J . Phys. Chem. 1993,97, 12725-12730
12725
Structure and Stability of Small TitaniudOxygen Clusters Studied by ab Initio Quantum Chemical Calculations Anden Hagfeldt,+aRobert Bergstriim,* Hans 0. G. SiegbahnJ and Sten LuneU'*i Department of Physical Chemistry, BOX532, University of Uppsala, S - 751 21 Uppsala, Sweden, Department of Physics, Box 530, University of Uppsala, S- 751 21 Uppsala, Sweden, and Department of Quantum Chemistry, Box 518. University of Uppsala, S-751 20 Uppsala, Sweden Received: September 28, 1993.
Ab Initio calculations have been performed on TinOzA clusters, with n = 1-3 and 6 = 0, 1, as well as some of their ionized species. The geometries of the different clusters were optimized, and vibrational analyses were performed at the ground-state equilibrium geometries. It was shown that the T i 4 bond lengths are essentially determined by the coordination numbers of the respective atoms, independent of n. Theclusters are characterized by having very low pendant oxygen vibrational frequencies. The ionization energies for the (TiOz),, clusters were significantly higher than those of oxygen-deficient, Tin0Z61, clusters. Some anomalous features of the n = 1 cluster, TiOz, as compared to clusters with n > 1, are discussed. Recent experimental mass spectrometric observations on titanium oxide clusters are discussed in the light of the present findings. Introduction Recent advances in the synthesis and characterization of nanometer-size semiconductor clusters have generated much interest in their physical properties.' The bands develop as a function of cluster size, and Brus2 concludes that states at the band extrema are more sensitive to size than are states near the band center, because of electron delocalization. This suggests that the band gap should increase with decreasing size. As one approaches the nanometer-sizedomain, however, a point is reached where the influence of surface atoms can no longer be ignored in describing the physics of these systems. Surface electronic states may have energy levels within the band gap, yielding a lowering of the band gap energy. Quantum chemical calculations using a molecular approach can provide useful information to elucidate the importance of the surface atoms. In a previous work, we applied this approach, using the semiempirical INDO method, tocalculate the densityof states for Ti02 (rutile) clusters containing up to 42 atoms.' Titanium dioxide has been studied as the photoanode in photalectrochemical cells for a long time in our laboratories. Following the production of an efficient solar cell based on a dye-sensitized sintered colloidal Ti02 film: this work has been intensified during the last year. The colloidal Ti02 particles constituting the film have a mean diameter of about 16 nm, yielding a high surface-to-volumeratio for the particles as well as a high roughness factor for the colloidal film, being typically lo00 for a 10 pm thick film. To avoid recombination processes in such films, a detailed understanding of the chemical nature of the photoformed electrons and holes, and the role these species play in heterogenous reactions at the TiOz/electrolyte interface, is required. A detailedtheoretical investigationon the importance of surface states of the Ti02 semiconductor should start with the use of as accurate quantum chemical methods as computer time permits on very small titanium/oxygen clusters. This is the p u r p e of the present paper. The results of these calculations can then be used for comparison with larger systems using more approximate methods. Small titanium/oxygen clusters, [Ti,,Ozd]+, where n equals 1-8 and 6 ranges from 0 to 4, have been produced by Yu and Freas.5 Titanium/oxygen cluster ions were produced by bom+ Department of Physical Chemistry. 8 Department of Physics. 8 Department of Quantum Chemistry. .Ahtract published in Advance ACS Absrrucrs, November 1, 1993.
barding either titanium foil or titanium dioxide powder with an energetic beam of xenon atoms. They were also formed by sputtering metal foil in a high-pressurefast-atom-bombardment (HPFAB) ion source, with 02 serving as a buffer/reactant gas to produce the observed metalloxygen cluster ions. Collisioninduced dissociation (CID) experiments of the produced titanium/ oxygen cluster ions were also performed. The overall observation by Yu and Freas was that the predominant cluster ions or fragments that were formed corresponded to 6 = 1. The HPFAB mass spectra showed that [(Ti02),,]+, 6 = 0, clusters were not abundant for n greater than 3. Some of the ions produced in the HPFAB source showed different CID fragmentations. It was suggested that, e.g., the [Ti204]+ cluster would fragment to lose an oxygen atom, forming a 6 = 1 cluster. Other 6 = 0 cluster ions were not sufficiently abundant to obtain their collision spectra. The low abundance of any [(TiOz),,]+ clusters was discussed in terms of loss of pendant oxygen atoms, either in a rapid decomposition after formation or by collisional activation within the ion source. Another possibility was proposed in terms of a large ionization potential for the (TiOz)" clusters. Thus, a high ionizationpotential of the 6 = 0 clusters may be determining the low abundance of their positive ions. In order to investigatethe geometrical structures of the cluster ions, Yu and Freas used a simple ionic pair-potentialmodel. With such a scheme, several low-energy isomers (local minima) were found for each homolog. It was observed that the most abundant clusters in the HPFAB mass spectra were the clusters with the lowest energy per atom within this ionic model ('energetically most stable"). Hence, the 6 = 1 clusters had the lowest energy per atom, except for n = 2 clusters, where the Ti204 structures were found to be as stable as the Ti203 clusters. To our knowledge only few ab initio quantum chemical calculations have been performed on small titanium/oxygen systems.6-8 Ramana and Phillips' performed an SCF-CI calculation on the Ti02 molecule with a polarized double-t basis set. The ground state of the Ti02 molecule was found to be partially . ionic with a configuration of T?+ (3d2) and 0- ( 2 ~ ~ )Their ground-state geometry agreed well with experiment, while the agreement between the computed vibrational frequency V I and the experimental value was less satisfactory. With the results from Yu and FreasS at hand, we have in this study applied ab initio calculationsfor optimizing the geometries of small neutral and singly ionized titaniumloxygen clusters. Calculations have been performed on (TiOz),,, n = 1-3, clusters
0022-365419312097-12725$04.00/0 0 1993 American Chemical Society
Hagfeldt et al.
12726 The Journal of Physical Chemistry, Vol. 97, No. 49, 1993 TABLE I: Calculated (at the SCF Level If Otherwise Not Indicated) and Experimental Values of Bond Length RO (A), Bond Angle 8 0 (deg), Total Energy -I& (hartrees), and the u1 Frequency (cm-l) for the Ti02 Molecule 102 151.024 ECP-MB 1.52 111 153.084 ECP-DZ 1s o 120 987.455 1331 1.63 STO-3G 122 992.746 1337 1.63 STO-4G 112 992.818 1292 STO-4G(*)‘ 1.58 114 998.624 810 1.65 D Z t P (CI)b 110f 15 962.0 1.62k0.08 exp‘ a The asterisk indicates a Ti 4d polarization function. These results are taken from ref 7. C The experimental values are taken from ref 10.
and on the oxygen-deficient configurations TiO, Ti203, and Ti30s. For n = 2, we obtained two stable isomers, denoted (TiO2)z:l and (Ti02)2:2.A vibrational analysis at the ground-state equilibrium geometry of the clusters has also been performed. On the basis of our calculations, we then discuss some of the observations of Yu and Freas. The effect of geometrical relaxation of these clusters on the band structure will be investigated in another paper.g
Figure 1. Structure of the (Ti02)2:1cluster geometry, optimized with the STO-4G(*) basis set, yielding a CZh symmetry. The smaller spheres denote Ti atoms, and the larger spheres, 0 atoms. Interatomicdistanccs: R12 = 2.61 A, Rl3 = 1.79 A, Rl, = 1.57 A, R34 = 2.45 A. Interatomic angles: (3)-(1)-(2) = (4)-(1)-(2) = 43.2’, (5)-(1)-(3) = 117.8’. 2
Computational Method Geometry optimizations were performed on the Ti,O2,,4 clusters, using as starting points the structuresthat were obtained with the pair-potential ionic model in ref 5 . The calculations were performed at the HF-SCF level. For the neutral clusters, spin-restricted (RHF) wave functions were used, while for the positively charged clusters the spin-unrestricted (UHF) method was adopted. The GAUSSIAN 90 programlo was used for the calculations. No symmetry restrictions were imposed for the geometry optimization of the clusters. A gradient optimization, using the Berny method,10 was used for the geometry optimization. First we made a steepest descent search, starting from the initial structure. Full convergence was never reached with this method, but the “best” geometry was used as an input for another optimization, now with NewtonRaphson steps in the Berny optimization. In order to obtain convergence, we needed to calculate the analytic Hartree-Fock force constants a t every point, which may be related to the fact that these types of clusters have potential energy minima with several very soft vibrational modes. A frequency analysis was then performed for each final optimized structure. One of the objectives of the present work is a comparison between a number of titanium/oxygen clusters. In order to do this in a consistent way, we need to apply the same basis sets for all clusters. The limitations on the basis set size were therefore set by the largest cluster considered, which was (Ti02)s. For this cluster, the largest tractable basis with the available computational resources turned out to be an STO-4G basis with an extra set of 4dpolarizationfunctionsaddedon theTiatoms. The performance of this basis, as well as some smaller ones, was tested on the Ti02 molecule. Since it was found to be quite satisfactory for our purposes (see below), it was used in the rest of the investigation.
Results and Discussion (1) Equilibrium Geometry for the Ti02 Molecule: Choice of Basis Set. As mentioned above, several different minimal or near-minimal basis sets were used to calculate the bond length, bond angle, total energy, and the vibrational frequency of the symmetric stretch, VI,for the Ti02 molecule. These parameters have also been measured experimentally11 for matrix-isolated TiOz, and a comparison of the calculated values with those experimentally observed is given in Table I. Included in Table I are also the results from a previously published ab initio calculation? using a polarized double-{ type basis set. The values of Ro and 8 0 are within the experimental errors for both minimal basis sets, with three or four Gaussian functions,
Figure 2. Structure of the (Ti02)2:2cluster geometry, optimized with the STO-4G(*) basis set, yielding a C30symmetry. Interatomic distances: Rl5 = 2.35 A, Rl3 = 1.93 A, R25 = 1.71 A, R16 = 1.57 A. Interatomic angles: (3)-(1)-(2) = 45.8’, (6)-(1)-(3) = 134.2’.
respectively. The calculated vibrational frequency is less satisfactory compared to the value observed experimentally. Ramana and Phillips performed an SCF-CI computation on the Ti02 molecule with a double-{type basis set with polarization functions added.’ They discussed their discrepancy in the V I value in terms of neglect of anharmonicity. They also found that thecalculated frequency of the symmetric stretch, i.e. the YI value, differed very little between an SCF treatment and a CI calculation, indicating that the SCF and CI potential energy surfaces were very nearly parallel near equilibrium. The VI frequency from our STO-4G(*) calculation is found to be much higher compared to the experimental one (34%), unusual for SCFcalculationsusing larger basis sets.12 This suggests that the minimal basis set together with the single-determinant wave function yields too steep a potential curve in the vicinity of the equilibrium structure. The frequency obtained with the STO-4G(*) basis is slightly better than that obtained with the STO-3G or STO-4G sets without 4d polarization functions. Geometry optimization of the Ti02 molecule using the standard effective core potential (ECP) available in the GAUSSIAN 90 program gave less satisfactory agreement of the equilibrium structure (cf. Table I) and was not considered further. The STO-4G(*) basis set yielded a good agreement for the bond length and the bond angle compared to the experimental values of the Ti02 molecule (cf. Table I) and was therefore used in the geometry optimization of the other clusters. (2) Equilibrium Geometries for T i 2 0 w ( a = 2,3; 6 = 0, 1) Clusters. It should be noted that no symmetry restrictions were imposed in the geometry optimizations of the different clusters. However, symmetric structures were, indeed, obtained after the optimizations, and the point group for each cluster will be given in the text. The optimized geometries for the ground states of (TiOZ),,, n = 2 and 3, are shown in Figures 1-3. For n = 2, two isomers (local minima) are found from our calculations. The lowest energy structure, (Ti02)Zlr has two pendant oxygenatoms, yielding a C2h symmetry (see Figure I), while in the other n = 2 structure, denoted (TiO2)z:. with a C3” symmetry, one of the pendant oxygen atoms is placed between the titanium atoms (see Figure 2). Yu and Freas found actually four isomers for the n = 2 clusters with the pair-potential ionic model (see Figure 6 in ref 5). For example, they found a structure with four oxygen
The Journal of Physical Chemistry, Vol. 97, No. 49, 1993 12727
Small Titanium/Oxygen Clusters
9
Y
F i p e 3. Structure of the (Ti02)3 cluster geometry, optimized with the STO-4G(*) basis set, yielding a C, symmetry. Interatomic distances: R12 = 2.62A, Rz4 = 1.78 A, Rl4 = 1.80 A, R28 = 1.57A. Interatomic angles: (2)-(1)-(3) = 176.1°,(2)-(1)-(4) =43.4',(2)-(1)-(5) = 43.2', 139.4', (4)-(1)-(6) = 122.0°, (3)-(1)-(4) = 133.8', (3)-(1)-(5) (5)-(1)-(6) 125.7', (4)-(1)-(7) = 118.1', (1)-(2)-(8) 130.8'.
4
Figwe4. Structureof the (Ti203):1 cluster geometry, optimized with the STO-4G(*) basis set, yielding a C, symmetry. Interatomic distances; R12 2.59 A, R13 1.81 A, Rls 1.57 A, R23 = 1.74 A, R34 2.41 A. Interatomic angles: (3)-(1)-(2) = 39.7O,(5)-(1)-(3) = 113.1'.
atoms between the two titanium atoms. We tried to optimize this structure, but a potential minimum was not obtained, indicating that the oxygen atoms repel each other so much that this cluster will probably not be stable. Figure 3 shows the structure of the (TiO2)3 cluster. This cluster can be viewed as an extension of the (Ti02)2:, structure. Geometry optimization yielded a C, structure for the (TiO2)3 cluster. To study oxygen-deficient clusters, Ti,02,,4, with 6 = 1, we started from the two Ti204 isomers. In Figure 4 one of the two pendant oxygen atoms in the structure of Figure 1 has been removed, leaving the (Ti203):l cluster with a C, symmetry. To optimize the oxygen-deficient(Ti02)2:2cluster, denoted (Ti203): 2, we removed the pendant oxygen atom in Figure 2, leaving three 0 atoms between the Ti atoms. With no symmetry restrictions applied during the geometry optimization, one of the 0 atoms moved away and became a pendant oxygen atom, yielding the (Ti203):l cluster (see Figure 4). While imposing C3" symmetry, we obtained imaginary frequenciesof 95i cm-1, indicating at least one negative force constant. Our calculations therefore indicate that the (TizO3):~cluster is not energetically stable, although this structure was found to have the lowest total potential energy with the ionic pair-potential model (cf. Figure 5 in ref 5). The instability of the (Ti203):~cluster found in our calculation can be related to large electronic repulsion terms associated with an excessive concentration of oxygen atoms in a small volume. We should, however, point out that the numerical value of the imaginary frequency is relatively small, so that it cannot be completely ruled out that also this isomer could be stable at a higher level of approximation, particularly if electron correlation is included.l3 In an earlier paper,3 we drew attention to the importance of thecoordination number distributionwithin different Ti02 clusters for density of states structures as a function of cluster size. It may be useful to discuss the equilibrium geometries obtained for the different clusters in this work in the same perspective. We also note that a model for the activity and selectivity of catalyst oxides was recently developed by A r n a ~ d . 1The ~ model is based on the formation enthalpy of the oxide and on the bulk and the surface coordination numbers. It turns out that the oxygen adsorption-
desorption process on catalyst oxides reveals a "quantification" of oxygen adsorption heats, due to the ratio between different surface coordination numbers and the bulk coordination number. For a comparison with the solid-state Ti02 crystal, we note that the Ti-0 bond lengths are 1.95 A for the rutile structure and 1.89 A for anatase. The shortest 0-0 distances are 2.54 and 2.68 A, and the shortest Ti-Ti distances, 2.96 and 3.02 A, respectively. Table I1 summarizes the Ti-0 bond lengths, 0-0 and Ti-Ti distances, and the ionization energies, as well as the SCF total energies for the different clusters. The Ti-0 bond lengths for the (Ti02)2:1and (TiO2)3 clusters in Figures 1 and 3 between an oxygen pendant atom, O(I) (( 1) indicates that this oxygen atom is coordinated to one Ti atom), and the corresponding titanium atom, Ti(3),are 1.57 A for both clusters. The bond lengths are increased when the coordination number for oxygen increases,being 1.79 A for Ti(3)4(2)in Figure 1. With n = 3, we obtained Ti(3)-0(2) = 1.78 A and Ti(4)4(2) = 1.80 A, Le., a slight increase in T i 4 distance with increasing coordination number for titanium. The Ti-Ti distances are also very similar for these two structures, being 2.61 and 2.62 A, for n = 2 and n = 3, respectively. The shortest 0-0distances are between atoms 3 and 4 in the (Ti02)2:1cluster and between atoms 5 and 4 (or 7 and 6) in the (TiO2)3 structure, being 2.45 and 2.44 A, respectively. The structures in Figure 1 and 3 can be viewed as building a Ti02 polymer chain with the Ti02 molecule as the monomer unit. Remembering that the Ti-0 distance obtained for the molecule, i.e., n = 1, was 1.58 A, we see that the distances between the atoms in these clusters are little affected upon going from n = 1 to n = 3. The atom-to-atom distance is thus mainly determined by the coordination numbers of the respective atoms. Figure 2, showing the (Ti02)2:2 cluster, reveals another kind of structure. Here we have two inequivalent Ti atoms, i.e, Ti@) and a pendant Ti(3),atoms 1 and 5 in Figure 2. Compared to those of the clustersdiscussedabove, the Ti atoms have approached each other with a metal-to-metal distance of 2.35 A. There is also a significant difference in Ti-0 bond lengths. The results are 1.93 and 1.57 A for Ti(4)-0(2)and Ti(4)4(l),respectively. The bond length is 1.71 A for Ti(3)-0(2). The 0-0 distances have also decreased slightly compared to those of the structures in Figures 1 and 3, yielding a distance of 2.40 A. Comparing the cluster in Figure 2 with the structures of Figures 1 and 3 with two pendant oxygen atoms and two oxygen atoms between the Ti atoms shows that when more 0 atoms are placed between the Ti atoms, shorter Ti-Ti and 0-0 distances are obtained. Removal of one of the terminal oxygen atoms in Figure 1 yields theoptimizedstructuredenoted (Ti203):l in Figure4. Theoverall impression is that the structure remains relatively unaltered upon going from Ti204 to Ti2O3. The pendant oxygen-titanium bond length remains the same, being 1.57 A in Figure 4. The Ti to 0 distance between atoms 1 and 3 in Figure 4, i.e. Ti(3)-0(2), is found to be 1.81 A, which is an increase of 0.02 A compared to the same distance in the corresponding cluster in Figure 1. The bond length between atoms 2 and 3 is contracted compared to that in the (Ti02)2 cluster by 0.05 A, yielding a value of 1.74 A. Ionization of the (Ti203):l cluster yields only a small change in the geometrical structure (see Table 11). From the above observations, we may conclude that the coordination number for both the oxygen and titanium atom affects the Ti-0 bond length and also the Ti-Ti and 0-0 distances. Increasing the cluster size while the same coordination number distribution is maintained, by letting n grow from 1 to 3, does not noticeably change the interatomic distances, and the latter are essentially determined by the coordination number of the atoms. However, changing the symmetry will also affect the geometry of the different clusters. This can be exemplified by comparing the ( T i 0 ~ ) 2and : ~ the (TiO2)z:z clusters. The latter cluster has a pendant Ti(3) atom, yielding different Ti(3)-O(2) bond lengths, 1.79 and 1.71 A, respectively (cf. Table 11). (3) Vibrational Frequencies, Oxygen-Binding Energies, and Ionization Energies. An important indication of the stabilities
Hagfeldt et al.
12728 The Journal of Physical Chemistry, Vol. 97, No. 49, 1993
(A)Shortest Ti-Ti rad 0-0 distance^ (A), TOWEwrgie~,-& ( h r r t r ~ )pnd , Ioai~rti~a TABLE 11: T i 4 Boad Energies (eV)for the Different Clusters, Calculated in the STO-&(*) Basis (TheFree 0 Atom Ground-State Energy in the %me Basis Being -74.33740 ha&-) Ti4 Ti-Ti 0-0 -Eo ionizn energy 1.58 1,57(WU b 1.79(33-(2) 1.57('H1)
(Ti031
(Ti02)2:1 (TiOd2:z
2.61
2.45
992.8 18 1986.002
6.5 (4.0) 7.8 (4.1)
2.35
2.40
1985.987
7.1
2.59
2.41
918.414 1911.510
4.8 (4.4) 3.8 (2.3)
2.62('H3) 2.60('H2)
2.43 2.3F
2904.682
3.9 (2.1)
2.63
2.60 2.37
992.671 1985.851
15.3 14.3
2.64, 2.62
2.38,2.40
2979.037
13.1
2.62
2.37
918.252 191 1.427
13.5 12.1
2.62('H3) 2.63(4H2)
2.40 2.3lC
2904.603
11.7
1.57(3H1)
a According to Koopmans' theorem. Within parenthcses are given adiabatic ionization energies calculated from total energy differences. b The superscript notation (3)-(1) indicates that this bond length is between a Ti atom and an 0 atom with coordination numbers of 3 and 1, respectively. CThe 0 atoms are coordinated to the pendant Ti atom.
TABLE III: Vibrational Frequencies (cm-1) for the Different Clusters (Boldface Numbers: Vibrational Stretch Modes) level
cluster (Ti0211 (Ti02121 (Ti02122 (Ti02)s (Ti~03):1
(Ti203+):l cluster (TiOd3
1
2
3
4
5
6
7
8
9
10
11
12
356 102 185 66 120 173
1247 180 185 66 256 275
1292 220 415 143 300 364
334 415 148 505 532
382 535 171 670 630
497 535 275 873 843
684
847 758
890 889 318 1309 1382
915 889 387
1293 1071 582
1319 1312 673
13 674
14 85 1
15
16
851
879
and rigidities of different clusters is given by their vibrational frequencies. These are summarized in Table 111. We also illustrate the lowest, bending type, vibrational mode as well as the lowest mode with predominant T i 4 stretching character for each neutral cluster in Figures 5-8. The symmetry characters of the vibrational modes are given in the figure captions. For then = 1 ton = 3 series,i.e. theTiO2 molecule,the (Ti02)kI and the (TiO2)3 clusters, some interesting trends can be observed. The lowest vibrational mode for these clusters show a decreasing trend in frequency going from 356 cm-l for n = 1 to 102 and 66 cm-1 for n = 2 and 3, respectively. These vibrational modes are displayed for the two latter clusters in Figures Sa and 7a. We have not found any experimental value of the bending vibration, ~ 2 of , the molecule. Ramana and Phillips' obtained a calculated value for Ti02 of 353 cm-1, which agrees well with our result. The low frequencies of these structures show that these small Ti02 clusters are characterized by flat potential surfaces with closely spaced vibrational levels. The lowest vibrational mode in the (TiO2)3 cluster, with Y = 66 cm-l, would actually be excited to its u = 3 state at room temperature. This is in agreement with the observation that some clusters exhibit, in a given energy
547 299 891 945
level 17 88 1
338 964 1081
18
19
20
21
918
957
1308
1314
interval, a two-phase existence.15 In such cases, the cluster spends long time intervals as a solid, then jumps into a liquid form for extended time periods, then turns back into a solid, etc. Clusters experiencingsuch transformationscan be characterized by a broad shallow potential surface with closely spaced vibrational levels, whereas steep potential curves with more widely spaced energy levels are the characteristic of solid clusters. Looking at the first vibrational stretching mode of the same clusters (see Figures 5b and 7b), we observe the same trend; Le., the frquencies are decreasing with increasing n. For the Ti02 molecule, the lowest stretching mode is the asymmetric stretch, ~ 3 at , 1247 cm-I. For the symmetric stretch, vl, we obtained a value of 1292 cm-I. Experimentally,lo the values of u3 and ~1 have been measured to 935 and 962 cm-1, respectively. This indicates that our calculated vibrational frquencies are significantly overestimated. The (Ti02)2:1cluster yields a symmetric stretch for the terminal oxygen atoms at 497 cm-', while a value of 338 cm-I is obtained for the (TiO2)3 cluster, for the same type of stretching mode. To the extent that stretching frequency can be used as a criterion of bond strength (cf., however, below), the fact that the pendant oxygen atoms exhibit such low stretching
Small Titanium/Oxygen Clusters
The Journal of Physical Chemistry, Vol. 97, No. 49, 1993 12729
Figure 7. (a) Lowest vibrational level, having A’ character (top), and (b) first vibrationalstretching mode, A’ character, for the (TiO2)lcluster
Figure 5. (a) Lowest vibrational level, having B, character (top), and (b) firstvibrationalstretching mode, A,character, for the(TiO2)z:Icluster
(bottom).
(bottom).
Figure 8. (a) Lowest vibrational level, having A’ character (top), and (b) first vibrational stretchingmode, A’character, for the (TizO3):lcluster
(bottom).
Figure 6. (a) Lowest vibrational level, having E character (top), and (b) first vibrational stretching mode, A1 character, for the (Ti02)22 cluster
(bottom). frequencies for (Ti02)2:1 and (TiO& (note that the calculated values are expected to overestimate these frequencies by at least 30%; see above, section 1) fits very well with the observations from the experiment by Yu and FreasS on (Ti,Om)+ clusters. They found that, by sputtering a titanium foil in an HPFAB ion source, clusterswith 6 = 0 and n > 3 were not sufficiently abundant to be detected by the mass spectrometer. This is consistent with the fact that the vibrational stretching frequencies decrease strongly when n goes from 1 to 3. Thus, pendant oxygen atoms should be more easily removed from the larger clusters by collisional activation within the ion source. In Figure 6a we visualize the lowest vibrational frequency of the (TiO2)22 cluster at 185 cm-’. If we compare this to the (Ti02)zl structure, we see an overall increase in the frequencies for the (Ti02)2:2 isomer. This shows the effect of moving one of the terminal oxygen atoms to a position between the Ti atoms,
yielding a higher coordination number for this atom and as a consequence a steeper potential curve. The frequenciesare 2-fold degenerate except for the stretching modes, where the lowest vibrational stretch has a frequency of 547 cm-1 (see Figure 6b). This is an increase of 50 cm-1 compared to the lowest vibrational stretching frequency of the (TiO&l cluster. The (Ti02)2:2cluster may therefore be interpreted as having a more rigid structure than the (Ti02)k1 isomer. The total energy (see Table 11) shows a significant difference of 0.4 eV, the (Ti02)2:1cluster having the lower energy. The calculated vibrational frequencies of the oxygen-deficient (Ti203):l cluster are given in Table 111. Compared to that of the ( T i 0 ~ )cluster, ~ : ~ the lowest vibrational frequency, displayed in Figure 8a, has increased slightly, yielding a value of 120 cm-1. The first stretching type mode occurs at v = 505 cm-’ (see Figure 8b), which is an increase of only 8 cm-l. Theeffect ofionizationof the (TizO3):l cluster on thevibrational frequencies can be studied in Table 111. It is seen that the vibrational frequencieshave increased noticeably after ionization. The lowest vibrational frequency has increased by 53 cm-l, and the first stretching mode, by 27 cm-1, indicating that the ionized cluster is more rigid than the neutral one. To study in more detail the strengths of the pendant T i 4 bonds, the binding energies of these 0 atoms were also calculated
Hagfeldt et al.
12730 The Journal of Physical Chemistry, Vol. 97, No. 49, 1993
directly, from the total eneigies of the different molecular and atomic species involved. The latter energies are given in Table 11. The 0-binding energy in TiO2, obtained as the total energy of TiOz minus the combined energies of T i 0 and 0, is obtained as 1.8 eV, whereas the corresponding binding energies in (Ti02)2:1 and (TiOz)3 are 4.2 and 4.1 eV, respectively. The trend in 0-binding energy thus parallels the decrease in Ti-0 stretching frequencies upon going from n = 2 to n = 3. For n = 1, however, the two quantities behave very differently, which at first sight may seem puzzling. The reason is that, in contrast to e.g. a (stretching) vibration, the dissociation process is qualitatively different for n = 1 and n > 1. The binding in (Ti~03):lor Ti305 is very similar to that in the corresponding stoichiometric clusters; Le., all Ti-0 bonds are in both cases ordinary (polarized) single bonds. In the diatomic molecule TiO, in contrast, the Ti-0 bond is best described as an intermediate between a single and a double bond.6 The extra stability given to the T i 0 molecule by the higher bond order facilitates the removal of an 0 atom from TiOz, as compared to the larger ( T i 0 ~clusters, )~ explaining the anomalously low 0-binding energy in TiOz. It can be observed that, while reflecting the same decreasing trend as the stretching frequencies for n = 2, 3, the 0-binding energies of around 4 eV are nevertheless substantial. Easy removal of a pendant oxygen from the stoichiometric clusters therefore seems less likely as the sole explanation of the predominant abundance of 6 = 1 clusters in the work of Yu and F r e a ~ The .~ other important variable, the ionization energies, can be found in Table I1 for the different clusters. As seen from this table, the vertical ionization energies of the 6 = 1 clusters, Ti203 and Ti305, are only about half as large as those of the 6 = 0 clusters (3.8-3.9 eV, compared to 7.7-7.8 eV). The same holds for the adiabatic ionization energies, although these for natural reasons are much lower in both cases, 2.1-2.3 and 3.7-4.1 eV, respectively. This will obviously affect strongly the relative abundances of these two types of clusters in the mass spectra, in the observed direction. Additional data which can be extracted from Table I1 are the 0-binding energies in the ionized (stoichiometric) clusters, which are found to be 2.2, 2.4, and 2.6 eV, respectively, for the ions Ti02+,Ti204+,and Ti3O6+. These are substantially smaller than those in the corresponding neutral molecules and show that ifa stoichiometric cluster is ionized, it will lose an 0 atom much easier than before ionization. The greater abundance in the mass spectra5 of Ti,Ozn-l+ ions can then be understood as a combined effect of at least two factors, namely (i) that the 6 = 1 clusters are more easily ionized than the 6 = 0 structures and (ii) that pendant oxygen atoms dissociate relatively easily from the (TiO&+ clusters if such have been formed. The first mechanism, i.e. formation of Ti,OZ,,-l+ ions through direct ionization of TinOl,,-l clusters, should be accentuated for larger clusters, since the pendant-0-binding energies decrease when n increases.
Conclusions Starting from some of the [ T i n 0 2 ~clusters ] obtained with the pair-potential ionic model in ref 5, we have performed ab initio SCF geometry optimizations using an STO-4G( *) basis set. However, for some of the isomers in ref 5 , we did not obtain minima in the potential energy surface, indicating that the total neglect of covalent forces and metal-metal and oxygen-oxygen interactions in the ionic model leads to too many isomers and that only some of them are true local minima. Keeping the same coordination number distribution for the clusters,i.e. comparing the (TiOz) I, (Ti02)2:1,and (Ti0z)s clusters, we observe that the Ti-0 bond length and the Ti-Ti and 0-0 distances are very little affected. The interatomic distances are then determined essentially by the coordination numbers of the atoms.
As mentioned above (see section 3 under Results and Discussion), our vibrational frequencies are significantly overestimated compared to experimental values. Our conclusions are therefore drawn from an upper limit of the frequencies. Since we discuss the clusters in terms of their very low vibrational frequencies, indicating broad potential curves with closely spaced vibrational levels, our conclusions should be accentuated in more accurate calculations. With n going from 1 to 3, the first vibrational level as well as the first stretching mode decreases in frequency, yielding a low frequency of 66 cm-I for the (TiOz)3 cluster. Both these modes originate from the pendant oxygen atoms with a coordination number of 1. It is interesting to note that a temperature-programmed desorption spectrum of TiOz(s) only revealed one peak for oxygen desorption, corresponding to a surface oxygen atom with a coordination number of 1.I4 These findings are consistent with the trend indicated by the present cluster calculations. One of the experimental observations by Yu and Freas was that the formation of the 6 = 1 series was predominant and that there were not detectable amounts of (TiOz), clusters for n > 3 in the HPFAB ion source. One of the possible explanations given in their paper was that the pendant oxygen atoms could be removed from the clusters by collisional activation within the ion source. The fact that the vibrational frequencies for the stretching modes of the pendant oxygen atoms decrease when n goes from 1 to 3 in our calculations supports such an argument for larger clusters, n > 3. However, in the present study, a significant energy of 4 eV was calculated for the oxygen pendant atom to dissociate in both the (Ti02)Z1and (TiO2)3 clusters, indicating that we cannot explain the predominant abundance of the 6 = 1 clusters from this argument alone. Another explanation for the low abundance of (TiOz)nstructures was also discussed in ref 5 in terms of these clusters having a higher ionization energy compared to that of the predominant Tin02,,-] clusters. This hypothesis is confirmed by our calculations. A third factor which could enhance the relative abundance of 6 = 1 ions in the mass spectra was found to be the relatively low energy required to remove a pendant 0 atom from the Tin02,+ ions, if such have been formed in the primary process.
Acknowledgment. This work was supported by the Swedish Natural Science Research Council (NFR), the Swedish Research Council for Engineering Sciences (TFR), and the National Supercomputer Centre (NSC). Olle Bystrom is gratefully acknowledged for the preparation of the drawings. References and Notes (1) For recent reviews, see: (a) Bawendi, M. G.; Steigerwald, M. L.; Brus, L. E. Annu. Rev. Phys. Chem. 1990, 4 1 , 477. (b) Henglein, A. Top. Curr. Chsm. 1988, 143, 113. (2) Brus, L. E. New J . Chem. 1987, 1 1 , 123. (3) Hagfeldt, A.; Siegbahn, H. 0. G.; Lindquist, &E.; Lunell, S. Inr. J . Quantum Chem. 1992,44, 477. (4) O’Regan, B.; Grltzel, M. Nuture 1991, 353, 737. (5) Yu, W.; Freas, R. B. J . Am. Chem. Soc. 1990,112, 7126. ( 6 ) Bauschlicher,C. W., Jr.; Bagus, P. S.; Nelin, C. J. Chem. Phys. Lrrr. 1983, 101, 229. (7) Ramana, M. V.; Phillips, D. H. J . Chem. Phys. 1988. 88, 2637. (8) Kobayashi, H.; Yamaguchi, M. Surf. Sci. 1989, 214,466. (9) Hagfeldt, A.; Lunell, S.; Siegbahn, H. 0.G., Int. J . Quunrum Chem., in press. (10) Frisch, M. J.; Head-Gordon, M.; Trucks, G. W.; Foresman, J. B.; Schlegel, H. B.; Raghavachari,K.; Robb, M. A.; Binkley, J. S.; Gonzalez, C.; Defrees, D. J.; FOX,D. J.; Whiteside, R. A.; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R. L.; Kahn, L. R.; Stewart, J. J. P.; Topiol, S.; Pople, J. A. GAUSSIAN 90; Gaussian, Inc.: Pittsburgh PA, 1990. (11) (a) McIntyre, N. S.; Thompson, K. R.; Weltner, W., Jr. J . Phys. Chem. 1971,75,3243. (b) Weltner, W., Jr.; McLeod, D., Jr. J. Phys. Chem. 1965, 69, 3488. (12) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab initio Molecular Orbital Theory; John Wilcy & Sons, Inc.: New York, 1986. (13) Cf.: Lunell. S.: Feller, D.; Davidson, E. R. Theor. Chim. Acto 1990, 77, 1 1 1 . (14) Arnaud, Y. P. Appl. Surf. Sci. 1992, 62, 21, 37, 47. (15) See, e&: Berry, R. S. Sci. Am. 1990, 263 (2), 50.
