Theoretical Modeling of Electronic Excitations of Gas-Phase and Solvated TiO2 Nanoclusters and Nanoparticles of Interest in Photocatalysis

The optical absorption spectra of (TiO2)n, nanoclusters (n = 1-20) and nanoparticles (n = 35, 84) have been calculated from the frequency-dependent dielectric function in the independent particle approximation under the framework of density functional theory. The PBE generalized gradient approach based functional, the so-called PBE+U method and the PBE0 and PBEx hybrid functionals containing 25% and 12.5% of non-local Fock exchange, respectively have been used. The simulated spectra have been obtained in gas phase and in water on previously PBE0 optimized atomic structures. The effect of the solvent has been accounted for by using an implicit water solvation model. For the smallest nanoclusters, the spectra show discrete peaks whereas for the largest nanoclusters and for the nanoparticles they resemble a continuum absorption band. In the gas phase and for a given density functional, the onset of the absorption band (optical gap, Ogap) remains relatively constant for all nanoparticle sizes although it increases with the percentage of non-local Fock exchange, as expected. For all tested functionals, the tendency of Ogap in water is very similar to that observed in gas phase with an almost constant up-shift. For comparison, the optical gap has also been calculated at the TD-DFT level with the PBEx functional in the gas phase and in water. Both approaches agree reasonably well although the TD-DFT gap values are lower than those derived from the dielectric-function. Overall, the position of the band maxima and the width of the spectra are relatively constant and independent of particle size which may have implications in the understanding of photocatalysis by TiO2.


Introduction
Titanium dioxide (TiO2) is a well-known semiconductor material which utilizes light to trigger photocatalytic reactions such as organic contaminant degradation in air or water 123-4 and, even more appealing from a sustainable energy source, water splitting. 5 In addition, TiO2 is attractive due to its abundance in nature, high stability and non-toxicity. 6 Roughly speaking, the mechanism behind photocatalysis consists in the absorption of light promoting the excitation of TiO2 electrons from the valence band to the conduction band. The resulting excited electron-hole pairs are the main players of the redox photocatalytic reactions. However, the most common TiO2 polymorphs (rutile and anatase) exhibit band gaps of or larger than 3 eV, 7-89 thus largely inhibiting their practical use as only ~10% of the sunlight incoming photons have enough energy to be absorbed and hence to participate in the photocatalytic process.
Recently, several studies have shown that the photocatalytic activity does not only depend on the composition of the photocatalytic material and present evidence that both particle size and shape can play a noted role in controlling the fate of excitons in TiO2. 1011-12 The effect of shape and size of TiO2 nanoparticles in defining the corresponding band gap and their concomitant photoactivity has been illustrated in recent work on bottom-up 13,14 and top-down 15 models of TiO2 nanoclusters and nanoparticles including explicitly over one thousand atoms. These realistic models of nanoparticles allow one to surmount the difficulties encountered by experiments to represent different morphologies for a given composition. 10,16 Using different state of the art computational techniques, the size-and shape-dependence of the electronic structure of TiO2 nanoclusters and nanoparticles has been clearly established. 17-18 19 Despite all engineering efforts, the main scientific goal remains to optimize the efficiency of solar energy conversion into readily available electricity. Therefore, an understanding of the optoelectronic properties of TiO2 is necessary to unravel many fundamental questions concerning the experimental results and in particular the band gap, one of the main properties governing their technological applications both in photocatalysis and in solar cells.Error! Bookmark not defined.
To avoid confusions related to the generic use of the band gap term, it is important to distinguish between the electronic or fundamental (Egap) and the optical (Ogap) band gap. 20 Egap is measured by (direct and inverse) photoemission techniques and thus involves charged states either of cationic (free extra hole, h + ) or anionic (free extra electron, e -) character. 21 On the other hand, Ogap is obtained from optical absorption experiments that generate an excited electron hole (e --h + ) pair, usually referred to as exciton pair. 22 More precisely, Ogap is measured by photoacoustic spectroscopy (PAS), a technique which has been used as a nondestructive method for analyzing the optical properties of semiconductors. 23,24 It is important to point out that there are two well defined but different ways to determine Ogap. In the first one, Ogap is obtained from the absorption edge from a linear fitting in the plot of the square of the product between the absorption coefficient and the photon energy versus the photon energy for direct band gap or from the plot of the square root of the product between the absorption coefficient and the photon energy versus the photon energy for indirect band gap. 25 In the second approach, Ogap is estimated from the change of the derivative of intensity signal of the absorption coefficient with respect to the photon energy near the fundamental absorption edge. 26 The difference between Egap and Ogap gaps becomes negligible in the limit of sufficiently large systems (including bulk or extended surfaces) since the addition (removal) of one electron to (from) fully delocalized states does not affect significantly its electronic structure.
However, in small finite systems Ogap is lower than Egap which may be used for band gap engineering purposes, for instance by nanostructuring processes. 27 In the present work we present a systematic density functional theory (DFT) based study of the optical absorption spectra and optical gaps of (TiO2)n nanoclusters (n = 1-20) and nanoparticles (n = 35 and 84). The simulated spectra are obtained from the frequency-dependent dielectric function using GGA, GGA+U, and hybrid exchange-correlation functionals and results are validated by comparing to results arising from the more accurate and but also computationally more demanding, time-dependent DFT (TD-DFT) calculations. The present results provide unbiased information regarding the intensity of electronic transitions in TiO2 nanoparticles potentially relevant to photocatalysis.

Nanoparticle models and computational details
The geometrical structure of the (TiO2)n nanoclusters (n = 1-20) and nanoparticles (n = 35 and 84) studied in the present work has been taken from previous work using bottom-up and topdown approaches, respectively.Error! Bookmark not defined. , Error! Bookmark not defined. In short, the optimized structures for the nanoclusters correspond to global minima determined using a two-step procedure. In a first step, Monte Carlo basin hopping 28 calculations with classical interionic potentials 29 and data mining 30 approaches were used to seek for low-energy isomers. In a second step, DFT based calculations with the PBE0 exchange-correlation potential 31 were carried out obtaining the final lowest-energy structures. For the crystalline nanoparticles, the initial geometry was obtained from a Wulff construction 32 and the resulting, PBE0 optimized anatase nanoparticles exhibit an octahedral shape with (101) facets. For completeness, the structures of the nanoclusters and nanoparticles used in the present work are represented in Figure 1, further details regarding these structures can be found in the original references.Error! Bookmark not defined. , Error! Bookmark not defined. For additional information regarding the modeling of oxide nanoparticles, the interested reader is referred to the review article by Bromley et al. 33 Using the atomic structure of the nanoclusters and nanoparticles as described above, the optical absorption spectra were calculated using different density functionals. These involve the PBE implementation 34 of the generalized gradient approach (GGA), the PBE0 hybrid functional containing 25% of non-local Fock exchange and a modification of the latter including 12.5 % of non-local Fock exchange hereafter referred to as PBEx. The PBEx functional was found to better reproduce the experimental band gaps of bulk rutile and anatase as well as the main features of oxygen vacancies in these polymorphs. 35 In addition, the optical spectra have also been computed using the PBE+U approach within the Dudarev approximation 36,37 involving an effective U value.
To avoid any bias arising from the semiempirical character of this parameter, two different U values (4.5 and 6.0 eV) were used. In the following, the corresponding series of calculations will be referred to as PBE+4.5 and PBE+6.0, respectively. Here it is worth pointing out that even if most previous calculations on TiO2 have used the U parameter for studies of the bulk phases for Ti (3d) levels only,  it has been shown that including this parameter for both Ti (3d) and O (2p) levels improves the description of both the band structure and band gap. 42 Consequently, the U term was applied to both Ti (3d) and O (2p) levels. A final remark is necessary regarding the estimate of the optical gap by means of periodic density functional theory calculations. The band gap of bulk solids is usually obtained from the band structure provided by the Kohn-Sham one electron levels whereas in discrete systems it is estimated from the HOMO-LUMO energy difference. In both cases, this is a serious approximation since the rigorous calculation involves the use of many body perturbation techniques as in the GW methods 43 common implementations possibly including the Bethe-Salpeter equation if exciton effects are dominant. 44 Therefore, the fact that a given functionals leads to a Kohn-Sham band gap in agreement with experiment needs to be regarded as fortuitous and taken as a practical choice rather than as a success of that particular functional. Note also that, at least in part, the success of band structure methods using GGA type functionals in predicting band gaps originates from the almost linear relationship between these and GW calculated values. 45 Apart from the computationally expensive GW methods, other procedures exist to approach the optical gap of solids and nanoparticles. In particular, the frequency-dependent dielectric function and the TD-DFT methods provide physically grounded approaches that go well beyond the use of Kohn-Sham orbital energies or standard band structure calculations. Following previous work regarding the optical spectra of bulk TiO2 polymorphs, 46,47 the gas phase optical spectra of the TiO2 nanoclusters and nanoparticles studied in the present work were obtained from the frequency-dependent dielectric function 48 as obtained from DFT based calculations carried out using the wellknown Vienna ab initio simulation package (VASP) 49-51 code which uses a plane wave basis set to expand the valence electron density and a projector augmented wave (PAW) 52,53 description of the effect of electron cores on the valence electron density. For the gas phase nanoclusters and nanoparticles, the optical spectra were computed starting from a ground state calculation with the chosen exchange-correlation functional and calculating the complex frequency-dependent dielectric function, where the imaginary part is computed as a sum over empty states and the real part is obtained by the standard Kramers-Kronig transformation. For the same nanoclusters and nanoparticles, the calculation of the dielectric function in aqueous medium was performed using VASPsol, 54,55 a modification of VASP which adds the implementation of an implicit solvation model accounting for the interaction between solute and solvent in the framework of a joint density functional theory (J-DFT). [56][57][58] In this framework, the free energy of the combined solute/solvent system is obtained from the sum of two terms, one including the total electron density plus a thermodynamic average of the atomic densities of the solvent species and a second term accounting for the electrostatic energy. An additional term describes the free energy contribution of cavitation and dispersion.Error! Bookmark not defined. ,59 In the present calculations a value of 78.40 was used for the dielectric constant of water.
Both gas phase and water solvation, DFT calculations were carried out for the closed shell electronic ground state, i.e. without spin polarization. In order to avoid spurious interactions resulting from the inherent periodic character of the VASP code, the (TiO2)n (n = 1-20) nanoclusters were placed into a cubic box with a 30.0 Å side which is then replicated in the three spatial directions. For the larger (TiO2)35, and (TiO2)84 nanoparticles, larger cubic boxes with 45.0 and 55.0 Å sides, respectively, were employed. In this way, the shortest distance between atoms belonging to different periodic replicas was larger than 20 Å for both nanoclusters and nanoparticles. Since the calculations involve discrete systems, calculations were always carried out at the Γ point. The valence states of oxygen (2s, 2p) and titanium (4s, 3p) were expanded in a plane wave basis set with an energy cutoff of 400 eV. Convergence was facilitated by using a σ = 0.1 eV Gaussian smearing although upon convergence this was removed.
The intensity of the optical absorption spectra α as a function of the frequency ω were derived from the real and imaginary parts of the computed frequency-dependent dielectric function ε (ω) in the independent particle approximation according to 60 where ( ) = 1 ( ) + 2 ( ). The same computational approach has been previously applied to the study of the optical spectra of fluorine-doped titania in anatase, rutile, and brookite.Error! Bookmark not defined. In the present application of the dielectric function to finite nanoclusters and nanoparticles, the absolute value of the dielectric function cannot be obtained. Nevertheless, the relative values of the absorption coefficient for different nanoclusters and nanoparticles are expected to be correct.
The frequency-dependent dielectric function of the nanoclusters and nanoparticles and the intensity of the optical absorption spectra derived as in Eq. (1) allow one to extract several pieces of information. To this end, the lowest-energy rising section of the imaginary part of the dielectric function of all nanoclusters and nanoparticles was fitted to a linear function, and the intercept with the abscissa axis was taken as Ogap. Note that this graphical procedure introduces some uncertainty in the calculated Ogap so that the values thus computed have to be taken within an error bar of roughly ±0.1 eV. Next, the intensity of the absorption spectra was fitted to Gaussian function as in where I is the intensity and a, b and c are the fitting parameters. In particular, the maximum of the

Results and discussion
To facilitate the discussion we present results for nanoclusters and nanoparticles separately. This is justified from the fact that the former exhibit structures which are usually far from that expected from anatase crystallites whereas the atomic structure of the two nanoparticles studied in the present work is closely related to anatase crystal structure. Unless otherwise stated, Ogap values reported in the present work are derived from the frequency-dependent dielectric function as described in the previous section. This remark is important to avoid any misunderstanding with previous works where Ogap is measured as the HOMO-LUMO Kohn-Sham energy levels difference.

Trends in the optical gap of the (TiO2)n (n = 1-20) nanoclusters
The main results of the present study are the quantities derived from the simulated optical absorption spectra. Figure 3 presents Ogap as a function of the nanocluster size at the PBE, PBEx, and PBE0 levels. The optical gaps were computed by means of linear fits to the onset of the imaginary part of the dielectric function for each one of the nanoclusters studied, as explained above.
The first clear feature emanating from Figure 3 is that the Ogap value behaves quite regularly as a function of size, except for the smallest nanoclusters (n = 3 or 4) for which relatively large fluctuations are observed. It is worth noting that relatively large fluctuations in the electronic band gap (Egap) have been observed previously in such small nanoclusters, 11  Let us now examine the values of the optical gap as predicted from the PBE+U (U = 4.5 and 6.0 eV) approach. The summary of results for all nanocluster sizes is presented in Figure 4 and the average and standard deviation for the different series are reported in Table 1 whereas a full list of Ogap values as obtained from calculations with the PBE, PBEx, PBE0, PBE+4.5 and PBE+6.0 functionals is reported in Table 2.
Comparing Figure 3 with Figure 4, one can see that for PBE+4. to produce a decrease in the gap larger than for bulk-like samples.

Simulated optical spectra of the (TiO2)n (n = 1-20) nanoclusters
In the view of the previous discussion, the simulated optical absorption spectra for the TiO2 nanoclusters considered in this section have been obtained using the PBEx functional and the results are presented in Figure 6. Nevertheless, the absorption spectra for all n = 1-20 nanoclusters obtained from the five functionals (PBE, PBEx, PBE0, PBE+4.5, PBE+6.0) considered in the present study in gas phase and in water, are shown in Figures S1-S10 in Supporting Information (SI). For the various nanocluster sizes, the spectra are qualitatively similar with an onset at about 4 eV and extending out to about 14 eV. The main difference one can observe concerns the spectral shape, which shows discrete peaks for n = 5 and 10 but an essentially continuous band for the nanoclusters with n = 15 and 20 having diameters of ~0.8-0.9 nm. It is remarkable that band-like spectra are obtained even for small nanoclusters, especially considering that Ogap changes slowly with particle size and that for exciton shift, nanoparticles of more than 10000 TiO2 units and a diameter of about 20 nm are still in the limit between discrete and bulk-like behaviour.Error! Bookmark not defined.
A more quantitative survey of the spectral shape can be obtained by calculating two characteristic properties such as the energy of the band maxima and the width of the spectra as a function of size of the nanoclusters. Results for PBEx and for gas phase and water are presented in Table 3, and results for the rest of methods employed here can be found in Tables S1-S4 Table 2 focusing on the n = 20, 35, 84 series showing a clear decrease of this quantity. Again, this is the expected trend considering that Ogap should converge to the bulk limit from above, as it is also the case for Egap.
It is also of interest to compare the position of the band maxima and the width of the spectra along the same n = 20, 35, 84 series. The maximum shifts towards lower energies and the bands tend to be a little wider as the size increases. The optical absorption spectra for the n = 35 and 84 nanoparticles are shown in Figure 7. The shape of the nanoparticle spectra is comparable to those of the larger nanoclusters, especially for n = 20, shown in Figure 6 although the nanoparticle spectra have a less regular shape at intermediate energy. We recall that the nanocluster geometries are obtained from unconstrained global minimizations for each cluster size and the final structure is often far from that of anatase cuts. On the contrary, the nanoparticle initial geometries are obtained from Wulff construction cuts of anatase TiO2 bulk and this different origin could justify the somewhat different absorption spectra of nanoclusters and nanoparticles.
Let us know discuss the TD-DFT results for the n = 35 and 84 nanoparticles. The first excited state is located at 3.12 and 3.14 eV in the gas phase, and 3.19 and 3.17 eV in water, respectively. This estimate of the Ogap values of the two particles are significantly lower (by about 0.7 eV) than those obtained from the frequency-dependent dielectric function. This is at variance of the values reported above for the n = 1-20 nanoclusters where TDDFT values were just slightly below the dielectric-function-derived ones (Table 2). Clearly, the difference between both methods is amplified for the rather large nanoparticles. This could be due to an underestimate of the excited state energy by TD-DFT with the PBEx functional, which has a relatively low amount of non-local

Reliability of the HOMO-LUMO estimate of Ogap
As described earlier, the distinction between fundamental (electronic) and optical gaps has

Supporting Information
The Supporting Information is available free of charge on the ACS Publications website at DOI: Tables S1-S4 show the location of band maxima and FWHM of the spectra of the nanoclusters with n = 1-20 with the PBE0, PBE, PBE+ 4.5 and PBE+6.0 functionals, respectively.
Figures S1-S10 show the optical absorption spectra of nanoclusters with n = 1-20 as predicted from calculations using the PBE, PBEx, PBE0, PBE+4.5 and PBE+6.0 density functionals in gas phase and in water. Figures S1 and S2

Notes
The authors declare no competing financial interest.       Upper and bottom panels correspond to TiO2 nanoclusters and nanoparticles, respectively.