 4coordinated empty polyhedra. Finally, we observed that Pauling's first and second rules do not seem to describe coordination and topology accurately.
The work in this thesis, particularly the methods and tools developed to provide connection and empty polyhedral information, has further applications to projects needing topological information. In particular, the tools used to calculate connectivity and identify empty polyhedra can be used to identify
45
possible diffusion paths. By tracing paths between Li polyhedra through empty polyhedra, we can identify migration paths between those Li polyhedra. These migration paths can then be combined to construct possible diffusion paths through the structure. Another possible application of the tools is as one of the attributes that contribute to the calculation of the similarity between structures. Current work on structural similarity considers substructures and polyhedra but topology specifically the numbers of connections between polyhedra and empty polyhedra can be another descriptor to be included in the structural similarity function. This addition would allow the function to consider not just the substructures but also how those substructures are connected to each other.
46
1. Pauling , L. (1929). The Principles Determining the Structure of Complex Ionic Crystals. J.
Am. Chem. Soc. , 10101026.
I. D. (1988). What factors determine cation coordination numbers? Acta Cryst B, 44,
3. Wenger, M., & Armbruster, T. (1991). Crystal chemistry of lithium : oxygen coordination and
, 387399.
Threedimensional Polyhedra and Networks. Act Crystallographica 85771.
6. Blatov, V. A., DelgadoFriedrichs, 0., O'Keeffe, M., & Proserpio, D. (2007). Threeperiodic nets and tilings: natural tilings for nets. Act Cryst. A , 418425.
7. Carlucci, L., Ciani, G., & Proserpio, D. M. (2006). Chapter 1.3 Networks, Topologies, and
Entanglements. In D. Braga, & F. Grepioni (Eds.), Making Crystals by Design: Methods,
Techniques, and Applications. Germany: WileyVCH Verlag GmbH & Co. KGaA.
8. Blatov, V. A., Shevchenko, A. P., & Proserpio, D. M. (2014). Applied Topological Analysis
, 35763586.
9. Ceder, G., & Persson, K. (2013). How supercomputers will yield a golden age of materials
science. Scientific American, 309, 3440.
10. Fischer, C., Tibbetts, K. J., Morgan, D., & Ceder, G. (2006). Predicting crystal structures by
11. Hautier, G., Fischer, C., Jain, A., Mueller, T., & Ceder, G. (2010). Finding nature's missing
Materials, 22 (12), 37623767.
13. Yang, L., Dacek, S., & Ceder, G. (2014). Proposed definition of crystal substructure and
15. Rong, Z., Malik, R., Canepa, P., Gautam, G. S., Liu, M., Jain, A., et al. (2015). Materials
16. O'Keeffe, M. (1979). A Proposed Rigorous Definition of Coordination Number. Acta Cryst A
,772775.
17. Hoppe, R. (1979). Effective coordination numbers (ECoN) and mean Active fictive ionic radii (MEFIR). Z. Kristallogr. , 150, 2352.
47
18. Momma, K., & Izumi, F. (2008). VESTA: a threedimensional visualization system for
19. Hautier, G. (2004). HighThroughput Data Mined Prediction of Inorganic Compounds and
American Mineralogist, 99, 10711084.
21. Pauling, L. (1927). The Sizes of Ions and the Structure of Ionic Crystals. Journal of
American Chemistry Society, 765.
23. Shannon, R. D. (1969). Revised effective ionic radii and systematic studies of interatomic
751767.
, 129155.
8.1 Appendix A: Uncommon Coordination Number and Outliers
Based on these analyses of Li oxides, we observed that the most preferred coordination numbers appear to be approximately 4fold and 6fold coordination. However, we found several outliers with coordination numbers that differed from what we expected. Specifically, we found materials with coordination numbers such as 12fold and 5fold coordination. We checked these outliers to ensure that the coordination numbers were not simply misclassifications by the O'Keeffe or effective coordination number methods.
8.1.1 Li Oxides Stable Outliers
Lithium ions in lithium oxides are expected to have a preferred coordination numbers of 4 or 6.
However, while investigating outliers to this preference rule, we found various lithium ions with uncommon 5fold coordination and others with unexpected polyhedron. We also observed lithium ions with a coordination number of 5, which seemed to be a result of distortion due to nearby cations rather than the natural coordination number. For all examples of structure, we identified the relevant structure with the lowest energy above hull value so that the given structure is the most stable.
Figure 21: Exampled of 5fold coordinated Li.
a. (left) Li2CeO3 (29980), a clearly 5coordinated Lioxide structure with energy above hull of 0 eV/atom, formation energy per atom of 2.996 kJ/atom, and calculated coordination numbers 2.82 (effective) and 4.3 (O'Keeffe). b. (right) Li6PBrOs
(134422), a potentially 5coordinated Lioxidebromide structure. It has energy above hull of 0 eV/atom, formation energy per atom of 2.424 kj/atom, and calculated coordination numbers 2.99 (effective) and 4.7 (O'Keeffe).
49
5coordinated:
The above structures, presented in Figure 21, show substructures that could be classified as 5 coordinated. While they have the appearance of 5coordinated structures, this appearance may be due to distortion of the polyhedron. The two structures seem to be correctly classified as 5coordinated. Note that both Figure 21 a and 2 lb have energy above hull values of 0 eV/atom. This indicates that these structures are stable and that Li is, unexpected, stable in these 5coordinated sites. Thus, while 4fold and
6fold coordinated sites seem to be much more preferred than other coordination numbers, true 5fold coordination does exist.
6coordinated polyhedra outliers:
In most 6coordinated lithium ion sites, we see an octahedron around the central lithium (specifically 4 planar oxygen atoms plus 1 above and 1 below). However, also observed some triangular prism outliers as seen in Figure 19.
The structure in Figure 22, Ba
2
Li(CuO
2
)
3 has an energy above 0 hull of 0, indicating that the triangular prism polyhedron (or at the very least the structure that the triangular prism exists in) is stable. Thus, while the 6fold coordinated Li sites tend to polyhedra.
Ba2Li(Cu02)3 (134611), a 6coordinated Lioxide structure with a triangular prism coordination be mostly octahedron, other coordination polyhedron such as polyhedron. It has energy above hull of 0 eV/atom, formation energy per atom of 1.775 triangular prism may also exist. kj/atom, and calculated coordination numbers
5.94 (effective) and 7.1 (O'Keeffe).
8.1.2 Unstable Li Oxide Outliers
12coordinated:
The structures in Figure 23 demonstrate the existence of 12coordinated Li. However, based on the structures' energy above hull values, they are also considered not stable (e above hull > 0.05 eV/atom).
50
b) a)
C)
/'WO
Figure 23: Examples of unstable 12coordinated Li.
a. (left) LiLaTi2O6 (216246), a 12coordinated Lioxide structure with energy above hull of
0.09434 eV/atom, formation energy per atom of 3.353 kj/atom, and calculated coordination numbers 11.83 (effective) and 10.4 (O'Keeffe). b. (center) Li3AI(B03)2
(174134), a 12coordinated Lioxide structure with energy above hull of 0.1582 eV/atom, formation energy per atom of 2.698 kJ/atom, and calculated coordination numbers 7.43 (effective) and 9.05 (O'Keeffe). c. (right) LiLaNb4012 (216215), an 12coordinated Lioxide structure with energy above hull of 0.05652 eV/atom, formation energy per atom of 3.084 kJ/atom, and calculated coordination numbers of 11.64 (effective) and 10.2 (O'Keeffe).
a) b d) C
0
0
'00
Figure 24: Examples of unstable 8coordinated Li.
a. (top left) LiAlNiO3 (41984), an 8coordinated Lioxide structure with energy above hull of 0.24441 eV/atom, formation energy per atom of 2.226kJ/atom, and calculated coordination numbers 7.55 (effective) and 8.8 (O'Keeffe).
b. (top right) LiIOV4Si4O17 (55977), an 8coordinated Lioxide structure with energy above hull of 0.325 eV/atom, formation energy per atom of 2.354 kJ/atom, and calculated coordination numbers 8.00 (effective) and 9.4
(O'Keeffe). c. (bottom left) Li4Fe4CosO81 (51053), an 8coordinated Lioxide structure with energy above hull of 0.0960 eV/atom, formation energy per atom of 1.458 kj/atom, and calculated coordination numbers of 7.01 (effective) and
7.55 (O'Keeffe). d. (bottom right) LiBi2Sb3OI1 (210401), an 8coordinated Lioxide structure with energy above hull of
0.0231 eV/atom, formation energy per atom of 1.896 kJ/atom, and calculated coordination numbers of 7.81
51
One reason these unstable 12coordinated Li sites may exist is that the materials with these sites were theoretically derived. Specifically, these structures may have been derived using substitution of Li for some other larger firstcolumn cation (such as cesium). These substitutions would result in Li in a site that is much more highly coordinated than what is usually found in nature.
8coordinated:
The structures in Figure 24 demonstrate the existence of 8fold coordination, although most of the structures are considered unstable. Most of the above structures do form a symmetric 8vertexed polyhedron, except for Figure 24c, Li
4
Fe
4
Co
5
O
18
, which appears to be distorted due to the presence of Fe cation.
52