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This work investigates crystal lattice, electronic structure, relative stability, and high pressure behavior of TiO(2) polymorphs (anatase, rutile, and columbite) using the density functional theory (DFT) improved by an on-site Coulomb self-interaction potential (DFT+U). For the latter the effect of the U parameter value (0 < U < 10 eV) is analyzed within the local density approximation (LDA+U) and the generalized gradient approximation (GGA+U). Results are compared to those of conventional DFT and Heyd-Scuseria-Ernzehorf screened hybrid functional (HSE06). For the investigation of the individual polymorphs (crystal and electronic structures), the GGA+U/LDA+U method and the HSE06 functional are in better agreement with experiments compared to the conventional GGA or LDA. Within the DFT+U the reproduction of the experimental band-gap of rutile/anatase is achieved with a U value of 10/8 eV, whereas a better description of the crystal and electronic structures is obtained for U < 5 eV. Conventional GGA∕LDA and HSE06 fail to reproduce phase stability at ambient pressure, rendering the anatase form lower in energy than the rutile phase. The LDA+U excessively stabilizes the columbite form. The GGA+U method corrects these deficiencies; U values between 5 and 8 eV are required to get an energetic sequence consistent with experiments (E(rutile) < E(anatase) < E(columbite)). The computed phase stability under pressure within the GGA+U is also consistent with experimental results. The best agreement between experimental and computed transition pressures is reached for U ≈ 5 eV.
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calculations of crystal lattice, electronic structure, and phase
stability under pressure of TiO2polymorphs
M. E. Arroyo-de Dompablo,1,a) A. Morales-García,2and M. Taravillo2
1Departamento de Química Inorgánica, MALTA Consolider Team, Facultad de CC. Químicas,
Universidad Complutense de Madrid, 28040 Madrid, Spain
2Departamento de Química Física I, MALTA Consolider Team, Facultad de CC. Químicas,
Universidad Complutense de Madrid, 28040 Madrid, Spain
(Received 21 January 2011; accepted 8 July 2011; published online 1 August 2011)
This work investigates crystal lattice, electronic structure, relative stability, and high pressure behav-
ior of TiO2polymorphs (anatase, rutile, and columbite) using the density functional theory (DFT)
improved by an on-site Coulomb self-interaction potential (DFT+U). For the latter the effect of the
Uparameter value (0 <U<10 eV) is analyzed within the local density approximation (LDA+U)
and the generalized gradient approximation (GGA+U). Results are compared to those of conven-
tional DFT and Heyd-Scuseria-Ernzehorf screened hybrid functional (HSE06). For the investigation
of the individual polymorphs (crystal and electronic structures), the GGA+U/LDA+Umethod and
the HSE06 functional are in better agreement with experiments compared to the conventional GGA or
LDA. Within the DFT+Uthe reproduction of the experimental band-gap of rutile/anatase is achieved
with a Uvalue of 10/8 eV, whereas a better description of the crystal and electronic structures is ob-
tained for U<5 eV. Conventional GGA/LDA and HSE06 fail to reproduce phase stability at ambient
pressure, rendering the anatase form lower in energy than the rutile phase. The LDA+Uexcessively
stabilizes the columbite form. The GGA+Umethod corrects these deficiencies; Uvalues between
5 and 8 eV are required to get an energetic sequence consistent with experiments (Erutile <Eanatase
<Ecolumbite). The computed phase stability under pressure within the GGA+Uis also consistent with
experimental results. The best agreement between experimental and computed transition pressures is
reached for U5eV.© 2011 American Institute of Physics. [doi:10.1063/1.3617244]
The overestimation of electron delocalization is a known
drawback of density functional theory (DFT) methods, in
particular, for systems with localized d-electrons and f-
electrons.13The DFT+Umethod, developed in the 1990s,1,4
combines the high efficiency of DFT with an explicit treat-
ment of electronic correlation with a Hubbard-like model5for
a subset of states in the system. Non-integer or double occu-
pations of these states is penalized by the introduction of two
additional interaction terms, namely, the one-site Coulomb
interaction term Uand the exchange interaction term J. Af-
ter the initial success with the rock-salt type MO family (M
=Mn, Fe, Co, Ni),4,6the DFT+Uhas been extensively ap-
plied in the last years to investigate a wide variety of tran-
sition metal oxides where electron correlation results in a
strong electron localization. It has been shown that DFT+U
improves the capacities of DFT when dealing with energetic,
electronic, and magnetic properties of insulating materials
based on 3dtransition metals (see, for instance, Refs. 2and
7). Phase stability of that type of materials can also be suc-
cessfully reproduced within the DFT+U, improving the re-
sults of the conventional DFT (see, for instance, the cases of
MnO (Ref. 8) and FePO4(Ref. 9). However, the adequacy
of the DFT+Umethod to investigate early transition metal
a)Author to whom correspondence should be addressed. Electronic mail: Tel.: +34 91 3945222. FAX: +34 91 3944352.
compounds (Ti, V), where the more extended orbitals de-
crease the electron correlations, might be controversial.1015
Insulating materials such as TiO2provide an interesting
TiO2exhibits a large number of polymorphs as a func-
tion of pressure and temperature;16,17 rutile, anatase, brookite,
columbite, cotunnite, baddeleyite, and fluorite. This rich poly-
morphism, together with the interesting optical, electrical, and
mechanical properties of TiO2polymorphs, has originated a
large number of ab initio investigations (see, for instance,
Refs. 13,14,16, and 1821). However, reproducing the phase
stability of TiO2forms is a remaining challenge for ab initio
methods. Calculations performed at the Hartree-Fock (HF),
density functional theory (generalized gradient approxima-
tion (GGA) and local density approximation (LDA) func-
tionals), and hybrid functional (B3LYP, PBE0) levels have
been shown to produce wrong relative stabilities for anatase
and rutile polymorphs.17,19 Correlation effects have been
identified as a key factor to correctly reproduce the phase
stability.17,22 This context makes appealing to analyze the ad-
equacy of the DFT+Umethod to study TiO2polymorphs. The
DFT+Umethod has been recently utilized to investigate elec-
tron transport in rutile-TiO2,15,21 reduced forms of TiO2,13,23
and ultrathin films of rutile-TiO2.6,12 In this work we focus
on bulk phase stability and pressure driven transformations.
Additionally, to complete previous investigations of TiO2
using the B3LYP and PBE0 hybrid functionals,19 we
have also investigated the performance of the hybrid
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054503-2 Arroyo, Morales, and Taravillo J. Chem. Phys. 135, 054503 (2011)
Anatase [100] Columbite [001]Rutile [001]
FIG. 1. Schematic crystal structure of the TiO2polymorphs under investigation; anatase, rutile, and columbite.
Heyd-Scuseria-Ernzerhof functional (HSE06) (Refs. 2426)
recently implemented in the Vienna ab initio simulation pack-
age (VA SP ).
Figure 1shows the crystal structure of the three
polymorphs considered in this work, rutile, anatase, and
columbite, which are all based on octahedral TiO6units. In
the rutile structure (hcp packing of oxygen atoms), each tita-
nium octahedral shares two opposite edges along the caxis
and vertices in the ab plane. In the anatase structure (ccp
of oxygen atoms) the octahedra share four adjacent edges,
forming zig-zag chains running along the aand baxis. The
anatase and rutile varieties can be formed at ambient pres-
sure. Early calorimetry works already determined the ru-
tile form as the most stable one [H298 (anatase rutile)
=–11.7 kJ/mol (Ref. 27)], being the anatase a metastable
form, at any temperature.28 The possibility of particle’s mor-
phology affecting the thermodynamic stability of the TiO2
polymorphs has been investigated.2830 It is nowadays rec-
ognized that anatase is the stable nanophase of TiO2, that
is, anatase gets thermodynamically stabilized by small par-
ticle size.28 Among the TiO2polymorphs stabilized under
high pressure conditions (columbite, baddeleyite, and cotun-
nite) we choose the columbite form (orthorhombic α-PbO2),
in which oxygen atoms assume a distorted hexagonal close
packing configuration. The TiO6octahedra form edge-sharing
chains parallel to the caxis. Individual chains are connected
by corner-sharing to form a three-dimensional framework.
High pressure-high temperature treatment of either anatase or
rutile TiO2yields the columbite modification, which can be
quenched to ambient conditions.3133
One problem encountered using the DFT+Umethod
is the determination of an appropriate Uparameter value
for each compound. The values of Ucan be determined
through a linear response method which is fully consistent
with the definition of the DFT+UHamiltonian, making this
approach for the potential calculations fully ab initio.34 An
alternative route consists of selecting these values so as to
account for the experimental results of physical properties:
magnetic moments,35 band gaps,11 redox potentials,2,11 or
reaction enthalpies.36 Previous DFT +Uworks found a
value of U=10 eV for rutile TiO2by fitting calculated
band gaps to the experimental value.15,21 In this work we
discuss the suitability of the DFT+Umethod utilizing the
GGA+Uand the LDA+Ufunctionals to investigate TiO2
polymorphs and how the Uparameter (3 eV <U<10 eV)
affects various properties of TiO2; crystal lattice (Sec. III A),
electronic structure (Sec. III B), relative stability (Sec. III
C) and pressure driven phase transformations (Sec. III D).
Results are compared to the performance of the conventional
DFT and the hybrid HSE06 functionals. We will show
that the DFT+Umethod and the hybrid HSE06 functional
are suitable to investigate crystal structure and electrical
properties of TiO2polymorphs, improving the accuracy
of conventional DFT for band gaps and lattice parameters
prediction. Further, while the conventional DFT, the hybrid
functionals, and the LDA+Ufail to reproduce the phase
stability at both ambient and high pressure, proper results
are obtained within the GGA+Umethod. To the best of
our knowledge, it is the first time that the suitability of the
DFT+Uto investigate TiO2-polymorphism is discussed and
The total energy calculations and structure relaxations
were performed with the VASP.37,38 First, calculations
were done within the DFT framework with the exchange-
correlation energy approximated in the GGA and in the LDA,
utilizing two sets of potentials; ultrasoft-pseudopotential (PP)
and projector augmented wave (PAW).39 For the latter we
tested two different GGA-functionals, PBE (Ref. 40) and
PW91,41 which yielded very similar results. Our results uti-
lizing the PP are consistent with those previously reported;
in short, the method fails to reproduce the relative energy of
polymorphs. Thus, for conciseness, in this work we will only
discuss the PAW results.39
The LDA and the PBE form of the GGA exchange-
correlation functionals have been used together with their
LDA+Uand GGA+Uvariants as implemented in VASP.
The Ti(3p,3d,4s) and O (2s,2p) were treated as valence
states. Test calculations performed including the Ti-3sas va-
lence states yielded equivalent results. DFT+Ucalculations
were performed following the simplified rotationally invariant
form proposed by Dudarev.38,42 Within this approach, the on-
site Coulomb term Uand the exchange term J, can be grouped
together into a single effective parameter (U-J) and this effec-
tive parameter will be simply referred to as Uin this paper.
The Jvalue was fix to 1 eV. Effective Uvalues of U=3,
5, 8, and 10 eV were used for the Ti-3dstates. The energy
cut off for the plane wave basis set was kept fix at a con-
stant value of 600 eV throughout the calculations. The recip-
rocal space sampling was done with k-point Monckhorst-Pack
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054503-3 DFT+
calculations of TiO2polymorphs J. Chem. Phys. 135, 054503 (2011)
gridsof6×6×8 for rutile, 8 ×8×8 for anatase, and 10
×10 ×8 for columbite. As a first step the structures were
fully relaxed (cell parameters, volume, and atomic positions)
and the final energies of the optimized geometries were re-
calculated so as to correct the changes in the basis set of the
wave functions during relaxation. Second, from the relaxed
structure, within the GGA+Uapproximation, calculations
were performed at various constant volumes and the energy-
volume data were fitted to the Birch-Murnaghan equation of
16 B0[x2/31]3
+[x2/31]2[6 4x2/3],(1)
where x=(V0/V), being Vand V0the volume at pressure pand
the equilibrium volume at ambient pressure, respectively. B0
and B
0are the bulk modulus at ambient pressure and its pres-
sure derivative, respectively, and E0is the equilibrium energy.
Hybrid calculations were performed using the HSE06
functional, as implemented in VASP, with a screening param-
eter μ=0.2 Å1.The NKRED parameter was set to 2 mean-
ing that a HF kernel was evaluated on a coarser k-point grid.26
Full structure optimization was performed (cell parameters,
volume, and atomic positions). The optimized structures of
anatase and rutile were used as input for accurate static cal-
culations (the NKRED parameter was omitted, and a normal
grid was utilized in the HF calculation).
A. Crystal structure
Table Icompares the calculated lattice parameters and
volume for the fully relaxed structures of TiO2polymorphs
(HSE06, GGA/LDA , and GGA+U/LDA+Uwith U=3,
5, 8, and 10 eV) with the experimental ones. The unit cell
volume is overestimated in about 3% within the GGA. The
overestimation enlarges within the GGA+U, ranging from
TABLE I. Unit cell parameters (in Å) and volume per formula unit (in Å3)forTiO
2polymorphs obtained after complete structural optimization with HSE06,
conventional GGA/LDA, and GGA+U/LDA+Ucompared to experiments. Numbers in parentheses indicate the percent of deviation from experimental data.
Polymorph Method U(eV) V3) a(Å) b(Å) c(Å) (c/a)
ANATASE HSE06 34.07 (0) 3.766 (–0.50) 9.609 (1.02) 2.551 (1.51)
GGA 35.13 (3.11) 3.807 (0.58) 9.693 (1.90) 2.545 (1.27)
GGA+U3 36.15 (6.10) 3.853 (1.79) 9.742 (2.42) 2.529 (0.64)
GGA+U5 36.81 (8.04) 3.881 (2.53) 9.774 (2.76) 2.518 (0.20)
GGA+U8 37.81 (10.9) 3.922 (3.62) 9.830 (3.34) 2.506 (–0.28)
GGA+U10 38.46 (12.9) 3.947 (4.28) 9.871 (3.77) 2.501 (–0.48)
LDA 33.29 (–2.28) 3.74 (–1.19) 9.521 (0.09) 2.546 (1.31)
LDA+U3 34.24 (0.50) 3.793 (0.21) 9.52 (0.08) 2.51 (–0.12)
LDA+U5 34.84 (2.26) 3.819 (0.89) 9.555 (0.45) 2.502 (–0.44)
LDA+U8 35.82 (5.14) 3.86 (1.98) 9.617 (1.10) 2.491 (–0.87)
LDA+U10 36.50 (7.13) 3.882 (2.56) 9.687 (1.84) 2.495 (–0.72)
Experiment31 34.07 3.78512(8) 9.51185(13) 2.513
RUTILE HSE06 31.08 (–0.45) 4.590 (–0.08) 2.950 (–0.29) 0.642 (–0.31)
GGA 32.10 (2.82) 4.650 (1.22) 2.968 (0.32) 0.638 (–0.93)
GGA+U3 32.87 (5.28) 4.671 (1.68) 3.012 (1.80) 0.645 (0.16)
GGA+U5 33.41 (7.01) 4.687 (2.03) 3.042 (2.82) 0.649 (0.78)
GGA+U8 34.17 (9.45) 4.709 (2.51) 3.081 (4.13) 0.654 (1.55)
GGA+U10 34.69 (11.1) 4.725 (2.86) 3.108 (5.05) 0.658 (2.17)
LDA 30.32 (–2.88) 4.556 (–0.82) 2.922 (–1.24) 0.641 (–0.46)
LDA+U3 31.12 (–0.32) 4.580 (–0.30) 2.967 (0.28) 0.648 (0.62)
LDA+U5 31.68 (1.47) 4.599 (0.11) 2.995 (1.23) 0.651 (1.09)
LDA+U8 32.33 (3.55) 4.619 (0.55) 3.031 (2.45) 0.656 (1.86)
LDA+U10 32.92 (5.44) 4.637 (0.94) 3.062 (3.49) 0.66 (2.48)
Experiment (Refs. 31 and 63) 31.22 4.5938(1) 2.9586(1) 0.644
COLUMBITE HSE06 30.52 (–0.23) 4.539 (–0.04) 5.495 (0.04) 4.893 (–0.26) 1.077 (–0.28)
GGA 31.51 (3.00) 4.579 (0.84) 5.576 (1.51) 4.936 (0.61) 1.078 (–0.18)
GGA+U3 32.23 (5.36) 4.607 (1.45) 5.585 (1.67) 5.008 (2.08) 1.087 (0.65)
GGA+U5 32.77 (7.13) 4.629 (1.94) 5.603 (2.00) 5.053 (3.00) 1.092 (1.11)
GGA+U8 33.56 (9.71) 4.658 (2.58) 5.635 (2.59) 5.115 (4.26) 1.098 (1.67)
GGA+U10 34.09 (11.4) 4.674 (2.93) 5.664 (3.11) 5.151 (4.99) 1.102 (2.04)
LDA 29.78 (–2.65) 4.506 (–0.77) 5.450 (–0.78) 4.850 (–1.14) 1.076 (–0.37)
LDA+U3 30.53 (–0.20) 4.526 (–0.33) 5.477 (–0.33) 4.925 (0.38) 1.088 (0.74)
LDA+U5 31.05 (1.50) 4.542 (0.02) 5.501 (0.02) 4.971 (1.32) 1.094 (1.30)
LDA+U8 31.83 (4.05) 4.572 (0.68) 5.537 (0.68) 5.030 (2.53) 1.1 (1.85)
LDA+U10 32.35 (5.75) 4.591 (1.10) 5.563 (1.10) 5.066 (3.26) 1.103 (2.13)
Experiment (Ref. 31.) 30.59 4.541(6) 5.493(8) 4.906(9) 1.080
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054503-4 Arroyo, Morales, and Taravillo J. Chem. Phys. 135, 054503 (2011)
5% for U=3 eV to 12.5% for U=10 eV. Deviation of the
lattice parameters with respect to the experimental values is of
the order of 1.5% within the conventional GGA, and increases
within the GGA+Uas a function of U, reaching a maximum
difference at U=10 eV of around 4%. Within the conven-
tional LDA the unit cell volume and lattice parameters are
underestimated. The effect of the Hubbard Uis to expand the
unit cell, and a good reproduction of the experimental crystal
cell is achieved at Uvalues below 5 eV. The HSE06 underes-
timates the unit cell volume by a small amount of the order
of 0.3%. As it is generally found,25,26,44 the HSE06 predicts
more accurate cell parameters, with average deviations below
0.4%. The good performance of the hybrid functionals to re-
produce the crystal structure of TiO2polymorphs was pointed
out by Labat and co-workers, who found a deviation of lattice
parameters of 0.3% for rutile and 1.1% for anatase utilizing
the PBE0 hybrid functional.19
In order to evaluate the effect of the Uparameter in the
crystal structure prediction, it is more effective to analyze lat-
tice parameters ratios. The c/a ratio is a crucial parameter for
the rutile structural type, since changes in the c/a ratio depend
on the formation of metal-metal bonds45,46 (see discussion in
Sec. III B). Figure 2plots the variation of the c/a ratio for the
different polymorphs versus the Hubbard-U. The experimen-
tal values are indicated by a horizontal line. HSE06 functional
Ex perimental
U (eV)
0.660 GG A
HSE0 6
Calculated c/a
FIG. 2. Calculated DFT+Uvalues for the c/a ratio for the optimized crystal
structures of TiO2polymorphs as a function of the Uparameter (for con-
ventional DFT U=0eV);LDA+U(circles) and GGA+U(squares). Stars
correspond to the HSE06 values. Lines are a guide to eye. Experimental data
are indicated by horizontal lines.
values are denoted by stars. For rutile and columbite, where
the TiO6octahedra share edges along the caxis, the c/a ratio
increases with the value of U. The opposite trend is observed
for the anatase polymorph, in which edge-sharing occurs in
the ab plane (see Fig. 1). It can be observed in Fig. 2that in-
dependent of the functional (LDA or GGA), introducing a U
correction term allows a better description of the c/a ratio for
the three polymorphs. Within the GGA+Ufor columbite and
anatase small Uvalues of 0.7 and 2.6 eV, respectively, match
the experimental value. The required Uvalues to reproduce
the experimental data are similar in the LDA+U(0.82 eV
columbite, 1.2 eV rutile). Yet, for these two polymorphs the
HSE06 provides a quite accurate value. For the anatase poly-
morph, HSE06 and GGA/LDA show larger deviations to the
experimental c/a ratio, and Uvalues of 6 eV (GGA+U) and
3eV(LDA+U) are required to match experimental data. The
worse performance of DFT/HSE06 for the anatase polymorph
falls in the context of the difficulties found by any first prin-
ciple methods to reproduce this structure, compared to the ru-
tile for which lower deviations to experimental values have
been observed.17,19 Worth mentioning, while adequate Uval-
ues improve the results obtained within the conventional DFT,
very large Uvalues lead to a poor crystal lattice prediction.
Note the large deviation for the columbite polymorph when U
B. Electronic structure of anatase and rutile
One of the most common approaches to determine the
appropriate value of Uis to compare the calculated band gaps
for a set of Uvalues with the experimental band gap. This
practice, however, should be exercised with caution, unless
computational methods designed for describing excited states
are utilized. Therefore, in this work rather than attempting
to extract a concrete Uvalue, we discuss the general fea-
tures observed in the calculated electronic structure when
correlation effects are considered. Experimental band gaps
are 3.03 and 3.2 eV for rutile47 and anatase,48 respectively.
Figure 3shows the band gap values extracted in this work
Anatase GGA
Rutile GGA
Anatase LDA
Rutile LDA
Calculated Band Gap (eV)
U (eV)
FIG. 3. Calculated DFT+Uband-gap for anatase (squares) and rutile (tri-
angles) TiO2polymorphs as a function of the Uparameter; LDA+Uhollow
symbols and GGA+Ufilled symbols. Lines are a guide to eye. Stars corre-
spond to the HSE06 values. Experimental data are indicated by horizontal red
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054503-5 DFT+
calculations of TiO2polymorphs J. Chem. Phys. 135, 054503 (2011)
dxz dyz
M- O σ*
M- O π*
M- M
M- O π
M- O σ
Calculated DOS
Energy (eV)
Ti (d)
O (p)
O (p)
Ti (eg)
Ti (t2g)
II c
FIG. 4. Calculated density of states (DOS) of rutile-TiO2within the conventional GGA method. The black lines denote the total DOS; the partial DOS of
titanium and oxygen are represented in green and red lines, respectively. The Fermi level has been arbitrarily chosen as the origin of the energy. The right panel
is an schematic representation of rutile-TiO2band structure adapted from Refs. 46 and 50.
from the calculated density of state (DOS) for rutile (squares)
and anatase (triangles). Calculated band gaps within the con-
ventional DFT (rutile 1.86 eV, anatase 2.07 eV) are
similar to those obtained in previous DFT studies.19,20 Hy-
brid functionals yielded too large band-gaps, for instance, in
Ref. 19 the PBE0 values are 4.02 eV and 4.50 eV for anatase
and rutile, respectively. Interestingly, Janotti et al. reported a
HSE06 band gap of 3.05 eV after they reduced the fraction of
exact exchange in the hybrid functional to 20% from the orig-
inal 25%.49 Using the HSE06 functionals we obtained band
gap values of 3.6 eV for anatase and 3.2 eV for rutile (see
stars in Fig. 3). Within the DFT+Uthe band gap increases
with the value of the Uparameter, confirming the fact that a
Hubbard-like correction term (U) substantially improves the
accuracy of the calculated band-gap, compared to the con-
ventional DFT. In good agreement with other authors,15,21
values of 10 and 8.5 eV are required to match the experi-
mental band gap of rutile and anatase, respectively. However,
these values seem too large if one takes into account Uval-
ues extracted for other transition metal oxides where electron
localization effects are more relevant (for NiO U=6.4 eV
(Ref. 6), for MnO U=4.4 eV (Ref. 6)). Therefore, other fea-
tures in the electronic structure than the band gap should be
Figure 4shows the total calculated DOS of rutile to-
gether with the partial DOS of Ti and O in green and red,
respectively. The Fermi level is arbitrarily chosen as the ori-
gin of the energy. The DOS can be interpreted according to the
schematic band structure depicted on the right panel (adapted
from Refs. 46 and 50). An isolated Ti4+ion contains no oc-
cupied dorbitals, and therefore for a purely ionic Ti–O in-
teraction no Ti-dstates should be occupied. This means that
any dcharacter in the DOS below the Fermi level is a direct
result of oxygen–titanium covalent interactions. The σover-
lap between Ti-3d(eg) and O-2porbitals results in the bonding
σband, which appears below the Fermi level (predominantly
oxygen-2pin character), and the σ*band above the Fermi
level (mostly consisting of titanium-3dstates). The significant
mixing of O-2pstates and Ti-3dstates in both the σand σ*
bands supplies direct evidence for the strong covalent interac-
tion of the Ti–O bonds. The titanium t2gorbitals (dxy,dxz, and
dyz) are involved in πinteractions with the filled oxygen por-
bitals of appropriate symmetry. The bonding π-states appear
in the valence band and the antibonding π*in the conduction
band. In a perfectly cubic octahedral environment the t2gor-
bitals are degenerate. In rutile, however, the TiO6octahedra
share edges along the caxis, and vertex in the ab plane. This
distortion breaks the degeneration of the t2gorbitals. Given
the short Ti–Ti distance along the [001] axis (d(Ti–Ti) =c
=2.958 Å), the dxy orbitals interact forming a metal σ-band
along the shared edges of the octahedra.
Figure 5shows the evolution of the DOS with the Upa-
rameter within the GGA+U. A similar evolution is observed
within the LDA+U(see Ref. 51). Increasing the Uvalue has
two effects: (i) opening the energy gap between the valence
and conduction bands and (ii) a progressive merging of the t2g
and egderived bands, which do not differentiate any longer
at U=10 eV. The Uparameter keeps the dorbitals atomic
like, diminishing the effective overlapping of O-2pand Ti-3d
orbitals. In rutile, the downshift of the σ*band is related to
a reduction in the Ti(eg)-Opoverlapping (weaker Ti–O bond,
longer alattice parameter), while the up shift and narrowing
of the t2g-derived band is associated to a weaker Ti–Ti inter-
action (longer clattice parameter). Changes of the t2gband as
afunctionofUare more pronounced that those of the egband
as the clattice parameter increases more than the aparame-
ter (c/a ratio increases). Therefore, the DOS modifications as
a function of the Uparameter are consistent with the evolu-
tion of the lattice parameters and the c/a ratio (see Table Iand
Fig. 2).
To further analyze the adequacy of the DFT+Umethod,
in a first approximation, the calculated DOS could be
qualitatively compared with experimental x-ray absortion
spectra and electron energy loss spectra. It is, however,
important to point out that such comparison neglects the
excitation aspects, which can be taken into account by means
of quasiparticle calculations in the GW-approach (see, for
instance, Ref. 51 and references therein). The measured
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054503-6 Arroyo, Morales, and Taravillo J. Chem. Phys. 135, 054503 (2011)
-6 -4 - 2 0 2 4 6 8
Energy (eV)
U = 8 eV
U = 10 eV
U = 3 eV
U = 5 eV
Calculated DOS of
Ti O2
-6 -4 - 2 0 2 4 6 8
CFS = 2.4 eV
CFS = 1.9 eV
CFS = 1.6 eV
CFS = 1.2 eV
CFS = 0.9 eV
CFS = 2.6 eV
FIG. 5. DOS of rutile TiO2calculated using the HSE06, the conventional
GGA method, and the GGA+Uwith Uvalues of 3, 5, 8, and 10 eV. Guide-
lines show the displacement of the bands as the value of Uincreases. CFS
refers to the crystal field splitting between averaged energies of t2gand eg
derived bands. The DOS calculated using the LDA+Uis shown in the sup-
plementary information.
energy difference within the average energies of the t2g
and egbands (crystal field splitting, CFS) varies from 1.8
to 3.0 eV.53,54 It can be seen in Fig. 5that a too large
Uvalue (U>5 eV) dramatically reduces the CFS (for
LDA+Uresults see Ref. 51). Similarly, for anatase-TiO2
(experimental CFS value 2.6 eV (Ref. 54)), the introduc-
tion of the Uparameter in the calculation reduces the CFS
value; 2.2 eV for GGA, 1.7 eV for GGA+U,U=5eV
(see Ref. 51). In short, the DFT+Umethod allows a good
qualitative description of the electronic structure for moder-
ate Uvalues, below 5 eV. Using these results as a starting
point, the Uvalue could still be optimized to give the best
agreement with available experimental data utilizing the GW
method based on the DFT+U.52
C. Phase stability
Rutile is the stable bulk phase of TiO2at ambient
pressure,29 with the enthalpy of the anatase to rutile transition
taking values ranging from +0.42 kJ/mol (Ref. 55) to –11.7
kJ/mol.27 However, previous DFT works found anatase as the
most stable phase.1719 This discrepancy could not be solved
by testing different exchange-correlation functionals.17,19 Hy-
brid functionals also produce wrong phase stability, regard-
less the amount of HF exchange included.19 Figure 6repre-
sents the calculated total energy differences, referred to the
rutile polymorph, for anatase and columbite within the DFT
and DFT+U(U=3, 5, 8, and 10 eV), and using the HSE06
Rao 1961
Levchenko 2006
JANAF 1973
Ranade 2002
Calculated energy difference (eV/f.u.)
U (eV)
Anatase GGA
Columbite GGA
Anatase LDA
Columbite LDA
Anatase HSE06
Columbite HSE06
Exp. Anatase
Less stable than Rutile
Erut< E anat < E col
FIG. 6. Calculated DFT+Utotal energy differences for TiO2polymorphs as
a function of the Uparameter; LDA+Uhollow symbols and GGA+Ufilled
symbols. Data are referred to the rutile polymorph. EanataseErutile is denoted
by squares (DFT+U) and star (HSE06). ErutileEcolumbite is denoted by tri-
angles (DFT+U) and pentagon (HSE06). Solid lines are a guide to eye. The
red horizontal bars correspond to experimental data for –H of the anatase
to rutile transformation taken from the references. Also see Refs. 29,30,61,
and 62. The light grey area indicates the Uvalues for what the GGA+Ure-
produces the experimental phase stability.
functional. Positive energy differences indicate that a poly-
morph is less stable than the rutile form. Experimental data
of H (rutile anatase) are indicated by horizontal red
lines. The conventional GGA gives a relative energy Eanatase
<Erutile, which is in clear disagreement with experiments.28
The LDA produces a complete wrong stability sequence,
Ecolumbite <Eanatase <Erutile, with the columbite form as the
most stable polymorph. The HSE06 functional predicts that
the anatase form is 0.086 eV/f.u. more stable than the ru-
tile form; this energy difference is reduced to 0.055 eV/f.u
in more accurate calculations (see Sec. II).
Next, we analyze the effect of the Uparameter on phase
stability. It can be seen in Fig. 6that the relative energy of the
polymorphs strongly depends on the value of U. Noteworthy,
the LDA and GGA functionals predict different energetic sta-
bility. Within LDA+Uthe stability sequence is dominated by
the strong stabilization of the columbite form, a phase only
reachable at high pressure of the order of 6 GPa. Even though
at U=2.5 eV the rutile becomes the most stable form, the
columbite remains more stable than the anatase form. This
is to say, for U>2.5 eV the energetic sequence is Erutile
<Ecolumbite <Eanatase, which is not consistent with the ob-
served stability for the bulk phases. It can be seen in Fig. 6
that the experimental results (Erutile <Eanatase <Ecolumbite)are
well reproduced within the GGA+Uin the range of 5 eV <U
<8 eV. For values of U>8 eV, the rutile is the most stable
polymorph but the columbite form gains in stability to the
anatase form (Erutile <Ecolumbite <Eanatase).
To investigate phase stability under pressure, several
fixed-volume calculations were performed starting from the
GGA and GGA+Uoptimized structures. The strong failure
of the LDA and hybrid functionals to reproduce phase sta-
bility at ambient pressure, at the expense of a large compu-
tational effort for the latter, discourages the investigation un-
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054503-7 DFT+
calculations of TiO2polymorphs J. Chem. Phys. 135, 054503 (2011)
26 28 30 32 34 36 38 40 42
26 28 30 32 34 36 38 40 42
26 28 30 32 34 36 38 40 42
26 28 30 32 34 36 38 40 42
Volume ( Å
Calculated Energy (eV/f.u.)
U = 5 eV U = 10 eV
U = 3 eV
FIG. 7. Total energy vs. volume curves for TiO2polymorphs. Symbols correspond to the conventional GGA and GGA+Ucalculated data, and lines show the
fitting to the Birch-Murnaghan equation of state.
der pressure. Figure 7shows the calculated total energy as a
function of the volume for the different polymorphs within the
GGA and GGA+U(U=3, 5, and 10 eV), together with the
corresponding fit to the Birch-Murnaghan equation of state.
Table II lists the parameters of the fits of the ab initio energy-
volume data for the investigated TiO2forms. The bulk mod-
uli are clearly underestimated, independent of the choice of
U. Note the major differences with experimental bulk mod-
uli are observed for U=10 eV. For anatase and rutile in-
creasing Uvalues lead to lower bulk moduli, a trend observed
in other systems: Lu2O3,56 FeS2,57 and MnO.8The fluctua-
tions of the bulk moduli as a function of Uobtained for the
Columbite phase are not surprising neither (see, for instance,
Ce2O3(Ref. 58)orFe
In Fig. 7it can be observed that, within the GGA, the
global energy minimum, this is to say the polymorph stable
at ambient pressure, corresponds to anatase, with rutile be-
ing a metastable phase at any pressure. The GGA+Ucorrects
TABLE II. Calculated equation of state parameters for TiO2polymorphs (conventional GGA vs. GGA+U). E0,
0are the zero-pressure energy, volume, bulk modulus, and its pressure derivative, respectively.
Polymorph Method U(eV) E0(eV/f.u.) V03/f.u.) B0(GPa) B
0rms (eV/f.u.)
ANATASE GGA –26.9045 35.272 169.9 2.27 0.2
GGA+U3 –24.7312 36.239 164.9 2.53 0.2
GGA+U5 –23.3733 36.902 162.3 2.61 0.3
GG+U10 –20.2905 38.572 157.6 2.67 0.4
Experiment (Ref. 31.) 34.07 179 ±24.5
RUTILE GGA –26.8128 32.186 200.4 4.98 0.7
GGA+U3 –24.6944 32.946 199.5 4.77 0.6
GGA+U5 –23.3727 33.458 198.7 4.62 0.6
GGA+U10 –20.3735 34.740 195.3 4.40 0.5
Experiment (Ref. 63.) 31.22 211 ±76.76
COLUMBITE GGA –26.8288 31.651 195.9 4.22 1.1
GGA+U3 –24.6718 32.334 206.8 3.62 0.3
GGA+U5 –23.3363 32.866 202.3 3.69 0.4
GGA+U10 –20.3171 34.179 193.9 3.75 0.3
Experiment (Ref. 31.) 30.59 258 ±84.1
rms =(EEfit)2
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054503-8 Arroyo, Morales, and Taravillo J. Chem. Phys. 135, 054503 (2011)
0 2 4 6 8 1012141618
0 2 4 6 8 1012141618
0 2 4 6 8 1012141618
U = 3 eV
tr = 6.1 GPa
tr = 2.5 GPa
Calculated Enthalpy difference (eV/f.u.)
Pressure (GPa)
tr = 1.8 GPa
U = 5 eV
tr = 10.1 GPa
tr = 1.5 GPa
tr = 0.03 GPa
U = 10 eV
tr = 16 GPa
FIG. 8. Calculated Enthalpy vs. pressure for TiO2polymorphs within the
GGA+U(U=3, 5, and 10 eV).
these discrepancies with experiments. For U=3eVtheru-
tile form is stabilized by pressure, and for U=5, 10 eV the
rutile becomes the stable polymorph at ambient pressure, in
good agreement with experimental results. The curve of the
columbite polymorph crosses that of the most stable phase
(anatase for GGA and U=3 eV, rutile for U>5eV)ata
certain volume, indicating that columbite becomes more sta-
ble at a sufficient pressure. Comparing the predicted pressure
of the anatase columbite and rutile columbite transfor-
mations with experimental data is another way to determine
appropriate Uvalues.
D. Transition pressures
Figure 8shows the calculated enthalpy-pressure variation
for the TiO2polymorphs (at 0 K), within the GGA+U. The ef-
fect of Uis to extend the stability field of the rutile phase, rais-
ing/decreasing the pressure of the rutile/anatase to columbite
transformations. Experimentally, it has been observed that
anatase transforms to columbite at 2.6–7 GPa,31,32 being the
pressure of the transformation highly dependent on the parti-
cle size. Rutile undergoes the transformation to columbite at
about 10 GPa.33,60 Therefore, the GGA+Uwith U=5eV
provides the best agreement with experiments (PA-C =1.5
GPa, PR-C =10 GPa). At U=10 eV the rutile transforma-
tion occurs at too high pressure (16 GPa), and the anatase to
columbite transformation is not feasible.
The suitability of DFT+Umethodology to investigate
TiO2polymorphs is discussed and compared to conven-
tional DFT and hybrid HSE06 functionals. The GGA+U
and LDA+Umethods improve the prediction of ground state
properties and electronic structure of the investigated TiO2
polymorphs (anatase, rutile and columbite), with respect to
the conventional GGA/LDA. As expected, it is not possible
to extract an universal value of the Uparameter to repro-
duce all TiO2properties. Reproducing the experimental band
gap of rutile-TiO2requires a Uparameter of 10 eV, which in
turn produces a band structure in disagreement with experi-
ments. In addition, such large Uvalues worsen the reproduc-
tion of crystal structure. Generally speaking, we found that
the DFT+U(U5 eV) method is well suited to investigate
the properties of individual TiO2polymorphs (anatase, rutile,
and columbite). Yet, the hybrid HSE06 performs equally well
to accurately reproduce the crystal and electronic structures
of TiO2polymorphs.
We found that conventional GGA/LDA and HSE06 fail to
reproduce TiO2-phase stability, yielding a too low energy for
the anatase polymorph (Eanatase <Erutile). Introducing a Hub-
bard correction term (U) in the conventional LDA (LDA+U)
has the effect of excessively stabilizing the columbite poly-
morph, which is predicted as more stable than the anatase
form at any Uvalue. Note that contrary to any experimen-
tal observation columbite is even more stable than rutile for
U<2.6 e V. The GGA+Umethod yields an ener-
getic sequence consistent with experiments, Erutile <Eanatase
<Ecolumbite,forUvalues between 5 and 8 eV. The GGA+U
method also allows a correct prediction of phase stability at
high pressure. Phase transformations under pressure are the
best reproduced for Uvalues of the order of 5 eV. For U=
5 eV predicted transition pressures are rutile columbite 10
GPa (expt. 10 GPa) and anatase columbite 1.5 GPa (expt.
2.5–7 GPa). The present results indicate that the treatment
of correlation effects is important to investigate the phase-
stability of TiO2.
We conclude that computational results for titania poly-
morphs are very sensitive to the choice of the functional
(LDA/GGA) and the treatment of correlation effects (U
value); extreme care should be exercised when selecting a
computational methodology to investigate this system, in par-
ticular, for phase stability.
Financial resources for this research were provided by the
Spanish Ministry of Science (MAT2007–62929, CSD2007–
00045, CTQ2009–14596-C02–01) and Project No. S2009-
PPQ/1551 funded by Comunidad de Madrid. A.M.-G. ac-
knowledges a grant from the FPI Program of Spanish Ministry
of Science. Valuable comments from D. Morgan, J. Tortajada,
and V. G. Baonza are greatly appreciated. M.E.AdD. thanks
R. Armiento and G. Ceder for their kind help with the HSE06
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... For the Hubbard U we choose a value of 3.5 eV since it provides a compromise between the increasing band gap and preserving lattice parameters comparable with experiment. 39,40 The core electrons are accounted by ultrasoft pseudopotentials and we set the energy cutoff to 50 Ry for the wavefunctions and 400 Ry for the electron density. We obtained the theoretical lattice parameters of anatase by performing a variable cell optimisation with a 10×10×10 k-points mesh obtaining a lattice parameter a=3.838 ...
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Nanostructured oxide semiconductors are widely used in energy conversion, catalysis, sensing and environmental applications, due to their high stability, commercial availability, efficiency and low cost. Despite its crucial importance for the design of more efficient materials, the interplay between intrinsic and extrinsic defects is yet to be clarified. For example, oxygen vacancies (VO’s) can be either beneficial or detrimental to the desired performances, depending on a variety of factors. Here, we synthesize TiO2-x samples by the addition of three different N chemical sources (NH3, triethylamine, urea). X-ray absorption spectroscopy, confocal microscopy, UV–vis absorbance and fluorescence, are employed to explore the occurrence and location of VO’s both in real and energy spaces. High–grade bulk DFT simulations complement the experimental picture. Synergy between theory and experiment, on the one hand, estimates the relative VO’s content in the different samples from local structural information. On the other hand, a sharp optical transition at ~2.7 eV serves an unequivocal spectral signature of VO’s, allowing a semi-quantitative analysis by confocal microscopy. Fluorescence quenching of this feature is observed to a different degree in each sample under UV pumping and is attributed to the reaction of surface defects with atmospheric O2. Thus, we demonstrate that confocal microscopy can discriminate surface-localized VO’s if coupled with the detection of O2–induced fluorescence quenching. Concurrently, UV-induced photochromism and visible light photodegradation shed light on the most effective reactive defects. Eventually, surface-localized oxygen vacancies are predominant where actual N substitutional doping occurs, leading to materials exhibiting visible-light activity and characteristic photochromic behaviour. Implications on strategies for concomitant VO engineering and extrinsic doping are discussed.
... In addition, onsite Ti 3d electrons Coulombic repulsion (U) was considered by PBE+U. The relationship between computational approaches and electronic properties of dye@TiO 2 systems has been studied [64][65][66]. Concerning the Coulombic repulsion potential, it was discovered that U values between 3.5 and 10 eV generated band gaps in realistic agreement with the experiment. Additionally, U = 8.5 eV was chosen for titanium atoms in previous studies [67][68][69][70]. ...
... eV in previous studies. 53,54 With the energy converged to 10 −6 Ry in each self-consistent field (SCF) iteration step, the convergence criterion for the maximum force is 0.02 eV/Å in geometry optimization. The ab initio molecular dynamics (AIMD) simulations are performed to get the equilibrium configuration of catalysts in aqueous phase, while the metadynamics 55,56 (MTD) simulations are used to get the free energy change and barrier of each elementary step. ...
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The aldehyde hydrogenation for stabilizing and upgrading biomass is typically performed in aqueous phase with supported metal catalysts. By combining density functional theory calculations and ab initio molecular dynamics simulations, the model reaction of formaldehyde hydrogenation with a Pt/TiO2 catalyst is investigated with explicit solvent water molecules. In aqueous phase, both the O vacancy (Ov) on support and solvent molecules could donate charges to a Pt cluster, where the Ov could dominantly reduce the Pt cluster from positive to negative. During the formaldehyde hydrogenation, the water molecules could spontaneously protonate the O in the aldehyde group by acid/base exchange, generating the OH* at the metal-support interface by long-range proton transfer. By comparing the stoichiometric and reduced TiO2 support, it is found that the further hydrogenation of OH* is hard on the positively charged Pt cluster over stoichiometric TiO2. However, with the presence of Ov on reduced support, the OH* hydrogenation could become not only exergonic but also kinetically more facile, which prohibits the catalyst from poisoning. This mechanism suggests that both the proton transfer from solvent water molecules and the easier OH* hydrogenation from Ov could synergistically promote aldehyde hydrogenation. That means, even for such simple hydrogenation in water, the catalytic mechanism could explicitly relate to all of the metal cluster, oxide support, and solvent waters. Considering the ubiquitous Ov defects in reducible oxide supports and the common aqueous environment, this synergistic effect may not be exclusive to Pt/TiO2, which can be crucial for supported metal catalysts in biomass conversion.
The development of vision bionic systems is indispensable for the perception, memory, and processing of optical signals, which promotes the exploration of efficient visual perception systems. In this work, a simple and novel two‐terminal optoelectronic memristor based on the CuAlAgCr/TiO2/W (CTW) structure is prepared, where the CuAlAgCr high‐entropy alloys are employed as the top electrode for the first time. Before annealing, the CTW optoelectronic memristor exhibited fascinating performance, including uniformly distributed operating voltage, reliable data retention, and a higher switching ratio. Moreover, the optoelectronic memristor can be reversibly switched between volatile and nonvolatile memories by adjusting compliance currents. The CTW optoelectronic memristor annealed in air exhibits various artificial synaptic functions, such as short‐term memory, optical learning, and forgetting behavior under the illumination of the laser. The photo‐response current is increased from nano‐ampere to micro‐ampere level. Furthermore, a logic function unit based on CTW optoelectronic memristor is proposed, which realizes “AND” operation. Furthermore, first‐principles calculations of the CTW structure are performed to describe the influence of photocarriers on the barrier height at the CuAlAgCr‐TiO2 interface, revealing the working mechanism of the CTW optoelectronic memristor. This work has greatly facilitated the development of optically operated artificial synaptic devices and vision bionic systems. In the human cognitive system, light information is perceived by visual organs and transmitted to the visual cortex of the brain through neurons. To build a visual bionic system, efficient artificial synapses are crucial. The CuAlAgCr/TiO2/W optoelectronic memristor, which integrates sensing, storage, and computing functions under light stimulation, is a promising technology for optical synaptic function in artificial vision systems.
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Hydrogen incorporation in native surface oxides of metal alloys often controls the onset of metal hydriding, with implications for materials corrosion and hydrogen storage. A key representative example is titania,...
The electronic structures of S/Se atom-doped BiVO4 (010) surfaces and their photocatalytic behavior are studied with the DFT + U approach. Our theoretical results showed that the S/Se atoms preferentially occupy the position of O atoms on the outer surface of BiVO4 (010). Such S atoms generate a nearly delocalized (0.15–0.22 eV) and a delocalized occupying defect energy state (0.32–0.48 eV) above the valence band maximum (VBM), while the Se atoms produce two delocalized occupying defect energy states above the VBM at 0.35–0.44 eV and 0.60–0.65 eV. The delocalized defect states are equivalent to the bridge of electron transition from the VBM to conduction band minimum (CBM) and form an intermediate band semiconductor. These intermediate bands can lead to a strong absorption from visible light to UV regions, resulting in a redshift of the reflectance spectra. The position of the S/Se atoms occupying the O atoms at the outermost position of the (010) external surface plays a major role in improving its photocatalytic performance.
2D rare-earth metal carbides (MXenes) are attractive due to their novel electronic and magnetic properties and their potential as scalable 2D magnets. In this study, we used density functional theory with the Hubbard U correction to characterize the structure, termination, and magnetism in an out-of-plane ordered rare-earth containing M 3 C 2 T x MXene, Mo 2 NdC 2 T 2 (T = O or OH). We investigated the effect of the U parameter on the stability and magnetism of two possible termination sites: the hollow sites aligned with the inner Nd atoms (Nd-hollow sites) and those aligned with the closest C atoms (C-hollow sites). We found that increasing U Mo stabilized the Nd hollow sites, which minimized electrostatic repulsion between C and O atoms. Using U Mo = 3.0 eV and U Nd = 5.6 eV, obtained via the linear response method, we found that the energetically preferred termination site was C-hollow in Mo 2 NdC 2 O 2 and Nd-hollow in Mo 2 NdC 2 (OH) 2 . Regardless of termination and the Hubbard U value, we found Mo 2 NdC 2 O 2 and Mo 2 NdC 2 (OH) 2 to be magnetic. The C-hollow termination resulted in ferromagnetic states for all Hubbard U tested with no magnetic moment in Mo. In the case of Nd-hollow, Mo became magnetic for U Mo ≥ 4 eV. The difference of Mo magnetism in Nd-hollow and C-hollow was explained by crystal field splitting of the Mo d orbital caused by a distorted ligand.
The introduction of defects is one of the most recurrent pathways to generate modifications to materials’ electronic structure and surface reactivity. In this work, calculations based on the density functional theory (DFT) were applied to study the electronic properties of pristine and reduced TiO2(B)(100) ultrathin sheets to evaluate their potential as a semiconductor material for dye-sensitized solar cells (DSSCs). It was carried out by introducing vacancy defects on these surfaces and then adsorbing a catechol molecule, used as a model of a direct electron injection sensitizer (type-II dye), in different interaction configurations. Geometric, energetic, and electronic analyses were performed, focusing on the electronic structure changes and charge transfer between the dye and surface during molecular adsorption. The obtained results seem to indicate that a thickness of four layers is adequate to obtain a satisfactory slab model approximation of the TiO2(B)(100) surface. The presence of oxygen vacancy states among the majority of the reduced surfaces was observed as well as a reduction of the band gap energy value. Additionally, the adsorption of catechol in the reduced surface induced an increase in light absorption compared to the pristine model. These attributes suggest that reduced ultrathin sheets of TiO2(B) could be a suitable candidate as a photoelectrode for DSSC applications.
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Prior works have shown that density functional theory (DFT) with the DFT+U method resolves the underestimation of redox potentials calculated by conventional functionals for a number of transition metal compounds relevant for battery applications, including the olivine LixMPO4 (M = Fe, Mn, Co, Ni), layered LixMO2 (M = Co, Ni) and spinel-like LixMn2O4. We show that the redox potentials of these compounds are also well reproduced by the hybrid density functional by Heyd-Scuseria-Ernzerhof (HSE06). Hybrid functionals combine a conventional DFT functional with a part of Hartree-Fock (HF) exchange. While the HF part increases the computational expense by at least one order of magnitude, it provides, in contrast to DFT+U, a correction for the self-interaction error that does not rely on special treatment of the occupancies of the orbital states of ions or species-specific parameters. We compare the accuracy of regular DFT, DFT+U and HSE06 for the redox potentials, lattice constants, and other properties. Examples of electron delocalization problems connected to the self-interaction error in the systems are discussed, and shown to be resolved both by the hybrid functional and DFT+U methods. Comments are made on the possibility to approach the delocalization problem with a semi-local functional.
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We present ab initio calculations of the zero-temperature composition dependent spin transition pressures in rocksalt (B1) (Mg1−x,Fex)O. We predict that the spin transition pressure decreases with increasing Mg content, consistent with experimental results. At high-pressure, we find that the effective size of Mg is smaller than high-spin Fe but quite close to low-spin Fe, consistent with a simple compression argument for how Mg reduces the spin transition pressure. We also show that the spin transition is primarily driven by the volume difference between the high-spin and low-spin phases, rather than changes in the electronic structure with pressure. The volume contraction at the transition is found to depend non-monotonically on Fe content. For FeO we predict a B1 → iB8 transition at 63 GPa, consistent with previous results. However, we also predict an unexpected reverse transition of high-spin iB8 → low-spin B1 at approximately 400 GPa.
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At high temperature, coarse-grained (bulk) rutile is well established as the stable phase of TiO2, and nanophase anatase, thermodynamically stable relative to nanophase rutile, transforms irreversibly to rutile as it coarsens. The lack of experimental heat-capacity data for bulk anatase below 52 K]ends uncertainty to its standard entropy and leaves open a slight possibility that anatase may have a thermodynamic stability field at low temperature, as suggested by some theoretical calculations. In the present Study, the molar heat capacities of rutile and anatase were measured from 0.5 K to about 380 K. These data were combined with previously measured high-temperature heat capacities, and fits of the resulting data set were used to generate C-P,C-m degrees, Delta H-T(0)m degrees, and Delta(T)(0)G(m)degrees values at smoothed temperatures between 0.5 and 1300 K for anatase and 0.5 and 1800 K for rutile. Using these new data and the enthalpy of transformation between anatase and rutile at 298 K, the change in Gibbs free energy for the transition between anatase and rutile from 0 to 1300 K was calculated. These calculations reveal that the transformation from bulk anatase to bulk rutile is themodynamically favorable at all temperatures between 0 and 1300 K, confirming that bulk anatase does not have a thermodynamic stability field. Implications for the natural occurrence of these two minerals in terrestrial, lunar, and planetary settings are discussed. In particular, anatase requires low-temperature aqueous conditions for its formation and may be a reliable indicator of such conditions in both terrestrial and extraterrestrial settings.
We compare experimentally observed electron energy loss spectra (EELS) of uranium dioxide UO2 and nickel monoxide NiO with the results of ab-initio calculations carried out by using a method combining the local spin density approximation and the Hubbard U term (the LSD A + U method). We show that by taking better account of strong Coulomb correlations between electrons in the 5f shell of uranium ions in UO2 and in the 3d shell of nickel ions in NiO it is possible to arrive at a better description of electron energy loss spectra, cohesive energies and elastic constants of both oxides compared with local spin density functional theory. For NiO we also compare the LSDA + U results and EELS spectra with a self-interaction corrected LSDA calculation.
We investigate the relative energetic stability and the magnetic properties of MnO in the rocksalt (rs) , wurtzite (wz) , and zinc-blende (zb) structures using density-functional theory with different approaches to exchange and correlation: the semilocal generalized-gradient approximation (GGA), the GGA+U method with an additional on-site interaction U , and the spatially nonlocal hybrid functional HSE03 that accounts for screened exchange. In contradiction to experimental observations, GGA predicts the fourfold coordinated zb and wz structures to be energetically favorable in comparison to the sixfold-coordinated rs geometry. The use of the hybrid functional HSE03 improves the energetic stability of the rs structure but still fails to determine the correct ground state of MnO. This deficiency can be overcome by applying the GGA+U method with U&gsim;4eV . The computed total energies are used to fit the nearest- and next-nearest-neighbor exchange coupling constants of a Heisenberg model Hamiltonian. Only for the GGA+U functional with U&gsim;4eV , the coupling constants as well as the resulting critical temperature for the magnetic phase transition are in agreement with measured quantities. Therefore, we conclude that an appropriate treatment of the correlation effects in MnO and similar compounds is necessary not only for the electronic but also for the structural properties.
Clear, synthetic rutile (TiO2) single crystals have been investigated by electrical and optical methods. It seems possible to correlate the high temperature conductivity (EG=3.05 ev) with the threshold of optical absorption at low temperatures (3.03 ev) and with the maximum of the photoconductivity (3.03-3.06 ev). This evidence indicates an energy gap ca 3.05 ev for rutile as an insulator. Semiconducting rutile, prepared by hydrogen reduction at temperatures ≤800 °C, shows a blue color arising from an optical absorption maximum at ca 1.7 μ (0.73 ev). Conductivity-temperature plots for slightly reduced specimens indicate an optical activation energy of 0.68 ev. A theoretical calculation for the ionization of the first electron from an oxygen vacancy indicates 0.74 ev as the expected value, in good agreement with the experimental results. At room temperature the mobility of electrons in slightly reduced single crystals is ca 10-4 m2 /v-sec. Strongly reduced rutile is opaque; a comparison of electron concentrations calculated from weight loss and Hall coefficient data shows that for samples in which the electron concentration is 1026 /m3, all contribute to conduction at room temperature.
We compare experimentally observed electron energy loss spectra (EELS) of uranium dioxide UO2 and nickel monoxide NiO with the results of ab-initio calculations carried out by using a method combining the local spin density approximation and the Hubbard U term (the LSDA + U method). We show that by taking better account of strong Coulomb correlations between electrons in the 5f shell of uranium ions in UO2 and in the 3d shell of nickel ions in NiO it is possible to arrive at a better description of electron energy loss spectra, cohesive energies and elastic constants of both oxides compared with local spin density functional theory. For NiO we also compare the LSDA + U results and EELS spectra with a self-interaction corrected LSDA calculation.
The energetic stability of FePO4 polymorphs (berlinite, heterosite, monoclinic, and high-pressure forms) is investigated combining first-principles and experimental methods. Calculations at the density functional theory (DFT and DFT+U) level performed using the local density approximation (LDA) and the generalized gradient approximation (GGA and GGA+U) yield different relative energetic stability for those forms of iron phosphate. To discern the appropriate computational methodology, we have measured the transport and magnetic properties of high pressure (HP)-FePO4; we found that it is an insulating compound with a room-temperature resistivity of 2.107, and Fe3+ ions in high spin configuration (t2g2eg3). Both LDA and GGA methods fail to reproduce the physical properties of HP-FePO4, which are well-predicted within the GGA+U framework. This method predicts that berlinite-FePO4 is the most stable form at ambient pressure, whereas HP-FePO4 is the stable form at pressures above 2 GPa. Heterosite and monoclinic FePO4 are predicted as metastable phases in the whole pressure range. Accordingly, we found that both monoclinic and berlinite transform to HP-FePO4 under pressure (pressure range 2-6 GPa, 900 oC). Differential thermal analysis (DTA) and temperature-XRD measurements reveal that HP-FePO4 reverts exothermically and irreversibly to the berlinite form at about 700 °C. Discrepancies of our combined experimental-computational results with a previous high-temperature oxide melt solution calorimetry investigation are discussed.
TiO2−B is a highly promising anode material for rechargeable lithium batteries. Computational studies based on density functional theory (DFT) have been carried out on this material focusing on key issues relating to lithium insertion sites and lithium diffusion paths. Our simulation model shows good reproduction of the observed crystal structure of TiO2−B. Electronic structure calculations suggest that the lowest energy lithium site is a slightly off-center position in the b-axis channel for low lithium concentration (x < 0.125 for LixTiO2−B). Our calculated cell voltages are compatible with values from electrochemical measurements. Low Li migration energies are found for pathways along the b-axis channel and the [001] c-axis direction, suggesting significant Li ion mobility in this anode material.
Given the interest of silicates as potential electrode materials for lithium batteries, it is critical to fully understand the role that the inductive effect of the polyoxyanionic group, (XO4)n-, plays on the electrochemical performance of polyoxyanionic compounds. In this work we have combined experiments and first-principles methods to investigate to which extent the inductive effect of the X−O bond within the XO4n- polyanion (X = Si, Ge0.5Si0.5, Ge, Si0.5As0.5, Si0.5P0.5, As, P) modifies the redox energy of the V5+/V4+ couple in the LiyVOXO4 family of compounds. Calculations using the GGA+U method evidence a nice correlation between the X electronegativity (i.e., the magnitude of the XO4 groups' inductive effect) and details of the crystalline and electronic structures of Liy+1V4+OXO4/ LiyV5+OXO4 phases such as bond distances or band gaps. Besides the inductive effect of the polyanionic group, we found that the chemistry of these 2D compounds is also correlated to the Li+ site occupancy in the interlayer space. The calculated lithium insertion voltages display an almost linear dependence on the Mulliken X electronegativity; this offers promising prospects in the design of novel polyoxyanionic electrode materials. The new electrode materials Li2VOGeO4 and Li2VOSi0.5Ge0.5O4 have been electrochemically tested, providing good agreement between experimental and calculated lithium insertion voltages. Activation energies of the prepared compounds follow the band gap trend provided by the calculated data as well. Thus, experimental evidence support the computational results and the conclusions presented here.