The Pennsylvania State University
The Graduate School
MATERIALS DESIGN OF SUBSTRATES FOR GAS ADSORPTION
AND STORAGE
A Dissertation in
Physics
by
ZhaoHui Huang
c 2010 ZhaoHui Huang
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
December 2010
The dissertation of ZhaoHui Huang was reviewed and approved∗ by the following:
Vincent H. Crespi
Professor of Physics
Professor of Materials Science and Engineering
Downsbrough Professor
Dissertation Advisor, Chair of Committee
Jorge O. Sofo
Associate Professor of Physics
Associate Professor of Materials Science and Engineering
Nitin Samarth
Professor of Physics
Kristen Fichthorn
Merrell Fenske Professor of Chemical Engineering
Professor of Physics
Jayanth R. Banavar
Distinguished Professor of Physics
George A. and Margaret M. Downsbrough Department Head of Physics
∗
Signatures are on file in the Graduate School.
Abstract
All the three chapters in the thesis are originated from the efforts of hydrogen storage, although the metal stabilization can be beyond that. It is well known that
hydrogen storage is the bottleneck problem of a grand hydrogen economy, but its
notorious low volumetric density creates an overwhelming challenge in storage.
Materials-based storage might make it possible to store large quantities of hydrogen in small volume at practical temperature and pressure. However, the actual
experiments are hard to perform and even to explain the results, which gives computer simulations a big chance to investigate the storage. In this thesis, I use the
Density Functional Theory based software to explore the new materials. A brief
introduction of DFT theoretical background is given in the first chapter.
In Chapter 2, we study how to prevent the metal atoms from aggregating. In
nature, low-coordinated metal atoms can provide new opportunities for gas storage and catalysis when they are exposed to their environment. But unfortunately,
they are generally unstable against aggregation. We demonstrate that electron deficiency in an sp2 carbon layer, induced by heavy (but realistic) boron doping, can
stabilize sparse metal layers (Be, Mg, Sc and Ti) against aggregation thermodynamically. If the atomically dispersed metals are not thermodynamically favored,
take Pd as an example, local inhomogeneities in boron density will create large
kinetic barriers against aggregation, so Pd layer can be kinetically stable.
In Chapter 3, we first introduce the two new classes of materials. The compelling Ammonia Boron based materials used for chemical storage attract interests
because of their high hydrogen content, but the overly stable products seriously
eliminate any possibility of reversible storage. On the other hand, the organic
frameworks exhibit strong structural stability and accessibility. Almost all of the
atoms are on the surface. These amazing properties look like exclusively being tailored for the hydrogen release in one of the products, PAB polymer. We combine
the advantages of the polymer and framework, then design four series of frameiii
works. They are proven stable through molecular dynamics simulations.
Chapter 4 is focused on the essential issue of hydrogen applications: the release kinetics. We present a novel idea to tune the activation barrier by coupling
the polymer with an external framework spring, and then apply it to the simple
PAB – PAB/H2 transformation. Our results for planar polymer transformations
show that this coupling indeed change both the hydrogen binding energy EB and
its release barrier ∆EK , although the lowest barrier by this tuning is still too
large. When the polymer is compressed, the EB and ∆EK are lowered; when the
polymer is stretched, the EB and ∆EK are raised. Finally, we test a non-planar
transformation and achieve a great improvement in ∆EK , so the non-planar polymer transformation might be an much more effective way in hydrogen release.
iv
Table of Contents
List of Figures
vii
List of Tables
xi
List of Symbols
xii
Acknowledgments
xiii
Chapter 1
DFT Introduction
1.1 Hohenberg-Kohn Theorems . . .
1.2 Kohn-Sham Equation . . . . . . .
1.3 Exchange Correlation Functional
1.4 Pseudopotential . . . . . . . . . .
1.5 Plane Wave Basis Set . . . . . . .
1.6 DFT Molecular Dynamics . . . .
Chapter 2
Metal Atoms Stabilization
2.1 Metal Dispersal . . . . .
2.2 Calculation Detail . . . .
2.3 Charge Transfer . . . . .
2.4 Kinetic Barrier . . . . .
2.5 Summary . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
. . . . 1
. . . . 4
. . . . 6
. . . . 13
. . . . 18
. . . . 20
.
.
.
.
.
23
23
25
28
30
32
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Chapter 3
Boron Nitrogen Framework for Hydrogen Storage
33
3.1 Chemical Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
v
3.2
3.3
3.4
3.5
Organic Framework . . . . . . . . . . . . . .
Design of BN Framework . . . . . . . . . . .
Relaxed Structures . . . . . . . . . . . . . .
3.4.1 Wurtzite . . . . . . . . . . . . . . . .
3.4.2 Zincblende with Pure BN Vertex . .
3.4.3 Zincblende with One-Carbon Vertex
3.4.4 Nine-carbon Vertex . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . .
Chapter 4
Mechanical Tuning on Thermodynamics and
gen Release
4.1 Release Thermodynamics . . . . . . . . . . .
4.2 Nudged Elastic Band Method . . . . . . . .
4.3 Spring Model . . . . . . . . . . . . . . . . .
4.4 NEB Results . . . . . . . . . . . . . . . . .
Bibliography
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. 37
. 41
. 48
. 49
. 52
. 55
. 57
. 59
Kinetics of Hydro.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
61
61
63
66
71
80
vi
List of Figures
2.1
2.2
3.1
3.2
3.3
The energy gained by depositing metals onto various boron-doped
sp2 sheets, (a) referenced to a lifted unrelaxed mtal sheet or (b)
the isolated metal atom. For comparison, the bulk metal cohesive
energy is marked on the right-hand edge of the (b). Boron-carbon
sheet geometries are given in (c). Geometry 0 is a pure carbon
sheet. Increasing boron content stabilizes the dispersion of metals
across the boron carbon surface. For four metals: Be, Mg, Sc and
Ti, their highest binding energies are greater than the respective
cohesive energy, so the dispersed phases are favored over the bulk
metals. Bulk Pd is fcc; the others are hcp. . . . . . . . . . . . . . .
The evolution in the Sc-derived bands near the Fermi level with
increasing boron content in the boron carbon sheet. The bands
shown in black are for the structure with lower boron content, while
the red bands for the structure with higher boron doping. In each
panel, the red bands are raised by ∼0.4 eV to align the Fermi levels
and hence compensate for the electron deficiency of the higher boron
system. Boron doping depletes charge from the Sc levels. . . . . . .
The breakdown of borazine ring and intermolecular dehydrogenation
provide two mechanism in borazine pyrolysis to develop into boron
nitride like frameworks. . . . . . . . . . . . . . . . . . . . . . . . . .
Several kinds of vertices (Color scheme: red for oxygen, grey for
carbon, light pink for boron and the other color for metal atoms.
All hydrogen atoms are omited.) . . . . . . . . . . . . . . . . . . . .
The prototype cyrstals for the design of BN framework. Each of
bond is supposed to be replaced by a polymer segment (pink for
boron and blue for nitrogen). . . . . . . . . . . . . . . . . . . . . .
vii
27
29
36
39
42
3.4
3.5
3.6
3.7
3.8
3.9
Tested vertices for wurtzite and zincblende framework design (pink
for boron, blue for nitrogen and grey for carbon). 3.4(a) and 3.4(b)
both are failed sets of vertices for wurtzite framework. The stable
set of vertices for wurtzite is shown at 3.4(c). 3.4(d) is a failed set
for zincblende construction, while 3.4(e), 3.4(f) and 3.4(g) are stable
for zincblende. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Side(left) and top(right) views of the relaxed conformers containing six B-N bonds. From top to bottom: TC, TT, Deformed TC
and COIL. Boron atoms are in pink, nitrogen atoms in blue, and
hydrogen atoms are not shown above for clarity. . . . . . . . . . .
Relaxed wurtzite structures (pink for boron, blue for nitrogen, grey
for carbon and white for hydrogen): (a) BH2 NH2 linker with two
pairs of BN bonds; (b) The linker in (a) loses one H atom from every
B/N and turns into BHNH linker; (c) Dramatic structural change
after losing all the hydrogen with three-pair long BN linker; (d)
For the structure with four-pair long BN linker, losing all H atoms
creates a vertex made of eight carbon atoms. . . . . . . . . . . . .
Relaxed zincblende structures with pure B/N vertices (pink for
boron, blue for nitrogen, grey for carbon and white for hydrogen):
(a) BH2 NH2 linker with two pairs of BN bonds between a boron
vertex and a nitrogen vertex; (b) The structure with three-pair long
BHNH linker; (c) After all the hydrogens are released, the two-pair
long BN linker evolves into a BN atom chain. Notice all the vertices keep sp3 character; (d) After all the H atoms are released, the
three-pair long BN linker becomes a BN chain similar to the case in
(c), but nitrogen vertex is now a BN dimer vertex with sp3 character.
Relaxed zincblende structures with one-carbon vertex (pink for boron,
blue for nitrogen, grey for carbon and white for hydrogen): (a) The
linker is BH2 NH2 type with five atoms; (b) The connection relation with seven-atom long linker, all hydrogen atoms are omitted
for clarity; (c) The linker is BHNH type, and deforms much more
than does the previous series, see Fig. 3.9; (d) Removing all the H
atoms leads to a CB dimer vertex connecting four linkers. . . . . .
BHNH linker is distorted in the framewoks with one carbon atom as
vertex (pink for boron, blue for nitrogen, grey for carbon and white
for hydrogen). (a) the plane of the linker is bent as a whole. (b)
The dihedral angle B-N-B-N is flipped from π to 0. (c) the linker
loses planar character in out- of-plane distortion. . . . . . . . . . .
viii
46
47
51
54
55
57
3.10 Relaxed zincblende structures with nine-carbon vertex (pink for
boron, blue for nitrogen, grey for carbon and white for hydrogen):
(a) All the BH2 NH2 linkers in this series are TT type; (b) BHNH
linkers are less deformed than those in the previous series; (c) After
all the hydrogen are removed, 9-carbon vertices keep well their sp3
character. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.11 (a) Local DOS of the contribution from all the hydrogen atoms
attached to boron or to nitrogen at a given energy. The state distribution of B-attached H atoms is much narrower than that of
N-attached hydrogen. (b) The integated total local DOS at a given
energy is calculated by summing all the local states up to that energy. It accounts for what percentage of local states has been occupied by that energy. Here the reference is taken as the total local
states at the Fermi level. H atoms bond more strongly to N atoms.
4.1
4.2
4.3
58
59
The energy difference was affected by the polymer conformation
somewhat (pink for boron and blue for nitrogen). In planar transformation, it is exothermically 0.08eV per hydrogen molecule release;
in non-planar form, it is endothermically 0.05ev per H2 release. All
the hydrogen atoms are omitted. . . . . . . . . . . . . . . . . . . . 63
NEB method is often used to find a MEP starting from the initial
state and ending at the final state. In this method, a chain of
images (small dark circles) represents the MEP. This discrete NEB
path gradually move downward on the potential energy surface until
all the images satisfy force balance. The inset shows the projection
scheme taken by NEB: only the perpendicular component of the
true force F⊥
i and the parallel component of the spring force are
included in the NEB force FNEB
. . . . . . . . . . . . . . . . . . . . 67
i
An external spring can change both the thermodynamics and the
kinetics of dehydrogenation for PAB and PIB polymers. The original PAB and PIB polymers have the relation of energy vs length
shown by curves Eini and Efinal , respectively. Once the polymers
are coupled to the spring, the equilibrium positions move to where
solid blue lines are. The blue dashed line is the assumed position
of the transition state. The location where the blue spring energy
curves, centered at the natural spring length, intercept the state position lines gives the destablization amount for the initial, final and
transition states. (a) In the case L0 < x < x0 , ∆i > ∆ts , the activation barrier is lowered; (b) In the case L0 > x > x0 , the activation
barrier is raised. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
ix
4.4
4.5
4.6
4.7
The adjustment of the binding energy and activation barrier predicted by the simple model of a polymer coupled with a spring. (a)
Given a spring with force contant K (in unit eV/Å2 ) and natural
length L0 , the change in binding energy EB with respect to L0 ; (b)
the change in activation energy ∆Ek with respect to L0 . . . . . . .
The activation barrier is 2.17eV if a PAB segment releases one hydrogen molecule without mechanical spring coupling. All the data
in table 4.1 and 4.2 share this overall appearance. The seven images
are spline fitted to predict the barrier height. For the highest image,
the true potential gradient approaches zero, so it is almost at the
saddle point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The predicted transition state from the NEB calculation for a planar
transformation of PAB into PAB/H2 without mechanical coupling
(pink for boron, blue for nitrogen and white for hydrogen). After the
H2 is released, the rest of the backbone remains planar. The released
hydrogen molecule is pushed away from the polymer. The transition
structure shows the hydrogen atom has to break the bonding to a
N atom then form a H2 near a B atom. The distance between two
H atoms supposed to come off is 0.99Å. . . . . . . . . . . . . . . . .
The NEB computes a planar PAB-nonplanar PAB/H2 transformation (pink for boron, blue for nitrogen and white for hydrogen). (a)
The relation of energy against the unit cell length of polymers. It
is clear that they have different equilibrium lengths. (b) The initial structure is simplay a planar PAB polymer. (c) The transition
structure predicted again shows that the hydrogen release must start
from a broken N-H bond. The H2 molecule forms near the boron
atom and escapes from between two pairs of H-B-H triples, rather
than from the side as in the planar-planar transformation. (d) The
final structure is a nonplanar PAB/H2 polymer. The B-N bond losing two H atoms takes the horizontal position in the final polymer
segment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
71
74
75
78
List of Tables
2.1
3.1
3.2
3.3
4.1
4.2
The kinetic barriers against metal aggregation, based on a coarsegrained percolation analysis, for a range of overall boron concentrations in the sp2 lattice. Middle columns give the fractions of
coarse-grained cells with the indicated numbers of boron atoms. . .
30
The average binding energy (eV) per hydrogen molecule in both release steps: BH2 NH2 −→ BHNH+H2 and BHNH −→ BN+H2 . For
those unit cells deviating much from cubic, their lattice parameters
are also given in column (a, b, c) and (α, β, γ) . . . . . . . . . . . . . 50
The bond length alternation in wurtzite frameworks with BH2 NH2
type linker. Every linker contains two-, three- or four-pair BN bonds. 53
The bond length (Å) change in the one-carbon vertex frameworks
after all the hydrogen atoms are removed. The linker is either 7- or
9-atom long. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
The modification of thermodynamics and kinetics when the
pair long BN polymer is coupled with a spring K=2.5eV/Å2
polymer is in force balance with the spring.) . . . . . . . .
The modification of thermodynamics and kinetics when the
pair long BN polymer is coupled with a spring K=15eV/Å2
polymer is in force balance with the spring.) . . . . . . . .
xi
four(The
. . . .
four(The
. . . .
76
76
List of Symbols
DFT Density Functional Theory
GGA Generalized Gradient Approximation
LDA Local Density Approximation
MEP Minimum Energy Path
NEB Nudged Elastic Band
PAB Polymeric Aminoborane [-BH2 NH2 -]n
PIB Polymeric Iminoborane [-BHNH-]n
PAW Projeted Augmented-wave
VASP Vienna ab initio Simulation Package
xii
Acknowledgments
I feel grateful to my advisor, Prof. Vincent Crespi, for the discussion, support and
patience that he has shown during my time at Penn State. His incredible scientific
knowledge and research philosophy have always inspired me. Had I have another
chance to choose the research field, definitely I would have chosen hydrogen storage
project.
I warmly thank Dr. Paul Lammert for his help and indispensable assistance
during my study. Many discussions and his knowledge have greatly contributed to
the success of my work.
At the time of graduation, when I look back the past eight years on the Ph.D
study, it was a painful road full of frustration and excitment. My parents always
trusted me and encouraged me. Their supports are essential part of this degree.
xiii
Dedication
To my parents.
xiv
Chapter
1
DFT Introduction
1.1
Hohenberg-Kohn Theorems
For a given many electron system, traditonal quantum mechanics describes the
electronic state with a multi-variable wavefunction Ψ(r1 , r2 , . . . , rN ), and then
solves the Schrödinger eqution (1.1), where T̂ , Û and V̂ee are kinetic energy, external potential and electron-electron interaction operators, respectively. To solve
this equation, the hamiltonian matrix is diagonalized in the infinite Hilbert space.
With the number of electrons N increasing, the dimension of the space increases
exponentially and turns out to be an impossible task if N > 20.
h
i
ĤΨ = T̂ + Û + V̂ee Ψ = EΨ
N
N
N
X
1
1X 2 X
∇i +
U(ri ) +
Ĥ = −
2 i
|ri − rj |
i
i
(1.1)
Density Functional Theory (DFT) offers an alternative way to solve the manybody problem. The fundamental physical quantity for the system is now the charge
density of the ground state instead of an unwieldy multi-variable wave function,
which does not need to be found explicitly. All the observables can be calculated
through a neat formalism of functional derivatives. The DFT method can greatly
reduce the computational burden and increase the scalability in the system since
DFT can deal with up to several hundreds of atoms.
2
The theoretical groundings of DFT is two Hohenberg-Kohn theorems [1]. The
first theorem states that the external potential U(r) is determined by the charge
density of the ground state n(r) up to an arbitrary constant. Since the Hamiltonian
is defined by the external potential, it follows that all the states including excited
states are completely determined.
For a given density n(r), define the universal functional F [n] as a constrained
search over all possible wave functions that yield the density n(r) and minimizes
T + Vee ,
F [n] = min hΨ T̂ + V̂ee Ψi
(1.2)
Ψ→n
which follows the Levy and Lieb’s definition of general functional F [n(r)] [2]. It
automatically includes the degenerate ground states in the search. The minimizing
wave function can then be denoted by Ψ[n]. In an alternative view, the exact
ground state wavefunction of density n(r) can be considered as the wavefunction
that yields the density n(r) and minimizes T + Vee . The total energy functional
can then be written as:
E[n] = F [n] +
Z
U(r)n(r)dr = T [n] + Vee [n] +
Z
U(r)n(r)dr
(1.3)
The exact kinetic energy functional T [n] and the exact electron-electron repulsion
functional Vee [n] are,
T [n] = hΨ[n] T̂ Ψ[n]i and Vee [n] = hΨ[n] +V̂ee Ψ[n]i
(1.4)
The second theorem states that the functional E[n(r)] has its minimum at
the true ground state density n0 (r). With the aid of a functional derivative, the
theorem can be expressed as Eq. (1.5). This property is a direct consequence of
the general variational principle. By minimizing the functional E[n], with respect
to n(r), the energy of the ground state is obtained. Finding the ground state is
actually through a two-step minimization procedure.
δE =0
δn(r) n0 (r)
(1.5)
The HK theorem for the ground state can be generalized to the case of an
inhomogeneous electron gas in thermal equilibrium [3]. In the grand canonical
3
ensemble at fixed temperature T 6= 0 and chemical potential µ, the grand potential
functional on density matices ρ̂ also has a minimum property analogous to that of
the ground state energy. This grand potential functional is constructed,
1
Ω[ρ̂] = Tr ρ̂(Ĥ − µN̂ ) + ln ρ̂
β
(1.6)
whose minimum is the equilibrium grand potential Ω[ρ0 ] ≤ Ω[ρ]
1
Ω[ρˆ0 ] = − ln Tre−β(Ĥ−µN̂ )
β
(1.7)
where ρˆ0 is the grand canonical equilibrium density matrix
e−β(Ĥ−µN̂ )
ρˆ0 =
Tre−β(Ĥ−µN̂ )
(1.8)
As an analogy to the first HK theorem, the equilibrium electronic density n(r)
also uniquely determines an external potential U(r) which in turn determines ρˆ0 ,
so ρˆ0 is a functional of density n(r). Now the universal functional of the density
n(r) valid for all U(r) can then be taken as
F [n(r)] = Tr ρˆ0
1
T̂ + V̂ee + ln ρˆ0
β
(1.9)
For a given potential U(r), the functional
Ω[n(r)] =
Z
U(r)n(r)dr + F [n(r)]
(1.10)
equals the equilibrium grand potential when n0 (r) is the correct equilibrium density. Any other density n(r) will cause
Ω[n(r)] > Ω[n0 (r)]
(1.11)
This follows the above minimum property of Ω[ρ̂] since Ω[n0 (r)] is the right grand
potential Ω[ρˆ0 ]. The statement can be called the Mermin theorem. It is an analogy
of the second HK theorem.
The Mermin theorem leads to even more powerful conclusions than the HK
4
theorems, namely that not only the energy, but also the entropy, specific heat, etc.
are functionals of the equilibrium density. However, the Mermin functional has not
been widely applied. The simple fact is that it is much more difficult to construct
useful, approximate functionals for the entropy (which involves sums over excited
states) than for the ground state energy.
1.2
Kohn-Sham Equation
The Hohenberg-Kohn theorems provide the grounding of DFT calculation, but
they do not explicitly specify how to construct the universal functional F [n], which
is not necessarily in the formalism of Hartree-Fock and can be orbital free. In the
most popular scheme, however, Kohn-Sham formalism [4] takes such an orbital
strategy. Since for multi-electron system, the kinetic energy equals the absolute
value of the total energy, any meaningful improvement must decrease significantly
the error in T [n(r)]. A major breakthrough in Kohn-Sham formalism is building a
fictitious non-interacting many electron system which has the same ground state
density as the original interacting system. The introduced error will be lumped
into the potential term. The advantage of such a treatment is that the kinetic
energy is accurate.
Kohn and Sham proposed the rearranngement of universal functional F [n] into
three parts,
F [n] = Ts [n] + EH [n] + Exc [n]
(1.12)
where Ts [n] is the kinetic energy for a non-interacting system with single electron
orbital φi ,
N
1 X 2 hφi ∇ φi i
Ts [n] = −
2 i
(1.13)
EH [n] is the classical Coulomb static repulsion called Hartree energy,
1
EH [n] =
2
ZZ
n(r)n(r′ )
drdr′
|r − r′ |
(1.14)
and Exc [n] is the exchange-correlation energy,
Exc [n] = T [n] + Vee [n] − Ts [n] − EH [n]
(1.15)
5
Exc is just the difference of kinetic and full interaction energies of the true system
from those of the fictitious independent particle system whose electron-electron
interaction is just the classical Hartree energy. In the Hartree-Fock approximation,
there is no correlation term. The exchange term can be express explicitly with
orbital φi (The summation is over only same spin electrons),
N
N
1 XX
Ex = −
2 i j
ZZ
φ∗i (r)φ∗j (r′ )
1
φj (r)φi (r′ )
′
|r − r |
(1.16)
The expression of the total energy in Eq. (1.3) can then be expanded,
E[n] = Ts [n] + EH [n] + Exc [n] + EN e [n]
ZZ
Z
1
n(r)n(r′ )
′
= Ts [n] +
drdr + Exc [n] + UN e (r)n(r)dr
2
|r − r′ |
N
N
N ZZ
1 X 2 1 XX
1
2
= −
hφi ∇ φi i +
|φi (r)|2
|φj (r′ )| drdr′
′
2 i
2 i j
|r − r |
N Z X
M
X
ZI
+Exc [ρ] −
|φi (r)|2 dr
(1.17)
|r
−
R
|
I
i
I
E[n] is minimized with respect to Kohn-Shan single electron orbital φi subject
to the orthogonality constraints < φi |φj >= δij . This can be done by simply
introducing a Lagrange multiplier ǫi . The Kohn-Sham equation can be derived as
follows,
1
− ∇2 +
2
Z
′
n(r )
dr′ + Vxc −
′
|r − r |
N
X
I
ZI
|r − RI |
!
φi = ǫi φi
(1.18)
the exchange-correlation potential and the density are obtained by,
δExc
Vxc (r) =
δn(r)
and
n(r) =
N
X
i
|φi (r)|2
(1.19)
The Hamiltonian in Eq. (1.18) is an effective single particle Hamiltonian with
Kohn-Shan orbital φi as its eigenvectors.
Ĥ
KS
1 2
φi = − ∇ + Veff φi = ǫi φi
2
(1.20)
6
The effective potential Veff is,
Veff =
Z
N
X ZI
n(r′ )
′
dr
+
V
−
xc
|r − r′ |
|r − RI |
I
(1.21)
The antisymmetrized electron many-body wavefunction can be expressed with
φi in the form of a Slater determinant,
ΦKS
φ1 (r1 ) φ2 (r1 ) · · · φN (r1 )
1 φ1 (r2 ) φ2 (r2 ) · · · φN (r2 )
=√ ..
..
..
N! .
.
.
φ1 (rN ) φ2 (rN ) · · · φN (rN )
(1.22)
Since the effective potential depends on one electron orbital, which is unknown,
so the solution has to be found iteratively until the difference between intial charge
input and resulting charge output is within tolerable error.
The Kohn-Sham orbitals are not eigenstates of the original Hamiltonian, so
they do not have apparent physical meaning. However, they usually represent well
molecular orbitals in crystals. In many cases, the Kohn-Sham eigenvalues give an
adequate description of the band structure of the real crystal.
1.3
Exchange Correlation Functional
Since the form of exchange-correlation functional is unknown, it is pivotal for successful DFT calculation to evaluate Exc correctly. To better understand the properties of Exc , the concept of pair density [5] is introduced since electron interactions
always involve pairs of electrons. It originates from many-body wavefunctions.
The first order density matrix is defined as:
′
ρ(r, r ) = N
Z
3
d r2 · · ·
Z
d3 rN Φ(r, r2 , · · · , rN )∗ Φ(r′ , r2 , · · · , rN )
(1.23)
the diagonal elements of this density matrix are just densities,
ρ(r, r) = n(r)
(1.24)
7
and the exact kinetic energy,
1
T [n] = −
2
Z
d3 r′ ∇2 ρ(r, r′ ) r=r′
(1.25)
For a single Slater determinant, it can be rewritten with Kohn-Sham orbitals
ρ(r, r′ ) =
N
X
φi (r)∗ φi (r′ )
(1.26)
i=1
The pair density is defined as:
′
P (r, r ) = N(N − 1)
Z
3
d r3 · · ·
Z
d3 rN |Φ(r, r′ , r3 , · · · , rN )|
2
(1.27)
this quantity has a clear physical meaning: P (r, r′) d3 r d3 r′ is the probability to find
an electron in d3 r around r and another electron in d3 r′ around r′ . Thus it contains
information on the correlations among the electrons. There is an important integral
based on the fact that if one electron is found already, only the other N −1 elctrons
can be found in the space.
Z
d3 r′ P (r, r′) = (N − 1)n(r)
(1.28)
P (r, r′) can also be expressed in terms of density,
P (r, r′) = n(r) [n(r′ ) + nxc (r, r′ )]
(1.29)
where Hartree energy is written out explicitly and nxc (r, r′) is the potential exchange correlation hole density around an electron at point r. nxc (r, r′ ) should
obey the constraint Eq. (1.28),
Z
d3 r′ nxc (r, r′ ) = −1
(1.30)
nxc (r, r′) can be further divided into the exchange hole and the correlation hole.
The exchange hole is a special case and can be calculated directly from KS orbitals
if the wavefunction assumes a single Slater determinant. For states φi and φj , only
8
the terms with same spin survive integration in Eq. (1.27),
′
Px (r, r ) =
N
X
i,j=1
φ∗i (r)φ∗j (r′ ) [φi (r)φj (r′ ) − φi (r′ )φj (r)]
2
= n(r)n(r′ ) − |ρ(r, r′ )|
(1.31)
leading to the relation
2
nx (r, r′) = − |ρ(r, r′ )| /n(r)
and
Z
d3 r′ nx (r, r′) = −1
(1.32)
(1.33)
the exchange hole acts only for electrons with the same spin and is negative everywhere. The correlation hole is everything left not that is in the exchange hole,
which must integrate to zero, so it just redistributes the charge and has both positive and negative parts. Correlation is most important for electrons of opposite
spin,
Z
d3 r′ nc (r, r′ ) = 0
(1.34)
The potential energy of an interacting electron system is,
1
Vee =
2
Z
3
dr
Z
d3 r′
P (r, r′)
|r − r′ |
(1.35)
Thus from the view of pair density, the potential energy part of exchange correlation is simply the Coulomb interaction between the charge density and its surrounding exchange correlation hole. The full exchange-correlation energy including kinetic terms can be found from the “coupling constant integration formula”
that is based on Feynman-Hellman theorem [6]. It states that for a parameter
λ-dependent Hamiltonian T̂ + λV̂ , the derivative of eigenvalue with respect to this
λ equals to the expectation of the derivative of Hamiltonian with respect to λ
∂ Ĥ λ dE λ
λ
= hψ λ ψ i
∂λ dλ
(1.36)
9
ψ λ is the eigenfunction of H λ , thus
E=E
λ=1
=E
λ=0
+
Z
0
1
D
E
dλ ψ λ |Vˆee |ψ λ
(1.37)
As an analogy [7], we introduce a coupling constant λ into the univeral functional F [n] and broaden its definition,
D
E
F λ [n] = Ψλ [n]|T̂ + λVˆee |Ψλ [n]
(1.38)
To be consistent, Ψλ [n] denotes the minimizing function of T̂ + λVˆee . For λ = 1,
we have the real physical system, for λ = 0, we have the Kohn-Sham system. Let
Φλ [n] denote the Kohn-Sham function. For all values of λ, the density remains
that of the physical system. Notice Fλ=1 [n] = F [n] and Fλ=0 [n] = Ts [n], and then
apply the Feynman-Hellmann theorem to this λ–dependent Hamiltonian,
F [n] = Ts [n] +
Z
0
1
D
E
dλ Ψλ [n]|Vˆee |Ψλ [n]
(1.39)
The next step is to rewrite the exchange-correlation functional as Exc [n] = T [n] −
Ts [n]+Vee [n]−EH [n]. The Kohn-Sham quantities are independent of the parameter
λ, so λ-dependent Exc
λ
Exc
[n]
D
E D
E
λ
λ
λ
ˆ
ˆ
= Ψ [n]|T̂ + λVee |Ψ [n] − Φ [n]|T̂ + λVee |Φ [n]
λ
(1.40)
Inserting the definition of the exchange-correlation energy to Eq. (1.39),
Exc [n] =
=
1
Z
Z0 1
D
E
dλ Ψλ [n]|Vˆee |Ψλ [n] − EH [n]
dλ
0
λ
Uxc
[n]
λ
(1.41)
λ
λ
Here Uxc
= Veeλ − EH
is the scaled
potential exchange-correlation,
and the coordiD
E
λ
λ
λ
λ
nate scaling relation Vee = λ Ψ [n]|Vˆee |Ψ [n] as well as EH = λEH are used in
the derivation. The above expression for Exc can be further written as,
Exc
1
=
2
Z
3
dr
Z
n(r)
dr
|r − r′ |
3 ′
Z
0
1
dλnλxc (r, r′)
(1.42)
10
In the form of potential energy, the full exchange-correlation energy is evaluated at all intermediate coupling constants. This actually can be considered as
the interaction between the electron density and its coupling constant averaged
av
hole density nav
xc [8]. nxc is represented by the third integral in Eq. (1.42). The
convention here is that the electron density is positive, while the hole density is
negative.
The simplest model to calculate the exchange-correlation functional Exc is the
Local Density Approximation(LDA) in which the functional depends only on the
values of electronic density at each point in space. The most successful LDA are
those based on the homogeneous electron gas(HEG) [9], in which the nuclei are
replaced by a uniform positively charged background. This development is natural
since the HEG is the simplest system to demonstrate the characteristics of an
interacting electron system. The exchange hole of the HEG can be calculated
analytically from the wavefunction. The exchange energy density is
ǫx = −
3 3
3
kF = − ( n(r))1/3
4π
4 π
(1.43)
The correlation hole tends to weaken the interaction strength and is much more
complicated. There is no analytical form for the correlation hole. Gellmann and
Beruckner [10] gave the correlation energy at high density limit rs → 0 for an
unpolarized gas,
ǫc (rs ) → 0.311 ln(rs ) − 0.048 + rs (A ln(rs ) + C)
(1.44)
rs is another measure of the density, satisfying the relation 4π/3rs3 n(r) = 1. At
low density the system can be considered a Wigner crystal with zero point motion
leading to [11]
ǫc (rs ) →
a2
a1
a3
+ 3/2 + 2 + · · ·
rs rs
rs
(1.45)
The most accurate results for the correlation energy of the HEG are found from
quantum Monte Carlo(QMC) calculations at many values of rs . They are then fitted into an analytic form Ec (rs ) and extrapolated to meet the high- and low-density
limits mentioned in the above. Widely used formula are Perdew and Zunger [12]
(PZ), Ceperly-Alder [13] and its Vosko, Wilkes and Nussair [14] (VWN) interpo-
11
lation.
Back to LDA, the Exc is simply an integral over the whole space with the energy
density ǫxc at each point assumed to be the same as in a HEG with that density,
LDA
Exc
[n(r)]
=
Z
d3 rn(r)ǫHEG
xc [n(r)]
(1.46)
LDA approximates the hole at any point as the hole of a HEG with the electron
density n(r) at the center,
′
HEG
′
nLDA
xc (r, r ) = nxc (n(r), r − r )
(1.47)
This approximation on ǫxc actually implies an average over the true exchange correlation hole density. The exchange-correlation hole in the HEG must be spherical,
but the true hole is often highly aspherical. Luckily, this is not a serious problem.
LDA does not replicate the details of the hole, but it does approximate well the
system-averaged hole (spherical average). For small separation u = r − r′ , a local
approximation can be very accurate, as the hole density cannot be very different
at r′ from its value at r. As the separation increases, the density at r′ could differ
greatly from that at r, especially in a highly inhomogeneous system, for example,
regions of large gradient as near a nucleus or in the tail of a density. Overall,
the short-range hole is well-approximated, but the long-range hole is not. The
constraint of Eq. (1.30) and the system average make LDA still successful even at
large deviation.
It was not until the early 90’s that LDA became the standard approach for all
DFT calculations. Although LDA might not be accurate enough for most chemical
reaction purposes, the errors it makes are very systematic and rarely large. The
true hole decays exponentially while the HEG hole decays slowly in a power law
with an oscillation. This rapid decay of the exact hole leads to a more negative
energy density than the quantity predicted by LDA, hence LDA underestimates
Exc in magnitude. In solid context, LDA eigenvalues are often plotted as the band
structure. One famous failure of LDA is smaller band prediction than the actural
value, for example, some semiconductors are gapless in LDA. For cohesive energies
of solids, LDA tends to overbind.
The efforts to improve LDA started as early as in the original Kohn-Sham
12
paper,where they were worried that LDA might not be a good approximation and
thought that gradient expansion should increase the accuracy. Such a ”gradient
expansion approximation” (GEA) with the low order expansion of Exc , however,
does not lead to consistent improvement over LDA [15]. On the contrary, it often
produces worse results. The most serious problem is that gradients in real materials
are so large that the expansion breaks down. GEA indeed deepens the hole at small
separation u, but contains large positive oscillation at large separation u. That is
the reason why GEA does not work.
Modern Generalized Gradient Approximations(GGA) refer to a variety of ways
to modify the functional behavior at large gradients to preserve the constraint
Eq. (1.30). The improvement are based on the analysis of exchange- correlation
hole. A generalized form of GGA can be defined as [16]
GGA
Exc
[n(r)]
=
=
Z
Z
d3 rn(r)ǫxc [n(r), ∇n(r)]
d3 rǫHEG
(n)Fxc [n(r), ∇n(r)]
x
(1.48)
where Fxc is the enhancement factor and ǫHEG
is the exchange energy of the HEG.
x
It is convenient to use s = |∇n(r)|/((2kF )n(r)), dimensionless reduced density
gradient, for discussion. In the relevant range for most materials, 0 < s < 3, the
magnitude of the exchange hole is increased by a factor 1.3-1.6. and results in an
exchange energy lower than the LDA, leading to the decrease of binding energy and
correction of LDA overbinding. The GGA generally provides better description of
inhomogeneous system, and significantly extends the DFT to chemical reaction.
The construction of GGA uses a real-space cutoff, that is, the exchange hole
of GGA includes only the contributions from the GEA exchange hole that are
negative, and truncates the resulting hole at the first value of separation u to satisfy
the Eq. (1.30). The exchange hole is zero in the region beyond this separation u.
The region where the GEA hole becomes positive are simply removed from GGA,
leading to a more negative system-averaged hole.
Numerous forms of Fxc (n, s) have been proposed. There are three widely used
forms: Becke [17] (B88), Perdew and Wang [18] (PW91) and Perdew, Burke and
Enzerhof [19] (PBE). All of them give the similar result if s < 3. For larger gradient
13
s > 3, their different asymptotic behaviours result from choosing different physical
conditions for s → ∞.
Beside LDA and GGA, hybrid functionals are also used to approximate the
functional Exc . They incorporate a portion of exact exchange from Hartree-Fock
theory with exchange and correlation from other sources such as LDA. This hybridization scheme provides a simple way to improve many molecular properties
like atomization energy, bond length and vibration frequency. A notable example
here is B3LYP [20] exchange correlation functional, which can be decomposed into
B3LY P
LDA
Exc
= Exc
+a0 (ExHF −ExLDA )+ax (ExGGA −ExLDA )+ac (EcGGA −EcLDA ) (1.49)
where a0 = 0.20,ax = 0.72 and ac = 0.81 are the three empirical parameters
determined by fitting the predicted values to a set of experiemt data. EGGA
is
x
from B88 exchange functional, EGGA
from the correlation functional of Lee, Yang
c
and Parr [21], and ELDA
from VWN correlation functional.
c
1.4
Pseudopotential
When DFT calculations come into practice, a major simplication is the pseudopotential method, which is based on the fundamental knowledge: in solids, the
bonding behaviour is decided by the valence electrons, and in most cases, core electrons do not participate in the process of forming bonding. These core electrons
are quite strongly bound, and do not respond effectively to the motion of valence
electrons, hence they can be treated as essentially fixed, and only the chemically
active valence electrons need to be dealt with explicitly. This is called the frozen
core approximation.
Pseudopotentials (PP) are a type of frozen core approximation [22]. The essence
of this method is: the core electrons are considered together with the nucleus as a
rigid non-polarizable ion core, and this ion core potential is replaced by an effective
potential, whose ground state wave function Φ mimics the all electron (AE) wave
function Ψ outside a core radius. Inside the core, Φ is quite smooth and wiggles
in Ψ due to the orthogonalization requirement are removed. For a plane-wave
basis set, only low frequency planewaves are needed to represent Φ. This makes
14
planewaves a simple and effficient basis in a wide range of application, and the
complexity in the AE calculation is transferred to the generation of PP package.
The PP can be conceptually constructed as follows: Let H be the Hamiltonian with core and valence wavefunctions, ψc and ψv , respectively. Establish a
pseudostate φv by
|φv i = |ψv i +
X
hψc(i) |φv i|ψc(i) i
H|φv i = ǫv |ψv i +
= ǫv |φv i +
i
X
i
X
i
(i)
(i)
ǫ(i)
c hψc |φv i|ψc i
(i)
(i)
(ǫ(i)
c − ǫv )hψc |φv i|ψc i
(1.50)
(i)
ǫc and ǫv are the eigenvalues of the ith core and valence electrons. The summation
runs over all the ions.
"
H+
X
i
#
(i)
(i)
(ǫv − ǫ(i)
c )|ψc ihψc | φv (r) = ǫv φv (r)
(1.51)
The pseudostate φv satisfies a Schrödinger-like equation with an additional
summation term in the square brace on the left side. Adding this term to the origP
(i)
(i)
(i)
inal potential V, V + i (ǫv ) −ǫc )|ψc ihψc |, creates the effective pseudopotential.
(i)
Here the summation is dependent on eigenvalue ǫv . Because ǫv > ǫc , the effective
pseudopotential is weaker than V.
A good pseudopotential should have a good balance between three aspects:(1)
it should be as soft as possible so that the expansion of the pseudo-wavefunction
uses as few planewaves as possible. Increasing the core radius would bring more
freedom to choose a function inside the core region; (2)The transferability is also
a concern. The psuedopotential created for a given reference configuration should
replicate as many other configurations as possible. In solids, the actual crystal
potential might be very different from that in the reference atomic configuration.
Since we make a frozen core approximation, the core should be kept as small as
possible to increase the similarities among configurations; (3) There should exist a
direct connection between the pseudo charge density and the AE charge density.
For the quality of transferrability of pseduopotentials, the logarithmic deriva-
15
tives at core cut-off radius Rc can be used as an indicator,
1
dΦ(Rc , E)
dΨ(Rc , E)
1
=
Φ(Rc , E)
dr
Ψ(Rc , E)
dr
(1.52)
Better transferrability is then associated with a larger range of reference energy E,
over which the above equation holds adequately. For a good pseudopotential, it
should approximate well over the energy range including valence bands and lower
conduction bands in solids. Using Green’s theorem, it can be further proven that
the change of logrithmic derivative with respect to the reference energy is related
to the norm of wave function [23],
Z Rc
1
∂ ∂
Ψ2L (r, E)r 2 dr
ln ΨL (r, E)
=− 2 2
∂E ∂r
R
Ψ
(R
,
E)
c
0
c L
r=Rc
(1.53)
L is angular momentum number. This equation shows that if the transformation
from the AE wave function to the PP wave function conserves the norm inside
the core,their logrithmic derivatives match the first order derivative with respect
to energy as well, so the difference between AE- and PP- logrithmic derivatives is
a second order deviation from the reference energy, which is the essence of normconserving pseudopotential.
The methods to generate norm-conserving pseudopotentials were developed by
Hamann and Bachelet et al. [24]. They have a semil-local form in which a different
potential is created for each atomic angular momentum, and the form is as follows,
V PS =
X
Vl P̂l
(1.54)
l
For each angular momentum component, we select a different cut-off core radius Rl , which must be between the outmost node in the AE valence wavefunction
and its final extremum, since only there are the charge distribution and moments
of the AE wave functions well reproduced by the PP wavefunctions. The PP wavefunctions are constructed to equal to the AE wavefunctions outside the given core
radius Rl . When r < Rl ,the PP wavefunctions differ from the true wavefunctions,
but the norm is conserved.
In practice, first-row elements, transition metals and rare earth elements with
16
strongly localized orbitals are computationally demanding to describe with standard norm-conserving pseudopotentials. They require a large plane wave basis set.
To generate softer PPs, the first successful attempt is the ultrasoft PP (US-PP)
aproach introduced by Vanderbilt [25]. In his method, the core radius is increased
significantly beyond the outermost extremum and can be taken as half the distance between neighbor atoms. The constraint of norm conservation is lifted at
the cost of worse transferrability. To complement the resulting charge difference,
local atom-centered augmentation charge has to be used.
The simple procedure to creat ultrasoft PP is as follows: For each angular
momentum, choose a set of atomic reference energies Elj spanning over the energy
range of band states of interests, typically j is 1–3, and then the radial Schrödinger
equation is solved within the core radius Rl at each Elj . The AE partial wave ψlj (r)
will be smoothed to create pseudo wavefunction φlj (r) with the constraint that ψlj
and φlj and their derivatives match at Rl . Similarly, the local potential will also
be smoothed to match the AE potential outside Rl .
Several auxiliary functions need to be built. Qmn (r) are constructed as the
difference between AE charge density and pseudo-charge density.
∗
Qmn (r) = ψm
(r)ψn (r) − φ∗m (r)φn (r)
(1.55)
the index m and n run over all channels {lj}, the moment can also be defined as
qmn =
Z
Rl
Qmn (r)dr
(1.56)
0
For each reference energy, the pseudo-state φj (r) is the solution of the generalized eigenvalue problem
Ĥφj (r) = ǫj Ŝφj (r)
(1.57)
with
Ĥ = T + VH (r) + Vxc (r) + VL (r) +
X
mn
and
Ŝ = Iˆ +
X
mn
qmn |βm ihβn |
Dmn |βm ihβn |
(1.58)
(1.59)
17
|βm > is the set of projection operators, that is, the product of the spherical har-
monics and radial function. Ŝ is an overlap operator, which is different from the
unity operator only inside the core sphere. The local component VL and the coefficients Dmn of the non-local part VN L , the last term in Eq. (1.58), are determined
from the above secular equation. In ready-made PP package, VL is the bare local
potential. The Hartree and exchange correlation contribution have to be removed
from the local potential. The nonlocal part VN L of pseudopotential is also bare
potential. Dmn need to be unscreened using the following relation,
Dmn =
(0)
Dmn
+
Z
V (r)Qmn (r) d3 r
(1.60)
V is the local potential including bare local potential VL , Vxc and VH . The summaP
(0)
tion of VL + mn Dmn |βm ihβn | is then encapsulated into a package for application
use. This pseudopotential is then characterized by the set of projectors |βm i, the
(0)
coefficient Dmn and the local potential part VL .
The charge density n(r) is not equal to the square of the pseudowavefunctions,
but instead to the summation of this square and an augumentation charge inside
the sphere, Eq. (1.61). When the self-consistent iterations proceed, the contribution of the augmentation charge inside the sphere changes along with the pseduowavefunctions. This charge yields the potential and may be thought of as part of
the pseudopotential, so the ultrasoft PP is evolving during the calculation.
n(r) =
X
occ
"
φ∗i (r)φi (r) +
X
mn
Qmn (r)hφi|βm ihβn |φi i
#
(1.61)
Blöchl developed an alternative method [26], the projector augmented wave
method (PAW) to lift the norm conserving constraint, which combines the aspects
of pseduopotential and augmented plane wave(APW) approaches. Very similar
to US-PP, PAW method is also a core frozen approximation, and uses a set of
finite size psueudo partial waves mixed with plane waves to construct pseudo-wave
functions. These partial waves are centered at every atom.
Unlike in US-PP, where the augmentation charges are computed on a regular
grid in real space, the PAW method works directly with the full AE wavefunction
and potentials. Using radial grids centered at each atom to represent rapidly
18
changing functions is a major improvement from US-PP since within the US-PP
method, hard augmentation charges are generally expensive to decide in terms of
memory and computing time. In the framework of PAW, the true charge density
n(r) is given by [27]
n(r) = ñ(r) + n1 (r) − ñ1 (r)
(1.62)
where ñ(r) is the pseudo-charge-density derived from pseudo-wavefunction on a
plane-wave grid, n1 and ñ1 are onsite charge densities treated on the support radial
grids around each ion and calculated on AE and pseudo partial waves, respectively.
In fact, there exists a close connection between US-PP and PAW methods. The
US-PP can be obtained readily from the PAW total energy functional formalism
by linearizing two onsite functionals around a reference density. For materials in
which charge distribution resembles that of a reference atom within the core region,
the results of US-PP are almost the same as those achieved from PAW. But for
the system with strong charge transfer, changes of atomic orbital occupation (such
as change of hybridization), strong polarization or large local magnetic moments,
the augmentation charge in the US-PP can be hard to determine, while PAW
potentials are easier to define and show advantages over the US-PP scheme.
The other advantage of PAW over US-PP lies in dealing with alkali, alkali-earth
and transition metals to the left side of the Periodic Table. Whenever necessary,
it is always possible to unfreeze semicore states and turn them into valence states.
Fortunately, it is very simple and straightfoward to unfreeze semi-core states in the
PAW scheme because only one partial wave must be added. For metal compounds,
the energy cutoff and hence the size of plane wave basis set are usually decided
by harder nonmetal elements, so the unfreezing semi-core states of metal does not
increase the size of the basis set.
1.5
Plane Wave Basis Set
Now we will discuss how the single-particle Kohn-Sham equation is solved. When
DFT is applied to periodic crystals, the effective potential in the Kohn-Sham
equation must obey the crystal periodictiy. The Kohn-Sham orbital has the form
19
of Bloch wave due to the translational symmetry [28],
φnk (r) = unk (r)eik·r
(1.63)
where unk (r + R) = unk (r), R is any Bravais lattice, n is a band index, k is
Bloch wave vector that is confined to the first Brillouin Zone (BZ). For analytic
convenience, a Born- von Karman boundary condition of macroscopic periodicity
is introduced,
φ(r + Ni ai ) = φ(r)
i = 1, 2, 3
(1.64)
where ai are three primitive vectors and N = N1 N2 N3 is the total primitive cells
in the crystal. Thus the general form for allowed Bloch wave vector is as follows,
k=
m2
m3
m1
b1 +
b2 +
b3
N1
N2
N3
(1.65)
bi are three reciprocal vectors and mi are integers from 1 to Ni .
Both the Kohn-Sham orbital φ(r) and effective potential Veff (r) can be expanded with plane waves that satisfy the boundary condition
unk (r) =
X
Ck+G eiG·r
and
Vef f (r) =
X
VG eiG·r
(1.66)
The plane-wave expansion only contains plane waves with the periodicity of the
lattice and therefore with reciprocal lattice vectors as wave vectors. This leads to
the secular equation,
X
~2
2
ǫ(k) −
(k + G) Ck+G =
VG−K Ck+K
2m
K
(1.67)
In actual applications, the expansion cannot be infinite and must be truncated.
Given a threshold Gcut , all the higher frequency planewaves G > Gcut are discarded, so all the planewaves have the kinetic energy (~2 /2m)|k + G|2 ≤ Ecut . If
Veff (r) is smooth enough and not rapidly changing, this approximation is fine.
For many quantities such as density of states (DOS), total energy and matrix
elements, the integral over given states in the first BZ is needed. To evaluate this
integral numerically, the integral has to be transformed to a weighted sum over a
20
set of discrete reciprocal k-points.
I=
Z
f (k)dk =
BZ
1X
ωi f (ki )
Ω i
(1.68)
This k-point set is generated based on the crystal symmetry. Although these
k-points are not necessarily uniformly spaced, but the most common scheme by
Monkhorst and Pack [29] indeed uses a uniform mesh in k-point sampling. First
define the sequence of numbers,
ur =
2r − qr − 1
2qr
r = 1, 2, · · · , qr
(1.69)
where qr is an integer that determines the number of k-points in a lattice direction.
The k-point on the mesh are then defined as
kprs = up b1 + ur b2 + us b3
(1.70)
This gives q 3 distinct points uniformly spaced in the first BZ.
The lattice symmetry can significantly reduce the number of k-points by using
only k-points in the irreducible BZ wedge. The other k-points can be transformed
through symmetry operations on the one in the subset of irreducible k-points. In
addition, The weight factor is actually determined by the ratio of the order of the
entire group to the order of the group of the wave vector at the given k-point.
With plane wave basis set, the self-consistent iteration procedure is transformed
to repeatedly solve algebraic equation. Once the electronic ground state is found,
the ionic configuration is relaxed to find a new electronic ground state until all of
ions are in equilibrium position. After forty years of development, DFT has now
become a full-fleged tool to solve many-body Schrödinger equation and provides
scientists a useful view to microscopic world.
1.6
DFT Molecular Dynamics
Molecular Dynamics (MD) is a simulation tool used to probe the features of atomic
motion with high temporal and spatial resolution. It needs to define the way by
21
which a pair of particles interacts each other. Classical MD employs empirical
interatomic potentials which may fail for covalent and/or metallic system, while
DFT MD simulation takes advantage of its capacity to provide accurate description
of chemical bonding and derives the interatomic forces directly from the electronic
ground state (calculated within DFT).
The common DFT MD method was based on the scheme of Car and Parrinello [30](CP). CP introduces explicitly the electronic degree of freedom and
creates a fictitious system in which electron and ion are coupled together through
an equation of motion (EOM). With this scheme, diagonalization, self-consistency,
ionic relaxation, and volume and strain relaxation are achieved simultaneously.
This fictitious system is described by the Lagrangian in Eq. (1.71), and the
potential energy E contains the parameters KS orbitals φi, the nuclear coordiates
RI and all the possible external constraints imposed on the system αi , like the
volume Ω, the strain ǫµν , etc.
L = K −EZ
X1
X1
X1
µ
d3 r|φ̇i |2 +
MI ṘI2 +
µν α˙ν 2 − E[φi , RI , αν ] (1.71)
=
2 Ω
2
2
ν
i
I
where the φi are subject to the orthogonal constaints,
Z
Ω
d3 rφ∗i (r, t)φi (r, t) = δij
(1.72)
The Lagrangian in Eq. (1.71) generates a dynamics by variations on the parameters φi’s, RI ’s and αν ’s through the equation of motion (EOM):
MI R̈I = −∇RI E
X
δE
µφ̈i(r, t) = − ∗
+
Λij φj (r, t)
δφi (r, t)
j
µα¨ν = −
δE
δαν
(1.73)
where Λij are Lagrangian multipliers introduced in order to satisfy the orthonomality constraints. µ is a fictitious mass for the electronic degree of freedom. The ion
22
dynamics in Eq. (1.73) are the direct result of the Hellmann-Feynmann theorem,
while the dynamics associated with the φi ’s and the αν ’s are fictitious and serve
only as a tool to perform the dynamical simulation.
The summation of the first three terms in Eq. (1.71) defines a classical kinetic
energy K. K is related to the temperature of the system. It can be a measure of the
departure of a system from the minimum of its total energy within the electronic
self-consistent framework. By variation of the velocities, i.e., the {φ̇}’s, {ṘI }’s, and
{α̇}’s, the temperature of the system can be slowly decreased and the equilibrium
state of minimal E is reached when T approaches zero. At equilibrium φ̈i = 0, the
EOM of the electrons in Eq. (1.73) is identical within a unitary transformation to
the KS equation, and the eigenvalues of the Λ matrix coincide with the occupied
KS eigenvalues.
For a given molecular configuration RI , the Born-Oppenheimer (BO) potential
energy surface can be obtained by minimizing the total energy functional E[RI ,φi ]
with respect to the Kohn-Sham φi , but in the CP method, the simultaneous time
evolution of ionic and electronic degree of freedom is determined by integrating
the above EOM equation. This integration is started after initially the electronic
wave functions have been relaxed to their ground state. At successive steps, total
energy minimization is not necessary. The dynamics of electrons keeps them on the
electron ground state which corresponds to the new ionic configuration at each step.
In other words, if the ions are allowed to move at a given temperature, the classical
kinetic energy of the electrons remains zero. At any time, the electrons are in their
ground state and the ions move under the action of BO forces. Accordingly, any
significant energy transfer between the ionic and the electronic degree of freedom
should not happen so that electronic states do not deviate away much from BO
surface. The optional procedures to achieve this goal include [31]: (1) performing
periodic energy minimization to “bring the system back to the BO surface”; (2)
making the physical system contact a Nosé thermostat [32] to keep the average ionic
temperature constant. A Nosé thermostat is actually an extra particle contributing
an additional degree of freedom in the extended system. This extra particle has a
Nosé mass that determines the response of the thermostat to the physical system.
The energy exchange between the thermostat and the physical system simulates a
canonical ensemble for the latter system.
Chapter
2
Metal Atoms Stabilization
2.1
Metal Dispersal
In nature, most metals exist in the form of bulk compounds since metal atoms
always tend to form close-packed structures. In such a bulk material, except the
atoms on the surface, most atoms are present in high coordinate environment
and bound to the neighbour atoms. Exposure of highly dispersed metal atoms
to the ambient environment are generally subject to metal-metal aggregation, so
they cannot be found individually. If a stable structure of under-coordinate metal
atoms could be made, it will provide many novel physical and chemical properties,
including strong spin-dependent scattering, novel modes of catalysis from unsaturated frontier orbitals, and enhanced binding of closed-shell species through charge
donation to open orbitals on the metal. Standard metal surfaces already exhibit
some of these effects, such as catalysis, but the extension to even lower-coordinate
atomically dispersed metals promises to greatly expand the scope for novel physics
and chemistry.
One important application in which the under-coordniate metal atoms are
salient is hydrogen storage. Molecular hydrogen is an attractive fuel, with a higher
energy density by weight than gasoline and no CO2 emission. However, gaseous
hydrogen’s volumetric energy density, even in pressurized tanks, is very low, while
liquefaction is thermodynamically expensive. Storage in non-molecular form may
provide a solution, but most metal hydride hosts are too heavy and chemical storage confronts kinetic barriers in removing hydrogen from the covalent bond. Also,
24
the lightest metal hydrides, MgH2 and Li2 NH, bind hydrogen too tightly to release
it under practical conditions. Physical adsorption to high-surface -area substrates
preserves the H-H bond and hence avoids issue of reaction barriers and irreversibility. However physisorption is generally too weak for practical room-temperature
storage [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]. Recently, attention has turned
to a hybrid approach that combines elements of physical and hydride mechanisms,
where a low-dimensional carbon substrate is coated with a sparse layer of metal
(Sc, Ti) atoms [44, 46, 47]. Such a design could combine the large specific area
of sp2 carbon with the stronger binding of a hydride. It is natural that exposed
metal atoms with unfilled orbitals produce exceptionally “sticky” systems with
binding via charge donation [45] to empty d-orbitals and molecular polarization;
the greater challenge is to prevent highly dispersed under-coordinated metal atoms
from sticking to themselves [49], so that they are available to bind other species of
interest for catalysis, gas storage, etc.
First-principles density functional calculations [46, 47] have already shown that
isolated Ti or Sc atoms bound to a graphenic surface (i.e. C60 ) ,assuming without
seeing each other, avidly adsorb molecular hydrogen at a density and binding
energy appropriate to meet the aggressive DOE storage goals, that is, 5.5% of the
storage system weight and 0.040 kg H2 /L by 2015 [48]. For example, Yildrim’s work
[46] showed that a Ti atom bound to C60 surface can adsorb up to four hydrogen
molecules, either by physisorption or chemisorption, with binding energy varying
in 0.325∼1.162eV, depending both on the position of Ti atom and the order for H2
to be adsorbed. Zhao’s work [47] has similiar results for Sc to stand on five-fold
axis of C60. The binding energy of physical adsorption ranges from 0.26eV to
0.42eV. Unfortunately, all the above local minima are not a global minimum as
demonstrated in the further calculation [49]. The clustering phase of Ti atoms
on C60 surface is energetically much perferred, so these structures are kinetically
and thermodynamically unstable against clustering of metal atoms, and hence
aggregation will happen. If the aggregation happens, the metal atoms inside cannot
bind hydrogen molecules, leading the total number of hydrogen molecules adsorbed
will greatly decrease. Can exposed open-shell metal atoms be stabilized against
aggregation?
Since a stable sp2 sheet is difficult to activate towards metal carbide forma-
25
tion, the primary mechanism of aggregation for dispersed metals on carbons is
metal-metal binding. Therefore atomically dispersed metals can be stabilized in
two ways: (1) kinetically, through large activation barriers against their motion
and (2) thermodynamically by improving the stability of the dispersed state relative to competing phases. Dispersal of alkali metals onto carbons is easy, due to
their low ionization energies and weakly bound bulk phases [50]. Other metals are
more cohesive in the bulk and transfer charge to carbon less readily. Late first-row
transition metals such as chromium and iron can coordinate to electron-deficient
π complexes, as in ferrocene, where charge transfer from the metal facilitates aromaticity. We take ferrocene as a guide, but apply the lesson to the early transition
metals Sc and Ti. The extended sp2 sheet is made electron deficient through
substitutional boron doping to facilitate charge transfer.
Inhomogeneous boron doping of an sp2 carbon sheet could thus simultaneously
drive up the kinetic barriers against metal aggregation (if the motion of metal
atoms between boron sites is impeded) and drive down the energetic cost of metal
dispersal (if charge transfer from the metal ameliorates an electron deficiency in
the underlying sheet). The highly directional nature of the carbon-carbon and
carbon-boron bonds stabilizes the electron-deficient sp2 host against structural rearrangement, i.e. the strong kinetic stability of boron-carbons provides a context
in which to thermodynamically stabilize dispersed metals. Although direct incorporation of boron from the vapor phase in electric arc [51] or laser ablation [52]
integrates only a few atomic percent boron into carbon nanotubes before losing
the graphenic growth habit, the chemical reaction [53] of a mixture of BCl3 and
benzene produces well-organized graphitic materials with extremely high boron
content, up to BC3 . The pyrolysis of boron-containing polymeric precursors [54]
can produce high surface area sp2 boron-carbons with substantial (∼ 7 atomic %)
boron content.
2.2
Calculation Detail
We studied five metal species: Be, Mg, Sc, Ti and Pd. Magnesium is a close
cousin of beryllium and forms the well-known compound MgB2 and a tightly bound
metal hydride, MgH2 . Pd is a classic bulk hydriding metal and catalyst. Ti and
26
Sc are early transition metals with small ionization energies, weak cohesion in
the bulk material, and numerous empty d orbitals for Kubas binding or novel
catalytic interactions. For simplicity, we take a planar boron-carbon sheet, with
no pentagonal or heptagonal rings, in an orthorhombic unit cell with a large 10
or 12 Å spacing between layers. The metals are distributed in a sparse atomic
layer directly above the boron-doped graphene layer. Each unit cell contains eight
carbon or boron atoms and one metal atom.
The total energy calculations use a projector augmented wave [55] pseudopotential density functional [56] with a plane-wave energy cutoff of 400 eV and a
3×3×1 Monkhorst-Pack mesh centered at the Γ point [29]; (higher cutoffs gave
very similar results). A Gaussian smearing[57] of 0.02 eV controls the partial occupation of the electron wave function. The generalized gradient approximation [16]
that we use yields reasonably accurate binding energies for low-coordinate system,
although the van der Waals component is lacking. Relaxation of both internal
coordinates and transverse unit cell dimensions proceeds until the force on each
atom is less than 0.02 eV/Å.
The total energies were calculated for several possible arrangements of boron
and carbon atoms at each of several B:C stoichiometries: 0:8 (i.e. pure carbon) 1:7,
1:3 and 3:5; (the highest boron concentration exceeds that obtained experimentally
so far and hence models a local boron-rich inclusion). The areal density of metal
atoms is high enough that metal atoms are not fully isolated from each other [58],
hence a careful comparison is necessary to properly understand the energetics. We
define two distinct energies to characterize the metal binding to the boron carbon
sheet. The deposition energy Edeposit is the energy difference between the fully
relaxed metal-boron-carbon structure (EBC−M ) and the summed energies of the two
components: an isolated relaxed boron-carbon sheet (EBC ) and an isolated rigid
metal layer (EM ) at the coordinates that it assumes in the combined relaxed system:
Edeposit = EBC + EM − EBC−M . The isolated metal layer cannot be independently
relaxed, since it would simply collapse. The analysis treats the metal layer as
a collective rigid quasi-atom that is adsorbed onto the sp2 sheet. This notional
exfoliation isolates the component of the total energy associated with just metalsheet interactions. Taking an isolated metal atom as the reference point instead of
the lifted metal layer, we obtain the binding energy, Ebinding = EBC +EMa −EBC−M .
27
7
6
Ti TiSc
Ti Sc
Be
Sc
Ti
Be
Be
Sc
5
Deposition 4
Energy[eV]
3
Ti
Sc
Be
Ti
Ti Sc
Sc
Be Be
Ti
Sc
1
2
3
4
5
6
Geometry of Unit Cell
Be
5
Binding 4
Energy[eV]
3
Pd Pd
7
Ti TiSc
Sc
6
Pd Pd Pd
Ti
Pd Pd
Mg
Mg
Mg Mg
2
Sc BePd
Mg
Mg
Mg
Mg
1
Pd
BeMg
BeMg
0
0
Ti
7
Ti
Sc
0
8
Pd
Sc
Pd
BePd
Pd
BeMg
0
(a)
Be
Be
Ti
2
1
Ti
Sc
Mg
1
Mg
Mg
Be
Pd
Mg
Ti
Sc
Be
Pd
Mg
Ti
Sc
Sc
Ti
Sc
Ti
Be
Be
Pd
Mg
Be
Sc
Pd
Be
Pd
Mg
Pd
Mg
Mg
2
3
4
5
6
Geometry of Unit Cell
7
8
(b)
(c)
Figure 2.1. The energy gained by depositing metals onto various boron-doped sp2
sheets, (a) referenced to a lifted unrelaxed mtal sheet or (b) the isolated metal atom.
For comparison, the bulk metal cohesive energy is marked on the right-hand edge of the
(b). Boron-carbon sheet geometries are given in (c). Geometry 0 is a pure carbon sheet.
Increasing boron content stabilizes the dispersion of metals across the boron carbon
surface. For four metals: Be, Mg, Sc and Ti, their highest binding energies are greater
than the respective cohesive energy, so the dispersed phases are favored over the bulk
metals. Bulk Pd is fcc; the others are hcp.
This quantity is to be compared to Ecohesive to determine thermodynamic stability.
Here the cohesive energy Ecohesive is referred to as the energy required to atomically
disperse metal atoms from the bulk state into isolated atoms. In our convention, a
boron carbon system that strongly binds metal atoms has a large positive binding
energy.
After an initial relaxation without metal present, the boron-doped graphene
28
remains planar with carbon-boron and boron-boron bond lengths ranging from
1.49 to 1.58 Å and 1.64 to 1.76 Å respectively. Hence the unit cell expands
with boron doping. The relaxed boron carbon structures is then decorated with
metal atoms and relaxed again. Several different starting positions for the metal
atoms are tested, and the most stable such position is taken for further analysis.
Figure 2.1 shows the deposition energy (2.1(a)) and binding energy (2.1(b)) for five
metals above the respective boron-doped sheets at B:C stoichiometries of 0:8 (case
0), 1:7 (case 1), 1:3 (cases 2, 3 and 4) and 3:5 (cases 5, 6, 7 and 8). For comparison,
the binding energy plot also gives each metal’s bulk cohesive energy. The main
trend is, for deposition and binding energies, both increase with boron content. For
example, the deposition energy of Ti is 2.39 eV per metal atom for pure carbon,
but increases to 3.82 eV/atom for 12.5% boron, 4.84 eV/atom for 20% boron, and
6.24 eV/atom for the highest boron concentration. The binding energy shares the
similar trend, and the binding energy at this concentration exceeds the cohesive
energy of bulk Ti 6.54eV/atom. Strikingly, the binding energies of the other three
metals are already higher than their bulk cohesive energies (3.77 eV, 1.47 eV and
4.51 eV for Be, Mg and Sc, respectively). Hence they are stable thermodynamically
against aggregation when dispersed onto a boron-doped carbon sheet.
2.3
Charge Transfer
The chemical bonding is always related to the charge transfer, but the direct qualitative analysis is difficult to do because the continuous electronic charge has to be
devided and assigned to a set of atoms in the system. The common Mulliken analysis is not applicable for plane wave basis wavefunctions. The Bader analysis [59]
devides the real space by zero flux surface perpendicular to which the gradient of
the charge density is zero. It has nothing to do with the physics origin. Although
it can sometimes be used to obtain qualitative information as to general trends,
The absolute allocation of charge to atoms via real-space partition is problematic.
And moreover there exists an essential problem in dealing with coreless hydrogen
region.
Since PAW pseudopotentials always introduce a set of localized projection operators decomposing wavefunction at each atom by angular momentum and then
29
E@eVD
E@eVD
0
Ef
E@eVD
0
-1
-2
-3
1
Ef
-1
0
-2
-1
-3
-4
-2
-4
-5
G
X
L
G
(a) ScB0 C8 vs ScB1 C7
k
Ef
-3
-5
G
X
L
G
(b) ScB1 C7 vs ScB2 C6
k
G
X
L
G
k
(c) ScB2 C6 vs ScB3 C5
Figure 2.2. The evolution in the Sc-derived bands near the Fermi level with increasing
boron content in the boron carbon sheet. The bands shown in black are for the structure
with lower boron content, while the red bands for the structure with higher boron doping.
In each panel, the red bands are raised by ∼0.4 eV to align the Fermi levels and hence
compensate for the electron deficiency of the higher boron system. Boron doping depletes
charge from the Sc levels.
create auxiliary charge for each channel inside a sphere, this can be used to determine qualitatively the bonding characteristics for every band. But this projection
method implies to define the radius for each type of atom. This fixed radius is
not very appropriate for our structure since with boron doping increasing, the distance beteen metal and graphene also decreases, and hence some carbon or boron
components will be counted in as the metal contribution. So the simple idea that
integrating all the site-projected DOS up to the Fermi level is not a good idea to
decide who owns the charge and who does not.
Here we give up the effort to dertmine quantitatively the amount of charge
transfer and turn to the comparison strategy. After the Kohn-Sham equation is
solved, we have eigenstates at hand, we will compare how boron doping changes the
eigenstates. We analyze the charge transfer by examining trends in the occupation
of the Kohn-Sham eigenstates as a function of boron content. Taking for illustrative purposes the scandium system with the largest deposition energy at each
B:C stoichiometry, Figure 2.2 compares band structures pair-wise for successively
higher boron concentrations. Only bands with a significant scandium component
are shown; these are the highest occupied bands. Black dashed curves are the
lower boron concentration, the solid red line the higher. The (electron-deficient)
red bands are raised by ∼0.4 eV to align the Fermi levels. The occupied Sc-derived
states of the more highly boron doped system have lower electron occupation than
those in the lower-boron system. Hence boron doping pulls charge from the Sc
onto the sp2 plane to ameliorate the charge deficiency of the π complex.
30
Table 2.1. The kinetic barriers against metal aggregation, based on a coarse-grained
percolation analysis, for a range of overall boron concentrations in the sp2 lattice. Middle
columns give the fractions of coarse-grained cells with the indicated numbers of boron
atoms.
Boron
density
5%
10%
15%
20%
25%
30%
31.5%
2.4
Number of borons in
0
1
2
0.651 0.300 0.046
0.394 0.430 0.156
0.216 0.432 0.288
0.102 0.348 0.398
0.037 0.222 0.444
0.008 0.096 0.384
0.004 0.065 0.339
grain
3
0.002
0.019
0.064
0.152
0.296
0.512
0.593
Kinetic
Ti
3.85
2.41
2.41
1.25
1.25
1.25
0.00
barrier
Pd
2.42 eV
1.31 eV
1.31 eV
0.49 eV
0.49 eV
0.49 eV
0.00 eV
Kinetic Barrier
Compositional variations across the boron carbon sheet create preferred locations
for adsorbing metal atoms. Metal atoms placed at these sites via nonequilibrium
techniques (e.g. electrochemical deposition) may be kinetically trapped even if
atomic dispersal is not thermodynamically favored. The energy barrier against
two metal atoms, each bound at locally boron rich regions, traversing the borondeficient region between them can be estimated from percolation theory, in the limit
of low metal density. Consider a coarse-graining of the boron carbon sheet into
distinct metal binding sites, with a range of local binding energies (due to variations
in local boron concentration). We coarse-grain on the eight-boron/carbon unit cell
already defined, producing a square lattice topology. Should the density of any
given set of weaker-binding coarse-grained sites exceed the percolation threshold for
the coarse-grained square lattice, then two far-separated metals atoms must cross
this percolating barrier in order to reach each other. Hence the kinetic barrier is
defined by the strongest binding site that must be included in this set in order to
attain a percolating barrier against aggregation.
Assuming for simplicity that the substitution of boron in the carbon lattice is
random, the relative densities fi of each local B:C stoichiometry i can be estimated
from a simple binomial expansion, given a certain overall boron concentration.
These values are given in table 2.1. The percolation threshold P for a square
lattice is 0.593 [60]. If f0:8 > P , then a percolating barrier of pure carbon sites
31
exists and the kinetic barrier against aggregation is Emax − E0:8 , since one must
cross a pure-carbon site in order to connect two far-separated metals that are each
bound to the sheet by Emax . Emax is defined as the largest binding energy of a
metal-occupied site. For concreteness, we assume that all of (and only) the 3:5 sites
are occupied with metal, so that Emax = E3:5 . If f0:8 < P but f0:8 + f1:7 > P , then
the percolating barrier must contain sites with one boron, so the kinetic barrier
is Emax − E1:7 . At 20% boron substitution, the condition f0:8 + f1:7 + f2:6 > P
is required, so a metal atom bound to a 3:5 site must cross a 2:6 site to reach
another (distant) 3:5 site, yielding a kinetic barrier of 1.25 eV for Ti or 0.49eV for
Pd. If no percolating barrier forms even including all sites less binding than those
metal-occupied, then a complementary analysis can be performed which considers
not a percolating barrier which the metal atoms must cross, but a percolating
path through which the metal atoms can move. In this complementary analysis,
the kinetic barrier is set by the energy difference between the largest and smallest
binding sites that must be included in a set that forms a favorable percolating path
connecting the two metal atoms. When the boron loading exceeds 31.5%, the 3:5
regions alone form percolating cluster, so there is no kinetic barrier against metal
motion in this simple model.
Although this percolation analysis is crude, it clearly demonstrates that the
kinetic barrier against metal motion in an inhomogeneous boron-substituted carbon sheet can greatly exceed kT at room temperature. The precise kinetic barriers
obtained are sensitive to the topology of the chosen coarse-grained lattice. A triangular lattice with a percolation threshold of 0.50, for example, may better respect
the underlying symmetry of the sp2 bond. As an illustration, consider this percolation threshold, but use the same binding energies for Ti as before, for convenience.
One then obtains kinetic barriers of 3.85, 2.41, 2.41, 1.25, 1.25, 0.0 and 0.0 eV
respectively for the boron concentrations given in table 2.1. In addition, this percolation analysis assumes that the metal atoms are far-separated from each other,
so that a percolation path can fit in between. At more realistic metal densities,
the paths connecting metal atoms will be shorter, so the kinetic barriers will be
lower. On the other hand, the coarse graining ignores any intra-grain variations in
metal binding; these would tend to increase the kinetic barrier.
32
2.5
Summary
First-principles density functional calculations demonstrate that increasing boron
content in an sp2 boron carbon sheet favors atomic dispersion of several metals.
For the highest boron:carbon ratio, indicative of a locally boron-rich inclusion,
atomically dispersed beryllium, magnesium, scandium and titanium are all more
stable than the respective bulk metals and aggregation is suppressed. For lower
boron concentrations, kinetic barriers due to local variations in boron concentration
(and the surface energy penalty of incipient metal nanoclusters) can provide a
substantial kinetic stability for the dispersed phase. This is the case of palladium.
Strikingly, the boron-doped carbon sheet can be a more optimal environment for
the metal atom than is a surround of bulk metal. By destabilizing the boron
carbon sheet – which is possible due to the strong kinetic stability of the sp2 bond
– one stabilizes the dispersed metal. On physical grounds, one anticipates that the
result can be generalized to many other metals.
Chapter
3
Boron Nitrogen Framework for
Hydrogen Storage
3.1
Chemical Storage
Practical, reversible hydrogen storage at high volumetric and gravimetric capacity
remains a stark challenge to the materials community. Generally, storage materials
fail on one of two criteria: the accessible capacity is too low (bad thermodynamics)
or the system is not reversible (bad kinetics). Chemical storage, in which the
hydrogen is sequestered in the covalent bonds of host molecules, can provide very
high apparent capacities. Unfortunately, the large energy scale of a covalent bond
(∼1–3eV) compared to the natural energy scale of reversible room temperature
adsorption and release (∼0.1–0.3eV) demands a very precise free energy balance
between the hydrogenated and dehydrogenated phases of the host molecule. Even
more challenging, the release of hydrogen molecules often involves drastic covalent
bond rearrangement, hence substantial energy barriers. From the view of reaction
rate, the hydrogen storage/release kinetics of most chemical storage materials are
slow and often irreversible.
Chemical storage generates hydrogen through a chemical reaction [61]. Common reactions are hydrolysis or alcoholysis of chemical hydrides and pyrolysis dehydrogenation. Hydrolysis involves the oxidation of chemical hydrides with water
to produce hydrogen, for example,
34
NaBH4 + 2H2 O −→ NaBO2 + 4H2
MgH2 + 2H2 O −→ Mg(OH)2 + 2H2
The hydride generally needs to be protected from contact with moisture. At
the point of use, the reaction can be controlled in an aqueous medium via pH
adjustment and the use of catalyst. Sodium borohydride (NaBH4 ) and magnesium
dihydride (MgH2 ) are classified in this category. Both hydride regenerations must
take place off-board. Water needs to be carried on-board and by-products must
be removed from the vehicle.
Methanol or ethanol react with lightweight metal hydride such as LiH, NaH
and MgH2 in an alcoholysis reaction. This reaction can be controlled to occur at
room temperature and below. However, as is the case with hydrolysis, alcoholysis
reaction products must be recycled off-board the vehicle. The alcohol must also be
carried on-board. A recently filed patent uses ethylene glycol to react with sodium
borohydride [62]
NaBH4 + 2HOCH2 CH2 OH −→ NaBO2 (OH)2 + 4H2
Saturated hydrocarbons natually attract attention because of their large hydrogen content, but unfortunately, they are relatively inert in nature since hydrogen
is too tight covalently bound to carbon. Still there are some efforts in this area, for
example, the decalin dehydrogenation/naphthalene hydrogenation reaction. Naphthalene is an aromatic, white solid hydrocarbon crystal with formula C10 H8 that
readily sublimes at room temperature. Its structure is derived by the fusion of
a pair of benzene rings. Decahydronaphthalene(known as decalin) is an industial solvent and has high hydrogen content (7.3wt%, 64.8kgH2 /m3 ). The decalin
structure can be thought of as a fusion of two cyclohexane rings. Decalin does
not have serious evaporative loss because of it high boiling points [63] (187 ◦C for
trans-decalin, 196 ◦ C for cis-decalin and 218 ◦C for naphthalene).
The dehydrogenation reaction occurs at 210 ◦ C and is represented in the following reaction. A platinum-based or noble-metal-supported catalyst is required
to enhance the kinetics of hydrogen production.
35
C10 H18 −→ C10 H8 + 5H2
An alternative approach is to extract hydrogen from Ammonia Borane (AB)
related materials. These materials are isoelectric to hydrocarbons but less stable, with a more favorable free energy blalance towards hydrogen release. The
decomposition process consists of a series of pyrolysis reaction,
NH3 BH3 −→ NH2 BH2 + H2 −→ NHBH + H2
AB is a white crystalline solid with 19.5 wt% of hydrogen. Its decompostion
involves a two-step thermal process, and moreover, these steps can be overlapped or
separated depending on the heating rate [64]. When AB is heated up very quickly,
the first step of decomposition does not finish before the further decompositon
of products begins. When it is heated up slowly, AB can finish the first step of
decomposition completely even below the melting temperature 385K [65]. At the
first step, the decomposition is accompanied by a simultaneous release of hydrogen
at 1.1±0.1 mol H2 per mol AB as well as the formation of polymeric aminoborane
([-BH2 NH2 -]n ) and small amount of volatile borazine(N3 B3 H6 ) [64]. A serious
concern is too slow release kinetics. The typical time scale is 3-10 hours to finish
all hydrogen release.
Polyaminoborane (PAB) is a white non-crystalline solid at room temperature. In the range of 380-500K, PAB undergoes a thermal decomposition [66].
The residue of PAB decomposition is a white powder and can be represented by
(BNHx )z -polymer, where x,z are dependent on the duration of heating. Ideally,
if every boron and nitrogen atom lose a hydrogen in a PAB chain, then the final product is [-BHNH-]n (polyiminoborane, PIB), hence x = 2. When x < 2, the
residue might be a framework structure since it shares some infrared characteristics
with that of pure B-N bonding in boron nitride. Possibly the formation of boron
nitride lattice will continue to proceed above 500K. In this second decomposition
step [67], different gaseous products are also observed by mass and infrared spectrometry. They include monomeric aminoborane BH2 NH2 , borazine (BHNH)3 ,
small amounts of diborane and hydrogen. BH2 NH2 and (BHNH)3 mainly appear
in the temperature range 420-500K, and their amount increases with rising heating
36
(a) B3 N5 H8
(b) B5 N5 H8
(c) B6 N6 H10
Figure 3.1. The breakdown of borazine ring and intermolecular dehydrogenation provide two mechanism in borazine pyrolysis to develop into boron nitride like frameworks.
rate. BH2 NH2 monomer is unstable at room temperature and will get back to PAB.
For hydrogen release, both gas volumetric and thermal gravimetric investigation
indicate that it starts at approximately 395K [66] and continues up to 453K. The
release amount of hydrogen is at 1.1±0.1 mol H2 per mol BH2 NH2 unit. The total
release amount does not increase at higher temperature and is independent of the
heating rate. Overall, the hydrogen available in these two reactions is 6.5 wt% and
6.9 wt%, respectively.
Among the products of AB pyrolysis, borazine is very stable and should be
avoided since its existence eliminates any possibility of reversible hydrogen storage.
From the view of structure, borazine is both isoelectronic and isosteric to benzene.
Three lone pairs of electrons on nitrogen atoms tend to delocalize and form an aromatic stable ring. This aromatic stabilization makes the borazine moleculre much
more stable than linear AB. By heating it at 380 ◦ C [68], hydrogen atoms gradually come off from borazine. Analysis on intermediate compounds indicates that
on pyrolysis borazine must first form dicyclic boron-nitrogen compounds by losing
hydrogen. These compounds then in turn lose more hydrogen to form an larger
and larger extended planar framework like that of boron nitride, which possess
considerable aromatic similarities and delocalized bonding. And moreover, there
exist at least two types of reaction to proliferate such frameworks (see Fig 3.1):
the breakdown of borazine ring, as shown in the products B3 N5 H8 and B5 N5 H8 ,
and intermolecular dehydrogenation with the elimination of H2 and the formation
of a B-N bond as in B6 N6 H10 .
The thermal decomposition of borazine is not completely analogous to the thermal decomposition of benzene. Intermolecular dehydrogenation without breaking
37
aromatic rings is the main mechanism for benzene decomposition, since all products of benzene pyrolysis are derived from different fusions of benzene rings [69],
but for borazine decomposition, besides the above mechanism, there also exists
ring cleavage mechanism as illustrated in the products B3 N5 H8 and B5 N5 H8 . Once
the boron nitride frameworks begin to be produced, they are notoriously stable up
to 1000 ◦C and make the system reversibility much worse.
The hydrogen release in most cases is quite slow. For the practical application
of storage, it seems inevitable that heating must be used to speed up the reaction.
A serious side effect is that with a high heating rate, the production of boron nitride
framework also increases, making the storage device gradually lose its functionality,
so it is an essential problem to prevent the BN sheet like structure from being
produced. Recently, a new class of material, organic frameworks, has drawn a great
deal of attention and lends us the creative idea to prevent BN network creation.
In the next section, we will give a brief introduction to this class of material.
3.2
Organic Framework
Generally, creating porous rather than close-packed structures is, if not impossible,
very difficult to achieve since the crystalization requires complete control of local
kinetics, which is generally lacking within a multistep synthesis. Recently, the
design and synthesis of crystalline extended solids have been developed by linking
building blocks (0D molecules or 1D polymer chain) together through strong covalent bonds or metal clusters. The point where cross-connection happens is called a
vertex, the part between two vertice is called a linker. For a given lattice topology,
a vertex can simply be an atom or, can be replaced with an extended unit that has
the same geometry and connection relation as the single atom, which is generally
called a secondary building unit (SBU). This SBU unit with same points of extension for linkers functions much better as a vertex than do single atoms, since in
porous framework it offers better bond bending stiffness and thermal stability. A
vertex can be a metal ion at the center coordinated with strongly bonded organic
ligands or formed by a group of atoms of light elements(H,B,C,N and O). They
are well-known to form strong covalent bonds.
Several common vertices are shown in Fig. 3.2. A B3 O3 ring has three points on
38
B atoms for connection that are reserved for linkers [70]. Another triangular vertex
has three extension points on a B atom of the five-membered ring BO2 C2 [70]. Both
of these can be used to create extended two dimensional sheets with the same lattice
connectivity as graphene or boron nitride. The third vertex is a paddle -wheeled
square SBU with extension falling on C atoms. Depending on dihedral angles
between the two SBUs, this vertex can be incorporated into a one-dimensional
linear rod [71], a two-dimensional square grid [72] and three-dimensional NbO
network [73]. This vertex can even be used to produce a large single polyhedral
molecule [74]. The tetrahedral vertex is very common in the cubic system. It
can be realized in COF with TBPM [75], or zinc/cobalt nitrite in ZIFs (Zeolitic
Imidazolate Framework, a MOF series) [76]. In all of the ZIFs, a Zn or Co atom
is connected to four imidazolate(IM) or substituted IM linkers to create a tetrahedral vertex. The octahedral unit is most commonly used to form a 3D framework
with cubic symmetry. One such realization is based on oxygen-centered ZnO4 [77]
tetrahedron surrounded by six carboxylates that are transformed by a 4-fold rotation inversion axis. More types of vertices can be found in a review article [78].
The molecules that can be used as linker include: cyanide, formate, di-/tri-/tetracarboxylates, imidazolate and so on as well as their derivatives. They generally
contain ring structures to keep the linker very stable and stiff to resist deformation.
This is the essence of successful synthesis of framwork struture: well-defined and
rigid molecular building blocks will maintain their structural integrity throughout
the construction process.
On the conceptual level, the synthesis of frameworks starts with choosing the
type of network and then decomposing it into several building blocks. The geometry of vertex and linker must be consistent with the chosen network, leaving
only a few vertex candidates qualified. These molecular building blocks are then
assembled into a framework structure with the given topology. Sometimes a compromise has to be made between keeping high symmetry and using complicated
vertices. High symmetry constraints lead to a simple vertex with fewer atoms,
on the contrary, complex molecular building blocks are needed for low symmetry
framework structures. What framework will be produced out of synthesis? Experiments suggest that the most symmetric nets are the most likely to result after the
reaction reaches the equilibrium [75]. For example, cubic structures are the most
39
(a) B3 O3
(b) BO2 C2
(d) TBPM
(c) Cu2 (CO2 )2
(e) Zn4 O
Figure 3.2. Several kinds of vertices (Color scheme: red for oxygen, grey for carbon,
light pink for boron and the other color for metal atoms. All hydrogen atoms are omited.)
common, and structures with just one kind of linker are preferred, but recently
the realization of ZIF series succeeded in taking two different kinds of linkers to
weaken the symmetry constraint.
The synthesis of COF and MOF needs to determine the reaction condition necessary to yield desired vertex and linker in situ. Surprisingly, although the number
of frameworks that can be made is actually infinite, the universal method to attain them is a simple one-step process. For instance, heating in a closed vessel an
N,N′ -diethylformamide (DEF) solution mixture of Zn(NO3 )2 ·4H2 O and the acid
form of 1,4-benzenedicarboxylate (BDC) gives crystalline IRMOF-1. Similarly,
COFs involve single step self-condensation or co-condensation reaction of discrete
molecules leading to the desired vertex such as stable tetrahedral vertex units and
three-fold triangular units in [111] for cubic system. Experiments have shown that
under the closely related reaction conditions, many frameworks of the same type
such as the above metioned IRMOF series can be produced with diverse pore sizes
and functionalities depending on what substituted functional groups are used as
the linker. From the view of linker, these derivatives have the relevant geomet-
40
rical attributes, varying the component in the middle while keeping the bonding
character unchanged at the ends. After they are assembled into the same type of
the designed skeleton structure, the change in functional groups does not change
much the local kinetics at the ends of linkers, therefore the reaction condition
does not affect seriously the geometry of resulted products if a linker is replaced
with its derivatives. This tunability of framework potentially creates for us the
opportunity to produce many new frameworks to meet specific requirements. If
the framework for hydrogen storage could be made, it should resort to a simple
process to synthesize.
The typical properties of framwork structures include stability and porositiy.
Since both covalent vertices and metal cluster vertices are strongly chemically
bound, there is no surprise that COF and MOF have good thermal stability.
For example, COF-1 and COF-5 both are thermally stable up to 500 ◦ C, and
3D COFs(COF-102/103/105/108) are stable below 450 ◦ C. Similarly, IRMOF-6
keeps its rigid structure intact up to 400 ◦ C. For ZIF-68/69/70, thermal gravimetric analysis (TGA) revealed that they are thermally stable up to 390 ◦ C. The
stability of the ZIFs has also been tested by heating them in boiling benzene,
methanol and water for seven days corresponding the extreme industrial conditions. All of ZIFs retain their structure under these conditions.
The porosity results from two aspects: a linker is always much longer than
any single atomic bond since a bond is replaced by a sequence of bonds. This
increases the void space between two vertice; On the other hand, the large size of
non-compact vertex inevitably leads to large void spaces within framework [77].
The direct consequence of impressive porosity is large specific surface area that
is in accordance with the fact that most of atoms in a framework structure are
located on the surface. The atoms on a linker can be fully accessed from within the
pores during the synthesis and later during adsorption. For COFs, their BrunauerEmmett-Teller (BET) surface area are 711, 1590, 3472 and 4210m2 /g for COF1,COF-5, COF-102 and COF-103,respectively. The pore volumes are 0.32, 1.00,
1.35 and 1.66cm3 /g in the same order. Such is the case for MOFs as well. ZIF68/69/70 all have respective large BET areas: 1090, 950 and 1730 m2 /g. The
percentage of free volume for IRMOF series varies from the lowest, 55.8%, for
IRMOF-5 to the highest 91.1% for IRMOF-16. As a result of large pore volume,
41
the immediate application for COF and MOF is storage devices. IRMOF-6 [79]
has a remarkable uptake of 240cm3 CH4 at standard pressure and temperature
per gram at 298K and 36 atm, which corresponds 70% of the amount stored in
compressed tank with much higher pressure (205atm). ZIF-68/69/70 [76] exhibit
a high capacity of CO2 retention. According to calculation, one liter of ZIF-69
can store 82.6 liters of CO2 at 273K. When they are exposed to a mixture stream
containing both CO2 and CO, CO passes by very smoothly and CO2 is highly
selected to stop. The property of porosity will be inherited also by the hydrogen
storage frameworks.
In this section, we mainly discuss a new class of material that has been synthesized: Organic Framework. By precisely choosing the chemical building blocks and
controlling the reaction conditions, the simple heating or condensation procedure
yields amazingly pure crystalline framework structures. The concept of vertexlinker assembling immediately lends us the solution in prevention of the formation
of boron-nitride network. We can build up the frameworks with hydrogen containing polymer segments. As a consequence of the framework connectivity, these
polymer segments cannot come together. In COF and MOF frameworks, the stability is also a main concern. Experimentalists answer this question with strong
stable vertices and linkers. The most common used linkers consist of many five- or
six-member rings. From the view of hydrogen charge/release, this type of linker
seems overly stable. Our framework in hydrogen storage needs to go further and
allow linker deformation during dehydrogenation.
3.3
Design of BN Framework
The formation of a monolithic BN network eliminates the possibility for reversible
hydrogen storage, so it should be prevented. Here we propose a combination of
B-N-H based materials with framework structures, whose network connectivity
is designed to resist formation of an extensive pure BN network. This more open
structure could retain reactive sites for hydrogenation or dehydrogenation throughout many hydrogen storage/release cycles. We aim at the second step of pyrolysis
wherein PAB transforms to PIB, in other words, every B/N in BH2 NH2 -polymer
loses one hydrogen atom and transforms into BHNH-polymer.
42
(a) wurtzite
(b) zincblende
Figure 3.3. The prototype cyrstals for the design of BN framework. Each of bond is
supposed to be replaced by a polymer segment (pink for boron and blue for nitrogen).
The key issue to set up framework structure is to choose the right vertex.
It must be chemically stable during many hydrogenation/release cycles. In the
context of efforts to increase hydrogen content as much as possible, the vertex
should be as light in weight as possible, so here we mainly focus on fairly simple
vertices using light atoms. The convenient choices are carbon, boron and nitrogen
atoms.
To be consistent with sp3 character of both ends of PAB polymer, surely tetrahedral vertices should be used. If boron and nitrogen atoms are simply used as
vertices, there are two crystalline forms in which boron and nitrogen atoms are connected as tetrahedra: zincblende and wurtzite [80]. They are the two-component
analogs of cubic and longsdaleite diamond. In the wurtzite crystal (Fig. 3.3(a)),
the layers of atoms are stacked in the “ABAB” sequence, while in the zincblende
crystal (Fig. 3.3(b)), the layers are stacked “ABCABC”.
Wurtzite boron nitride has two simple components: boron atoms and nitrogen
atoms. The space group of wurtzite crystal is #186 P 63mc. In Fig. 3.3(a), boron
and nitrogen alternatively sit at the vertex position. Both of them take the wyckoff
positon (2b) but with different height in the unit cell. All the sites has symmetry
3m and the same type of atoms can be connected by a screw axis 63 . In the
framework design, we need two component parts acting like boron and nitrogen
atoms in the simple wurtzite boron nitride. These two parts have spatial or internal
structure instead of a single point. As a whole, they will be put in the same wyckoff
position and, from the view of topology, function as “vertices” as do the B and
43
N atoms in Fig. 3.3(a). But from the view of connectivity, only one atom in each
component part acts as a real vertex (where the four linkers meet together) in the
framework structure. One consists of a vertex with three boron-nitrogen polymer
linkers and will be put in (1/3,2/3,u) and (2/3,1/3,1/2+u); the other one is a
strut linker that separates two layers, here we use a carbon chain, and will be put
in (1/3,2/3,0) and (2/3,1/3,1/2). u is a parameter, for closed packed structure,
u=3/8 [81]; In the context of framework, u is set at the length of the strut.
Zincblende boron nitride has also two simple components: boron atoms and
nitrogen atoms. The symmetry group of zincblende is space group #216 F43m.
One component, say boron atom, occupy wyckoff position (4a) (0,0,0) and the
other component, nitrogen atom, occupy (4c) (1/4,1/4,1/4). Both positions have
the site symmetry 43m.
In the design of our framework, we strictly impose symmetry constraints only
on vertices, not on linkers and especially not for hydrogen atoms on the linkers,
since those hydrogen atoms surrounding an atom vertex always relax to avoid
getting too close to each other. The other reason why it is not possible to keep
symmetry is the linker has spatial structure and cannot be treated as a straight line
only. In simple crystal lattice, the connection between two vertices is represented
by a straight line that has no internal structure, but after this line is replaced by a
concrete polymer segment, the internal points of the polymer do not take positions
as high symmetric as are the end points. The set of symmetry operations applicable
for internal points is small than the set of symmetry operations for the end points.
For example, those zincblende frameworks are created through three-fold rotation
in [111], generally, they do not keep symmetry 4 in [001] direction. Sometimes,
the reflection symmetry can be kept by coinciding this reflection mirror m with
BH2 NH2 linker plane.
Molecular dynamics (MD) simulation is used to determine whether or not a
vertex can keep its integrity under heating. For wurtzite structure, there are two
types of tetrahedral vertices in a unit cell, each type realized by two real vertices.
Every type of vertices needs to be MD tested (one connected to boron, one to
nitrogen directly or through carbon), so we cut out the two vertices, one for each
type, from periodic crystal to reduce the computational cost and terminate the
ends with hydrogen; For zincblende framework, there are also four tetrahedral
44
vertex sites in a unit cell, but only four linkers connected to such a vertex are
contained in the unit cell, so it is necessary to add the extra linkers for another
type of vertex, and terminate the ends with hydrogen too. This process creates the
“vertex moleclue”. It contacts a Nosé thermalstat to make the simulation run in a
canonical ensemble. The Nosé thermostat is actually an extra particle with Nosé
mass interacting with the real physical system. The Nosé mass controls how fast
the thermostat responds to the change in the real physical system. The simulation
is meaningful only if the frequency of the induced temperature fluctuation is in
the same magnitude of vibrational frequency of the framework.
In each ionic loop, the electronic charge density is achieved through a selfconsistent iteration. The electronic charge density predicted for the next step is
calculated by mixing the charge density at the present step with the density in
the previous step. Often the charge density change in the previous step is not
fully included in strength, but partially included in the mixing. This is called
damping. The optimal mixing parameters are chosen from several static runs to
make the convergence fastest. In general, these vertex molecules have long linkers
attached and are sensitve to the mixing parameters. Softer damping leads to faster
convergence.
The sets of vertices tested for wurtizte structures are shown in Fig. 3.4. As
mentioned above, each unit cell contains two types of vertices, either extended or
connected at boron or at nitrogen atom of a linker. At the center of the vertex
pair in Fig. 3.4(a), there is a boron or nitrogen atom with extention points at three
linkers of the same type. The last extention is located at carbon atom to create a
single strut that is all-carbon. In such a crowded environment, one linker is forced
not to carry hydrogen. Unfortunately, this pair can not pass the MD test even
for a short duration at very low 100 ◦C heating in which one of nitrogen atoms
disconnects with the boron center. When the central part of the vertex is replaced
with a nine-carbon complex as shown in Fig. 3.4(b), the whole vertex still cannot
pass the MD test, but this time the breakdown occurs inside the linker. BH2 NH2
chain breaks and the atoms at the extension point still attach to carbon atoms.
The successful wurtzite design uses a center carbon surrounded by four carbon
atoms, three of which are extended to BH2 NH2 linkers and the forth of which is
reserved for a carbon chain strut (Fig. 3.4(c)). With this vertex, a “wurtzite” type
45
structure can be created using linkers of successively greater lengths, containing
two-,three- or four- B-N dimer units. These structures do not meet all the symmetry requirements imposed by the #186 space group, since two different groups
occupy symmetry-equivalent positions (one group is rotated by π with respect to
another), so the word “wurtzite” here should be taken literally. We use it just as an
indicator that these structures follow the same general connectivity as a wurtzite
crystal.
As for the sets of vertices used for zincblende structures, each unit cell also
contains two types of vertices. The pair of the vertices in Fig. (3.4(d)) is easily broken under 200 ◦ C heating. The simplest vertex set in the Fig. 3.4(e), a
boron or nitrogen atom with all the nitrogen or boron neighbors can stand 3.5ps
500 ◦C heating. The vertex in Fig. 3.4(f) and 3.4(g) uses a different strategy in
the cubic design. The single carbon and nine-carbon complex are alternatively
surrounded by boron and nitrogen atoms to act as a vertex. Every linker contains
an odd number atoms, for example, three boron atoms and four nitrogen atoms
or vice versa. Every pair of B-N forms dative N→B bonding, making boron partially carry negative charge and nitrogen partial positive charge. The unpaired
atom is forced to find another atom on the other side across the centered carbon.
This carbon acts like the intermediary to charge transfer. Accordingly, enhanced
bonding is expected for the carbon and its four neighbours. With these vertices,
a zincblende type structure can be created using five-, seven- or nine-atom long
linkers. The three fold rotation axis along [111] is used to produce three linkers
between (1/4,1/4,1/4) and either one out of (1/2,1/2,0), (1/2,0,1/2) or (0,1/2,1/2).
For the linker pair, (1/4,1/4,1/4)–(0,1/2,1/2) and (1/4,1/4,1/4)–(1/2,0,1/2), this
procedure violates the symmetry requirement of 2-fold rotation implied by the four
fold inversion rotation axis at (1/4,1/4,z). This can be an example that when a
straight line between two vertices is replaced by a concrete polymer, the orignal
higher symmetry cannot be kept. Our zincblende structures are constructed with
vertices in Fig. 3.4(e), 3.4(f), 3.4(g). As in the case of wurtzite, here the word of
“zincblende” is just an indicator of the overall lattice connectivity.
These vertices are connected by PAB or PIB linkers, i.e. BH2 NH2 or BHNH
polymers. PAB exisits in many conformers because the N-B-N-B rotation barrier
is relatively small (7kcal/mol per dihedral angle [82]) , so it is easy to modify the
46
(a)
(b)
(d)
(e)
(c)
(f)
(g)
Figure 3.4. Tested vertices for wurtzite and zincblende framework design (pink for
boron, blue for nitrogen and grey for carbon). 3.4(a) and 3.4(b) both are failed sets
of vertices for wurtzite framework. The stable set of vertices for wurtzite is shown at
3.4(c). 3.4(d) is a failed set for zincblende construction, while 3.4(e), 3.4(f) and 3.4(g)
are stable for zincblende.
geometry of PAB. Fig. 3.5 shows four conformers containing six pairs of boronnitrogen bonds. When the dihedral angle N-B-N-B is either 0 or π, the polymer
assumes a planar structure. Planar conformers include trans-cisoid(TC) and transtransoid(TT). TT oligomers are less stable than TC chains based on the total energy comparison, but TT’s dihedral angle, π, is at the minimum of a rotational
profile. For small perturbations to deviate from π, this rotational barrier drives
the oligomer back to the minimum, TT conformer. The TC chain is similiar to a
transition state between two nonplanar structures [82]. This picture is confirmed
by a vibrational analysis since TC chain presents one imaginary frequency. Deformed TC and COIL are two examples of nonplanar conformers. As shown in the
Fig. 3.5, the total energy in descending order is (assuming the reference energy
at TC): 0.36eV for TT, 0.00eV for TC, -0.16eV for deformed TC and -0.40eV for
COIL. This is the accumulated difference for six pairs of B-N bonds. Obviously,
the energy difference for a single pair of B-N bond is comparable to the thermal
kinetic energy at room temperature, so the difference is acturally small, and every
conformer might co-exist at room temperature.
During relaxtation, the length of polymer unit cell varies in a large range,
while the total energy changes only slightly, so there are many states with roughly
47
(a) TT
(b) TC
(c) Deformed TC
(d) COIL
Figure 3.5. Side(left) and top(right) views of the relaxed conformers containing six
B-N bonds. From top to bottom: TC, TT, Deformed TC and COIL. Boron atoms are
in pink, nitrogen atoms in blue, and hydrogen atoms are not shown above for clarity.
the same energy, and the conformation entropy contribution can be important
in favoring disordered linker conformations especially at high temperature, but
the qualitative analysis of conformation entropy is beyong our DFT calculation,
so we do not try to determine which conformer is favored for the linker. Our
calculations show that the average release energy per hydrogen molecule changes
48
little for different types of conformer. For simplity we just use TT chain as linker,
the conclusions as regards hydrogen binding should be applicable to the other
conformations.
The stability of the linker is also a concern. Since the length of every linker is
finite, the strength for a B-N monomer to bind its neighbour is position dependent.
The binding energy for the monomer at the ends of a linker is different than that
for the monomer in the middler of the linker. This makes it impossible to give
a simple conclusion. So we turn to a much simpler case, an infinite PAB or PIB
polymer, where every B-N monomer has the same binding energy to its neighbours.
In such case, the notion of “polymerization energy” is introduced and represents
how much energy is lowered for an isolated BH2 NH2 or BHNH monomer to be
incorporated into an infinite chain. The PAB polymer is only weakly bound against
its monomer BH2 NH2 [83](∆Epol ∼ 0.6eV). This is confirmed in MD simulation.
In many 500 ◦C runs, the bond length between boron and nitrogen atoms in a
finite linker is stretched above its normal value 1.60Å and can be more than 1.70Å
or even break down. On the other side, PIB is quite stable with respect to the
monomer BHNH (∆Epol ∼ 2.3eV).
3.4
Relaxed Structures
In the last section, the stability of four series of vertices is proven in MD tests. We
construct four series of frameworks with these vertices as well as different length of
linkers including PAB, PIB and backbone without hydrogen existence. These series
are (1) wurtzite with five-carbon vertex; (2) zincblende with single boron/nitrogen
vertex; (3) zincblende with single carbon vertex and an odd number of atoms in
the linker; (4) zincblende with nine-carbon vertex and an odd number of atoms
in the linker. They are optimized through DFT based volume relaxation with the
Vienna Ab-initio Simulation Packeage (VASP).
In the volume relaxation, the stress tensor is calculated. Both cell volume and
cell shape are allowed to change. Because of the finite size of the plane wave basis
set, the diagonal components of the stress tensor are not correct. This error is
often called “Pulay stress”.
The Pulay stress and related problems affect the behavior of the density func-
49
tional calculation since it indirectly changes the energy cutoff. In VASP, all volume/cell shape relaxations use a constant basis set, that is, the number of plane
waves is fixed. Initially all G-vectors within a sphere are included in the basis.
As the cell shape relaxation proceeds, the direct and reciprocal lattice vectors
will change accordingly. Although the number of reciprocal G-vectors in the basis
is kept fixed, the length of the G-vectors changes, hence the energy cutoff also
changes. Or to be more precise, the shape of reciprocal region to be integrated
becomes an elipsoid. Restarting VASP after a volume relaxation causes VASP to
adopt a new “spherical” cutoff sphere and thus the energy changes discontinuously.
To decrease the Pulay stress, we use a larger basis set, hence a higher energy cutoff at 500eV in volume relaxation. This value is a compromise between
computational cost and the effort to get accurate diagonal elements in the stress
tensor.
The tested vertices form four structural series, each of which contains linkers
of three different lengths. Following the two pyrolysis steps of AB, one half of
hydrogen atoms is first removed and then the other half. So each series consists of
nine frameworks. The average binding energy per hydrogen molecule released at
the step is
EB1 = (E[BHNH]n + nEH2 − E[BH2 NH2 ]n )/n
EB2 = (E[BN]n + nEH2 − E[BHNH]n )/n
(3.1)
All the results are collected in table 3.1. EB2 is much larger than EB1 in all cases.
3.4.1
Wurtzite
The three wurtzite structures with differing linker lengths are all stable, with the
carbon chain struts perpendicular to the horizontal plane. Lattice vectors exactly
√
are of the form of 1/2a~i ± 3/2a~j, where ~i and ~j are horizontal cartesian lattice
vectors. This set of vectors and the carbon strut at (1/3,2/3,0) create a hexagonal
pattern in the horizontal plane. a is the distance between the centers of such two
hexagons, 16.101Å, 20.233Å and 24.886Å, respectively. All the linkers are of planar
TT type (Fig. 3.6(a)). They are perpendicular to the horizontal plane and B-N
bond length in table 3.2 alternates. This contradicts the result of reference [83],
50
Table 3.1. The average binding energy (eV) per hydrogen molecule in both release
steps: BH2 NH2 −→ BHNH+H2 and BHNH −→ BN+H2 . For those unit cells deviating
much from cubic, their lattice parameters are also given in column (a, b, c) and (α, β, γ)
Design Origin
Wurtzite
Zincblende
BN Vertex
Unit Cell
B12 N12 C28 H72
B18 N18 C28 H96
B24 N24 C28 H120
B5 N5 H16
B9 N9 H32
B13 N13 H48
Zincblende
1C Vertex
B10 N10 C2 H40
B14 N14 C2 H56
B18 N18 C2 H72
Zincblende
9C Vertex
B10 N10 C18 H40
B14 N14 C18 H56
B18 N18 C18 H72
EB1
EB2
a, b, c(Å)
-0.2442 0.9288
-0.2666 0.8867
-0.2248 0.5701
6.45
-0.0246 2.2990
7.84
6.41
10.22
0.4260 2.0603
10.75
10.00
14.17
13.70
0.2465 1.5312
14.13
11.75
10.38
0.1795 1.1982
11.26
12.96
10.61
0.2495 1.1521
12.43
11.37
0.2156 1.1845
12.27
12.61
0.4051 1.3366
0.3441 1.2811
0.2461 1.2187
α, β, γ(◦)
52.30
58.15
52.58
55.59
61.28
53.69
60.82
57.45
60.68
64.86
62,67
60.90
74.54
48.38
56.03
69.18
91.15
45.92
where the alternation does not show up. The difference between the average longer
bond length and the average smaller bond is 0.097Å, 0.058Å and 0.050Å for two-,
three- and four-pair long linkers, respectively.
In the process of BH2 NH2 transformation into BHNH, every boron/nitrogen
atom loses one hydrogen, but carbon atoms in the vertex keep their hydrogen
since their bonding with hydrogen is much stronger. For instance, consider the
case with a linker of three pairs of BN, each of which loses one H2 molecule.
After it loses twelve hydrogen atoms more at the carbon vertices, the total energy
increases sharply by 50.83eV, that means, the energy needed per hydrogen molecule
51
(a)
(b)
(c)
(d)
Figure 3.6. Relaxed wurtzite structures (pink for boron, blue for nitrogen, grey for
carbon and white for hydrogen): (a) BH2 NH2 linker with two pairs of BN bonds; (b)
The linker in (a) loses one H atom from every B/N and turns into BHNH linker; (c)
Dramatic structural change after losing all the hydrogen with three-pair long BN linker;
(d) For the structure with four-pair long BN linker, losing all H atoms creates a vertex
made of eight carbon atoms.
to release from the carbon vertices is as high as 1.71eV! Similarly, for the same
framework, removing all the hydrogen atom needs 55.30eV more than removing
most of hydrogen atoms while keeping twelve hydrogen atoms at carbon vertex
sites. Hence, in the first step, we only remove hydrogen atoms from boron/nitrogen
sites. In the second step of release, all the hydrogen atoms even attached to carbon
52
are removed to calculate EB2 to explore the possible maximum release. Although
keeping twelve hydrogens at carbon vertices could bring minor reduction on EB2 ,
here the reduction is 0.21eV for the above framework, it is still far beyong the
reach of any practical application.
EB1 does not depend much on the length of the linker: 0.244eV, 0.267eV and
0.225eV for two-, three- and four-pair BN structures. What’s more important, this
step of release is exothermic. For the framework structures with BHNH linker, after
losing hydrogen, they deviate a little bit from perfect hexagonal symmetry. The
angle between two horizontal lattice vectors is not kept at 2π/3, while varying
from 118.17◦ to 120.23◦ . The vertical lattice vector tilts slightly away from the
perpendicular postion (see Fig 3.6(b)). That is because the carbon chain struts
are not straight any more. The BHNH linker plane are not any more perpendicular
to the horizontal plane, moreover they are warped into a curved shape.
If all the hydrogen atoms are removed, the boron-nitrogen linkers simply relax
into a linear chain, and the structure still keeps a framework topology, but the
carbon tetrahedral vertex loses much of its sp3 character, becomes unstable and
can evolve in many ways, thus resulting in a final geometry strongly depending on
the length of the linker. Examples are frameworks with three- and four-pair of BN
bonds shown in Fig. 3.6(c), 3.6(d). One case has both the linker and the carbon
chain strut warped into a curving shape. In another case, upper and bottom vertex
get closer and form a stable structure. The second step is an endothermic process.
EB2 in the second step to release hydrogen becomes much larger, at 0.929eV,
0.884eV and 0.570eV, respectively. The last value is lower because the formation
of a particularly stable carbon vertex compensates partially for the high energy
cost of removing hydrogen atoms.
3.4.2
Zincblende with Pure BN Vertex
The vertices in this series are boron and nitrogen atoms, each of which is surrounded by four other type of atoms. Every boron vertex has four nitrogen vertices as its neighbors, and vice versa. The linker consists of one-, two- or three-pair
B-N bonds. The structure design starts from zincblende, ending up a triclinic unit
cell (see an example in Fig. 3.7(a)). For structures with PAB linkers, all the B/N
53
Table 3.2. The bond length alternation in wurtzite frameworks with BH2 NH2 type
linker. Every linker contains two-, three- or four-pair BN bonds.
Two
C-B 1.635
B-N 1.653
N-B 1.553
B-N 1.646
N-C 1.486
Three
C-B 1.639
B-N 1.634
N-B 1.575
B-N 1.636
N-B 1.573
B-N 1.625
N-C 1.490
Four
C-B 1.638
B-N 1.632
N-B 1.577
B-N 1.627
N-B 1.582
B-N 1.626
N-B 1.574
B-N 1.625
N-C 1.492
atoms on a linker carry two hydrogen atoms, so at a vertex, the space is really
crowded. Four atoms around a vertex have to adjust themselves to avoid colliding
each own’s hydrogen atoms, hence assuming new positions other than in special
points imposed by the space group #216. This adjustment causes all the linkers to
assume a non-planar form, and therefore the unit cells deviate from perfect cubic
symmetry. With the longer linker, this deviation is gradually diminished (see table 3.1 for lattice parameters). There, a,b,c are the lengths of three lattice vectors,
and α,β,γ are the angles among them.
For PIB structures, all of the vertices keep sp3 character after one half of the
hydrogen atoms are removed (see Fig. 3.7(b)). The PIB lattice system does not
change much with respect to that of the respective PAB structure, so the unit
cell volume does not change much when compared to the volume of PAB unit
cell. In this series, all the PAB linker are slightly distorted away from their planar
configuration, which makes a sharp contrast against the next series. In the next
series, the distortion of PAB linkers are significantly larger than those occured in
this series. For the binding energy, EB1 varies with respect to linker length, while
EB2 is too high.
After the three PAB structures lose all of their hydrogen atoms, all the linkers
relax into a linear chain geometry, resulting in larger unit cells when compared
to the volume of PAB unit cells. This results from the B-N-B-N zigzag in both
PAB and PIB linkers being straightened out, therefore the distance between two
vertices is increased. The average B-N bond length is 1.30Å, much shorter than
54
(a)
(b)
(c)
(d)
Figure 3.7. Relaxed zincblende structures with pure B/N vertices (pink for boron, blue
for nitrogen, grey for carbon and white for hydrogen): (a) BH2 NH2 linker with two pairs
of BN bonds between a boron vertex and a nitrogen vertex; (b) The structure with threepair long BHNH linker; (c) After all the hydrogens are released, the two-pair long BN
linker evolves into a BN atom chain. Notice all the vertices keep sp3 character; (d) After
all the H atoms are released, the three-pair long BN linker becomes a BN chain similar
to the case in (c), but nitrogen vertex is now a BN dimer vertex with sp3 character.
that in PAB linker, 1.60Å. These frameworks without hydrogen then are made of
more tightly bound pure B-N chain as linkers. The structures with one- or twopair of BN bonds keeps every vertex still in sp3 type in Fig. 3.7(c) , while in the
structure with three-pair BNs, the boron vertex does not lose its sp3 character,
but the nitrogen vertex in Fig. 3.7(d) evolves into an equivalent sp3 vertex with a
BN dimer at the center.
55
(a)
(b)
(c)
(d)
Figure 3.8. Relaxed zincblende structures with one-carbon vertex (pink for boron, blue
for nitrogen, grey for carbon and white for hydrogen): (a) The linker is BH2 NH2 type
with five atoms; (b) The connection relation with seven-atom long linker, all hydrogen
atoms are omitted for clarity; (c) The linker is BHNH type, and deforms much more
than does the previous series, see Fig. 3.9; (d) Removing all the H atoms leads to a CB
dimer vertex connecting four linkers.
3.4.3
Zincblende with One-Carbon Vertex
This series uses a centered carbon with two pairs of boron and nitrogen as its
neighbour. This type of vertex stands 4.5ps heating at 500 ◦C. That assures the
stability of the vertex. The linker contains an odd number of B/N atoms, five,
seven or nine atoms altogether. After volume relaxation, PAB, PIB and pure
BNC frameworks are all stable (Fig. 3.8). In the PAB frameworks, the adjustment
of hydrogen near a vertex is a big perturbation and causes all the BH2 NH2 linkers
56
to be non-planar, furthermore, results in much stronger deformation of the unit
cell shape than does the previous series. The basic parameters of the unit cell are
listed in the Table 3.1. They significantly deviate from cubic type a = b = c and
α = β = γ = 90◦ . Although PAB linkers are hardly deformed, the B-N bond
length does not show the alternation of bond lengths that appeared for the planar
linker in the wurtzite structure, and all the bonds are quite near their average
value of 1.60Å. This seems to imply that there is no obvious weak spot that could
lead to potential breakdown.
The direct consequence of non-planar PAB linkers is that after they transform
themselves into PIB linkers, PIB linkers are also non-planar. When they are incorporated into the frameworks, many geometries with roughly the same energy are
possible. If the orientation of PIB linker remains the same as that of the transforming PAB linker, therefore the positions of vertices do not change much, the volume
of PIB unit cells does not change much. This is the case for the structures with
five- or seven-atom long PIB linker. If the orientation of PIB linker changes much
as occured in the structure with nine-atom long PIB linker, the volume of PIB
unit cell changes much too. In this case, the B-N backbone of PIB linker becomes
less twisted and more extended, so the distance between two vertices connected
by a PIB linker is increased. The PIB unit cell volume increases from 1084.26Å3,
the respective PAB unit cell volume, to 3036.97Å3. Overall, in this series, the PIB
linker is distorted much more than in the previous series (see Fig. 3.9). The distortion includes the whole plane bent towards a cylindrical shape, the curving linker
axis accompanying B-N-B-N dehedral angle flip and simple out of plane rotation.
In the pure BNC structures, every vertex is made of a B-C dimer and behaves
like a sp3 vertex with four bonds projecting outward at roughly tetrahedral angles.
There are still four linkers connected to this dimer, two nitrogen atoms at the
boron end of the dimer, the third nitrogen and a boron atom at the carbon end
of the dimer. B/N atoms now are connected into linear chain with bent axis. The
interesting thing is, the B-N bond length exhibits alternation pattern (see table 3.3)
where we listed the frameworks with seven- and nine-atom long linkers. For the
former case, four linkers at a vertex all exhibit bond length alternation, but for
the latter case, only two linkers do. Once the alternation happens, the B-N chain
should be regarded as a chain of B-N monomer linkered together by dative N→B
57
(a)
(b)
(c)
Figure 3.9. BHNH linker is distorted in the framewoks with one carbon atom as vertex
(pink for boron, blue for nitrogen, grey for carbon and white for hydrogen). (a) the
plane of the linker is bent as a whole. (b) The dihedral angle B-N-B-N is flipped from π
to 0. (c) the linker loses planar character in out- of-plane distortion.
Table 3.3. The bond length (Å) change in the one-carbon vertex frameworks after all
the hydrogen atoms are removed. The linker is either 7- or 9-atom long.
C-N
N-B
B-N
N-B
B-N
N-B
B-N
1.298
1.334
1.302
1.339
1.297
1.344
1.289
7-Atom
C-B 1.532
B-N 1.280
N-B 1.343
B-N 1.297
N-B 1.350
B-N 1.297
N-B 1.416
B-C 1.612
Linker
B-N 1.383
N-B 1.288
B-N 1.342
N-B 1.297
B-N 1.336
N-B 1.301
B-N 1.334
B-N
N-B
B-N
N-B
B-N
N-B
1.420
1.297
1.348
1.294
1.343
1.278
9-Atom
B-N 1.369
N-B 1.290
B-N 1.341
N-B 1.295
B-N 1.340
N-B 1.299
B-N 1.338
N-B 1.286
Linker
C-B 1.526
B-N 1.283
N-B 1.339
B-N 1.299
N-B 1.342
B-N 1.293
N-B 1.341
B-N 1.287
N-B 1.369
B-C 1.613
bond rather than a pure covalent bond. The reference [84] calculated an infinite
B-N chain and found no alternating pattern at all. Here, the nine-atom long linkers
have exhibited some characteristics of an infinite chain, and the influence of carbon
at vertices gradually decreases.
3.4.4
Nine-carbon Vertex
This series is constructed with nine-carbon vertices and linkers containing an odd
number of atoms. The four triple bonds defined by the eight carbons that surround
the central sp3 carbon act like a buffer to increases the distances between hydrogen
atoms on the B or N atoms closest to each vertex, so they do not need too much
58
(a)
(b)
(c)
Figure 3.10. Relaxed zincblende structures with nine-carbon vertex (pink for boron,
blue for nitrogen, grey for carbon and white for hydrogen): (a) All the BH2 NH2 linkers in
this series are TT type; (b) BHNH linkers are less deformed than those in the previous
series; (c) After all the hydrogen are removed, 9-carbon vertices keep well their sp3
character.
adjustment to achieve equilibrium and thus give unit cells just slightly deviating
from the perfect cubic cell. MD simulation confirmed its stability during 2ps
heating at 500 ◦C.
The BH2 NH2 linkers are all of the TT type, and BHNH linkers are also planar.
BN chains are bent less than those in the previous series. Since the linker is similar
in all three frameworks, it is meaningful to compare the frameworks as a function
of linker length. From table 3.1, the tendency is clear: increasing the length of
the linker causes both EB1 and EB2 to decrease. For practical applications, EB1
at 0.3∼0.4eV is acceptable, but EB2 , greatly than 1.0eV, thermodynamically will
make it impossible to make use of the hydrogen in the second release step. The
three previous series share the similar conclusion.
Finally, we analyze how the locally projected electronic eigenstates onto hydrogen are distributed in these framework structures. In Fig. 3.11(a), all the local
electronic DOS on hydrogen atoms attached to boron or nitrogen are drawn against
energy. And the integrated local DOS up to a given energy is shown in Fig 3.11(b).
Both local integrated DOS are scaled by the total local states at the Fermi level.
Hydrogen atoms attached to B are located at higher levels than their counterparts
at N atoms and their local DOS distribution is narrower. This is consistent with
the knowledge that the bonding energy of B-H is weaker than that of N-H.
59
(a)
(b)
Figure 3.11. (a) Local DOS of the contribution from all the hydrogen atoms attached
to boron or to nitrogen at a given energy. The state distribution of B-attached H atoms
is much narrower than that of N-attached hydrogen. (b) The integated total local DOS
at a given energy is calculated by summing all the local states up to that energy. It
accounts for what percentage of local states has been occupied by that energy. Here
the reference is taken as the total local states at the Fermi level. H atoms bond more
strongly to N atoms.
3.5
Summary
Ammonia borane based materials are promising on hydrogen storage because of
its high hydrogen content. The hydrogen can be released through pyrolysis in
the range 340–500K, but after the hydrogen molecules are released, the remaining
B and N atoms tend to form a boron-nitride like framework. This framework is
notoriously stable and kills any possiblity in reversibility of hydrogen storage. To
prevent this intractable BN solid from forming, we take the idea of vertex-linker
assembly and design the framework to prevent the polymer segments from meeting
each other.
Our design is based on two crystal prototype: wurtzite and zincblende. The
four series of frameworks are (1) wurtzite with five-carbon vertex; (2) zincblende
with single boron/nitrogen vertex; (3) zincblende with single carbon vertex and
an odd number of atoms in the linker; (4) zincblende with nine-carbon vertex and
an odd number of atoms in the linker. The linker could be either PAB, PIB or
backbones without hydrogen. They vary in length. All the frameworks are stable
structures, though the relaxed zincblende structures lose symmetry and come to
be triclinic, but all of them meet the goal to prevent the formation of a pure, dense
BN solid.
60
The wurtzite structures deforms under with hydrogen release. The deformation
is small after the first half of the hydrogen is removed, while it can be drastic after
all the hydrogen is removed. The deformation includes two aspects: the vertex
loses its sp3 character and the linker axis is bent greatly. There is an adjustable
parameter to determine the distance between two layers, the length of the carbon
strut linker, to affect the structure geometry. It is possible that, after all the
hydrogen atoms are removed, the left framework will form very stable adamatanelike structure.
The zincblende structures fails if a carbon vertex is connected to four atoms
which are all uniformly either B or N. The series (3) and (4) use a new strategy:
alternative arrangement of B and N atoms around a vertex as well as a linker with
an odd number of atoms. This significantly increases the stability of the vertex,
hence helps stablize the zincblende structures. No matter how much hydrogen is
removed, half or all, the remaining frameworks are stable and does not fall apart.
For hydrogen release, EB1 is much smaller than EB2 in all cases. In terms of
weight percentage, the first step release accounts for 3.40∼4.57%, 5.71∼6.48%,
6.41∼6.64%, 3.97∼4.90% for the above series (1), (2), (3), (4), respectively. Obviously, there exist carbon atoms in the series 1 and 4 that do not carry hydrogen
atoms for release, which significantly decreases the weight percentage.
Chapter
4
Mechanical Tuning on
Thermodynamics and Kinetics of
Hydrogen Release
4.1
Release Thermodynamics
The boron-nitrogen polymer polyaminoborane (BH2 NH2 )n (PAB) and its dehydrogenated polymer polyiminoborane (BHNH)n (PIB) are isoelectronic to polyethlene
(PE) and polyacetylene (PA), respectively. Every pair of C-C bond in PA and PE
is replaced by a B-N bond. In the transformation of PAB into PIB, experiments
measure the decomposition enthalpy at the range 14.2∼26.3 kJ (0.147∼0.273 eV)
per mole PAB sample, and it is exothermic nature [66]. This is an overall consequence of several decomposition reactions proceeding simultaneously rather than
the PAB decomposition only. The higher value might be more close to the true
enthalpy since it was achieved at the lower heating rate and the creation of other
volatile products was controlled to be as small as possible. This is quite different
from physical adsorption of hydrogen and most chemisorption which are deemed to
energy intake. Our DFT calculations show that the groud state energy difference
between PAB polymer and PIB polymer with released hydrogen molecules is somewhat related to the conformation, but the difference is small. In Fig. 4.1(a), for
the transformation between planar polymers, the energy difference is 0.08eV per
62
H2 molecule in an exothermic release process; In Fig. 4.1(c), for the non-planar
case, the energy difference is 0.05eV per H2 from an endothermic release. The
latter value is actually similar to the value 1.50kcal/mol (0.065eV) from another
group [87], where solid vibartional correction (Zero point correction) is not included. If ZPE is included, the value of 1.50kcal/mol will be modified downward
to -9.57kcal/mol (0.415eV) [87]. For that enthalpy, the exothermic nature might
be overestimated by this calculation, but whatever the exact value is, the enthalpy
is small.
From the view of thermodynamics, the equilibrium of the reaction is determined by the free energy. Temperature influence on the equilibrium constant K
of this reaction is through ∆ ln K = −∆G0 ∆( T1 ). Since there is no significant
increase in hydrogen release by increasing the heating temperature, the free energy change must be very small. Among many possible reaction products, the
pathway toward hydrogen release is thermodynamically favored since the reaction
is exothermic and the product, hydrogen gas and PIB polymer, are more stable
than alternative reaction pathways such as PAB breakdown into mono BH2 NH2
units (In the previous chapter, the polymerization of PAB polymer is about 0.6
eV per monomer unit, hence monomer BH2 NH2 tends to form polymer). But the
reaction does not proceed spontaneously, instead, dynamically, it needs very long
time to reach equilibrium, in the scale of tens of hours. So this hydrogen release
is most likely controlled by kinetics. A high kinetic barrier prevents the reaction
from proceeding quickly and producing more stable products. The conclusion is
supported by the fact that in the experiment, other volative products except H2
and PIB increase with heating rate. Those products are supposed to have smaller
activation barrier.
Here we will focus on the simpler case, planar conformation, since the number
of reaction coordinates here is much less than that in the non-planar case. This
makes it very simple to analyze how the activation barrer is affected by the length of
the planar polymer. If the analysis starts otherwise with the non-planar polymer,
every reaction coordinate needs to be dealt with independently. Except the total
length, many bond angles and dihedral angles also enter into play which makes
the analysis too complicated.
The unit cell of BH2 NH2 or BHNH polymer contains four pairs of B-N dimers.
63
(a)
(b)
(c)
(d)
Figure 4.1. The energy difference was affected by the polymer conformation somewhat
(pink for boron and blue for nitrogen). In planar transformation, it is exothermically
0.08eV per hydrogen molecule release; in non-planar form, it is endothermically 0.05ev
per H2 release. All the hydrogen atoms are omitted.
The periodic structures are relaxed at different fixed length, then all the data of energy against the length of the unit cell are fitted into a harmonic relation. This gives
the unit cell equilibrium length at 10.62Å(Fig. 4.1(a)) and 10.15Å(Fig. 4.1(b)), respectively. The total energy as a function of unit cell length closely follows a
quadratic dependence, hence they can be considered as a spring with the respective force constants of 2.39 and 3.54 eV/Å2 . If their lengths can be changed by
an external force, then their optimal lengths will move accordingly away from
the natural equilibrium. This deformation may change both the binding energy
and activation barrier if the magnitude of the deformation is different for the initial, final and transition states. In the following sections, we will first discuss the
method used to determine the activation barrier, then made a rough analysis to
see if it is possible to tune the reaction barriers and reaction enthalpies through
this mechanism.
4.2
Nudged Elastic Band Method
For chemical reaction, a common problem is the time evolution of a system undergoing bond breaking and forming or atomic rearrangement. This problem generally
requires finding the lowest energy path from one stable configuration to another.
Such a path is often called the minimum energy path (MEP). It is convenient to
64
define a reaction coordinate to describe where the state is on the potential energy
surface and the degree to which the reaction has proceeded. This coordinate can
be a bond length, bond angle, dihedral angle or a combination of the above quantities. Essentially, the reaction coordinate is the distance along the reaction path.
The potential energy maximum along the MEP is the saddle point energy which
gives the activation energy barrier, a quantity of central importance in estimating
how fast a reaction proceeds.
The most common means of finding the saddle point is the chain-of-states
method [85]. A chain of images or replicas of the system is generated between
the end-point configurations and all the intermediate images are optimized simultaneously. The distribution of the images represents a discrete representation of
the reaction path. Each image corresponds to a specific geometry of the distinguishable atoms on their way from the initial to the final state, a snapshot along
the reaction path. Generally, the initial images are located along a line joining the
initial and final states. These images are connected together with springs that offer
the forces that prevent intermediate images from relaxing back to the end-point
configurations. In the context of finding the MEP, a object function defined in
Eq.( 4.1) will be minimized with respect to the positions of intermediate images,
R1 ,R2 ,· · · , RP−1 while keeping R0 , RP fixed. Once the enery of this chain of
images has been minimized, the true MEP is revealed. This method is called the
plain elestic band method (PEB).
S(R1 , R2 , · · · , RP−1 ) =
P
X
i=0
V (Ri ) +
P
X
kP
i=1
2
(Ri − Ri−1)2
(4.1)
The motion of each image is determined by a Newton-like equation with the force
Fi acting on the image i,
∂Ri
= Fi = −∇V (Ri ) + FSi
∂t
(4.2)
where the spring force FSi is
FSi = ki+1 (Ri+1 − Ri ) − ki (Ri − Ri−1 )
(4.3)
65
The deficiencies of PEB lie in two aspects: first, if the elastic band is too stiff,
the predicted path will cut the corner of the saddle point region and overestimate
the activation barrier; second, the images will tend to move towards the end regions
and avoid the barrier region, hence decreasing the resolution in the barrier region.
Overall, PEB is not a good method to find MEP and transition states.
An important algorithm to improve MEP finding is the Locally Updated Planes
method (LUP) [86]. LUP introduces a creative idea to overcome the above shortcomings rooted in PEB. It introduces a notion of projection onto the reaction path.
A local tangential unit vector to the path is estimated by the tangent to the line
segment connecting two consecutive images in the chain
qi =
Ri+1 − Ri
|Ri+1 − Ri |
(4.4)
the method then minimizes the energy of each image within the hyperplane with
normal qi . The system can only move in the direction perpendicular to qi . The
MEP estimation is improved by updating the tangential vectors after a given number of steps of relaxation. One remaining concern is that LUP does not guarantee
equal distances between images and can even give a discontinuous path.
Now we will see how the LUP’s projection scheme can help remove the prolems
inherent in the PEB method. The problem with corner cutting results from the
component of the spring force which is perpendicular to the reaction path and
tends to pull images off the MEP. The problem with the resolution decrease comes
from the component of the true force ∇V (Ri ) along the path. The true potential
force makes distance between images unequal so the net spring force along the
path can cancel the parallel component of the true potential force.
The Nudged Elastic Band method (NEB) incorporates the essence of both
PEB and LUP and gives a satisfactory solution for finding the MEP. In the NEB
method, a minimization of an elastic band is carried out where the perpendicular
component of the spring force and the parallel component of the true potential
force are discarded. The force Fi then takes the form (see the Fig 4.2),
FNEB
= −∇V (Ri )|⊥ + FSi · qi qi
i
(4.5)
66
where ∇V (Ri )|⊥ = ∇V (Ri )(1 − qi qi ). This projection of the perpendicular component of the true potential force and the parallel component of the spring force is
referred to as ‘nudging’. These projections decouple the dynamics of the path itself
from the specific distribution of images. The relaxed configuration of the images
satisfy ∇V (Ri )|⊥ = 0, so the images lie on the MEP. What is more important,
because the spring force only affects the distribution of images within the path,
the choice of the spring constant is quite arbitrary. Since it is always perferrable
to have more images in the barrier region, we want to somehow produce a high
density of images in the vicinity of the barrier. This can be achieved by adjusting
the spring stiffness as a function of position along the chain. In the barrier region,
harder springs should be used while near the end region, softer springs. Although
the implementation of NEB within VASP does not allow one to change the spring
constant, it is worth pointing out this freedom.
In a practical application where the energy changes quickly along the path but
the perpendicular restoring force is weak, a small amount of the spring force in the
perpendicular direction should be retained but not so much in order to straighten
out the ’kinky’ path.
4.3
Spring Model
For a real reaction, there are many paths to connect the intial and final states
on the potential energy surface. If the multi-dimensional potential energy surface
changes, the transitional state accordingly changes too. An external force can bring
about such a change. This is our starting point. Here we will investigate how the
activation barrier changes if an external force is applied to the polymer chain.
The external force may result from a framework structure to which the polymer
segment is attached. Here it is simply represented by a spring with constant K
and equilibrium length L0 . This simplication helps the investigation focus on the
polymer transformation itself, while the structural change on the source of the
external force is ignored. The external spring modifies the system by adjusting the
equilibrium length and total energy of the polymer.
The system is now the polymer coupled to an external spring. In our specific
case, it consists of four pairs of B-N bonds and an external spring with periodic
67
Figure 4.2. NEB method is often used to find a MEP starting from the initial state
and ending at the final state. In this method, a chain of images (small dark circles)
represents the MEP. This discrete NEB path gradually move downward on the potential
energy surface until all the images satisfy force balance. The inset shows the projection
scheme taken by NEB: only the perpendicular component of the true force F⊥
i and the
NEB
parallel component of the spring force are included in the NEB force Fi
.
boundary conditions. This roughly simulates a four-pair long linker in the framework. The initial and final states are PAB and PIB segments, respectively. The
unit cell made of four pairs of B-N bonds is in force balance with the external
spring for that cell. In other words, this polymer segment coupled with the spring
is an isolated system that satisfies the force equilibrium condition,
dE(x)
− K(x − L) = 0
dx
(4.6)
where x is the new equilibrium length of the segment. If the natural equilibrium
68
length of the polymer is x0 , then only two cases are possible: L0 < x < x0 or
x0 < x < L0 .
To begin, we make a rough estimate of the magnitude of the spring-tuning
effect, based on two approximations: first, the position of the transition state is
assumed to be midway between the initial and final states. Here each state is
considered to be represented by its equilibrium unit cell length, so the “midway”
is referred to the length of the unit cell of the transitional structure as the average
equilibrium length of the unit cell of PAB and PIB. This is a different concept
than the “distance” between images which is generally defined as the common
N-dimensional Euclid metric,
v
u N
uX
(R
d=t
im
m=1
− Rjm )2
(4.7)
where Ri and Rj are two N-dimensional vectors corresponding to the position of
image i,j. The change of an image position not only includes the change in the
length of the unit cell, but also the changes in the cross-section perpendicular to
the polymer axis. Second, the initial estimate assumes that the external spring
does not change much the internal reaction coordinates except for the total cell
length of polymer, thus the original dependence of activation barrier on those
internal coordinates remains the same as it was before the polymer states are
adjusted. Under these approximations, the energy of the polymer plus spring
system is simply a summation
1
EP = EP0 + K(x − L0 )2
2
(4.8)
where P stands for the initial, final and transition states, and P0 stands for the
states without mechanical spring coupling. Each of the three states is destabilized
by an amount 1/2K∆2 , where ∆ is the magnitude of the respective deformation.
Because the equilibrium lengths of the PAB and PIB segments are different from
each other, one obtains different destabilizations for all three states. Suppose the
new equlilibrium lengths are x1 and x2 for PAB and PIB segment, respectively,
69
the energy difference between new initial and final states is then
1
1
EB = EPIB + K(x2 − L0 )2 + 4EH2 − (EPAB + K(x1 − L0 )2 )
2
2
1
= EB0 + K (x2 − L0 )2 − (x1 − L0 )2
2
(4.9)
The change in thermodynamics EB is the difference between destablization amount
of the initial and final states. Similarly, the change in activation barrier takes the
form,
1 ∆Ek = ∆Ek0 + K (x3 − L0 )2 − (x1 − L0 )2
2
(4.10)
Since there is freedom to choose both K and L0 , conceptually, we can move both
binding energy and activation barrier toward our preference.
The Fig. 4.3 illustrates the above principle for mechanically coupled spring to
modify the binding energy and the activation barrier. Eini and Efinal are the one
dimensional curves of energy against unit cell length for both polymer segments.
After spring adjustment, the new equilibrium states are represented by solid blue
lines. In the middle is the transition state (dashed blue lines). The blue harmonic
curves are centered at the natural length of the external spring. Their action on
EB and ∆Ek is determined by how much the spring destablizes the three states.
Here ∆i , ∆ts and ∆f stand for the destabilization of the initial, transition and final
states, respectively. In the case of L0 < x < x0 , ∆i > ∆ts , the initial state is more
destabilized than is the transition state, so the activation barrier is decreased.
EK = ET S + ∆ts − (EIN I + ∆i )
EK = EK0 + ∆ts − ∆i < EK0
(4.11)
In the case of x0 < x < L0 , the situation is reversed. The transition state is
more destabilized by the external spring than is the initial state, accordingly, the
activation barrier is increased. The overall effects in change of EB and ∆EK are
shown in Fig. 4.4(a) and 4.4(b). A spring with larger force constant causes larger
changes in binding energy EB and smaller changes in activation energy ∆Ek .
The purpose of this simplified model is to explore means to modify the hydrogen
release kinetics, lowering the release barrier. Since both PAB and PIB polymers
can be thought of as springs, both of them can be coupled with an external struc-
70
E@eVD
1.5
Efinal
1
0.5
Eini
0
Df
9.5
E@eVD
1.5
1
Di
Dts
10
10.5
11
(a)
Eini
Df
0.5
0
L@ÅD
Di
Efinal
9.5
10
10.5
11
Dts
L@ÅD
(b)
Figure 4.3. An external spring can change both the thermodynamics and the kinetics
of dehydrogenation for PAB and PIB polymers. The original PAB and PIB polymers
have the relation of energy vs length shown by curves Eini and Efinal , respectively. Once
the polymers are coupled to the spring, the equilibrium positions move to where solid
blue lines are. The blue dashed line is the assumed position of the transition state.
The location where the blue spring energy curves, centered at the natural spring length,
intercept the state position lines gives the destablization amount for the initial, final
and transition states. (a) In the case L0 < x < x0 , ∆i > ∆ts , the activation barrier is
lowered; (b) In the case L0 > x > x0 , the activation barrier is raised.
ture acting like an outside spring. With this mechanical coupling, energy can be
transferred from and to this coupled structure. This redistribution of the total
energy causes the internal energy stored in the polymers to change; accordingly,
71
E@eVD
1
DEK @eVD
0
-1
9.5
10
10.5
11
(a)
K=15
K=0
0
K=2.5
K=0
-0.5
K=2.5
0.1
K=15
0.5
-0.1
L0 @ÅD
9.5
10
10.5
11
L0 @ÅD
(b)
Figure 4.4. The adjustment of the binding energy and activation barrier predicted by
the simple model of a polymer coupled with a spring. (a) Given a spring with force
contant K (in unit eV/Å2 ) and natural length L0 , the change in binding energy EB with
respect to L0 ; (b) the change in activation energy ∆Ek with respect to L0 .
the activation barrier changes too. In the next section, we will show the results
from more accurate NEB calculations. Qaulitatively, the kinetic barrier change
predicted by this simplified model is not correct, since the energy is assumed to
be dependent only on the length of the polymer. For large deformation, obviously
it is not true, but this model does shed some light on how to tune the activation
barrier.
4.4
NEB Results
In our calculations, a quasi-Newton algorithm is used to relax the ions into their
ground states in the hyper-plane perpendicular to the normal of every image.
The forces and the stress tensor are calculated and used to determine the search
directions for finding the instantaneous equilibrium positions. This algorithm is
an improvement of the original Newton algorithm. The Hessian matrix, made
of second order derivatives, does not need to be found explicitly any more. The
Hessian or directly its inverse matrix is updated instead by analyzing successive
gradient vectors in the previous steps, and then an iterative scheme like Rk+1 =
Rk − Hk−1 ∇f (Rk ) is performed, where Rk , Rk+1 are successive sets of variables
(simply atom coordinates or their linear combinations), Hk is the Hessian matrix at
the kth step and f (Rk ) is the object function to be optimized. How many previous
iterations should be included? First of all, the information in these included steps
72
should be linearly independent or otherwise the inverse matrix is ill-defined. The
number of vectors kept in the iteration history (which corresponds to the rank of
the Hessian matrix) must not exceed the number of degrees of freedom. Naively
the number of degrees of freedom is 3*( NIONS-1), NIONS is the total number of
the ions.
Another algorithm used in finding transition states is damped molecule dynamics. The ion position is updated by a revised equation of motion where an artificial
damping force is added into the force on each ion,
R̈ = −µṘ + αF
(4.12)
µ determines the strength of damping. When µ = 0, there is no damping; when
µ = 2, the algorithm is equivalent to a simple steepest descent algorithm and
corresponds to maximum damping. VASP also provides a simple velocity quench
option: when the force and velocity are in opsite directions, the velocity is set
to zero; when they are along the same direction, the magnitude of velocity is
proportional to the force.
A NEB calculation starts with equidistance images. This equidistance tends to
be violated significantly at the beginning of the calculation and the geometry can
change dramatically. In order to avoid instability, the optimization will be done
only for a small subset of variables at each step. Here we set a very low dimensionality NFREE=2, that is, only two steps of results are kept in the relaxation
iteration history. The rank of the inverse Hessian matrix is two, so only two variables or two freedoms of atom coordinate combination are optimized at this step.
Once the search approximates the minima for Eq.( 4.1), this low dimensionality
constraint will be lifted. This strategy actually addresses issues of stability at the
cost of convergence speed.
All of the NEB calculation here use seven images. The atomic coordinates
in every image are initiatized from a linear coordinate interpolation between the
initial position and final position of the respective atoms. Sometimes, this interpolation can cause a few atoms to approach each other too closely in a certain
image. For the planar transformation, because the hydrogen release is arranged
to occur on the side of a planar PAB polymer, there is no such a problem, but
73
for the nonplanar transformation, the atom positions change greatly, and atom
approaching too closely can happen often. At that time, the coordinates need to
be adjusted manually to make the positions reasonable.
For the PAB polymer system, since there are many conformers, the local minima can be many. If the initial and final states are not next to each other along the
reaction path, VASP most possibly finds another local minimum rather than the
transition state of interest. This matches our actual calculations when we try to
locate the reaction path between the PAB and PIB polymers. VASP simply gives
us a nonplanar polymer backbone with one hydrogen molecule released, which is
not a transition state. So here the final state is set as a PAB segment losing the
first hydrogen molecule, PAB/H2 , rather than a simple PIB polymer. After losing
the hydrogen molecule, the backbone of the polymer remains planar. This stable
state PAB/H2 is expected to be next to the PAB polymer state along the reaction
path. The change on the final state to PAB/H2 does not mean that the idea in the
simple model does not work, but is just motivated by a need to simplify the NEB
calculations while at cost of the tuning on a kinetic barrier predicted by the simple
model. Later we will work on the remedy. The new PAB/H2 polymer has a very
similar harmonic behaviour to that of the PAB polymer since its force constant,
2.25eV/Å2 , is almost the same as PAB’s 2.39eV/Å2 . The equilibrium lengths of
unit cells of PAB and PAB/H2 are also very close, 10.62Å and 10.52Å, respectively. By the similar analysis in the last section, the destabilization amounts for
the initial and TS states are quite the same, so the change of the kinetic barrier
cannot be large.
From the view of NEB calculations, even this retraction from the above simple
model cannot guarantee to find the right transition state. In many runs, we simply
have a distorted PAB polymer or a PAB polymer losing one pair of hydrogen atoms.
This pair of hydrogen atoms is easily pushed away from the polymer backbone.
In such a case, the normal of hyperplanes in a NEB calculation is mainly decided
by this H2 , since its coordinate changes most. Once this happens, this H2 is not
located in the hyperplane and all the rest of atoms are located in the hyperplane.
Remeber in a NEB calculation, only atoms in the hyperplane are allowed to relax.
Then these atoms are relaxed to a local minima without the released H2 , which
continues to be pushed further away.
74
2.5
Energy(eV)
2.0
1.5
1.0
0.5
∆Ek = 2.1677eV
0.0
−0.5
0
1
2
3
4
5
6
7
Distance(Å)
Figure 4.5. The activation barrier is 2.17eV if a PAB segment releases one hydrogen
molecule without mechanical spring coupling. All the data in table 4.1 and 4.2 share
this overall appearance. The seven images are spline fitted to predict the barrier height.
For the highest image, the true potential gradient approaches zero, so it is almost at the
saddle point.
The other important point is, due to the lengths of initial and final states are not
equal to each other, we assume the unit cell length of every image gradually changes
from the length of the initial structure to the length of the final structure. This is
not a serious problem for planar transformation because the equilibrium lengths
of unit cells of PAB and PAB/H2 are almost equal. For nonplanar transformation
case, The difference of the lengths of the unit cells can be large. Our NEB test
calculations show the equilibrium length of the TS unit cell is roughly at 40% away
from the length of the inital PAB cell if the calculation converges.
When the spring associated with the putative external framework is switched
on, the spring constant of the framework is chosen as K=2.5 or 15.0 eV/Å2
with several different spring equilibrium lengths for each case. As a reminder,
a K=2.5eV/Å2 spring is roughly as stiff as the initial PAB polymer, while an
external structure with K=15.0eV/Å2 is much stiffer than a PAB polymer.
For the fictitious NEB spring connecting the seven images together, its spring
constant varies within 3∼8eV/Å2 . The data of energies against positions of relaxed
images are spline fitted to predict the activation barrier. The result is included in
75
(a) Initial
(b) TS
(c) Final
Figure 4.6. The predicted transition state from the NEB calculation for a planar
transformation of PAB into PAB/H2 without mechanical coupling (pink for boron, blue
for nitrogen and white for hydrogen). After the H2 is released, the rest of the backbone
remains planar. The released hydrogen molecule is pushed away from the polymer. The
transition structure shows the hydrogen atom has to break the bonding to a N atom
then form a H2 near a B atom. The distance between two H atoms supposed to come
off is 0.99Å.
the fitting only if it satisfies that at the highest image, the spring force and true
force both approach zero. With this criteia, the highest image is very near the
saddle point. The popular climbing NEB is supposed to get the same behaviour.
All the data are collected in tables 4.1 and 4.2. Every energy quoted includes a
spring energy following Eq. (4.8), i.e. they are the sum of the polymer energy and
the external spring energy.
Fig. 4.5 shows the fitting of the activation barrier calculation for the planar PAB
transformation. In this case, there is no external framework spring. The NEB
spring constant is set at 8eV/Å2 . Obviously, the images are not equal-distance
distributed (here the distance is defined as in the Eq. 4.7). We also calculate the
activation barrier with a softer NEB spring chain K=3.0eV/Å2 . The new result,
2.1423eV, is almost the same as the data in the Fig. 4.5, 2.1677eV, which supports
the reliability of our data.
Fig. 4.6 shows the structure of the transition state for the above planar polymer
transformation. The polymer backbone does not change too much. One hydrogen
atom leaves its original host, a nitrogen atom, moves toward a boron and increases
the N-H distance to 1.44Å from 1.02Å (the normal N-H bonding length). At the
boron place, there accumulate two hydrogen atoms, 1.37Å and 1.38Å away from
the boron. The distance between these H atoms is 0.99Å. When compared to the
normal B-H bonding length 1.21Å, the B-H bond length changes much less than
that in the N-H bond length. Here the trend is very clear: the nitrogen has to
lose one hydrogen atom first and this hydrogen atom will form a molecule near the
76
Table 4.1. The modification of thermodynamics and kinetics when the four-pair long
BN polymer is coupled with a spring K=2.5eV/Å2 (The polymer is in force balance with
the spring.)
Spring Equilibrium
Length L0 (Å)
9.70
10.20
10.70
11.10
H2 Binding
Energy (eV)
0.6026
0.6789
0.7482
0.7582
Kinetic Barrier
Against H2 Release (eV)
2.0603
2.1244
2.0855
2.1995
Table 4.2. The modification of thermodynamics and kinetics when the four-pair long
BN polymer is coupled with a spring K=15eV/Å2 (The polymer is in force balance with
the spring.)
Spring Equilibrium
Length L0 (Å)
9.70
10.20
10.70
11.10
H2 Binding
Energy (eV)
0.5184
0.6242
0.7689
0.8001
Kinetic Barrier
Against H2 Release (eV)
1.9571
2.0771
2.1300
2.2255
neighbor boron atom. Since the N-H bond is stronger than the B-H bond (see the
analysis in the last chapter), it explains why hydrogen release has a large kinetic
barrier.
In the previous section, a simplified model on the transformation from PAB to
PIB has shown the possibility to change both the binding energy and the kinetic
barrier by coupling with an external spring. When L0 < x0 , the PAB polymer is
compressed, the kinetic barrier is lowered and the reaction becomes more exothermic; when x0 < L0 , the PAB polymer is stretched, and the kinetic barrier is raised
and the reaction becomes endothermic. The more deformed, either compressed or
stretched, the larger the modification of both the binding energy and the activation barrier. This model is based on the fact that the equilibrium length of PAB
and PIB are quite different from one another, and that their harmonic behaviours
are also very different each other. Hence, the destabilization amount for the intial,
TS, and final states can be quite different.
But for the real process of hydrogen release, this cannot be true. The hydrogen
molecules always come off one by one, so there is no a transition state that directly
77
connects PAB and PIB. Our NEB calculations try to find the TS state between
a planar PAB polymer and a planar PAB losing one H2 molecule, PAB/H2 . As
mentioned earlier, because of their similar relations of energy vs. polymer length
and almost the same equilibrium lengths, hence their similar deviations from the
natural equilibrium lengths, then the introduced destablization difference cannot
be large by the simple model. This conclusion matches the results in tables 4.1
and 4.2. The largest destabilization value in both tables is just 0.03eV.
Accordingly, if the simple model stands right, all the data in table 4.1 and 4.2
should be roughly the same, but they are not. Even without large destabilization,
the accurate NEB calculations replicate well the qualitative trend from the above
simplified spring model. When the polymer is compressed, the binding energy
decreases and the activation barrier is lowered; when the polymer is stretched,
the binding energy increases and the barrier is raised. Furthermore, a stiffer external spring gives a broader adjustment range for both EB and ∆Ek , although
the VASP barrier results do not monotonically increase as predicted by the simple
spring model, especially when the polymers are stretched. The increase in the
activation barrier under tension is not as rapid as the decrease in the barrier under
compression. These changes in the kinectic barrier is mainly not caused by different destablization on the inital, transiton and final states, but by the change of
internal coordinates. The simple assumption that only the length of the polymer
determines the height of the barrier is not a good approximation, but it provided
the impetus to perform a more detailed calculation that provides a similar ultimate
result, albeit with a modified mechanism.
Now we will turn to a planar PAB-nonplanar PAB/H2 transformation. This is a
preliminary step to realize the principles behind the simple spring model. When the
compression of PAB/H2 continues to decrease the unit cell length of this polymer,
it will buckle and assume a nonplanar conformation. The B-N bond rotates by
π/2 to a horizontal position (see Fig. 4.7(d)). The nonplanar polymer PAB/H2 is
very soft and experiences much larger deformation compared to planar PAB (see
Fig. 4.7(a)), and moreover, the equilibrium length of the nonplanar PAB/H2 is
quite different from that of PAB, 9.94Åand 10.62Å, respectively. So it is possible
to create different amounts of destabilization for the inital and transition states.
When this PAB polymer is coupled with an external spring of the force contant
78
−120.8
Energy(eV)
−121.0
−121.2
−121.4
−121.6
PAB/H2
−121.8
PAB
−122.0
9.5
10.0
10.5
11.0
11.5
Cell Length(Å)
(a)
(b)
(c)
(d)
Figure 4.7. The NEB computes a planar PAB-nonplanar PAB/H2 transformation (pink
for boron, blue for nitrogen and white for hydrogen). (a) The relation of energy against
the unit cell length of polymers. It is clear that they have different equilibrium lengths.
(b) The initial structure is simplay a planar PAB polymer. (c) The transition structure
predicted again shows that the hydrogen release must start from a broken N-H bond.
The H2 molecule forms near the boron atom and escapes from between two pairs of
H-B-H triples, rather than from the side as in the planar-planar transformation. (d)
The final structure is a nonplanar PAB/H2 polymer. The B-N bond losing two H atoms
takes the horizontal position in the final polymer segment.
K=2.5eV/Å2 and equilibrium length L0 =9.8Å, the initial PAB polymer is selfbalanced at a unit cell length of 10.20Å, the final PAB/H2 polymer has the unit
cell length of 9.83Å. For the binding energy, the final state is 0.1852eV higher than
the initial state. The destablization amounts for the initial and transtion states
are 0.1984eV and 0.0917eV. This is an example of the transformation in which the
polymer length changes greatly. The NEB calculation starts with an one-image
run. The designed reaction path is as follows: the B-N bond gradually rotates
by π/2 about the polymer axis. The two H atoms supposed to come off increase
their bond lengths while rotating. The dihedral angle H-B-N-H quickly decreases
toward 0. Finally, they form a molecule and escape from the open space between
two pairs of H-B-H triples, facing the H-B-N-H dihedral angle. Once the one-image
79
run converges, more images can be included.
The barrier calculated by NEB is 1.5690eV, so the adjusted barrier height is
1.5690 − 0.1984 + 0.0917 = 1.4623eV. This value is much lower than the lowest one
from planar-planar transformation. The result gives us a strong implication that
nonplanar transformation is very promising in reduction of the barrier. Since nonplanar structures should be represented by more than one geometrical parameters
besides the polymer length, we will not make an assertion based on only one data.
In this project, we set up a very general model of a polymer mechanically
coupled with an external spring, and its ends are attached onto this external spring.
Based on this simple model, we demonstrate that the binding energy of H2 to the
polymer can be tuned by controlling the effective stiffness and preferred equilibrium
length of this external spring. At the beginning of this chapter, we have mentioned
that the reaction of PAB into PIB after all the hydrogen are released is almost
heat neutral, but for the first H2 release, the reaction needs large energy input,
so with more hydrogen molecules removed from PAB, the reaction becomes more
exothermic to make the whole series heat neutral. As for the kinetic barrier,
although the planar-planar transformation with an external coupling shows some
ability in tuning, 1.96∼2.23eV with a stiffer external spring, this range is still far
beyond the scope of practical application. But a transformation of planar PAB to
nonplanar PAB/H2 shows a very promising reduction in the kinetic barrier. In the
future, we will continue working on the kinetic barrier improvement and design
real framework to realize the external spring.
Bibliography
[1] P. Hohenberg and W. Kohn, Phys. Rev. 136 B864 (1964)
[2] M. Levy, Proc. Nat. Acad. Sci. USA 76 6062 (1979); M. Levy, Phys. Rev. A
26 1200 (1982); M. Levy and J.P. Perdew, in Density Functional Methods in
Physics, edited by R.M. Dreizler and J. da Providencia, Plenum, New York,
1985, p.11; E. Lieb, in Physics as Natural Philosophy, edited by A. Shimony
and H. Feshbach, MIT Press, Cambridge, 1982, p.111; E. Lieb, Int. J. Quant.
Chem. 24 243 (1983); E. Lieb, in Density Functional Methods in Physics,
edited by R.M. Dreizler and J. da Providencia, Plenum, New York, 1985,
p.31;
[3] N.D. Mermin, Phys. Rev. 137, A1441 (1965)
[4] W. Kohn and L. J. Sham, Phys. Rev. 140 A1133 (1965)
[5] R.M. Martin in Electronic Structure Basic Theory and Practical Methods,
p.65, Cambridge, 2004
[6] R.P. Feynman, Phys. Rev. 56, 340 (1939); D.J. Griffiths in Introduction to
Quantum Mechanics, Prentice Hall, NJ, 1995, Prob. 6.27
[7] K. Burke, http://dft.uci.edu/book/gamma/g1.pdf, in The ABC of DFT, 2007,
Chap.13
[8] P.Gori-Giorgi, F. Sacchetti and G.B. Bachelet, Phys. Rev. B 61, 7353 (2000)
[9] G.F. Giuliani and G. Vignale, in Quantum Theory of the Electron Liquid,
Cambridge University Press, UK, 2005, p.13
[10] M. Gellmann and K.A. Brueckner, Phys. Rev. 106, 364 (1957); W.J. Carr
and A.A. Maradudin, Phys. Rev. 133, 371 (1964)
[11] W.J. carr, Phys. Rev. 122, 1437 (1961); D. Pines and P. Nozières, in The
Theory of Quantum Liquids, Vol.1, Addison-Wesley, 1989
81
[12] J.P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981)
[13] D.M. Ceperley and B.J. Alder, Phys. Rev. Lett. 45, 566 (1980)
[14] S. Vosko, L. Wilk and M. Nusair, Can. J. Phys. 58, 1200 (1983)
[15] P.S. Svendsen and U. von Barth, Phys. Rev. B 54, 17402 (1996)
[16] J.P. Perdew and K. Burke, Int. J. Quant. Chem. 57, 309 (1996); J. P. Perdew
and Y. Wang, Phys. Rev. B 33, 8800 (1986); J. P. Perdew, Phys. Rev. B 33,
8822 (1986)
[17] A.D. Becke, Phys. Rev. A 38, 3098 (1988)
[18] J.P. Perdew and Y. Wang, Phys. Rev. B 45,13244 (1992)
[19] J.P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996)
[20] A.D. Becke, J. Chem. Phys. 98, 1372 (1993); K. Kim and K.D. Jordan, J.
Phys. Chem. 98, 10089 (1994); P.J. Stephens, F.J. Devlin, C.F. Chabalowski
and M.J. Frisch, J. Phys. Chem. 98, 11623 (1994)
[21] C. Lee, W.T. Yang and R.G. Parr, Phys. Rev. B 37, 785 (1988)
[22] D.J. Singh and L. Nordstrom in Planewaves, Pseudopotentials, and the LAPW
Method, Springer, 2nd Ed. 2006, Chap. 3
[23] W.C. Topp and J.J. Hopfield, Phys. Rev. B 7, 1295 (1973), Appendix part
[24] D.R. Hamann, M. Schlüter, and C. Chiang, Phys. Rev. Lett. 43, 1494 (1979);
G.B. Bachelet, D.R. Hamann and M. Schlüter, Phys. Rev. B 26, 4199 (1982)
[25] D. Vanderbilt, Phys. Rev. B 41, 7892 (1990); K. Laasonen, R. Car, C. Lee and
D. Vanderbilt, Phys. Rev. B 43, 6796 (1991); K. Laasonen, A. Pasquarello,
R. Car, C. Lee and D. Vanderbilt, Phys. Rev. B 47, 10142 (1993)
[26] P.E. Blöchl, Phys. Rev. B, 50, 17953 (1994)
[27] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999)
[28] N.W. Ashcroft and N.D. Mermin in Solid State Physics, Saunders College,
1976, Chap. 8 Electron Levels in a Periodic Potential: General Properties
[29] H.J. Monkhorst and J.D. Pack, Phys. Rev. B 13, 5188 (1976)
[30] R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985)
[31] G. Kresse and J. Hafner, Phys. Rev. B 49,14251 (1994)
82
[32] S. Nosé, J. Chem. Phys. 81, 511 (1984); S. Nosé, Mol. Phys. 52, 256 (1984);
P. E. Blöchl and M. Parrinello, Phys. Rev. B 45, 9413 (1992); D.M. Bylander
and L. Kleinman, Phys. Rev. B 46, 13756 (1992)
[33] A. Ansón, M.Benham, J. Jagiello, M.A. Callejas, A.M. Benito, W.K. Master,
A. Züttel, P. Sudan and M.T. Martı́nez, Nanotechnology 15 1503 (2004)
[34] M. Shiraishi, T. Takenobu, M. Ata, Chem. Phys. Lett. 367 633 (2003)
[35] H. Kajiura, S. Tsutsui, K. Kadono, M. Kakuta, M. Ata and Y. Murakami,
Appl. Phys. Lett. 82 1105 (2003)
[36] C.M. Brown, T. Yildirim, D.A. Newmann, M.J. Heben, T. Gennett, A.C.
Dillon, J.L. Alleman, J.E. Fischer, Quantum rotation of hydrogen in singlewall carbon nanotubes, Chem. Phys. Lett. 329 311 (2000)
[37] B.K. Pradhan, G.U. Sumanasekera, K.W. Adu, H.E. Romero, K.A. Williams,
P.C. Eklund, Physica B 323 115 (2002)
[38] D.G. Narehood, J.V. Pearce, P.C. Eklund, P.E. Sokol, R.E. Lechner, J. Pieper,
J.R.D. Copley, J.C. Cook, Phys. Rev. B 67 205409 (2003)
[39] J.S. Arellano, L.M. Molina, A. Rubio, J.A. Alonso, J. Chem. Phys. 112 8114
(2000)
[40] Y. Okamoto, Y. Miyamoto, J. Phys. Chem. B 105 3470 (2001)
[41] N. Jacobson, B. Tegner, E. Schroöder, P. Hyldgaard, B.I. Lundqvist, Comp.
Mater. Sci. 24 273 (2002)
[42] J.Zhao, A. Buldum, J. Han, J.P. Lu, Nanotechnology 13 195 (2002)
[43] J. Li, T. Furuta, H. Goto, T. Ohashi, Y. Fujiwara, S. Yip, J. Chem. Phys.
119 2376 (2003)
[44] I. Suarez-martinez, A. Felten, J.J. Pireaux, C. Bittencourt, C.P. Ewels, J.
Nanosci. Nanotech. 9, 6171; H. Johll and H.Ch. Kang, Phys. Rev. B 79,
245416; G.Kim, S.H. Jhi, S. Lim and N.Park, Appl. Phys. Lett. 94, 173102
[45] Kubas, G.J. J. Organomet. Chem. 635, 635,37 (2001)
[46] T. Yildirim, J. Iniguez and S. Ciraci Phys. Rev. B 72, 153403 (2005).
[47] Yufeng Zhao, Yong-Hyun Kim, A. C. Dillon, M. J. Heben, S.B. Zhang, Phys.
Rev. Lett. 94, 155504 (2005)
[48] http://www1.eere.energy.gov/hydrogenandfuelcells/storage/pdfs/targets
onboard hydro storage.pdf
83
[49] Q. Sun, Q. Wang, P. Jena and Y. Kawazoe, J. Am. Chem. Soc. 127 14582
(2005)
[50] K. Rytkönen, J. Akola and M. Manninen, Phys. Rev. B 75 075401 (2007)
[51] B. Wang, Y.F. Ma, Y.P. Wu, N. Li, Y. Huang, Y.Sh. Chen, Carbon. 47 2112
(2009); S.M. Vieira, O. Stéphan, D.L. Carroll, J. Mater. Res. 21,3058 (2006)
[52] J.L. Blackburn, Y. Yan, C. Engtrakul, P.A. Parilla, K. Jones, T. Gennett,
A.C. Dillon and M.J. Heben, Chem. Mater. 18, 2558 (2006); K. McGuire,
N. Gothard, P.L. Gai, M.S. Dresselhaus, G. Sumanasekera and A.M. Rao,
Carbon 43, 219 (2005); P.L. Gai, O. Stephan, K. McGuire, A.M. Rao, M.S.
Dresselhaus, G. Dresselhaus and C. Colliex, J. Mater.Chem. 14 669 (2004)
[53] J. Kouvetakis, R. B. Kaner, M. L. Sattler, and N. Bartlett, J. Chem. Soc.,
Chem. Commun. 1758 (1986).
[54] T.C. Mike Chung, Y. Jeong, Q. Chen, A Kleinhammes and Y. Wu, J. Am.
Chem. Soc. 130, 6668 (2008)
[55] P. E. Blöchl, Phys. Rev. B 50, 17953 (1994); G. Kresse, and D. Joubert, Phys.
Rev. B, 59, 1758 (1999)
[56] G. Kresse, and J. Hafner, Phys. Rev. B 47, 558 (1993); G. Kresse, and J.
Hafner, Phys. Rev. B 49, 14251 (1994); G. Kresse, and J. Furthmüller, Comput. Mat. Sci. 6, 15 (1996); G. Kresse, and J. Furthmüller, Phys. Rev. B 54,
11169 (1996)
[57] M. Methfessel, and A.T. Paxton, Phys. Rev. B 40, 3616 (1989)
[58] This interaction between metal atoms could be direct, via orbital overlap or
Coulombic forces, or indirect, via relaxations in the intervening boro-carbon
sheet.
[59] R.F.W. Bader, Atoms in Molecules–A quantum Theory, Oxford University
Press, New York, 1990
[60] M.J. Lee, Phys. Rev. E 78, 031131 (2008); M.E.J. Newman and R.M. Ziff,
Phys. Rev. Lett. 85, 4104 (2000)
[61] http://www1.eere.energy.gov/hydrogenandfuelcells/storage/chem storage.
html
[62] J.S. Zhang, T.S. Fisher, J.P. Gore, D. Hazra, P.V. Ramachandran, Inter. J.
Hydrogen Engy. 31, 2292 (2006)
84
[63] D. Sebasitián, E.G. Bordejé, L. Calvillo, M.J. Lázaro and R. Moliner, Inter.
J. Hydrogen Energy 33, 1329 (2008)
[64] G. Wolf, J. Baumann, F. Baitalow, F.P. Hoffmann, Thermochim. Acta 343,
19 (2000).
[65] M.G. Hu, R.A. Geanangel, W.W.Wendlandt, Thermochim. Acta 23, 249
(1978)
[66] J. Baumann, F. Baitalow, G. Wolf, Thermochim. Acta 430,9 (2005).
[67] F. Baitalow, J.Baumann, G. Wolf, K. Jaenicke-Rößler and G. leitner, Thermochim. Acta 391, 159 (2002)
[68] A.W. laubengayer, P.C. Moews Jr. and R.F. Porter, J.Am. Chem. Soc. 83,
1337 (1960)
[69] C.R. Kinney and E. DelBel, Ind. Eng. Chem. 46, 548 (1954)
[70] A.P. Côté, A.I. Benin, N.W. Ockwig, M. O’Keeffe, A.J. Matzger, O.M. Yaghi,
Science 310, 1166(2005)
[71] H. Furukawa, J. Kim, N.W. Ockwig, M. O’Keeffe and O.M. Yaghi, J. Am.
Chem. Soc. 130, 11650(2008)
[72] O.M. Yaghi, H. Li and T.L. Groy, Inorg.Chem. 36 4292 (1997)
[73] M. Eddaoudi, J. Kim, M. O’Keeffe and O.M. Yaghi, J.Am.Chem.Soc. 124,
376 (2002)
[74] Zh. Ni, A. Yassar, T. Antoun and O.M. Yaghi, J.Am.Chem.Soc. 127,
12752(2005); M. Eddaoudi, J. Kim, J.B. Wachter, H.K. Chae, M. O’Keeffe
and O.M. Yaghi, J.Am.Chem.Soc. 123, 4368(2001); B. Moulton, J.J. Lu, A.
Mondal and M.J. Zaworotko, Chem.Commun. 863(2001); O.M. Yaghi, H.L.
Li, C. Davis, D. Richardson and T.L. Groy, Acc.Chem.Res. 31, 474(1998)
[75] H.M. El-Kaderi, J.R. Hunt, J.L. Mendoza-Cortés, A.P. Côté, R.E. Taylor, M.
O’Keeffe, O.M. Yaghi, Science 316,268(2007)
[76] R. Banerjee, A. Phan, B. Wang, C. Knobler, H. Furukawa, M. O’Keeffe and
O.M. Yaghi, Science 319, 939(2008)
[77] M. Eddaoudi, D.B. Moler, H.L. Li, B.L. Chen, T.M. Reineke, M. O’Keeffe
and O.M. Yaghi, Acc.Chem.Res. 34, 319(2001); H.L. Li, M. Eddaoudi, M.
O’Keeffe and O.M. Yaghi, Nature 402, 276(1999)
[78] O.M.Yaghi, M.O’Keeffe, N.W.Ockwig, H.K.Chae, M.Eddaoudi and J.Kim,
Nature, 423, 705 (2003)
85
[79] M.D. Hollingsworth, Science 295, 2410 (2002); M. Eddaoudi, J. Kim, N. Rosi,
D. Vodak, J. Wachter, M. O’Keeffe and O.M. Yaghi, Science 295, 469(2002)
[80] International Tables for Crystallography, Vol.A, Space Group Symmetry, 5th
Ed. Reprinted with corrections, Springer (2005), p584, p658
[81] E.H. Kisi and M.M. Elcombe, Acta Cryst. C45, 1867 (1989)
[82] D. Jacquemin, E.A. Perpète, V. Wathelet and Jean-Marie André, J. Phys.
Chem. A 108, 9616 (2004)
[83] A. Abdurahman, M. Albrecht, A. Shukla and M. Dolg, J. Chem. Phys. 110,
8819 (1999)
[84] A. Abdurahman, A. Shukla and M. Dolg, Phys. Rev. B 65, 115106 (2002)
[85] H. Jonsson, G. Mills and K. W. Jacobsen, in Nudged Elastic Band Method
for Finding Minimum Energy Paths of Transitions, in Classical and Quantum
Dynamics in Condensed Phase Simulations, Chap.16, ed. B. J. Berne, G. Ciccotti and D. F. Coker, World Scientific, 1998; G. Henkelman, B.P. Uberuaga,
and H. Jonsson, J. Chem. Phys. 113, 9901 (2000); G. Henkelman, and H.
Jonsson, J. Chem. Phys. 113, 9978 (2000)
[86] C. Choi and R. Elber, J. Chem. Phys. 94 751 (1991)
[87] C.R. Miranda and G. Ceder, J. Chem. Phys. 126, 184703 (2007)
Vita
ZhaoHui Huang
Education
2010 Ph.D., Condensed Matter Physics, Pennsylvania State University
1999 M.S., Astrophysics, Peking University
Awards
2009 Duncan Graduate Fellowship, PSU
2007 Duncan Graduate Fellowship, PSU
2006 Duncan Graduate Fellowship, PSU
2005 Duncan Graduate Fellowship, PSU
2002-2003 Braddock Graduate Fellowship, PSU