ORIGINAL RESEARCH
Size-dependent electronic properties of nanomaterials: How this
novel class of nanodescriptors supposed to be calculated?
Karolina Jagiello
1
•
Bartłomiej Chomicz
1
•
Aggelos Avramopoulos
2
•
Agnieszka Gajewicz
1
•
Alicja Mikolajczyk
1
•
Pierre Bonifassi
1
•
Manthos G. Papadopoulos
2
•
Jerzy Leszczynski
3
•
Tomasz Puzyn
1
Received: 20 July 2016 / Accepted: 26 August 2016 / Published online: 3 September 2016
Ó The Author(s) 2016. This article is published with open access at Springerlink.com
Abstract In this study, the influence of the size on the
electronic properties (e.g. electronic energy) of three
nanometal oxides: ZnO, TiO
2
, and Al
2
O
3
were investi-
gated. The wurtzite, rutile and corundum type of clusters
were selected to represent ZnO, TiO
2
, and Al
2
O
3
, respec-
tively. To study the effect of the size on the property, we
have build several molecular cluster models with different
number of atoms and performed for those clusters quan-
tum–mechanical calculations. For small clusters, up to 40
atoms, the calculations at different levels of theory,
including: density functional theory (DFT), Hartree–Fock
method, and the semi-empirical PM6 method were carried
out. The results from ab initio and DFT calculations were
utilized to validate the less time-consuming PM6 approach.
The PM6 method was then employed for larger clusters.
Linear regression models were developed to describe the
relationships between size (number of atoms in cluste r) and
the electronic properties. The developed and validated
methodology is transferable and could be applied for other
type of nanosized clusters to calculat e properties that are
considered as potential nanodescriptors for nano-QSAR
modelling.
Keywords Nanodescriptors Electronic properties
Nanometre-sized metal oxides Nano-QSAR
Introduction
During the last 15 years, the number of studies devoting to
the investigation of the influence of size of nanoparticles
(NPs) on the biological response and their physical/chem-
ical properties has been significantly increased [
1–4]. The
main conclusion from those studies is that size of the NPs
is an important factor that determines both biological
effects of nanoparticles and their properties [
5, 6]. Other
parameters that have been proved to influence the activity
and properties of NPs include : shape, composition, surface
structure, and the ratio of the surface area to volume [
6].
Properties, for which the size influence was investigated,
mostly included the physical characteristic of NPs, such as:
their magnetism [
7], photodegradation efficiency [8], and
optical properties [
1]. However, since various computa-
tional approaches (e.g. quantitative structu re–activity
relationships and read-across approach) that utilized the
electronic properties, such as: the HOMO–LUMO gap
energy, the heat of formation, and the total energy are
currently more often applied to nanomaterials [
9–12], there
is an urgent investigate the impact of their size also on
those properties. There are few papers considering this
subject [
5, 13–16]. Authors of those studies noticed that
electronic properties can change with size according to two
main schemes: (1) increase/decrease linearly or (2)
increase/decrease nonlinearly up to the saturation points
[
5]. Since there is a relationship between size and the
Electronic supplementary material The online version of this
article (doi:
10.1007/s11224-016-0838-2) contains supplementary
material, which is available to authorized users.
& Tomasz Puzyn
t.puzyn@qsar.eu.org
1
Laboratory of Environmental Chemometrics, Faculty of
Chemistry, University of Gdansk, Wita Stwosza 63,
80-308 Gdan
´
sk, Poland
2
Institute of Biology, Pharmaceutical Chemistry and
Biotechnology, National Hellenic Research Foundation, Vas.
Constantinou Ave. 48, 11635 Athens, Greece
3
Interdisciplinary Nanotoxicity Center, Jackson State
University, 1400 Lynch St., Jackson, MS 39217, USA
123
Struct Chem (2017) 28:635–643
DOI 10.1007/s11224-016-0838-2
property at least for some characteristics, the question is
raised if there is possibility to describe this relationship by
means of chemometrics methods. Consequently, assuming
that there would be the possibility to develop the linear
relationships model applying the results computed for
series of small molecular clusters one could by means of
these models estimate the property of a given, larger size
nanoparticle. In this way the computing resources
employed and the time required for performing theoretical
studies for large clusters be significantly reduced.
The goal of our study is twofold: (1) first to provide a
methodology on how the size-dependent electronic prop-
erties for nanoclusters should be efficiently calculated and
(2) to develop and describe proper linear regression models
(LRs) allowing to predict electronic property for particular
size of the nanoparticle. We have selected three nanosized
metal oxides that received sign ificant interest, particularly
by their wide and still growing range of applications. The
selected oxides include: ZnO, TiO
2
, and Al
2
O
3
[17, 18].
Moreover, this selection allowed us to make the compar-
ison between size-depende nt electronic properties obtained
for metal oxides that differ in stoichiometry (different
metal oxidation numbers: Zn
2?
,Al
3?
,Ti
4?
) and symmetry.
The calculated properties could be utilized in nano-QSAR
models as a new class of nanodescriptors.
Methodology
Cluster construction
Based on the experimental crystal lattice parameters taken
from the literature, we have generated a series of molecular
clusters for three nanometre-sized metal oxides, ZnO,
TiO
2
, and Al
2
O
3
[19–21]. Appropriate size of clusters was
obtained by subsequently increasing the lattice parameters
in three dimensions. Consequently, supercells differing in
size, ranging from 12 to 99 atoms, from 30 to 39 atoms,
and from 10 to 40 for ZnO, TiO
2
, and Al
2
O
3,
respectively,
were constructed. This procedure was described and
applied in our previous studies [
5]. Molecular clusters were
generated by means of Mercury software packages [
22].
Quantum–mechanical calculations
The coordinates of the experimental crystal lattice param-
eters propagated in three dimensions were utilized as inputs
for quantum–mechanical calculations. For small clusters,
up to 40 atoms, single point-type cal culations were per-
formed by employing several theory levels including: (1)
semi-empirical Parameterized Model 6 (PM6) method
[
23]; (2) density functional theory (DFT) using B3LYP—
Becke, three-parameter Lee–Yang–Parr functional [24] and
M06—the hybrid Meta Density functional [
25] with the
following basis sets: 3-21??G** [
26] and aug-cc-pVDZ
[
27] and (3) Hartree–Fock method followed by Moller–
Plesset perturbation theory (MP2) [
28] in order to account
for electron correlation effects. Calculations of various
electronic properties for larger clusters were conducted
with single point-type semi-empir ical PM6 method. We
calculated prope rties that are proved to have application in
QSAR studies as descriptors [
29–31]. All calculated
properties are summarized in Table
1. Calculations were
performed by employing the MOPAC 2012 [
32] and
Gaussian [
33] software packages.
Relationships between size and properties
(chemometrics analysis)
We have used electronic properties calculated for various
sizes of molecular clusters to investigate the influence of
size (expressed in total number of atoms per cluster) on the
properties of nanosized metal oxides: ZnO, TiO
2
, and
Al
2
O
3
.
Validation of PM6 methods: analysis of small clusters data
Results obtained for small clusters (up to 40 atoms) were
applied to validate the less time-consuming method of
calculation: the semi-empirical PM6 method. In this case,
for each data set (data calculated at different theory levels:
PM6; DFT; HF), we developed a linear regression model
(LR model), according to the formula:
Y
X
P
¼ A
X
nAt þ B
X
ð1Þ
where Y
X_P
electronic property predicted at X level of
theory (X meaning PM6; DFT; HF; etc.); nAt size
expressed by total number of atoms per cluster; A
X
regression coefficient; B
X
intercept. To estimate the values
of A
X
and B
X
, we utilized the approach that minimizes the
sum of squared residuals [
35]. The goodness-of-fit of each
model was evaluated by calculating the determination
coefficient (R
2
)[36]. To estimate errors of the models, we
calculated for each cluster the relative errors, according to
Eq.
2:
% Error
jj
¼
ðY
X
Y
X
P
Þ
Y
X
100 % ð2Þ
where Y
X
property calculated at X level of theory; Y
X_P
property predicted at X level of theory.
The model developed for data calculated with the PM6
method was further applied to predict property at different
levels of theory, e.g. at DFT level, according to formula:
Y
Xvs:PM6
¼ a
Xvs:PM6
A
PM6
nAt þ b
Xvs:PM6
B
PM6
ð3Þ
636 Struct Chem (2017) 28:635–643
123
where Y
Xvs.PM6
re-calculated electronic property from PM6
method using X method; nAt size expressed in total
number of atoms per cluster; A
PM6
regression coefficient of
model developed for data calculated using PM6 method;
B
PM6
intercept for model developed for data calculated
using PM6 method; a
Xvs.PM6
and b
Xvs.PM6
correction
coefficient that are the ratios of equations’ coefficients
obtained in LR models (Eq.
1).
To statistically compare the calculated properties with
those predicted according to two strategies (Eqs.
1 and 3),
we have applied the pairwise Student’s t test (defined
pairwise are as follow: Y
X
/Y
X_P
; Y
X
/Y
Xvs.PM6
, where Y
X
property cal culated at X level of theory; Y
X_P
property
predicted at X level of theory; Y
Xvs.PM6
property re-calcu-
lated from PM6 to X level of theory).
Relationships between size of wurtzite-type clusters of ZnO
and the selected properties: the key study
The validated PM6 method was then utilized to investigate
the influence of size on the properties for clusters con-
taining more then 40 atoms. We applied this for wurtzite-
type clusters of ZnO. The relationships between size and
each particular property were investigated. In the case of
linear type of changes the LR models were developed and
evaluated according to the same protocol described for
small clusters.
The diagram that illus trates methodology applied is
presented in Fig.
1.
Results and discussion
Cluster construction
We studied the influence of size on the electronic proper-
ties of three nanometre-sized metal oxides, ZnO, TiO
2
, and
Al
2
O
3
. Molecular clusters various in size were constructed
for each oxide. In the case of zinc oxide, there are several
existing crystal structures: (1) wurtzite; (2) zinc blend and
(3) rocksalt [
37]. Rutile and anatase are two most common
and widely used polymorphs of TiO
2
[38, 39]. Taking into
account the thermodynamic stability, we considered wurt-
zite-type, rutile-type, and corundum-type clusters for ZnO,
TiO
2
, and Al
2
O
3
[20, 37, 39], respectively. Clusters
ranging from 12 to 99 atoms, from 30 to 39 atoms, and
from 10 to 40 atoms for ZnO, TiO
2
, and Al
2
O
3
, respec-
tively, were constructed. In order to obtain hexagonal-ZnO
structures, tetragonal-TiO
2
, and trigonal-Al
2
O
3
structures
of appropriate size, the crystal lattice parameters were
increased in all three dimensions. Examples of generated
clusters are shown in Fig.
2.
Quantum–mechanical calculations
The coordinates of the experimental crystal lattice param-
eters propagated in three dimensions were used as the
inputs for quantum–mechanical calculations. For small
clusters (up to 40 atoms), we performed calculations at
several theory levels: (1) semi-empirical, i.e. PM6 method;
(2) DFT method using B3LYP and M06 functionals; and
(3) ab initio methods.
Comparison of the results obtaine d for small-size clus-
ters (SI: Table S.1; Table S.2 and Table S.3) indicated that
there are no significant differences in DFT-based outcomes
produced with different functionals applied. More signifi-
cant differences are observed between resu lts obtained
from PM6 method and calculations performed by means of
other methods, which is reasonable according to the
Table 1 Symbols and
definitions of all calculated
molecular descriptors
Symbol Definitions of molecular descriptors Units Ref.
EE Electronic energy eV [
34]
TE Total energy eV [
34]
HOMO Energy of the highest occupied molecular orbital eV [
34]
LUMO Energy of the Lowest unoccupied molecular Orbital eV [
34]
Fig. 1 Proposed methodology diagram
Struct Chem (2017) 28:635–643 637
123
differences in each of the methods [40]. However, com-
parison of the Hartree–Fock method relying on the single-
electron approximation with the DFT technique that
includes electron correlation suggests that DFT produces
better estimates of molecule structural parameters [
40].
One can notice that results obtained at PM6 level differ
from DFT ones by approximately constant values, e.g.
TE
PM6
TiO
2
=TE
B3LYP=321þþG
TiO
2
¼ 0:025;
regardless of total number of atoms included in rutile-type
clusters of TiO
2
. It could suggest that calculations at PM6
level corrected by this constant value coul d be comparable
with those obtained by more sophisticated and computa-
tionally more expensive methods (at least with DFT ones).
Relationships between size and properties
(chemometrics analysis)
Validation of PM6 methods: analysis for small clusters
In the main text, we presented figures and tables related to
one property per metal oxide; other detail information
could be found in the supplementary materials.
Results obtained for small-size clusters, Table S.1,
Table S.2 and Table S.3 in SI, indicated that in this range
of sizes, total energy (TE) and electronic energy (EE)
changed linearly, regardless of method of calculations.
This is in agreement with our previous studies [
5]. In case
of HOMO and LUMO energy, the changes with increas-
ing number of atoms included in cluster are not linearly
size-dependent. Considering that, these properties will not
be analysed and further used in order to validate the PM6
computations. Consequently, linear regression models
were developed only for size changes of TE and EE,
according to Eq.
1; results are summarized in Table 2.
Determination coefficient (R
2
) of each model that is close
or even equal to 1 (see Table
2) indicates that there is
good fit of models to calculated data. Moreover, we have
also investigated the relative errors for each prediction
performed, calculated according to Eq.
2. Relative low
values of errors that in most cases do not exceed 10 %
confirm the good quality of the models (See Table S.4 in
SI).
However, it needs to be pointed out that electronic
properties for small-size clusters might be computed with
the approximate high error. It is related to their high
reactivity. Decreasing size of nanomat erials causes
increase in surface-to-edge ratio, which, in other words
means that atoms at the surface (which are scaled with n
2/3
,
where n is the number of atoms) are in close neighbour-
hood with the smaller number of atoms compared to macro
scale materials [
41, 42]. Thus, the small clusters of
nanoparticles have higher bond energy per atom that could
result in lower accuracy of calculations performed for
them. In this contribution, we took into account small
clusters, for which fraction of atoms included at the surface
is equal to about 30 % of all atoms, regardless of studied
cluster. This means that these small-sized clusters can be
scaled up to larger clusters, for which this ratio is more or
less similar (*30 %).
We applied here a model developed for data computed
at PM6 level of theory to predict properties for more
sophisticated methods, according to Eq.
3 (model devel-
oped for data computed with PM6 method corrected by
means of corrections coefficients). In order to validate PM6
method, we statistically compared the properties calculated
at DFT level (applying the following functional and basis:
B3LYP/3-21??G** and M06/3-21??G** for ZnO and
Al
2
O
3
and B3LYP/aug-cc-pVDZ and M06/aug-cc-pVDZ
for TiO
2
) with those predicted by means of the PM6-cor-
rected model (property calculated by means of Eq.
1 and
property calculated by means of Eq.
3). To do so, we
applied the pairwise Student’s t test. The calculated values
of p [ 0.05 indicate that the results from each of the
compared models do not differ significantly, Table
3.
Therefore, PM6-corrected model might be applied to
Al
1.98 Å
O
1.66 Å
1.94 Å
Ti
1.98 Å
O
1.99 Å
1.97 Å
Zn
O
(c)(b)(a)
Fig. 2 Examples of model
structures of a (Al
2
O
3
)
n
,
b (TiO
2
)
n
, and c (ZnO)
n
clusters, respectively
638 Struct Chem (2017) 28:635–643
123
predict total energy and electronic energy with good
approximation to those calculated at higher level of theory.
This conclusion was confirmed in relative residual
investigation performed for developed models, Fig.
3 and
Fig.
4. There are not significant differences between
residuals obtained with strategy 1 (%|Error|
X
, where X
means the method of calculation) and strategy 2 (%|Er-
ror|
Xvs.PM6
). Moreover, residuals obtained by employing
PM6-corrected method to re-calculate properties at higher
levels of theory are in many cases lower than residuals
obtained by employing model developed for particular
method o f calculations, e.g. relative errors obtain for total
energy computed for ZnO cluster containing 36 atoms are
as follows: %|Error|
B3LYP/3-21??G**
= 3.8 in comparison
to %|Error|
B3LYP/3-21??G**vs.PM6
= 2.9 ( Fig. 3 upper, left
panel). Thus, based on the computed results and consid-
ering the time and resources employed to perform calcu-
lations, PM6 method can be used for the qualitative
calculation of the selected elect ronic properties of the
studied metal oxides. Therefore, this method was applied
to calculate properties of larger clusters including more
than 40 atoms, for which fraction of atoms at the surface do
not exceed 30 % of all atoms.
Relationships between size of wurtzite-type clusters of ZnO
and the selected properties: the key study
The validated PM6 method was applied in order to study
the influence of size on the total energy and electronic
energy of wurtzite-type clusters of ZnO (Table S.5 in SI).
We have noticed that these properties of ZnO decreasing
linearly with increasing size of cluster, which is in agree-
ment with our previous studied [
5].
Additionally, we developed linear regression models for
the total energy, Fig.
5a, and the electronic property,
Fig.
5b, that allow predicting these properties based on the
size of the cluster, at least in the investigated range of size.
Developed models, see Fig.
5 and Table S.5 in SI, are
characterized by high determination coefficient and rela-
tively low errors.
Conclusions
A novel methodology that facilitates predictions of size-
dependent electronic properties of nanometre-sized metal
oxides was developed. The application of this methodol-
ogy was tested and confirmed for metal oxides containing
metals at different oxidation states (Zn
2?
,Al
3?
,Ti
4?
),
which means that they differ in stoichiometry and sym-
metry. This allows one to assume that it could be employed
for other oxides.
Table 2 Linear regression models used to predict total energy for clusters up to 40 atoms
Semi-empirical methods Density functional theory Ab initio
PM6 B3LYP/3-21??G** for ZnO and Al
2
O
3
B3LYP/
aug-cc-pVDZ for TiO
2
M06/3-21??G** for ZnO and Al
2
O
3
M06/
aug-cc-pVDZ for TiO
2
HF/3-21??G** for ZnO and Al
2
O
3
HF/aug-
cc-pVDZ for TiO
2
ZnO
Model TE
PM6
=-151.6nAt - 1.5
R
2
= 0.99
TE
B3LYP/3-21??G**
=-26,700.0nAt ? 2678.3
R
2
= 0.99
TE
M06/3-21??G**
=-26,699.0nAt ? 2679.0
R
2
= 0.99
TE
HF
=-29,661.0nAt ? 32,090.0
R
2
= 0.98
TiO
2
Model TE
PM6
=-8.04nAt - 0.23
R
2
= 1.0
TE
B3LYP/aug-cc-pVDZ
=-333.3nAt ? 0.32
R
2
= 1.0
TE
M06/aug-cc-pVDZ
=-333.3nAt ? 0.44
R
2
= 1.0
TE
HF
=-332.1nAt - 23.1
R
2
= 0.99
Al
2
O
3
Model TE
PM6
=-196.0nAt ? 29.1
R
2
= 0.99
TE
B3LYP/3-21??G**
=-3836.0nAt - 172.2
R
2
= 0.99
TE
M06/3-21??G**
=-3846.1nAt ? 12.78
R
2
= 1.0
TE
HF
=-3906.8nAt ? 273.9
R
2
= 0.99
Struct Chem (2017) 28:635–643 639
123