titanium 
 dioxide ( tio2 ) nanostructures have attracted 
 great interest in the past few decades due to their low cost , environmental 
 compatibility , and experimentally proven potential for photocatalytic and photovoltaic applications . in particular , 
 the use of such nanostructures as heterogeneous catalysts for the 
 photocatalytic splitting of water to produce renewable hydrogen and in dye - sensitized solar cells has 
 been the subject of intense research . recent studies suggest that the size and the shape of tio2 nanostructures 
 directly influence their performance in these applications and hence 
 must have a clear effect on the microscopic electron  hole pair 
 production , recombination , separation , and diffusion rates . to understand the physics and chemistry underlying the application 
 of tio2 nanostructures in photocatalysis and photovoltaics 
 from a theoretical point of view , tio2 nanostructures and 
 extended systems have been computationally extensively studied using 
 a variety of methods . most of these studies employ either ground state density functional 
 theory ( dft ) or its excited state variant time - dependent density functional 
 theory ( td - dft ) . in the former case , optical excitations are generally 
 assumed to map onto excitations from occupied to unoccupied kohn alternatively , 
 for the lowest triplet excited state , the excitation is modeled as 
 a state obtained self - consistently by using a scf approach . td - dft , in contrast , is a genuine excited state method that can be 
 used for any number of excitations of any multiplicity . however , just 
 like dft in the case of the ground state , td - dft ( and scf ) 
 suffers from the fact that the results are , to a smaller or larger 
 degree , dependent on the exchange - correlation ( xc ) potential used . 
 this might be especially true for tio2 and other transition 
 metal containing systems . on the other hand , correlated wave function 
 methods used in quantum chemistry , e.g. , coupled cluster ( cc ) and 
 complete active space second order perturbation theory ( caspt2 ) , do 
 not suffer from this problem and are perhaps the gold standard for 
 excitation calculations . however , the poor scaling of such methods 
 with the number of electrons in the system means that they can only 
 be used for very small nanoclusters containing merely a few transition 
 metal atoms . for example , the largest system studied to date for tio2 using correlated wave function methods is the ( tio2)3 trimer . the majority of 
 cc and caspt2 simulations focused on the description of the excited 
 states of the tio2 monomer . finally , there have 
 also been studies on tio2 nanostructures using green s 
 function based many - body perturbation theory methods : gw and bse . here , we perform a study where we compare the performance 
 of td - dft 
 using different xc potentials with correlated wave function methods 
 for a number of relevant nanoparticle structures , sampling a range 
 of titanium environments such as 3-fold , 4-fold , and 5-fold coordination . we consider correlated wave function results obtained with the equation 
 of motion  coupled cluster ( eom - cc ) methods and td - dft results 
 obtained with the pbe , b3lyp , cam - b3lyp , and bhlyp xc potentials . we focus here on the performance of these methods when calculating 
 the vertical singlet excitation spectra of the nanoparticles , the 
 equivalent of the experimental uv  vis absorption spectra . we 
 pay special attention to the lowest singlet excitation ( s1 ) , as this 
 excitation , following kasha s principle , is the likely source of fluorescence ( luminescence ) and 
 the state relevant to applications such as photocatalysis and photovoltaics . an accurate description of this state at the ground state geometry 
 is , therefore , a crucial staring point for future computational work 
 on tio2 nanoparticles that will focus on modeling phenomena 
 that involve excited state relaxation , following the ideas developed 
 by us and others . in this paper , we will demonstrate that , for most of the tio2 nanoparticles , td - dft , with all the tested xc potentials 
 and eom - cc calculations give qualitatively similar results . moreover , 
 we will show that td - dft calculations , using the b3lyp and cam - b3lyp 
 xc potential , give the best quantitative fit to eom - cc excitation 
 energies . importantly , however , we will also show that for an important 
 subset of structures td - dft can give qualitatively different results 
 depending on the xc potential used and that , in this case , only td - cam - b3lyp 
 and td - bhlyp calculations yield results that are qualitatively consistent 
 with those obtained using eom - cc theory . we will demonstrate that 
 the discrepancies for these structures arise from a particular combination 
 of defects , excitation involving which are poorly described by td - pbe 
 and td - b3lyp . finally , we demonstrate that these defects react exothermically 
 with water to form hydroxyl groups and show that for hydrated tio2 nanoparticles the qualitative discrepancies between the different 
 methods , observed for naked particles , disappear . the geometry of each structure used 
 in this work was optimized 
 at the dft level with the hybrid b3lyp xc potential in conjunction with the triple- def2-tzvp basis 
 set . the harmonic frequencies at the 
 geometries of the stationary points obtained in the dft optimizations 
 were calculated using the same dft setup ( b3lyp / def2-tzvp level ) to 
 verify that the optimized structures correspond to proper minima on 
 the ground state potential energy surface . the electronic ground state 
 of all the clusters studied in this work is assumed to be a closed - shell 
 singlet . for the naked ( tio2)n clusters , where n = 18 and 10 , we used 
 the global minimum ( gm , the lowest energy structure for a given cluster - size ) 
 geometries reported by a number of groups , while for the clusters n = 9 and 
 1113 , we used the recent structures obtained by chen and dixon . to investigate a larger range of titanium and 
 oxygen coordination environments , we also examined a number of metastable 
 isomers ( i.e. , local minima for a given cluster size that lie higher 
 in energy than the gm ) . for the ( tio2)2 dimer , 
 this included two metastable structures that we labeled  cis  and  club ,  with c2v and cs symmetry , respectively , while for the trimer ( tio2)3 we included one 
 higher metastable isomer with c1 symmetry , 
 labeled as  alt .  the geometries of the stable and metastable 
 structures for the n = 15 subset are shown 
 in figure 1 , while the remaining n = 613 structures are given in section esi-1 of the supporting information . global minimum ( gm ) atomic 
 configurations for ( tio2)n clusters 
 with n = 15 . 
 for the dimers and trimers , metastable clusters are also shown . labels 
 include point group symmetry , and for the non - gm structures , the energy 
 difference in ev relative to the gm energy calculated at the b3lyp / def2-tzvp 
 level . the hydrated systems , 
 ( tio2)n(h2o)m where n and m range from 1 to 3 , were obtained through 
 the saturation of all the under - coordinated titanium and oxygen atoms 
 present in the naked clusters with hydroxyl groups and protons , respectively . 
 here , we assumed that titanium atoms are normally coordinated by at 
 least four oxygen atoms and oxygen atoms form at least two bonds . the harmonic frequencies were also calculated on the dft optimized 
 geometries of each hydrated nanoparticle , in order to verify that 
 they correspond to proper minima . the dft optimized structures include 
 ti(oh)4 , with s4 symmetry ; 
 ( tio2)2(h2o)2 , with c2v symmetry ; and ( tio2)2(h2o ) , ( tio2)3(h2o)2 , and ( tio2)3(h2o)3 , all with c1 symmetry . the coordinates of all the structures ( naked and hydrated ) used in 
 this study are listed in esi-1 and esi-3 , respectively . ( tio2)n(h2o)m hydrated clusters , with n and m ranging between 1 and 3 . in order to saturate 
 relevant defects , one water molecule is added per singly coordinated 
 oxygen atom in the originally naked clusters [ with the ( tio2)3(h2o)3 cluster as the only exception , 
 where one additional water molecule was added to generate the same 
 coordination environment for all the titanium atoms ] . for each cluster for all of the b3lyp optimized geometries , the 
 energies of the 
 lowest singlet excited states were obtained at the td - dft / def2-tzvp 
 level with different xc potentials ( the pure gga pbe , the hybrids b3lyp and bhlyp , and the range - separated hybrid xc potential cam - b3lyp ) . for selected b3lyp optimized geometries , the 
 four lowest energy excited states were also calculated with two different 
 coupled cluster ( cc ) approaches : eom - ccsd and eom - ccsdt , methods already discussed in the context of tio2 clusters 
 in our previous work . due to the 
 high computational cost required , we were not able to 
 optimize the structures of clusters larger than the monomer with the 
 cc approaches . however , for some clusters , we have also obtained the 
 minimum energy geometries with the pbe and cam - b3lyp xc potentials 
 and calculated the respective vertical excitations . the use of b3lyp 
 optimized structures , when compared to their fully optimized ground - state 
 counterparts for each different xc potential , was found to introduce 
 an average difference of 0.08 ev in the calculated excitation energies 
 for the smaller ( tio2)n naked 
 clusters , where n = 15 . vertical excitations 
 at these optimized geometries are listed in the supporting information esi-2 . while for all td - dft calculations , 
 we employed the def2-tzvp basis 
 set , in the case of the eom - cc methods , we used two different basis 
 sets : the split - valence def2-sv(p ) basis - set and the larger triple- def2-tzvpp basis set . from now on , these two basis sets will be referred to as 
 sv and tz , respectively . all the coupled cluster calculations , for 
 reasons of computational tractability , employed the frozen core approximation 
 where only the valence electrons are correlated ( i.e. , the 1s orbitals 
 of the oxygen atoms and the 1s to 3p orbitals of the titanium atoms 
 are frozen ) . our previous work showed , 
 in the case of the tio2 monomer , that the use of the frozen 
 core approximation introduces a typical downward shift of 0.1 ev in 
 the calculated energy relative to the all - electron cc results . the dft / td - dft calculations employing the pbe , b3lyp , and bhlyp 
 xc potential were performed with the turbomole 6.4 code , while the td - dft results obtained for the cam - b3lyp 
 xc potential were calculated with the gamess us code ( version 26 october 
 2012 ) . the coupled cluster calculations 
 employed the tensor contraction engine ( tce ) module of the nwchem 6.1 package . for the graphic representations of the 
 clusters studied , we used 
 the pymol molecular graphics system , while 
 vmd was employed for the visualization 
 of the differences between the ground state and excited state density . in this section , we will first compare the 
 lowest excitation energies 
 of the ( tio2)n gm clusters 
 obtained with the four different td - dft xc potentials ( pbe , b3lyp , 
 bhlyp , and cam - b3lyp ) . we will then investigate how the choice of 
 the xc potential affects the shape of the td - dft optical spectrum , 
 and in particular we compare the spectra of the ( tio2)n clusters that show different trends in their 
 excitations . next , to further understand what is the origin of the 
 differences between excitation energies calculated with various xc 
 potentials , we investigate the excitation energies of a selection 
 of small clusters ( n = 15 ) with relevant 
 defects using eom - cc and compare the td - dft results ( td - b3lyp and 
 td - camb3lyp ) with the eom - cc benchmark values . moreover , through the 
 calculation of the  value we try to investigate the origin 
 of the trends shown by the different xc potentials , further confirming 
 the usefulness of this diagnostic test for the detection of possible 
 charge transfer ( ct ) problems in td - dft . finally , we define some eom - cc 
 benchmarks for hydrated structures and try to understand if after 
 hydration the differences between the values predicted with the various 
 xc potentials still persist . as shown 
 in figure 3 , the four different xc potentials 
 exhibit a similar 
 trend in the description of the lowest excitation energy ( s1 , the 
 absorption on - set ) for the ( tio2)n gm clusters . there appears to be a constant energy shift that 
 is dependent upon the amount of exact hf - like exchange ( hfle ) included 
 in each of the xc potentials employed . for example , pbe with 0% hfle 
 gives the lowest excitation energies , while bhlyp with 50% of hfle 
 yields the highest values . excitation energies obtained using b3lyp 
 that has 20% hfle , and , cam - b3lyp , with 19% at short - range and 65% 
 at long - range , lie in between the pbe and bhlyp values . however , aside 
 from the similarities observed among the four data series , it is easy 
 to see that the trend for td - pbe and td - b3lyp has two clear dips of 
 ca . 0.5 ev , at ( tio2)3 and ( tio2)10 , which are absent in the cam - b3lyp and bhlyp results . in 
 addition , the pbe xc potential results show a similar but smaller 
 dip for the ( tio2)6 structure , while td - b3lyp 
 places this excitation in between the energies obtained for ( tio2)5 and ( tio2)7 , and td - cam - b3lyp 
 and td - bhlyp predict a local maximum in the trend for this geometry . 
 both of the lowest excitations for the ( tio2)3 and ( tio2)10 clusters correspond to states 
 with weak absorption intensity ( i.e. , low oscillator strength ) . however , 
 it is important to stress that they are not dark states ( excitations 
 with zero absorption intensity ) ; with td - b3lyp the oscillator strengths 
 are 9  10 ( although 2 orders of magnitude 
 weaker than the most intense excitation among the hundred lowest excitations ) 
 and 3.6  10 ( 1 order of magnitude weaker ) , 
 respectively . lowest singlet excitation energies calculated with different 
 td - dft 
 xc potentials for the b3lyp / def2-tzvp optimized ground state ( tio2)n gm structures . pbe values are 
 represented by blue diamond markers , b3lyp red squares , camb3lyp green 
 circles , and bhlyp excitations are displayed as purple triangles . in agreement with the observations 
 above , figure s2.4 in section 
 esi-2 of the supporting information shows 
 that there is a good linear fit between the td - cam - b3lyp and td - bhlyp 
 predicted absorption on - set values of the gm clusters ( r of 0.99 , with r being the 
 coefficient of determination of the fit , which ranges between 0 and 
 1 ; as r approaches 1 , the quality of 
 the fit improves ) , while the correlation between td - cam - b3lyp and 
 td - b3lyp absorption on - sets is much weaker ( r of 0.80 ) . not surprisingly , the clearest outliers in the 
 latter case are the absorption on - set values for ( tio2)3 and ( tio2)10 gm clusters . we 
 also investigated the trends for the next two excited states 
 ( the second and third lowest singlet excitations , s2 and s3 ) for the 
 same set of clusters ( see values and graph in esi-2 ) . the trends found for these higher excited states are 
 very similar to the ones observed for the lowest excited state shown 
 in figure 3 . the two dips for the ( tio2)3 and ( tio2)10 structures 
 are still there for the s2 and s3 states calculated with td - pbe and 
 td - b3lyp . the s2 and s3 states for ( tio2)3 exhibit 
 absorption intensities ( as calculated with td - b3lyp ) 1 order of magnitude 
 more intense than the lowest excitation , while for ( tio2)10 the s2 and s3 states have a weaker intensity compared 
 to the s1 state . the underestimation of the excitation energies observed 
 when using pure gga xc potentials or hybrid xc potentials with a small 
 hfle contribution ( e.g. , b3lyp ) is thus not limited to the lowest 
 excited state , but it has an influence on higher energy states as 
 well and therefore on the overall shape of the optical spectrum . in 
 order to understand the size of this underestimation on the whole 
 spectrum , we compared the shape of the optical spectra calculated 
 with different xc potentials for the anomalous ( tio2)3 and ( tio2)10 clusters and the ( tio2)6 cluster as shown in figure 4a  c ( for all of the xc potentials employed , only the 
 26 ev excitation range is shown ) . td - dft calculated optical 
 spectra ( 100 lowest excitations , or 50 
 lowest in the case of td - cam - b3lyp for reasons of computational tractability ) 
 for ( a ) ( tio2)3 gm , ( b ) ( tio2)6 gm , and ( c ) ( tio2)10 gm optimized structures 
 at the b3lyp / def2-tzvp level . in the top left of each spectrum , as 
 insert is shown , with a zoom in of the low - energy part of the spectrum . the black line represents the td - pbe results , the red one the td - b3lyp , 
 and the green one td - cam - b3lyp , while the blue one corresponds to 
 the td - bhlyp spectra . all excitations plotted are represented as gaussians 
 with a standard deviation of 0.03 ev . the different shapes of the spectra for ( tio2)3 emphasize the influence of the chosen xc potential . td - pbe 
 and td - b3lyp 
 xc potentials predict a weak shoulder at low energies ( 1.97 and 2.96 
 ev , respectively ) , followed by an approximate 0.7 ev gap to the next 
 peak . in the case of td - cam - b3lyp and td - bhlyp , no such large gap 
 is observed , and the most intense peaks , for both of these xc potentials , 
 seem to roughly agree after being rigidly shifted by ca . 0.5 ev . when 
 the td - b3lyp spectrum is shifted upward by ca . 1 ev and compared to 
 the td - cam - b3lyp spectrum , we can clearly see that the two xc potentials 
 show a poor agreement in the lower energy range of the spectrum , with 
 b3lyp underestimating the first excitation energy and completely missing 
 the lower energy features predicted by td - cam - b3lyp . the two xc potentials 
 show a much better agreement at higher energies , and although the 
 intensities do not match perfectly , all the peaks fall within the 
 same range of energies . the td - b3lyp , td - cam - b3lyp , and td - bhlyp calculated 
 spectra for the n = 6 gm cluster , after a rigid shift 
 is applied , show a very good agreement on the position of the excitation 
 peaks . a suitable shift could not be found in order to match the spectrum 
 generated using the pbe xc potential . finally , for the spectrum of 
 the ( tio2)10 structure shown in figure 4c , as expected , a similar behavior to the ( tio2)3 structure is observed . for example , the td - pbe 
 and td - b3lyp calculated spectra show a very weak shoulder at lower 
 energies , which , even after a rigid shift in energy , can not be made 
 to coincide with any of the peaks predicted by td - cam - b3lyp or td - bhlyp . as 
 observed in the previous section , the lowest energy excitation 
 for the two ( tio2)3 and ( tio2)10 structures is described differently depending on the xc 
 potential employed in the td - dft calculation . to benchmark the td - dft 
 results , , we confine our investigation to clusters 
 of a similar size to the ( tio2)3 cluster . because 
 the gm ( tio2)3 cluster exhibits a three - coordinated 
 titanium atom and a three - coordinated oxygen atom , structural elements 
 not present in the other gm for n = 15 , we 
 also consider metastable clusters of two and three tio2 units reported in the literature , which have similar coordination 
 environments for titanium and/or oxygen atoms . this led us to include 
 the so - called club structure , a ( tio2)2 isomer that is 0.73 ev ( b3lyp / def2-tzvp ) higher in energy 
 than the n = 2 gm and a trimer isomer that lies 0.32 
 ev higher than the n = 3 gm . the latter has a three - coordinated 
 titanium atom , and in contrast with the trimer gm , no three - coordinated 
 oxygen atoms . for completeness , we also compared td - dft and eom - cc 
 excitations for another ( tio2)2 isomer , which 
 is the cis version of the trans gm 
 and lies 0.25 ev ( b3lyp / def2-tzvp ) in energy above it . all of these 
 structures are shown in figure 1 . in 
 figures 5 and 6 , the 
 four lowest excitation energies of the ( tio2)2 and ( tio2)3 global minima , respectively , are 
 plotted as a function of the chosen energy definition , td - dft ( b3lyp 
 and cam - b3lyp ) and different flavors of eom - cc : eom - ccsd / tz , eom - ccsd / sv , 
 and eom - ccsdt / sv ( with the exception of eom - ccsd / tz excitation energies 
 for the ( tio2)3 gm cluster , eom - cc data were 
 taken from our previous work ) . care should 
 be taken , when comparing absolute excitation energies , as the convergence 
 with respect to basis sets ( tz or better ) and cc excitation level 
 ( eom - ccsdt or better ) are currently only numerically feasible for 
 the tio2 monomer , and even then supercomputing facilities 
 are required ( 4002000 cores per run with each core having 
 at least on the order of 2 gb of memory ) . we thus focus instead primarily 
 on the relative ordering of the different excitations , exploiting 
 the fact that the clusters are highly symmetric and the excited states 
 thus span a number of different irreducible representations . trend in the 
 four lowest excitation energies of the ( tio2)2 gm dimer cluster as calculated with different method 
 combinations ( 1bg red diamonds , 1au green squares , 1bu blue circles , 
 and 2ag purple triangles ) . sd and sdt stand 
 for eom - ccsd / t , while sv and tz correspond to the def2-svp and def2-tzvpp 
 basis sets . all of the td - dft calculations ( b3lyp and cam - b3lyp ) employed 
 the def2-tzvp basis set . all of the eom - cc data shown were taken from 
 our previous work . the inset shows , for 
 all the methods employed , the difference between the higher excited 
 states ( e.g. , s4 , s3 , and s2 ) compared to the lowest excited energy 
 s1 . trend in the five lowest excitation energies 
 of the ( tio2)3 gm trimer as calculated with 
 different method combinations 
 ( 2a red diamond , 1a green 
 square , 3a blue triangle , 4a 
 purple circle , and 2a black diamond ) . sd and sdt 
 stand for eom - ccsd / t , while sv and tz correspond to the def2-svp and 
 def2-tzvpp basis sets . all of the td - dft calculations ( b3lyp and cam - b3lyp ) 
 employed the def2-tzvp basis set . all of the eom - cc data bar the eom - ccsd 
 tz ( sd tz ) excitation energies where taken from our previous work . the 
 same energetic 
 ordering of the lowest excited states is observed for td - dft and the 
 different eom - cc flavors , with roughly similar energy differences 
 between the different excited states ( except that the gap between 
 the s2 and s3 excited state for td - b3lyp is much reduced ; this is 
 clearly shown in the inset of figure 5 , where 
 for each method we plotted the energy of the sx state against the 
 energy of the lowest excited state s1 ) . the ( tio2)3 data in figure 6 paint , however , a 
 different picture , as there are a number of crossovers between the 
 lowest excited states . we first compare the td - b3lyp and td - cam - b3lyp 
 results , which are , in line with the discussion above , very different . 
 not only is the energy difference between the s1 and s2 states reduced 
 from 0.6 ev for td - b3lyp to only 0.1 ev for td - cam - b3lyp , but the 
 next couple of excited states lies much closer in energy for td - b3lyp 
 than for td - cam - b3lyp . some of the higher lying excited states effectively 
 become degenerate in the case of td - b3lyp and show a different ordering 
 than found with td - cam - b3lyp . the td - cam - b3lyp and all eom - cc results , 
 in contrast , are very similar ( with the exception of an interchange 
 between the close - lying s1/s2 states for both sets of eom - ccsd results ) . having compared the energies of different excited states of individual 
 clusters , we now compare the excitation energies between the ground 
 and lowest excited states for sets of clusters again , we perform 
 calculations both with td - dft ( td - b3lyp and td - cam - b3lyp ) and eom - cc 
 methods ( eom - ccsdt / sv up to ( tio2)3 , eom - ccsd / tz 
 up to ( tio2)4 , and eom - ccsd / sv up to ( tio2)5 ) . figure 7a contains 
 the td - dft results and figure 7b and c the 
 eom - ccsd / t tz and eom - ccsd / t sv data , respectively . overall , td - dft 
 and eom - cc predict similar magnitudes of excitation energies as well 
 as a similar ordering of the lowest excited states of the different 
 clusters . for example , all methods employed here predict the same 
 ordering of the lowest energy excitations of the dimer and monomer 
 ( trans > cis > club 
 > monomer ) . td - dft and eom - ccsd / sv also agree on where 
 the lowest 
 excitation energies of ( tio2)4 and ( tio2)5 global minima lie relative to that of the dimer . however , for the ( tio2)3 gm cluster , the different 
 methods give widely different results . in agreement with eom - ccsd , 
 td - cam - b3lyp predicts that the lowest excitation energy of the ( tio2)3 gm to be slightly higher than that for the ( tio2)2 gm . in contrast , td - b3lyp places the ( tio2)3 gm lowest excitation in a similar energy range 
 as the lowest excitation of the ( tio2)2club isomer and below the alt trimer . eom - ccsd / tz 
 and eom - ccsd / sv predict that the lowest excitation energy of the ( tio2)3 gm lies slightly higher than that of the ( tio2)2 gm , while eom - ccsdt / sv puts the lowest excitation 
 energy of the ( tio2)3 gm slightly below that 
 of the ( tio2)2 gm . comparison of td - dft 
 excitation energies with eom - cc results for 
 a series of selected ( tio2)n clusters , with n = 15 . the global minima 
 ( gm ) are represented by diamond markers , the metastable cis and club dimers by squares and triangles , respectively , 
 and the metastable trimer ( alt ) by circles . ( a ) comparison 
 of td - camb3lyp ( green markers ) and td - b3lyp ( red ) excitations . ( b ) 
 comparison between eom - ccsd and eom - ccsdt results with the high quality 
 tz basis set . ( c ) comparison between eom - ccsd and eom - ccsdt results 
 with the small sv basis set . overall , both in terms of the relative energies of different 
 excitations 
 of the clusters and between different clusters , eom - cc calculations 
 suggest that td - dft does a reasonable job in describing electronic 
 excitations of tio2 clusters . however , xc potentials with 
 no or low percentages of hfle , such as pbe and b3lyp , struggle for 
 specific structures . td - cam - b3lyp ( and td - bhlyp ) , in contrast , yields 
 lowest excitation energies qualitatively consistent with eom - cc for 
 all structures studied . finally , when considering absolute excitation 
 energies , taking into account the caveats discussed above , those obtained 
 with td - cam - b3lyp lie generally quantitatively closest to the eom - cc 
 data . we , therefore , would recommend using this xc potential , where 
 possible , to model excited state processes in tio2 nanoparticles . it is well - known that the use of standard xc 
 potentials with no hfle ( i.e. , gga ) or a low percentage of hfle ( e.g. , 
 b3lyp ) in td - dft can result in the underestimation of charge transfer 
 ( ct ) excitations , where the origin and final destination of the excited 
 electron are separated spatially . it is appealing to suppose that 
 this erroneous energetic stabilization of ct states might be the origin 
 of the discrepancy between td - pbe and td - b3lyp on one hand and td - cam - b3lyp , 
 td - bhlyp , and eom - cc on the other hand . hence , it is important to 
 try to characterize the ct character of the excitations for the different 
 nanoclusters , as calculated with td - dft . in this section , this diagnostic quantifies the spatial overlap 
 between the ( unperturbed ) occupied and virtual orbitals involved in 
 the td - dft excitation and hence is an estimate of the ct character 
 of a specific excitation . the  value corresponds to a dimensionless 
 number , which varies from 0 ( no overlap between the occupied and virtual 
 orbitals responsible for a td - dft excitation ) to 1 ( complete overlap 
 between the occupied and virtual orbitals ) . in the original 
 paper by peach , it is suggested 
 that a td - dft excitation with small  may be associated with 
 large ct errors , while in contrast small ct errors are associated 
 with large . in this fashion , the authors imply that , for example , 
 td - pbe excitations with  < 0.4 or td - b3lyp excitations with < 0.3 are likely to contain a significant error , while 
 for td - cam - b3lyp no correlation is observed between errors and spatial 
 overlap , as measured by . these 
 limits , however , have been obtained for a set of organic molecules , 
 and to our knowledge no one has yet published similar thresholds for 
 inorganic systems , e.g. , metal oxides . figure 8 shows  values calculated 
 with td - b3lyp for the three lowest excitations obtained for the n = 113 gm ( tio2)n clusters . we calculated the  values also for the pbe , 
 cam - b3lyp , and bhlyp xc potentials for all the gms and the metastable 
 isomers , see the supporting information esi-2 . for the lowest excited state ( s1 , blue diamonds in figure 8) , all of the td - b3lyp values lie below the threshold 
 ( 0.30 ) defined by peach et al . for hybrid xc potentials . however , 
 among the  values calculated for the s1 state using td - b3lyp , 
 perhaps not surprisingly , both the ( tio2)3 and 
 ( tio2)10 gm clusters are clearly the lowest , 
 with some other clusters , ( tio2)7 , ( tio2)11 , and ( tio2)9 , showing 
 moderately low values as well . for the non - gm structures , the alt trimer has the lowest  value . values 
 for the higher excited states ( s2 and s3 ) , as shown in figure 8 , can be , depending on the particular cluster , both 
 larger and smaller than that of s1 . examples of clusters with  
 values for higher excited states that lie below their s1 counterparts 
 include ( tio2)5 ( s2 ) , ( tio2)9 ( s2 ) , and ( tio2)13 ( s3 ) . values 
 for the three lowest excited states ( lowest state 
 shown by blue diamonds , second lowest state by red squares , and third 
 lowest state by green circles ) of ( tio2)n clusters , with n = 113 , calculated 
 with b3lyp xc potential . the dashed black line shows the  threshold 
 defined by peach et al . for organic systems . the  data for our titania 
 clusters yield a number of interesting 
 observations . first , all of the  values for the tio2 clusters are smaller than typical  values for organic systems , 
 i.e. the tio2 clusters consistently have a smaller overlap 
 of the orbitals involved in the excitation . we believe that this arises 
 from the fact that excitations in inorganic systems , such as tio2 , typically involve the displacement of an electron from one 
 sublattice ( using the term lattice loosely ) to another sublattice , 
 here from orbitals based on oxygen atoms to those based on titanium 
 atoms . in such a scenario , even local excitations ( i.e. , excitations 
 where both centers involved are spatially close ) might have low  
 values , especially if the material is rather ionic and there is thus 
 only limited overlap between orbitals on either sublattice . the good 
 qualitative fit between the td - dft and eom - cc results for excitations 
 with  values between 0.15 and 0.3 strongly suggests that these 
 are not excitations which are badly described by xc potentials due 
 to ct related problems . a more interesting observation is that the 
 excitations for which the description is problematic with td - pbe and 
 td - b3lyp , both relative to that in td - cam - b3lyp and td - bhlyp and to 
 that by eom - cc , are the same excitations that have very low . 
 this observation suggests that the reason why the description of these 
 specific excitations is problematic within td - pbe and td - b3lyp is 
 that they have an especially strong ct character , and hence the states 
 involved are erroneously stabilized , i.e. , lie lower in energy relative 
 to the s0 ground state than they should . for example , as outlined above , the second excitation ( s2 ) of the 
 ( tio2)5 structure has a rather low  value , 
 substantially lower than that of s1 and s3 , and it is interesting 
 to compare in this context the predictions of td - b3lyp with those 
 obtained with other methods . the s2 excited state in td - b3lyp belongs 
 to the a irreducible representation ( just as s0 and s1 , and 
 hence it can be labeled as the 3a state ) , while 
 in eom - cc ( and in td - cam - b3lyp and td - bhlyp ) s2 belongs to the a 
 irreducible representation ( the 1a state ) and 
 s3 in contrast is the 3a state . this swap in the 
 energetic ordering of the lowest two states is probably a direct result 
 of the erroneous ct related stabilization of the 3a 
 state in td - b3lyp , which shifts it below the 1a 
 state . a close inspection 
 of the atomic structures of the clusters studied , the orbitals involved 
 in the excitations and the ground state excited state density 
 differences , suggests that the problematic ct excitations typically 
 involve one or more singly coordinated oxygen atoms as the origin 
 of the excited electron and hence the location of the formed hole . 
 the structural fragment on which the excited electron localizes varies 
 and can be , for instance , a titanium atom with nominally only three 
 oxygen atoms in its first coordination sphere ( as found for the ( tio2)3 gm and ( tio2)2club structures , see figure 9 ) or 
 a fragment constituted of three four - coordinated titanium atoms surrounding 
 a common three - coordinated oxygen atom ( as found for the ( tio2)10 gm structure , see figure 10 ) . the latter combination of atoms is superficially similar 
 to that of one - half of the valence alternation pair ( vap ) defect - pair 
 in silica nanostructures . the presence 
 of these structural fragments , however , is no guarantee that the lowest 
 excitations will have ct character . this is illustrated , for example , 
 by the case of the ( tio2)2club isomer ( see figure 9 ) . this structure has 
 both a singly coordinated oxygen atom and a triply coordinated titanium 
 atom , but its lowest three excitations all have reasonable td - b3lyp 
  values ( see esi-2 ) , and their td - b3lyp excitation energies 
 qualitatively agree with eom - cc results . td - b3lyp lowest singlet 
 excited state density difference and  
 value obtained for ( a ) the club dimer and ( b ) the 
 gm trimer . in both pictures , the green lobes represent regions of 
 excess electron density ( where the excited electron component of the 
 excited state is located , e ) , whereas the blue lobes represent regions deficient in electron 
 density ( where the hole component is found , h ) . td - b3lyp lowest singlet 
 excited state density difference and  
 value obtained for the ( tio2)10 structure . the 
 green lobes represent excess regions of electron density ( where the 
 excited electron component of the excited state is located , e ) , whereas the blue lobes 
 represent regions deficient in electron density ( where the hole is 
 found , h ) . when rationalizing the above observations , we have to make 
 a choice 
 between a semi - ionic or a semicovalent reference frame . in the latter 
 reference , one can think of low - energy excitations as the transfer 
 of an electron between two well - defined centers that deviate electronically 
 and structurally from the bonding pattern in the remainder of the 
 particle ( e.g. , the silanone , si o , nonbridging oxygens , si  o * , and siloxy , si  o centers 
 in silica ) . in contrast , in the semi - ionic reference , the  centres  
 involved in low - energy excitations are much less well - defined and 
 importantly much less obvious upon inspection . the hole and electron 
 component of an excitation could be smeared out over a number of ions , 
 with the precise ions involved ( i.e. , localization sites ) governed 
 by a subtle interplay between the on - site electrostatic potential 
 ( the energetic cost of adding or removing an electron from a specific 
 ion ) and the electrostatic interaction between the excited electron 
 and the remaining hole ( i.e. , the exciton binding energy ) . intuitively , 
 the latter picture seems closer to what we observe for ( tio2)n nanoparticles . o distances in the clusters , and the fact that 
 the ti  o distances for the singly coordinated oxygen atoms 
 are always virtually the same ( 1.6  ) , independent of the coordination 
 number of the nearest titanium atom , also suggests that the semi - ionic 
 reference frame is the most apt description of tio2 nanoparticles . while in this picture , it is relatively difficult to link structure 
 to excitations , there is one important thing we can note . while singly 
 coordinated oxygen atoms have generally the lowest on - site electrostatic 
 ( or madelung ) potential and are the centers from which an electron 
 gets removed if one does a single - point calculation for the cation , 
 this does not mean , however , that it will be involved in the lowest 
 energy excitation ( or more generally in low energy excitations ) . for 
 example , for ( tio2)4 the first two lowest excitations 
 do not involve the singly coordinated oxygen atoms , while for ( tio2)5 only the second excitation ( with td - b3lyp , see 
 discussion in section 3.3 and figure s-2.3 
 of the esi-2.5 ) involves the singly coordinated 
 oxygen atom . the lowest excitation of the ( tio2)2club dimer , also , does not involve the singly coordinated 
 atom ( see figure 9 ) . the hole and excited electron 
 in this case are localized in close proximity , maximizing excited 
 electron hole overlap (  , see above ) and minimizing the 
 charge - transfer character of the excitation . clearly , there is thus 
 a subtle balance between the on - site electrostatic potential and the 
 electrostatic interaction between electron and hole , where the latter 
 in practice is sufficiently strong to change which parts of the cluster 
 are involved in low energy excitations . taking into account 
 the discussion above , it is likely that the 
 location of the excited electron and hole will change during excited 
 state processes that involve nuclear displacement ( e.g. , relaxation ) . 
 as a result , the electronic character of an excitation , its  
 value , and the ability of td - dft to correctly describe the excitation 
 might also change . in future work , we will , therefore , study this 
 explicitly . here , however , it is important to note that without a 
 correct description of the ( energetic ordering of the ) lowest excited 
 states at the ground state geometry , one has no hope of describing 
 what happens upon moving away from this starting point on the excited - state 
 energy landscape . upon hydration , all the centers discussed above 
 ( i.e. , singly coordinated 
 oxygen atoms and under - coordinated titanium atoms ) are saturated by 
 the addition of hydroxyl groups ( oh ) and 
 protons ( h ) . kj/(mol h2o ) in the case of ( tio2)2 , in line with previous estimates from the literature ) . as structural and electronic features of the 
 naked systems are modified , so are the orbitals involved in the optical 
 excitation . in figure 11 , we present the comparison 
 between the td - b3lyp , td - cam - b3lyp , and eom - ccsd sv absorption on - set 
 for a series of hydrated ( tio2)n(h2o)m systems , where n and m range from 13 . the addition 
 of water to the naked clusters results in a blue shift of their optical 
 spectra . figure 11 also shows , in line with 
 that discussed above , how the cam - b3lyp excitations always lie at 
 higher energies than their b3lyp counterparts , with a mean difference 
 of 0.29 ev . it is important to notice that , unlike what is observed 
 for specific naked structures ( e.g. , ( tio2)3 and ( tio2)10 ) , we found a generally good agreement 
 between b3lyp and cam - b3lyp excitations for all hydrated clusters . 
 in this case , the  values computed for the b3lyp excitations 
 for the hydrated clusters are generally higher than those calculated 
 for the naked clusters ( values listed in the esi-4 ) . moreover , the td - dft results with both xc potentials show a generally 
 good agreement with those obtained using eom - ccsd / sv , with an average 
 absolute difference of 0.27 ev for b3lyp and 0.20 ev for cam - b3lyp . thus , hydration seems to effectively remove the centers responsible 
 for the problematic ct excitations and , more importantly , suggests 
 that td - b3lyp in general will give good results for hydrated systems . comparison 
 between td - dft ( b3lyp , red markers and cam - b3lyp , green 
 markers ) lowest singlet excitations and eom - ccsd def2-sv(p ) ( blue 
 markers ) for hydrated ( tio2)n(h2o)m systems , where n and m range between 1 and 3 . the diamond 
 markers represent the ti(oh)4 , ( tio2)2(h2o)2 , and ( tio2)3(h2o)2 structures , respectively ; the triangle marker , 
 the ( tio2)2h2o cluster ; and the square , 
 the ( tio2)3(h2o)3 cluster . in this paper , we have studied the suitability of td - dft to describe 
 low - energy excitations in tio2 nanoparticles through a 
 comparison with eom - cc quantum chemistry calculations . we find that 
 td - dft generally gives a good qualitative and also , in the case of 
 td - b3lyp and td - cam - b3lyp , a quantitative fit to excitation energies 
 predicted by the more accurate , but computationally much more expensive , 
 eom - cc method . however , for selected particles , e.g. , the ( tio2)3 gm cluster , we observe large deviations from 
 eom - cc results when using the td - pbe and td - b3lyp xc potentials . calculation 
 of  , which provides a measure of the orbital overlap between 
 the orbitals involved in the excitation , and visual inspection of 
 the excited - state density differences in these cases suggests that 
 problems arise for these systems when computing charge - transfer excitations , 
 the energy of which relative to local excitations is underestimated 
 by td - pbe and td - b3lyp . such problems are completely absent for td - cam - b3lyp 
 and td - bhlyp . on the basis of this and the overall good quantitative 
 fit of td - cam - b3lyp and eom - cc results , we recommend using this xc 
 potential , where possible , when studying excitations in tio2 nanoparticles . finally , hydration of these particles removes the 
 structural centers responsible for the problematic charge - transfer 
 excitations , and all xc potentials studied here yield a good qualitative 
 fit to eom - cc results for hydrated particles .