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

Подождите немного. Документ загружается.

8.4 Calculation of Surface Structure 375

Figure 8.21 c and d shows the density of states decomposed by layer for the ﬁ rst

three surface planes. At the surface the octahedral environment of the bulk Ni

ions is lost and so the e

orbitals are no longer degenerate. This is seen as a split-

ting of the minority spin states at the bottom of the conduction band in the PW91

calculation with the d

orbital ( z perpendicular to the surface) moving to lower

energies, giving a further reduction in the estimated e

to just 0.1 eV. A similar

effect is seen for the PW91 + U model, which gives a surface layer band gap of

2.9 eV. By the second sub - surface layer the symmetry of the cation sites appears

to be O

as the splitting is removed.

The adsorption of both CO and NO on the NiO(100) surface has been investi-

gated using photoelectron diffraction [115] giving a tilted geometry with Ni

–

C and

–

N distances signiﬁ cantly longer than predicted using gradient - corrected DFT.

Rohrbach and coworkers showed that the PW91 + U approach is able to give much

better agreement with the experimental data for CO and NO. [116] . The observed

tilted geometries are unexpected from comparable metal complex chemistry, for

which π back - donation from metal to ligand would favor a linear geometry.

However, in the metal oxide surface we have seen how exchange splitting shifts

the occupied metal d orbital bands to lower energies and so their mixing with

adsorbate orbitals becomes less efﬁ cient in a PW91 + U model. Molecular tilting

allows σ donation of charge from the diatomic to the surface d

from both the

5 σ and occupied π orbitals, which would be symmetry forbidden in an upright

adsorption mode. For the NO case, the partially occupied 2 π * orbital is hybridized

with the d

and the 5 σ contribution is almost negligible. Similar geometries for

NO adsorption on MgO, where there is no option for back - donation, have been

found in the PW91 calculations of Schneider and coworkers [117] .

The use of HF - like terms in Equation 8.40 suggests that just using HF itself for

these types of material may be preferable. Indeed Towler and coworkers have

shown that UHF correctly predicts the anti - ferromagnetic state to be lower in

energy than either the ferromagnetic state or a solution with spins paired on a

site - by - site basis. However, their estimated band gap of around 13.6 eV is a signiﬁ -

cant overestimate of the experimental value of 3.8 eV [58] . A much better agree-

ment between calculated and experimental band gaps can be obtained with the

hybrid B3LYP function, which gives 3.9 eV in this case [57] .

Indeed, in these problems involving electron localization, hybrid functionals

provide an alternative approach that has the advantage that exchange is applied

uniformly to the electron density and not just to the transition metal centers. In

the next section we look at examples using either DFT + U or hybrid functionals

to examine defect states on oxide surfaces.

8.4.5

Defects on the Surfaces of Transition Metal Oxides

Anion defects on oxide surfaces are important centers for adsorption and activa-

tion of molecules in catalysis. For example, oxidation reactions following a

Mars – van Krevelen mechanism require oxygen transfer from a surface to an

376 8 Theory: Periodic Electronic Structure Calculations

adsorbate, a process that creates an anion defect. Reduction of a metal oxide in its

simplest form can be thought of as the liberation of oxygen:

MO MO e O g

mn mn

=++

()

−12

(8.42)

In a calculation, m and n deﬁ ne the stoichiometry of the simulation cell, and

the two electrons that remain after a surface O

2 −

ion has formed part of the neutral

(g) molecule have been written explicitly. Using a periodic approach it should

be remembered that the defect will be repeated in neighboring cells. To obtain

results relevant to isolated defect sites requires cell sizes large enough to allow

interactions between the defect and its own images to be neglected.

In alkaline earth oxides such as MgO, the two electrons of Equation 8.42 are

known to be localized at the defect site by the Madelung potential of the solid [55] .

However, in reducible oxides such as TiO

there is the alternative of localizing the

electrons at metal sites to give two Ti

centers as shown schematically in Figure

8.22 . Reduction of a stoichiometric oxide in this way is a local event giving an

electronic structure in which the reduced metal centers are mixed in with fully

oxidized ions. As we have seen, the correct modeling of charge and spin states in

such a system requires a balanced treatment of exchange and correlation energies

and is currently best achieved using either hybrid functionals or a DFT + U

scheme.

Figure 8.22 also shows how heterolytic cleavage of water at such a defect site

appears to repair the structure of the surface but leaves the two metal sites reduced.

For the resulting hydroxylated surface it is possible to push electrons around to re -

oxidize one center, creating an alternative description of the surface with a hydroxy

radical. EELS experiments show little difference between the defective and hydroxyl-

ated surfaces [118] suggesting that water adsorption does not oxidize the defective

surface. However, ultraviolet photoemission spectroscopy ( UPS ) does distinguish

between them, suggesting that changes in the electronic energy levels at the vacancy

do occur [119] . Surface hydroxyls are known to be important in the photocatalytic

applications of TiO

, so Valentin and Pacchioni set out to investigate the electronic

Figure 8.22 Reaction scheme for the adsorption of water at

an O

vacancy on TiO

(110). After adsorption there is the

possibility of electron reorganization to give OH radical

species.

8.4 Calculation of Surface Structure 377

nature of the surface hydroxyls and oxygen defect structures to attempt to clarify

their electronic structure [120] . Their work compares the gradient - corrected DFT

functional PBE and the hybrid B3LYP methods using the CRYSTAL localized basis

set code. In the calculations the full electronic structure, including core states, is

optimized self - consistently, avoiding the use of pseudopotentials.

Figure 8.23 compares the calculated band structures from PBE optimized geom-

etries with B3LYP results both at the PBE geometry and after re - optimization of

the atomic co - ordinates using the hybrid functional. In the PBE/PBE result the

bands due to the hydroxylated Ti atom orbitals are contained within the conduction

band states, much as the surface states of the perfect surface. Density and spin

plots also show that the two electrons associated with the Ti

centers in Figure

8.22 are spread over the Ti ions of the entire slab. The geometries obtained from

singlet and triplet electronic states are also identical and show only minor changes

of bond lengths around the defect site. The Fermi level in this PBE calculation lies

in the conduction band and so the pure functional is unable to reproduce the

electronic structure even qualitatively. Fixing the PBE calculated geometry and

carrying out a B3LYP calculation (PBE/B3LYP) does move the defect states out of

the conduction band but the separation is small. The B3LYP geometry optimiza-

tion gives different geometric structures for the singlet and triplet states, with the

triplet 0.6 eV lower in energy. This B3LYP/B3LYP calculation gives a larger band

gap than PBE as expected from the work of Zhang and coworkers on the

dependence of E

on the amount of exact exchange presented in Figure 8.6 . The

calculated adsorption energy for water at the vacancy site following the scheme

in Figure 8.22 is 1.8 eV, which is in reasonable agreement with the experimental

heat of adsorption of H

O to this surface estimated using temperature programmed

desorption (1.4 eV). The band structure from the B3LYP optimization shows two

bands around 1 eV below the bottom of the conduction band. These bands are

Figure 8.23 The calculated band structure for the O

defective TiO

(110) surface with the geometry optimized

using PBE and bands calculated at the PBE (PBE/PBE) or

B3LYP (PBE/B3LYP) levels and with the geometry and bands

calculated using B3LYP (B3LYP/B3LYP). From ref. [120] .

378 8 Theory: Periodic Electronic Structure Calculations

narrow, indicating electron localization, and closely associated with two particular

Ti surface ions. The ﬁ rst is the Ti

between the two hydroxyl groups, while the

second is a nearby Ti

. The d orbitals involved are shown along with the calculated

density of states plot of Figure 8.24 . The lower of the two bands is for the Ti

ion

(Figure 8.24 a) while the singly occupied d orbital of the under coordinated Ti

ion gives a gap state nearer the conduction band (Figure 8.24 b). The geometry

relaxation of the surface leads to elongation of the Ti

–

O bonds by up to 0.1 Å

owing to the charge localization at these metal centers.

Figure 8.24 Calculated density of states for the O

defective

TiO

(110) surface (lower plot) and the same surface after

hydroxylation (upper plot). Only the majority spin DOS is

shown in each case. The metal orbitals responsible for the

gap states are picked out in images (a) – (d). From ref. [120] .

8.4 Calculation of Surface Structure 379

Calculations at the B3LYP level on the defective surface also show two localized

d orbital states in the band gap. These are associated with the Ti

site formed by

removal of the O

atom (Figure 8.24 d) and a nearby Ti

ion (Figure 8.24 c) with

the former higher in energy than the latter. This arrangement gives a smaller

repulsion between the two electrons left by the vacancy formation than would

placing them either side of the defect site as suggested in Figure 8.22 . Comparing

the calculated density of states in Figure 8.24 for the defective and hydroxylated

surface conﬁ rms that there is a shift in the occupied energy levels, with the level

associated with the Ti coordinated by O

oxygen atoms shifting down in energy

by around 0.6 eV on adsorption of water. However, both structures indicate that

the electrons left at the defect site by Equation 8.42 reside at metal centers, so that

water does not act as a oxidizing agent in this case.

As an example application of the DFT + U method to the study of anion vacan-

cies we turn to molybdenum oxide, MoO

. Oxides of molybdenum have been

known for some time to show activity for oxidation catalysis in which the oxide

surface itself acts as the oxygen donor in Mars – van Krevelen - based mechanisms

[121 – 123] . The structure of molybdenum trioxide is shown in Figure 8.25 and

consists of bi - layers formed from edge - sharing MoO

octahedra, although there is

considerable distortion away from O

symmetry at Mo. There are three structurally

distinct oxygen atoms: asymmetric, symmetric and terminal. The asymmetric

bridging oxygen atoms are twofold coordinate, forming one long (2.25 Å ) and one

short (1.73 Å ) bond with two Mo atoms in the same layer. The symmetric bridging

oxygen atoms are threefold coordinate, with two equal bonds (1.95 Å ) to Mo atoms

in the same layer and one much longer interaction (2.33 Å ) with a Mo atom of the

Figure 8.25 Two bilayers of bulk α - MoO

. The three different

types of O atoms are indicated (terminal, symmetric and

asymmetric). Oxygen, red; molybdenum, blue.

380 8 Theory: Periodic Electronic Structure Calculations

other sub - layer. Finally, the terminal oxygen is bonded to only one Mo atom,

forming the shortest Mo

–

O bond in the system (1.67 Å ). The crystal unit cell is

orthorhombic, with parameters a = 3.9628, b = 13.855 and c = 3 6964.Å

′

for a unit

cell containing Mo

[124] . The bi - layers are formed parallel to the (010) plane

with no chemical bonds between them, making this the easiest surface to

cleave.

MoO

is actually the fully oxidized end - member of a series of materials with

stoichiometries from MoO

to MoO

. For example, Wang and coworkers [125]

identiﬁ ed two molybdenum sub - oxides Mo

and Mo

by means of electron

diffraction and high - resolution transmission electron microscopy ( HRTEM ) in

combination with image simulation. These are derived from MoO

by crystallo-

graphic shearing to accommodate the large number of oxygen vacancies. However,

the materials are related to the basic structure of MoO

with the Mo

(100)

surface built up of MoO

(010) terraces with MoO

(100) edges and MoO

(001) kinks

[126] . Since the unit cells of these types of materials are complex, modeling studies

have concentrated on the perfect MoO

(010) and its point defects as representative

of the terrace regions in the sub - stoichiometric material.

Different theoretical approaches have been used to study MoO

surfaces. Two

major ab initio periodic HF studies were published almost at the same time by

Papakondylis and Sautet [127] in 1996 and Cor à and coworkers [128] in 1997.

Papakondylis and Sautet used the CRYSTAL program to carry out HF - level calcula-

tions, with correlation energy estimates added using the PW91 functional with the

HF density. They showed how the structure can thought of as built up from an

MoO

molecule to the three - dimensional structure via a chain polymer and ribbon

based on the components of the crystallographic bi - layers. The evolution of the

band structure from the MoO

molecular orbitals could be followed through this

sequence. The (100) surface of MoO

contains penta - coordinated Mo atoms, which

are accessible Lewis acid centers. At the time automated geometry optimization

was not possible, but the interaction of adsorbed H

O with the ﬁ xed surface were

optimized “ by hand ” based on the position and orientation of the molecule over

the surface. This showed that the adsorption of H

O molecules on these sites is

favorable but the electron transfer between the surface and adsorbate was actually

quite small.

Cor à and coworkers also used HF and the localized basis set approach of

CRYSTAL with a posteriori correlation corrections from PW91 [127] . They opti-

mized the bulk structure by systematically varying each degree of freedom inde-

pendently, starting with the weakest interaction ﬁ rst. This is the interlayer spacing

for which they found a minimum, both at the correlated and the HF level. The

GGA method gave an underestimated interlayer spacing while HF gave an over-

estimate, consistent with the expected under - binding in GGA approaches. The fact

that minima are found at all indicates the presence of a weak attractive Coulombic

force between bi - layers since the long - range correlation responsible for van der

Waals interactions are absent in both approaches. They then analyzed the ground -

state electronic properties of MoO

, giving the electron density map shown in

Figure 8.26 a. Here the difference between the calculated density and that of a set

8.4 Calculation of Surface Structure 381

of non - interacting ions placed at the lattice positions is plotted to show where

charge has shifted to because of bonds formed between the ions. There is a net

loss of charge from the interlayer region and a particularly notable increase in

charge density between the Mo at the center of the ﬁ gure and the associated ter-

minal O atom. This, along with a Mulliken charge analysis, shows that the nature

of the Mo

–

O interaction changes considerably according to the oxygen type, from

strongly covalent for the shortest bond (terminal oxygen) to a predominantly ionic

interaction for the longest bonds (interlayer Mo

–

O(symmetric)). The oxygen

atoms denoted O(2) and O(2 ′ ) in Figure 8.26 a are asymmetric, and this is clearly

reﬂ ected in the asymmetry of the accumulation of charge density between the

central Mo and O(2) or O(2 ′ ). The DOS plot from our own work [129] using PW91

functionals in the VASP code is shown in Figure 8.26 b. The calculated valence

band region has a width of 5.8 eV. The computed band gap is 2.1 eV and, as

expected for a GGA method, this is an underestimate of the experimental value.

MoO

is an n - type semiconductor with an indirect band gap between 2.9 eV and

3.15 eV [130] . The DOS shows that the valence band contains both metal and

oxygen contributions, conﬁ rming that there is a considerable covalent character

to the Mo

–

O bonds. Even so, the top of the valence band is largely O(2p) in

Figure 8.26 (a) Difference electronic charge

density map (calculated bulk minus isolated

2 −

ion densities), displayed in the

crystallographic plane containing the central

Mo and the vertices O(1)O(2 ′ )O(3 ′ )O(2) of

one isolated MoO

octahedron. Continuous,

dashed and dot - dashed lines correspond

to positive, negative and zero difference

respectively. The interval between the

isodensity lines is 0.005 a.u. (electrons a

−3

The map extends beyond the central

octahedron and includes part of the interlayer

space, denoted by A. Taken from ref. [128] .

(b) Total DOS curve (black solid) with

decomposition into molybdenum (dashed)

and oxygen (dot - dashed) contributions, for

the MoO

bulk system. The energy origin is

chosen as the highest occupied state and a

Gaussian smearing width of 0.2 eV is applied.

382 8 Theory: Periodic Electronic Structure Calculations

character and the bottom of the conduction band consists largely of Mo orbitals

so the band gap is of a charge - transfer type.

Equation 8.42 indicates that the removal of a surface oxygen ion leaves two

electrons to be accommodated by the surface. For the terminal oxygen vacancy on

the MoO

(010) surface we have compared PBE and PBE + U calculations using

the VASP code [129] . In the PBE calculations the two electrons move into states

in the conduction band and are spread evenly between the Mo centers of the simu-

lation slab. In these calculations the surface structure is changed considerably,

with one of the asymmetric oxygen atoms neighboring the defect moving to

replace the lost terminal oxygen atom (Figure 8.27 a). In the PBE + U case the two

electrons are localized, reducing the single Mo ion from Mo

to Mo

and giving

a triplet ground state. This is conﬁ rmed by a plot of the spin density difference

( ρ

( r ) − ρ

( r )) which shows the two electron spins localized at the metal center

where the terminal oxygen atom was removed (Figure 8.27 c). The corresponding

surface relaxation is considerably reduced, with the bridging symmetric and asym-

metric oxygen atoms staying in plane (Figure 8.27 b). As far as the rest of the lattice

is concerned, the electronic structure of the Mo

center must closely resemble the

original Mo

O species. The value of U in these calculations was set to a value

of 6.3 eV by comparing the calculated spin density with a reference cluster calcula-

tion at the B3LYP hybrid functional level.

Testing the accuracy of calculations on defects is more difﬁ cult, since experi-

mentally the defects are a minority and randomly distributed species. However,

we have been to make comparisons with surface spectroscopy results. Queeney

and Friend have synthesized oxidized Mo(110) surfaces with speciﬁ c types of

oxygen coordination and characterize them using infrared and electron energy loss

techniques [131, 132] . During the oxidizing process, two distinct vibrational fre-

quencies υ (Mo

O) peaks were resolved at 992 and 1016 cm

− 1

using infrared spec-

troscopy. These are higher in frequency than any other surface vibrations owing

to the high bond order between Mo and terminal oxygen atoms. When total oxida-

tion was achieved by maintaining a ﬂ ow of O

following high - temperature oxida-

tion, all terminal (Mo

O) sites were populated, and a single υ (Mo

O) at 996 cm

− 1

was observed. Using the numerical second derivative approach the vibrational

frequencies of M

term

were calculated for our perfect surface and for the terminal

oxygen groups neighboring the defect site. This gave values of 1023 cm

− 1

and

995 cm

− 1

suggesting that the high - frequency peak seen experimentally is in areas

that are relatively defect free while the lower frequency may be assigned to termi-

nal oxygen atoms near to defects in the partially oxidized ﬁ lm.

Our ﬁ nal example is a study using DFT + U of the oxidation of CO over ceria.

This concerns the reduction of surface sites with the simultaneous oxidation of

an adsorbate. According to PW91 calculations [108] the most stable surfaces of

fully oxidized CeO

are (111), (110) and (100), with calculated surface energies

of 0.68, 1.01 and 1.41 J m

− 2

respectively. Nolan and Watson [133] have carried out

calculations on CO adsorption over these preferred surfaces using PW91 coupled

with the Hubbard model using a value of U = 5 eV based on studies of the

defective CeO

surface [134] . The adsorption of CO on the (111) surface is found

8.4 Calculation of Surface Structure 383

Figure 8.27 (a) Chain of sites along the asymmetric oxygen

direction taken from the PBE relaxed MoO

(010) surface

with a terminal oxygen atom vacancy; the structure of the

perfect surface is overlaid as a line drawing. (b) As (a) for the

PBE + U optimization. (c) The calculated spin density for the

PBE + U calculation showing the full simulation cell for parts

(a) and (b).

384 8 Theory: Periodic Electronic Structure Calculations

to be typical of physisorption, with an interaction energy of only − 25 kJ mol

− 1

and

no structural alteration of either CO or the surface. Over the other two planes,

initial placement of the CO molecule in a surface site bridging two oxygen ions

resulted in the spontaneous formation of a carbonate

2−

species. This involves

removal of two O atoms and two electrons from the surface and so two Ce centers

are reduced from Ce

to Ce

at the same time. In the calculated spin densities,

Figure 8.28 , the two reduced Ce centers are clearly seen with the additional elec-

trons entering f states.

8.5

Conclusions

We have tried to highlight in this chapter the differences between the major

computational methods for the simulation of the electronic structure of materials

based on periodic boundary conditions. HF theory has a well deﬁ ned theoretical

framework but ignores correlation energy from the outset. This is introduced

via the density in LSDA and GGA methods but at the expense of a less well

deﬁ ned treatment of exchange. Even so the surface structure of stoichiometric main

group oxides is well described by either approach. The electronic states are less sat-

isfactory, with HF tending to severely overestimate and DFT to underestimate band

gaps. For catalysis, the empty states in the conduction band or in gap states are

responsible for Lewis acidity and so an accurate description of the energies of

these states is desirable. We have seen how hybrid functionals can correct for the

underestimation of the band gap in DFT by including some proportion of exact

exchange.

For transition metal oxides with partially ﬁ lled d states, the balance between

Coulomb and exchange interactions is even more critical. DFT strongly underes-

timates the exchange gap present in materials with magnetic order such as NiO,

while it is again overestimated in HF. This can be corrected either through hybrid

functionals or with the DFT + U approach.

Figure 8.28 The spin density (blue) for CO

chemisorption on the (100) surface of ceria.

Taken from ref. [133] . Oxygen, red; Carbon,

grey; Cerium, light yellow.