# Structure and optical properties of polymeric carbon nitrides from atomistic
simulations

Changbin Im  [  Björn Kirchhoff  [  Igor Krivtsov  [  [  Dariusz Mitoraj  [
Radim Beranek  [  Timo Jacob  [<EMAIL_ADDRESS>
###### Abstract

Detailed understanding of the structural and photophysical properties of
polymeric carbon nitride (PCN) materials is of critical importance to derive
future material optimization strategies towards more desirable optical
properties and more photocatalytically active materials. However, the wide
range of structural motifs found in synthesized PCNs complicates atomistic
simulations that rely on well defined models. Performing hybrid DFT studies,
we systematically investigate formation energy trends and optical properties
of PCNs as a function of dimensionality, going from molecular oligomers over
periodic sheet models to stacked crystals. Thermochemical calculations that
take into account vibrational enthalpy and entropy contributions predict that
a mixture of structural motifs including the melon string structure,
poly(heptazine imide), and g-C3N4 motifs is stable under typical synthetic
conditions. The degree of lateral condensation as well as stacking can reduce
the bandgap while out-of-plane corrugation of the material increases both
stability and the optical gap. The key result of this work is that already
small domains of strongly condensed PCN are calculated to give rise to
favorable optical properties. This result reconciles conflicting literature
reports indicating that the thermodynamically favorable melon motif has a too
large bandgap compared to experiments, while the g-C3N4 structure, for which
bandgap calculations are in better agreement with experiments, does not agree
with measured chemical compositions of PCNs. Finally, we postulate a new
computational model for carbon nitride materials that encompasses the most
important structural motifs and shows a bandgap of ca. 2.9 eV.

###### keywords:

American Chemical Society, LaTeX

Ulm University] Institute of Electrochemistry, Ulm University, Albert-
Einstein-Allee 47, 89081 Ulm, Germany Ulm University] Institute of
Electrochemistry, Ulm University, Albert-Einstein-Allee 47, 89081 Ulm, Germany
Ulm University] Department of Organic and Inorganic Chemistry, University of
Oviedo-CINN, 33006 Oviedo, Spain University of Oviedo-CINN] Institute of
Electrochemistry, Ulm University, Albert-Einstein-Allee 47, 89081 Ulm, Germany
Ulm University] Institute of Electrochemistry, Ulm University, Albert-
Einstein-Allee 47, 89081 Ulm, Germany Ulm University] Institute of
Electrochemistry, Ulm University, Albert-Einstein-Allee 47, 89081 Ulm, Germany
Ulm University] Institute of Electrochemistry, Ulm University, Albert-
Einstein-Allee 47, 89081 Ulm, Germany [Helmholtz-Institute-Ulm] Helmholtz-
Institute-Ulm (HIU), Helmholtzstr. 11, 89081, Ulm, Germany [Karlsruhe
Institute of Technology] Karlsruhe Institute of Technology (KIT), P.O. Box
3640, 76021, Karlsruhe, Germany IR,NMR,UV

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##  1 Introduction

Polymeric carbon nitrides (PCNs) have recently gained considerable attention
as a class of non-metallic photocatalytic materials.1 Among many potential
applications, this new material class is currently being explored in
particular for photo-induced hydrogen production,2, 3, 4, 5, 6, 7 alcohol
oxidation,8, 9, 10, 11 and the CO2 reduction reaction12, 13 due to their
intriguing photo-physical properties. PCNs feature optical gaps of ca. 2.8 eV.

Typical PCN materials obtained by straight-forward thermal synthesis are semi-
crystalline and contain a multitude of micro-structural motifs.14, 15, 16, 17,
18 This lack of crystallinity complicates structural characterization.
However, if controlled reaction conditions can be enforced, certain motifs can
become more prominent. For example, the ”melon” structure, first described by
Berzelius and Liebig19, 20 consisting of strands of linearly polycondensed
melamine monomers interconnected via hydrogen bonds, was successfully
synthesized by Lotsch et al. by thermal treatment of melamine in a closed
reaction vessel at 630 ∘C.21, 22, 14 A mesoporous, non-hydrogenated graphitic
carbon nitride (g-C3N4) was postulated and explored for photocatalytic
hydrogen production.23, 24 High condensation g-C3N4 model systems, where
heptazine units are linked by tertiary amines into a 2D sheet structure, were
suggested as theoretical models for PCNs by Lie and Cohen as early as 1989.25
Since then, calculations explored models based on g-C3N4 that exhibit
desirable optical properties which are in good agreement with experimentally
measured PCN samples.18 However, the low hydrogen content of this structure is
incompatible with elemental analysis of PCN samples, which are usually in
better agreement with melon-type model systems. To this date, the idealized
g-C3N4 structure has not been unambiguously identified in any synthesized
samples. As a result, the exact structural characterization of PCNs is still
an open question, and no single computational model on its own so far was able
to fully explain the measured properties.

Combined theoretical and experimental efforts were able to shed light on some
of the structural parameters of PCNs. Gracia et al. for example found that
corrugation can stabilize heptazine-based PCNs.26 Li et al. found that micro-
structural features of PCNs such as H-bonding, interlayer spacing, and
stacking configuration are susceptible to temperature variations during
synthesis.27 Salt melts were explored for PCN synthesis and the templating
effect from these salts was found to facilitate the synthesis of PCN
structures with large cavities, such as poly(triazine imide) (PTI) and
poly(heptazine imide) (PHI).28, 29, 30

Uncertainty with respect to the structure of PCNs is one of several issues
that theoreticians face when simulating this new material class. Further
complications include the importance of accurately describing the optical
properties, which necessitates the use of costly hybrid density functional
theory (DFT) methods that augment regular generalized-gradient approximation
(GGA) calculations with exact exchange contributions derived from Hartree-Fock
wave function theory calculations on the Kohn-Sham orbitals.31, 32
Unfortunately, large system sizes are often required to model PCNs, which can
make hybrid DFT calculations computationally unfeasible. Finally, PCNs are
characterized by a multitude of intricate structural features such as
stacking, edge boundaries, and corrugation which further increases the
complexity of model systems.33, 34, 35, 32 The structural complexity of PCNs
therefore hampers computational efforts aimed at improving our understanding
of their fascinating properties, including strong excitonic effects,36 charge
accumulation,37, 38, 39, 40, 41 and tuning of the optical properties towards
more efficient visible light absorption.16, 42

Most notably, there is significant disagreement in literature between
theoretical bandgap predictions and experimentally measured bandgaps. In
particular, models for which bandgaps of 3 eV and lower are calculated
typically feature a high degree of condensation while elemental analyses of
synthesized materials with similar bandgaps show hydrogen contents that are
incompatible with the theoretical models.43, 44, 45, 46 The authors are
convinced that this discrepancy is the result of the structural complexity of
PCNs which is hard to represent properly in simulations.

In this study, we aim at postulating a more complete model system for PCNs. To
this end, the relationship between structures and optical properties of PCNs
are systemically investigated. Calculations initially focus on exploring
individual parameters such as the degree of condensation, increase of
dimensionality, and corrugation. To investigate these parameters and to model
the polycondensation reaction from monomer to various PCN structures, the
thermochemical approach introduced by Botari et al.47 is adopted and extended
by incorporating vibrational enthalpy and entropy contributions of the solid-
state materials. By varying the chemical potential of NH3,
$\Delta\mu_{\ce{NH3}}$, refined thermochemical maps are obtained and reveal a
thermodynamic route for the formation of the fully-deaminated PCN structure,
typically referred to as g-C3N4, via PHI as an intermediate.

The key finding of the present work is that already small domains of g-C3N4
embedded in less condensed PCN structures are sufficient to obtain bandgaps of
3 eV and lower. Thus, a comprehensive model system is postulated from a
combination of the most important structural features explored in this work
that affect the stability and optical properties of PCNs. This structure is
thermodynamically feasible, shows a bandgap of ca. 2.9 eV while retaining a
low degree of condensation, and is predicted to form under experimentally
accessible conditions. This model constitutes an important step towards a more
holistic description of this fascinating material class in atomistic
simulations.

##  2 Methodology

###  2.1 Computational Details

All calculations were performed using the Vienna Ab initio Simulation Package
(VASP) version of 5.4.4 which employs the projector-augmented wave (PAW)
method.48, 49, 50, 51, 52 A plane wave energy cut-off of 400 eV was used and
the wavefunction was optimized to an accuracy of $10^{-6}$ eV. Atomic
coordinates were relaxed until forces reached below $5\times 10^{-2}$ eV/Å.
Gaussian-type finite-temperature smearing was employed with a smearing width
of 0.01 eV. Long range interactions were treated with DFT-D3 dispersion
correction.53, 54

The atomic and cell coordinates were relaxed using the exchange-correlation
functional by Perdew, Burke, and Ernzerhof (PBE) within the generalized
gradient approximation (GGA).55 Accurate final energy results were then
obtained by performing single-point calculations on the PBE-optimized
structures using the hybrid functional by Heyd, Scuseria, and Ernzerhof with
25 % exact exchange and a screening factor of 0.2 Å-1 (HSE06).56 The HSE06
functional was found to deliver a good description of the electronic
properties of PCNs.31, 35, 32

Atomic coordinates of molecular models were optimized using the ORCA package
version 4.2157, 58 using the PBE0 functional,55, 59, 60 def2-TZVP basis set,
def2/J auxiliary basis set, and DFT-D3 dispersion correction. Then, single
point calculations were performed in VASP using the HSE06 functional to obtain
final energy results. The pre-optimization in ORCA was found to be necessary
because geometry relaxations in VASP regularly did not find the energetic
minimum structures for molecules. In particular, corrugation is more
pronounced in the ORCA optimized structures, which in fact also give lower
total energy results after subsequent HSE06 single-point calculations in VASP.
We assume that ORCA, with which we used finite basis sets and a hybrid
functional for the relaxation, enabled symmetry breaking more easily compared
to the GGA-based VASP plane wave calculations. To avoid spurious interactions
between molecules and in case of 2D surface calculations, a vacuum layer of
($\sim 20$ Å) was used and dipole correction was enabled in the relevant
directions (molecules: all directions, surfaces: perpendicular to surface).
The Monkhorst-Pack scheme was adopted to create k point grids for Brillouin
zone integration.61 A single, $\Gamma$-centered k point was used for molecular
structures. The vibrational contribution was calculated from force constant
matrices using the VASP-Phonopy interface.62, 63, 64 A data set of atomic
coordinate files of the investigated structures in the VASP-POSCAR file format
has been made available.65

###  2.2 Thermochemistry

For thermochemical analysis, we consider a simple thermal synthesis which uses
melamine as the monomer. The thermochemical stability for each structure is
assessed by the formation free energy $\Delta G_{\text{F}}(T,p)$ using the
following equation:

$\displaystyle\Delta G_{\text{F}}(T,\emph{p})=G_{\text{structure}}(T,p)-(n\
G_{\text{melamine}}(T,p)-m\ \mu_{\ce{NH3}}(T,p)).$ (1)

Temperature ($T$) and pressure ($p$) are taken into account for the solid-
state structures. The free energy of any particular structure under
investigation, $G_{\text{structure}}$, and of a melamine monomer,
$G_{\text{melamine}}$, are calculated as follows:

$\displaystyle G=E^{\text{total}}+F^{\text{vib}}+pV$ (2)

where $E^{\text{total}}$ is total energy, $F^{\text{vib}}$ is vibrational free
energy. The contribution of vibrational degree of freedom can be obtained from
phonon density of states $\sigma(\omega)$.

$\displaystyle F^{\text{vib}}(T,\omega)$ $\displaystyle=$
$\displaystyle\int\,d\omega F^{\text{vib}}(T,\omega)\sigma(\omega)$ (3)
$\displaystyle=$
$\displaystyle\frac{1}{2}\hbar\omega+k_{\mathrm{B}}T\ln\bigl{[}1-\exp\Big{(}\frac{-\hbar\omega}{k_{\mathrm{B}}T}\Big{)}\bigr{]}$

The gas phase chemical potential of ammonia, $\mu_{\ce{NH3}}(T,\emph{p})$,
which depends on temperature and partial pressure, is an important term to
connect simulations with experimental conditions. We adopt the expression for
$\Delta\mu_{\text{\ce{NH3}}}(T,p)$ from Botari et al.47 This expression for
the calculated $\mu_{\text{\ce{NH3}}}(T,\emph{p})$ is based on the polyatomic
ideal gas approximation composed of translational, rotational, vibrational
terms:66, 67, 68

$\displaystyle\mu(T,p)$ $\displaystyle=$ $\displaystyle
E_{\ce{NH3}}^{\text{total}}+E^{\text{ZPE}}_{\ce{NH3}}+\Delta\mu(T,p)+pV$ (4)
$\displaystyle\Delta\mu(T,p)$ $\displaystyle=$
$\displaystyle\mu_{\text{trans}}+\mu_{\text{rot}}+\mu_{\text{vib}}$ (5)
$\displaystyle=$
$\displaystyle-{\frac{1}{2}}k_{\text{B}}T\Big{\\{}\ln\Big{[}\Big{(}\frac{2\pi
m}{h^{2}}\Big{)}^{\frac{3}{2}}\frac{(k_{\text{B}}T)^{\frac{5}{2}}}{p}\Big{]}+\ln\Big{(}\frac{k_{\text{B}}T}{{\sigma}^{\text{sym}}B_{0}}\Big{)}-\ln\Big{[}1-\exp\Big{(}{-\frac{\hbar\omega}{k_{\text{B}}T}\Big{)}\Big{]}}\Big{\\}}$

where $k_{\text{B}}$ is the Boltzmann constant, $m$ is the mass of ammonia,
$\sigma^{\text{sym}}$ is the symmetry number, $B_{0}$ is the rotational
constant, $\omega$ is the fundamental modes of the particle, $\hbar$ is the
reduced Planck constant.

To validate our approach, we successfully reproduced published formation free
energy trends for PCNs, see Figure S2 and S3.47 The published trends, however,
neglect the influence of vibrational contributions on the free energy of both
oligomers and solids. Significant changes can be observed when vibrational
contributions are included (see Figure S4).

Finally, the degree of condensation (DoC) is used as a descriptor throughout
this work. It is calculated as the percentage quotient between the number of
condensation points found in a particular structure and the product of the
number of melem units - which are stable intermediates formed from the
precursor melamine - that the structure is made up of times the maximum number
of condensations that each melem unit in the structure can support (6):

$\text{DoC}=\frac{\text{\\# of condensations}}{N_{\text{melem}}\times 6}\times
100.$ (6)

###  2.3 Model Systems

In order to simulate the thermodynamics of PCN formation, we simulate the
polycondensation thermodynamics from the melamine monomer up to the tetramer
phase. The investigated model systems are illustrated in Figure 1.

Figure 1: Illustration of molecular oligomer model systems investigated in
this study. The models are named after a hxy naming scheme, where $x$ is the
number of heptazine units and $y$ is an increasing number that indexes the
oligomers.

The investigated 2D- and 3D-periodic models are illustrated in Figure 2, while
the mixed-motif structures postulated in section 4.5 are shown in Figure S1.

Figure 2: Illustration of periodic 2D surface and 3D crystal models
investigated in this study. Structures are composed of s-melon, melon, PHI,
C9N13H3, cg-C3N4(corrugated), and pg-C3N4(planar) structural motifs.

The atomic coordinates of all structures were relaxed using the methodology
outlined in section 2.1. Even though no constraints or restraints were placed
on the models, the melon, pg-C3N4, and C9N13H3 structures retained an
idealized flat configuration, which is known to be a local minimum.26, 69, 34
The other periodic structures show various degrees of corrugation. Corrugation
is notorious due to the large number of local minima in the coordinate space.
While this limitation affects our ability to predict absolute stability values
with high confidence, it is only of minor importance when looking at larger
scale stability trends, which is the focus of the present work.

##  3 Results

###  3.1 Formation Energy Trends for 1D Materials

To understand the condensation process, formation free energy ($\Delta
G_{\text{F}}$) phase diagrams are constructed using the melamine monomer and
NH3 molecules as reference. Since there is only one possible configuration to
form a dimer, different trimer and tetramer configurations are explored first.
All calculated molecular structures are illustrated in Figure 1. The relative
stability trends for the trimer and tetramer phases are shown in Figure 3(a)
and 3(b), respectively.

Figure 3: The free energy of formation for all (a) trimer and (b) tetramer
structures are shown as function of $\Delta\mu_{\text{\ce{NH3}}}$. The free
energy of formation is normalized to eV/atom to be able to compare structures
with different total number of atoms. Free energy results are then expanded
into $T$-$p_{\ce{NH3}}$-dependent phase diagrams for the most stable (c)
trimer and (d) tetramer structures, respectively. The experimental temperature
range for synthesis of these molecular structures is given between the dotted
and the dashed line.27, 70, 21, 22, 71

The phase diagram reveals that two competing condensation processes exist
during the molecular oligomerization stage: non-cyclic and cyclic
condensation. Non-cyclic condensation is the dominant process at more positive
values of $\Delta\mu_{\ce{NH3}}$ while cyclic condensation is dominant below
$\Delta\mu_{\ce{NH3}}$ values of ca. -1.0 eV, both in case of trimers and
tetramers.

In case of the tetramer structures, the cyclic h45 and the h43 configurations
(see Figure 1) are found to be similarly favorable. Both of these structures
contain a condensed ring of three heptazine units (h32) with an additional
dangling heptazine unit, which is either attached at a terminal ($\ce{-NH}R$,
where $R=\text{heptazine},\text{h}_{45}$) or at a bridging group (-NR, h43).
On the other hand, among the non-cyclic configurations, h42 and h48 compete as
the most stable configurations with only small deviations of $\Delta
G_{\text{F}}$ of $10^{-4}-10^{-3}$ eV/atom. These structures as well as h41
and h42 actually constitute rotamers which can be converted into each other by
rotation around bridging R-NH-R moieties. Structure h48 appears to be slightly
more stable at $\Delta\mu_{\text{\ce{NH3}}}$ values close to zero, likely due
to stabilizing van-der-Waals interactions that result from this configuration.
Both of the angled structures (h42 and h48) are more stable than the linear
structure h41. However, the energy differences are small enough that a mixture
of all of the discussed configurations may be formed from a thermodynamic
point of view.

The $\Delta G_{\text{F}}$ differences between the most stable trimer and
tetramer structures can be illustrated in a more practically-oriented way by
expanding $\mu_{\text{\ce{NH3}}}$ into $T$-$p$ phase diagrams, see Figures
3(c) and (d). At reported synthetic conditions,21, 22, 71 the formation of
melem (h1) begins at ca. 350–370 ∘C (Figure 3 (c) and (d)). From gravimetric
analysis it is commonly assumed that formation of molecular oligomer
structures is the dominant process until a temperature of ca. 500 ∘C.70 At
higher temperature, more condensed structures (i.e., 2D-condensed polymers)
are expected to form. From this analysis, it can be inferred that the more
condensed structures h32 (trimer) and h45 (tetramer) are most likely to form
in the typical experimental temperature regime (500–630 ∘C)14 and low NH3
partial pressure ($p_{\ce{NH3}}$), whereas the non-cyclic h34 and h42
structures are thermodynamically preferred at higher $p_{\ce{NH3}}$.

###  3.2 Formation Energy Trends for 2D and 3D Materials

The formation free energy trends for 2D and 3D structures are shown in Figure
4.

Figure 4: The free energy of formation for all (a) 2D and (b) 3D structures is
shown as a function of $\Delta\mu_{\ce{NH3}}$. The free energy of formation is
normalized to eV/atom to be able to compare structures with different total
number of atoms. Then, free energy results are expanded into a
$T$-$p_{\ce{NH3}}$-dependent phase diagrams for the most stable (c) 2D and (d)
3D structures, respectively. The experimentally measured temperature range for
the formation and the decomposition of PCNs are given as dotted black and
yellow lines.27, 70 The dotted grey line indicates the stability region for
the fully flat pg-C3N4-2D/-3D phases which constitutes a local minimum
structure and lays behind the more stable corrugated cg-C3N4-2D/-3D structure.

The melon-2D structure accounts for the most stable structure up to
$\Delta\mu_{\text{\ce{NH3}}}=-1.49$ eV. The PHI-2D structure appears as an
intermediary phase between $\Delta\mu_{\text{\ce{NH3}}}=-1.49$ and $-1.68$ eV.
For more negative $\Delta\mu_{\text{\ce{NH3}}}$, the highly-condensed
cg-C3N4-2D structure becomes the most stable phase. By deconvoluting
$\Delta\mu_{\text{\ce{\ce{NH3}}}}$ into a $T$-$p$ diagram, Figure 4 (c), a
similar trend regarding the $p_{\ce{NH3}}$ is obtained as for the molecular
phases, i.e., higher degree of condensation (DoC) can be achieved by reducing
$p_{\text{\ce{NH3}}}$ at the experimental temperature conditions (500–630
∘C).14

In the 3D phase, cg-C3N4-3D is the most stable structure at
$\Delta\mu_{\text{\ce{NH3}}}\leq-1.33$ eV. At more positive values of
$\Delta\mu_{\text{\ce{NH3}}}$, the melon-3D structure becomes more stable. The
$T$-$p$ phase diagram (Figure 4 (d)) reveals that there is thermodynamic
competition between the formation of the cg-C3N4-3D and melon-3D structures -
or, in fact, their coexistence - in the experimental temperature range
(500–630 ∘C) and at a $p_{\ce{NH3}}$ of 1 atm. This phase border shows that
lowering $p_{\ce{NH3}}$ will make the more condensed cg-C3N4-3D structure
thermodynamically more favourable.

###  3.3 Optical Property Trends

Calculated trends of the HOMO-LUMO gaps for the molecular structures as well
as bandgap trends for the periodic structures are listed in Table S1. Figure 5
summarizes the calculated values.

Figure 5: Summary of the calculated optical gap trends; HOMO-LUMO gaps are
given for molecules and bandgaps are given for the periodic structures. For
simplicity, the averaged HOMO-LUMO gaps are presented for the dimer up to
tetramer structures. A color palate is given for reference to roughly indicate
the energy range corresponding to visible light.

Note that HOMO-LUMO gaps for the molecular models are given as average values
over all dimers, trimers, and tetramers, respectively.

The optical property results give rise to several noteworthy trends: first,
the optical gap decreases as the DoC increases. This trend is particularly
apparent for the 2D periodic models. For instance, the melon-2D model, which
shows a DoC of 50%, has a ca. 0.7 eV larger bandgap than the highly-condensed
cg-C3N4-2D model (DoC = 100%).

Second, as the dimensionality of the system increases from 1D molecules to 2D
sheet models to 3D crystal models, the optical gap decreases. This trend is
noteworthy in particular for the 2D to 3D transition where the growth
mechanism is stacking of the 2D layers, i.e. no additional covalent bonds are
formed.

Third, corrugation of the structures increases the optical gap slightly. For
example, the completely-flat pg-C3N4-2D model, which constitutes a local
minimum configuration of the system, has a ca. 0.2 eV smaller bandgap than the
corrugated and thermodynamically more stable ($\Delta E=0.7$ eV/heptazine
unit) cg-C3N4-2D system. This effect is even more pronounced for the 3D
crystal models, where the non-corrugated system has a narrower bandgap of ca.
0.6 eV. At the same time, the interlayer spacing of the non-corrugated model
decreases by ca. 0.17 Å compared to the corrugated model, indicating
significantly stronger interaction between layers.

Lastly, we studied the bandgap as function of an increasing number of
heptazine units for hydrogen-terminated nanoribbons. The studied model
systems, bandgap results, and a comparison to the optical properties of some
key structures investigated above are given in Figure 6.

Figure 6: Calculated bandgap trends as a function of the number of heptazine
monomers in hydrogen-terminated ribbon models of increasing width. Band gaps
of key 2D structures are given in light red for reference (see Figure 5). The
ribbon models neglect the influence of corrugation, see text for more details.
As the number of heptazine units increases from 2 to 7, the bandgap decreases
from 3.6 eV to 2.9 eV, with the most significant decrease observed already
from 2 to 3 heptazine units (ca. 0.6 eV).

Calculations show that bandgaps converge to a value close to that of the 100 %
DoC pg-C3N4-2D model for ribbons that are 5$-$7 heptazine units wide. Already
at a width of 3 heptazine units, a significant reduction of the bandgap from
ca. 3.6 to 3.0 eV is observed. This result suggests that already small areas
of g-C3N4 are sufficient to produce the bandgaps observed for the fully-
condensed 2D materials.

Note that corrugation is not taken into account in this test set and thus,
values are closer to the flat pg-C3N4-2D model rather than the corrugated
cg-C3N4-2D model. For larger ribbons (5$-$7 units), it is expected that
bandgap values would shift to larger values if corrugation was accounted for,
analoguous to the difference between cg- and pg-C3N4-2D.

###  3.4 The Influence of Hydrogen Bonding on the melon Structure

The melon-2D and melon-3D models are set apart from the other structures
because their individual strings interact via hydrogen bonds exclusively
rather than via any bridging covalent bonds. The investigated melon structures
exhibit high thermodynamic stability both in the literature47, 35 and our
calculations, as shown in Figure 4. However, the formation energy trends were
obtained by optimizing the atomic configurations at a temperature of 0 K; any
temperature dependence of the calculated Gibbs free energy values is accounted
for using the approximations described in the method section. Notably, it is
known from atomistic and experimental studies on the structure of water that
hydrogen bond lengths elongate quadratically as a function of increasing
temperature.72 The calculations presented here therefore likely underestimate
the separation distance between the hydrogen-bonded melon strings and
overestimate their stability at ambient temperature and particularly at
elevated temperature during synthesis.

To account for this shortcoming, another melon-based periodic model is
investigated: the single-stringed model, s-melon-2D, as presented in Figure 2.
This structure constitutes an extreme case, i.e. where melon strings are
separated far enough that no more interaction via hydrogen bonds takes place
between them. This model is therefore likely more representative for the
string separation distance at high temperatures, such as during synthesis.
Figure 7 compares the $T$-$p$-dependent stability of the favourable PHI and
cg-C3N4 structures with the s-melon structure instead of the hydrogen-bonded
melon-2D structure shown in Figure 4.

Figure 7: Formation free energy phase diagrams of 2D and 3D structures updated
from Figure 4. The hydrogen-bonding stabilized melon-2D/-3D structure are
excluded, instead the single-string s-melon-2D/-3D structures are shown. The
dotted and the dashed line indicated the experimentally observed temperature
where formation of 2D and 3D structures begins27, 70, 18 and where the
melon-3D structure was found to form,14 respectively.

By comparing Figure 7 and Figure 4, it becomes clear that hydrogen bonding
significantly contributes to the stability of the melon models. The exclusion
of this contribution shifts the stability region of s-melon-2D structure in
Figure 7 a) to a lower temperature and the PHI-2D structure now appears to be
the most favourable configuration at reported experimental synthetic
conditions as indicated by the dotted line.27, 70, 18 In case of the 3D
models, Figure 7 (b) shows that now, the cg-C3N4-3D and PHI-3D structural
motifs are in thermodynamic competition, which means that a mixture of both
structure can be expected to form under typical synthetic temperature
conditions (500–630 ∘C).

##  4 Discussion

The purpose of this study is to elucidate the relationship between the
structure of PCN materials and the corresponding optical properties from a
thermodynamic point of view and as a function of material dimensionality. In
the following, three key structural parameters that greatly affect the optical
properties will be discussed in more detail based on the presented calculation
results: (i) degree of condensation, (ii) stacking, and (iii) corrugation.

###  4.1 Correlation between Degree of Condensation and Synthetic Conditions

This section discusses in detail how different synthetic conditions can
facilitate different degrees of condensation (DoC) of the synthesized
material, according to the calculated formation energy trends. Figure 8
summarizes the formation free energy trends for the most stable structures
presented in Figures 3, 4, and 7. An unfiltered overview containing all
structures is shown in Figure S5. In this depiction, emphasis is placed on
highlighting the change of trends of the thermodynamically preferred structure
as a function of $\Delta\mu_{\text{\ce{NH3}}}$, i.e. at various $T$ and $p$
conditions.

Figure 8: Free energy of formation for key 2D and 3D structures as a function
of a selected set of $\mu_{\text{\ce{NH3}}}$ conditions. The free energy of
formation is normalized to eV/atom to be able to compare structures with
different total number of atoms.

The formation free energy, $\Delta G_{\text{F}}$, is calculated for each
structure from NH3 and melamine as the educts. The middle-ranged cyan, black,
yellow, blue, and orange lines represent the $T$-$p$ parameter space that can
be achieved under typical laboratory conditions.

From the formation energy trends, it is evident that the DoC is proportional
to $\Delta\mu_{\text{\ce{NH3}}}$; structures with a higher DoC become more
favourable as $\Delta\mu_{\text{\ce{NH3}}}$ becomes more negative. In the
aforementioned middle range (blue to cyan lines), all structure show similar
thermodynamic stability. We can therefore conclude that from a thermodynamic
perspective, the experimentally synthesized material most probably is a
mixture of these different structural motifs with no strong preference for any
one particular motif.

In addition, the results suggest that certain structural motifs may become
dominant when specific synthetic conditions are enforced. On one extreme end
of the $T$-$p$ spectrum, we predict that the less-condensed melon string
structures can be enforced by performing the synthesis at low temperature
conditions and high $p_{\text{\ce{NH3}}}$, for example in a closed reaction
vessel (red and green lines). On the other extreme end, if the synthesis is
performed at high temperature and low $p_{\ce{NH3}}$, for example in a
reaction vessel with an applied vacuum (magenta line), the more condensed
cg-C3N4 and PHI structures become favourable. While these extreme cases are
not representative of conditions that can be readily achieved in the
laboratory, they serve to illustrate the influence of the parameters of
temperature and pressure on the expected composition of the PCN material.

Therefore, in order to obtain a PCN with a high DoC and a optical gap of 3 eV
or less, these computational trends suggest that high temperature and low
$p_{\ce{NH3}}$ conditions should be maintained during the synthesis. These
calculated trends are corroborated by experiments by Inoki et al.73, who
reported successful synthesis of a g-C3N4 structure under controlled low
$p_{\ce{NH3}}$ (ca. $9.0~{}\times~{}10^{3}$ Pa) at 550 ∘C. Their synthesized
material shows a bandgap of 2.03 eV with an average crystal size of less than
3 nm. On the other hand, Lotsch et al. showed that formation of the melon
structural motif can be promoted by conducting synthesis in a closed ampule at
630 ∘C.14 Further condensation of the melon strings is likely hindered by the
relatively high NH3 $p_{\ce{NH3}}$ within the ampule. Both of these
experimental observations are in good agreement with the predicted trends
presented in this work (Figure 4(d)).

Finally, the presented results may help to reconcile conflicting observations
from theory and experiments. While published theoretical results predict that
high DoC is a requirement to achieve the desired bandgap, experiments measure
bandgaps of 2.6–2.7 eV for materials where elemental analysis shows high H
content that would indicate a low DoC.22, 74, 18 According to the presented
trends, the synthesized material likely contains a mixture of different more
or less condensed structural motifs, giving rise to desirable optical
properties in g-C3N4-like domains embedded in a matrix of less-condensed PCN
such as the melon string motif. Consequently, this conclusion raises a follow-
up question: how large do these highly-condensed domains need to be in order
to exhibit the desired properties? This question will be addressed in section
4.3.

###  4.2 Interconversion

It is clear from the presented computational trends that there is
thermodynamic competition between the melon-3D structure and the idealized,
fully-condensed cg-C3N4-3D structure for the end product. Consequently, the
question arises if and how interconversion can take place between these
structure. More specifically, in this section we discuss whether a
thermodynamic pathway exists to convert these phases into each other, for
example by annealing the synthesis product under specific conditions.

From a structural point of view, direct conversion from melon to the fully-
condensed structure is not straightforward (DoC = 50% for melon, DoC = 100%
for cg-C3N4). It is therefore reasonable to assume that the PHI structure (DoC
= 75%) may be an intermediate structure connecting them. In fact, the PHI
structure has already been successfully synthesized, indicating that this
structure is at least metastable, if not thermodynamically favourable under
specific conditions.75, 76, 77, 78, 79, 80 We therefore hypothesize that the
large pores in the PHI structure may be filled with heptazine monomers under
certain conditions to give the cg-C3N4 structure. Selected hypothetical
pathways and their associated reaction free energy trends at different $T$-$p$
conditions are summarized in Figure 9.

Figure 9: Reaction flow chart connecting structures via the calculated
reaction free energy, $\Delta G_{\text{r}}$, which depends on
$\Delta\mu_{\text{\ce{NH3}}}$ (-2.6 eV $<$ $\Delta\mu_{\text{\ce{NH3}}}$ $<$
-1.9 eV). Temperature is fixed to 628 ∘C, which constitutes the maximum
temperature before carbonization of the material sets in, so that the reaction
free energy trends depend solely on $p_{\ce{NH3}}$. Two reaction mechanisms
are suggested: i) The non-deamination pathway involving stacking and hydrogen
bonding and ii) the deamination pathway involving release of NH3. The
structures are classified by their degree of condensation (DoC). The ”pore-
filling” route is suggested to connect PHI structure to cg-C3N4 structure; see
text for more details.

More detailed $p$-$T$ phase diagrams from which these trends were summarized
are shown in Figure S6.

Here, the reaction free energy $\Delta G_{\text{r}}$ between any product and
an educt state is calculated as

$\displaystyle\Delta G_{\text{r}}(T,p)=n\ G^{\,\text{product}}(T,p)-m\
G^{\,\text{educt}}(T,p)\ -o\ G^{\text{\ce{NH3}}},$ (7)

where $G^{\,\text{product}}(T,p)$, $G^{\,\text{educt}}(T,p)$, and
$G^{\text{\ce{NH3}}}(T,p)$ constitute the Gibbs free energy of the product and
educt configurations as well as the NH3 molecule, respectively, and $n$, $m$,
and $o$ are stoichiometric factors.

The trends shown in Figure 9 were calculated at 628 ∘C, which is close to the
decompostion temperature of ca. 630 ∘C.70 The reaction profile can then be
controlled by $p_{\ce{NH3}}$. The proposed reaction profile conveys two
important thermodynamic factors that determine the end structure at the
respective conditions:

  1. 1.

The non-deamination pathway is thermodynamically more favourable than the
deamination pathway because stabilization occurs via hydrogen bonds and/or
stacking. However, this process does not increase the DoC.

  2. 2.

$p_{\ce{NH3}}$ plays a pivotal role in controlling the reaction free energy of
the deamination pathways, which increases the DoC.

From these trends, it is clear that the melon structure represents a
thermodynamic sink. The conversion from melon-3D to the idealized cg-C3N4-3D
structure only becomes exothermic at drastic conditions of $p_{\ce{NH3}}=10$
Pa. The same is true for the deamination pathway leading from s-melon-3D to
PHI-2D. However, if the PHI-2D stage can be reached, which will likely be in
equilibrium with the melon-2D structure due to similar formation free energy
trends from s-melon-2D, the subsequent stacking to PHI-3D and pore filling
reaction to cg-C3N4-3D, which already becomes spontaneous close to atmospheric
pressure, are significantly downhill.

To develop a conversion strategy, it is crucial to determine from which point
in this flowchart one starts. For example, if synthesis of a PHI material has
already succeeded, for example with the help of templating effects,78, 77
further condensation due to pore filling is possible at comparatively mild
conditions. However, converting a melon-3D material into cg-C3N4-3D may
require drastic conditions. When synthesis starts from the educts, a mixture
of melon-3D and more condensed material, as signified by the idealized
cg-C3N4-3D system, can be expected due to thermodynamic competition of the
s-melon-2D $\rightarrow$ s-melon-3D and the s-melon-2D $\rightarrow$ melon-2D
pathway.

###  4.3 Influence of Stacking vs. Lateral Condensation

The presented results show that stacking, i.e. the 2D $\rightarrow$ 3D
transition, is thermodynamically more favorable than condensation or pore
filling reactions. Furthermore, a central conclusion of sections 4.1 and 4.2
is that the synthesized material is most likely a mixture of structures,
particularly of the melon-3D structure and partially pore-filled PHI-3D. As
mentioned at the end of section 4.1, we now face an important question: how
big do these highly-condensed domains, which are likely interspersed
throughout a host matrix of less strongly condensed melon- or PHI-type
structures, need to be in order for the material to exhibit the bandgaps
commonly measured in experiments?

In order to investigate this question, we performed a convergence study for
strongly condensed carbon nitride nanoribbons of increasing width, see Figure
6. Results show that the optical gap of a ribbon structure composed of 7 or
more heptazine units ($E_{\text{gap}}=2.85$ eV) has reached that of the fully-
condensed structures (cg-C3N4-2D: 2.99 eV, pg-C3N4-2D: 2.79 eV). This result
underlines that small domains of strongly-condensed PCN of ca. 3–4 nm2 can
already exhibit the desired optical properties. Even though our computational
predictions show that the material prefers to grow via stacking rather than
laterally, condensation of only 7 heptazine units should be readily possible.
This extrapolated minimum crystalline size is in good agreement with
experiments by Inoki et al., who calculated a bandgap of 2.06 eV for PCN
crystals of ca. 3 nm size.73

Furthermore, the average HOMO-LUMO gap of all studied tetramers was computed
to be 3.61 eV, which is comparable to that of the melon 2D structures (3.62 eV
and 3.68 for s-melon-2D and melon-2D, respectively; see Table S1). While the
absolute values of molecular HOMO-LUMO gaps and bandgaps from periodic
calculations are not directly comparable, this trend is still noteworthy. It
is also corroborated by a combined experimental$-$theoretical study by Li et
al., who report that the optical gap of heptazine oligomers rapidly converges
with size.27 Therefore, this result not only emphasizes again that a high DoC
is a key driving force for red shifting of the optical gap, but is also in a
good agreement with the ribbon model convergence study which shows that only
small domains of high condensation PCNs are required to produce a desirable
bandgap.

###  4.4 Influence of Corrugation

Corrugation of carbon nitride materials is a significant factor in terms of
thermodynamic stability and optical properties. Using GGA-DFT and MP2 methods,
Gracia and Kroll found that corrugation enhances the thermodynamic stability
of condensed carbon nitride structures by relieving the repulsive interaction
between adjacent lone-pair electrons of the nitrogen atoms.26 This conclusion
is in agreement with many other published studies81, 26, 34, 69 as well as the
simulation results presented here. For example, our calculations show that the
corrugated fully-condensed cg-C3N4-2D model is more stable than its fully flat
counterpart pg-C3N4-2D ($\Delta E=0.7$ eV/heptazine unit). In terms of optical
properties, corrugation increases the bandgap, see Table S1. For example, the
flattened pg-C3N4-3D structure has a 0.6 eV narrower bandgap compared to the
corrugated — and thermodynamically more stable — cg-C3N4-3D structure.

Out-of-plane corrugation of the material occurs in order to minimize repulsive
interactions between adjacent N atoms.26 Here, structural factors contributing
to corrugation are examined in more detail, namely the DoC, lateral
crystallite size, and stacking. To this end, we compare the calculated root-
mean-square deviations (RMSD) of the out-of-plane shift of atomic coordinates
for different 1D, 2D, and 3D structures, see Table S2.

First, lateral crystallite size is examined. The RMSD shift increases from
0.01 Å for the heptazine monomer (h1) up to 0.25 Å for the pentamer (h51).
This result suggests that larger structures corrugate more easily. Section 4.1
already established that the experimental material is most likely a mixture of
different structural motifs and section 4.3 established that already small
areas of g-C3N4 can exhibit bandgaps of 3 eV and less. In light of these
trends, the present results suggest that in order to achieve a desirable
bandgap, a material with only small connected patches of the same structural
motif, thereby avoiding corrugation, is desirable. A recent study by Mitoraj
et al. has used this principle to good effect.17 The group employed
functionalization with a small fraction of azo-linkers in the inner domain to
trigger local flattening of the material, thereby redshifting the absorption
spectrum.

The DoC can have an even more profound impact on corrugation compared to
crystallite size. For example, comparing RMSD shifts for the melon-2D (0.00
Å), PHI-2D (0.57 Å), and cg-C3N4-2D (0.80 Å) structures in Table S2, it
becomes apparent that corrugation increases as the DoC increases, which
counteracts the trend of narrowing bandgaps with increasing DoC (see Table
S1). This result once again emphasizes that if the goal is to obtain a PCN
material with low bandgap, connected areas of the same structural motif should
be kept small since corrugation becomes more pronounced as these areas
increase in size.

Finally, stacking appears to inconsistently affect the RMSD shift of the
atomic coordinates. For example, the out-of-plane RMSD shifts of atoms in the
2D and 3D cg-C3N4 models are close to identical (see Table S2). The bandgaps
of these models are close to identical, see Table S1. On the other hand, the
bandgap of the fully-flattened pg-C3N4 model is reduced by 0.44 eV due to
stacking. Going from PHI-2D to PHI-3D, the extent of corrugation decreases,
thereby slightly reducing the bandgap. This result indicates that at least
based on the present data set, there is no obvious systematic trend connecting
stacking and corrugation. Therefore, any new model system requires detailed
investigation in this regard.

To summarize, corrugation has a profound impact on the stability and
electronic structure of carbon nitride materials. In calculations, corrugation
is found to stabilize structures thermodynamically and to increase the
bandgap. The latter point is two-pronged: corrugation weakens the $\pi$-$\pi$
interactions in the lateral direction but also weakens interlayer interactions
perpendicular to the layers. From these results, we hypothesize that if the
goal is to achieve a material with a small bandgap, synthetic conditions
should be optimized to yield a PCN consisting of small domains of the same
structural motif since corrugation becomes more pronounced with increasing
domain size. As discussed in section 4.3, small domains of g-C3N4 material
interspersed in a host matrix of less-condensed material are sufficient to
obtain a desirable bandgap.

###  4.5 Postulation of a More Complete Computational Carbon Nitride Model

In this final section, we aim to combine the insights gained throughout this
study to postulate a more complete model for the computational investigation
of PCNs. While the experimental material is arguably too complex to be treated
with any one separate model, we are convinced that there is nevertheless great
value in taking steps towards representing as many of the intricacies of this
fascinating material class as possible in simulations.

The new model, herein referred to as the ”mixed” model, should satisfy the
following conditions:

  1. 1.

The melon structure (50% DoC) should be represented in the model as it
constitutes one of the thermodynamically most stable structures found in the
present study.

  2. 2.

The PHI model (75% DoC) is another thermodynamically stable candidate
structure that warrants representation, also because successful synthesis of
this motif has been reported using a templating method with salt melts.28, 82,
80

  3. 3.

From bandgap results in this study, only the fully condensed cg- and pg-C3N4
structures can sufficiently explain the experimental bandgap. The planarized
pg-C3N4-3D structure gives a bandgap as low as 2.35 eV. Calculated free energy
trends suggest that these structures form under low $p_{\text{NH3}}$
conditions. This 100% DoC structure should therefore be represented as well.

  4. 4.

Results on the influence of corrugation as well as the presented lateral size
vs. bandgap data show that small domains of g-C3N4 with low corrugation are
sufficient to achieve a narrower bandgap. The model size, therefore, can
remain overall small, with rapid alternations between the above structural
motifs.

The resulting model as well as a comparison between its DoC and optical
properties with those of its three parent structures is illustrated in Figure
10.

Figure 10: (a) Illustration of the postulated ”mixed” PCN model as well as the
parent structures it is composed of. (b) Calculated bandgaps for the mixed
structural motif as well as for the parent structures as a function of the
DoC. The oval shape indicates the range of DoCs and bandgaps obtained for the
four tested mixed models; the yellow star indicates the most promising model
that is also depicted in (a).

Shown here is only the most promising mixed model with 57% DoC. Three other
structures with DoC up to 71 % were investigated as well and are shown in the
Supporting Information, see Figure S1 for illustrations of their structures
and Table S1 for the optical properties.

Because the fully-condensed structural motif (red hexagon) is confined in a
small domain, only mild corrugation occurs (0.573 Å vs. 0.804 Å for cg-C3N4)
and the resulting optical bandgap is ca. 2.9 eV, see Table S1. It is
noteworthy that the optical bandgap of the mixed structures is lower than that
of cg-C3N4-3D, likely due to the aforementioned encapsulation and local
flattening. This result confirms the initial hypothesis that extended
condensed structures are not necessary to achieve a desirable bandgap, and
highlights that mixed structures can easily give rise to low bandgaps.

To the best of our knowledge, this is the first time that a model system
capturing this many important facets of PCNs at once is presented in
literature. This study, however, is limited to the structure of the pure PCN
materials and does not take into account functional end groups such as –N–H
and –N–CN which may be able to further adjust the local electronic
structure.83 The influence of functional groups on electronic properties and
reactivity will be explored in the future using this model system as basis.

##  5 Conclusion

The thermodynamic stability and the optical properties of heptazine-based
carbon nitride materials were investigated using accurate hybrid density
functional theory calculations. Three structural parameters were identified
that strongly affect the optical properties:

  1. 1.

degree of condensation (DoC; decreases bandgap and decreases stability),

  2. 2.

stacking (decreases bandgap and increases stability),

  3. 3.

and corrugation (increases bandgap and increases stability).

Materials with a high DoC also show pronounced corrugation, which presents a
dilemma if material optimization towards narrow bandgaps is the goal.
Materials with high DoC and pronounced corrugation form at conditions where
the chemical potential of NH3, $\mu_{\ce{NH3}}$, is strongly negative, i.e. at
high temperature and low $p_{\ce{NH3}}$. The melon string structure is
preferred at less negative $\mu_{\ce{NH3}}$ but exhibits a large bandgap. We
identified interconversion pathways starting from the melon string structure
towards more condensed structures. However, from the relatively small
differences in thermochemical stability between these structures, it is
expected that materials synthesized under typical experimental conditions
consist of a mixture of more and less condensed structural motifs. The
thermodynamic preference for stacking over lateral condensation likely further
discourages formation of extended areas of the same motif.

From the calculated trends, we postulate a complex computational model that
combines features of the melon string, poly(heptazine imide) (PHI), and g-C3N4
model systems. This mixed structure shows a similar thermodynamic stability
region as its parent molecules and exhibits, despite its lower DoC of 57%, a
narrower bandgap of 2.9 eV compared to the fully condensed but strongly
corrugated cg-C3N4-3D model. The bandgap is likely a result of local
flattening of the small, encapsulated, strongly-condensed motif featured in
the model.

The present study therefore illustrates that the experimentally well
established favorable optical properties of PCNs can emerge from comparatively
small g-C3N4 domains embedded in a framework of thermodynamically more stable
and less condensed motifs like the melon string or PHI structures. In future
studies, the presented computational model will be used to explore the effects
of functionalization and templating ions on the stability, catalytic activity,
and optical properties of PCNs.

This work was funded by the Deutsche Forschungsgemeinschaft (DFG — German
Research Foundation) through TRR 234 CataLight (project no. 364549901) as well
as JA 1972/27-1 (project no. 428764269). Computational resources were provided
by the state of Baden-Württemberg through bwHPC and the German Research
Foundation (DFG) under Grant No. INST 40/467-1 FUGG. C.I. acknowledges the
German Academic Exchange Service (DAAD, Ref. No. 91676720). I.K. acknowledges
the support of the Alexander von Humboldt Foundation through the Humboldt
Research Fellowship.

##  6 Supporting information

Figure S1: Proposed structures of the mixed motifs used in the study. The structures are composed of the PHI and cg-C3N4 structure because of their high DoC (79%–94%). The structures of Mixed1-3D and Mixed2-3D have the edge boundaries which reduce the corrugation. The both Mixed1-3D and Mixed2-3D are distinct from the sequence of the structural motifs. The Mixed3-3D is made of inclusion of PHI structure within cg-C3N4 domain to both layers and the Mixed4-3D is constructed of cg-C3N4 layer (upper) and PHI layer (bottom), respectively. Figure S2: The evaluation of the calculated chemical potential of ammonia used in this study. The reference plots are taken from the NIST-JANAF84, 85 and the literature.86, 47 The dotted points ($\ce{NH3}$ NIST-JANAF and $\ce{NH3}$ exp. data) and small dotted lines ($\ce{NH3}$ ideal gas) are taken from the experimental and the literature, respectively. The bold dashed line illustrates the calculated $\Delta\mu_{\text{\ce{NH3}}}(\text{T},p)$ in this study. Then we added two $p_{\ce{NH3}}$ conditions of 0.001 MP and 50 MP to further trace the variations. Figure S3: The contour plot of $\Delta\mu_{\text{\ce{\ce{NH3}}}}$. The plot is derived from the equation (4). The black points is sampled the respective $\Delta\mu_{\text{\ce{\ce{NH3}}}}$. (See Figure 8 of red, green, blue, and purple. Figure S4: The calculated phase diagram of the normalized free energy of formation between melon-3D and cg-C3N4-3D. The inclusion of the vibrational contributions increases the relative stability of of cg-C3N4-3D (blue area) to $\Delta\mu_{\text{\ce{\ce{NH3}}}}$ direction. The phase diagram of melon structure from the literature is shown in gray area.47 Figure S5: Free energy of formation for 2D and 3D structures with regards to various $\mu_{\text{\ce{NH3}}}$. The $\mu_{\text{\ce{NH3}}}$ is differentiated from the combination of $T$ and $p_{\ce{NH3}}$. The free energy of formation is normalized to eV/atom due to the different numbers of elements. Figure S6: Respective phase diagram of the free energy reaction for 2D and 3D structures. The contour map is used to find spontaneous $\Delta\mu_{\text{\ce{\ce{NH3}}}}$ for temperature and $p_{\ce{NH3}}$ conditions. Table S1: Summary of Optical property Structure | H-L gap (eV) | Structure | bandgap (eV)
---|---|---|---
h1 | 4.62 | s-melon-2D | 3.62
h2 | 3.98 | melon-2D | 3.68
h31 | 3.64 | PHI-2D | 3.32
h32 | 3.56 |  C9M13H3-2D | 2.95
h33 | 3.92 | cg-C3N4-2D | 2.99
h34 | 3.80 | pg-C3N4-2D | 2.79
h41 | 3.52 | s-melon-3D | 3.39
h42 | 3.71 | melon-3D | 3.12
h43 | 3.52 | PHI-3D | 3.18
h44 | 3.61 |  C9M13H3-3D | 2.86
h45 | 3.46 | cg-C3N4-3D | 2.95
h46 | 3.65 | pg-C3N4-3D | 2.35
h47 | 3.59 | Mixed1-3D | 2.95
h48 | 3.72 | Mixed2-3D | 2.88
|  | Mixed3-3D | 2.99
|  | Mixed4-3D | 2.89
Figure S7: The degree of condensation and RMSD value are plotted with 3D structures.The Mixed structures are composed of PHI and g-C3N4 structures (see Figure S1). The RMSD value of 3D structure is obtained from the average of the each layer. Table S2: Summary of Degree of Corrugation Structure | RMSD of | Structure | RMSD of
---|---|---|---
out-of-plane (Å) | out-of-plane (Å)
h1 | 0.01 | melon-3D | 0.00
h2 | 0.08 | PHI-3D | 0.33
h31 | 0.20 | cg-C3N4-3D | 0.80
h41 | 0.20 | pg-C3N4-3D | 0.00
h51 | 0.25 | Mixed1-3D | 0.57
Mixed1-2D | 0.58 | Mixed2-3D | 0.57
Mixed3-2D | 0.35 | Mixed3-3D | 0.31
PHI-2D | 0.57 | Mixed4-3D | 0.37
cg-C3N4-2D | 0.81 |  |

##  7 Reference

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# What can human minimal videos tell us about dynamic recognition models?

Guy Ben-Yosef1,2,5, Gabriel Kreiman2,3, Shimon Ullman2,4
1Computer Science and Artificial Intelligence Laboratory , Massachusetts
Institute of Technology, USA
2Center for Brains, Minds, and Machines, Massachusetts Institute of
Technology, USA
3Children’s Hospital, Harvard Medical School, USA
4Department of Computer Science and Applied Mathematics, Weizmann Institute of
Science, Israel
5GE Research, Artificial Intelligence, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In human vision objects and their parts can be visually recognized from purely
spatial or purely temporal information but the mechanisms integrating space
and time are poorly understood. Here we show that human visual recognition of
objects and actions can be achieved by efficiently combining spatial and
motion cues in configurations where each source on its own is insufficient for
recognition. This analysis is obtained by identifying minimal videos: these
are short and tiny video clips in which objects, parts, and actions can be
reliably recognized, but any reduction in either space or time makes them
unrecognizable. State-of-the-art deep networks for dynamic visual recognition
cannot replicate human behavior in these configurations. This gap between
humans and machines points to critical mechanisms in human dynamic vision that
are lacking in current models.

##  1 The role of temporal information in dynamic visual recognition

Previous behavioral work has shown that human visual recognition can be
achieved on the basis of spatial information alone Potter & Levy (1969);
Ullman et al. (2016), and on the basis of motion information alone, as in the
case of identifying human activities from biological motion Johansson (1973).
At the neurophysiological level, neurons have been identified that respond
selectively to objects and events based on purely spatial information, or
motion information alone Sáry et al. (1993); Perrett et al. (1985). However,
several behavioral studies have also provided strong support suggesting that a
combination of spatial and temporal information can aid recognition, e.g.,
Parks (1965); Morgan et al. (1982), but whether and how space and time may be
integrated remain unclear.

One of the domains in which temporal information is particularly relevant is
action recognition. While several approaches and computational models have
been developed to recognize actions from videos – and recent models are based
on late or early integration of the spatial and temporal information processed
by deep networks (e.g., Giese & Poggio (2003); Tran et al. (2015); Donahue et
al. (2015); Karpathy et al. (2014); Simonyan & Zisserman (2014); Hara et al.
(2018)) – it remains unclear whether current models make an adequate and
human-like use of spatiotemporal information. In order to evaluate the use of
spatiotemporal integration by computational models, it is crucial to construct
test stimuli that ‘stress test’ the combination of spatial and dynamic
features. A difficulty with current efforts is that in many action recognition
data sets (e.g. Soomro et al. (2012); Kay et al. (2017)) high performance can
be achieved by considering purely spatial information, and so they are not
ideally set up to rigorously test spatiotemporal integration.

Here we developed a set of stimuli that can directly test the synergistic
interactions of dynamic and spatial information, to identify spatiotemporal
features that are critical for visual recognition and to evaluate current deep
network architectures on these novel stimuli. We tested minimal spatiotemporal
configurations (also referred to below as minimal videos), which are composed
of a set of sequential frames (i.e., a video clip), in which humans can
recognize an object and an action, but where further small reductions in
either the spatial dimension (i.e., reduction by cropping or down sampling of
one or more frames) or in the time dimension (i.e., removal of one or more
frames from the video) turns the configuration unrecognizable, and therefore
also uninterpretable, for humans.

##  2 A search for minimal videos

The search for each minimal video started from a short video clip, taken from
the UCF101 dataset Soomro et al. (2012), in which humans could recognize a
human-object interaction. The search included different video snippets from
various human-object interaction categories (e.g., ‘a person rowing’, ‘a
person playing violin’, ‘a person mopping’, etc). The original video snippets
were reduced to a manually selected 50x50 pixel square region, cropped from 2
to 5 sequential non-consecutive frames, and taken at the same positions on
each frame. These regions served as the starting configurations in the search
for minimal video configurations described below. In the default condition,
frames were presented dynamically in a loop at a fixed frame rate of 2Hz. An
illustration of the starting configuration for one of these examples is shown
in Fig. 1A (We highly encourage the reader to view the dynamic version of Fig.
1 that is attached to this submission). The starting configuration was then
gradually reduced in small steps of 20% in size and resolution (same procedure
as in a previous study Ullman et al. (2016)). At each step, we created reduced
versions of the current configuration, namely five spatially reduced versions
decreasing in size and resolution, as well as temporally reduced versions
where a single frame was removed from the video configurations. Each reduced
version was then sent to Amazon’s Mechanical Turk (MTurk), where 30 human
subjects were asked to freely describe the object and action. MTurk workers
tested on a particular video configuration were not tested on additional
configurations from the same action type. The success rates in recognizing the
object and the action were recorded for each example. We defined a video
configuration as recognizable if more than 50% of the subjects described both
the object and the action correctly.

The search continued recursively for the recognizable reduced versions, until
it reached a video configuration that was recognizable, but all of its reduced
versions (in either space or time) were unrecognizable. We refer to such a
configuration as a ‘minimal video’. An example of a minimal video is shown in
Fig. 1B, and the reduced sub-minimal versions are shown in Fig. 1C-I. As shown
in Fig. 1B-D, the spatial content of the minimal and temporal sub-minimal
configurations was very similar: there were only minor spatial content added
from frame 1 to frame 2. Despite the similarity of the two frames, there was a
large difference in human recognition due to the removal of the motion signal.
As shown in Fig. 1E-H, in the tested cases the motion content of the minimal
and (spatial) sub-minimal is very similar, that is, the pixels that are
cropped out do not remove a large amount of image motion. This implies that
the motion signal alone is not a sufficient condition for human recognition of
minimal videos.

A prominent characteristic of minimal video configurations was a clear and
consistent gap in recognition between the minimal configurations and their
sub-minimal versions. The mean recognition rate was $0.71\pm 0.11(mean\pm SD)$
for 20 minimal video configurations (such as the one in Fig. 1B), $0.29\pm
0.15$ for the spatial sub-minimal configurations (such as the ones in Fig.
1E-I), and ($0.16\pm 0.14$) for the temporal sub-minimal configurations (such
as the ones in Fig. 1C-D). The difference in recognition rates between the
minimal and sub-minimal configurations were statistically highly significant.
The minimal videos included 2 frames of n x n pixels, where $n=20\pm 7.1$ on
average. Although highly reduced in size, the recognition rate for the minimal
videos was high, and was not too far from the recognition rate of the original
UCF101 video clips (mean recognition = $0.94\pm 0.07$), even though the
original clips had an average of 175 frames (versus 2 frames), each with
320x240 colored RGB pixels. Recognition rates for the minimal videos were also
close to the recognition rates for the level above it in the search tree (the
‘super minimal video configuration’, mean recognition = $0.81\pm 0.07$).

Figure 1:  Example of a minimal video. A short initial video clip showing
‘mopping’ activity (A) was gradually reduced in both space and time to a
minimal recognizable video configuration (B). The numbers on the bottom of
each image show the fraction of subjects who correctly recognized the action
(each subject saw only one of these images). The spatial and temporal trimming
was repeated until none of the spatially reduced versions (E-I, solid
connections) or temporally reduced versions (C,D, dashed connections) reached
the recognition criterion of 50$\%$ correct answers. Spatial reduced versions:
In E each frame was cropped in the top-right corner, leaving 80$\%$ of the
original pixel size in B. F,G,H are similar versions where the crop is on the
top-left, bottom-right, and bottom-left corners, respectively, I is a version
where the resolution of each frame was reduced to 80$\%$ of the frame in B.
Temporal reduced versions: A single frame was removed, resulting in static
frame#1 in C, and static frame#2 in D. Follow this link for the animated
version of this figure.

##  3 Convolutional network models for recognizing minimal videos

To further understand the mechanisms of spatiotemporal integration in
recognition, we tested current models of spatiotemporal recognition on our set
of minimal videos, and compared their recognition performance to human
recognition. Our working hypothesis was that minimal videos require
integrating spatial and dynamic features, which are not used by current
models. The tested models included the C3D model Tran et al. (2015); Hara et
al. (2018), the two-stream network model Simonyan & Zisserman (2014), and the
RNN-based model Donahue et al. (2015), three popular network deep models that
represent three different modern approaches to spatiotemporal recognition. Our
computational experiments included three types of tests with increasing amount
of specific training, to compare human visual spatiotemporal recognition with
existing models.

Testing pre-trained DNNs on minimal videos: In the first test, models were
pre-trained on the UCF-101 dataset for video classification. We tested such
pre-trained models on our set of minimal videos, to explore their capability
to generalize from real-world video clips to minimal configurations.
Classification accuracy by the C3D model for minimal videos was significantly
lower than the classification accuracy achieved for the original full video
clips, from which we cropped the minimal videos. As example for a typical
minimal video, 75% of humans could correctly identified the action while the
correct answer was not even among the top 10 for the network model.

Testing fine-tuned DNNs on minimal video: Next, we evaluated whether training
the models with minimal videos (fine-tuning) could help improve their
performance. We used a binary classifier based on the convolutional 3D network
model Tran et al. (2015); Hara et al. (2018), which was pre-trained on the
large SportM dataset Karpathy et al. (2014). The network was then fine-tuned
on a training set including 25 positive examples similar to a minimal video
from a single category and type (the ‘rowing’ minimal video, see examples in
Supplementary Fig. 2A, all positive examples were validated as recognizable to
humans), as well as 10000 negative examples (e.g., Supplementary Fig. 2B).
Data balancing techniques were used to ensure that the results would not be
biased by the imbalance between positive and negative examples. The binary
classifier was then tested on a novel set of positive and negative examples
similar to the ones used during training. Since our set of positive examples
was constrained to specific body parts and specific viewing positions in
‘rowing’ video clips, the fine-tuned classifier was able to correctly classify
most of the negative test examples; the Average Precision (AP) was 0.94.
Still, a non-negligible set of negative examples was given high positive score
by the fine-tuned model, from which we composed a new set that we refer to as
‘hard negative video configurations’ for further analysis. The hard negative
configurations included 30 negative examples that were erroneously labeled as
positive by the fine-tuned network model. Comparing accuracy of human and
network recognition for the set of hard negative configurations further
revealed a significant gap: humans were not confused by any of the hard
negative examples, while the fine-tuned network scored the hard negatives
higher than most positive examples.

Testing DNNs on minimal vs. sub-minimal videos: A distinctive property of
recognition at the minimal level is the sharp gap between minimal and sub-
minimal videos. We therefore further compared recognition by the binary CNN
classifier and human recognition by testing whether the network model was able
to reproduce the gap in human recognition between the minimal configurations
and their spatial and temporal sub-minimal ones. For this purpose, we
collected a set of minimal and sub-minimal videos showing a large gap in human
recognition, which did not overlap with the training set for the network
model. We tested the fine-tuned network model from above on a set containing
20 minimal videos, 20 temporal sub-minimal configurations and 20 spatial sub-
minimal configurations, all from the same category of ‘rowing’ in a similar
viewing position and size. Although this network model performed well on
normal action classification (0.94 AP, see above) it was not able to replicate
human recognition performance over this test set. While there was a clear gap
in human recognition between minimal and spatial sub-minimal videos (average
gap in human recognition rate 0.63), and between minimal and temporal sub-
minimal videos (average gap in human recognition rate 0.68), the differences
in recognition scores given by the network model for the minimal and sub-
minimal examples were significantly smaller. This discrepancy between human
behavior and the models persisted even after using standard data augmentation
techniques.

##  4 Spatiotemporal interpretation and future models

Subjects who could recognize the minimal videos could also identify internal
parts in them (e.g., body parts, action objects, etc.), and we hypothesized
that action recognition is accompanied by detailed interpretation of the image
parts as well as spatiotemporal relations between parts. To test this
conjecture we ran a new series of experiments where subjects were instructed
to describe internal components of the videos. MTurk subjects were presented
with the minimal videos, along with a probe pointing to one of its internal
spatial components. We defined a component as ’recognized’ if it was correctly
labeled by more than 50% of the subjects. Average recognition for 31
components in 5 different types of minimal videos was 0.77$\pm$0.17. For
example, in the “mopping” minimal video (Fig. 1) most of the subjects were
able to correctly label the arm, legs, stick and vacuum. To assess whether the
dynamic video configurations were necessary for interpretation, we repeated
the experiment using the spatial sub-minimal and temporal sub-minimal
versions, using the same procedure of inserting a probe in the frames. In
contrast to the reports obtained from the minimal videos, subjects
consistently struggled to recognize the parts in the sub-minimal videos.

We conclude that since minimal videos are limited in their amount of visual
information, and require efficient use of the existing spatial and dynamic
cues, comparing their recognition by humans and existing models uncovers
differences in the use of the available information. Future studies could
extend recent modeling of full interpretation of spatial minimal images Ben-
Yosef et al. (2018), to the modeling of detailed spatiotemporal
interpretation, including modeling of internal parts and their spatiormeporal
features and relations. The spatiotemporal features and relations can be
complex (e.g., a spatiotemporal relation of ‘parts containment’) to allow
human-like detailed spatiotemporal interpretation. Triggering these more
complex features will therefore be done in a selective, top-down manner, and
complementary to a first-stage forward aggregation of spatiotemporal filters (
Tran et al. (2015); Hara et al. (2018)). Focusing on a new type of models that
cannot only label a particular action, but can also perform spatiotemporal
interpretation will lead to a better understanding and more accurate modeling
of spatiotemporal integration and human recognition. Full report is in Ben-
Yosef et al. (2020). Code and data are available at
https://github.com/guybenyosef/introducing_minimal_videos.git

### Acknowledgments

GBY, GK, and SU were supported by the National Science Foundation, Science and
Technology Center Award CCF123121. SU was supported by EU Horizon 2020
Framework 785907 and ISF grant 320/16.

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##  Appendix A supplementary material

Figure 2: [supplementary] Positive and negative examples for training and
testing classification models. Positive examples are minimal videos from a
single specific category and specific viewing position, while negative
examples are non-category videos at the same style and resolution as minimal
videos.