Jackson S.D., Hargreaves J.S.J. Metal Oxide Catalysis

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8.4 Calculation of Surface Structure 365

of the water O atom (O

) with a low coordination Al ion through a dative bond in

which charge is donated from the adsorbate into the Lewis acid center. This leads

to a calculated binding energy of 97 kJ mol

− 1

which increases by a further 30 –

40 kJ mol

− 1

on heterolytic dissociation of an O

–

H bond to form two surface hydroxyl

groups. The range of energies stems from the choices possible for the surface

oxygen that receives the H

in the ﬁ nal state.

The free energy of dissociation based on constrained dynamics also showed that

the calculated barrier depends on the geometry of the ﬁ nal state. If an O neighbor

of the Al

to which molecular oxygen was adsorbed is used to receive H

, the

transition state is a strained four - membered ring and a barrier of 28 kJ mol

− 1

obtained. This reduces to only 9 kJ mol

− 1

when the pathway to place H

on a second

neighbor oxygen is used. The second neighbor has the advantage that the transi-

tion state formed is a six - membered ring and so the O

–

surf.

angle is nearer

to linear at 166 ° , compared to 135 ° in the case of the nearest neighbor.

The geometry of the transition state has also been found to be important for the

dissociation of HF on α - Al

(0001) and the subsequent halogen exchange with

based on DSOLID calculations [102] .

One of the easiest cleavage planes on rutile - structured TiO

corresponds to

the (110) surface. In the bulk structure [103] all Ti atoms are six coordinate with

four equivalent equatorial Ti

–

O bonds (1.97 Å (PBE)) slightly shorter than the

two axial (2.00 Å (PBE)). The preferred termination of the (110) surface is shown

in Figure 8.17 a and contains a mixture of sixfold (Ti

) ions joined by surface

bridging oxygen atoms (O

) and ﬁ vefold (Ti

) coordinate surface cations. Com-

pared to bulk centers the Ti

atoms have lost one axial Ti

–

O bond. The line of

bridging oxygen atoms on the bulk termination is mirrored below the surface

plane and so a neutral stacking unit consists of an atomic tri - layer. For the simula-

tion of this surface the choice of the number of these tri - layers in the slab is

important. Figures 8.17 b and c show the result of relaxations using the SIESTA

code with three and four tri - layer slabs respectively. In both cases the Ti

atoms

move down into the surface toward their remaining axial oxygen atom, but this is

a small effect compared with the surface ion movement seen for α - Al

(0001).

In addition, the four tri - layer slab is deformed and the surface appears to optimize

to a rumpled structure. In the thinner three tri - layer slab the surface is less dis-

torted from the bulk termination. Pacchioni and coworkers [104] have pointed out

that slabs with an odd number of tri - layers have a symmetry plane in the middle

of the slab that is absent for even numbers of stacking units. To investigate this

difference they used the PW91 functional, and compared results from the localized

basis set code CRYSTAL and plane - wave basis set code VASP. The calculated bulk

band gap from the plane - wave approach is around 0.2 eV lower than with the local-

ized basis set and this is ascribed to the use of ultra - soft pseudopotentials in the

VASP calculations. The different behavior for odd and even numbers of tri - layers

also results in oscillations in the band gap with slab thickness, which are mainly

due to changes in the calculated position of the bottom of the conduction band.

These states are associated with the coordinatively unsaturated Ti

surface atoms

and are composed largely of the d orbitals with lobes perpendicular to the surface.

366 8 Theory: Periodic Electronic Structure Calculations

However, unlike the α - Al

(0001) example, these are not separate gap states at

the surface but simply the lowest lying states of a band that also contains mixed

–

O crystal orbitals.

In Pacchioni ’ s calculations it was found that the occupied bonding Ti

–

O hybrid-

ized orbitals tend to form bonds that are stronger between the surface tri - layer and

the ﬁ rst sub - surface one than are found between tri - layers in the bulk. For the

even slabs this leads to a tendency to form a set of paired tri - layers, which give

rise to the observed rumpling, whereas for odd slabs a more regular spacing of

the tri - layers in the slab center is achieved.

The (110) surface of TiO

has also been subjected to LEED - IV experiments,

giving direct information on the surface structure [80] . Thompson and Lewis have

used the odd - numbered tri - layer slabs recommended by Pacchioni to consider the

Figure 8.17 (a) The TiO

(110) surface with surface Ti

and

atoms. Titanium, silver; oxygen, red; other atoms in the

slab, green. (b) The relaxed structure of the three tri - layer slab

viewed in cross - section. (c) The relaxed structure of the four

tri - layer slab. Atom colors for the surface layer of (a) are used

throughout (b) and (c).

8.4 Calculation of Surface Structure 367

accuracy of the surface geometry from calculations compared to this data [105] .

They employed the plane - wave basis set code VASP with the PW91 functional.

Their results, for the surface layer atoms, are reproduced in Table 8.6 and shows

that the surface relaxations are converged for slabs of nine tri - layers. This table

also includes results of earlier calculations by Gale and Harrison, who compared

LDA and HF methods for the same surface and who employed seven tri - layer slabs

[106] . Gale and Harrison found that the bridging O

atoms remain at their posi-

tions in the simple bulk termination whereas the experimental data and the more

recent calculations indicate a signiﬁ cant outward movement. However the metal

ion positions are better reproduced in the earlier work.

8.4.3

Inﬂ uence of Environment on Surface Structure

It is clear from experiments that the character of an oxide surface is dependent

on its environment. For example, surfaces may be prone to hydroxylation and

many catalytic processes require reduction/oxidation cycles that must create/

replenish oxygen ion defect sites at the surface. If the surface composition of an

oxide is different to the bulk, the surface energy can no longer be deﬁ ned as simply

the difference between the bulk and slab calculations per unit surface area.

However, the equilibrium surface energy can still be obtained from calculated

energies if we turn to the idea of the chemical potential [94] . The surface of an

oxide should be in equilibrium with both the oxide bulk and the atmosphere above

the oxide and so the chemical potential of all species must be constant in the three

regions, bulk, surface and atmosphere. The simplest case to consider is the effect

of oxygen partial pressure on the surface stoichiometry. For an oxide of formula

the formation of a non - stoichiometric surface in a slab simulation can be

thought of as the chemical reaction

mn N n

AO O A O

AeO

+→

(8.32)

Table 8.6 The displacement of surface atoms from their bulk positions in TiO

(110).

Expt [79] Method (Number of tri - layers in slab)

PW91 (5)

[105]

PW91 (7)

[105]

PW91 (9)

[105]

PW91 (11)

[105]

LDA (7)

[106]

HF (7)

[106]

0.25 ± 0.03

0.29 0.33 0.42 0.43 0.22 0.25

− 0.19 ± 0.03 − 0.10 − 0.07 − 0.03 − 0.03 − 0.17 − 0.17

0.25 ± 0.03

0.09 0.13 0.23 0.23 0.01

− 0.01

0.27 ± 0.08

0.24 0.27 0.32 0.32 0.13 0.11

368 8 Theory: Periodic Electronic Structure Calculations

where the formula on the right hand side is that of a simulation slab and the

additional term on the left, required to balance the equation, can be thought of as

coming from gas - phase oxygen. The term N

deﬁ nes the oxygen excess in the

slab and can be negative.

The surface free energy, γ , is just the free energy for the reaction of Equation

8.32 per unit surface area of the slab. If a slab simulation geometry gives two sur-

faces of area S :

γµ=

−−

()

slab

AO eO O

(8.33)

where G

slab

is the free energy of the slab representation of the surface and g

the free energy per stoichiometric unit of the bulk material. The chemical poten-

tial, µ

, is used to obtain the free energy of the excess O species. It is useful to

deﬁ ne the oxygen excess per unit surface area, Γ

, when Equation 8.33 becomes:

γµ=

−

()

−

slab

AO O O

(8.34)

To link this deﬁ nition of surface energy to calculations on relaxed slabs we now

make the assumption that the entropy in the slab and bulk oxide free energy

terms can be neglected. Then the surface energy based on calculated quantities

gives

γµ=

−

()

−

slab

AO O O

(8.35)

in which E

slab

is the energy of the geometry optimized slab deﬁ ned by the right

hand side of Equation 8.32 and e

is the energy per formula unit of a relaxation

of the bulk. Although Equation 8.35 deﬁ nes the surface energy, it still requires the

chemical potentials for the excess oxygen to be set. The oxygen chemical potential

under any given conditions of temperature T and partial pressure P

can be

estimated using:

µµ

(

)

(8.36)

where the superscript 0 is used to indicate standard conditions and R is the molar

gas constant. Usually the zero of chemical potential would be taken as

the value for dioxygen gas under such standard conditions. However, using

calculated data the energy is referenced to a state in which all atom cores and

valence electrons are isolated and so an estimate of the energy of the dioxygen

molecule is required to continue the calculation. The assumptions used by DFT

and other ﬁ rst principles methods that can be applied to the solid state mean that

8.4 Calculation of Surface Structure 369

the accuracy of the calculation of such absolute values of molecular energy is not

great. It is better to estimate the standard state chemical potential using a cycle

which gives µ

as the difference between more reliable quantities such as lattice

energies.

For example the standard energy of formation of the oxide is the energy for the

reaction:

As O g A O s

2mn

()

→

()

(8.37)

for which,

∆Ggmn

mnf

()

=− −µµ

(8.38)

Since the metal will usually have a standard state which is solid the values

and

can be obtained from extrapolation of DFT optimization calculations.

Accurate values of ∆G

mnf

()

are tabulated for the common oxides [56] , for

example the value for Al

is − 1576 kJ mol

− 1

and so Equation 8.38 provides a route

without explicit calculations on O

Equation 8.38 also sets the lower limit of the range of values accessible for µ

since reduction of the oxide will occur spontaneously into the pure metal and

dioxygen gas if the free energy for Equation 8.37 is positive. This means for a

stable oxide:

AO A

−gmg

(8.39)

An upper bound for µ

can be estimated from the point at which O

(g) begins

to condense on the surface. The thermodynamically stable composition of the

surface can now be predicted over the range of accessible partial pressures of O

by using the pressure dependence of the chemical potential from Equation 8.36

in Equation 8.35 .

This approach was ﬁ rst applied to α - Al

(0001) by Finnis and coworkers [107]

who demonstrated that the Al termination which forms the type II non - polar

surface is stable under normal conditions. Elevated oxygen partial pressures

(several tens of atmospheres) would be required to make an oxygen - terminated

surface the preferred form. Reduction of the surface will not be favorable until the

lower limit of oxygen partial pressure is met, at which point bulk reduction is

favorable anyhow.

These thermodynamic arguments can also consider the hydroxylation of sur-

faces by including the chemical potential of hydrogen and/or water [108] . This type

of analysis shows that Al

surfaces of the α - phase are either Al or OH terminated

under all accessible conditions. However, the inﬂ uence of the environment is

different for each Miller index owing to variation in the calculated surface energies.

370 8 Theory: Periodic Electronic Structure Calculations

Phase diagrams speciﬁ c to each experimentally observed face have been derived,

which show that conditions should exist for which the (0001) surface is dehydroxyl-

ated, to give an Al termination, while other surfaces remain fully hydroxylated.

For transition metal oxides the surface composition can be altered more readily

by the partial pressure of oxygen. For example, RuO

(110) has been shown to move

from an over - oxidized state for high O chemical potential to a reduced form at the

lower end of its accessible range [109] . In such cases the electrostatic arguments

against “ polar ” surfaces have to be adapted in the light of surface cation reduction.

It should also be remembered that these calculations give the thermodynamic

stability of a given surface composition and do not take account of any kinetic

barriers to the required changes.

8.4.4

Transition Metal Oxides with Partially Filled d Bands

The band picture of electronic states in solids works well when there is strong

overlap between neighbors in the lattice. This leads to broad bands and delocalized

electronic states. The early successes of band theory were in the description of the

physical properties of metals such as electrical and thermal conductivity [110] .

However, in transition metal oxides band structure alone is often not sufﬁ cient.

Consider a d

system such as Ni

in NiO. The usual picture would be to draw

localized orbitals at each Ni site which are split into three t

and two e

energy

levels by the crystal ﬁ eld. The eight d electrons per ion would then ﬁ ll the energy

levels as shown on the right in Figure 8.18 . This is a highly localized picture of

the electronic structure since the sets of d electrons are associated strongly with

particular nuclei. If movement of electrons occurs between centers we have to

remove an electron from one of the d

centers, leaving a d

, and shift it to a neigh-

bor which becomes d

. The electron – electron repulsion between the d electrons

will be proportional to the number of pairs of electrons present at each center.

With two d

ions we have 56 electron pairs, while the d

+ d

in isolation have a

total of 57 electron pairs. So the movement of electrons between centers requires

energy, effectively creating an energy gap. This type of argument was developed

by Mott and Hubbard to explain the insulating properties of transition metal

oxides with partially ﬁ lled d shells. They introduced a parameter, U , which is used

to represent the Coulomb term as a function of d - orbital occupation so that the

relative energies of alternative arrangements can be estimated.

Figure 8.18 The localized electron populations of the t

and E

states for two Ni

ions in NiO. The average

conﬁ guration is d

but transport of electrons through the

material would produce d

and d

centers.

8.4 Calculation of Surface Structure 371

A purely band structure - based treatment of this problem results in very narrow

bands for the d orbitals because of the weak overlap between neighboring Ni

centers. However the band would be partially ﬁ lled and so it would be concluded

that the system is metallic. This is the result generated by a simple LSDA approach,

as shown early on by Terakura and coworkers [111] . In the Mott – Hubbard analysis,

the localized picture becomes important when U is greater than the band width

predicted by band theory. In addition the electron conﬁ guration affects the

exchange energy. For example, the two electrons in the triplet d

conﬁ guration will

also have an exchange interaction and this is lost if one spin is reversed to produce

a singlet. This means that the e

energy levels appear lower in the triplet state than

the singlet so that an “ exchange gap ” can be introduced.

For transition metal oxides such as NiO the Mott – Hubbard model gives much

better agreement with experiment; localization of the electrons in this way explains

the anti - ferromagnetic ordering observed in NiO and its electrical insulator proper-

ties [112] . The terms required to calculate the electron – electron interaction energy

on a site, E

UHF

, can be written in terms of the d - electron occupations, using a UHF

approach:

UHF

−

()

′

−

′

≠

′

∑∑

ωω

(8.40)

where the sum over ω is over the spin states α and β , with − ω signifying the

opposite spin state to ω . The sum over m and m ′ considers each of the d states at

the site. In this way the ﬁ rst term takes into account the Coulomb interaction

between electrons of opposite spin while the second term is for Coulomb interac-

tions between electrons of the same spin, with a correction for the exchange

interaction (rather confusingly given the symbol J ).

Of course some of these interaction terms are already present in an LSDA cal-

culation and so to incorporate this into a general DFT scheme requires these

duplicate contributions to be removed. One way to do this has been proposed by

Dudarev and coworkers [113] . They have derived a functional incorporating the U

and J parameters of the Mott – Hubbard model which is written:

LSDA U LSDA U++

−

()

−

()

∑

ωω

(8.41)

The advantage of this expression is that U and J appear only as their difference

( U − J ) and so it requires only a single parameter to be determined for any particu-

lar transition metal oxide under consideration. The J parameter shows little

system - to - system variability and is taken to be around 1 eV in most cases. Based

on a comparison of the calculated DOS and experimental electron energy loss

spectra ( EELS ) the U parameter for NiO was set at 6.2 eV. This gave a ground state

in which the Ni spins are aligned within (111) planes and neighboring planes have

372 8 Theory: Periodic Electronic Structure Calculations

opposite spin as shown in Figure 8.19 . This anti - ferromagnetic ordering and the

calculated site magnetization are in good agreement with experimental data. On

a practical level, the ordering of spins in this way means that the crystallographic

fcc unit cell cannot be used for the simulation as it contains an odd number of

(111) planes, and a supercell has to be employed which is commensurate with the

spin system. The calculated density difference between the LSDA + U and LSDA

calculations is plotted in Figure 8.20 . This shows that the inclusion of the Hubbard

model leads to a reduction of density in the interatomic region between Ni

and

2 −

ions and increased localization of charges at the metal centers. The LSDA cal-

culation therefore overemphasizes the importance of covalent bonding between

the ions compared to the LSDA + U results.

More recently Rohrbach and coworkers have used NiO as an example to test

the implementation of the Hubbard model in the VASP code [114] . Their work

allows the combination of the PW91 gradient corrected functional with the on - site

Coulomb and exchange corrections given in Equation 8.41 . To set the U parameter

Rohrbach and coworkers plotted the bulk lattice parameter, metal center magnetic

moment and band gap over the range U = 1 to 9 eV. The best compromise for

accurate agreement with experiment on these three properties was U = 6.3 eV,

practically the same value as used in the earlier calculations of Dudarev and

coworkers [113] . The effect of the U term can be quantiﬁ ed by comparing results

of calculations based on PW91 alone with those at this optimal value of U. Figures

8.21 a and b show an overlay of the DOS from unrestricted calculations for the

majority and minority spin states, respectively. At the PW91 level the band gap is

underestimated at 0.5 eV compared to the experimental values of between 4.0 and

4.3 eV. The calculations also indicate a simple Mott – Hubbard band gap with the

top of the valence band consisting of metal majority spin states of e

symmetry

Figure 8.19 The antiferromagnetic AF

state of NiO. The (111)

planes through Ni

centers are shown with arrows indicating

the orientation of the majority spin for each plane. Note that

the atoms at the top and bottom corners of the cell are

crystallographically, but not magnetically, equivalent. Nickel,

blue; oxygen, red.

8.4 Calculation of Surface Structure 373

along with minority t

states and the bottom of the conduction band being just

minority spin e

states. The O(2p) contribution at these energies is relatively minor.

This means that the energy gap is created by the difference in energy levels for

different spin states. The majority degenerate e

states are occupied by two elec-

trons and so have an exchange contribution to their energy whereas the minority

states are empty. Thus it is an “ exchange splitting ” of the e

states that creates

the gap, which is the classic Mott – Hubbard argument for insulating behavior in

this type of oxide. In the PW91 + U this exchange splitting of the metal d orbitals

is increased to around 9.5 eV, because the electron localization increases the sta-

bilizing effect of the exchange interaction on the occupied majority e

states. Now

the top of the valence band has a largely O(2p) character, with contributions from

both majority and minority Ni t

states. The lowest lying conduction band states

are mainly from the minority spin e

states, suggesting that the band gap is now

of mixed charge - transfer and Mott – Hubbard type. The band gap has also widened

to 3.2 eV, in much closer agreement with the experimental measurements.

Rohrbach and coworkers went on to consider the NiO(100) surface. Since NiO

is isostructural with MgO, this is a non - polar ﬂ at surface corresponding to one of

the faces of the cubic unit cell. Reference to Figure 8.19 shows that the surface

will also have an anti - ferromagnetic structure with rows of surface cations having

parallel spins along the lines in which the (111) planes intersect the surface.

Relaxation with PW91 + U shows a very small (1%) inward movement of the

surface layer, but no buckling of the type observed for MgO(100) is reported.

Buckling only occurs at the LSDA level and has not been observed experimentally.

Figure 8.20 Charge density difference [LSDA + U ] − LSDA

(U = 6.2 eV) for bulk NiO taken through a (100) plane of the

rock salt cubic unit cell, centered on a Ni

ion. Taken from

ref [113] .

374 8 Theory: Periodic Electronic Structure Calculations

Figure 8.21 Calculated densities of states for NiO, (a) bulk

structure at PW91 (solid lines) and PW91 + U (dashed lines)

for the majority spin, (b) as (a) for the minority spin.

PW91 calculation and (d) breakdown by layer for PW91 + U .

In each case energies are referenced to the top of the valence

band. Taken from ref. [114] .