REVISITING CHEMICAL SYSTEM SIZE USING STREUSEL (SURFACE
TOPOLOGY RECOVERY USING SAMPLING OF THE ELECTRIC FIELD):
THE ROLE OF SIZE IN ATOMIC, MOLECULAR, AND SOLID-STATE
PROPERTIES
by
AUSTIN M. MROZ
A DISSERTATION
Presented to the Department of Chemistry and Biochemistry 
and the University of Oregon Division of Graduate Studies 
in partial fulfillment of the requirements
for the degree of
Doctorate of Philosophy
June 2022
DISSERTATION APPROVAL PAGE
Student: Austin M. Mroz
Title: Revisiting Chemical System Size Using STREUSEL (Surface Topology
REcovery Using Sampling of the ELectric field): the role of size in atomic,
molecular, and solid-state properties
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctorate of Philosophy degree in the Department of
Chemistry and Biochemistry by:
James Prell Chair
Christopher H. Hendon Core Member
Jeffrey Cina Core Member
Gregory Bothun Institutional Representative
and
Krista Chronister Vice Provost for Graduate Studies
Original approval signatures are on file with the University of Oregon Division of 
Graduate Studies.
Degree awarded June 2022
ii
© 2022 Austin M. Mroz
 All rights reserved.
iii
DISSERTATION ABSTRACT
Austin M. Mroz
Doctorate of Philosophy
Department of Chemistry and Biochemistry
June 2022
Title: Revisiting Chemical System Size Using STREUSEL (Surface Topology 
REcovery Using Sampling of the ELectric field): the role of size in atomic, 
molecular, and solid-state properties
Atomic, molecular, and porous material void space shape and size play a 
critical role in chemistry. The most familiar method was formalized by Bondi: the 
van der Waal radii and volume of atoms and molecules. However, the rigid sphere 
approximation of atoms fails to describe highly polarized chemical systems. To 
overcome this challenge, numerous other approaches based on electron density have 
been presented, but these approaches intrinsically struggle to describe the surface 
area and volume of cations. Herein, we revisit the timeless problem of assessing 
sizes of atoms, molecules and porous material void spaces, through examination
of the electric field produced by these chemical systems. In this way, we are able 
to recover chemical volumes and surface areas from simple DFT calculations.
We perform a series of benchmarks to ensure the generality of our approach and 
demonstrate that electric field calculations provide unique insights into quantifying 
sizes of ions and polar molecules. Our method further lays the foundation for the 
development of analytical interaction energies based on assessment of the Coulomb 
potential produced by molecules systems, while providing a rigorous approach
to study size dynamics as a function of chemical environment. Beyond these 
advancements, we demonstrate the advantageous role of electric field-derived size in 
determining the void space characteristics of nanoporous, crystalline materials.
iv
CURRICULUM VITAE
NAME OF AUTHOR: Austin M. Mroz
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, OR, USA
Rose-Hulman Institute of Technology, Terre Haute, IN, USA
DEGREES AWARDED:
Doctor of Philosophy, Physical Chemistry, 2021, University of Oregon
Master of Science, Physical Chemistry, 2018, University of Oregon
Master of Science, Chemistry, 2017, Rose-Hulman Institute of Technology
Bachelor of Science, Mechanical Engineering, 2016, Rose-Hulman Institute of
Technology
AREAS OF SPECIAL INTEREST:
Electronic Structure Theory
Computer Informatics in Chemistry
Metal-Organic Frameworks
PROFESSIONAL EXPERIENCE:
Graduate Teaching Fellow, University of Oregon, 2016-2019
Graduate Teaching Assistant, Rose-Hulman Institute of Technology, 2016
Product Engineering Intern, Alison Transmission, 2014
Design Intern, Rose-Hulman Ventures, 2013
GRANTS, AWARDS AND HONORS:
Graduate Student Award for Excellence in Teaching of Chemistry,
University of Oregon, 2019
Deans First Year Merit Award, University of Oregon, 2018
v
PUBLICATIONS:
Mroz, A. M.; Hendon, C.H. ”Revisiting the van der Waals radii: molecular
volumes and surface areas of molecules derived from the electric field,”
Submitted.
Mroz, A. M.; Hendon, C.H. ”Pore Topology Characterization using the
Electric Feidl” In preparation.
Mroz, A. M.; D’Antona, N.; Sikma, E.; Cohen, S. M.; Hendon, C.H.
”Engineering Conductive Pathways in Metal-Organic Frameworks via
p-type Redox-Active Interstitials,” In preparation.
Mancuso, J. L.; Mroz, A. M.; Le, K. N., Hendon, C.H. ”Electronic
Structure Modeling of Metal-Organic Frameworks,” Chem. Rev., 2020,
16, 8641.
LeRoy, M. A.; Mroz, A. M.; Mancuso, J. L.; Miller, A.; Van Cleve,
A.; Check, C.; Heinz, H.; Hendon, C. H.; Brozek, C. K. ”Post-
synthetic modification of ionic liquids using ligand-exchange and redox
coordination chemistry,” J. Mater. Chem. A, 2020.
Hamann, D. M.; Bardgett, D.; Bauers, S.R.; Kasel, T. W.; Mroz, A.
M.; Hendon, C. H.; Medlin, D. L.; Johnson, D. C. ”Influence of
Nanoarchitecture on Charge Donation and Electrical Transport
Properties in [(SnSe)1 + δ][TiSe2]q Heterostructures,” Chem. Mater.,
2020, 32, 5802.
Donor, M. T.; Mroz, A. M.; Prell, J. S. ”Experimental and theoretical
investigation of overall energy deposition in surface-induced unfolding of
protein ions,” Chem. Sci., 2019, 10, 1497.
Sutton, E.C.; McDevitt, C. E.; Prochnau, J.Y.; Yglesias, M. V.; Mroz,
A. M.; Yang, M. C.; Cunningham, R.M.; Hendon, C.H.; DeRose, V.J.
”Nucleolar Stress Induction by Oxaliplatin and Derivatives,” J. Am.
Chem. Soc., 2019, 141, 18411.
McDevitt, C. E.; Yglesias, M. V.; Mroz, A. M.; Sutton, E.C.; Yang, M. C.;
Hendon, C.H.; DeRose, V.J. ”Monofunctional platinum(II) compounds
and nucleolar stress; is phenanthriplatin unique?,” J. Biol. Inorg. Chem.,
2019, 24, 899.
vi
Kasel, T. W.; Deng, Z.; Mroz, A. M., Hendon, C. H.; Butler, K. T.;
Canepa, P., “Metal-free perovskites for non linear optical materials,”
Chem. Sci., 2019, 10, 8187-8194.
Mroz, A. M., “Pro- and Antioxidant Activity of Selenomethionine:
Preventative Measures against Metal-Mediated DNA Oxidation,” M.
Sc., Rose-Hulman Institute of Technology, Terre Haute, IN, 2017.
vii
ACKNOWLEDGEMENTS
This work would not have been possible without the support of my advisor,
Prof. Christopher H. Hendon, as well as my committee: Prof. James Prell, Prof.
Jeffrey Cina, and Prof. Gregory Bothun.
viii
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . 1
II. DENSITY FUNCTIONAL THEORY: AN OVERVIEW . . . . . . . . 7
Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . 8
Perturbation Theory . . . . . . . . . . . . . . . . . . . . . 10
Variational Method . . . . . . . . . . . . . . . . . . . . . 11
The Many-Body Hamiltonian and the Born-Oppenheimer Approximation 12
Thomas, Fermi, and Dirac . . . . . . . . . . . . . . . . . . . . 15
Hartree, Hartree-Fock, and the Slater Determinant . . . . . . . . . 16
The Birth of Density Functional Theory . . . . . . . . . . . . . . 19
1. The total energy of the system is a functional of
νext, and, by extension, n(r) . . . . . . . . . . . . . . . . . . 19
2. The electron density that yields the minimized total
energy is the ground state electron density, and may
be solved for using the variational principle . . . . . . . . . . . 21
Kohn-Sham Density Functional Theory . . . . . . . . . . . . . . 22
Exchange and Correlation . . . . . . . . . . . . . . . . . . . . 24
The Exchange-Correlation Hole . . . . . . . . . . . . . . . . 24
Defining Exchange-Correlation . . . . . . . . . . . . . . . . 25
Pair Correlation Function . . . . . . . . . . . . . . . . . . . 28
Self-Interaction Effects . . . . . . . . . . . . . . . . . . . . 29
Exchange-Correlation Approximations (i.e. the
functionals of DFT) . . . . . . . . . . . . . . . . . . . . . 29
General Calculation Considerations . . . . . . . . . . . . . . . . 33
ix
Chapter Page
Functional Selection . . . . . . . . . . . . . . . . . . . . . 36
Electronic structure modeling of extended solids . . . . . . . . . . 41
Periodic Models and k -Points . . . . . . . . . . . . . . . . . 42
Electronic Band Gap . . . . . . . . . . . . . . . . . . . . . 48
Electronic Band Structures . . . . . . . . . . . . . . . . . . 51
Electronic Band Dispersion . . . . . . . . . . . . . . . . . . 54
Density of States . . . . . . . . . . . . . . . . . . . . . . 55
Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
III.CHEMICAL SIZE OF ATOMS . . . . . . . . . . . . . . . . . . . 59
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Method Description . . . . . . . . . . . . . . . . . . . . . . . 59
Volumetric pixel resolution . . . . . . . . . . . . . . . . . . 60
Functional Benchmarking . . . . . . . . . . . . . . . . . . . . 61
Neutral and Neutral Bonded Systems . . . . . . . . . . . . . 69
Ionic and Ionic Bonded Systems . . . . . . . . . . . . . . . . 73
Ionic Liquids . . . . . . . . . . . . . . . . . . . . . . . . 80
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 82
IV.CHEMICAL SIZE OF PORES . . . . . . . . . . . . . . . . . . . 85
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Results and Discussion . . . . . . . . . . . . . . . . . . . . . 88
Pore Volumes and Surface Areas . . . . . . . . . . . . . . . . 88
Imaging Linker Vacancies using the Electric Field . . . . . . . . 92
x
Chapter Page
V. COVALENT BRIDGE INTERSTITIALS AS A ROUTE
TOWARDS CONDUCTIVE FRAMEWORKS . . . . . . . . . . . . . 97
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Results and Discussion . . . . . . . . . . . . . . . . . . . . . 101
Infinite wire models . . . . . . . . . . . . . . . . . . . . . 103
Calculation Details . . . . . . . . . . . . . . . . . . . . 103
Bulk wire formation simulations . . . . . . . . . . . . . . . . 108
Determining the degree of linker rotational freedom . . . . . . 109
Generalized statistical model . . . . . . . . . . . . . . . . . 112
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 114
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . 118
xi
LIST OF FIGURES
Figure Page
1. A timeline featuring the history of the atomic model (navy,
top panel), experimental size quantification methods
(periwinkle, middle panel), theoretical size quantification
methods (teal, lower panel), and major milestones in
computational chemistry (purple). . . . . . . . . . . . . . . . . . 2
2. Radii (y-axis) of Mg in two oxidation states (x-axis) for a
series of experimental size methods (CN=coordination number). . . . . 4
3. There exist several tiers of computational practices within
theoretical chemistry. These range in their description of
chemical system, and selection depends on system size, as
well as the target properties. . . . . . . . . . . . . . . . . . . . . 7
4. Jacob’s ladder of functionals where accuracy increases
with computational cost. Some common functionals
corresponding with each rung are listed on the right. . . . . . . . . . 38
5. Linear correlation between computed electronic band gap
and bipyridine dicarboylate (bipydc) incorporation in Zr-
UiO-67. The HSE06 (top) and PBEsol (bottom) functionals
show qualitatively the same trend, but PBEsol consistently
underestimates the band gap by 1 eV. The gray star
represents the extrapolated value derived from the best
fit line to account for a computational limitation. Data
obtained from ref (1). Figure reprinted with permission from ref. (2). . . . 40
6. Computed electronic band gaps of HKUST-1, UiO-67,
UiO-66, Zn-BTC, and linker functionalized derivatives.
HSE06 systematically predicts larger band gap energies
than PBE, more closely matching the experimental values.
Data obtained from ref (3). Figure reprinted with permission
from ref. (2). . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
xii
Figure Page
7. As size of a chemical system increases toward an infinite
solid, the number of molecular orbitals becomes very
large. For high symmetry materials, the crystallite can be
described using a smaller, repeating unit, whose projection
through reciprocal space generates electronic bands. Figure
reprinted with permission from ref. (2). . . . . . . . . . . . . . . . . 42
8. Although the smallest computational cell can save resources,
an improper model may be recovered if a chemical
interaction permeates beyond a single geometric cell.
For example, (a) structural distortions are sometimes
captured in temperature independent DFT (the ground
state may feature titling of nodes); (4) and (b) magnetic
ordering can be challenging if the unit cell contains
an odd number of coupling metals (for example, the
unit cell of Co2Cl2bis(1H-1,2,3- triazolo[4,5-b],[4,5-
i ])dibenzo[1,4]dioxin), (BTDD), contains 3 Co- (II),
which do prefer to order antiferromagnetically (5)). Figure
reprinted with permission from ref. (2). . . . . . . . . . . . . . . . . 44
9. General computational approach used to obtain electronic
structure properties from solid-state structures. (a)
Beginning with an experimentally obtained crystal
structure, partial occupancies must be resolved. Then,
where applicable, protons must be added to charge balance
the cell, in effect establishing the oxidation state of the
metals. Extrinsic solvent may be removed to simulate
the activated MOF. Symmetry can then be enforced
and a computational primitive cell may be available. (b)
The structure can then be equilibrated, and higher level
electronic structure properties can be obtained by sampling
the first Brillouin zone, including electronic band structures
and corresponding density of states. Figure reprinted with
permission from ref. (2). . . . . . . . . . . . . . . . . . . . . . . 49
10. An illustration of the band gap in a semiconductor as a
function of reciprocal space when the CBM and VBM are
at the same point in reciprocal space providing a direct
band gap and when they are at different points in reciprocal
space creating an indirect minimum absorption energy.
Figure reprinted with permissions from ref. (2). . . . . . . . . . . . . 51
xiii
Figure Page
11. Node or linker vacancies result in charge balancing ions
bound to the vacant sites (e.g., a proton or formate).
Both nodes and linkers can also be redox active forming
structurally bound charge carriers. Figure reprinted with
permissions from ref. (2). . . . . . . . . . . . . . . . . . . . . . . 57
12. Decreasing the volumetric pixel volume increases the time-
to-solution (dashed curves) exponentially, yet minimizes the
accuracy of the calculated volume (solid curves). Thus, we
take the kneepoint of the time curve as the ideal volumetric
pixel volume, 0.008 Å3. . . . . . . . . . . . . . . . . . . . . . . 61
13. Calculated STREUSEL volumes of H2 and CH4 optimized
with each basis set (x-axis), normalized to the volume
obtained from the aug-cc-pVTZ (y-axis). The normalized
volumes represent the precision of each basis set, a metric
of how close the returned electronic structure is to that
obtained using aug-cc-pVTZ. aug-cc-PVTZ is considered
sufficiently large – as demonstrated by the convergence of
normalized volumes with larger basis sets (i.e. cc-pVQZ and
aug-cc-pVQZ. . . . . . . . . . . . . . . . . . . . . . . . . . . 62
14. Calculated STREUSEL volumes of a series of industrially
relevant small molecules optimizes with each functional
(y-axis), normalized to the volumes obtained from the
CCSD-full functional (x-axis). The spread of normalized
volumes represents the precision of each functional, a metric
of how close the returned electronic structure is to that
obtained using CCSD-full, which is assumed to yield the
exact geometry. The forty-nine functionals are grouped by
functional type (generalized gradient approximation (GGA),
generalized gradient exchange (GGE), hybrid (H), meta
(M), hybrid-meta (HM), and local-density approximation
(LDA). To increase resolution, Ne and F2 are not included
in this plot and are discussed extensively in the main text. . . . . . . . 63
xiv
Figure Page
15. Calculated STREUSEL volume of a series of industrially-
relevant small molecules optimized with each functional
(y-axis), normalized to the volume obtained from teh
CCSD-full functional (x-axis). The spread of normalized
volumes represents the precision of each functional, a metric
of how close the returned electronic structure is to that
obtained using CCSD-full, which is assumed to yield the
exact geometry. The forty-nine ffunctionals are grouped by
functional type (generalized gradient approximation (GGA),
generalized gradient exchange (GGE), hybrid (H), meta
(M), hybrid-meta (HM), and local-density approximation (LDA). . . . . 66
16. The mean absolute deviation (y-axis) of two volume
methodologies (STREUSEL, van der Waals) across all
of the studied functionals (x-axis). The mean absolute
deviation is averaged across the mean absolute deviation
for industrially relevant small molecules was calculated
assuming the volume obtained from the CCSD-full
optimized geometry as the chosen measure of central
tendency. The mean absolute deviation is a metric of
how close the returned electronic structure is to that
obtained using CCSD-full, which is assumed to yield the
exact geometry. The forty-nine functionals are grouped by
functional type. . . . . . . . . . . . . . . . . . . . . . . . . . 68
17. Atomic radii for elements 1-96 of the periodic table
calculated using STREUSEL are presented. Neutral atoms
were modeled using functional CCSD-full with the SDDAll
basis set. The radii are calculated from the STREUSEL-
obtained volumes under the spherical approximation. . . . . . . . . . 70
18. A comparison of atomic radii recovered using various
conventional size metrics. a) Sizes computed from
STREUSEL are similar to those presented by Alvarez, and
Rahm and Boyd (top panel). b) Alternative size metrics are
presented as the upper and lower limits. . . . . . . . . . . . . . . . 72
19. Calculated STREUSEL ionic radii (y-axis) for relevant
oxidation states of atoms 1-86 (x-axis). . . . . . . . . . . . . . . . 74
xv
Figure Page
20. Comparison of proton count (nprotons)-to-electron count
(nelectrons) ratio (x-axis) for the first row transition metals
(y-axis) across a series of oxidation states (colors) featuring
data points correlated with calculated STREUSEL ionic
radii (data point size). . . . . . . . . . . . . . . . . . . . . . . 76
21. STREUSEL volumes (y-axis) for a series of six diatomic
molecules were calculated for the optimized structures, and
a series of models with expanded bond lengths (x-axis).
Individual ion volumes (i.e. Li+, F−, Cl−, and Br−) were
calculated and summed to obtain the maximum volume for
each ionic compound. . . . . . . . . . . . . . . . . . . . . . . . 78
22. Comparison of molecular volumes for a series of
geometrically equilibrated diatomic molecules with varying
dipole moments. Divergence between electronic density-
based size metrics and STREUSEL depends on the system
polarity and atomic electronegativity. . . . . . . . . . . . . . . . . 79
23. The Boyd (periwinkle), Bader (mint), and STREUSEL
(charcoal) surfaces of a series of diatomic molecules. All
surfaces are plotted using VESTA. The Boyd surface is
depicted at an isosurface value of 0.00015 eÅ−3. The Bader
surface is depicted at an isosurface value of 0.00030 eÅ−3.
The STREUSEL surface is dipicted at an isosurace value
relative to the minimum value of the electric field surface
mapped on the electrostatic potential map. . . . . . . . . . . . . . 81
24. Connolly surface area lacks information. van der Waals
surface area does not account for polarized bonds. . . . . . . . . . . 88
25. There is good agreement between the pore volumes obtained
using Platon and STREUSEL. Both void space calculation
metrics are performed using pristine structures. The solid
black line represents the values where Platon = STREUSEL. . . . . . . 89
26. Comparison of experimental pore volumes and STREUSEL
pore volumes for a series of MOFs. Data points are
colored according to the probe gas molecules used in
the experimental measurement. there appears to be
no correlation with the experimental probe gas; this is
likely due to unknown differences in sample preparation
and measurement conditions. The full dataset with
corresponding references is presented in Appendix B. . . . . . . . . . 90
xvi
Figure Page
27. Comparison of experimental surface areas calculated
via BET (green) and Langmuir (blue) isotherms and
STREUSEL surface areas for a series of MOFs. . . . . . . . . . . . 91
28. Comparison between published Langmuir surface areas for
experimental samples of MOF-5, the Connolly surface area
(purple), and the STREUSEL surface area (green). . . . . . . . . . . 93
29. Three passivated models of UiO-66 are examined in this
study; H2O, Cl
−/H2O, F
−/H2O, and formate. . . . . . . . . . . . . 96
30. The two examined positions for the covalent bridging
motifs are labelled within the MOF-5 subunit. Five
covalent bridging motifs were examined – polyaniline
(PANI), polyphenylene sulfide (PPS), two thienothiophene
derivatives (TT1 and TT2), and poly(p-phenylene vinylene)
(PPV). Covalent bridging motif lengths are displayed. . . . . . . . . . 102
31. The Fermi-aligned band structure of pristine MOF-
5 features flat bands with a large band gap (3.50 eV).
MOF-5 models featuring polyaniline (MOF-5-PANI) and
polyphenylene sulfide (MOF-5-PPS) interstitials reveal
increased band dispersion and smaller band gaps than their
pristine counterpart. Oxidized models of MOF-5-PANI and
MOF-5-PPS are also presented and feature bands that cross
the Fermi level. . . . . . . . . . . . . . . . . . . . . . . . . . 105
32. The Fermi-aligned band structure of pristine MOF-5
features flat bands with a large band gap (3.50 eV). MOF-5
models featuring two thienothiophene derivatives (MOF-
5-TT1, and MOF-5-TT2) reveal increased band dispersion
and smaller band gaps than their pristine counterpart.
Spin-polarized electronic band structures of the oxidized
models of MOF-5-TT1 and MOF-5-TT2 are also presented
and feature bands that cross the Fermi level. Spins are
separated by band color within the oxidized models. . . . . . . . . . 106
33. The vacuum-aligned band structure of pristine MOF-5
features flat bands with a large band gap (3.50 eV). MOF-5
models featuring polyaniline (MOF-5-PANI) and poly(p-
phenylene vinylene) (MOF-5-PPV) interstitials reveal
increased band dispersion and smaller band gaps than
their pristine counterpart. . . . . . . . . . . . . . . . . . . . . . 107
xvii
Figure Page
34. Two SH group orientations were sampled to determine the
degree of linker rotational freedom; 1) toward, in which the
SH groups of adjacent linkers point toward each other, and
2) away, in which the Cl groups of adjacent linkers point
away from one another. . . . . . . . . . . . . . . . . . . . . . . 109
35. a The bulk wire length distribution totaled over 100
simulations. One run resulted in a wire length of 15 units,
“model-15.” b The bulk wire length distribution for model-
15. The MOF-5 depiction of model-15 with the longest wire
highlighted in yellow. . . . . . . . . . . . . . . . . . . . . . . . 111
36. a Net theory is used to translate the crystalline structure
of MOF-5 to a tensor. b The tensor elements may then
be iterated to represent bulk wire formation in four
distinct steps; 1) the initial tensor is initialized with linker
orientations, 2) paths are then identified, 3) linkers that are
not in a wire are reoriented, steps 1) – 3) are repeated until
no linker reorientation will result in a new wire formation,
at this point, 4) the macroscopic connectivity is assessed.
For visual simplicity, we do not show the layers in the third
dimension – although our statistical model does take these
into account. . . . . . . . . . . . . . . . . . . . . . . . . . . 113
37. Unconnected linkers as a function of algorithmic step for a
series of model sizes using the generalized statistical model. . . . . . . 114
38. The bulk wire length distribution for a 6-2 net topology
allowing connections between 90deg linkers, totaled over 100
iterations of the general statistical model. . . . . . . . . . . . . . . 115
39. The bulk wire length distribution for a 6-2 net topology
allowing connections between 180deg linkers, totaled over
100 iterations of the general statistical model. . . . . . . . . . . . . 116
xviii
LIST OF TABLES
Table Page
1. Summary of properties embodied by five functional types:
local-density approximation (LDA), generalized gradient
approximation (GGA), meta-GGA (M-GGA), hybrid-GGA
(H-GGA), weighted density approximation (WDA). . . . . . . . . . . 33
2. A comprehensive list of the representative functionals from
each level of theory (ab initio, generalized gradient exchange
(GGE), local-density approximation (LDA), range separated,
generalized gradient approximation (GGA), hybrid-GGA
(H-GGA), hybrid-meta-GGA (HM-GGA), and meta-GGA
(M-GGA)) examined in this work. . . . . . . . . . . . . . . . . . 65
3. Molecular volumes for the relevant small molecules examined
in this study calculated using the van der Waals (vdW) and
the STREUSEL volume calculation methods. . . . . . . . . . . . . . 69
4. Molecular volumes (Å3) comparison between Bader, Boyd,
van der Waals and STREUSEL size quantification methods. . . . . . . 71
5. Boyd, Bader, Batsanov and STREUSEL-derived
molecular volume (Å3) comparison for two ionic liquids
possessing perhalometallate anions. The sum of single
molecule volumes for the individual constituents
([BMIM ]+, [V Cl − 2−4] , [CoCl4] ) are presented, as well as
the fully geometry-optimized ion pair. The percent difference
(%) is presented for each ionic liquid model. BMIM = 1-
butyl-3-methylimidazolium . . . . . . . . . . . . . . . . . . . . . 81
6. Comparison of electronic properties for the infinite wire
models listed in order of decreasing band dispersion (meV). . . . . . . 104
xix
CHAPTER I
INTRODUCTION
Chemical size is a fundamental property that governs a wealth of chemical
interactions, including steric and non-covalent interactions, (6;7;8;9;10;11;12) enzyme
docking, (13) and electrochemical properties, (14) in addition to the application
performance of porous solids, such as heterogeneous gas storage, (15;16;17), separation
technologies, (18;19) and catalytic activity. (20;21;22;23)
Quantifying chemical system size, however, is convoluted by the definition of
the atomic “surface,” since calculated sizes depend both on the model and chemical
environment. (14;24) For example, conventional size metrics predicated on the familiar
van der Waals radii are not directly applicable for ions in molecular electrolyte
systems because a sum of rigid ionic spheres poorly represents chemical regions of
high polarization. (14) Indeed, alternative size quantification methodologies, both
experimental and theoretical, are necessary.
Size quantification metrics have developed through a synergy of both
experiment and theory, (24) Figure 1. Initial experimental measurements of atomic
size were performed by Meyer in 1870, where he used a relationship between
material density and atomic size, and obtained a periodic trend in atomic
volumes. (25) These values were later refined by Bragg (26) and Pauling (27), who
developed methods for assessing atomic radii through X-ray scattering. Bond then
revisited the radii presented by Pauling and Bragg, and it is Bondi’s work that
underpins and is synonymous with the “van der Waals radii”. (28;29) Batsanov then
expanded the van der Waals radii for elements 1-96 through examination of single
crystal X-ray diffraction, molar volumes and crystal packing. (30) A more modern
approach was taken by Alvarez, (31;32) and Biswas and Ghosh, (33) who extracted
1
Figure 1. A timeline featuring the history of the atomic model (navy, top panel),
experimental size quantification methods (periwinkle, middle panel), theoretical size
quantification methods (teal, lower panel), and major milestones in computational
chemistry (purple).
atomic radii using a statistical analyses of online databases. Marrying all of these
approaches is atomic size derived from solid phase experiments and/or molecular
spectroscopic data. This, however, limits the ability to derive accurate atomic
and molecular volumes for environments not represented in empirical data, (24)
posing challenges for size quantifications of molecules with high polarity (34;35;36) and
unusual elongated bonds, (37;38;39) among others. These systems also pose problems
for size metrics that estimate covalent interactions, e.g. Pyykkö radii, (40) for the
same reasoning.
2
Independent of precise methodology, all size calculation metrics work to
identify the extent to which a chemical system permeates in space. Indeed, as
illustrated by the variety of methodologies presented above, there are different
ways of defining the surface of a chemical system. The advent of inexpensive
computational developments and hardware advancements has enabled a density
functional theory (DFT)-derived size calculation methods, particularly useful
for emerging exotic molecules. (41) Theory-based size metrics proved an accurate
path towards assessing atomic and molecular volumes of novel chemical systems,
assuming the theory is exact. Such calculations were pioneered by Slater who
employed the maximum radial density of outermost single particle wavefunctions
to define atomic radii. (42;43;44) Further theoretical size quantifications explored
deriving size from potential minima, (45;46;47) gradients of Tomas-Fermi kinetic
energy functional and exchange-correlation energy, (48) electronegativity derived
using DFT, (49) and electronic moment calculations. (50)
Bader (51) first identified the utility of quantum chemical simulation and
computed size at the Hartree-Fock level of theory, in defining the surface of a
chemical system and ultimately calculating the size by defining an electron density
cutoff of 0.002 e bohr−3. This cutoff was further refined by Boyd (52) to be 0.001
e bohr−3 with the justification that any smaller value of electron density would
result in a negligible change in calculated radii. (52) Recently, Rahm and Hoffmann
applied the Boyd cutoff to anions and cations using electron densities obtained
from DFT (24) with a hybrid GGA functional, PBE0 (53;54;55), and a sufficiently large
basis set, ANO-RCC. There, the authors highlight that since DFT is founded on
the Hohenberg-Kohn theorems, which state that the ground state energy of a
many electron system is defined by the distribution of the electron density, (56;57)
3
Figure 2. Radii (y-axis) of Mg in two oxidation states (x-axis) for a series of
experimental size methods (CN=coordination number).
DFT itself is an approximation of an exact exchange-correlation functional. Thus,
simply using DFT to compute size presents its own challenges as there are many
available functional types, which differ in their treatment of electron density, and
yield differing resultant properties. (58;59;60) Yet, the key benefit of theory-based
methods makes them attractive, as they are inherently general and may be applied
to both molecules and materials that are not presented in online repositories.
Further, the use of electron density poses problems for modeling cations,
who certainly interact with their surroundings bar beyond their electron cloud
via strong Coulombic interactions. (61) Thus, instead of examining the density of
electrons, this work revisits the size quantification problem through examination
of the electrostatic potential, and associated electric field. This method is, in
principle, robust even for polarized and ionic systems, in addition to enabling
quantification of atomic, molecular, and porous scaffold void size, and possibly even
analytical interaction energies.
4
While previous metrics are concerned with the space that a chemical system
resides in – occupied space – this work focuses on the volume that a chemical
system affects – affected space. This is best demonstrated by examining Mg
and Mg2+, Figure 2. Owing to the increased oxidation relative to the neutral
counterpart, cations are expected to possess a smaller occupied space; this is
reflected in published ionic size metrics, (26;62;63;64;65;66;67) which are compiled for Mg,
Figure 2. These metrics are inherently dependent on coordination number, resulting
in several accepted values for Mg, and reveals how close other chemical species may
bond with Mg. Yet, this does not reveal how close chemical species need to be to
be attracted to Mg. Perhaps counter-intuitively, electric field-based sizes increase
with increasing oxidation.
Within neutral atoms, the number of protons is equivalent to the number
of electrons; therefore, the net electric field produced is 0. Ions, however, possess
an inequivalent number of electrons and protons, thus, there exists a net field that
extends beyond the neutral atom – ultimately resulting in a larger affected size.
Thus, the electric field-based size metric presented in this work reveals that Mg2+ is
larger than Mg, Figure 2. In this way, we assess the known long-range interactions
associated with cations. (68;69)
The utility of this methodology is facilitated by the culmination of this
thesis – a post-DFT software termed STREUSEL (Structure Topology REcovery
Using Sampling of the ELectric field). (70) The validity of the approach facilitated by
this software is assessed via several atomic, molecular, and solid-state experiments
detailed in the following chapters.
Chapter 2 presents the background and subsequent derivations leading to
the eventual development of DFT. Chapter 3 outlines the atomic studies used
5
to validate this approach, as well as several molecular experiments depicting the
utility of the electric field-based size approach. Chapter 5 outlines the necessity of
electric field-based size in calculating the void space properties of porous scaffold,
namely metal-organic frameworks (MOFs). Chapter 6 expands on the application
of the presented size metric to MOFs, in which the presented size metric is used to
assess the likelihood certain MOF scaffolds will accommodate a variety of redox-
active interstitials. Chapter 7 outlines future applications, developments and
directions for the STREUSEL software package.
6
CHAPTER II
DENSITY FUNCTIONAL THEORY: AN OVERVIEW
The history of chemical modeling finds its foundation in the evolution of the
atomic model, Figure 1. While theories concerning the connectivity and shapes of
atoms date back to the 17th century, (71) we begin our discussion with the emergence
of the solid sphere atomic model postulated by John Dalton in 1808 (72) and the
eventual development of quantum mechanics in 1926. Within this time frame,
the scientific foundation of modeling was refined, eventually leading to the vast
computational toolbox available today. The simplified schematic presented in
Figure 3 features the models primarily applied within chemistry, which range in
level of theory.
Figure 3. There exist several tiers of computational practices within theoretical
chemistry. These range in their description of chemical system, and selection
depends on system size, as well as the target properties.
The highest level of theory, ab initio, is derived from quantum mechanics
and rely on varying methods of determining a solution to the Schrödinger Equation
7
and provide electronic insights into chemical phenomena. While the most accurate,
quantum mechanics-based methods are largely limited by system size.
Within the context of this thesis, all chemical modeling was performed using
density functional theory (DFT), as implemented in the Vienna Ab Initio Software
Package (VASP), (73;74;75;76) and Gaussian09. (77)
The chapter is organized as follows: We start with the six tenets of quantum
mechanics, followed by a discussion of the many-body Hamiltonian and the pivotal
role of the Born-Oppenheimer Approximation. We then move into a discussion
of the contributions of Tomas, Fermi, and Dirac; followed by the developments
presented by Hartree and Fock, and the role of the Slater Determinant. The
derivation of theory is culminated in a discussion of Kohn-Sham density functional
theory (DFT), which includes a practical discussion of exchange and correlation.
We conclude this chapter with a practical discussion of general calculation
considerations when using electronic structure theory, as well as a discussion of
electronic structure theory applied to the solid-state.
Quantum Mechanics
Several experiments performed in the late 19th and early 20th centuries
revealed the wave-particle duality of matter, leading scientists to eventually develop
the more general theory of quantum mechanics to describe these newly observed
phenomena. The birth of quantum mechanics was spurned by the discovery of
the photon credited to Planck, (78), notion of quantized light from Einstein, (79),
and the subsequent presentation of the atomic model presented by Bohr. (80) The
field was furthered by the introduction of the spin quantum number, (81) and the
development of the de Broglie wavelength, (82), which ultimately supported the
wave-particle duality hypothesis that is critical to modern quantum mechanics.
8
These developments enabled Heisenberg (83) and Born (84;85) to present a non-
relativistic theory of quantum mechanics. This was followed by the presentation
of the wave-mechanical description of the atom by Schrödinger, (86) and the
development of the Heisenberg uncertainty principle. (87) These developments
ultimately led to the tenets of quantum mechanics, (88) with which we will begin
our discussion of ab initio computational methods. We will follow the description
presented by McQuarrie: (89)
1. The exact state of any quantum mechanical system is described by its
wavefunction, Ψ(r, t), which is dependent on the positions (r) of the particles
in the system at a specific time (t). While Ψ(r, t) often contains imaginary
components, Ψ∗(r, t)Ψ(r, t) is real and represents the probability a particle
resides in r at t. Thus, by extension, the likelihood of a particle occupying
any region of space is ∫ ∞
Ψ∗(r, t)Ψ(r, t)dr = 1 (2.1)
−∞
2. There exists a quantum mechanical Hermitian operator whose expectation
value is real that describes every experimental observable.
3. Any measurement of an observable described by operator  will only be
eigenvalues (a) that satisfy
ÂΨ = aΨ (2.2)
Thus, a measurement of a system in eigenstate  will yields its eigenvalue,
a. The state of a system is not required to be an eigenstate of Â, and can be
expanded to a complete set of eigenvectors of Â,
ÂΨi = aiΨi (2.3)
9
as ∑n
Ψ = ciΨi (2.4)
i
However, if the observable is measured, only one of the eigenvalues (ai)
is observed, the identity of which is described as a probability. Upon
measurement of Ψ that returns an eigenvalue, ai, Ψ collapses to the
corresponding eigenstate (Ψi) associated with ai.
4. The average value of an observ∫able can be calculated using∞
< A >= Ψ∗ÂΨdτ =< Ψ|Â|Ψ > (2.5)
−∞
if the system is in a state that can be described by a normalized
wavefunction.
5. The time-dependent Schrödinger equation
∂Ψ
ĤΨ(r, t) = i~ (2.6)
∂t
describes the evolution of the wavefunction of a system in time.
6. With respect to all coordinates, the total wavefunction (Ψ(r, t)) must be
antisymmetric. Lastly, electron spin is a required coordinate.
There does not exist an exact solution to the Schrödinger equation for
systems larger than the Hydrogen atom. Approximate methods must be used to
obtain solutions; these include perturbation theory and the variational method.
Perturbation Theory. The major assertion of Perturbation Theory
is that the Hamiltonian (Ĥ, operator describing the total energy of a system) may
be split into two components, i) the “unperturbed” Hamiltonian, which can be
solved, and ii) the “perturbed” Hamiltonian, which cannot be solved. Contingent
on a small perturbation, this theory is used to continually refine the “unperturbed”
10
Hamiltonian such that the contributions of the “perturbed” Hamiltonian are
realized.
Variational Method. The Variational Method is an analytical
procedure whereby “trial” wavefunctions are continually guessed for the system.
These wavefunctions include a series of variable (“variational”) parameters, which
are iterated until the energy of the trial wavefunction is minimized. This method
is based on the Variational Principle, which states that any energy obtained using
the Variational Method is greater than the true total energy of the system. The
Variational Principle is derived as follows.
Let φ represent the trial wavefunction described by a linear combination of
exact eigenfunctions Ψi ∑
φ = ciΨi (2.7)
i
Thus, the approximated energy (E) of φ is∫
E[φ] = ∫φ∗Ĥφ
φ∗
(2.8)
φ
Substituting (2.7) into (2.8) yields ∑∑ ∗ ∫c c ∫Ψ∗ij i j i ĤΨjE[φ] = ∗ ∗ (2.9)
ij ci cj ΨiΨj
Recall, Ψj are exact eigenfunctions of Ĥ, thus
ĤΨj = εjΨj (2.10)
Using(2.10), and the considering that eigenfunctions of Ĥ form an orthonormal set,
we find ∑∑c∗i i ciεiE[φ] = (2.11)
i c
∗
i ci
Subtracting the exact ground state energ∑y, ε0 from (2.11)∗
E[φ]− ε i ci ci(εi − ε0)0 = ∑ ∗ (2.12)
i ci ci
11
Since ∑
c∑∗i i ci(εi − ε0)∗ ≥ 0 (2.13)
i ci ci
then
E[φ]− ε0 ≥ 0 (2.14)
E[φ] ≥ ε0 (2.15)
Thus any calculated energy of a trial wavefunction is always greater than or equal
to the exact ground state energy.
The Many-Body Hamiltonian and the Born-Oppenheimer
Approximation
We examine the formulation of the many-body Hamiltonian using the
organization presented by Szabo and Ostlund. (90) The many-body Hamiltonian,
Ĥ, accounts for the kinetic energy of the electrons (T̂e) and the nuclei (T̂n), as well
as the potential energy of electron-electron repulsion (Ûee), nuclei-nuclei repulsion
(Ûnn), and electron-nuclei Coulombic attraction (Ûen)
Ĥ = T̂e + T̂n + Ûe + Ûn + Ûen (2.16)
The kinetic energy terms take the form∑ ~2
T̂ 2n = − 5
∑ 2M Rii i (2.17)~2
T̂ 2e = − 5
2m ri
i e
where Mi is the mass of nucleus i, me is the mass of an electron, R represent the
nuclear coordinates, and r represents electronic coordinates. The potential energy
12
terms take the form
1 ∑∑ Z 2iZje
Ûnn =
2
1 ∑∑ 4π0|Ri −Ri 6 j|j=i e2
Ûee =
2 ∑∑ 4π0|ri − r |
(2.18)
i 6 jj=i
2
− ZieÛen =
4π0|R − r |i j i j
where Zx is the nuclear charge of nuclei x, and e is the elementary charge.
Consider a Hamiltonian that is a function of only two coordinate terms, q1
and q2; then
Ĥ = Ĥ1(q1) + Ĥ2(q2) (2.19)
By extension, the Schrod̈inger equation becomes
ĤΨ(q1, q2) = EΨ(q1, q2) (2.20)
Assuming that Ψ(q1, q2) is separable such that Ψ(q1, q2) = ψ1(q1)ψ2(q2), and ψa(qa)
is an eigenfunction of Ĥa possessing eigenvalue Ea, then
ĤΨ(q1, q2) = (Ĥ1 + Ĥ2)ψ1(q1)ψ2(q2)
= Ĥ1ψ1(q1)ψ2(q2) + Ĥ2ψ1(q1)ψ2(q2)
= E1ψ1(q1)ψ2(q2) + E2ψ1(q1)ψ2(q2) (2.21)
= (E1 + E2)ψ1(q1)ψ2(q2)
= EΨ(q1, q2)
Therefore, the product of the individual eigenfunctions yields the total
eigenfunction, and the individual eigenvalues are additive.
The many-body Hamiltonian, (2.16), is also a function of two coordinates,
nuclear (R), and coordinates of the electrons (r)
Ĥ(R, r) = T̂e(r) + Ûe(r) + T̂n(R) + Ûn(R) + Ûen(R, r) (2.22)
13
Problematically, Ûen(R, r) prevents the separation of the nuclear and electronic
components of the Hamiltonian and subsequently the separation of the
wavefunction.
Thus, we use the Born-Oppenheimer Approximation, which posits that
nuclei are stationary with respect to the reference frame of an electron, due to the
large difference in masses between these two particles. In this way, we examine a
fixed nuclear configuration, which leads to the electronic Schrod̈inger equation
Ĥeψe(R, r) = Eeψe(R, r) (2.23)
where Ĥe = T̂e(r) + Ûee(r) + Ûen(R, r) excludes the nuclear components. The total
energy (Etot) is, thus, described using the original Hamiltonian presented in (2.16),
Ĥψe(R, r)ψN(R) = Etotψe(R, r)ψN(R) (2.24)
Equation 2.24 is afforded only after invoking the Born-Oppenheimer approximation,
which allows us to separate the electronic (ψe(r)) and nuclear (ψN(R)) components
of the wavefunction (Ψ). Th{e elect∑ronic wav∑efunction∑is solve}d using1 Zn 1
Ĥeψe(R, r) = − 52i − + ψe(R, r)2 r r
i n,i ni i>j ij (2.25)
= Ee(R)ψe(R, r)
where the summation terms are iterated over electrons (i, j) and nuclei (n,m). The
resulting electronic energy (Ee) is then employed in the nuclear Hamiltonian to
solve for the total system{ener∑gy (Etot) ∑ }1 ZnZm
ĤNψN(R) = − 52n +Ee(R) + ψN(R)2M R
n n n>m nm (2.26)
= EtotψN(R)
Often commercial electronic structure theory software packages, such as Gaussian90
and VASP, solving the many-electron wavefunction return the total energy at a
14
fixed geometry, i.e. Ee, solving only the electronic Schrod̈inger equation. We turn
now to the varying approximations made so that the many-electron wavefunction
may be solved.
Thomas, Fermi, and Dirac
The culmination of contributions from Thomas, Fermi, and Dirac, termed
Thomas-Fermi-Dirac theory, yielded a theory that only depends on the total
electron density (ρ(r)). This simplifies the problem because the density is only a
function of three variables, the (x, y, z,) coordinates. The total energy of a system
(ETF ) is written as ∫ ∫ ∫ ∫ ′
ETF [ρ(r)] = A ρ(r)5/3
1 ρ(r)ρ(r )
k dr + ρ(r)vext(r)dr + ′ drdr
′ (2.27)
2 |r − r |
where the first term is the electronic kinetic energy associated with a system of
non-interacting electrons in a homogeneous gas, the second term is the classic
electrostatic energy of attraction between nuclei and electrons, and the third term
is the electronic Coulombic repulsion term. Thus, to assess the ground state energy
of a system within Thomas-Fermi theory (2.27) is minimized, while keeping the
number of electrons constant.
Defining the energy in terms of the electron density is powerful, yet there
are several disadvantages to the Thomas-Fermi theory; i) this theory does not
predict the bonding of atoms, thus solids and molecules cannot form, ii) the kinetic
energy is a gross approximation, in which small errors have a large impact on
the final computed value, and iii) the description of the Coulombic repulsion is
classical, neglecting the exchange-correlation interaction term. Moreover, owing to
its inability to self-consistently reproduce atomic shell structure, this theory fails.
To mediate the last disadvantage of the Thomas-Fermi approximation,
Dirac proposed an approximation for the exchange-correlation term based on the
15
homogeneous electron gas ( )1/3 ∫
−3 3Exc[ρ(r)] = ρ(r)3/4dr (2.28)
4 π
The exchange correlation term may then be written in terms of the exchange-
correlation energy(εxc[ρ(r)]), which is dependent on the Seitz radius (rs) and is
defined as ( )1/3
−3 9 1 ≈ −0.4582εx[ρ(r)] = (2.29)
4 4π2 rs rs
and the total energy is ∫
ELDAxc [ρ(r)] = ρ(r)εxc[ρ(r)]dr (2.30)
The Dirac exchange-correlation term, (2.30), was added to the Thomas-Fermi
theory, (2.27), yet this did not improve the Thomas-Fermi theory. The subsequent
improvement to this theory was the Hartree Approximation, which replaces the
many-electron wavefunction with a product of single-electron wavefunctions.
Hartree, Hartree-Fock, and the Slater Determinant
We turn our attention to solving the electronic wavefuntion (ψe(r)), where
we exclude the nuclear coordinates (R) for the remaining discussion. Thus far we
have neglected the spin component of the electrons; to account for the spin (ω),
we introduce an additional term, x = {r, ω}. The electronic wavefunction is a
function of xN , ψe(x1, x2, ..., xN), where N is the number of electrons in the system.
Including electronic spin accounts for the sixth postulate of quantum mechanics
(the precursor to the Pauli exclusion principle), which posits that a wavefunction
must be interchangeable across coordinates between fermions. This is represented
mathematically
ψe(x1, ..., xx, ..., xy, ..., xN) = −ψe(x1, ..., xy, ..., xx, ..., xN) (2.31)
16
The Hartree Approximation is the simplest approximation of electron-
electron interactions. Within this approximation, the true N -electron wavefunction
is replaced by a product of single particle orbitals
1
Ψ(r1s1, r2s2, ..., rnsn) = √ ψ1(r1, s1)ψ2(r2s2)...ψn(rnsn) (2.32)
N
where the single particle orbitals are comprised by a spatial function (φi(ri)) and an
electron spin function (σi(si))
ψi(risi) = φi(ri)σi(si) (2.33)
The electron spin function takes values of α, β, or , . Yet (2.33) does not satisfy
(2.31), thus the Hartree Approximation does not consider exchange interactions,
which is otherwise required by the Pauli exclusion principle.
Exchange interactions are accounted for in the Hartree-Fock theory, which
defines the wavefunction as an asymmeterised product of orbitals. Thus, the
Hartree-Fock wavefunction is reminiscent of a linear combination of terms in (2.33).
1
ΨHF = √ [ψ1(r1s1)ψ2(r2s2)...ψN(rNsN)−ψ1(r2s2)ψ2(r1s1)...ψN(rnsn)+ ...] (2.34)
N !
ΨHF can easily be represented∣∣ using the Slater determinant∣∣ ∣∣ ∣∣ψ1(r1s1) ψ1(r2s2) · · · ψ1(rNsN) ∣∣∣1 ∣∣∣ψ2(r1s1) ψ2(r2s2) · · · ψ2(r s ∣N N)ΨHF = √N ! ∣∣
∣
.. .. . . ..
∣∣ . .
. . ∣∣∣∣∣
ψ ∣N(r1s1) ψN(r2s2) · · · ψN(rNsN)
The orbitals within ΨHF fo∫llow
ψ∗i (r)ψi(r)dr = 〈ψi|ψi〉 = δij (2.35)
which is otherwise known as the orthonormal constraint.
17
The Hartree-Fock total energy (EHF ) is obtained by taking the expectation
value of the Hamiltonian (Ĥ),
EHF = ∑〈ΨH∫F |Ĥ|ΨHF(〉N )1
= ψ∗i (r) − ∇2 + νext(r) ψi(r)dr
i
N N ∫ ∫ 2
1 ∑∑ |ψ (r)|2|ψ (r)|2 (2.36)i j
+ ′ drdr
′
2 |r − r |
∑iN ∑jN ∫ ∫ ψ∗(r)ψ (r)ψ∗(r)ψ (r)
− 1 i i j j ′
| − ′
δ
| sjs
drdr
2 r r i
i j
The final term of (2.36) is the exchange energy, and is a product of the Pauli
exclusion principle. To complete (2.36), an ionic repulsion term must also be
included.
Minimizing (2.36) subject to (2.35) yields the Hartree-Fock ground state
energy (EHF0 ) and is accomplished by taking advantage of the Euler Lagrange
method. The stationary e(quation g∑iven∑by )N N
δ EHF0 − εij (〈ψi|ψj〉 − 1) (2.37)
i j
Importantly, εij represents a Hermitian matrix, and the Hartree-Fock equations are
s(ubsequently represented )
N ∫ N ∫
−1
∑ |ψ ′ 2 ∑ ′ ∗ ′j(r )| ψi(r )ψj (r )ψj(r)∇2 + νext(r) + ′ ψi(r)− ′ δs s dr′ = εiψi(r)2 |r − r | |r − r | i j
j j
(2.38)
Generally (2.38) are solved using the self-consistent field procedure. ψi(r) of (2.38)
are the ground state orbitals, also known as the self-consistent orbitals. Within
the self-consistent procedure, the set of orbitals is iterated until the energy is
minimized.
18
Yet, HF theory involves only one determinant of the electronic wavefunction,
neglecting a large number of atlernative, possible wavefunctions. It is statistically
unlikely that the true wavefunction for the chemical system is contained within
the set of iterative wavefunctions. Beyond this, single determinant is only truly
accurate for a non-interacting set of electrons; thus, an additional interaction
energy term is necessary and is achieved by including the correlation energy (Ec),
where
Ec = E0 − EHF (2.39)
and EHF ≥ E0, according to the variational principle. HF includes correlation
energy by using a linear combination of Slater determinants to account for excited
state configurations. This requires a lot of computing resources, which is feasible
for only small systems.
The Birth of Density Functional Theory
DFT addresses the the disadvantages of HF theory and is founded on the
Hohenberg-Kohn (HK) theorems, (91) which ultimately builds on the Tomas-Fermi
use of electron density (n(r)) to solve the quantum many-body problem. Consider
a system composed of electrons moving under the influence of an external potential
(νext(r)). HK-DFT posits that
1. The total energy of the system is a functional of νext, and, by
extension, n(r). The total energy functional (E[n(r)]) can thus be presented in
terms of νext ∫
E[n(r)] = n(r)νext(r)dr + F [n(r)]
(2.40)
= 〈Ψ|Ĥ|Ψ〉
19
where F [n(r)] is an unknown, univeral functional describing electron density, and Ĥ
is a function of the electronic Ĥ (F̂ ) and the external potential (V̂ext)
Ĥ = F̂ + V̂ext (2.41)
and F̂ relies on the kinetic energy operator (T̂ ) and the interaction operator (V̂ee)
F̂ = T̂ + V̂ee (2.42)
Since F̂ is consistent across all N-electron systems, Ĥ depends solely on the number
of electrons and V̂ext.
We can prove this theorem by reductio ad absurdom.
Consider the case where there are two external potentials (ν1, and ν2) that
yield the same electron density (n0(r)). Thus the Hamiltonian describing each
system (Ĥ1 and Ĥ2) each possessing a ground state wavefunction (Ψ1 and Ψ2).
By the variational principle and (2.40), we can generate an inequality relating the
ground state energies of each system (E01 and E
0
2)
E01 < 〈Ψ2|Ĥ1|Ψ2〉 = 〈Ψ2|Ĥ∫2|Ψ2〉+ 〈Ψ2|Ĥ1 − Ĥ2|Ψ2〉 (2.43)
= E02 + n0(r)[νext1(r)− νext2(∫r)]dr
Notably, upon exchanging the subscripts, (2.43) still holds. Let n0(r)[νextx(r) −
νexty(r)]dr = c. Then
E01 < E
0
2 + c (2.44)
E02 < E
0
1 + c (2.45)
Summing (2.44) and (2.45) yields
E0 + E0 < E01 2 2 + E
0
1 (2.46)
20
a direct contradiction. Therefore, the ground state electron density determines
the position of the nuclei within a chemical system, and give rise to the electronic
properties.
2. The electron density that yields the minimized total energy
is the ground state electron density, and may be solved for using the
variational principle. Considering that the electron density gives rise to the
external potential, and the Hamiltonian depends on the external potential and the
number of electrons, then the wavefunction is a functional of the electron density
(Ψ[n(r)]). Subsequently,
F̂ [n(r)] = 〈Ψ|F̂ |Ψ〉 (2.47)
We can then write the total energy functional in a ν-representable form (Eν [n(r))
in terms of another electron densit∫y (n′(r))
Eν [n(r)] = n
′(r)ν ′ext(r)dr + F̂ [n (r)] (2.48)
By the variational principle
〈ψ′|F̂ |ψ′〉+ 〈ψ′|V̂ ′ext|ψ 〉 > 〈ψ|F̂ |ψ〉 〈ψ|V̂ext|ψ〉 (2.49)
From (2.47) and (2.47), we arrive at the final representation of the second
Hoh∫enberg-Kohn theorem ∫
n′(r)ν ′ext(r)dr + F [n (r)] > n(r)νext(r)dr + F [n(r)]E [n
′
ν (r)] > Eν [n(r)]
(2.50)
The Hohenberg-Kohn theorems, however, do not provide a method of solving the
electron density of a ground state system. Instead, they provide the foundation to
the conventional DFT that is employed today – Kohn-Sham DFT.
21
Kohn-Sham Density Functional Theory
The Kohn-Sham (KS) formulation maps the system containing the
interacting electrons onto a fictitious system of non-interacting electrons whose
movement is described by a series of KS single particle potentials (νKS(r)). From
the Hohenberg-Kohn theorems, we may achieve the ground state energy of a system
by minimizing the energy functional, (2.40), while keeping the total number of
electrons consta[nt, ∫ (∫ )]
δ F [n(r)] + νext(r)n(r)dr − µ n(r)dr −N = 0 (2.51)
where µ is
δF [n(r)]
µ = + νext(r) (2.52)
δn(r)
By considering a non-interacting system of electrons, the exact kinetic
energy can be calculated directly using the KS formalism.
Similar to HF theory, the KS ground state wavefunction (ΨKS) is a
combination of single-particle orbitals (ψi(ri))
ΨKS = √
1
det [ψ1(r1)ψ2(r2) · · ·ψN(rN)] (2.53)
N !
The unknown, universal functional describing the electron density, F [n(r)], is
composed of three terms
F [n(r)] = Ts[n(r)] + EH [n(r)] + Exc[n(r)] (2.54)
Here, Ts[n(r)] is the kinetic energy of the non-interacting electron∫g∫as, EH is the
classical electrostatic energy of the electrons where E [n(r)] = 1 n(r)n(r
′)
H | − ′| drdr
′,
2 r r
and Exc is the exchange-correlation energy that speaks to the difference between
the non-interacting kinetic energy and the exact kinetic energy, as well as the non-
classical contribution of the electron-electron interactions.
22
Expanding (2.52) leads to
δTs[n(r)]
µ = + νKS(r) (2.55)
δn(r)
The KS potential, νKS9(r), is a function of the external potential (νext(r)), Hartree
potential (νH(r)), and the exchange-correlation potential (νxc(r))
νKS(r) = νext(r) + νH(r) + νxc(r) (2.56)
where ∫
δEH [n(r)] n(r
′)
νH(r) = = dr
′
δn(r) |r − r′|
(2.57)
δExc[n(r)]
νxc(r) =
δn(r)
Similar to HF theory, we obtain the ground state electron density and total energy
by solving for the N 1-elec[tron Schrödinger]equations
−1∇2 + νKS(r) ψi(r) = εiψi(r) (2.58)
2
where εi is the Lagrange multiplier associated with the orthonormality of the single
particle states, ψi. The electron density ∑(n(r)) is thenN
n(r) = |ψi(r)|2 (2.59)
i=1
and the non-interacting kinetic energy (Ts[∫n(r)]) isN
−1
∑
Ts[n(r)] = ψ
∗
i (r)∇2ψi(r)dr (2.60)2
i=1
Similar to the HF scheme, the Kohn-Sham equations, (2.56) - (2.60), are solved
self-consistently. As opposed to solving one equation for the electron density
directly (as is done in Tomas-Fermi theory), the N one-electron equations must
be solved. This is quite advantageous; when system complexity increases (i.e.
size increases), only the number of equations that must be solved changes. Yet,
KS-DFT is still an approximate method – the exact exchange correlation energy
23
(Exc[n(r)]) is not known. The implicit form,
Exc[n(r)] = T [n(r)] + Ts[n(r)] + Eee[n(r)]− EH [n(r)] (2.61)
depends on the exact kinetic energy (T [n(r)]), the KS kinetic energy (Ts[n(r)]),
the exact electron-electron interaction energy (Eee[n(r)]), and the Hartree energy
(EH [n(r)]). Subsequent developments to DFT focus on developing an increasingly
accurate description of this exchange-correlation energy term.
Exchange and Correlation
The Exchange-Correlation Hole. Consider the non-relativistic many-
body electronic Hamiltonian,
Ĥ = T̂ +∑V̂ext + V̂eeN1 ∑N ∑N ∑N 1 (2.62)
= − ∇2i + νext(r) +2 |r − r′|
i=1 i=1 i=1 j>i
which depends on the kinetic energy (T̂ ), the external potential (V̂ext), and the
electron-electron interaction energy (V̂ee). The expectation value of V̂ee takes the
form ∫ ∫
〈 | | 〉 1 P (r, r
′)
Ψ V̂ee Ψ = dr
′dr (2.63)
2 |r − r′|
where Ψ is the normalised antisymmetric ground state wavefunction of the system,
and P (r, r′) is the pair d∫ensity∫ function,
P (r, r′) = N(N − 1) · · · |ψ(rs, r′s′, r3s3, · · · , rNsN)|2dr3s3 · · · drNsN (2.64)
The pair density function describes the probability that a pair of electrons reside at
points r and r′, respectively. Perhaps more useful, we can write the electron density
as a function of the pair density function, ∫
1
n(r) = P (r, r′)dr′ (2.65)
N − 1
24
∫
Since n(r)dr = N , ∫ ∫
P (r, r′)dr′dr = N(N − 1) (2.66)
Within the classical description, electrons are not correlated. Thus, the pair density
function takes the form
P (r, r′) = n(r)n(r′classical ) (2.67)
Using (2.67) would be incorrect, however, because electrons are subject to
Fermi statistics. We must account for quantum mechanical forces, such as non-
Coulombic interactions and the Pauli exclusion principle. The exchange-correlation
interactions ultimately reduces the electron density at r due to its pair at r′; in this
way, each electron is associated with a volume of decreased electron density (e.g.
hole) surrounding itself. This can be accounted for by including another term in
the pair density function, which becomes
P (r, r′) = n(r)n(r′) + n(r)nxc(r, r
′) (2.68)
where nxc(r, r
′) is the exchange-correlation hole density, containing the quantum
mechanical effects surrounding each electron. Importantly, nxc(r, r
′) must follow the
normalisation condition ∫
n ′ ′xc(r, r )dr = −1 (2.69)
which asserts that the hole corresponds with a decrease in electron density and
cancels out the charge of one electron.
Defining Exchange-Correlation. The exact definition of Exc[n(r)] can
be obtained using coupling constant integration. Within this method, a coupling
constant (λ) that describes the relationship between the non-interacting system
and the interacting system in defined. At its core, this term describes the strength
of the electron-electron interactions within the system. The exchange-correlation
25
energy is thus ∫ ∫
1 nxc(r, r
′)
Exc[n(r)] = n(r)dr dr
′ (2.70)
2 |r − r′|
where nxc(r, r
′) is the exchange-correlation hole, which is averaged over the coupling
constant-dependent hold (nλxc(r, r
′)) ∫ 1
n (r, r′) = nλxc xc(r, r
′)dλ (2.71)
0
from which the exchange-correlation energ∫y density (εxc[n(r)]) can be defined
1 nxc(r, r
′)
εxc[n(r)] =
′
2 |r − r′
dr (2.72)
|
While this definition holds, we do not know the exact form of nxc(r, r
′), and, thus,
cannot solve the many-body Schrödinger problem. Yet, this method provides the
relationship between the interacting and non-interacting kinetic energies, as well
as the connection between the electron density pivotal to DFT and the many-body
wavefunction with the definition of the exchange-correlation hole.
Importantly, we can show that the magnitude of the exchange-correlation
hole density will always be smaller than the electron density, from (2.69) with the
pair density at zero, leading to
nxc(r, r
′) ≥ −n(r′) (2.73)
An important characteristic of the pair density is that it is symmetric under
coordinate exchange,
P (r, r′) = P (r′, r) (2.74)
In combination with (2.68), we can define
P (r, r′) = n(r)n(r′) + n(r)nxc(r, r
′)
(2.75)
P (r′, r) = n(r′)n(r) + n(r′)nxc(r
′, r)
26
using (2.74), it follows that the equations of (2.75) are equivalent
n(r)n(r′) + n(r)nxc(r, r
′) = n(r′)n(r) + n(r′)nxc(r
′, r) (2.76)
solving for n (r, r′xc ) yields
n(r)n (r, r′xc ) = n(r
′)n(r) + n(r′)n ′xc(r , r)− n(r)n(r′)
(2.77)
= n (r′, r′xc )n(r
′)
leading to
′
n (r, r′) = n (r′
n(r )
xc xc , r) (2.78)
n(r)
The exchange and correlation components of the exchange-correlation hole
density are independent,
n (r, r′) = n (r, r′xc x ) + n
′
c(r, r ) (2.79)
where the exchange (Fermi) hold (nx(r, r
′)) is
nx(r, r
′) = nxc,λ=0(r, r
′) (2.80)
and the correlation (Coulomb) hole (nc(r, r
′)) is
nc(r, r
′) = n ′xc,λ=0(r, r )− n (r, r′x ) (2.81)
Fortunately, the exchange hole may be known exactly using the Hartree-Fock
definition of the exchange correlati∫on energy∫(Ex),
1 nx(r, r
′)
E = n(r)dr dr′x (2.82)
2 |r − r′|
where nx(r, r
′) is a fucntion of the spin∑orb[i∑tals (ψj(r, s)), ]N 21
nx(r, r
′) = − |ψ∗j (rs)ψ ′j(r s)| (2.83)n(r)
s j
Ultimately resulting in, ∫
nx(r, r
′)dr′ = −1 (2.84)
27
From (2.69) and (2.79) we can ass∫ert
nc(r, r
′)dr′ = 0 (2.85)
Pair Correlation Function. For simplicity, we can redefine n ′xc(r, r ) in
terms of a pair-correlation function (g ′xc(r, r )),
nxc(r, r
′) = n(r′)[g ′xc(r, r )− 1] (2.86)
and ∫ 1
g ′ λxc(r, r ) = gxc(r, r
′)dλ (2.87)
0
Thus, far away from electrons, the pair correlation function tends towards unity
and we know that nxc(r, r
′) decreases with increasing r′. For continuity, we can also
show that the pair-correlation function is, in fact, a probability. Let P (r, r′) = 0 of
(2.68), which leads to
0 = n(r)n(r′) + n(r)nxc(r, r
′) (2.88)
Solving for nxc(r, r
′),
′ −n(r)n(r
′)
nxc(r, r ) =
n(r)
(2.89)
= −n(r′)
Expanding (2.86) leads to
nxc(r, r
′) = n(r′)g (r, r′xc )− n(r′) (2.90)
Thus, from (2.89) we know that when P (r, r′) = 0, then gxc = 0. This essentially
tells us that the probability of an electron at r′ given an electron is located at r is
dictated by g ′xc(r, r ).
While a universal model for gxc(r, r
′) is non-existent, we may calculate the
exact pair-correlation function for specific cases. This method is routinely employed
for quantum Monet Carlo techniques.
28
Self-Interaction Effects. Another important attribution of the
exchange-correlation hole is that it cancels the self-interacting effects that are
installed by the Hartree term. We can demonstrate this effectively by observing
a 1-electron system, which we can model by setting one of the spin densities to 0
(nβ = 0), which leads to ∫ ∫
nα(r)dr = n(r)dr = 1 (2.91)
Thus, we obtain the exact kinetic and potential energies of the system and
P (r, r′) = 0 due to the lack of electron-electron interactions. Further, gxc(r, r
′) = 0
because there is only one electron in the system; (2.86) thus becomes
nxc(r, r
′) = n(r′)[gxc(r, r
′)− 1]
(2.92)
= −n(r′)
In this way, the exchange-correlation energy cancels the Hartree Energy of the
system, thereby eliminating the self-interaction effects.
We can also define the exchange (νx,α) and correlation (νc,α) potentials,
νx,α([nα(r)]) = −νH([n(r)]) + C1
(2.93)
νc,α([nα(r)]) = 0 + C2
where νH is the Hartree potential, and νx,α and νx,α are only definable up to a
constant, C1, C2, respectively. Most functions are not sufficient, resulting in a sel-
interaction error (ESIE) in the total energy,
ESIE = EH [n(r)] + Exc[nα(r), 0] (2.94)
In this way, the self-interaction error allows us to assess the degree of self-
interaction still present with a given exchange-correlation approximation.
Exchange-Correlation Approximations (i.e. the functionals of
DFT). Functionals are employed that model the exchange-correlation hole to
29
varying degrees of accuracy. The general form of an exchange-correlation functional
(Exc[n(r)]) is ∫
Exc[n(r)] = n(r)εxc(r)dr (2.95)
where εxc is the energy density. Essentially, functionals differ in the way that they
determine εxc(r) surrounding each electron. We will discuss five functional types:
1. Local Density Approximation (LDA) – Within this framework, the true
exchange-correlation energy is approximated locally as the exchange-
correlation energy of a homogeneous electron gas of the same density. This is
because the exchange-correlation energy is known exactly for a homogeneous
electron gas. Thus, LDA only depend∫s on the local density, leading to
ELDAxc [n(r)] = n(r)ε
homo
xc [n(r)]dr (2.96)
where εhomoxc is the energy density of the homogeneous electron gas.
ELDAxc [n(r)] may be separated into exchange E
LDA
x [n(r)] and correlation
ELDAc [n(r)] contributions. E
LDA[n(r)] is known, whereas ELDAx c [n(r)] is solved
for in an iterative manner. LDA is known to perform better for solids than
for molecules.
2. Generalized Gradient Approximation (GGA) – GGA is built from the
gradient expansion approximation (GEA), which was originally suggested
by Hohenberg and Kohn as an extension to LDA. Within this approximation,
additional higher order density gradient terms are included in the definition
of the exchange-correlation energy. Yet, first-order GEA does not follow
the non-positivity constraint, where the exact on-top conidtion for a pair of
opposite spin electrons nx(r, r
′) = −n(r), nor does GEA follow the sum rule of
30
(2.85). GGA is an expansion of GEA – ultimately solving the aforementioned
challenges with GEA.
Perdew presented a procedure that cuts off the GEA exchange hole in real
space. (92) This remedies the original challenges faced with GEA. The GGA
exchange-correlation energy (EGGAxc [n(r)]) is commonly written in terms of
the enhancement factor (Fxc[n(r),∇n(r)]), which describes the differences
between the present system a∫nd a homogeneous electron gas,
EGGAxc [n(r)] = n(r)ε
homo
xc [n(r)]Fxc[n(r),∇n(r)]dr (2.97)
where Fxc is commonly written as a function of the seitz radius (rs) and the
dimensionless reduced density gradient (s(r)),
|∇n(r)|
s(r) = (2.98)
2kF (r)n(r)
kF is the Fermi wavevector,
k 2 1/3F = [3π n(r)] (2.99)
Fxc(rs, s) takes different forms for each GGA functional. Often GGA
functionals are compared by plotting the enhancement factors for a series
of rs values.
3. meta-GGA (M-GGA) – M-GGA functionals differ from conventional GGA
functionals in that they are composed of additional semi-local terms, such as
additional higher order density gradients and kinetic energy density terms
(τ(r)), ∑occ.1
τ(r) = |∇ψi(r)|2 (2.100)
2
i
Integrated τ(r) yields the non-interactin∫g kinetic energy (Ts[n(r)])
Ts[n(r)] = τ(r)dr (2.101)
31
The general form of∫the M-GGA exchange-correlation energy is
EMGGAxc [n(r)] = f [n(r),∇n(r),∇2n(r), τ(r), µ(r), · · · , γ(r)]dr (2.102)
where µ(r) · · · γ(r) represent additional semi-local quantities. M-GGA
functionals are typically constructed using empirical molecular data, which
introduces an inherent bias.
4. hybrid-GGA (H-GGA) – As their name suggests, H-GGA functionals are
a hybrid of exact Hartree-Fock exchange and conventional GGA. H-GGA
functionals commonly take the general form,
EHGGA = α(EHF − EGGA) + EGGAxc x x xc (2.103)
where EHFx is the Hartree-Fock exchange using Kohn-Sham orbitals, and α
is the exchange mixing coefficient, which is typically fitted semi-empirically.
This was further supported by Becke, CITE THIS who approximated the
adiabatic connection integral fo∫r the exact exchange-correlation as1 1 1
Exc = U
λdλ = U0 + U1 (2.104)
0 2 2
λ = 0 is the exchange-only limit, thus, λ = 1 is the most local part of the
electron interactions.
5. Non-Local – Fully non-local approximations employ the exact density
functional expression for Exc[n(r)], in addition to describing the exact
exchange-correlation hole using analytical functions. Two examples of fully
non-local approximations are the Average Density Approximation (ADA),
and the Weighted Density Approximation (WDA). The general expression
takes the form ∫ ∫
NL 1 n
model(r, r′)
Exc [n(r)] = n(r)dr
xc dr (2.105)
2 |r − r′|
32
There are both advantages and disadvantages to using fully non-local
approximations. Notably, these functional types do not involve as many
assumptions as their counterparts; in this way they achieve forms that are
closer to the exact functional. While this is powerful, these functionals are
highly computationally intensive, and intractable for many systems.
Table 1 summarizes properties of the discussed exchange-correlation
functional types.
Table 1. Summary of properties embodied by five functional types: local-density
approximation (LDA), generalized gradient approximation (GGA), meta-GGA (M-
GGA), hybrid-GGA (H-GGA), weighted density approximation (WDA).
Property LDA GGA M-GGA H-GGA WDA Exact
non-empirical  †  −−
Localityα L SL SL NL/SL NL NL
explicit local
  
exchange hole
explicit εxc(r)   
limr→∞ ε
−αr −αr −αr 1 1 1
xc(r) −e −e −e − − −2r 2r 2r
limr→∞νxc(r) −e−βr −e−βr −e−βr −− − 1 −12r r
HEG limitβ     
self-interaction
‡ 
correction
† - GGAs may also be defined semi-empirically
α - Local (L), Semi-Local (SL), Non-Local (NL)
‡ - typically only valid in principle
β - Homogeneous Electron Gas (HEG)
General Calculation Considerations
This section borrows heavily from my published work, (2) in which we discuss
the application of electronic structure theory to metal-organic frameworks (MOFs).
Electronic structure calculations provide insight into chemical systems at
a fidelity that may be evasive with experimentation alone. The reliability and
robustness of the calculations depend not only on the structure and identity of the
33
chemical system but also on the functional and basis set. To judge the accuracy
of a model, calculated results are compared with experimental or higher level
computational data (93) (e.g., band gap for electronic properties, bond length/angles
and lattice parameters for structure, or vibrations and formation enthalpies to
investigate stability). (94;95) The interdependence of functional and structure poses
a multifaceted challenge of functional selection to obtain both realistic physical
parameters and electronic wave functions.
By far the most important factor in obtaining reliable computational
data is using reliable atomic coordinates. As such, it is good practice to perform
electronic calculations on structures equilibrated at that level of theory; small
deviations in atomic positions can result in large electronic disparities. (96) Yet, it is
routine to report high-level electronic structure calculations on geometries obtained
using structures equilibrated at lower levels of theory; (97;98) we will later discuss
the implications of doing so, and why it has been so successful. Still, there are
reports of practitioners using experimental crystallographic positions in density
functional theory (DFT) models for large and complex systems when computational
equilibration exceeds the abilities of the computational resources available; this
has proven useful in rationalizing experimental phenomena. (99;100) While none of
these approaches provide the true ground state electronic density associated with
the terminal functional used for the reported the electronic properties, (101) the
models themselves are reproducible and consistently provide useful insights into
the chemistry of both molecules and MOFs alike. (1;97;102)
Although one may become disenchanted with computational approaches
because, at face value, it would appear one could “turn knobs” to recover
essentially any desired result, responsible benchmarking and control studies curtail
34
problematic model reductions and parametrizations. In reality, no DFT approach
is perfect because the nature of the functional that recovers the exact density and
energy is unknown and a vast array of functionals and corrective terms exist simply
to account for deficiencies in the mathematical description of electron exchange
and correlation. But it is worth remembering what modelers are trying to achieve;
fundamentally, the primary objectives of an electronic structure model are 2-fold:
(i) be reproducible, such that future studies invoking a published methodology may
build from past results, and (ii) effectively describe the realistic chemical system in
light of the various approximations and additions that were chosen during model
construction. The many knobs thus exist because of the breadth of physics present
in MOFs. For example, it may be necessary to add additional dispersion terms
to recover computational lattice parameters in highly conjugated frameworks
that more closely match experiment and spin-orbit coupling (SOC) may not be
important for tetrahedral Zn(II), but will certainly be needed for tetrahedral
Co(II). (103)
Of course, the basis set also plays a role in the accuracy of computed
properties. Here we will only make a few general remarks but refer the interested
reader to an excellent series of previous works. (104;105;106;107;108) For polar
compounds, and MOFs in particular, it is important to describe the localization
of electrons and their polarization and diffusivity. To do so requires a basis set of
“sufficient size”, which refers to the minimum number of functions necessary to
accurately describe the shape of the electron density. Basis functions themselves
come in many forms, the most intuitive being “atom-centered”, (104;109;110;111)
which construct molecular orbitals or bands from some linear combination of
atomic orbitals. In addition to inclusion of valence orbitals, other functions
35
can be added to describe diffuse electron density far from the nuclei (useful for
noncovalent interactions and anions) and electronic polarization (arising from
chemical interactions between atoms with different electronegativities). The most
common basis in solid-state calculations is a series of plane-waves, (73;112) which
simultaneously enable the description of the bonding interactions within the unit
cell while also enabling sampling of long-range interactions beyond the periodic
boundary condition. The basic assumption for the remainder of this Review is
that a sufficiently large basis set has been used to properly account for diffuse and
polarized electron density. (113)
One further assumption for charge analysis and other postprocessing
techniques is a preconverged wave function generated from a self-consistent field
(SCF) routine, (114) i.e., for a given geometry, the electronic component of the
Hamiltonian has reached the minimum value within a certain tolerance. The
following subsection will serve to detail functional and basis set selection when
applying electronic structure theory to MOFs.
Functional Selection. As discussed previously, DFT is founded on the
theorems of Hohenberg and Kohn, which purport that the energy of a chemical
system is a direct function of the ground-state electron density. (91) In practice,
the Kohn-Sham (56) formalism is used to construct the ground-state electron
density of interacting electrons in a static external potential from density functions
describing individual, noninteracting electrons. The main shortcoming of DFT,
however, is that there is no known functional that satisfies the Hohenberg and
Kohn theorem, and hence we are unable to compute the exact energy or ground-
state density of a system. Instead, a range of available functionals exist that vary
in their approximate treatment of the exchange-correlation term of the Hamiltonian
36
(where exchange and correlation are purely quantum mechanical effects that can
be thought of as a repulsive and attractive term, respectively); the challenge lies
in identifying which functional provides the best electron density or energy for a
particular chemical system based on the parameters considered during functional
construction and the dominant physics present in the MOF. There is ongoing
dialogue about the relative advantages of employing a functional parametrized to
better approximate the “real” electron density, or “real” energy; however, there is
room in electronic structure modeling for both philosophies. (115)
Within the DFT construct, functionals are divided into broad classes based
on their treatment of the electron density gradient; this hierarchy of electronic
structure approaches is sometimes referred to as Jacob’s ladder, Figure 4. (116;117)
The bottom rung of the ladder begins with the local (spin) density approximation
(LDA, LSDA), which assumes a homogeneous distribution of electrons throughout
the material. LDA functionals therefore are readily applied to metals, i.e., materials
with nonzero density of states (DOS) at the Fermi level (EF ), such as bulk
platinum. Some common LDA functionals include VWN (118) and PZ81. (119)
However, it would be misguided to apply any variation of LDA to the subset of
metallic MOFs (120;121;122;123) (e.g., Ni3(HITP) ),
(124;125)
2 despite their nonzero DOS
at EF . The electron density is not homogeneously distributed throughout the
scaffold; the electron density on the Ni and N are vastly different and the bond
will be improperly described.
Instead, the generalized gradient approximation (GGA) (126;127) and related
methods are applied to systems with varying electron density (e.g., molecules,
MOFs, inorganic clusters, etc.). GGA considers both the electron density and
the gradient of the electron density when recovering energetic values and hence
37
Figure 4. Jacob’s ladder of functionals where accuracy increases with
computational cost. Some common functionals corresponding with each rung are
listed on the right.
enables better descriptions of systems with inhomogeneous charge density, such
as metal-organic hybrids. Familiar “pure” GGA functionals include PBE (54)
and PW91, (127) and these have been widely invoked in both the molecular and
solid-state communities for both structure and derivative electronic property
analysis. (128;129)
Still, the core shortcoming of all standard GGA approaches (and DFT
approaches in general) is their inability to correctly describe both electron exchange
and correlation; (130) a variety of more extravagant GGA methods combining
progressively more complicated mathematics have been implemented in order
to better describe these two key components of electronic structure theory. So-
called meta-GGA functionals (such as M06-L (131) and TPSS (132) historically
invoke both the first and second derivative of the electron density to describe the
system, while hybrid GGA functionals include some amount of exact electron
exchange as computed with the Hartree-Fock exchange functional (HF). The
GGA functional PBE, for example, becomes the hybrid functional PBE0 (133) upon
addition of 25same functional can have a screened potential added, becoming
38
HSE06. (134) Conversely, the most ubiquitous hybrid GGA functional, B3LYP, (135)
has components of exchange computed with HF, LDA / VWN, and GGA / B, (136)
in addition to the exchange and correlation computed at the GGA / LYP (53) level.
Given the diverse library of functionals to choose from, it is unsurprising
that both experimentalists and theorists can feel overwhelmed in the early stages
of model construction. Indeed, the most common practice is to invoke a well-
documented functional known to recover the appropriate electron density or
energy for similar chemical systems. With this in mind, hybrid- GGA methods
are widely viewed as the minimum level of theory necessary to describe the
electronic structure of systems where exchange and correlation play a major role
in the electronic structure. This is particularly pertinent in MOFs containing
spin-polarized transition metals, (137) where exchange interactions not only play a
leading role in the energy of the material, but also define the nature of the frontier
orbitals/bands, as well as the band gap of semiconducting materials. In light of
these considerations, by far the most common hybrid functionals used in MOF
modeling are PBE0, HSE06, B3LYP, M06, and variations thereof, (138) despite their
significant increase in computational demand. (139)
Rather than boldly sticking with one functional, systematic functional
analyses may be performed to identify the optimal exchange-correlation treatment
for each material or study. DFT results can benchmarked to quantities derived
from experiment or higher level ab initio calculations such as band gap energy, (140)
ionization potential, (98) lattice parameters, (93) and formation enthalpies. (3) One
study monitored the incorporation of bipyridine dicarboxylate (bipydc) linkers into
UiO-67 (a MOF formally made from biphenyldicarboxylate, bpdc) by comparison
of computed band gap and experimental lattice parameter contraction. There,
39
Figure 5. Linear correlation between computed electronic band gap and bipyridine
dicarboylate (bipydc) incorporation in Zr-UiO-67. The HSE06 (top) and PBEsol
(bottom) functionals show qualitatively the same trend, but PBEsol consistently
underestimates the band gap by 1 eV. The gray star represents the extrapolated
value derived from the best fit line to account for a computational limitation. Data
obtained from ref (1). Figure reprinted with permission from ref. (2).
they found excellent agreement between experimental lattice parameters and unit
cells optimized with the PBE functional corrected for solids (PBEsol), Figure 5. (1)
In the same study, the PBEsol (141) yielded consistently smaller band gaps than
HSE06 (HSE06 was routinely greater by 1 eV), but qualitatively comparable
trends in electronic differences as a function of substitution are observed. A similar
conclusion was reached in another study examining a larger selection of MOFs
with an analogous metal and ligand chemistry, Figure 6. (3) The data in Figures
5 and 6 also highlight a common theme in the literature: GGA functionals tend
to underestimate both the experimental and hybrid-computed band gap, and the
addition of exact (HF) exchange is necessary to recover more realistic electronic
band gaps. (142;143;144) In sum, assuming a sufficiently high-fidelity basis set, pure
GGA functionals are the lowest level of theory required to provide a reasonable
electronic description of most MOFs. However, the recovery of experimental
semiconducting properties requires the incorporation of some exact exchange.
40
Figure 6. Computed electronic band gaps of HKUST-1, UiO-67, UiO-66, Zn-BTC,
and linker functionalized derivatives. HSE06 systematically predicts larger band
gap energies than PBE, more closely matching the experimental values. Data
obtained from ref (3). Figure reprinted with permission from ref. (2).
Sometimes, however, the disagreement between experiment and theory is
more subtly hidden in poorly described interactions such as long-range dispersion;
the addition of correctional terms can also help empirically tune a DFT output to
match an experimental data set or simply add additional physics that may better
describe the system. These additional Hamiltonian terms were not necessary in the
models presented in this thesis, and, thus, are outside the scope of this discussion.
Electronic structure modeling of extended solids
With the exception of rare amorphous variants, (145;146;147;148;149;150;151)
MOFs are ordered solids. Thus, conventional solid-state modeling techniques
can be applied where MOF crystals possess symmetry operations that enable
the description of the extended material using a repeating unit. This section is
intended to provide a background and some general calculation considerations when
41
Figure 7. As size of a chemical system increases toward an infinite solid, the
number of molecular orbitals becomes very large. For high symmetry materials,
the crystallite can be described using a smaller, repeating unit, whose projection
through reciprocal space generates electronic bands. Figure reprinted with
permission from ref. (2).
modeling periodic MOF properties, including their electronic band gaps, electronic
band structures, and DOS, as well as their lattice vibrations. Finally, we discuss
MOFs with imperfections, those that host vacancies and interstitials.
Periodic Models and k-Points. Viewing MOFs as an extended array
of molecules, each component (i.e., the linker and the prenucleated node) contains
a family of molecular orbitals. Upon self-assembly of the MOF these electrons
mix to form new molecular orbitals that expand over larger and larger regions of
space. In principle, one could model a complete MOF crystallite (i.e., a very large
chemical system as a molecule) and obtain the exact electron energetics and spatial
distributions for all possible molecular orbitals in the material. Owing to symmetry,
however, several of the molecular orbitals in the crystallite will be very similar in
energy and centered on the same atoms. These geometric degeneracies enable a
reduction in computational cost by modeling the associated electrons as interacting
periodically in bands, Figure 7.
42
Electronic bands can be thought of as delocalized molecular orbitals whose
energy depends on the extent of electronic interaction in crystallographic directions
within the crystal. Indeed, Bloch’s Theorem purports that for a nondefective
material, the periodicity of the lattice describes the periodicity of the overall wave
function. (152) Information on both the structural and electronic properties of a bulk
sample can therefore be gleaned from a single unit cell (although the approach does
not provide information about the surface chemistry at the grain boundaries of a
crystallite). One way of thinking about the construction of bands in solids is to
first create molecular orbitals for all atoms contained within the computational
cell, then, using harmonics, the electronic interactions with neighboring cells are
computed. (153) By doing so, both the electronic properties of the discrete unit
cell are recovered, as well as the influence of longer ranged (de)localization and
electrostatics, which can only be observed by sampling beyond one computational
cell. (154)
While it is convenient to visualize bands as large delocalized molecular
orbitals in real space, they are often computed in reciprocal space (i.e. k -space)
using Bloch’s theorem paired with a basis set constructed from interfering
planewaves to determine the combinations and populations of electronics bands.
k -space vectors can be loosely thought to sample long-range interactions in the
crystallographic directions defined by the k -vector.
From experimental convention, a crystallographic unit cell (or conventional
cell) contains the smallest chemical representation of a system that exhibits the
same overall symmetry as the pristine lattice. Calculations using this geometry,
composition, and associated lattice parameters will certainly be suitable for direct
calculation of electronic properties, however, some space groups offer symmetry
43
Figure 8. Although the smallest computational cell can save resources, an improper
model may be recovered if a chemical interaction permeates beyond a single
geometric cell. For example, (a) structural distortions are sometimes captured
in temperature independent DFT (the ground state may feature titling of nodes); (4)
and (b) magnetic ordering can be challenging if the unit cell contains an odd
number of coupling metals (for example, the unit cell of Co2Cl2bis(1H-1,2,3-
triazolo[4,5-b],[4,5-i ])dibenzo[1,4]dioxin), (BTDD), contains 3 Co- (II), which do
prefer to order antiferromagnetically (5)). Figure reprinted with permission from
ref. (2).
related primitive cells that contain a smaller repeating crystalline unit of lower
symmetry. (155) Typically, calculations are run on the primitive cell unless the unit
cell is required to capture magnetic ordering or structural deformations that cannot
be described using the smallest repeating unit, some examples of which are shown
in Figure 8.
Every ordered MOF crystallizes into one of 230 unique space groups, (156)
each containing a series of high symmetry k-points. (157) The lowest crystal
symmetry, P1, features no internal symmetry operations, all atoms in the cell
44
are unique and must be explicitly computed, and hence there is only one high
symmetry k -point in the first Brillouin zone, Γ. (157) Chemically, the Γ-point is the
equivalent of projecting the molecular orbitals contained within the unit cell in
three dimensions: the electronics of the material are governed by the interactions
captured explicitly within the computational cell rather than their interactions with
neighboring cells. MOFs typically crystallize in high symmetry space groups, (152)
and often feature more than one symmetrically unique k -point that will contribute
to the total energy of the system, and hence should be sampled during the SCF.
Within crystals that feature anisotropic bonding (for example, graphene,
2D MOFs, etc.) certain crystallographic directions contribute more significantly
to the bonding and stabilization of the system than others. In the 2D examples,
the in-plane and out-of-plane interactions will contribute differently and sometimes
seemingly unpredictably to the total energy of the system. It is hence good practice
to sample as many k -points as possible to ensure a more tightly converged total
electronic energy.
The selection of k-points depends on the crystal symmetry, and these k-
points traverse the first Brillouin zone. (158) In practice, the contribution of each
sampled k -point is taken as a weighted average (determined by symmetry) to
recover the total system energy. The position of sampled k -points through the first
Brillouin zone is computed with one of two philosophically dissimilar approaches:
a Γ-centered Monkhorst-Pack, or a non- Γ-centered Monkhorst-Pack k -grid. The
main difference is that the former forces sampling of the Γ, while the latter does
not guarantee that any high symmetry points are sampled. (159) Crystal symmetry
thus helps inform two key computational considerations: (i) the feasibility of
using a computational primitive cell, and (ii) the k -path to explicitly examine how
45
electron energies changed in the extended solid. Almost all of these considerations
are enabled in freely available software, and high symmetry points of the first
Brillouin zone can be found in online databases like the Bilbao Crystallographic
Server. (157)
In principle, one would compute the total energy of a system by integrating
over the entire first Brillouin zone or approximating the integration with the
summation of a very large number of k -points. In practice, the total energy is
asymptotically related to the number of k -points and a convergence test is required
to elucidate when a sufficiently dense grid has been invoked. We can distill the key
considerations for k -grid generation and their impact on the material properties
down to the following:
– In Bloch’s theorem, planewave cutoff (i.e., the basis set) and the k -points are
independent. Yet both affect the total energy of the system. Hence, to obtain
converged results, it is good practice to benchmark the k-grid and basis set
separately.
– Generally, sampling additional k -points provides better descriptions of long-
range orbital interactions within a crystal system. Increasing k -grid density
does not always reduce the total energy of the system that depends on the
nature of the bonding and antibonding orbitals. (160) It is therefore difficult to
predict the impact on total energy without first running the calculation.
– Empirically, a k-grid with a density of 25x (lattice parameter)−1 provides
an estimate of the k -point density required for a reasonable sampling of
electronic interactions within the crystal. (155) Computational feasibility of the
resultant grids relies on large lattice parameters, which typically infer large
46
numbers of explicitly defined chemical interactions within the unit cell, the
extent of electronic interactions rapidly diminishes in real space, hence why
large crystals often show little dependence on increased k-point density. There
is a trade-off between modeling large systems with many explicit electrons,
versus small systems with many explicit k-points. Moreover, while a reduction
to the primitive cell vastly reduces the Slater determinant’s size, it comes
at the penalty of requiring several additional k -points (which contribute
significantly to the computational time).
– In MOF chemistry, it is common to perform geometric optimizations with a
sufficient k -grid at a lower level of theory, then refine the electronic structure
with a singlepoint energy calculation at a higher level of theory; the size
of the k -grid in either case is determined by MOF symmetry and resource
availability.
– Importantly, there is no formal relationship between lattice parameter and
k-point density; rather, large unit cells often contain sufficient descriptions of
strong interactions, and hence long-range sampling does not overly alter the
energetics of the system. Regardless of whether this is true for all MOFs, this
procedure produces a highly reproducible systematic error in the total energy
of the system that is acceptable in many cases.
– Most MOFs crystallize in high symmetry space groups, and it is good practice
to include all high symmetry kpoints in the computation of the total energy
of the system, and ideally during the optimization routine.
Regardless of how many k -points are selected for the optimization routine,
the material must be geometrically equilibrated (and the SCF must be converged)
47
to ensure that the energetics are computed without spurious numerical fluctuation.
A schematic of the general computational approach for solid-state materials is
shown in Figure 9. Assuming that a structure has been optimized, the following
sections discuss the computable properties and utility of using a solid-state model
for MOF systems.
Electronic Band Gap. The ionic lattices of most MOFs feature a
discrete electronic band gap, while some can be described as metallic (classified
as having nonzero DOS at the Fermi level). (161) Within the subgroup of materials
that feature electronic band gaps, their conductive properties depend on numerous
other properties (e.g., the material’s defect chemistry, the mobility of the charges,
the electronic band gap, etc.). Because standard solid-state computations omit
temperature, and without an exhaustive defect analysis, we are unable to assign the
position of the Fermi level. In light of this, MOFs are largely insulating materials,
but we note that they are interchangeably referred to as semiconductors in the
literature. (162;163;164;165)
The fundamental energy gap, Eg, is defined by the lowest energy occupied-
to-unoccupied transition in a system (independent of whether it is symmetry
allowed); (166) its magnitude is the difference in energy between the VBM and
CBM. Because of a derivative discontinuity and other localization errors associated
with the exchange potential in semiconducting systems, GGA functionals
systematically underestimate MOF band gaps. (167;168;169) Empirically, this
systematic underestimation can be improved by incorporating a component of exact
HF exchange into the Hamiltonian. (170;171) However, the computational expense
associated with hybrid functionals (172) prompts the recovery of semiconducting
48
Figure 9. General computational approach used to obtain electronic structure
properties from solid-state structures. (a) Beginning with an experimentally
obtained crystal structure, partial occupancies must be resolved. Then, where
applicable, protons must be added to charge balance the cell, in effect establishing
the oxidation state of the metals. Extrinsic solvent may be removed to simulate
the activated MOF. Symmetry can then be enforced and a computational primitive
cell may be available. (b) The structure can then be equilibrated, and higher level
electronic structure properties can be obtained by sampling the first Brillouin zone,
including electronic band structures and corresponding density of states. Figure
reprinted with permission from ref. (2).
49
properties from single-point calculations at this level of theory on a structure
obtained using GGA. (173)
MOFs with bands that exhibit different energies as a function of k -space
(i.e., band dispersion) may exhibit an indirect band gap, schematically presented in
Figure 10, where the VBM and CBM occur at different points in reciprocal space.
In such cases, Γ-point sampling is insufficient to capture the dominant electronic
influences. The lowest energy transition in an indirect band gap material requires
the coupling of photon absorption/ emission to a phonon mode in order to conserve
energy and momentum because the wave vector at the top of the valence band
does not match the wave vector at the bottom of the conduction band. (174) Indirect
band gaps thus diminish the intensity but extend the lifetime of formed excitons by
preventing recombination; indirect band gaps are desirable in photoactive solids for
the enhancement of quantum efficiency. (175) Although there are almost no reported
indirectband MOFs, (176;177) the changing band gap energy in reciprocal space is a
phenomenon ripe for exploration.
Certainly, most MOFs feature wide band gaps and highly localized
electronic structures that give rise to flat bands with well-defined orbital
contributions to the frontier states and direct gaps. The localization of electrons
makes MOFs intriguing for photocatalytic applications, (178;179;180;181;182;183) as
it affords one route toward accessing dense populations of high energy electrons
localized on transiently reduced motifs. (184;185) Moreover, electronic structure may
be tuned to desirable bulk properties, facilitated by the inherent modularity of
MOFs.
Although the nature of band edges is material/composition dependent, the
electronic band gap of a MOF can be modified by metathesis and functionalization
50
Figure 10. An illustration of the band gap in a semiconductor as a function of
reciprocal space when the CBM and VBM are at the same point in reciprocal space
providing a direct band gap and when they are at different points in reciprocal
space creating an indirect minimum absorption energy. Figure reprinted with
permissions from ref. (2).
of the linker and/or inorganic node. There are four possible frontier orbital
orientations in MOFs, wherein the band edges are defined by (i) ligand-to-ligand,
(ii) ligand-to-metal, (iii) metal-to-ligand, or (iv) metal-to-metal excitations,
ultimately dictated by the material composition. Countless examples of DFT
screenings are reported for band gap modulation via metal (97;186;187;188;189;190;191)
and linker exchange (102;192;193;194;195;196;197;198;199;200) across the gamut of MOFs.
MOF-5, which is a pivotal part of Chapter 5 for example, has been subject to
numerous theoretical studies systematically exchanging components of the inorganic
node (188;191;201;202;203;204) or the linker. (205;206;207;208)
Electronic Band Structures. Electronic band structures are a plot of
electronic energy as a function of k -space (i.e., electron momentum). The shape of
the bands provides tremendous amounts of information about material properties,
including the potential to transport charges, overall material stability and, to the
trained eye, even the composition. Given MOFs are made from discrete molecules,
51
it is somewhat unsurprising most feature localized (i.e., flat) electronic bands
that facilitate the use of cluster models. However, through development with a
complement of experiment and theory, there are increasing examples of MOFs
featuring curved (i.e., dispersive) bands and interesting MOF applications that
rely on the description of bulk electronic structure. Thus, this section will discuss
the calculation and implications of MOF electronic band structures.
The electronic band structure is simply constructed by sampling the energy
of electrons at various k -points. The difference in energy between the bottom
and top of the band at two dissimilar k -points is referred to as the bandwidth,
or band dispersion, and is essentially a measure of the “curviness” of the bands.
Higher band dispersion means more mobile charge carriers and also indicates
something about the extent of orbital overlap and long-range interactions in that
crystallographic vector. Because these plots ultimately depict electron momentum,
the second derivative of a band near a high symmetry k -point (sometimes called
special points) yields the effective mass of a charge carrier in that band.
Perhaps it is somewhat intuitive that a highly symmetric crystal would
feature a large degree of electronic degeneracy within the unit cell; it is less
intuitive to imagine how crystal symmetry affects the bulk electronic properties,
i.e., those that extend beyond a single computational cell. As mentioned in the
previous section, the material should be properly equilibrated using all of the high
symmetry k -points in addition to a dense grid between them. Practically, however,
this is often impossible, but also unnecessary; generally flat-banded materials (like
MOFs) can be described with Γ-only sampling because the insulating interface
between the metal and ligand is sufficiently accounted for by explicit orbital
interactions within the computational unit cell. (209;210;211) It is, however, typically
52
good practice to sample all labeled high symmetry k-points for the given space
group, (156;212;213) as presented in the Bilbao Crystallographic Server, (157) ensuring
that the total energy has included all important long-range interactions. (214)
Although the chemical connectivity in MOFs is usually highly symmetric,
MOFs themselves have highly anisotropic electron density through the
crystallographic unit cell: there are pores! Thus, it is expected that the electronic
band structures should contain some interesting information, albeit subtle, arising
from the periodic absences of electron density. Indeed, one paradigm in the
conductive MOF literature is whether the charge carriers are more delocalized
through-space or through-bonds. (215;216) The band structure sheds light on this.
Consider two similar scaffolds made from π-stacked triphenylene-based linkers:
in the case of Ln(hexahydroxytriphenylene), the inorganic nodes form continuous
ionic bonds throughout the material, (217) whereas in Ni3(HITP)2 the sheets are
nonbonded. In both cases, the greatest band curvature is found to be centered
on the linker, but associated with the out-of-plane direction. In other words, the
conductivity mode is “through space”. (215;216) The delineation of “through-bond”
or “through space” should not be confused with that between band or hopping
conduction, which are separate mechanisms that can be distinguished by the
presence of an activation energy associated with the conduction. (215;218;219)
Electronic band structure calculations primarily serve to graphically identify
crystallographic directions in which electrons are highly delocalized and strongly
interacting. Such plots contain a significant amount of information when paired
with the crystal structure and a map of the high symmetry kpoints for the MOFs
parent space group. The analysis can be further complimented by quantitative
53
analysis of the electronic band dispersion (a concept we have already introduced),
and most importantly, the density of states.
Electronic Band Dispersion. Band dispersion is the difference in
energy of a band between two high symmetry points in reciprocal space. Greater
dispersion is thought to indicate the material may be a better electrical conductor
in that direction. (220) Perhaps a more apt description is to think of dispersion (or
curvature) as an indicator of electron and hole delocalization. In this mindset,
the “effective mass” of a charge carrier can be computed to quantify the extent
of delocalization associated with a band. In doing so, the mass of a free charge
carrier in vacuum, m0, is renormalized to describe its behavior in the periodic
potential of the MOF. The second derivative of a band near a special point or
crossing the Fermi-level yields the effective mass, m∗, of the electron or hole that
would populate that band via
1 1 ∂2E(k)
∗ = (2.106)m ~ ∂k2
where ~ is Planck’s constant. (221) In practice, this is readily solved by fitting a
parabola to a series of k-points very near the point of interest. Often, practitioners
will present both “heavy” and “light” effective masses, corresponding to the bands
with the least and most curvature, respectively.
A final consideration in the computation of the electronic band structure
is that while the computed band dispersion is recovered from temperature-
independent DFT, the experimental observation of the band dispersion itself
does depend on temperature. Thus, dispersion energies less than kT (i.e., the
Boltzmann distribution at the operating temperature) are lost by the thermal
smearing of states and can be considered essentially flat; the precision enabled
by DFT may cause the appearance of subtle electronic phenomena, such as an
54
indirect band gap, that are inconsequential under realistic thermal conditions. This
particular issue was highlighted in a recent publication, where the authors predict
band dispersion on the order of kT (<30 meV) and attributed the observable bulk
properties of Zn- SURMOF-2 (222) to this dispersion and its indirect band gap.
While it is difficult to imagine a dispersion of this magnitude being a factor at
room temperature, it does highlight that periodic DFT is precise and does enable
high fidelity studies of dissimilar electronic states in materials with subtly changing
energetics.
Density of States. DOS plots are critical to assessing electronic
band parentage, providing information about the type of charge transport that
may exist, be it ligand-to-ligand, metal-to-ligand, ligand-to-metal, or metal-to-
metal fundamental gap. Hence, together with the electronic band structure and
analysis of band dispersion, the DOS paints a comprehensive picture of the MOF’s
anticipated function. The integral of the DOS indicates how many electrons
occupy that energy level, and its parentage informs which atoms/ orbitals are
contributing to that energy level. Often, DOS plots are partitioned into atomic
specific contributions (atom-projected DOS, pDOS) or orbital specific contributions
(orbital-projected DOS), providing tremendous amount of insight into the nature
of the bands in a material, as well as a lens through which the effects of linker
functionalization, compositional substitutions, and interstitial influences may be
assessed. (194) (192;223;224) The DOS is computed for all sampled k -points, enabling
practitioners to examine the DOS at a single k -point, or the sum of them. The
former can be helpful when MOF bands cross one another in k -space and has been
used fruitfully inMOF calculations for systems with large unit cells. (225) In spin
polarized materials, the DOS representation may be further partitioned by spin.
55
This knowledge is valuable because the spin bias afforded by the high degree of
spinsplitting is advantageous for spintronic devices that require precise control over
electron dipole orientation. (226)
DOS plots have also been helpful in determining the directionality of charge
transport in conductive MOFs. Charge hopping is likely the mechanism of electron
transport in most MOFs, commonly promoted accessing mixed valency through
either partial redox of the metal/linker or via photoexcitation. (227;228;229;230;231)
In sum, the electronic DOS provides a detailed view of the atomistic
contributions to energy levels, which can be quite complex for large chemical
systems. Information gleaned from these analyses gives modelers the ability
to predict the locality and direction of electron transfer, which are important
in catalytic applications. The DOS also helps classify a material as a metal or
semiconductor/insulator based on the absence or presence of density at the Fermi
level. However, this section only focused on the utility and application of electronic
DOS, which can be identified at a single k -point or summed over the entire kgrid.
A similar sampling and DOS analysis can be performed on vibrational band
structures (i.e., phonon bands), where the DOS then describes the population of
oscillators, a dynamic phenomenon.
Defects. MOFs are largely been treated as pristine, defect-free
extended solids. Indeed, periodic boundary conditions demand crystallinity. This
approximation is deeply rooted in the characterization of MOFs, as evidenced
by the procedures for obtaining the crystal structure, pore volume, and gas
uptake properties, which typically assume a pristine architecture. (232) However,
defect formation is entropically favorable, (233) thus all materials contain defects.
Indeed, significant effort has been invested in minimizing experimental defect
56
Figure 11. Node or linker vacancies result in charge balancing ions bound to
the vacant sites (e.g., a proton or formate). Both nodes and linkers can also be
redox active forming structurally bound charge carriers. Figure reprinted with
permissions from ref. (2).
concentrations, (234) but defects can also give rise to highly desirable properties
(including the formation of charge carriers via doping pathways, open metal sites
for catalysis, creating chromophores, (235) etc.), (236) and as such are an emerging
area of interest to the MOF field. (237)
In conventional crystalline materials (e.g., Si) there are three general
types of defects: (i) vacancies, where an atom is missing from the lattice, (ii)
substitutions, where a lattice atom is exchanged for one that does not normally
occupy that lattice site, and (iii) interstitials, where an additional atom is in a
typically unfilled lattice site. (234;238) MOFs regularly exhibit vacancies in the form
of missing nodes or linkers, (239) and substitutions through isoreticular chemistry
or postsynthetic metathesis, Figure 11. (240) Further, MOFs can play host to a
variety of species within their pores, which can be considered interstitials. Each
of these defect types may spawn additional functionality and modularity to the
parent framework, the most catalogued of which is the liberation of active sites for
catalysis via linker vacancies. (241)
Pairs or groups of these lattice point defects yield further terminology:
Frenkel and Schottky defects. A Frenkel defect is essentially a vacancy and
57
compensatory interstitial that yield a charge neutral material, while Schottky
defects are defined by multiple vacancies that sum to zero charge. Because MOFs
are ionic, both Frenkel and Schottky defects are common. For example, a combined
node and linker vacancy may be considered a Schottky defect and a linker omission
passivated by an additional formate may be considered a Frenkel defect. Moreover,
exotic Frenkel defects may be installed by including a charge balancing ion within
the pore after the redox of a MOF component (Figure 11); this is uniquely enabled
in MOFs considering they contain large amounts of vacuous space to support the
ions. (242;243;244;245;246)
Because defects are often enthalpically disfavored, and installation is
driven by entropy, there is a concentration limit at which the free energy of the
formation of additional defects becomes positive. The defect concentration limit
depends on the host material, but conventional semiconductors are limited to
only a few percent. (247) Because of their low lattice density and limited electronic
delocalization, MOFs can uniquely exceed these concentrations by at least an order
of magnitude, sustaining very high vacancy populations. (248) For example, MUF-32,
can reversibly reach linker vacancy defect concentrations of 80%. (249) These high
defect concentrations come as a mixed blessing because on one hand we are able
to model them in a crystallographic unit cell and not have to worry about their
effective concentration, but we must worry about the impact of artificial defect
periodicity. Hence, one challenge with pristine models is that they neglect the
electronic impact of defects present in real systems.
Defects are discussed in further detail in the following chapters. Specifically,
vacancies are explored in Chapter 4, and interstitials are discussed in Chapter 5.
58
CHAPTER III
CHEMICAL SIZE OF ATOMS
Introduction
Atomic and molecular shape and size play a critical role in chemistry.
The most familiar method was formalized by Bondi: the van der Waal radii and
volume of atoms and molecules. However, the rigid sphere approximation of atoms
fails to describe highly polarized chemical systems. To overcome this challenge,
numerous other approaches based on electron density have been presented, but
these approaches intrinsically struggle to describe the surface area and volume
off cations. Herein, we revisit the timeless problem of assessing sizes of atoms
and molecules, through examination of the electric field produced by atoms and
molecules. In this way, we are able to recover chemical volumes and surface areas
of simple density functional theory (DFT) calculations. We perform a series
of benchmarks to ensure the generality of our approach and demonstrate the
electric field calculations provide unique insights into quantifying sizes of ions
and polar molecules. Our method further lays the foundation for the development
of analytical interaction energies based on assessment of the Coulomb potential
provided by molecular systems, while providing a rigorous approach to study size
dynamics as a function of chemical environment.
Method Description
To begin, we have developed a post-processing DFT software, STREUSEL
(Surface Topology REcovery Using Sampling of the ELectric field), to recover
volumes and surface areas of atoms, molecules, and materials, from their electric
field. Since electric field is defined as the negative gradient of the electrostatic
potential, the electric field embodies the direction of greatest increase in
59
electrostatic potential. This is significant because the increased slope of the electric
field enables us to more clearly define the edge of a chemical system in space, while
additionally accounting for the long-range interactions observed with cations.
We define the edge of a chemical system as the point in space where there is
“near-zero” variance in the electric field magnitude. Due to computation efficiency,
DFT calculations return the electrostatic potential values on the order of 10−6
eV (2.306x10−5 kcal mol−1), thus we consider a change of less than 10−5 eV
(2.306x10−5 kcal mol−1) to be the most precise we can obtain. This definition is
justifiable thermodynamicall – a typical van der Waals interaction is on the order of
0.9560-1.912 kcal mol−1. (250;251) out cutoff is significantly smaller and thus able to
assess these weak interactions. Indeed, the utility of “cutoffs” within chemical size
quantification methodologies is not unheard of. (24;51;52)
Volumetric pixel resolution. In order to employ this definition of
size and identify the chemical surface we use the electrostatic potential provided by
DFT at a discrete number of volumetric pixels (voxels). The result of this sampling
procedure is thus dependent on the size of the voxels (i.e. resolution of the
electrostatic potential calculation), at the tradeoff of an exponential relationship
between time-to-volume/surface area solution and voxel resolution.
The electrostatic potential tensors are extracted from the Gaussian
optimization calculations. The dimension of the tensor (number of elements per
side) and the length (in Euclidean space) of each side is variable; the resolution is
best represented as the real space volume that each element in the tensor represents
(holding the total length of the computational box constant). At large (> 0.02
Å3) voxel volume is on the order of the volume of the chemical species, resulting in
artificially large calculated volumes.
60
1000 35
Volume N2 O2
Time N O 30
800
25
600 20
400 15
10
200
5
0 0
0.000 0.005 0.010 0.015 0.020
Volumetric Pixel Volume (Å3)
Figure 12. Decreasing the volumetric pixel volume increases the time-to-solution
(dashed curves) exponentially, yet minimizes the accuracy of the calculated
volume (solid curves). Thus, we take the kneepoint of the time curve as the ideal
volumetric pixel volume, 0.008 Å3.
To identify the optimal resolution (maximizing the voxel volume, while
minimizing the time-to-solution) the time-to-solution and calculated volumes were
calculated for four representative models (N2, O2, N, and O), Figure 12.
From these data, we find a voxel volume of Å3 to yield a desirable trad-off
between computation time and volumetric property resolution. This voxel volume
is reasonable, given a typical chemical system is on the order of Å.
Functional Benchmarking
As with all electronic structure methods, the size and shape of molecules
depends on both the functional and basis set used in the model development.
Recently, a systematic assessment of 128 DFT functionals canvassed their
performance for the recovery of total energy, and, separately, electron density. (115)
While that paper revealed a significant energetic and structure dependence on
functional, it did not directly emphasize the functional also changes the shape of
the molecules. Since atomic position is determined by electron density, and electron
61
Time to solution (s)
Volume (Å3)
Figure 13. Calculated STREUSEL volumes of H2 and CH4 optimized with each
basis set (x-axis), normalized to the volume obtained from the aug-cc-pVTZ (y-
axis). The normalized volumes represent the precision of each basis set, a metric of
how close the returned electronic structure is to that obtained using aug-cc-pVTZ.
aug-cc-PVTZ is considered sufficiently large – as demonstrated by the convergence
of normalized volumes with larger basis sets (i.e. cc-pVQZ and aug-cc-pVQZ.
density is determined by atomic position (i.e. the self-consistent field and geometry
optimization routine), there is therefore an inherent dependence on functional
selection in our size quantification approach. (252)
To elucidate these dependencies, and arrive at an ideal method to recover
size using STREUSEL, we examine fifteen small, neutral molecules (Br2, C2H2,
CH4, Cl2, CNCl, CO2, F2, H2, H2S, N2, NH3, O2, SO2, CO, and H2O) using forty-
nine functionals and a large basis set, aug-cc-pVTZ. (253) We benchmarked the
basis set to ensure that it was sufficient to describe polarized and diffuse electronic
systems, Figure 13. Each molecules’ size is computed using the geometry obtained
from equilibration using the stated functional. These values are compared to the
geometry, volume, and surface area computed with CCSD-full (254;255;256;257) – which
we use as our reference for the exact solution. (115) We also examined the use of
62
Figure 14. Calculated STREUSEL volumes of a series of industrially relevant
small molecules optimizes with each functional (y-axis), normalized to the volumes
obtained from the CCSD-full functional (x-axis). The spread of normalized volumes
represents the precision of each functional, a metric of how close the returned
electronic structure is to that obtained using CCSD-full, which is assumed to yield
63
the exact geometry. The forty-nine functionals are grouped by functional type
(generalized gradient approximation (GGA), generalized gradient exchange (GGE),
hybrid (H), meta (M), hybrid-meta (HM), and local-density approximation (LDA).
To increase resolution, Ne and F2 are not included in this plot and are discussed
extensively in the main text.
each functional applied to the structure obtained from the CCSD-full calculation
in order to isolate the impact of structure on functional performance (since it is
becoming increasingly routine to perform geometry optimizations using a low level
functional, and higher fidelity electronic structure single point calculations on these
geometries). Individual molecular volumes are presented, Figure 14
While our approach is limited to the upper rungs of Jacob’s ladder, (2;117)
we canvas eight electronic structure classes (ab initio, generalized gradient
approximation (GGA), generalized gradient exchange (GGE), hybrid-GGA (H-
GGA), hybrid-meta-GGA (HM-GGA), local-density approximation (LDA), meta-
GGA (M-GGA), and range separated functionals. A comprehensive list of studied
functionals is presented, Table 2
Perhaps surprisingly, there appears to be no clear trend in prediction of
electric field-based molecular size as we ascend Jacob’s Ladder, Figure 34, i.e.
higher-level DFT functionals do not necessarily outperform lower level ones,
excluding ab initio methods, which are highly accurate. Yet, GGE, HM-GGA, and
M-GGA functionals appear to systematically underestimate molecular volumes,
while the commonly invoked PBE (GGA) functional performs similarly to other
functionals, such as B97-D, X3LYP, BLYP, B3LYP, and TPSSLYP1W.. With
reference to Ref. (115), out approach offers no opinion on whether the theoretical
method should be selected to provide accurate density, or energy, but rather prefer
functionals that correctly recover geometry closest to that obtained from CCSD-
full.
We compare the ability of two size methodologies (Batsanov expanded set of
atomic van der Waals radii and electric field) to recover small molecule volumes
similar to those obtained using CCSD-full, Figure 33. Here, we plot the mean
64
Table 2. A comprehensive list of the representative functionals from each
level of theory (ab initio, generalized gradient exchange (GGE), local-density
approximation (LDA), range separated, generalized gradient approximation (GGA),
hybrid-GGA (H-GGA), hybrid-meta-GGA (HM-GGA), and meta-GGA (M-GGA))
examined in this work.
ab initio GGE LDA
CCSD-full (254;255;256;257) BVWN5 (118;258) Xalpha (56;91;259)
MP3 (260;261) G96VWN5 (118;262) LSDA (56;91;118;259)
MP4 (263;264) OVWN5 (118;265;266) SVWN5 (56;91;118;259)
Range Separated GGA H-GGA
wB97XD (267) PBE (92;268) APFD (269)
CAM-B3LYP (270) B97d (271) B1LYP (53;258;272)
BLYP (53;258;272) B3LYP (53;258;272)
BPBE (54;258;268) B3P86 (258)
BPL (258) BHandH (273)
G96LYP (53;264;272) BHandHLYP (273)
HCTH (274;275;276) HSE06
HFB (258) M06 (138)
LC-wPBE (277;278;279) M11 (280)
PBELYP (53;54;268;272) O3LYP (265;266)
PBEPW91 (54;127;268;281;282) HSE03 (134;139)
SLYP (56;91;259) PBE0 (133;283)
SPBE (56;91;259) SOGGA11 (284)
TPSSLYP1W (53;132;272) X3LYP (53;272;285)
mPWPBE (54;268;286) mPW1PBE (286)
mPW3PBE (286)
HM-GGA M-GGA
B1B95 (258) M06L (131)
BMK (287) M11L (288)
TPSSh (132;289;290) TPSS (132)
VSXC (291)
tHCTH (292)
65
Figure 15. Calculated STREUSEL volume of a series of industrially-relevant
small molecules optimized with each functional (y-axis), normalized to the volume
obtained from teh CCSD-full functional (x-axis). The spread of normalized volumes
represents the precision of each functional, a metric of how close the returned
electronic structure is to that obtained us6i6ng CCSD-full, which is assumed to yield
the exact geometry. The forty-nine ffunctionals are grouped by functional type
(generalized gradient approximation (GGA), generalized gradient exchange (GGE),
hybrid (H), meta (M), hybrid-meta (HM), and local-density approximation (LDA).
absolute deviation across the small molecule volumes for each functional, where the
volume obtained using the CCSD-full optimized geometry is treated as the chosen
measure of central tendency. This provides a statistical metric of similarity between
the volumes recovered using each functional and CCSD-full for two size metrics.
Owing to the rigid sphere approximation and negligible changes in bond lengths
between functionals, we observe van der Waals volumes are least affected by the
functional geometry. The mean absolute deviation of the electric field-based sizes
is larger than those returned by van der Waals; this ultimately indicates that the
electric field is more sensitive to small changes in geometry.
Indeed, deviations in van der Waals volumes for different functionals, Table
3, arises solely from variations in the resulting geometries (i.e. bond lengths),
as opposed to the varying treatments of the electronics. Therefore, it would be
misleading to present an average difference between STREUSEL and van der Waals
derived volumes; the variability of STREUSEL and van der Waals volumes arises
from diferences in atomic radii, which are exacerbated in polyatomic systems.
Unsurprisingly, differences between van der Waals and STREUSEL computed
volumes or the fifteen industrially relevant molecules are not predictable, Table
3.
With respect to Figure 33, we are guided to favor functionals and ab initio
methods displaying the lowest mean absolute deviation from each functional class,
including: MP3 and MP4 (ab initio); CAM-B3LYP (range separated); BMK (HM-
GGA); BHandH, BHandHLYP, and X3LYP (H-GGA); TPSS and VSXC (M-GGA);
BPL and G96LYP (GGA); BVWN5 (GGE); Xalpha (LDA). Generally, however,
higher level functionals return volumes closer to those obtained from CCSD-full
across the three examined size quantification methodologies.
67
Figure 16. The mean absolute deviation (y-axis) of two volume methodologies
(STREUSEL, van der Waals) across all of the studied functionals (x-axis). The
mean absolute deviation is averaged across the mean absolute deviation for
industrially relevant small molecules was calculated assuming the volume obtained
from the CCSD-full optimized geometry as the chosen measure of central tendency.
The mean absolute deviation is a metric of how close the returned electronic
structure is to that obtained using CCSD-full, which is assumed to yield the exact
68
geometry. The forty-nine functionals are grouped by functional type.
Table 3. Molecular volumes for the relevant small molecules examined in this study
calculated using the van der Waals (vdW) and the STREUSEL volume calculation
methods.
Molecule vdW volume (Å3) STREUSEL volume (Å3) Difference (%)
Ne 11.25 8.260 26.58
F2 22.03 17.25 21.70
O2 22.97 18.77 18.28
H2O 19.28 23.43 -21.52
N2 23.49 19.69 16.18
CO 26.68 22.81 14.51
NH3 22.68 26.16 -15.34
CH4 28.15 25.08 10.91
Kr 26.52 24.26 8.522
H2 10.49 9.690 7.626
C2H2 34.56 32.30 6.539
CO2 33.21 31.49 5.179
Cl2 39.54 37.50 5.159
SO2 39.02 39.94 -2.358
Br2 47.87 46.99 1.838
H2S 29.43 29.49 -0.2039
Neutral and Neutral Bonded Systems. For illustrative purposes,
we use our general approach to predict the radii for the periodic table of non-
interacting atoms, Figure 35. The atomic radii recovered by STREUSEL are
comparable with those presented by Alvarez, (31;32) Boyd, (52) and Rahm, (24) Figure
36. The related Rahm and Boyd radii from electron density cutoff values show
excellent agreement in relative size trends. The largest disagreement occurs at high
atomic numbers where the diffusivity of electrons is often poorly described due
to partial occupancy of higher energy orbitals, and basis set error. (293) Yet, close
similarity of STREUSEL radii with both Rahm and Boyd radii is unsurprising,
considering the shared dependence on a cutoff value derived from DFT.
69
1 18
1 2
H He
1.24
hydrogen 2 13 14 15 16 17 1.05helium
3 4 5 6 7 8 9 10
Li Be B C N O F Ne
2.06 1.92 1.75 1.61 1.49 1.46 1.37 1.26
lithium beryllium boron carbon nitrogen oxygen neon
11 12 13 14 15 16 17 18
Na Mg Al Si P S Cl Ar
2.13 2.15 2.14 2.03 1.93 1.86 1.75 1.69
sodium magnesium 3 4 5 6 7 8 9 10 11 12 aluminium silicon phosphorus sulfur chlorine argon
19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr
2.35 2.49 2.40 2.34 2.29 2.24 2.04 2.00 1.99 1.95 2.02 1.98 2.11 2.07 2.00 1.96 1.89 1.84
potassium calcium scandium titanium vanadium chromium manganese iron cobalt nickel copper zinc galium germanium arsenic selenium bromine krypton
37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe
2.42 2.59 2.52 2.47 2.42 2.34 2.30 2.24 2.09 2.04 2.01 2.10 2.23 2.21 2.17 2.16 2.09 2.03
rubidium strontium yttrium zirconium niobium molybdenum technetium ruthenium rhodium palladium silver cadmium indium tin antimony tellurium iodine xenon
55 56 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86
Cs Ba Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn
2.54 2.75 2.42 2.38 2.29 2.26 2.19 2.17 2.14 2.07 2.03 2.21 2.24 2.23 2.22 2.14 2.11
cæsium barium hafnium tantalum tungsten rhenium osmium iridium platinum gold mercury thallium lead bismuth polonium astatine radon
87 88
Fr Ra
2.55 2.74
francium radium
57 58 59 60 61 62 63 64 65 66 67 68 69 70 71
La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
2.60 2.57 2.58 2.65 2.64 262 2.60 2.51 #### #### 2.54 2.53 2.51 2.51 2.45
lanthanum cerium praseodymium neodymium promethium samarium europium gadolinium terbium dysprosium holmium erbium thulium ytterbium lutetium
89 90 91 92 93 94 95 96
Ac Th Pa U Np Pu AmCm
2.69 #### 2.58 #### 2.60 2.58 2.55 2.53
actinium thorium protactinium uranium neptunium plutonium americium curium
Figure 17. Atomic radii for elements 1-96 of the periodic table calculated using STREUSEL are presented. Neutral
atoms were modeled using functional CCSD-full with the SDDAll basis set. The radii are calculated from the
STREUSEL-obtained volumes under the spherical approximation.
70
Table 4. Molecular volumes (Å3) comparison between Bader, Boyd, van der Waals
and STREUSEL size quantification methods.
Bader Boyd STREUSEL van der Waals
O2 14.93 17.18 18.81 22.91
N2 17.09 19.66 19.69 23.49
Separately, the empirical dependence of both the Alvarez and Pyykkö radii,
which are self-consistently derived bying empirical and calculated data of atoms
in molecules, . (40) It is, thus, unsurprising that STREUSEL radii are considered
larger than Pyykkö radii, which considers atoms in bonded environments. Further,
unscaled Boyd radii represent the upper bound in accepted atomic radii, deriving
radii from vacuous, neutral atomic calculations. (52) The STREUSEL radii yield
neither the upper, nor lower bounds of accepted atomic radii values, indicating that
our approach is not too farfetched.
The reported van der Waals and Pyykkö radii are consistently lower than
those obtained using STREUSEL, Boyd, Rahm or Alvarez. On average, the
resultant volumes computed with STREUSEL are 5% larger than those obtained
from the conventional van der Waals approach. This may be due, in part, to the
crystallographic basis for the atomic radii, which inherently considers atoms in
bonded environments. van der Waals radii are obtained from crystallographic data
under two assumptions: i) contact distances obtained from X-ray diffraction data
are temperature dependent, and ii) atoms are approximated as spheres. (32)
The difference between van der Waals volumes, the electron density
cutoffs presented by Bader and Boyd, and the electric field cutoff recovered using
STREUSEL is illustrated via a volumetric comparison of O2 and N2, Table 4. We
use the example in part to demonstrate that STREUSEL is yielding comparable
size results to other approaches, but also to emphasize that the approach is
71
3.75 a
3.25
2.75
2.25
1.75 Alvarez
This Work
1.25 Rahm
Boyd - scaled
0.75
b Boyd - unscaled
4.00 van der Waals
Pyykkö
3.00
2.00
1.00
0.00
0 20 40 60 80 100
Atomic number
Figure 18. A comparison of atomic radii recovered using various conventional size
metrics. a) Sizes computed from STREUSEL are similar to those presented by
Alvarez, and Rahm and Boyd (top panel). b) Alternative size metrics are presented
as the upper and lower limits.
72
Atomic Radii (Å)
sensitive to composition, geometry, and electronic structure. In this case, the
diatomic molecule geometries were obtained from CCSD-full (254;255;256;257) paired
with the aug-cc-pVTZ (253) basis set, as implemented by Gaussian09. (77) Often
O2 and N2 are considered to be near-equivalent in size,
(294;295) posing historical
challenges with size-based separations. (296) While greater difference in volume
between O2 and N2 recovered by electron density-based size metrics is expected
considering the dissimilar bonding and electronic configurations of these two gases,
the STREUSEL and Batsanov approaches obtain values aligning with experimental
trends – they are difficult to separate.
Ionic and Ionic Bonded Systems. The primary shortcoming
of the van der Waals size method is the inability to accurately recover sizes
of polar chemical systems. Indeed, polar molecules are instrumental in
essentially all chemistry, but particularly operative in both electrochemistry
and condensed matter systems, such as ionic liquids, where size plays an
important role in determining many physically relevant properties. (297;298)
Beyond molecular examples, size has also been reported to contribute to charge
mobility in transparent oxides, (299) as well as ion mobility in batteries. For
example, INSERT. Indeed, there are several examples of ionic size metrics in
literature, (24;26;27;62;63;64;65;67) each reporting ionic radii of similar magnitude.
73
Figure 19. Calculated STREUSEL ionic radii (y-axis) for relevant oxidation states of atoms 1-86 (x-axis).
74
We present the electric field-derived radii or relevant ions, Figure 23.
Unsurprisingly, the STREUSEL ionic radii are larger than the Shannon-Prewitt
radii – since STREUSEL radii consider free ions (i.e. not associated with a
coordination number). As such, divergence between the Shannon-Prewitt size
metric and STREUSEL relates to the type of size that is addressed by each metric
– Shannon-Prewitt focuses on the space that each ion occupies, while STREUSEL
accesses the space that each ion affects.
Interestingly, the electric field-derived ionic radii are highly dependent on
oxidation state. First row transition metals are presented, Figure 38, for a more
detailed examination of competing factors. From general physics we know that
electric fields are generated by charged particles – in the case of ions, there are two
oppositely charged, interacting particles (protons and electrons), which produce a
net field. As oxidation state is changed, the ratio of protons to electrons changes
(since oxidation state is correlated with this charge ratio) and the field propagates
further/closer to the ion center. Thus, ionic radius increases with increasing
number of protons relative to electrons, demonstrated by STREUSEL in Figure
38.
Yet, the STREUSEL ionic radii are independent of coordination. Indeed,
the size of atoms change depending on the field that they are subjected to; this is
evident from the coordination number specifications on many empirically-derived
ionic radii. (32;62;63;64;67) One of the key aspects of our electric field-derived approach
is the ability to theoretically assess a change in size based on proximity to other
electric fields. For example, the field produced by a Li+ should be larger than Li+
in an ion pair, LiCl, simply because the field is proportional to the charge – they
passicate one another. We can demonstrate that the size of ions depends on atomic
75
Figure 20. Comparison of proton count (nprotons)-to-electron count (nelectrons) ratio
(x-axis) for the first row transition metals (y-axis) across a series of oxidation states
(colors) featuring data points correlated with calculated STREUSEL ionic radii
(data point size).
76
proximity, through progressive increase in bond length in a series of diatomic
molecules (HF, HCl, HBr, LiF, LiCl, LiBr), Figure 21. Here, the volumes of LiF,
LiCl, and LiBr increase with increasing bond length. interestingly, the volumetric
trend of the dissociated diatomics (Li+ + F− > Li+ Cl− > Li+ + Br−) presents
a size trend in opposition with the periodic table atomic size trends. We find
F− is larger than Cl−, which is larger than Br−, due to the efffective core charge
relationship. This is at odds with the generally accepted metric that Br− is the
largest of the three, which is certainly true from an electron density and bonding
perspective, but orthogonal to the inherent reactivity and affected size of the free
ions themselves. From this perspective, the larger of the three anions is expected to
be the most electronegative, F−. We also present the same series of experiments for
gas phase acidic halides, HF, HCl, and HBr, Figure 21. Generally, as the ions are
separated the size increases to a point. However, in the case of a fully dissociated
H+ and X−, the proton no longer possesses electron density, resulting in a terminal
decrease in size after approximately 3-4 Åseparation. These experiments point to
two key observations; i) that size depends on interatomic proximity, and ii) that
the size of ionic systems deviates form rigid size metrics, simply because of charge
(de)shielding.
The effect of interatomic proximity and charge (de)shielding on chemical
system size is further highlighted by the comparison of electric field-derived
and electron density-derived volumes of a series of geometrically equilibrated
diatomic molecules (HF, HCl, HBr, LiF, LiCl, LiBr), Figure 22. Size dependency
on increasing dipole moment is only exhibited by electric field-based size metrics.
Across the gas phase acidic halides, the increase in dipole moment leads to an
increased deviation between the electric field-derived and electron density-derived
77
Figure 21. STREUSEL volumes (y-axis) for a series of six diatomic molecules
were calculated for the optimized structures, and a series of models with expanded
bond lengths (x-axis). Individual ion volumes (i.e. Li+, F−, Cl−, and Br−) were
calculated and summed to obtain the maximum volume for each ionic compound.
volumes; the studied gase phase acidic halides are more similar in size than
conventional metrics purport.
While a similar relationship is exhibited in the Li-based diatomic molecules,
there are additional factors at play. The higher valence-to-core charge of the Li+-
based diatomic molecules introduces competing effects, which results in a larger
field-based size. The increased electronegativity of the F− in this instance is
exacerbated, leading to an amplified electric field-derived size – a phenomenon
that is neglected in the electron density-based size calculations. This is further
demonstrated by the relationship between electric field and electronegativity.
Since STREUSEL assigns size based on the electric field produced by the
atoms, which is determined by the Coulombic charge of the atoms, the higher
valence-to-core charge should yield a larger ield. Indeed, we may derive the
mathematical relationship between electric field (E) and electronegativity (χ̄)
78
Figure 22. Comparison of molecular volumes for a series of geometrically
equilibrated diatomic molecules with varying dipole moments. Divergence between
electronic density-based size metrics and STREUSEL depends on the system
polarity and atomic electronegativity.
defined by Rahm as the average energy of the electrons, (300)
1 ∑n
χ̄ = niεi (3.1)
n
i=1
where n is the total number of electrons, εi is the energy of the electrons in the
ith level, and ni is the occupancy of the i
th level. We can expand this equation to
describe the electronegativity of a region in space, (χ̄(r)), where the total number
of electrons becomes the number density of electrons within region r (n(r)) and the
summation of energy is the energy density of that region (µE(r)). Thus,
µE(r)
χ̄(r) = (3.2)
n(r)
from electromagnetics, the energy density in free space is occupied by an electric
field (E) is
1
µ = ε E2E 0 (3.3)
2
where ε0 is the permittivity of free space. Combining (3.2) and (3.3) and taking
advantage of the relationship between charge density (ρ(r)) and number density
79
(ρ(r) = qn(r)) √
2ρ(r)χ̄(r)
E(r) = (3.4)
qε0
√
thus, E ∝ χ̄; as χ̄(r) increases, so does E.
This is further demonstrated by comparing the recovered electron density-
based and electric field-based surfaces for this series of diatomic molecules, Figure
23. Owing to the polarity of these molecules, the recovered surfaces are noticeably
aspherical. From an electronic structure perspective, polarity necessitates the use
of basis sets composed of multiple functions. Moreover, while two molecules may
take up a similar amount of space, the shape may be inherently different. The gas
phase acidic halides specifically demonstrate that the cationic components of these
molecules, which are protons and do not possess electron density, generate a field.
We show that the majority of the space occupied by the gas phase acidic diatomic
molecules is centered over the proton – in direct opposition o the electron density-
based surface.
Ionic Liquids. As a more sophisticated example of the utility of
STREUSEL, we can use our approach to assess sizes of molecules and their
ions. Specifically with regard to molecular ions, for example those found in ionic
liquids, molar volume is a critical parameter which is thought to govern a wealth
of physical properties including density, viscosity, and so forth. (301) However,
the density and molar volume should depend on the size of the ions, which itself
depends on whether they are computed together or separately (see the relationship
presented in Figure 21). This is inpart why the van der Waals approach seems
to be widely employed for ion size estimates, (14) because the method implicitly
accounts for intermolecular attractions. Yet, ions are thought to interact via their
fields, which is neglected by van der Waals.
80
Figure 23. The Boyd (periwinkle), Bader (mint), and STREUSEL (charcoal)
surfaces of a series of diatomic molecules. All surfaces are plotted using VESTA.
The Boyd surface is depicted at an isosurface value of 0.00015 eÅ−3. The Bader
surface is depicted at an isosurface value of 0.00030 eÅ−3. The STREUSEL surface
is dipicted at an isosurace value relative to the minimum value of the electric field
surface mapped on the electrostatic potential map.
Table 5. Boyd, Bader, Batsanov and STREUSEL-derived molecular
volume (Å3) comparison for two ionic liquids possessing perhalometallate
anions. The sum of single molecule volumes for the individual constituents
([BMIM ]+, [V Cl − 2−4] , [CoCl4] ) are presented, as well as the fully geometry-
optimized ion pair. The percent difference (%) is presented for each ionic liquid
model. BMIM = 1-butyl-3-methylimidazolium
Batsanov Boyd Bader STREUSEL
[BMIM ]+ + [V Cl4]
− (Å3) 268.6 139.3 124.9 246.7
[BMIM ][V Cl 34] (Å ) 217.6 114.8 105.1 247.1
Percent Difference (%) 20.98 18.57 17.22 0.1620
2[BMIM ]+ + [CoCl ]+4 (Å
3) 430.2 216.3 159.6 404.2
[BMIM ]2[CoCl4] (Å
3) 293.6 173.1 195.2 390.1
Percent Difference (%) 37.75 22.19 20.07 3.550
81
Consider recently reported ionic liquids featuring inorganic perhalometallate
anionic constituents, Table 5. (302) The rigid sphere approximation returns larger
volumes than the electron density and electric field-based methods. The Bondi
presentation of the van der Waals radii excludes atoms possessing an atomic
number > 10; (28) thus, the revised van der Waals radii presented by Batsanov (30)
were used for these volume calculations of the ionic liquid constituents. Intuitively,
we would expect anions to be larger in field than their neutral counterparts
(Figure 23), thus, it is slightly unexpected that the van der Waals volumes
are the largest across all studied species. This is likely due to the derivation o
Batsanov radii from experimental contact distances coupled with the spherical
atom approximation. (28;30) Indeed, the electron density of the halides in these
anionic compounds is likely drawn towards the transition metal, which would result
in an efectively smaller volume, yet produces a larger field.
Conclusions
From Figure 21, we are initially inclined to think that as molecules are
pulled apart their volume should increase, Yet, this is not the case for polynuclear
systems (e.g. systems containing more than one molecule) presented in Table 5.
Within polynuclear systems there are two competing phenomena determining
the size of the ions, i) anionic and cationic electric fields interacting, leading to
a decrease in size, and ii) electric fields of the anionic and cationic components
are stabilizing each other, ultimately changing each other’s shape. To explore
this nuance, we further examine two ionic liquid systems, [BMIM ][V Cl4] and
[BMIM ]2[CoCl4], and the corresponding geometries of the equilibrated anionic
([V Cl4]
−, [CoCl4]
2−) and cationic ([BMIM ]+) constituents, Table 5 (here BMIM
denotes 1-butyl-3-methylimidazolium). We present the electron density-derived,
82
STREUSEL-derived, and Batsanov-derived volumes for the geometry-optimized
polynuclear system, individual components, and the sum of the individual
components.
As a result of the electron density stabilization, the optimized ionic liquid
systems possess a smaller electron density-derived volume than the summed
volumes of their individual counterparts. While it is not due to electron density-
stabilization, the rigid sphere volumes exhibit a similar trend. In this case, it is
because of the rigid sphere overlap between anionic and cationic constituents of
the polynulear system is accounted for in the fully optimized model. STREUSEL,
on the other hand, returns the contact surface (e.g. the extent to which each
component affects its surroundings) of the ionic components. This is evident from
the significantly smaller percent differences between the sum of single molecule
volumes and fully geometry-optimized ion pairs, Table 5. These results indicate
that the STREUSEL size metric is a viable option for assessing size-dependent ionic
liquid properties using free-ion molecular calculations.
These studies reinforce the notion that there is not an ideal size metric for
all chemical systems; rather, compatibility between size metric and chemical system
is paramount. This is exemplified by Figures 21 and 22, where it is evident that
the electric field, and resultant volume, depends on the proximity of the atoms
within the molecule. When the proximity is changed, bonding changes as well,
increasing the complexity of calculating size and necessitating electronic structure-
based methods. And in order to properly compute the size of molecules, from
Figure 33, we observe an exchange-correlation functional dependence – revealing
that ab initio methods, MP3 and MP4, with a minimum triple-zeta basis set return
geometries closest to CCSD-full. However, the more efficient functionals, B1LYP,
83
B3LYP, BHandH, BHandHLYP, and CAM-B3LYP, return geometries of adequate
volume with respect to CCSD-full. It is ultimately the duty of the theorist to select
a suitable functional for their system.
84
CHAPTER IV
CHEMICAL SIZE OF PORES
Abstract
Thus far, we have discussed the role of size in terms of the volume of
occupied space. As demonstrated in Chapter 3, this of particular interest
to atomic and molecular studies. Yet, the interactions between molecules and
materials is also dependent on size. Consider the binding sites in pharmacology,
or the role of porous materials in gas separation and storage. Indeed, void space
properties (i.e. pore volumes and surface areas) are pivotal to our fundamental
understanding of these processes and, for example, our abilities to predict ideal
host-guest pairs. Within this chapter we discuss the expansion of the STREUSEL
size calculation methodology to that of void spaces within porous materials, namely
MOFs. Here, we examine the disadvantages of the current computational porous
characterization techniques, and verify the performance of STREUSEL within this
arena.
Introduction
Since their conception, (303) metal-organic frameworks (MOFs) have attracted
practitioners from organic, inorganic, and materials disciplines, (304;305;306) each
with a shared interest in physical properties afforded by mixing multitopic
organic linkers and metal ions or clusters (secondary building units, SBUs,
or nodes). (307;308) The combination of SBUs with the multitude of linkers has
enabled a seemingly infinite landscape of materials with modular properties, (309)
all featuring one commonality: crystallographically ordered void spaces. (310;311)
Porosity in conjunction with functional MOF components catalyzed interest
in these materials as atomically precise pores enable a series of heterogeneous
85
gas storage (15;16;17) and separation technologies (18;312) as well as access to
extremely high loadings of catalytically active centers. (20;21;22;23) In more recent
embodiments, exotic electronic structures have been discovered, (120;209;313) providing
a foundation for applications in high surface area electrodes in electrochemical
devices (314;315;316;317;318;319;320;321) and as ordered arrays of qubits for MOF-mediated
quantum computing. (322;323;324) Clearly, their diverse composition and structure
enable a wealth of possible applications.
Many of these applications depend on the accessibility of the pores;
pore topology is largely determined by the shape, size, and composition of
the linkers. (325;326) Of course, the node composition and topology also play a
determining role in the structure, (308;327;328) but there are far fewer synthetic
handles available to modify and create novel inorganic clusters. Thus, the
prediction of MOF properties is well-suited to computation, (329) as we can rapidly
construct a large family of MOFs using only the geometry of the SBUs and a
nearly endless selection of linkers. (330) Indeed, one of the most elegant aspects of
MOF chemistry is the ability to transmetallate (331;332;333;334;335) and geometrically
substitute one linker for another of similar geometry (336;337;338) enabling a broad
gamut of structurally related materials. (339) Such isoreticularity (340;341;342) is a
cornerstone of the field (343;344;345) and has been a key focus for several years with
notable successes (e.g., efficient absorption of water, (173;346;347;348) and as site
isolated catalysts (349;350;351)).
The chemical and physical properties of a MOF are derived from
the identity of its two building blocks: the metal center and organic
linkages. (216;313;352;353) These two components generate the pore, which is the feature
that imparts the macroscale material properties and functionality. (354)
86
The pore is thought to provide additional physical influence on properties
such as site adsorptivity, (355) enantioselectivity, (356) and size selectivity (357;358)
as functions of the pore volume, aperture, and composition. In such cases, a
complete description of the pore aperture and/or cage created within may be
necessary to explain intrapore reactivity. Such was the case for a borylation
catalyst incorporated as the linker of a UiO-67-analogue. (359) The heterogenized
system exhibited >99% chemoselectivity toward a monoborylated product, far
exceeding the selectivity of its homogeneous counterpart. (357) Ultimately, this study
concluded that selectivity of their catalytic reaction resulted from pore confinement
restricting access to the diborylated product rather than local sterics or electronic
interactions. (359)
The best structural characteristics describing the pore are the pore
volume and surface area, which can be determined experimentally and
computationally. (353;360) However, the experimental and computational methods use
inert gas molecules to probe pore structure, yielding only the accessible surface area
and neglecting the true pore topology, Figure 24. (232;360) Experimentally, Brunauer-
Emmett-Teller (BET) models are used to derive surface area from adsorption-
desorption isotherms and relies on multilayer surface coverage. (232;352;360) This is
problematic because pore filling and formation of the monolayer and multilayer
occur simultaneously. (232;360) Computational methods mimic experimental methods
using a Grand Canonical Monte Carlo (GCMC) approach, (352;353;361) which is
computationally expensive and relies on a rigid sphere approximations to depict the
void space (362;363;364;365) (which, from Chapter 3, we know are not suitable for highly
ionic materials, such as MOFs). From both the computational and experimental
approaches, it is apparent that pore features with a radius of curvature less than
87
Figure 24. Connolly surface area lacks information. van der Waals surface area
does not account for polarized bonds.
that of the gas molecule are neglected. Consequently, the resolution of the SA is
limited by the radius of the chosen gas molecule. (360;366)
We can envision a method to calculate void space properties using
STREUSEL, which would yield values that are independent of a probe gas
molecule. We perform our solid-state DFT calculations using the popular software
VASP; to accommodate this, STREUSEL has added functionalities that enable it
to process VASP output files and calculate void space properties. Thus, we obtain a
more precise image of the pore volume and surface area.
Results and Discussion
Pore Volumes and Surface Areas. Since the logic of the STREUSEL
size determination algorithm discussed in Chapter 3 relies only on the electrostatic
potential obtained from DFT calculations, the presented electric field cutoff holds
for solid-state calculations. To verify our approach we generate electrostatic
potential maps from single-point calculations for the MOFs contained within
the CoRE MOF database, (367) and calculate the STREUSEL pore volumes and
surface areas. (70) In order to verify that the values we obtain from the electric
field are realistic, we compare our calculated pore volumes with those obtained
88
Figure 25. There is good agreement between the pore volumes obtained using
Platon and STREUSEL. Both void space calculation metrics are performed using
pristine structures. The solid black line represents the values where Platon =
STREUSEL.
using the PLATON software, which is the industry standard for void space
calculations, (362;363;364;365) Figure 25. Indeed, there is good agreement between
the pore volumes obtained using Platon and those obtained using STREUSEL,
suggesting that the STREUSEL methodology is a valid approach.
With the knowledge that STREUSEL yields realistic pore volumes, we
compare these values with experimental measurements of pore volume and surface
area for a dataset of 100 MOFs, Figures 26 and 27. Unsurprisingly, there is less
agreement between experimental and STREUSEL pore volumes than seen with
the Platon pore volumes, Figure 25; this is likely because the theoretical values
89
Figure 26. Comparison of experimental pore volumes and STREUSEL pore
volumes for a series of MOFs. Data points are colored according to the probe
gas molecules used in the experimental measurement. there appears to be no
correlation with the experimental probe gas; this is likely due to unknown
differences in sample preparation and measurement conditions. The full dataset
with corresponding references is presented in Appendix B.
are obtained using pristine models, which is not representative of an experimental
sample. (2) This is further reinforced by the overestimation of surface area by
STREUSEL, Figure 27.Here, we observe that experimental samples are far less
porous than their theoretical counterparts. A similar overestimation of pore volume
is also observed. Indeed, this is expected; the STREUSEL methodology accesses
topological features unobtainable by methods relying on gas probe molecules.
Logically, additional topological features would increase calculated pore volumes
and surface areas relative to experimental measurements.
In addition to the unavoidable presence of defects, (233) experimental
preparation is known to have large impacts on measured void characteristics. (368)
For example, Kaye et al. observed a 1500 m2g−1 decrease in measured surface area
90
Figure 27. Comparison of experimental surface areas calculated via BET (green)
and Langmuir (blue) isotherms and STREUSEL surface areas for a series of MOFs.
91
for MOF-5 prepared in air vs. air-free. (368) Consider eight published, experimental
surface areas for MOF-5, Figure 28. While each value is valid, the large range in
measured surface areas is likely due to three competing factors: (17;369)
1. Collapsed pores within the samples – This is likely due to the vacuous
conditions that are required by pore volume and surface area measurements.
2. Occluded pores within samples – This is likely due to leftover reagent
molecules that were not fully washed away after synthesis.
3. Linker and/or metal center vacancies – This is due to the entropic favorability
of defect formation.
These factors have predictable effects on measured pore volumes – namely, the
presence of occluded and collapsed pores will result in an artificial decrease in
measured pore volume, while vacancies will result in an artificial increase. Effects
on surface area are far more complex. While collapsed pores will understandably
result in a decreased surface area, the effect of occluded pores notably depends on
the pore aperture:pore surface area ratio. Abstractly, there may exist a topology
whose pore aperture area is larger than the surface area of the chemical motif
within the pore. In this instance, occluded pores would result in an artificially
larger measured surface area. The effect of linker vacancies will be discussed in
detail in the following section. Comfortingly, the STREUSEL surface area falls
within the range of published surface areas for MOF-5.
Imaging Linker Vacancies using the Electric Field. Vacancies
occur as either missing linkers or metal clusters, as well as single atom omissions
from the inorganic clusters. (375) The elucidation of experimental vacancies is
extremely challenging in MOFs and other solids, often requiring a potpourri of
92
Ref. (306)
Ref. (374)
Ref. (373)
Ref. (372)
Ref. (371)
Ref. (370)
Ref. (368) (in air)
Ref. (368) (air-free)
Figure 28. Comparison between published Langmuir surface areas for experimental
samples of MOF-5, the Connolly surface area (purple), and the STREUSEL surface
area (green).
93
experimental methods (PXRD, EXAFS, NMR, IR, etc.) to extract even an average
defect concentration per node. (376;377;378) As a result, we are still learning about
their formation in extremely well-studied materials and only have a rudimentary
understanding of their formation. (379;380;381;382)
To completely characterize a defect in MOFs, a combined theory and
experimental approach is often required. For example, the identity of anionic
species coordinated to the node at a linker vacancy in UiO-66 and -67 was obtained
by IR and NMR experiments; the catalytic mechanism occurring at active sites
defined by these defects was then studied with an appropriate cluster model to
assess reagent adsorption in the presence and absence of linkers. (383) More advanced
experimental methods, such as diffuse scattering, electron microscopy, transmission
electron microscopy, and electron crystallography, have also been paired with
theory to refine structural motifs of defects. (384) (385)
Periodic models, however, still face challenges for defective systems because
of the cost associated with forming a unit cell sufficiently large enough to mitigate
artificial defect ordering. (386) Furthermore, vacancies are realistically either anionic
or cationic and often compensated by an adventitious counterion, mandating some
charge passivation routine. (387) In experiment, these ions originate from the salts
or solvents used in the synthesis, and these salts can be used to inform model
construction. (388)
Modulators can be used to deliberately install linker vacancies as they
compete with the multitopic linker to bind to the metal node either during
synthesis or postsynthetic modification. (240;377;389) Cluster vacancies may be
targeted by achieving multiple linker vacancies; this causes either an absence of
a linker for the cluster to bind to or the stability of an already bound cluster to
94
be compromised. (390) Linker vacancy concentration may be tuned using a variety
of synthetic conditions ranging from modulator concentration and acidity (391)
to solvent selection. (392;393) Selectively altering linker vacancy concentration not
only enables cluster vacancy formation but also affords defect engineering, which
can be used to intentionally alter macroscopic properties. For example, increased
modulator concentration during synthesis correlated to increased gas uptake in
defective UiO-66 due to additional missing linkers, as experimentally demonstrated
by Wu et al. with N2 uptake isotherms.
(394)
Most computational defect studies have been performed on the UiO-series
of materials, as they are known to feature high concentrations of defects. Here, we
will use UiO-66 as a model to explore the effects of modulators on calculated pore
volumes and surface areas, Figure 29.
95
Figure 29. Three passivated models of UiO-66 are examined in this study; H2O,
Cl−/H −2O, F /H2O, and formate.
96
CHAPTER V
COVALENT BRIDGE INTERSTITIALS AS A ROUTE TOWARDS
CONDUCTIVE FRAMEWORKS
Abstract
The high porosity, extended connectivity, and tunability of metal-
organic frameworks (MOFs) makes them ideal candidates for the development of
electrocatalysts, charge storage materials, and chemiresistive sensors. However,
the highly ionic nature of the inorganic-organic interface inhibits the realization of
high electrical conductivity thereby limiting their utility for these applications. In
this project, we explore a retrofitting strategy to post-synthetically install redox-
active interstitials that covalently interconnect MOF linkers as one avenue towards
augmenting the electrical properties of these otherwise insulating frameworks. The
formation of retrofitted organic linkages within a given scaffold is governed by a
statistical distribution of the orientation of linkable units as demonstrated by an
atomic level simulation examining the kinetics of wire formation and the resulting
distribution within an extended unit cell of MOF-5. Periodic DFT calculations
quantify the conductive capacity o the newly formed charge transport pathways.
Examining other common scaffolds, we find that the macroscopic topology of post-
synthetically installed wires (i.e. extended wires vs. conductive loops) formed
through new covalent bonds, and their resultant capacity for electronic conduction,
is dictated by the effect of MOF symmetry on the statistical distribution of linker
orientations. By combining statistical models o wire formation with electronic
structure theory applied to periodic models of the retrofitted MOFs we show that
post-synthetic formation of pseudo-polymeric wires is a promising strategy to instill
electrical conductivity in MOF scaffolds.
97
Introduction
Tunable materials design is poised to solve a series of device performance
challenges within the electrochemical device arena. (395;396) One of the material
properties pinnacle to electrochemical device performance is surface area; (397)
increased surface area is responsible for increased number of active sites in
electrocatalysts, (398) increased capacitance in the case of supercapacitors, (399)
increased rate of electrochromic switching, (400;401) and improved performance of
resistivity-based chemical sensors (402) within the energy storage device realm,5
among other benefits. (403;404;405) Currently, the leading class of materials for
high surface area are metal-organic frameworks (MOFs). (15;406) The simple
building block construction (metal nodes connected by organic linkers) further
offers additional routes towards bulk property tuning via post-synthetic
modification (407;408;409;410;411) and retrofitting. (412;413;414) Despite the compositional
diversity of MOFs afforded by the simple construction, the majority of scaffolds
within this class of nanoporous, crystalline materials are insulating. (217;415;416) This
limits their application in the electrochemical industry, which would otherwise
benefit from the large surface areas offered by MOFs. (217;313)
The insulating nature of MOFs can, in part, be attributed to their
construction from redox-inactive organic linkers and closed shell metal
centers. (415;416) Yet, due to the tunable nature of MOFs, redox chemistry may
be promoted within these scaffolds by introducing redox-active components
in several ways; (415;416) i) during synthesis, (417;418) ii) via post-synthetic
modification, (419;420;421;422) or iii) as guest molecules within the pores. (404;423;424;425)
Indeed redox-activity in MOFs has been shown to enhance gas uptake and
98
binding, (417), engender electrical conductivity, (402), as well as exhibit reversible
electrochromism. (401)
Of the varying methods of inducing redox activity in MOFs, (416) this work
focuses on installing redox-active covalent bridging motifs between aromatic linkers.
This is similar to the host-guest interaction of HKUST-1 infiltrated with 7,7,8,8-
tetracyanoquinododimethane (TCNQ) (0.07 S cm−1), which yielded a million-
fold increase in electrical conductivity over pristine HKUST-1 (10−8 S cm−1). (423)
TCNQ bridges Cu(II) paddlewheels, enabling improved electronic coupling between
the these dimeric Cu subunits. In this way, Allendorf et al. not only render and
otherwise insulating MOF conductive, but coordination between TCNQ and the
Cu(II) paddlewheels results in a continuous charge transport path throughout the
scaffold.
The TCNQ-Cu(II) paddlewheel coordination complexes within the scaffold
are inherently composed of repeating subunits throughout the material. In this
way, we can consider the TCNQ-Cu(II) subunit chain a monomer within the
scaffold. This not unheard-of in the MOF community. Indeed, MOF-polymer
hybrid materials have been synthesized and are advantageous for a range of
applications, (426;427) including; improving the handling and implementation of
the hybrid material over MOFs, (428) and improved gas storage a separation
capabilities. (429) For instance, tetracyanoquinodimethane (TCNQ) is a tetratopic
redox active molecule that fortuitously can bridge adjacent Cu-paddlewheels in
Cu3(BTC)2. The extent of bridging was shown to cause an increase in electrical
conductivity (6 orders of magnitude, up to 0.07 S cm−1). (423) Comparing periodic
models of the pristine framework and with the TCNQ interstitial showed strong
binding (84 kJ mol−1) and cluster models consisting of two benzoate-capped
99
paddlewheels saturated with water molecules and bridged by TCNQ, showed
new conduction band states localized on the TCNQ molecule. (423;430) It was thus
reasoned that thermally promoted charge transfer between the framework and
guest molecule enhanced charge mobility. Later, scanning electron microscopy and
porosimetry supported the bridging arrangement of TCNQ proposed by theory. (431)
Nonnative organic linkers bridging metal sites also impacted charge mobility
in Ni3(HITP) .
(412)
2 Band structures, calculated using the B3LYP hybrid functional
and the triple-ζ basis set, pob-TZVP, were constructed for different forms of
Ni3(HITP)2 including a variant with layers separated by 4,4’-bipyridine bridging
the nickel sites and pristine Ni3(HITP)2 with the same interlayer spacing as the
bridged derivative. Bulk Ni3(HITP)2 is found to be metallic, but the band gap
widens with increasing interlayer separation, and upon inclusion of the bipyridine
interstitial band dispersion is seen to decrease, effective mass values increase, and
the band gap widens; Ni3(HITP)2 is rendered a semiconductor. The authors note
that square-planar Ni(II) bound to four nitrogen atoms results in filled dz2 orbitals,
which do not readily bond to additional out of plane organic linkers. Potential
energy curves for pyridine coordination to this square planar motif showed the
hypothetical semiconducting structure was unlikely to form, however Cr3(HITP)2
with bridging 4,4’-bipyridine may be an alternative, stable structure.
We present and validate a post-synthetic avenue towards electrically
conductive MOFs by including redox active covalent bridging motifs between
linkers, leading to bonded, redox non-innocent linkers that form a pseudo-polymer
within the scaffold. Electronic properties of MOF-5-covalent-bridge pairs are
assessed using periodic density functional theory (DFT) calculations. Wire
formation pathways are examined using python-driven atomistic simulations.
100
Macroscopic wire conformations are investigated using an abstract statistical
model.
Results and Discussion
The tractability of this approach is established using MOF-5 (Zn4O(BDC)3
(BDC = 1,4-benzenedicarboxylate)) (17) for three main reasons: i) the simple 6-
2 connected net, which results in cubic pores that are readily abstracted for the
statistical model, ii) demonstrated propensity for redox reactivity, (420) and iii) prior
evidence of linker crosslinking. (432;433)
Selection of the covalent bridging motif invited two design considerations; i)
Fermi level shift, and ii) geometric constraints. Installation of the covalent bridging
motif introduces additional states, resulting in a shift to the Fermi level; n-type
interstitials shifts the bands up because additional electrons are introduced to the
system, while p-type shifts bands down due to the inclusion of holes. Considering
the mid-gap states, we focus our search on p-type interstitial bridging motifs, whose
downward shift of the bands accommodate the additional states.
Taking into consideration the octahedral configuration of the inorganic
MOF-5 node, we examine covalent bridging motifs that may fit within the 5.21
Åand 9.90 Ågaps, Figure 30. This leads us to a series monomers from redox-
active electrically conductive organic polymers; polyphenylene sulfide (PPS), (434)
polyaniline (PANI), (435;436) poly(p-phenylene vinylene) (PPV), (437) and two
thienothiophene derivatives (TT1 and TT2). (438;439;440) For example, PANI-based
electrode materials boast high specific capacitance (700 F g−1), (436) as well as
several accessible oxidation states. (441) The protonated form of polyaniline is
electrically conductive, (435) featuring values on the order of 2 – 10 S cm−1 for
pressed pellets. (442) Thus, we install the reduced form of PANI in MOF-5. In
101
Figure 30. The two examined positions for the covalent bridging motifs are
labelled within the MOF-5 subunit. Five covalent bridging motifs were examined –
polyaniline (PANI), polyphenylene sulfide (PPS), two thienothiophene derivatives
(TT1 and TT2), and poly(p-phenylene vinylene) (PPV). Covalent bridging motif
lengths are displayed.
addition to, Thienothiophene derivative TT1 featuring aromatic caps offer charge
mobilities on the order of 0.1 cm2 (V s)−1. (443)
Three modeling approaches are used to assess the efficacy of each
MOF-bridge pair; i) infinite wire models where we examine the bridging motif
conformation that infinitely spans reciprocal space to obtain the ideal electronic
properties, ii) atomistic simulations of bulk wire formation where we examine the
subtle changes in scaffold geometry upon iterative inclusion of covalent bridges
within a bulk model, and iii) statistical models where we assess the macroscopic
wire topology yielded by each crystallographic configuration.
102
Infinite wire models. In order to identify the validity of electrical
conductivity induced by covalently-bridged interstitials, we probe the electronic
structure and properties of the ideal linear wire configuration within the MOF-
5 scaffold for five bridging motifs: polyphenylene sulfide (MOF-5-PPS), poly(p-
phenylene vinylene) (MOF-5-PPV), polyaniline (MOF-5-PANI), and two
thienothiophene derivates (MOF-5-TT1 and MOF-5-TT2). Calculation details
provided in the following subsection. Results are summarized in Table 6.
Calculation Details. Density functional theory (DFT) was used
to evaluate the structural and electronic properties of the pristine MOF-5 and
MOF-5 scaffold with five bridging motifs: polyphenylene sulfide (MOF-5-PPS),
poly(p-phenylene vinylene) (MOF-5-PPV), polyaniline (MOF-5-PANI), and two
thienothiophene derivates (MOF-5-TT1 and MOF-5-TT2).
Structural relaxation of pristine MOF-5 was performed using the PBESol
functional, (141) as implemented in the Vienna ab initio Software Package
(VASP). (73;74;75;76) The wavefunctions of the valence electrons were expanded
with plane-waves with a cutoff of 500.00 eV; core electrons were treated with
the projector augmented wave theory. (2) The total energy was converged within
10−5 eV and forces to 0.005 ev Å−1; the Brillouin-zone was integrated over a well
converged and symmetrized 2x2x2 k -point mesh.
The MOF-5-X (X = PPS, PANI, PPV, TT1, TT2) models were generated
by manually inserting the covalent bridging motifs within the fully-optimized
pristine MOF-5 structure such that they generate an infinite wire across the
periodic boundary conditions. The resulting structures were optimized to the same
criteria described above, and symmetry was not enforced.
103
Table 6. Comparison of electronic properties for the infinite wire models listed in
order of decreasing band dispersion (meV).
Model Band Gap (eV) Dispersion (meV) Work Function
MOF-5-PPV 1.26 313.5 -5.26
MOF-5-PANI 1.09 75.5 -4.13
MOF-5-TT1 2.07 69.0 -5.28
MOF-5-TT2 2.02 61.1 -5.15
MOF-5-PPS 1.44 29.6 -5.98
MOF-5 3.50 -6.58
Of the five models, MOF-5-PPV and MOF-5-PANI exhibited the smallest
band gaps, 1.26 eV and 1.09 eV, respectively, and largest band dispersion, 313.5
meV and 75.5 meV, respectively Figure 2. Electronic band structures and density
of states (DOS) for MOF-5-PPS, MOF-5-TT1, and MOF-5-TT2 are presented in
the Supporting Information Figures 31 and 32. The pristine MOF-5 band structure
is aligned to the background potential, (444) while the MOF-5-PPV and MOF-5-
PANI band structures are aligned to the pristine MOF-5 conduction band (CB),
highlighted in gray. Interestingly, the frontier bands of MOF-5-PPV are contributed
by the PPV interstitial, while only the MOF-5-PANI valence band (VB) is located
on the PANI interstitial. This is demonstrated by the atomic cntributions to the
VB in the DOS, Figure 33.
Introducing mid-gap states alters the atomic contribution to the band edges.
For example, MOF-5 VB is localized on the inorganic and organic components
of the scaffold, while the CB is localized solely on the organic linker (BDC). The
band edges of MOF-5-PPV are localized on the PPV interstitial, suggesting charge
transfer would occur within the covalent bridge upon photoexcitation. (189) The
VB of the MOF-5-PANI model is localized on the PPS interstitial, while the CB
is localized on the BDC of the MOF-5 scaffold. This suggests upon photexcitation
charge-transfer would occur between the linker and covalent bridge. (189)
104
Figure 31. The Fermi-aligned band structure of pristine MOF-5 features flat bands
with a large band gap (3.50 eV). MOF-5 models featuring polyaniline (MOF-5-
PANI) and polyphenylene sulfide (MOF-5-PPS) interstitials reveal increased band
dispersion and smaller band gaps than their pristine counterpart. Oxidized models
of MOF-5-PANI and MOF-5-PPS are also presented and feature bands that cross
the Fermi level.
105
Figure 32. The Fermi-aligned band structure of pristine MOF-5 features flat bands
with a large band gap (3.50 eV). MOF-5 models featuring two thienothiophene
derivatives (MOF-5-TT1, and MOF-5-TT2) reveal increased band dispersion
and smaller band gaps than their pristine counterpart. Spin-polarized electronic
band structures of the oxidized models of MOF-5-TT1 and MOF-5-TT2 are also
presented and feature bands that cross the Fermi level. Spins are separated by
band color within the oxidized models.
106
Figure 33. The vacuum-aligned band structure of pristine MOF-5 features flat
bands with a large band gap (3.50 eV). MOF-5 models featuring polyaniline
(MOF-5-PANI) and poly(p-phenylene vinylene) (MOF-5-PPV) interstitials reveal
increased band dispersion and smaller band gaps than their pristine counterpart.
107
Upon inclusion of the interstitials, there is an increase in band dispersion (or
band width), indicating more mobile charge carriers, (2) Table 6. Band dispersions
are recovered using temperature-independent DFT calculations; band dispersions
less than kT are considered flat because they are lost to thermal smearing. At
standard-state conditions, kT = 25.7 meV. Thus, the low band dispersion (29.6
meV) of the MOF-5-PPS model, may not be a factor in experimental observables
at room temperature.
The band edges of each model are localized on different components of
the scaffold. Notably, the wired models feature organic VB, while the VB of the
pristine MOF-5 model is localized on the Zn node. Thus, inclusion of the p-type
interstitials bypasses the insulating inorganic node of the MOF scaffold; this has
been shown to be beneficial in photocatalytic systems. (189)
The infinite wire models reveal the ideal electronic properties. Thus, we
use an atomistic simulation examine the necessary, minute conformational changes
upon covalent bridge insertion; in this way, we examine the effect of wire formation
on the MOF-5 scaffold.
Bulk wire formation simulations. Inserting a series of kinetically-
directed covalent bridges within a MOF scaffold breaks the symmetry of the
crystalline framework. (445;446) Moreover, linkers may need to rotate to accommodate
additional covalent bridges, which is difficult to capture within the infinite
wire models – an inherently static picture. We assess the resulting spatial
implications using an atomistic wire formation simulation; code is available
on GitHub. (447) The employed model is comprised of a 3x3x3 supercell of
MOF-5 consisting of dichloroterephthalic acid linkages in two configurations
(2,5-dichloroterephthalic acid, and 3,6-dichloroterephthalic acid). The initial
108
Figure 34. Two SH group orientations were sampled to determine the degree of
linker rotational freedom; 1) toward, in which the SH groups of adjacent linkers
point toward each other, and 2) away, in which the Cl groups of adjacent linkers
point away from one another.
conformation of the dichloroterephthalic acid linkers is determined by sampling a
Boltzmann distribution, described in the following subsection.
Determining the degree of linker rotational freedom. The
orientational favorability of the BDC linkers within the pristine MOF-5 scaffold
were determined using periodic DFT calculations. To assess the rotational
restriction between adjacent 1,4-benzenedicarboxylate (BDC) linkers within the
pristine MOF-5 primitive cell, the adjacent BDC linkers were thiolated, yielding
2,5-dithiol-1,4-benzenedicarboxylate linkers. Two models were constructed featuring
the chloro groups pointed toward and away from each other, Figure 34, simulating
the extrema of the rotational positions each linker samples.
Structural relaxation of the chlorinated MOF-5 models were performed using
the PBESol functional,1 as implemented in the Vienna ab initio Software Package
(VASP). (73;74;75;76) The wavefunctions of the valence electrons were expanded
109
with plane-waves with a cutoff of 500.00 eV; core electrons were treated with
the projector augmented wave theory. (2) The total energy was converged within
10−5 eV and forces to 0.005 ev Å−1; the Brillouin-zone was integrated over a well
converged and symmetrized 2x2x2 k -point mesh.
The absolute difference between the total energy for each optimized model
(Ediff ) is calculated,
Ediff = |Eaway − E 2toward| (5.1)
and compared with kT, the Boltzmann distribution at the operating temperature
(0.025 eV at standard state conditions), a viable assessment of the relevance for
energy differences. (2)
In the case of the MOF-5 model, an energy difference of INSERT was
obtained. Since this is less than 0.025 eV, we assume no favorability for linker
orientation, and sample an unbiased random Boltzmann distribution of linker
orientations for the atomistic simulation and general statistical model.
The atomistic wire formation simulation is composed of three iterative
algorithmic steps; i) initialize each linker with a dichloroterephthalic acid
configuration informed by sampling the Boltzmann distribution, ii) freeze the Cl
atoms that are pointed toward each other within the model cell, and iii) reorient
the free dichloroterephthalic acid linkers. These steps are repeated until no
additional wires may be formed. Finally, the covalent bridging motifs are inserted
between connected Cl atoms; the linkers of the MOF-5 scaffolds are rotated to
accommodate the covalent bridging motifs.
Owing to the spatial separation and rotational freedom of the linkers in the
octahedral, 6-2 connected net, we observe a preference for short (< 4 subunits)
linear wires across 100 wire formation simulations, Figure 35a. Only one of the 100
110
Figure 35. a The bulk wire length distribution totaled over 100 simulations. One
run resulted in a wire length of 15 units, “model-15.” b The bulk wire length
distribution for model-15. The MOF-5 depiction of model-15 with the longest
wire highlighted in yellow.
wire formation simulations resulted in a wire that spans the width of the 3x3x3
supercell (“model-15”), Figure 35b. Unsurprisingly, the wire length distribution of
model-15 mirrors that of the 100 simulations. Ultimately, the rotational freedom
and spatial separation of the organic linkages afforded by the 6-2 net topology of
MOF-5 results in only short, linear wires throughout the supercell.
While larger than the infinite wire models, the 3x3x3 model is not sufficient
enough to examine the bulk wire conformation. Thus, we use a generalized
statistical model and algorithm to probe bulk wire conformations within much
larger (e.g. 50x50x50) cells.
111
Generalized statistical model. The macroscopic wire morphology is
broadly directed by the rotational freedom of the linkers and the connectivity of
the crystallographic net. To identify the exact implications of these contributing
material features, we designed a statistical model to simulate wire growth and
assess bulk wire conformation. Here, we describe the statistical model via the
MOF-5 wired examples examined in this work; the code used to run the statistical
model is available on GitHub. (447)
MOF-5 is described by a 6-2 connected net; the inorganic node is connected
to six linkers, each of which only connect with two inorganic nodes. This net
is decomposed into a tensor where each tensor element represents a different
component of the MOF topology (i.e. metal, linker, pore), Figure 36a. The linker
orientations are sampled from a weighted Boltzmann distribution. In this way,
the linker orientation is directed by the degree of rotational freedom specific to
the MOF described in the net. The degree of linker rotational freedom is obtained
using the methodology described above.
There are two main algorithmic steps to the wire formation simulation,
Figure 36b; i) sample Boltzmann distribution to determine linker orientations,
ii) identify paths within the model. Importantly, once a linker is determined to
be a part of a wire, it is no longer allowed to rotate – effectively simulating the
presence of a covalent bridge. These two steps are repeated until no further linker
orientation may result in a wire. We ensure that this is the case for a series of
model sizes by plotting the number of unbridged linkers as a function of algorithmic
step, Figure 37; we observe a plateau in unbridged linkers, indicating the model
runs to termination.
112
Figure 36. a Net theory is used to translate the crystalline structure of MOF-5
to a tensor. b The tensor elements may then be iterated to represent bulk wire
formation in four distinct steps; 1) the initial tensor is initialized with linker
orientations, 2) paths are then identified, 3) linkers that are not in a wire are
reoriented, steps 1) – 3) are repeated until no linker reorientation will result in a
new wire formation, at this point, 4) the macroscopic connectivity is assessed. For
visual simplicity, we do not show the layers in the third dimension – although our
statistical model does take these into account.
113
Figure 37. Unconnected linkers as a function of algorithmic step for a series of
model sizes using the generalized statistical model.
Running a series of simulations on the same net topology reveals
macroscopic wire formation patterns. Unsurprisingly, the 6-2 connected net in
conjunction with no favorable linker orientation features a preference for short,
linear, in-plane wires, Figure 38. Moreover, the wire length distribution of the
general statistical model, Figure 38, is similar to that obtained using the atomistic
wire formation simulation, Figure 35; this is expected considering these two
models are describing the same MOF-5-polymer system. We modified the original
algorithm to simulate PPV covalent bridges (linkers that were 180deg apart were
allowed to bind), Figure 39. Unsurprisingly, the total number of wires decreases,
yet the same distribution is achieved.
Conclusions
The amalgamation of models employed within this study informs us of
three major findings. First, incorporation of p-type, redox-active organic linkers
within the otherwise insulating MOF-5 scaffold yield semi-conducting scaffolds
that turn metallic upon oxidation. Indeed the MOF-5-PPV and MOF-5-PANI
models outperform the other models explored in this study, suggesting they may
114
Figure 38. The bulk wire length distribution for a 6-2 net topology allowing
connections between 90deg linkers, totaled over 100 iterations of the general
statistical model.
115
Figure 39. The bulk wire length distribution for a 6-2 net topology allowing
connections between 180deg linkers, totaled over 100 iterations of the general
statistical model.
116
be promising for experimental synthesis. Yet, the spin polarization observed in
the thienothiophene derivatives is promising for chemical sensing applications. (448)
Second, while the inclusion of covalent bridges breaks symmetry, it does not
sufficiently distort the MOF-5 scaffold enough to prevent wire formation. Thus,
the spatial separation and rotational freedom of the 6-2 net topology is, ultimately,
beneficial to the overall integrity of the MOF scaffold. Third, agreement between
the atomistic simulation and the general statistical model indicates that the
statistical model is a promising route towards scanning diverse topological nets
– enabling efficient identification of net topologies that may yield interesting (i.e.
hoops, helices, etc.) wire conformations.
117
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