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THE JOURNAL OF CHEMICAL PHYSICS 135, 054503 (2011)

DFT+

U

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 U≈5eV.© 2011 American Institute of Physics. [doi:10.1063/1.3617244]

I. INTRODUCTION

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.1–3The 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:

e.arroyo@quim.ucm.es. Tel.: +34 91 3945222. FAX: +34 91 3944352.

compounds (Ti, V), where the more extended orbitals de-

crease the electron correlations, might be controversial.10–15

Insulating materials such as TiO2provide an interesting

case.

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 18–21). 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

0021-9606/2011/135(5)/054503/9/$30.00 © 2011 American Institute of Physics135, 054503-1

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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. 24–26)

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.28–30 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.31–33

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

demonstrated.

II. METHODOLOGY

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+

U

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

state:43

E(V)=E0+9V0B0

16 B0[x2/3−1]3

+[x2/3−1]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 Å−1.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).

III. RESULTS AND DISCUSSION

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) V(Å3) 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

-101234567891011

1.075

1.080

1.085

1.090

1.095

1.100

1.105

Experimental

Experimental

Rutile

Anatase

Columbite

Ex perimental

U (eV)

0.635

0.640

0.645

0.650

0.655

0.660 GG A

HSE0 6

LD A

2.49

2.50

2.51

2.52

2.53

2.54

2.55

-101234567891011

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

>6eV.

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

0246810

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

0246810

Anatase

Rutile

Rutile

Anatase

Anatase GGA

Rutile GGA

Experimental

HSE06

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

bars.

054503-5 DFT+

U

calculations of TiO2polymorphs J. Chem. Phys. 135, 054503 (2011)

dxy

dxz dyz

M- O σ*

M- O π*

M- M

M- O π

M- O σ

-6

-4

-2

0

2

4

6

8

Calculated DOS

t2g

eg

Energy (eV)

TOTAL

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

examined.

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

054503-6 Arroyo, Morales, and Taravillo J. Chem. Phys. 135, 054503 (2011)

-6 -4 - 2 0 2 4 6 8

Energy (eV)

HSE06

GGA

U = 8 eV

U = 10 eV

U = 3 eV

U = 5 eV

Calculated DOS of

Rutile

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.17–19 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

0246810

-0.09

-0.06

-0.03

0.00

0.03

0.06

0.09

0.12

0.15

0.18

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

GGA+U

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. Eanatase–Erutile is denoted

by squares (DFT+U) and star (HSE06). Erutile–Ecolumbite 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-

054503-7 DFT+

U

calculations of TiO2polymorphs J. Chem. Phys. 135, 054503 (2011)

26 28 30 32 34 36 38 40 42

-27.0

-26.8

-26.6

-26.4

-26.2

-26.0

-25.8

26 28 30 32 34 36 38 40 42

-24.8

-24.6

-24.4

-24.2

-24.0

-23.8

-23.6

26 28 30 32 34 36 38 40 42

-23.4

-23.2

-23.0

-22.8

-22.6

-22.4

26 28 30 32 34 36 38 40 42

-20.4

-20.2

-20.0

-19.8

-19.6

-19.4

ANATASE RUTILE COLUMBITE

Volume ( Å

3

/f.u.)

Calculated Energy (eV/f.u.)

DFT

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

1–xMgxO(Ref.59)).

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,

V0,B0,andB

0are the zero-pressure energy, volume, bulk modulus, and its pressure derivative, respectively.

Polymorph Method U(eV) E0(eV/f.u.) V0(Å3/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 =(E−Efit)2

n.

054503-8 Arroyo, Morales, and Taravillo J. Chem. Phys. 135, 054503 (2011)

-0.04

-0.02

0.00

0.02

0.04

0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0 2 4 6 8 1012141618

0 2 4 6 8 1012141618

0 2 4 6 8 1012141618

-0.04

0.00

0.04

0.08

0.12

U = 3 eV

Prut-col

tr = 6.1 GPa

Pana-col

tr = 2.5 GPa

Calculated Enthalpy difference (eV/f.u.)

Pressure (GPa)

Pana-rut

tr = 1.8 GPa

U = 5 eV

Prut-col

tr = 10.1 GPa

Pana-col

tr = 1.5 GPa

Pana-rut

tr = 0.03 GPa

U = 10 eV

Prut-col

tr = 16 GPa

Rutile

Anatase

Columbite

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.

IV. CONCLUSIONS

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(U≈5 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.

ACKNOWLEDGMENTS

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

calculations.

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