Improved Tight-Binding Charge Transfer Model

and Calculations of Energetics of a Step on

Rutile TiO2(110) Surface

Emile Maras,⇤,† Nicolas Salles,‡ Robert Tetot,‡ Tapio Ala-Nissila,† and Hannes

Jonssonk

Department of Applied Physics and COMP CoE, Aalto University School of Science,

Finland, SP2M/ICMMO, Universite Paris Sud, UMR 8182, F91405 Orsay Cedex, France.,

ICB, Universite Bourgogne, CNRS UMR 5209, Dijon, F-21078, Department of Physics,

Box 1843, Brown University, Providence RI 02912-1843, Department of Applied Physics,

Aalto University School of Science, Finland, and Faculty of Physical Sciences, University

of Iceland, Reykjavık, Iceland

E-mail: e1000.3000@gmail.com

⇤To whom correspondence should be addressed†Department of Applied Physics and COMP CoE, Aalto University School of Science, Finland‡SP2M/ICMMO, Universite Paris Sud, UMR 8182, F91405 Orsay Cedex, France.¶ICB, Universite Bourgogne, CNRS UMR 5209, Dijon, F-21078§Department of Physics, Box 1843, Brown University, Providence RI 02912-1843kDepartment of Applied Physics, Aalto University School of Science, Finland?Faculty of Physical Sciences, University of Iceland, Reykjavık, Iceland

1

Abstract

A second-moment, tight-binding charge equilibration (SMTB-Q) model for the

atomic interactions in TiO2 is refined by comparison with results of density functional

theory (DFT) calculations within the generalized gradient approximation, and used to

study the atomic structure of the h111i step on the rutile (110) surface. The model is

parametrized to reproduce the structure and energetics of a rutile crystal and its low

index surfaces, but also reproduces well anatase crystal and its surfaces. The applica-

tion to a stepped surface represents an extrapolation to lower coordinated atoms and

two extensions of the model were made. First, a continuous dependence of the e↵ective

radius of the O atoms on atom coordination was introduced and second, a covalent

O-O interaction was added. The revised SMTB-Q model is then found to reproduce

well the relative energy of local minima on the DFT energy surface. In particular, we

demonstrate that the refined model is useful in global optimization studies by using

it to search for reconstructions of the h111i step on the rutile (110) surface. The low

energy configurations generated with the model were used as input in subsequent DFT

calculations. A low energy reconstruction involving a large corrugation along the step

edge is found in this manner. When additional TiO2 units are introduced, a recon-

struction previously found by Martinez et al. (U. Martinez et al., Phys. Rev. B 2011,

84, 205434) is also successfully reproduced. However, in the present model the atomic

forces are not reproduced accurately as compared to DFT, which shows that additional

terms such as angular dependent terms may be needed to improve the accuracy.

————————————–

1 Introduction

Titania (TiO2) is of great interest for a variety of applications such as photocatalysis and

solar cells.1–4 The catalytic properties of TiO2 surfaces have been shown to be greatly en-

hanced by the presence of defects, such as steps and oxygen vacancies.5–8 To understand

2

the physical and chemical properties of these non-ideal surfaces, it is crucial to know their

atomic structure. Numerical calculations can provide important insights but require both

accurate representation of the interatomic interactions and exhaustive search for the low-

est energy arrangements of the atoms, which at low temperature represent the most likely

atomic structures. The atomic interactions can be accurately described by solving the elec-

tronic structure problem using, for example, density functional theory (DFT) within the

generalized gradient approximation (GGA), although this level of theory is known to have

problems describing the electronic properties and atomic structure at defects such as oxygen

vacancies.8 Furthermore, DFT calculations are computationally demanding and can only be

used for global optimization (GO) of the atomic structure of relatively small systems. To

overcome this limitation, it is advantageous to use faster although more approximate models

such as classical, empirical potentials in combination with GO methods.

The modeling of atomic interactions with empirical potentials in solids such as TiO2

which have mixed ionic and covalent bonding is a challenge, however. Several approaches

have been proposed, see for example a recent review by Liang et al.9 Such models have

mostly been developed and used for configurations with highly coordinated atoms, while low

coordinated defects have been less well characterized. A second moment tight binding charge

equilibration model (SMTB-Q10) for TiO2 has been presented and shown to successfully

reproduce the experimental properties of titania crystal as well as surface energy of low

index surfaces.11 This is a variable-charge model in which the iono-covalent metal-oxygen

bond is described by means of the tight-binding formalism for approximate description of

the electronic structure of the oxide.

A vicinal surface with h111i steps on the TiO2(110) surface (see Figure 1a and Figure

1b) was studied recently by Martinez et al. 12,13 and by Hardcastle et al. 14 who did DFT

calculations of the (451) surface unit cell shown in Figure 1a. Two systems were considered

with and without one extra TiO2 unit. They carried out a GO study of both systems. With-

out the extra TiO2 unit, no configuration of lower energy than that of the bulk truncated

3

[110]

[001]

(451)

111

(a)

(b)

Figure 1: Top (a) and perspective (b) view of the h111i step on the rutile TiO2(110) surface.The large grey spheres are Ti atoms, while the small red ones indicate the O atoms (theterrace atoms are light red and gray).

4

configuration (BT) was found. However, after adding an extra TiO2 unit, a reconstruction

lowering the energy of the h111i step by 0.33 eV was found (see Figure 2b). The correspond-

ing atomic coordinates can be downloaded from Ref. 15. In the reconstructed configuration

(hereafter referred to as h111iR1), the two extra oxygen atoms sit close to their crystalline

positions, whereas the extra Ti atoms sit in the oxygen octahedron halfway between the

upper and the lower terraces. The stability of this configuration can be explained from the

increase in coordination of step atoms compared with the bulk truncated configuration. As

illustrated in Figure 2, the coordination of two of the Ti atoms increases by one when adding

the extra TiO2, while the extra Ti atom gets a coordination of six as in the crystal. One

interesting feature of this reconstruction is its high corrugation which can be used as a seed

to grow one dimensional strands.16 The GO algorithm used by Martinez et al. 12,13 was based

(a)

5

46

5

(b)

66

6

55

Figure 2: Top view of (a) the bulk truncated h111iBT step and of (b) the reconstructionh111iR1 found by Martinez et al. 12,13 Colors are as in Figure 1, except that in (b) theextra TiO2 is emphasized by using orange and blue for O and Ti atoms, respectively. Thecoordination number of some of the Ti atoms is also indicated.

on a genetic algorithm17 generating new configurations which were systematically relaxed

using DFT. Due to the high computational cost of DFT, only three TiO2 units could be

included in the GO (the other atoms were only allowed to relax) thus limiting the range of

possible configurations.

The purpose of the present work is to benchmark the SMTB-Q model on defect structures

5

with lower coordinated atoms than in previous tests, in particular for the h111i steps on

rutile TiO2(110), and refine its applicability to such configurations. Compared to DFT,

this empirical potential is much less computationally demanding, allowing us to explore a

large set of atomic configurations. To this end, we have carried out extensive GO calculations

with the revised SMTB-Q model and subsequently relaxed the most promising configurations

using DFT. The previously found reconstruction of the step with additional TiO2 units12,13

(see Figure 2b) was reproduced in these calculations using the revised SMTB-Q model.

Additionally, we found a previously unreported energy lowering reconstruction of the step

without an additional TiO2 unit.

The article is structured as follows. The presentation of the methodology is in Section

2. The SMTB-Q model and its extended version are presented in Section 2.1. The DFT

calculations are described in section 2.2. A comparison of the revised SMTB-Q model and

DFT results is presented in section 2.3. The global optimization is presented in Section 2.4.

The results of the simulations are presented in Section 3. Finally, a summary and discussions

are in Section 4.

2 Methods

2.1 The SMTB-Q model

In this section, the SMTB-Q model is reviewed for completeness.10,18,19 The modifications

introduced to improve its accuracy at low coordination are then presented.

2.1.1 Review of previous version of the SMTB-Q model

The SMTB-Q model is based on a quantum description of oxides proposed by Goniakovski

and Noguera,20,21 in which the equilibrium charges are determined by a self consistent charge

equilibration, the so-called QEq approach.22 The QEq formalism allows ionic charges to vary

in response to changes in the local environment of the ions, which is crucial when considering

6

surfaces and defects. Moreover, in the SMTB-Q model, the ionic-covalent metal-oxygen bond

is described by means of the tight-binding formalism which takes into account the electronic

structure of the oxide, in contrast to other variable-charge models. The covalent bonding

energy is, for example, reduced as the ionic charges increase. This makes the model stable

with respect to charge transfer.

We consider an oxide CNCONO

, where NC and NO give the proportion of the cations, C,

and oxygen atoms, O. The total energy Etot of the system is the sum of the repulsive energy

Erep, the ionization energy Eion, the covalent energy Ecov, and the Coulomb energy Ecoul:

Etot = Erep + Eion + Ecov + Ecoul. (1)

Repulsive short range interactions. The repulsive energy is given by

Erep = E

CCrep + E

COrep + E

OOrep (2)

where E

CCrep and E

COrep represent the cation-cation and cation-oxygen repulsive interactions

and are described by a short range Born-Mayer pair potential (eq 3 and eq 4):

E

CCrep =

XC

i

XC

j<i

aCC exp

�pCC

✓rij

r

0CC

� 1

◆�(3)

E

COrep =

XC

i

XO

j

aCO exp

�pCO

✓rij

r

0CO

� 1

◆�(4)

where the sumXA

is carried out over all atoms of type A, rij is the distance between

atoms i and j, r0CC and r

0CO are reference atomic distances, and aCC, aCO, pCC and pCO are

adjustable parameters. The repulsive energy between a pair of O atoms is described by a

Buckingham potential of the form

E

OOrep =

XO

i

XO

j<i

↵ exp

✓�rij

⇢

◆(5)

7

where ↵ and ⇢ are adjustable parameters.

Ionization energy. The ionization energy is expanded up to second order with respect to

the atomic charges Qi:

Eion =X

i

EAi + �AiQi +1

2JAiQ

2i (6)

where Ai specifies the atom type (C or O) of the ith atom. The summation is over all atoms

of the system, and Qi corresponds to the charge of atom i. EA is the energy of a neutral

atom of type A and is zero since our energy reference is the state with isolated neutral atoms.

�A and JA are the electronegativity and hardness of species A, respectively.

Covalent energy. The covalent energy is divided into an anion-cation interaction and an

anion-anion interaction as

Ecov = E

COcov + E

OOcov . (7)

The term E

COcov describes the covalent bonding between the cations and anions :

E

COcov = �

X

i

0

@X

j,Aj 6=Ai

fcut(rij, rl1, rl2)⇠2COB

2NAi exp

�2qCO

✓rij

r

0CO

� 1

◆��QAi

1

A

12

(8)

where the sumP

j,Aj 6=Ai

is carried out over cations if atom i is an anion and over anions if

atom i is a cation. This expression of the covalent energy is derived from the quantum

model developed by Noguera and Goniakowski20,21 (for further details see Ref. 10). The

reference distance r

0CO is taken to be equal to the equilibrium first neighbor distance. The

parameter B =�p

NC +pNA

�/2, where NA and NC characterize the composition of the

CNCONO

oxide. ⇠CO corresponds to the hopping integral between an anion and a cation.

⇠CO and qCO are adjustable parameters (note that ⇠CO = NC

NO⇠OC). The term �QAi =

�QAi

⇣2 n0

NA� �QAi

⌘takes into account the e↵ect of charge transfer on the covalent bond,

where �QAi =��Q

formA

���|QAi | is the di↵erence between the formal charge of an atom of type A,

Q

formA , and the e↵ective charge of atom i (of type A), Qi. n0 = min (NCdC, NOdO) corresponds

8

to the number of shared electronic states between C and O atoms. The parameters dC and

dO correspond to the degeneracy of the outer electronic orbitals of C and O, respectively.

A Ti atom has a 3d24s2 electronic structure in its outer shells. We consider here that only

the d-electrons hybridize so dC = 5. The oxygen electronic structure is 2s22p4, but assume

only the p-electrons hybridize so that dO = 3. For TiO2 we have n0 = 5. The function

fcut(rij, rl1, rl2) smoothly terminates the short-range interactions within a region defined by

two cuto↵ radii rl1 and rl2.

fcut(rij, r1, r2) =

8>>>>>><

>>>>>>:

1 if rij < r1

dr

3�� 10

�3 � 15�4dr � 6

�5dr2�

if r1 rij r2

0 if rij > r2

(9)

where dr = rij � r2 and � = r2 � r1.

Coulomb energy. In order to minimize the error introduced when using a truncation

distance for the calculation of the long range Coulomb interaction, the Coulomb energy

Ecoul is calculated by applying the method of charge neutralization23 on the full Coulomb

interaction E

fullcoul, with a truncation distance of rcoulcut = 12.17 A. This truncation distance has

been chosen in order to be significantly di↵erent from any interatomic distance in the rutile

crystal. The full Coulomb energy is given by

E

fullcoul =

X

i

X

j<i

QiQjJcoulAiAj

(10)

where J

coulAiAj

is the Coulomb integral between charge densities ⇢i and ⇢j:

J

coulAiAj

=

Z Z⇢i(r1)⇢j(r2)

r12

d

3r1d

3r2. (11)

Here r12 is the distance between positions r1 and r2, ⇢i(r) = Mn |r � ri|n�1 exp ((�2n+ 1) / (4Ri))

corresponds to an s-type Slater orbital on atom i at position ri, where Mn is the normaliza-

9

tion constant, n is the quantum number of the outer valence orbital, and Ri is the covalent

radius of atom i. Ri was a fixed parameter in the original Qeq formulation.22 The signifi-

cance of Ri is more complex in a solid because it must depend on the surroundings of the

atom and in particular on its coordination. In previous studies using the SMTB-Q model, a

fixed radius of the cation was found to be satisfactory, while in order to obtain good surface

properties the radius of the oxygen had to vary with coordination. This is consistent with

the fact that the oxygen anion is more polarizable than cations. Each oxygen atom was

assigned a radius depending on its initial coordination.

Charge optimization. The total energy of the system depends on the charge distribution.

For a given atomic configuration, the charge distribution is optimized in order to minimize

the total energy. This is done by using a simulated annealing technique with a convergence

parameter of �c = 0.0002 eV or �c = 0.02 eV depending on the precision required.24 Note

that the charge optimization is the most time consuming part of the energy calculation. For

a typical atomic relaxation, the charge optimization slows down the calculation by roughly

a factor 200 and 20 with �c = 0.0002 eV and with �c = 0.02 eV, respectively. Using a

smaller value for �c thus significantly speeds up the calculation but decreases the precision

of the energy calculated. Nevertheless, the energy di↵erence is rather small and was found

to be systematically less than 0.1 eV for more than 50 configurations containing at least

100 atoms. In the context of global optimization, this precision is satisfying since the most

promising configurations can be refined afterwards and we thus used �c = 0.02 eV.

2.1.2 Revision of the SMTB-Q model

Interpolation of the Coulomb energy as a function of oxygen coordination. As-

signing a fixed radius to each oxygen atom depending only on its initial coordination cannot

be satisfactory in the context of GO since the coordination of an atom can change signif-

icantly in the process. The coordination of an atom should thus change continuously as

10

the atom is displaced. For a given configuration, the coordination, ci, of oxygen atom i is

calculated from

ci =XC

j

fcut(rij, rc1, rc2), (12)

where the sum is carried out over all cations and the function fcut(r, r1, r2) is defined in eq

9.

The contribution of a cation to the coordination of an oxygen atom decreases smoothly

from unity for rij < rc1 to zero for rij > rc2, where rij is the distance between the cation and

the oxygen atom. rc1 is chosen to be close to the nearest neighbor distance, and rc2 should be

smaller than the second nearest neighbor distance. For the present system we chose rc1 = 2

A and rc2 = 3.5 A. Note that the coordination as well as its first and second derivatives are

continuous functions of the atomic positions.

A change in the coordination of an oxygen atom leads to a change in the radius, which

in turn leads to a change in J

coulAiAj

and thus to a change in the Coulomb energy. For the

sake of simplicity, instead of interpolating the variation of the radius of an oxygen atom with

coordination, we directly interpolate the variation of JcoulAiAj

with coordination. The Coulomb

energy is thus expressed as

E

fullcoul =

X

i

X

j<i

QiQjJAiAj , (13)

where JAiAj is interpolated from the variation of JcoulAiAj

as a function of the coordination of

the oxygen atoms.

Table 1: Oxygen radius, RO, for di↵erent coordination numbers of the O-atoms.These values were obtained in previous studies in order to reproduce experimen-tal properties of the systems specified in brackets. The corresponding Coulombintegral Jcoul

OC between an O-Ti pair as computed from eq 11. The distance be-tween the atoms is 2 A.

coordination 2 3 4 6RO(A) 0.58 (TiO2) 0.543 (TiO2)10 0.537 (UO2)19 0.52 (SrO)10

J

coulOC (eV) 2.24140 2.29774 2.30653 2.33102

Several oxides with di↵erent coordinations of the oxygen atoms have been simulated with

11

the SMTB-Q model. The optimized values of the oxygen radius for various coordination

numbers are listed in Table 1. The general trend is that the radius of an oxygen atom

decreases with increased coordination. This is in good agreement with the prediction of

Wilson et al. 25 From the oxygen radius, we calculate J

coulAiAj

for an oxygen cation pair at a

given distance for each value of the oxygen coordination (see Table 1). Based on these values,

the following interpolation for JAiAj as function of the coordination was designed:

8>>>>>>>>>>>>>><

>>>>>>>>>>>>>>:

JCiCj = J

coulCiCj

JOiCj(ci) = J

coulOiCj

(c = 3) +hJ

coulOiCj

(c = 3)� J

coulOiCj

(c = 2)iFcut(ci)

+JcoulOiCj

(c=6)�JcoulOiCj

(c=3)

3(ci � 3� Fcut(ci))

JOiOj(ci, cj) = J

coulOiOj

(3, 3) + ci+cj�6

6(Jcoul

OiOj(6, 6)� J

coulOiOj

(3, 3))

+(Fcut(ci) + Fcut(cj))(JcoulOiOj

(3,3)�JcoulOiOj

(2,2)

2�

JcoulOiOj

(6,6)�JcoulOiOj

(3,3)

6)

(14)

where Fcut(c) is defined by

Fcut(c) =

8>>>>>><

>>>>>>:

c� 3 + � + Flc(3� �)� Flc(3), c < 3� �

Flc(c)� Flc(3), 3� � < c < 3 + �

Flc(3 + �)� Flc(3), c > 3 + �

(15)

Here Flc(c) = c [c3 � 12c2 � 4(3� 2�)(3 + �)2 + 6c(32 � �

2)] / (16�3). � is a parameter which

can be adjusted but which has to satisfy 0 < � < 1. In our simulations we use � = 0.3.

Note that for a pair of cations, JCiCj does not depend on the coordination since the

cations have a fixed radius. Equations 14 and 15 were designed to ensure that JAiAj and its

derivative are continuous functions of coordination. Figure 3 shows that eq 14 reproduces

well the behavior of JcoulOC as a function of coordination. For coordination larger than three,

J

coulOC increases almost linearly. Similar linear dependence is assumed for low coordination

12

with a continuous transition between both regimes from c = 3� � to c = 3+ �. The change

of the trend between low and high coordination as indicated in Figure 3 can be explained

from the fact that atoms having coordination larger (smaller) than three are bulk (surface)

atoms. Similarly, when considering a pair of oxygen atoms, the interpolation in eq 14 gives

satisfactory results. For instance for two atoms separated by 2 A and � = 0.3, we obtain a

square root of the sum of squared residuals S =qP

ci,cj[JOiOj(ci, cj)� J

coulOiOj

(ci, cj)]2 = 0.017

eV, where ci and cj take the values 2, 3, 4 and 6 in the summation.

Addition of attractive oxygen-oxygen interaction. As will be discussed in section

2.3, a comparison of the SMTB-Q model with DFT shows that the original SMTB-Q model

significantly overestimates the energy of configurations when two oxygen atoms are close to

each other. The agreement with DFT results was improved by adding an attractive oxygen-

oxygen interaction to include the possibility of covalent O-O bonding acting at short range

in the model. The additional term is

E

OOcov = �

XO

i

XO

j

fcut(rij, rOO1 , r

OO2 )aOO exp (�bOOrij) (16)

where r

OO1 , rOO

2 , aOO and bOO are adjustable parameters.

2.1.3 Parametrization of the revised SMTB-Q model

The goal of the parametrization is to reproduce properties of TiO2 oxide phases. For the

sake of consistency, the atomic parameters of the oxygen and the repulsive O-O interaction

terms were chosen to be the same as in previous studies19 without any adjustments. All the

other parameters of the model (except for the attractive O-O terms in eq 16) were adjusted

to minimize the chi-square function:

F =X

G

✓Gexp �Gcalc

�G

◆2

(17)

13

where G represents the physico-chemical properties to be reproduced, e.g. the lattice con-

stants, the cohesive energy and the bulk modulus of the rutile crystal, and �G the estimated

experimental errors which allow to give specific weight to each property. The experimental

values of the properties used in the parametrization as well as the corresponding estimated

experimental errors are given in Table 4. The minimization of the chi-square function is done

with regards to the adjustable parameters by means of the simplex method as implemented

in the MINUIT package from the CERN.26

The revised SMTB-Q optimized parameters are presented in Tables 2 and 3. The exper-

imental and calculated properties for rutile and anatase crystals are presented in Tables 4

and 5. The calculated properties match the experimental values well within 1% for both the

rutile and anatase crystals. We emphasize that the good agreement for anatase was obtained

without fitting to data on anatase, illustrating the transferability of the SMTB-Q model to

other crystalline structures.

Low index crystal surfaces of anatase and rutile crystal were simulated using a slab of at

least 22 A in thickness. The surface energy was calculated from

ES =1

2AS

(En + nEcoh) , (18)

where En is the total energy of a slab containing n TiO2 units, and AS is the area of the

surface of the sample. Ecoh is the cohesive energy of the corresponding crystal phase. The

factor 1/2 takes into account the existence of the two free surfaces of the slab. The calculated

surface energy is compared with DFT calculations in Table 6. Depending on the methodology

(i.e. functional, basis set, slab thickness), there are some di↵erences between the results of

di↵erent DFT calculations.11 The surface energy obtained with the revised SMTB-Q is in

the same range as the DFT results, unlike the results of some previous variable-charge

models.27,28

The parameters for the O-O covalent bonding when two oxygen atoms come close to-

14

gether, as given by eq 16, were obtained by fitting the energy of 28 configurations of the

h111i step on the (110) rutile surface where at least two oxygen atoms are at close distance.

These configurations were selected from a set of configurations provided by B. Hammer and

H. H. Kristo↵ersen from the GO study of Martinez et al. 12,13 The O-O attractive interac-

tion terms were obtained from a least squares fit to the total DFT energy and to the force

acting on the two oxygen atoms. The configurations had been relaxed using DFT but were

not further relaxed with the SMTB-Q model. Since the cuto↵ distance of the O-O covalent

interaction is smaller than the O-O distance in the crystals, this extra interaction (which

was not used in the crystal calculations) does not influence the calculated crystal properties

and no re-adjustment of the other parameters in Tables 4 and 5 was needed.

Table 2: Optimized atomic parameters of the revised SMTB-Q model.

Ti O�A (eV) 0 6.5719

JA (eV) 10.572 10.2219

RA(A) 0.734 See Table 1

Table 3: Optimized pair parameters of the revised SMTB-Q model. Note thata Ti-Ti interaction was not included in the model.

TiO bondA p ⇠TiO q rl1 rl2 r

0CO

0.134 eV 12.609 0.5434 eV 2.0965 3.6 A 6.0 A 1.95 A

OO bondaOO bOO r

OO1 r

OO2 ↵ ⇢

-20.86 eV 0.916 A�1

1.4 A 1.8 A 580.44 eV 0.354 A

2.2 DFT calculations

Our DFT calculations were performed at the generalized gradient approximation (GGA) level

using the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional35 as implemented

in CP2K.36 The pseudopotentials of Geodecker, Teter and Hutter37,38 were used in the

15

Table 4: Calculated and experimental properties used to fit the potential pa-rameters of the revised SMTB-Q model for the rutile TiO2 �G gives the errorparameter which was used for each property G in the minimized chisquare func-tion given in eq (17). a and c are lattice constants, u is an internal displacementparameter, Ecoh is the cohesive energy, B is the bulk modulus and Cij are theelastic constants.

G SMTB-Q Experimental �Ga(A) 4.615 4.59429 0.005c(A) 2.982 2.95929 0.005u 0.304 0.30529 0.005

Ecoh (eV) 19.98 19.930 0.1B (GPa) 210.6 21131 5C11 (GPa) 288 26831 13C33 (GPa) 375 48431 24C44 (GPa) 115 12431 6C66 (GPa) 154 19031 10C23 (GPa) 151 14731 7C12 (GPa) 172 17531 9

Table 5: Calculated and experimental properties for the anatase TiO2. a and c

are lattice constants, u is an internal displacement parameter, Ecoh is the cohesiveenergy and qOx is the charge of oxygen atoms.

Calculated Experimentala(A) 3.828 3.78529

c(A) 9.448 9.51429

u 0.2088 0.208129

Ecoh (eV) 20.0045qOx -1.26989

Table 6: Surface energy in J/m2 for the various low-index faces of rutile andanatase TiO2. The abbreviations under the DFT header refer to the functionalused (see the original references for details).

Surface SMTB-Q DFT DFT + UPBE32 LDA32,33 PW9111 PBE34

(001) 1.70 1.36 1.68Rutile (100) 0.67 0.68 1.04

(110) 0.42 0.31 0.84 0.48 0.86(001) 1.01 0.90 1.38

Anatase (100) 0.73 0.53 0.96(101) 0.58 0.44 0.84

16

calculations, with a cuto↵ energy of 800 Ry. The Kohn-Sham orbitals were expanded in a

Gaussian-type basis set (molopt ).39 The rutile lattice constants were calculated by relaxing

a supercell containing 2⇥ 2⇥ 4 unit cells. In order to model a slab, full periodicity was used

with a 24 A thick vacuum in the z direction. With this set-up, the calculations were fast

enough to study a large number of configurations generated during the GO procedure. Since

k points are not implemented in CP2K, we refined our calculations for the most promising

configurations by using GPAW40,41 with the PBE exchange-correlation functional and a real-

space grid. Periodic boundary conditions were applied only along the (x, y) surface plane,

whereas a vacuum of at least 10 A between the surfaces and the simulation box boundary

was included in the z direction. A grid spacing of 0.20 A was used. The area of the slab

surface times the number of k points in the Brillouin zone was always larger than 500 A2.

Note that unless otherwise stated, the DFT results presented here were obtained with CP2K.

Table 7 presents the lattice constant obtained with the di↵erent methods.

Table 7: Lattice constants of rutile TiO2 obtained with the revised SMTB-Qmodel and with DFT calculations using CP2K and GPAW. See text for details.

SMTB-Q CP2K GPAWa(A) 4.615 4.607 4.660c(A) 2.982 2.969 2.973

2.3 Comparison between SMTB-Q and DFT

Configurations generated in the GO search of Ref. 12 were used to check the accuracy of

the revised SMTB-Q potential. These configurations were relaxed with CP2K and then the

energy was evaluated using the revised SMTB-Q model without any further relaxation. The

energy relative to the bulk truncated configuration is calculated as

Erel = Econf � Eh111iBT+⇣n

TiO2conf � n

TiO2

h111iBT

⌘Ecoh, (19)

17

where Econf is the total energy of the given configuration, Eh111iBTis the total energy of the

bulk truncated configuration (Figure 2a). n

TiO2conf and n

TiO2

h111iBTare the number of TiO2 units

in the given configuration and in the bulk truncated configuration, respectively. Figure 4

shows that good agreement in relative energy is obtained between DFT and revised SMTB-Q

results after adding the O-O covalent bonding energy of eq 8 in the SMTB-Q model. The fact

that good agreement is obtained for a broad range of configurations shows that the model

can reproduce some of the main features of the DFT potential energy surface. However,

when a minimization is carried out using the revised SMTB-Q model, the agreement is lost,

as can be seen in Figure 5. The SMTB-Q energy typically becomes much lower than the

DFT value. The decrease is 2 eV on average with a standard deviation of 2 eV. For a few

configurations, the di↵erence in energy is larger than 5 eV.

Subsequent relaxation with DFT starting from the SMTB-Q relaxed configurations gives

the striking result that the final DFT energy after this cycle is di↵erent from the initial DFT

energy for more than half the configurations, as can be seen in Figure 5. This shows that

relaxation with SMTB-Q took the configuration from one basin of the DFT potential energy

surface (PES) to another one. As shown in Figure 5, in most of the cases the final DFT

configuration has a significantly lower energy than the initial one. While this represents a

shortcoming of the SMTB-Q model, it makes it e�cient for finding reasonable configurations

of low energy on the DFT PES.

2.4 Global optimization

The revised SMTB-Q model was used in combination with a GO algorithm which explores

the PES by finding saddle points.42–44 Given an initial local minimum, several searches for

first order saddle points are started after slightly displacing the undercoordinated atoms in

a random way. The climb up the energy surface from the vicinity of the minimum up to an

adjacent saddle point is then carried out using the dimer method.42 For each saddle point

found, the system is displaced slightly along the unstable mode away from the known local

18

minimum and a minimization carried out to identify the minimum on the other side of the

saddle point. One of the new local minima found in this way is then selected as the next state

of the system, etc. The algorithm is essentially a simulated annealing algorithm combined

with long time scale dynamics.44 The EON software45 was used for these calculations.

3 Optimisation results

A GO calculation was carried out for the (451) surface unit cell. This cell gives the smallest

possible periodicity along the h111i step on the TiO2(110) surface. First a simulation was

carried out to search for low energy reconstruction of the step without an additional TiO2

unit. Then one TiO2 unit was added and a GO search for a low energy reconstruction carried

out. If a second TiO2 unit is added then the original cell configuration is obtained again, so

further addition of TiO2 units was not pursued.

3.1 Reconstruction without additional TiO2

An energy lowering reconstruction of the h111i step on the TiO2(110) surface was found using

GO. The atomic structure is shown in Figure 6a and the corresponding atomic coordinates

can be downloaded from Ref. 15. We denote this structure as h111iR2 . The lowering of

the energy calculated with DFT using the GPAW software is 0.03 eV. This reconstruction

exhibits a remarkably large corrugation along the step edge with two TiO2 units forming

a strand stretching out from the step edge. To form this reconstruction, one of the TiO2

units which belonged to the step edge (see the color emphasized unit in Figure 6) moves

away from the step and rotates around the other Ti atom belonging to the step edge. The

two O atoms sit close to crystal sites, while the Ti atom sits in the oxygen octahedron

halfway between the upper and lower terraces. The oxygen atoms surrounding the four-fold

Ti atom rotate clockwise. The coordination of two Ti atoms increases from five to six in

this process, while the coordination of one other Ti atom decreases from six to five. The

19

reconstruction results in a net gain in coordination which explains why it lowers the energy.

Note that the underlying oxygen atom indicated by the white arrow in Figure 6b is displaced

significantly so as to maintain the five-fold coordination of the neighboring Ti atom. In the

GO study in Ref. 12, this oxygen atom was only allowed to relax and could not make this

large displacement, making this reconstruction out of reach.

3.2 Reconstruction with one additional TiO2 unit

We also carried out a GO study of the step in the presence of an extra TiO2 unit. The

most stable configuration found is the one previously identified by Martinez et al. 12 and is

shown in Figure 2b. The lowering of energy is 0.33 eV when the cohesive energy of the rutile

crystal is taken into account for the additional TiO2 unit (see eq 19). This reconstruction

is stable because the coordination of the two Ti atoms is increased by one compared to

the bulk truncated step. This reconstruction is similar to the h111iR2 configuration found

without the extra TiO2 (Figure 6b). In both cases the color emphasized TiO2 unit is at a

position where the oxygen atoms sit near the crystalline positions, and the Ti atom in an

oxygen tetrahedron. The main di↵erence between the reconstruction with and without the

extra TiO2 unit is that in the latter the color emphasized TiO2 atoms are taken from the

step edge.

4 Summary and Conclusions

In summary, the SMTB-Q model which is known to reproduce the properties of the crystalline

phases and low index surfaces of TiO2 has been modified to extend its applicability to lower

coordinated atoms such as surface steps. The accuracy of the modified SMTB-Q model was

checked by comparing with DFT results on a large number of configurations of the⌦111

↵

step on the rutile (110) surface. Our study shows that there is in general good agreement

between the energy obtained from the revised SMTB-Q model and DFT for configurations

20

which are local minima on the DFT PES, but the force evaluated from the SMTB-Q is not

zero and the configurations are unstable. The accuracy of the potential might improve by

readjusting the model parameters using DFT forces as input in the objective function. When

doing the parametrization one could thus fit not only the energy but also the forces for a set

of DFT configurations in the vicinity of the minima.46,47 A successful optimization would

thus ensure that a local minimum of the DFT PES also corresponds to a local minimum of

the revised SMTB-Q PES. It may however also be necessary to improve the SMTB-Q model

and take into account angular contribution of the covalent binding as is done in some other

approaches.48–50 This angular contribution is currently being implemented.

Despite these shortcomings, we have found that the modified SMTB-Q model is an

e�cient tool for generating relevant configurations that can be optimised further using DFT

based methods. In particular, by using the revised SMTB-Q potential in a GO scheme, a new

reconstruction for the⌦111

↵step on a rutile (110) surface was found. This highly corrugated

reconstruction is similar to the one found in Ref. 12, but does not require the addition of an

extra TiO2 unit. It is also evident that the relatively low computational cost of the SMTB-Q

model makes it possible to examine much larger systems than those amenable to DFT, thus

opening the possibility of finding previously unknown structures with larger unit cells.

Acknowledgement

We gratefully thank Bjørk Hammer and Henrik Høgh Kristo↵ersen for making their DFT

results available to us and for helpful discussions. We thank Andreas Pedersen and Kari

Laasonen for assistance with some of the calculations. This work was supported in part by

Aalto University School of Science through its Energy Science Initiative (ESCI), the Academy

of Finland through the FiDiPro program (H.J., grant no. 263294) and COMP CoE grant

(T.A-N., no. 915804). Support from COST Action CM1104 is also acknowledged, as well as

computational resources from CSC-Ltd.

21

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27

1 2 3 4 5 6

2.2

2.25

2.3

2.35

c

JOC

Figure 3: The magnitude of JOC in eV as a function of the coordination of oxygen atoms.The J

coulOC values from Table 1 are shown with filled circles, and the interpolation using

equation 14 with a solid line.

28

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Figure 4: Energy of various configurations of the h111i step on (110) rutile surface relative tothe bulk truncated step configuration. Results from DFT (gray squares), SMTB-Q withoutthe O-O attraction (open red circles), and the modified SMTB-Q with the O-O attraction(filled blue circles). The configurations are ordered according to the relative energy in theDFT calculations.

29

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Figure 5: Energy of various configurations of the h111i step on (110) rutile surface relativeto the bulk truncated step configuration. Results from DFT (grey squares), after relax-ation with the SMTB-Q model (filled blue circles), and subsequent DFT relaxation startingfrom the revised SMTB-Q relaxed configuration (open red squares). The configurations areordered according to the relative energy in the DFT calculations.

30

5

5

56

4

(a) (b)

66

45

5

Figure 6: Top view of (a) the bulk truncated h111iBT step, and of (b) a new reconstructionh111iR2 obtained from a global optimization algorithm and using the modified SMTB-Qmodel. This reconstruction exhibits a remarkable corrugation along the step edge. Largeatoms are Ti and small ones are O. The light colors indicate atoms on the top terrace. Thecoordination number of some of the Ti atoms is specified. The underlying oxygen atomindicated by the white arrow is displaced significantly in the reconstruction process so as tomaintain the five-fold coordination of the neighboring Ti atom.

31

Figure 7: TOC.

32