21648
J. Phys. Chem. C 2009, 113, 21648–21656
Surfaces and Clusters of Mg(NH2)2 Studied by Density Functional Theory Calculations
Torleif A. T. Seip,† Roar A. Olsen,‡,§ and Ole Martin Løvvik*,†,|
Department of Physics, UniVersity of Oslo, Gaustadalleen 21, N-0318 Oslo, Norway, Leiden Institute of
Chemistry, Leiden UniVersity, P.O. Box 9502, 2300 RA Leiden, The Netherlands, Akershus UniVersity College,
P.O. Box 423, N-2001 Lillestrøm, Norway, and SINTEF Materials and Chemistry, P.O. Box 124 Blindern,
N-0314 Oslo, Norway
ReceiVed: May 20, 2009; ReVised Manuscript ReceiVed: NoVember 18, 2009
Mg(NH2)2 was studied by bulk, slab, and cluster calculations based on density functional theory within the
generalized gradient approximation. Mg(NH2)2 is confirmed to have a tetragonal unit cell belonging to the
space group I41/acd. Five different slabs and their corresponding cleavage energies have been calculated.
The most stable surface was the (112) surface, supported by low cleavage energy, special symmetry properties,
and small structural changes found during ionic relaxation. Comparison of the density of states calculated
from bulk, slab, and clusters indicated that occupied states in the band gap of clusters can be one reason why
complex hydrides with nanoparticle structure have enhanced kinetics (in addition to the increased surface
area). Calculations of the energy change during removal of NH3 and H2 showed that it is energetically easier
to remove NH3 than H2 from Mg(NH2)2, confirming a general trend for metal-N-H systems.
1. Introduction
An interesting group of complex hydrides for hydrogen
storage are the metal-N-H systems. Studies of these systems
were initiated when Chen et al. demonstrated that 9.3 wt % of
hydrogen can be reversibly stored in Li3N by a two-step reaction
involving lithium imide (Li2NH) and lithium amide (LiNH2).1
However, the enthalpy of hydrogenation for the first step is
-148 kJ/mol H2,2 which means that a temperature of more than
430 °C is needed for a complete reversible hydrogen storage
cycle. The second reaction has a much lower and beneficial
enthalpy change, -44.5 kJ/mol H2.2 The hydrogen storage
capacity of this step is still rather large, 6.3 wt %, but the release
temperature for hydrogen is high.1
Compositional alteration can greatly improve the thermodynamic properties of metal-N-H systems.3 Examples include
Li-B-N-H,4,5 Mg-Ca-N-H,6 and Li-Mg-N-H,7-10 which
all are new systems developed since 2004. In the ternary
Li-Mg-N-H system, the thermodynamic properties are
considerably improved compared with the binary Li-N-H and
Mg-N-H systems.3 Ichikawa et al. found that a LiH/Mg(NH2)2
ratio of 8:3 is most suitable when it comes to hydrogen storage.11
The dehydrogenation reaction includes the decomposition from
Mg(NH2)2 to Mg3N2 through MgNH, leading to a theoretical
hydrogen capacity of 6.9 wt %.12 Leng et al. found that this
mixture desorbs more than 5 wt % at 150 °C.12 They also
confirmed the reversibility of the reaction by repeating the
dehydrogenation after a hydrogenation under 10 MPa at
200 °C.
One of the major problems with the Li-Mg-N-H systems,
and metal-N-H systems, in general, is that NH3 often is
included as one of the decomposition products. This is a serious
problem because most low-temperature fuel cells are poisoned
* To whom correspondence should be addressed. E-mail: o.m.lovvik@
fys.uio.no.
†
University of Oslo.
‡
Leiden University.
§
Akershus University College.
|
SINTEF Materials and Chemistry.
by only small amounts of NH3.13 Another problem is that, if
the decomposition of metal-N-H systems involves loss of
nitrogen, this induces a change in the chemical composition of
the material, which, consequently, will degrade the cycling
hydrogen storage capacity. During investigation of the equilibrium concentration of NH3 in the Li-Mg-N-H system, Lie
et al.13 found that the dominant source for NH3 formations stems
from Mg(NH2)2.
To improve the knowledge and understand the underlying
physics of the Li-Mg-N-H systems, thorough knowledge
about each component constituting these systems is important.
This study intends to improve our understanding of the
properties of Mg(NH2)2 using density functional theory calculations. The study includes an analysis of the crystal structure,
the electronic density of states, and five selected slabs of
Mg(NH2)2. Furthermore, a variety of clusters were built to
investigate the properties of nanoparticles, as well as the energy
barriers involved in the removal of H2 and NH3 from the clusters,
and to try and understand why Mg(NH2)2 primarily releases
NH3. A natural extension of the study will be to use the same
methods to understand why the Li-Mg-N-H system primarily
releases H2. It is also worth mentioning that cluster calculations
on complex hydrides is in an early phase, so this study also
aims to contribute to develop and increase the understanding
of this methodology.
2. Methodology
The majority of the calculations in this work has been
performed using the Vienna Ab -initio Simulation Package
(VASP)14,15 employing the projector augmented wave (PAW)
method,16 using the Perdew-Burke-Ernzerhof (PBE) functional17 at the generalized gradient approximation level. The
standard potentials with the valence configurations 2s22p0 for
Mg, 2s22p3 for N, and 1s1 for H were used. Self-consistency of
the electron density was defined to be reached when the total
energy of two consecutive cycles differed by less than 0.01 meV.
The single-particle orbitals were smeared by a Gaussian
convolution with a width of 0.01 eV (one exception is mentioned
in Figure 8).
10.1021/jp9047264  2009 American Chemical Society
Published on Web 12/08/2009
Surfaces and Clusters of Mg(NH2)2 Studied by DFT
The convergence of the VASP calculations with respect to
the cutoff energy was checked thoroughly. It was found that a
cutoff energy of 450 eV was necessary to obtain calculated
forces with errors less than 0.05 eV/Å; 550 eV was needed for
calculated pressures converged within 0.5 MPa, and 800 eV
was required for the total energy to have a numerical error due
to the cutoff energy less than 1 meV/formula unit. A k-point
mesh of 2 × 2 × 1 assured an error of less than 1 meV in the
total energy of one unit cell of Mg(NH2)2. During the slab
calculations, the same k-point density was used within the plane
of the slabs, while only the Γ point was used in the perpendicular
direction. In the cluster calculations, only the Γ point was used.
All calculations were spin-restricted, after testing that this did
not imply any significant changes in energies or geometric
structure. Dipole corrections were not added to any of the slabs.
The ionic relaxations in VASP were performed using the
residual minimization method with direct inversion in iterative
subspace.15 This is an implementation of the quasi-Newton
algorithm and the atomic positions were relaxed simultaneously
with the cell parameters for the bulk structure. The ionic
relaxation of the slabs was more difficult, so here, the conjugate
gradient algorithm was first used on the initial structure, before
the quasi-Newton algorithm was used to give the final structure.
For the slabs, the bulk relaxed lattice constants were kept
constant, and only the atomic positions were optimized.
To assist the ionic relaxation of the clusters, the Amsterdam
Density Functional (ADF)18 program was employed in addition
to VASP (see section 3.3 for a detailed description of the
procedure). The criterion for relaxation of bulk, slab, and clusters
was that the forces between two atoms in the unit cell were
less than 0.05 eV/Å. The calculations in ADF were performed
using the PBE functional17 at the GGA19 level. A triple-ζ plus
one polarization function (TZP) basis set18,20 with no frozen core
was used. Furthermore, the general accuracy parameter in
ADF18,20 was set to 4.0 based on initial convergence tests of
the total energy of the clusters.
3. Results and Discussion
3.1. Bulk Structure. The relaxed bulk structure of Mg(NH2)2
was found by ionic relaxation of the experimental bulk structure
of Mg(ND2)2 found by Sørby et al.21 Figure 1 shows one unit
cell of the relaxed Mg(NH2)2 bulk structure viewed from the
[100] direction, where the tetrahedral configuration of amides
around Mg is shown. The unit cell remains in space group I41/
acd21 during the relaxation. The tetragonal conventional cell
consists of 224 atoms, which means that the calculations
performed were computationally demanding.
To further explore some of the geometric properties relevant
for the slab and cluster calculations, selected parts of the relaxed
Mg(NH2)2 structure are shown in Figures 2 and 3. In Figure 2,
the bulk structure is viewed from the [111j] direction, providing
a side view of the (112) planes. This figure demonstrates how
the layers are arranged as pairs oscillating together along the
horizontal axis. There is thus an increased distance between
every second layer of atoms in the (112) plane, while the
interlayer distance is virtually constant within the pairs. This
indicates that the chemical bonds between every second layer
of atoms in the (112) plane are weak and that the (112) surface
should be among the most stable ones. This is further discussed
in section 3.2. In Figure 3, the bulk structure is viewed from
the [11j0] direction, giving a side view of both the (112) and
the (112j) planes. Comparison between this figure and Figure 2
shows that Mg(NH2)2 is organized in weakly connected branches
pointing in the [11j0] or the [1j10] direction and that each branch
is separated from the other branches by (112) and (112j) planes.
J. Phys. Chem. C, Vol. 113, No. 52, 2009 21649
Figure 1. One unit cell of the relaxed Mg(NH2)2 bulk structure,
visualizing the tetrahedral configuration of amide units around the
magnesium ions. The large, green spheres represent Mg atoms, the
medium, blue spheres represent N atoms, and the small, red spheres
represent H atoms. This representation of the atoms is used throughout
this article.
In Table 1, the structural parameters calculated in this work
are compared to those found experimentally by Sørby et al.21
and those calculated by Velikokhatnyi et al.22 Our calculated
values for the lattice constants a and c are 1.1% and 1.8% larger
than the experimental values found by Sørby et al.21 and
approximately 0.5% larger than those previously calculated.22
Here, it should be mentioned that the deuterium in the
experimental structure found by Sørby et al. is for the sake of
simplicity and is referred to as hydrogen in the remainder of
this article, both in the text and in the tables.
Table 2 shows a comparison of the measured21 and calculated22 interatomic distances. It is clear that our results are in
excellent agreement with those found by Sørby et al.;21 ranges
of interatomic distances are virtually the same. The distances
found by Velikokhatnyi et al.22 exhibit, in general, a smaller
spread than ours.
When turning to the H-N-H angles, however, there was a
notable spread (101-107°) in the experimental study,21 whereas
the angles found in our and Velikokhatnyi et al.’s22 models were
much more homogeneous (102.5-102.7° and 102.5-102.9°,
respectively). All in all, the differences between our calculated
and the previously published structural parameters21,22 are small.
The most notable difference was in the interatomic distances
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J. Phys. Chem. C, Vol. 113, No. 52, 2009
Seip et al.
TABLE 1: DFT Calculated and Experimental Structural
Parameters for Mg(NH2)2
Figure 2. Chosen part of the bulk structure of Mg(NH2)2 viewed from
the [111j] direction. Mg-N and Mg-H bonds are drawn for interatomic
distances between 1.4 and 2.8 Å, whereas N-H bonds are drawn for
interatomic distances between 0.6 and 1.2 Å. Notice the increased
distance between every second atom layer.
A (Å)
C (Å)
V (Å3)
Mg(32g)
x
y
z
N1(16e)
x
y
z
N2(16d)
x
y
z
N3(32g)
x
y
z
H1(32g)
x
y
z
H2(32g)
x
y
z
H3(32g)
x
y
z
H4(32g)
x
y
z
exptl21
calcd (this work)
calcd22
10.376
20.062
2160
10.494
20.426
2249
10.445
20.312
2216
0.373
0.105
0.188
0.373
0.109
0.188
0.289
0.750
0.375
0.289
0.750
0.375
0.000
0.000
0.381
0.000
0.000
0.382
0.228
0.488
0.248
0.226
0.486
0.251
0.229
0.813
0.403
0.228
0.806
0.402
0.058
0.953
0.351
0.058
0.951
0.351
0.287
0.434
0.225
0.286
0.430
0.224
0.286
0.537
0.276
0.289
0.539
0.278
TABLE 2: Selected Experimental and Calculated
Interatomic Distances in Mg(NH2)2
distances (Å)
Figure 3. Chosen part of the bulk structure of Mg(NH2)2 viewed from
the [11j0] direction, which is perpendicular to Figure 2. Hydrogen atoms
are hidden in order to elucidate the branch structure. The Mg-N bonds
are drawn for interatomic Mg-N distances from 1.4 to 2.8 Å.
found here and in ref 22. The most likely reason for this are
the different functionals used; Velikokhatnyi et al. used PW91,
whereas the PBE was used in the present work. Furthermore,
Velikokhatnyi et al. used a different experimental structure as
the starting point for their relaxations (the Mg(NH2)2 structure
from ref 23). DFT calculations within the GGA approximation
performed by Løvvik et al.24 on a number of hydrides showed
that different starting points gave significant variations in the
size of the calculated lattice constants when the force relaxation
criterion is in the order of 0.05 eV/Å (the same as in the present
study). The last important deviation was that between the
experimental and DFT predicted H-N-H angles, which most
readily may be explained by the different temperatures of the
studiessroom temperature for the experiment and 0 K for the
modeling. Also, the calculations did not include zero-point
energies, which can give significant contributions in hydrides.
The total and local electronic density of states for Mg(NH2)2
are plotted in Figure 4. The atomic covalent radii of 1.30, 0.75,
and 0.37 Å have been used as the Wigner-Seitz radii for Mg,
N, and H, respectively. The plot is very similar to that calculated
by Velikokhatnyi et al.,22 validating both calculations.
atoms
exptl21
calcd (this work)
calcd22
Mg-N (min)
Mg-N (max)
Mg-H (min)
Mg-H (max)
Mg-Mg (min)
Mg-Mg (max)
N-H (min)
N-H (max)
H-H (min)
H-H (max)
2.00
2.17
2.51
2.68
3.42
3.47
0.95
1.07
1.48
2.19
2.00
2.17
2.52
2.68
3.42
3.47
0.95
1.07
1.48
2.19
2.08
2.10
2.59
2.64
3.51
3.53
1.02
1.03
1.59
2.31
The DOS consists of three main regions below the Fermi
level. The lower region (from -7 to -4.5 eV) is almost entirely
populated by H s electrons and N p electrons, with a very small
contribution of Mg p and d electrons. The mid region (from
-4.5 to -3.5 eV) is only populated with Mg s, p, and d
electrons, while the upper region (from -3.5 to 0 eV) is mainly
populated with Mg s, p, and d electrons. In addition, there is a
major contribution of N p electrons, as well as some N s and H
s electrons in the upper region. The band gap is approximately
3 eV. This indicates that Mg(NH2)2 is an insulator since DFT
at the GGA level is known to systematically underestimate the
band gap,25 meaning that the actual band gap is probably larger.
In the conduction band, Mg once more contributes with all three
kinds of electrons, with some contributions from N p electrons
and H s electrons.
Surfaces and Clusters of Mg(NH2)2 Studied by DFT
J. Phys. Chem. C, Vol. 113, No. 52, 2009 21651
Figure 4. Total and local electronic density of states for Mg(NH2)2 as a function of the energy relative to the Fermi level in eV. s, p, and d states
are shown as shaded blue, solid green, and dotted red lines, respectively. The local DOS is shown for Mg (upper pane), N (mid-upper), and H
(mid-lower) atoms, whereas the total DOS is shown in the lower pane.
From the DOS, we can identify strong overlap between the
low-lying electronic states for N and H, which indicates a strong
bond between N and H. Higher in the conduction band, there
is an overlap between Mg and N, supporting the presumption
that the material consists of ionically bonded Mg2+ and NH2units.
3.2. Surface Calculations. One of the most important issues
in current hydride research is to understand the hydrogenation
kinetics of the different hydrides. To describe this theoretically,
it is crucial to know the properties of their surfaces. This is
even more important when the hydride is nanostructured. Each
particle has many different surface facets, and by identifying
the surfaces that are most stable, essential information about
the morphology of nanoparticles can be gained.
Five different slabs of Mg(NH2)2 have been investigated,
cleaved along the (100), (001), (110), (012), and (112) surfaces
(see Figure 5). Each slab was constructed by inserting a vacuum
layer with the preferred orientation into the unit cell. It is
important that this vacuum layer is thick enough to prevent
electronic interactions between the slabs but, at the same time,
small enough to avoid unnecessary computational time. Initial
tests showed that converged cleavage energies were reached
when a vacuum layer of 14 Å was used. This vacuum layer
thickness has, therefore, been used in all the slab calculations.
When a vacuum layer is inserted in the preferred orientation
of the unit cell, two surfaces are created. In three of the cases,
the two surfaces have similar composition: the (110), (012),
and (112) slabs. In these cases, the cleavage energy is the same
as the surface energy. The two remaining slabs exhibit surfaces
with different composition: the (100) and (001) slabs. For these
slabs, the cleavage energy represents a mixture of two different
surface energies and can thus not be used for direct comparison
of surface stability. It is, here, worth mentioning that the NH2
units were always kept intact when the slabs were constructed
since we assumed that a very high energy was needed for
splitting these units (this was later confirmed during the cluster
calculations (see section 3.3)). This means that only Mg-N
bonds were broken during the construction of the slabs.
Another important aspect during the construction of the slabs
is the number of layers. A thick slab reassures that there will
Figure 5. Figures on the left- and right-hand sides show, from top to
bottom, the initial and relaxed (100), (001), (110), (012), and (112)
slabs. Notice that Mg consistently moves into the surface during
relaxation, while the amides move outward with H pointing out.
be no electronic interaction between the two surfaces of a slab,
but it also means that the calculations are more demanding. In
this work, most of the slab calculations have been performed
on slabs containing three formula units (except the (112) slab,
which contained four formula units, because of the special
symmetry of this face). From previous calculations on another
complex hydride (NaAlH4, ref 29), we believe that the thickness
21652
J. Phys. Chem. C, Vol. 113, No. 52, 2009
Seip et al.
TABLE 3: Cleavage Energies, Number of Broken Mg-N
bonds/nm2, and the Corresponding Energy and Structure
Changes ∆E and ∆rav during Relaxationa
surface
Ecleav (J/m2)
broken bonds/nm2
|∆E| (eV)
∆rav(Å)
(100)
(001)
(110)
(012)
(112)
Z16st
Z16nst
1.30
1.26
0.51
0.53
0.26
6.1
5.9
4.9
5.2
2.2
0.80
0.53
0.46
0.76
0.20
0.48
0.23
3.72
2.37
0.30
0.63
0.11
0.63
0.44
a
Due to the limited number of layers, the uncertainty in the
calculated cleavage energies is unknown. We have, nevertheless,
presented three digits to elucidate the magnitude of their
differences.
is large enough to avoid interactions between the two surfaces
so that the cleavage energies are converged.
The cleavage energy is defined as half the energy needed to
cut a crystal in two and then separate the two parts at infinite
distance. There are several ways to determine it26,27 based on
the general formula
Ecleav(N) )
Eslab(N) - NEbulk
2A
(1)
where Ecleav(N) is the cleavage energy, Eslab(N) is the energy of
a slab consisting of N formula units, and A is the area of the
slab surface per slab unit cell. Ebulk represents the bulk energy,
which, in our case, is the energy taken from bulk calculations;
other options26,27 were not available with the limited number of
layers in this study. The uncertainty in cleavage energy from
the limited number of layers, using the bulk energy as reference,
is difficult to estimate but can, from previous studies, be inferred
to be in the order of 0.1 J/m. In addition to this uncertainty,
two of the slabs (the (001)- and (100)-terminated ones) consist
of multiple dipoles due to the alternating Mg2+ and NH2- layers.
There is no easy solution to this problem, and we can only
conclude that the cleavage energies of these slabs are highly
uncertain. They have been included for completeness, but the
values should be taken with a grain of salt.
The calculated cleavage energy for the various surfaces is
shown in Table 3. It is clear that the most stable slab is the
(112)-terminated one, with a cleavage energy only half of the
second most stable slabssthe (110)- and (012)-terminated ones.
It is also clear that the cleavage energy is correlated with the
number of broken bonds per square length unitsconsistent with
previous studies on complex aluminohydrides28,29sand with the
structural changes ∆rav and the energy changes ∆E resulting
from the surface relaxations. In addition to exhibiting the lowest
cleavage energy, the (112) slab has distinctly lower ∆rav and
∆E than any of the other slabs. This should be compared to
Figure 2, where it is evident that (112)-terminated slabs with
an even number of layers have a very low number of broken
bonds and form the basis of the “branches” of this material.
3.3. Cluster Calculations. This section presents cluster
calculations on Mg(NH2)2. So far, most DFT calculations on
such systems have been periodic, either bulk or slab calculations.
One exception are the studies by Marashdeh et al. on Ti-doped30,31
and TiH2-doped32 NaAlH4, where semispherical clusters served
as models for nanosized NaAlH4 particles. Those studies used
the ADF program, which is a molecular DFT code employing
Slater-type orbitals as basis functions. Cluster calculations can
provide improved insight in combination with bulk and slab
calculations. First, real materials exhibit a range of surface facets.
Figure 6. Bulk-cut (left) and the relaxed (right) stoichiometric Z16st
cluster.
A cluster model can represent this situation very well since it
usually exhibits different surface terminations as well as edges
and corners. Furthermore, there are no boundary conditions
stemming from periodicity in cluster calculations, which allows
direct comparisons between defects at specific surface positions.
Also, the lack of periodicity opens the possibility to represent
more than one phase in the same model, facilitating studies of
phase transitions, decompositions, etc. Nevertheless, the use of
cluster models is not straightforward. Calculated properties will,
to some degree, depend on both the size and the shape of the
cluster. This is clearly a problem since current cluster models
usually are limited to approximately 1 nm (see below), whereas
real nanoparticles have diameters ranging from 60 nm and larger.
One can hope that this gap soon will be filled, with improved
synthesis techniques and development in computer codes and
hardware. Thus, calculations on clusters with the same size as
real nanoparticles can probably soon be performed. The cluster
calculations performed during this work have, therefore, practical benefit (increased knowledge about the chemical/physical
properties of Mg(NH2)2) and can, in addition, serve as a
guideline for further cluster calculations. The present cluster
calculations may, hence, contribute to the development and
increased understanding of this methodology.
The construction of cluster models for the periodic calculations performed with VASP was done by inserting a vacuum
between the clusters in all three dimensions. A distance between
the clusters of 12 Å was found to give convergence of the total
energy of one formula unit within 5 meV.
Relaxation of the cluster models with VASP turned out to
be difficult. Both quasi-Newton and conjugate gradient algorithms were tried, but none of the methods were able to relax
the bulk-cut clusters directly; the residual forces were in the
order of 0.5-1.0 eV/Å, and the structures moved unreasonably
far away from the bulk-cut structures. This motivated the use
of ADF for the ionic relaxations. This was more successful,
and with the quasi-Newton algorithm there was no problem
relaxing the clusters. If a cluster was partially relaxed in ADF,
it was possible to complete the relaxation using VASP. Actually,
tests showed that VASP was equally effective as or more
effective than ADF when starting from a partially converged
structure. The change in total energy between the bulk-cut and
relaxed cluster was very similar for the two methods; the
difference was at most a few percent. Also, the relaxed structures
were similar, with the average and median differences of relaxed
atomic positions calculated with the two methods being within
0.1 Å. On the basis of this experience, all cluster relaxations in
this work were done using the following procedure: the initial
cluster was, first, partially relaxed in ADF with residual forces
less than 0.3 eV/Å; then relaxation was completed using VASP
with the final force convergence criterion being 0.05 eV/Å. The
problems with using VASP for relaxing some of the clusters
from their bulk-cut starting point have two possible reasons. It
Surfaces and Clusters of Mg(NH2)2 Studied by DFT
J. Phys. Chem. C, Vol. 113, No. 52, 2009 21653
Figure 7. Bulk-cut (left) and the relaxed (middle) nonstoichiometric Z16nst cluster. The relaxed cluster is also shown to the right, rotated by 90°
and with Mg-N inserted. The (112) plane is also drawn (light blue) to expose how the branch structure is represented in the cluster.
Figure 8. Total electronic DOS for the (from top) bulk structure, the (110) slab, the Z16st, and the Z16nst clusters (the single-particle orbitals were
smeared by a Gaussian convolution with a width of 0.2 eV).
could be due to the fact that Mg(NH2)2 consists of molecular
units, making a localized basis set more appropriate. However,
there is also a possibility that the chosen size of the clusters is
too small and that the relaxed structures are actually local
minima. This is supported by the total energy of the VASPrelaxed structures, which decreased even after several hundred
relaxation steps. Because the resulting geometric structures from
the VASP calculations were very far away from the bulk crystal
structure of Mg(NH2)2 (and because the relaxations were
extremely tedious), we chose to neglect this possibility and used
the above-mentioned method to obtain (possibly metastable)
clusters with a geometric structure resembling the bulk. Also,
using VASP makes it easier to compare the results from cluster
calculations with those obtained from bulk and slab calculations.
One important property in this respect is the electronic density
of states, presented below. This will serve as a validation of
using a plane-wave code for describing clusters of a molecular
solid.
Our initial calculations included a number of clusters with
different sizes ranging from Z ) 6 to Z ) 22 (Z is the number
of formula units), following the scheme of Marashdeh et al.30
The optimal cluster size turned out to be Z ) 16, which was
large enough to preserve the overall bulk geometrical structure
without leading to excessive consumption of computer resources.
It was also clear that the surfaces of the cluster should consist
of amide ions with H atoms pointing outward; this lead to
minimal structural changes and low energy. This surface
structure may be explained by the ionic nature of the Mg-N
bonds; when Mg and N atoms are present on the surface, they
will generate electrostatic fields that destabilize the surface.
Hydrogen forms strong, covalent bonds with nitrogen that, at
the same time, neutralize these fields. Moreover, if H points
out of the surface, the amides are oriented in such a way that
most of the broken bonds are weak hydrogen bonds (like the
H-H bonds between the branches; see Figure 2). When
magnesium is located in the surface, the bonds between Mg in
the surface and subsurface N will be strengthened, reducing the
interatomic distance between Mg and N (this was observed in
both the slab and the cluster calculations).
The first cluster meeting these requirements was the stoichiometric Z16st cluster, consisting of 16 formula units. This cluster
has, however, an uneven distribution of the amides. The initial
and relaxed Z16st clusters are shown in Figure 6. It is clear that
the considerable structural changes in the upper right-hand
corner of the cluster are related to the uneven distribution of
the amides. We, therefore, built a nonstoichiometric cluster,
Z16nst. Here, one of the amides in Z16st was removed, giving a
more even distribution of the amides. The initial and relaxed
Z16nst clusters are shown in Figure 7. From these figures and
Table 3, where the structural and energy changes during
21654
J. Phys. Chem. C, Vol. 113, No. 52, 2009
Seip et al.
Figure 9. Z16nst cluster on the left (right)-hand side has a hydrogen
atom removed from a “corner” (“center”) amide.
Figure 10. Figures show the two different ways of removing H2 from
the Z16nst cluster: H2.1 (left) and H2.2 (right).
relaxation are shown, it is obvious that Z16nst is the cluster with
the least changes during relaxation. Because NH2 was removed
to create Z16nst, it is timely to raise the question whether this
cluster is charged due to the missing NH2- ion. We will
investigate potential problems with this, in what follows, but
emphasize that only neutral atoms were removed. This means
that the removal of an NH20 unit leaves an excess electron with
the cluster, which otherwise would have been associated with
an NH2- ion.
The most important test of the Z16nst cluster is to compare
its electronic density of states (DOS) with that of the Z16st
cluster, the (110) slab, and the bulk structure. Figure 8 shows
the total DOS for all these structures, and we immediately note
some major trends. The bands are narrower for the slab and
clusters than for the bulk structure. This is a usual effect,
explained by surface states for the slabs. In the case of clusters,
they should ideally have band structures like large molecules,
that is, with discrete energy levels. This shows up as narrow
bands after the numerical approximations of the methodology.
Furthermore, there are states filling the band gap of the slab
and the clusters. These states may be a reason why complex
hydrides with a nanoparticle structure have enhanced kinetics,
in addition to their increased surface area. All in all, the DOS
of the slab can be said to be intermediate between that of the
bulk and of the clusters. Perhaps most important, the DOS for
the Z16nst cluster is almost identical to that of the Z16st cluster.
On the basis of this and the small structure and energy changes
found during relaxation, we consider Z16nst to be a good cluster
model for further calculations.
We also created a cluster with two NH2 units removed, Z16nst2.
This gives a broader background for judging the validity of a
nonstoichiometric model and could also represent the second
step in a decomposition of the Z16st cluster. The electronic DOS
for Z16nst2 was very similar to that of Z16nst, further supporting
that Z16nst is a good model for further calculations.
Our next task was to simulate the first decomposition steps
by removing different species from the Z16nst cluster. Thus, H,
H2, NH2, and NH3 were removed from different surface sites
to see if some positions were more favorable than others and
to compare the different molecules. Only local energy minima
have been considered. This means that, when a species was
removed from the relaxed Z16nst cluster, the resulting cluster
was relaxed again. The total energy of this cluster, Enew, was
compared to the initial energy of the cluster, Einitial, and adjusted
with the standard state (gas phase H2, NH2, and NH3) energy
of the removed species Efree, to give the energy difference ∆E
between local energy minima:
TABLE 4: Energies (∆E from eq 2) Involved in the
Removal of H, H2, NH2, and NH3 from Different Surface
Positions of the Z16st, Z16nst, and Z16nst2 Clusters
∆E ) Enew + Efree - Einitial
(2)
First, a hydrogen atom was removed from two different
surface positions: a “corner” position and a “center” position
∆E (eV) after removal of
Z16st
H (corner)
H (center)
H2.1
H2.2
NH2.1 (corner)
NH2.2 (asym corner)
NH2.3 (center)
NH2.4 (asym center)
NH3.1
NH3.2
NH3.3
1.92
2.32
4.79
2.02
1.49
1.71
2.35
Z16nst
0.96
2.02
3.66
4.78
1.38
Z16nst2
0.59
2.06
3.61
1.09
1.91
1.78
1.79
-0.65
1.29
1.87
-0.06
(see Figure 9). Table 4 shows the energies involved in the
removal of different species from the clusters. We can see that
it is energetically favorable to remove a hydrogen atom from a
corner amide compared with removing it from the center for
all the three clusters. The relatively large energy differences
between the different clusters probably stem from the different
number of excess electrons in the clusters. Another reason can
be that the Z16st and Z16nst2 clusters have more substantial
structural changes than the Z16nst cluster because of a more
uneven distribution of the amides.
The reason why it is easier to remove a hydrogen atom from
a corner amide than from a center amide is that the corner amide
is more strongly bonded to Mg than the center amide. This must
be due to larger electrostatic effects near the corners, making
the corner Mg-N bond stronger. The strong Mg-N bond will
thus weaken the N-H binding, making it easier to remove H
from the corner amide.
H2 was then removed in two different ways from the Z16nst
cluster. For H2.1, the H atoms were removed from two different
amides, whereas for H2.2, both H atoms were removed from
the same amide (see Figure 10). As expected, H2.1 was
energetically favorable compared with H2.2 (a difference of 1.12
eV; see Table 4). In the H2.1 cluster, two imide molecules were
left after removal of the H2 molecule, while a separate N atom
without any H bonds was left in the H2.2 cluster. This was not
tested for the other clusters, but we assume that this behavior
is general.
An amide radical was then removed from both a corner
(NH2.1 (symmetric) and NH2.2 (asymmetric)) and a center
position (NH2.3 (symmetric) and NH2.4 (asymmetric)) for the
different clusters (see Figure 11). Here, it was energetically
favorable to remove the center amides compared with the corner
amides. This confirms what we already assumed from the
experience gained from the removal of monatomic H: the corner
amide is more strongly bonded to Mg than the center amide.
Surfaces and Clusters of Mg(NH2)2 Studied by DFT
J. Phys. Chem. C, Vol. 113, No. 52, 2009 21655
Figure 11. Z16nst cluster on the left (right)-hand side has removed an
amide radical from a corner (center) position.
Figure 13. Energy change versus distance for removal of H2 and NH3
from Z16nst along a linear pathway. Both H2 and NH3 were moved 4 Å
outward compared to their initial position (six steps of 0.667 Å). The
red line and points are for H2, whereas the blue line and points are for
NH3. The lines are drawn as guides for the eye only.
Figure 12. Figures show the final cluster when ammonia has been
removed from the Z16nst cluster in one or two steps: the NH3.1 (left)
and the NH3.2 (right) clusters.
The energy difference for the thermodynamical equilibrium
states for the initial clusters and the new clusters plus the energy
of NH2 gas (from -0.65 to 1.71 eV; see Table 4) was notably
smaller than the energies involved in the removal of H2 gas; in
one of the cases, it was even negative, meaning that desorption
will be thermodynamically advantageous. The reason for this
stems from the geometric properties of Mg(NH2)2 (see section
3.1 and Figure 3). The N atom in the NH2.4 molecule is weakly
linked to its nearest neighbor Mg due to a large Mg-N distance;
it belongs to another “branch” of the crystal structure.
The last molecule removed from the clusters was ammonia
(NH3). First, it was removed from the Z16nst cluster in two
different ways: For NH3.1, a center amide plus a neighbor H
atom from a corner amide were removed at the same time and
the new cluster was then relaxed. For NH3.2, a H atom from a
corner amide was first moved to the center amide, constructing
an imide ion and an ammonia molecule; this cluster was then
relaxed. The ammonia molecule was then removed, and the
cluster was again relaxed. These two procedures should yield
the same result, and this test was done to ensure that the
relaxation procedure was robust and did not result in different
local minima for different starting points. The final relaxed
clusters are shown in Figure 12, and it is indeed clear that the
two relaxed clusters are very similar. In addition, Table 4 shows
that their energies are very similar; within 15 meV (the
remaining difference must be due to the relaxations being
incomplete since we are using a finite force as convergence
criterion). This supports that the relaxations performed on the
Z16nst cluster are reliable and unrelated to the relaxation
procedure.
There was also a possibility to remove an NH3 molecule
without breaking any strong Mg-N bonds; this was named
NH3.3. It is placed in a similar position as NH3.1, but the Mg-N
bond strength is so much smaller that the removal is exothermic,
similar to that of NH2.4 above.
Ammonia was also removed from the two other clusters using
the NH3.1 scheme since this can be seen as the most representative removal from the main constituentssbranchessof the
crystal structure. The energy difference for the thermodynamic
equilibrium states for the initial clusters and the new clusters
plus the energy of NH3 gas was within the range from 1.78 to
2.36 eV (see Table 4). This means that it is, in all cases, much
more difficult to remove pure H in the atomic or molecular form
than H bonded to N, that is, NH2 and NH3, from Mg(NH2)2.
This is not unexpected since the former case involves the
restructuring of multiple covalent bonds, whereas the removal
of NH2 can be seen as removing an ion from an ionic material.
Because there are two possibilities to remove molecules
exothermically from this cluster, one can ask whether this cluster
is a valid starting point for the decomposition. A better procedure
would probably be to start from the (nonstoichiometric)
cluster(s) remaining after all exothermic removals. Alternatively,
the starting cluster could be annealed out of the local minimum
with first-principles molecular dynamics. Nevertheless, we
believe that the contribution to removal energies of the other
species from the presence of NH2.4 and NH3.3 is relatively small
since they are not nearest neighbors of the other removed
species. Also, we think it is important to describe our methodology in detail since there are very few similar studies in the
literature.
Finally, the energy changes versus distance (a crude estimate
of the transition barriers) involved in the removal of H2 and
NH3 molecules from Mg(NH2)2 were calculated. A linear
reaction path without relaxation of the intermediate structures
was considered for simplicity. Both H2 and NH3 were moved 4
Å outward compared to their initial position (six steps of 0.66
Å). This procedure overestimates the energy barriers but forms
a basis for comparing different reaction paths. To find the energy
barriers using more sophisticated transition path searches, for
instance, by the nudged elastic band (NEB) method, would form
an interesting future study; preliminary calculations in this
direction support qualitatively the results presented in this paper.
In Figure 13, the energy changes versus distance for desorption of H2 and NH3 from Z16nst along such a linear pathway
are shown. The desorbed molecules were originally located at
the H2.1 and NH3.1 sites depicted in Figures 10 and 12. The
activation energy needed to remove H2 is clearly higher than
that needed to remove NH3 from Mg(NH2)2. This gives a further
indication of why it is much harder to remove pure H2 than
NH3 from Mg(NH2)2.
It is well-known that the prevalence of gas desorbing from
Mg(NH2)2 is reversed when mixing with a hydride, for example,
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J. Phys. Chem. C, Vol. 113, No. 52, 2009
LiH. This can be a thermodynamic or kinetic effect or a
combination of both. These possibilities could most likely be
probed by creating a cluster model containing both phases and
studying various decomposition paths from such a combined
model. This would form an exciting future study.
4. Conclusions
Density functional calculations have been performed on the
magnesium amide Mg(NH2)2 using bulk and slab geometries
as well as cluster models. The bulk calculations confirm previous
studies of the complex hydride, but we uncovered the necessity
to use a very high energy cutoff in the plane-wave expansion
(800 eV) to obtain converged total electronic energies.
Calculated cleavage energies of a number of slabs showed
that the (112) slab is the most stable one. The (112) surface
also exhibits particularly small structure and energy changes
during relaxation from the bulk-cut structure, reflecting the
branchlike structure of this compound, with long and presumably
weak Mg-N bonds perpendicular to the (112) surfaces (Figure
2).
When building cluster models of this compound, we found
that at least 16 formula units were needed to build a stable
cluster without unreasonable restructuring. The exposed surfaces
of the cluster should primarily consist of NH2 ions. Such a small
cluster was very sensitive to a noneven distribution of ions,
which led to instabilities and reconstructions. To avoid this, a
nonstoichiometric model with one lacking NH2 unit was
constructed. These cluster models facilitated direct comparisons
between removals of different species at various locations,
representing possible starting points for decomposition of the
clusters. Stronger Mg-N bonds at the corners led to easier
removal of hydrogen but more difficult removal of NH2 at the
corners; this was reversed when moving to the side centers.
The energy required to remove an NH3 molecule is much smaller
than that required to remove H2, which explains why primarily
NH3 is released when pure magnesium amide is heated. This is
supported by calculations of energy changes versus distance for
removal of NH3 and H2 molecules, which tentatively suggest a
large kinetic barrier for H2 release.
In general, we have demonstrated that a combination of bulk,
slab, and cluster calculations in a plane-wave code can be used
to provide new and important insights of complex hydrides, such
as Mg(NH2)2.
Acknowledgment. Funding from the Research Council of
Norway through the NANOMAT program is acknowledged by
O.M.L. The work was supported by a grant of computational
time on the Notur metacenter for supercomputing.
Seip et al.
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