Molecular Structures and Energetics of the (ZrO2)n and (HfO2)n (n = 1

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J. Phys. Chem. A 2010, 114, 2665–2683

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Molecular Structures and Energetics of the (ZrO2)n and (HfO2)n (n ) 1-4) Clusters and Their Anions Shenggang Li and David A. Dixon* Chemistry Department, The UniVersity of Alabama, Shelby Hall, Box 870336, Tuscaloosa, Alabama 35487-0336 ReceiVed: October 28, 2009; ReVised Manuscript ReceiVed: January 2, 2010

The group IVB transition-metal dioxide clusters and their anions, (MO2)n and (MO2)n- (M ) Zr, Hf; n ) 1-4), are studied with coupled cluster (CCSD(T)) theory and density functional theory (DFT). Similar to the results for M ) Ti, these oxide clusters have a number of low-lying isomeric structures, which can make it difficult to predict the ground electronic state especially for the anion. Electron affinities for the low-lying structures are calculated and compared with those for M ) Ti. Electron affinities of these clusters depend strongly on the cluster structures. Anion photoelectron spectra are calculated for the monomer and dimer and demonstrate the possibility for structural identification at a spectral line width of e0.05 eV. Electron excitation energies from the low-lying states to the singlet and triplet excited states are calculated self-consistently, as well as by the time-dependent DFT and equation-of-motion coupled cluster (EOM-CCSD) methods. The calculated excitation energies are compared to the band energies of bulk oxides, indicating that the excitation energy is not yet converged for n ) 4 for these clusters. The excitation energies of the low-lying isomeric clusters are less than the bulk metal oxide band gaps and suggest that these clusters could be useful photocatalysts with a visible light source. Introduction Transition-metal oxides form an important family of materials widely used as industrial catalysts or catalyst supports. Although considered less important than TiO2 for catalysis,1,2 ZrO2, HfO2, and materials using these oxides are also known for their catalytic activities.3,4 For example, aluminum-oxide-promoted sulfated zirconia is used in large-scale petrochemical processes for catalytic cracking, alkylation, and isomerization of hydrocarbons.3 Tiania, zirconia, or hafnium oxide embedded in a carbon matrix is used to catalyze the aromatization of C6+ alkanes.3 In contrast to TiO2, which is an important photocatalyst and exhibits strong metal-support interactions as a metal catalyst support, ZrO2 is a well-known solid acid catalyst.4 As a result, the titania-zirconia mixed oxides have excellent catalytic properties and exhibit interesting surface acid-base properties, high surface area, great thermal stability, and strong mechanical strength.4 Besides their importance in catalysis, ZrO2 and especially HfO2 have recently been used to replace the SiO2 gate dielectric due to their high dielectric constant (high κ) in order to reduce current leakage as the transistor size continues to shrink.5 At room temperature, bulk ZrO2 and HfO2 adopt the structure known as baddeleyite, which differs from the natural bulk phases of TiO2 (rutile, anatase, and brookite) in that Zr and Hf are seven-coordinated whereas Ti is six-coordinated.1 This is due to the much larger size of the Zr(IV) and Hf(IV) ions than the Ti(IV) ion.1 Both zirconium and hafnium oxide clusters have been studied by experimental methods. Earlier studies of these oxides involved high-temperature mass spectrometric measurement of the thermodynamic properties and dissociation energies of MO (M ) Zr, Hf) and MO2.6-8 Murad and Hildenbrand derived * To whom correspondence should be addressed. E-mail: dadixon@ bama.ua.edu.

ionization potentials of ZrO and ZrO2 from their measured appearance potentials.8 By measuring the deflection of a molecular beam of MO2 (M ) Si, Ti, Zr, Ce, Th, Ta, U) in an electrostatic quadrupole field, Kaufman et al. found all of these dioxides to be polar except for SiO2.9 Chertihin and Andrews studied the reaction products of laser-ablated M (M ) Ti, Zr, Hf) atoms with molecular oxygen in a condensing argon stream.10 MO, MO2, M2O2, and MO2- were identified by oxygen isotopic splittings and shifts in the measured infrared (IR) spectra. The OdMdO valence angle was estimated to be 113 ( 5° for MO2 and 128 ( 5° for its anion from the isotopic fundamental of the asymmetric stretch. The similarity of the Zr and Hf oxide spectra was attributed to the combined effects of lanthanide contraction and relativistic effects for Hf. Reactions of O2 with 29 transition-metal ions were studied with an inductively coupled plasma/selected-ion flow tube (ICP/SIFT) tandem mass spectrometer, and the early transition-metal ions were found to form oxide cations by oxygen atom abstraction and the late transition-metal ions by molecular oxygen addition.11 Rotational spectra of jet-cooled ZrO2 and HfO2 were obtained by Fourier transform microwave spectroscopy.12 The MdO bond length in MO2 was measured to be 1.7710 ( 0.0007 Å for M ) Zr, and 1.7764 ( 0.0004 Å for M ) Hf. The O-M-O bond angle in MO2 was measured to be 108.11 ( 0.08° for M ) Zr and 107.51 ( 0.01° for M ) Hf. Larger clusters of the zirconium and hafnium dioxides have also been studied experimentally. IR radiation from a free electron laser were employed to resonantly excite zirconium oxide clusters in the gas phase, and ZrnO2n-1+ ions were formed at resonance frequencies between 600 and 700 cm-1 for all clusters.13 Zirconium oxide clusters were also studied by timeof-flight mass spectrometry and photoionization spectroscopy.14 By using ArF laser irradiation at 193 nm, the formation of stoichiometric clusters ZrnO2n+ at low ionization laser intensity and fragmented clusters ZrnO2n-1+ at high ionization laser

10.1021/jp910310j  2010 American Chemical Society Published on Web 02/03/2010

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intensity was observed. By using the ninth harmonic of a Nd/ YAG laser at 118 nm, the observed clusters were mainly ZrnO2n+ and ZrnO2n+1+ due to much less fragmentation of the clusters. Raman scattering measurement for the HfxZr1-xO2 nanoparticles was carried out by Herman and co-workers.15 Zirconia-rich particles were found to be tetragonal, whereas hafnia-rich particles have mixed tetragonal and monoclinic phases. Computational studies of the zirconium and hafnium dioxide clusters were mostly carried out at the density functional theory (DFT)16 or the Hartree-Fock (HF) theory level in association with the above experimental studies to help interpret the experimental data.10,12-14 The monomers were studied by Chertihin and Andrews,10 Brugh et al.,12a and Lesarri et al.12b Von Helden et al. have calculated the structures of the ZrnO2n (n ) 1-4) clusters,13 and Foltin et al. have calculated those for the ZrnO2n (n ) 3-6) clusters.14 Woodley et al.17 predicted the structures of the MnO2n (M ) Zr, Hf; n ) 1-8) clusters at the DFT level based on their study for M ) Ti for n ) 1-15.18 We have recently studied transition-metal oxide clusters for group VIB, MnO3n (M ) Cr, Mo, W; n ) 1-6), and group IVB, MnO2n (M ) Ti; n ) 1-4), and their anions at the DFT and coupled cluster (CCSD(T))19-22 levels to predict their molecular structures, electron affinities, electron excitation energies, clustering energies, and heats of formation.23-29 In our previous study on the TinO2n (n ) 1-4) clusters and their anions,27 we found multiple low-lying structures for both the neutral and anionic clusters for n > 1. For the anions for n ) 2 and 4, near-degenerate electronic states were predicted at the CCSD(T) level. Although the calculated relative energies at the DFT level for the neutral clusters compare reasonably well with those calculated at the CCSD(T) level, those for the anions can be qualitatively different, which can result in the prediction of different ground electronic states in case of near degeneracy. In this report, we use the DFT and CCSD(T) methods to study the MnO2n (M ) Zr, Hf, n ) 1-4) clusters and their anions, and we provide accurate molecular structures, electron affinities, as well as electron excitation energies for these clusters. Clustering energies and heats of formation for these clusters were reported elsewhere.28 Computational Methods Equilibrium geometries and harmonic vibrational frequencies were calculated at the DFT level with the B3LYP30,31 and BP8632,33 exchange-correlation functionals for both the neutral and anionic clusters. We used the equilibrium geometries of the titanium dioxide clusters27 as the starting structures for the geometry optimizations. We used the aug-cc-pVXZ basis set for O34 and the pseudopotential (PP)-based aug-cc-pVXZ-PP basis sets for Zr and Hf35 in the DFT geometry optimization and frequency calculations with X ) D and in the DFT singlepoint energy calculations with X ) T. For simplicity, we denote these combined basis sets as aX. For the monomers and dimers, geometries were also optimized at the CCSD(T) level. The CCSD(T) calculations were performed with the sequence of basis sets of X ) D, T, Q, with the geometries optimized for X ) D and T. The CCSD(T) energies with X ) D, T, and Q were extrapolated to the complete basis set (CBS) limit using a mixed Gaussian/exponential formula.36 The cardinal numbers for the X ) D, T, Q basis sets that we used are 2, 3, and 4. Our recent studies on the group VIB transition-metal trioxide clusters have shown that the effect of the choice of the cardinal numbers in this extrapolation scheme is fairly small.24 Core-valence correlation corrections for the 1s2 electrons on O and (n - 1)s2(n - 1)p6 electrons for

Li and Dixon Zr (n ) 4) and Hf (n ) 5) were calculated at the CCSD(T) level with the aug-cc-pwCVXZ basis set for O37 and the augcc-pwCVXZ-PP basis set for Zr and Hf,35 with X ) D and T; these combined basis sets will be denoted as awCVXZ. Scalar relativistic corrections were calculated as the expectation values of the mass-velocity and Darwin operators (MVD) from the Breit-Pauli Hamiltonian38 for the CISD (configuration interaction with single and double excitations) wave function with the aT basis set. A potential problem arises in computing the scalar relativistic correction for the molecules in this study as there is the possibility of “double counting” of the relativistic effect on the metal when applying a MVD correction to an energy which already includes some relativistic effects via the relativistic effective core potential (RECP). Because the MVD operators mainly sample the core region where the pseudo-orbitals are small, we assume any double counting to be small. The above approach follows our and others’ work on the accurate prediction of the heats of formation for a wide range of compounds.39 In addition, we estimated the error in the calculated electron affinities due to the use of the RECP basis sets for the monomers and dimers on the basis of our recent work.28 For the monomers, geometries were also optimized at the CCSD(T)/awCVTZ level with the 1s2 electrons on O and (n 1)s2(n - 1)p6 electrons for Zr and Hf correlated. The (n - 1)p orbital energies for Zr and Hf are fairly close to the 2s orbital energy for O (∆ε ) 11-13 eV for M ) Zr, 14-15 eV for M ) Hf), and there is substantial mixing between these two sets of orbitals, especially for M ) Zr. As shown in our recent benchmark calculations on the M-NH3 (M ) Na, Al, Ga, In, Cu, Ag) complexes, core-valence correlation has more effect on the geometry than the energy in this circumstance.40 In addition, geometries were also calculated at the CCSD(T) level with the second-order Douglas-Kroll-Hess Hamiltonian41 and the aug-cc-pwCVTZ-DK basis set.35,42 The 1s2 electrons on O and (n - 1)s2(n - 1)p6 electrons for Zr and Hf are correlated. For Hf, the 4f14 electrons are also correlated by including additional high angular momentum functions (2f2g1h), as these 4f orbitals lie very close in energy to the 2s orbital for O, and there is substantial mixing between these two sets of orbitals.28 Following our previous work,28 the pseudopotential error, ∆EPP,corr, is defined as

∆EPP,corr ) ∆EawCVTZ-DK - (∆EawCVTZ + ∆EMVD)

(1) In eq 1, ∆EawCVTZ-DK and ∆EawCVTZ are calculated as valence electronic energy differences, except for M ) Hf. The core-valence correlation corrections were also calculated at the CCSD(T)-DK/awCVTZ-DK level, except for M ) Hf. Our recent work28 has shown that the core-valence correction calculated at the CCSD(T)/awCVTZ level can be quite different from those calculated at the CCSD(T)-DK/awCVTZ-DK level, especially for the calculations of the total atomization energies. Molecular spin-orbit (SO) effects were calculated at the SODFT level with the B3LYP functional using a pseudopotential basis set with a spin-orbit component by the variational treatment of the one-electron SO operator.43 For the trimers and tetramers, equilibrium geometries calculated at the B3LYP level were used in CCSD(T) calculations. The CCSD(T) calculations were carried out with the aD and aT basis sets. Core-valence correlation corrections were calculated at the CCSD(T) level with awCVDZ basis sets, and scalar relativistic corrections were calculated as the expectation values of the MVD operators for the CISD/aT wave function.

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Figure 1. Structures of the low-energy conformers of the singlet ground state of (ZrO2)n (n ) 2-4). The bond lengths (Å) are calculated at the CCSD(T)/aT level for n ) 2 and at the B3LYP/aD level for n ) 3 and 4. The relative energies (kcal/mol) are calculated at the CCSD(T)/CBS level for n ) 2 and at the CCSD(T)/aT//B3LYP/aD level for n ) 3 and 4.

Time-dependent DFT (TD-DFT)44 calculations at the B3LYP/ aD and BP86/aD levels were carried out to obtain the lowest 10 singlet and triplet excitation energies from the singlet groundstate equilibrium geometries of the low-lying neutral structures. For the B3LYP functional, an asymptotic correction44f,g was employed, which has no significant effect on the excitation energies for these low-lying excited states.26 In addition, adiabatic and vertical excitation energies from the singlet ground state to the first triplet excited state were calculated selfconsistently at the B3LYP/aT//B3LYP/aD, BP86/aT//BP86/aD, and CCSD(T)/aD//B3LYP/aD levels, and they are compared with the TD-DFT results. Excitation energies from the singlet ground states to the lowest few singlet excited states were also calculated with the equation-of-motion (EOM) approach at the EOM-CCSD level45 with the aD basis set. All DFT calculations were performed with the Gaussian 03 program package.46 For the pure DFT methods, the density fitting approximation was employed to speed up the calculations.47 The density fitting sets were automatically generated from the atomic orbital primitives.46 The SO-DFT and TD-DFT calculations were performed with the NWChem 5.1 program package.48 The CCSD(T) and EOM-CCSD calculations were performed with the MOLPRO 2008.1 program package.49 The open-shell calculations were done with the R/UCCSD(T) approach, where a restricted open-shell Hartree-Fock (ROHF) calculation was initially performed and the spin constraint was then relaxed in the coupled cluster calculation.50 The calculations were performed on the Opteron-based dense memory cluster (DMC) and Itanium 2-based SGI Altix supercomputers at the Alabama Supercomputer Center, the Xeon-based Dell Linux cluster at the University of Alabama, and the local Opteronbased and Xeon-based Penguin Computing Linux cluster and the Opteron-based Linux cluster at the Molecular Science Computing Facility from the Pacific Northwest National Laboratory. Multidimensional Franck-Condon factors (FCFs) for the vibronic transitions from the ground state of the anion to that of the neutral cluster were calculated within the harmonic

approximation to simulate the anion photoelectron spectrum. These FCFs were calculated51 using the recursion relations first derived by Doktorov et al.52 and recently rederived by Hazra and Noijima.53 The program was adapted from the work of Yang et al.54 The three-level binary tree algorithms of Ruhoff and Ratner,55 which were derived from the binary tree algorithm of Gruner and Brumer,56 were implemented. In addition, a modified version of the backtracking algorithm by Kemper et al.57 has been implemented to generate all of the vibrational states to calculate at a given level on-the-fly. The backtracking algorithm also allows for efficient searching and utilization of the calculated integrals. In simulating the spectra, the calculated equilibrium geometries, harmonic frequencies, and normal coordinates were used without further modification. Our program interfaces with Gaussian 03, MOLPRO, and NWChem. A Boltzmann distribution was used to account for the finite temperature effect with a Lorentzian line shape function and a typical experimental line width. Results and Discussion Equilibrium Geometries of Neutral Clusters. Optimized molecular structures for the low-lying electronic states of the (MO2)n (M ) Zr, Hf; n ) 2-4) clusters are shown in Figure 1 for M ) Zr and in Figure 2 for M ) Hf. Those for their anions are shown in Figures 3 and 4, respectively. Additional conformations (we use the term conformer for these structures to include the term isomer) with higher energies are given as Supporting Information for both the neutral and anionic clusters. Also given in these figures are the optimized metal-oxygen bond lengths and calculated relative energies at 0 K at the CCSD(T) level (see the exact basis set used in the figure caption). For monomers and dimers, the metal-oxygen bond lengths and bond angles calculated at the CCSD(T), B3LYP, and BP86 levels of theory are given as Supporting Information. The calculated relative energies for these low-lying states at the CCSD(T) and DFT levels are shown in Tables 1 and 2. Those for the additional conformations are given as Supporting Information. These results are, in general, similar to those for

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Figure 2. Structures of the low-energy conformers of the singlet ground state of (HfO2)n (n ) 2-4). The bond lengths (Å) are calculated at the CCSD(T)/aT level for n ) 2 and at the B3LYP/aD level for n ) 3 and 4. The relative energies (kcal/mol) are calculated at the CCSD(T)/CBS level for n ) 2 and at the CCSD(T)/aT//B3LYP/aD level for n ) 3 and 4.

Figure 3. Structures of the low-energy conformers of the doublet ground state of (ZrO2)n- (n ) 2-4). The bond lengths (Å) are calculated at the CCSD(T)/aT level for n ) 2 and at the B3LYP/aD level for n ) 3 and 4. The relative energies (kcal/mol) are calculated at the CCSD(T)/CBS level for n ) 2, at the CCSD(T)/aT levels for n ) 3, and at the CCSD(T)/aD levels for n ) 4.

M ) Ti from our previous work,27 and we used the same notations for these structures for the ease of comparison. The ground states of MO2 (M ) Zr, Hf) were predicted to be the 1A1 state of C2V symmetry, similar to that of TiO2.27 The MdO bond lengths calculated at the CCSD(T)/aT level for M ) Zr and Hf (1.802 and 1.798 Å; see Supporting Information) are nearly the same, and they are ∼0.13 Å longer than that for M ) Ti. A similar trend was found for MO3 (M ) Cr, Mo, W), where the MdO bond lengths for M ) Mo and W calculated at the same level of theory (1.719 and 1.732 Å) are 0.12 to 0.14 Å longer than that for M ) Cr.24 The MdO bond lengths for the group IVB metal oxides are longer than those for the group VIB metal oxides by 0.07 to 0.08 Å for the same transition row. This can be attributed to the fact that the group IVB metals are in the nominal +4

oxidation state and the group VIB metals are in the nominal +6 oxidation state in these oxides. The OdMdO bond angles (109.7 and 109.6°; see Supporting Information) in ZrO2 and HfO2 calculated at the CCSD(T)/aT level are very close to the ideal tetrahedral value of 109.5°. They are ∼3° smaller than that in TiO2.27 A similar observation was made for MO3 (M ) Cr, Mo, W), where the OdModO and OdWdO bond angles (110.9 and 108.4°) were predicted to be within 1° of the ideal tetrahedral angle, and the OdCrdO bond angle was predicted to be ∼6° larger than the tetrahedral angle at the same level of theory.24 The larger OdMdO angles for M ) Ti and Cr as compared to those for the heavier metals is likely due to the stronger electron repulsion from neighboring oxygen atoms resulting from the much shorter MdO bond lengths.

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Figure 4. Structures of the low-energy conformers of the doublet ground state of (HfO2)n- (n ) 2-4). The bond lengths (Å) are calculated at the CCSD(T)/aT level for n ) 2 and at the B3LYP/aD level for n ) 3 and 4. The relative energies (kcal/mol) are calculated at the CCSD(T)/CBS level for n ) 2, at the CCSD(T)/aT levels for n ) 3, and at the CCSD(T)/aD levels for n ) 4.

The calculated ZrdO and HfdO bond lengths at the CCSD(T)/aT level are 0.02 to 0.03 Å longer than those measured by Fourier transform microwave spectroscopy (1.7710 ( 0.0007 Å for M ) Zr, and 1.7764 ( 0.0004 Å for M ) Hf).12 At the CCSD(T)/awCVTZ level with the 1s2 electrons on O and (n 1)s2(n - 1)p6 electrons on Zr and Hf correlated, the calculated MdO bond lengths are 1.777 and 1.776 Å, which are now in excellent agreement with experiment. Calculations at the CCSD(T)-DK/awCVTZ-DK level give similar MdO bond lengths of 1.776 and 1.783 Å, which are also in excellent agreement with experiment. The OdMdO bond angles calculated at the CCSD(T)/aT level are larger than the experimental values of 108.11 ( 0.08° for M ) Zr and 107.51 ( 0.01° for M ) Hf by 1.6 and 2.1° for M ) Zr and Hf, respectively. The calculated bond angles at the CCSD(T)/awCVTZ (108.3 and 107.9°) and CCSD(T)-DK/awCVTZ-DK (108.1 and 107.4°) levels are in excellent agreement with experiment. A similar observation is made for M ) Ti. The MdO bond length and the OdMdO bond angle obtained from Fourier transform microwave spectroscopy are 1.651 Å and 111.57°.58 Our calculated values at the CCSD(T)/aT level are 1.666 Å and 112.4°.27 Those calculated at the CCSD(T)/awCVTZ level (1.653 Å and 112.0°) and at the CCSD(T)-DK/awCVTZ-DK level (1.652 Å and 111.7°) are in better agreement with experiment, although the differences in the calculated values are smaller than those for M ) Zr and Hf. In our recent benchmark calculations on the M-NH3 (M ) Na, Al, Ga, In, Cu, Ag) complexes, we found that inclusion of the core-valence correlation had significant effects on the calculated M-N bond length for M ) In, Ag, and especially Na and Ga.40 This is due to the fact that the energies of the valence orbitals on the more electronegative elements (N and O) are similar to those of the inner-shell orbitals on the metal atoms. The ground state of M2O4 (M ) Zr, Hf) was predicted to be the 1Ag state in C2h symmetry, similar to that for M ) Ti. They also have two low-lying conformers of C2V and C3V symmetry on the singlet surface. At the CCSD(T)/CBS level, the C2V structure was predicted to lie higher in energy than the C2h structure by ∼6 kcal/mol for M ) Ti27 and Zr and ∼8 kcal/mol

for M ) Hf. The calculated energy difference between the C2h and C3V structures is ∼13 kcal/mol for M ) Ti, ∼7 kcal/mol for M ) Zr, and ∼3 kcal/mol for M ) Hf, which varies more than that between the C2h and C2V structures. Thus, from Ti to Zr to Hf, the C3V structure of the dimer is stabilized relative to the C2h structure, whereas there is little change in the relative stability of the C2V structure to the C2h structure. The ground state of M3O6 (M ) Zr, Hf) was predicted to be the 1A′ state of the Cs structure, similar to that for M ) Ti.27 Two low-lying conformers, the C1 and C2 structures, were also predicted on the singlet surface. The C1 structures for M ) Zr and Hf are local minima at both the B3LYP and BP86 levels, in contrast to that for M ) Ti, where it is a local minimum at the B3LYP level but not at the BP86 level. At the CCSD(T)/ aT//B3LYP/aD level, the C1 structure is higher in energy than the Cs structure by 8-10 kcal/mol, whereas the C2 structure is higher by 10-15 kcal/mol. The ground state of M4O8 (M ) Zr, Hf) was predicted to be the 1A1 state of the (C2V a) structure, similar to that for M ) Ti.27 Two low-lying conformers, the C2h and (C2V b) structures, were also predicted on the singlet surface. At the CCSD(T)/ aT//B3LYP/aD level, the C2h structure was predicted to be higher in energy than the (C2V a) structure by ∼10 kcal/mol for M ) Zr and Hf but only by ∼5 kcal/mol for M ) Ti. The (C2V b) structure was predicted to be higher in energy than the (C2V a) structure by ∼23 kcal/mol for M ) Zr and Hf but only by ∼16 kcal/mol for M ) Ti. Thus, these two structures are destabilized relative to the (C2V a) structure from Ti to Zr and Hf. The T1 diagnostics59 from the CCSD(T) calculations (Supporting Information) range from 0.028 to 0.035 for the lowlying structures of the (ZrO2)n (n ) 1-4) clusters and from 0.025 to 0.031 for the (HfO2)n clusters. In comparison, the T1 values for the (TiO2)n clusters, where some experimental benchmark values are available and good agreement with experiment is found, are 0.036 to 0.043;27 therefore, we expect the CCSD(T) method to perform similarly or better for (ZrO2)n and (HfO2)n. As the T1 diagnostic is an average indicator of the quality of the CCSD treatment for the entire molecule, we also

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TABLE 1: Relative Energies at 0 K (∆E0K, kcal/mol) for the Low-Energy Conformers of the Ground Singlet State for (MO2)n and of the Ground Doublet State For (MO2)n- (M ) Zr, Hf; n ) 2 - 4) Calculated at the CCSD(T) Level molecule Zr2O4 Hf2O4 Zr2O4Hf2O4Zr3O6 Hf3O6 Zr3O6Hf3O6Zr4O8 Hf4O8 Zr4O8Hf4O8-

state

∆EaDa

∆EaTb

∆EaQb

∆ECBSc

∆ECV(D)b,d

∆ECV(T)b,e

∆ESRb,f

∆EZPEg

∆E0Kh

1Ag (C2h) 1 A1 (C2V) 1 A1 (C3V) 1 Ag (C2h) 1 A1 (C3V) 1 A1 (C2V) 2 A1 (C3V) 2 A1 (C2V) 2 Ag (C2h) 2 A1 (C3V) 2 A1 (C2V) 2 Ag (C2h) 1 A′ (Cs) 1 A (C1) 1 A (C2) 1 A′ (Cs) 1 A (C1) 1 A (C2) 2 A′ (Cs) 2 A (C2) 2 A′ (Cs) 2 A (C2) 1 A1 (C2V a) 1 Ag (C2h) 1 A1 (C2V b) 1 A1 (C2V a) 1 Ag (C2h) 1 A1 (C2V b) 2 A′ (Cs) 2 A1 (C2V b) 2 A (C2) 2 A′ (Cs) 2 A1 (C2V b) 2 A (C2)

0.00 6.02 6.41 0.00 3.35 7.15 0.00 3.11 5.89 0.00 7.92 9.59 0.00 9.66 12.39 0.00 9.64 11.50 0.00 38.12 0.00 40.65 0.00 8.88 24.28 0.00 6.96 25.55 0.00 8.67 19.78 0.00 12.05 17.86

0.00 5.88 6.55 0.00 3.67 6.99 0.00 2.65 5.53 0.00 7.23 9.05 0.00 8.32 11.73 0.00 7.82 10.02 0.00 36.87 0.00 38.85 0.00 11.29 22.00 0.00 9.60 22.62

0.00 5.85 6.43 0.00 3.45 7.05 0.00 2.85 5.75 0.00 7.78 9.50

0.00 5.84 6.33 0.00 3.30 7.09 0.00 3.00 5.91 0.00 8.17 9.82

0.00 +0.28 -1.58 0.00 -0.72 +0.33 0.00 +1.22 +0.79 0.00 +0.49 -0.23 0.00 +0.99 +0.93 0.00 +0.53 +0.34 0.00 +1.10 0.00 -0.02 0.00 +0.08 +1.81 0.00 +0.33 +0.58

0.00 +0.39 +0.30 0.00 +0.31 +0.45 0.00 -0.36 -1.03 0.00 -0.40 -1.19

0.00 -0.01 +0.04 0.00 +0.03 -0.01 0.00 -0.06 -0.06 0.00 -0.06 -0.07 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -0.03 0.00 -0.03 0.00 -0.01 -0.03 0.00 -0.02 -0.02

0.00 -0.04 -0.22 0.00 -0.21 -0.05 0.00 -0.07 -0.03 0.00 -0.14 -0.10 0.00 -0.20 +0.02 0.00 -0.18 0.00 0.00 -0.17 0.00 -0.14 0.00 -0.41 -0.60 0.00 -0.32 -0.45 0.00 -0.18 -0.75 0.00 -0.14 -0.61

0.0 6.2 6.4 0.0 3.4 7.5 0.0 2.5 4.8 0.0 7.6 8.5 0.0 9.1 12.7 0.0 8.2 10.4 0.0 37.8 0.0 38.7 0.0 10.9 23.2 0.0 9.6 22.7 0.0 8.5 19.0 0.0 11.9 17.3

a Geometries from CCSD(T)/aD for n ) 2 and from B3LYP/aD for n ) 3 and 4. b Geometries from CCSD(T)/aT for n ) 2 and from B3LYP/aD for n ) 3 and 4. c Extrapolated using the mixed Gaussian/exponential formula for the CCSD(T) energies with the aD, aT, and aQ basis sets. d CCSD(T)/awCVDZ. e CCSD(T)/awCVTZ. f CISD/aT. g BP86/aD. h ∆E0K ) ∆ECBS + ∆ECV(T) + ∆ESR + ∆EZPE for n ) 2, and ∆EaT + ∆ECV(D) + ∆ESR + ∆EZPE for n ) 3 and 4. For M4O8-, ∆E0K ) ∆EaD + ∆ZPE.

TABLE 2: Relative Energies at 0 K in kcal/mol for the Low-Energy Conformers of the Ground Singlet State for (MO2)n and of the Ground Doublet State for (MO2)n- (M ) Zr, Hf; n ) 2-4) Calculated at the CCSD(T), B3LYP/aT//B3LYP/aD, and BP86/aT//BP86/aD Levels state

CCSD(T)a,b

B3LYPa

BP86

CCSD(T)a,b

B3LYPa

BP86

Ag (C2h) 1 A1 (C2V) 1 A1 (C3V) 1 A′ (Cs) 1 A (C1) 1 A (C2) 1 A1 (C2V a) 1 Ag (C2h) 1 A1 (C2V b) 2 A1 (C3V) 2 A1 (C2V) 2 Ag (C2h) 2 A′ (Cs) 2 A (C2) 2 A′ (Cs) 2 A1 (C2V b) 2 A (C2)

M ) Zr 0.0 6.2 6.4 0.0 9.1 12.7 0.0 10.9 23.2 0.0 2.5 4.8 0.0 37.8 0.0 8.5 19.0

M ) Zr 0.0 6.2 9.2 0.0 5.1 6.8 0.0 12.6 13.3 0.0 –3.6 –3.4 0.0 23.6 0.0 –2.5 22.0

M ) Zr 0.0 6.2 8.5 0.0 5.2 5.4 0.0 12.0 13.9 0.0 –4.5 –5.1 0.0 17.9 0.0 –4.9 20.0

M ) Hf 0.0 7.5 3.4 0.0 8.2 10.4 0.0 9.6 22.7 0.0 7.6 8.5 0.0 38.7 0.0 11.9 17.3

M ) Hf 0.0 7.3 6.7 0.0 3.8 4.5 0.0 11.5 12.1 0.0 0.1 –0.7 0.0 23.8 0.0 –2.2 22.6

M ) Hf 0.0 7.2 6.3 0.0 3.6 2.9 0.0 11.3 12.5 0.0 –1.3 –2.9 0.0 17.8 0.0 –4.7 19.3

molecule M 2 O4 M 3 O6 M 4 O8 M2O4– M3O6– M4O8–

a

1

ZPEs from BP86/aD. b CCSD(T)/CBS for n ) 2 and CCSD(T)/aT for n ) 3 and 4 (CCSD(T)/aD for M4O8-) from Table 1.

report the D1 diagnostic (Supporting Information), which is an indicator for the most difficult part of the electronic structure of a molecule to describe.60 The D1 diagnostics from the CCSD(T) calculations are 0.068-0.078 for the (ZrO2)n clusters

and 0.059-0.073 for the (HfO2)n clusters, and these values are consistent with the size of the T1 values.60 Natural Charges of Neutral Clusters. Table 3 presents the natural charges on the O, Zr, and Hf atoms from the natural

Molecular Structures and Energetics of (ZrO2)n and (HfO2)n

J. Phys. Chem. A, Vol. 114, No. 7, 2010 2671

TABLE 3: Natural Charges on the Metal and Oxygen Atoms for the Low-Energy Conformers of the Ground Singlet State for (MO2)n (M ) Ti, Zr, and Hf; n ) 1-4) from the Natural Population Analysis at the B3LYP/aD Level molecule ZrO2 Zr2O4 Zr3O6 Zr4O8 HfO2 Hf2O4 Hf3O6 Hf4O8 TiO2 Ti2O4 Ti3O6 Ti4O8

a

state

M

Od(M)

A1 (C2V) Ag (C2h) 1 A1 (C2V) 1 A1 (C3V) 1 A′ (Cs) 1 A (C1) 1 A (C2) 1 A1 (C2V a) 1 Ag (C2h) 1 A1 (C2V b) 1 A1 (C2V) 1 Ag (C2h) 1 A1 (C2V) 1 A1 (C3V) 1 A′ (Cs) 1 A (C1) 1 A (C2) 1 A1 (C2V a) 1 Ag (C2h) 1 A1 (C2V b) 1 A1 (C2V) 1 Ag (C2h) 1 A1 (C2V) 1 A1 (C3V) 1 A′ (Cs) 1 A (C1) 1 A (C2) 1 A1 (C2V a) 1 Ag (C2h) 1 A1 (C2V b)

1.94 2.08(×2) 2.08(×2) 2.20(dO), 2.03 2.15(×2), 2.12 2.17(×2), 2.11 2.09(×2), 2.21 2.20(×2,dO), 2.21(×2) 2.16(×2,dO), 2.23(×2) 2.17(×2,dO), 2.20(×2) 2.02 2.14(×2) 2.14(×2) 2.29(dO), 2.08 2.23(×2), 2.18 2.24(×2), 2.18 2.14(×2), 2.30 2.27(×2,dO), 2.29(×2) 2.23(×2,dO), 2.31(×2) 2.24(×2,dO), 2.27(×2) 1.52 1.53(×2) 1.53(×2) 1.53(dO), 1.47 1.45(×2), 1.55 1.43, 1.54, 1.62(-O) 1.55(×2), 1.42 1.52(×2,dO), 1.53(×2) 1.50(×4) 1.43(×2,dO), 1.63(×2)

-0.97(x2) -0.96(×2) -0.94(×2) -0.99 -0.98(×2) -1.00, -0.96 -0.94(×2) -0.96(×2) -0.96(×2) -1.00(×2) -1.01(×2) -1.01(×2) -0.99(×2) -1.06 -1.04(×2) -1.01, -1.06 -0.99(×2) -1.02(×2) -1.02(×2) -1.06(×2) -0.76(×2) -0.67(×2) -0.65(×2) -0.64 -0.63(×2) -0.63, -0.65 -0.64(×2) -0.60(×2) -0.60(×2) -0.63(×2)

1

1

O-(M) -1.11(×2) -1.13(×2) -1.08(×3) -1.09(×2), -1.18, -1.11a -1.17, -1.19, -1.07, -1.08 -1.13(×4) -1.16(×4), -1.12, -1.13b -1.17(×4), -1.09(×2)a -1.10(×4), -1.18, -1.16 -1.13(×2) -1.15(×2) -1.11(×3) -1.12(×2), -1.21, -1.12a -1.19, -1.21,-1.09, -1.10 -1.15(×4) -1.20(×4), -1.15, -1.13b -1.20(×4), -1.11(×2)a -1.13(×4), -1.21, -1.18 -0.86(×2) -0.88(×2) -0.79(×3) -0.78(×2), -0.84, -0.81a -0.89(×2), -0.76, -0.77 -0.81(×4) -0.83(×4), -0.77, -0.81b -0.82(×4), -0.75(×2)a -0.78(×4), -0.85, -0.88

Tribridged oxygen atoms. b Tetrabridged oxygen atoms.

population analysis.61,62 Those calculated for the (TiO2)n (n ) 1-4) clusters are also given for comparison. For the monomer, the natural charge on the metal atom increases substantially from Ti (1.52e) to Zr and Hf (1.94e and 2.02e), indicating greater ionic character for the heavier metal oxides. A similar trend was predicted for MO3 from Cr (1.33e) to Mo and W (1.94e and 2.15e).23 Furthermore, the natural charges calculated for Zr and Hf are comparable to those for Mo and W, despite the difference in their formal oxidation state. The natural charge decreases from Ti to Cr, which is suggestive of greater ionic character in TiO2 than that in CrO3. From the monomer to the dimer, the natural charges on the metal atoms increase by 0.1-0.3e for M ) Zr and Hf but remain essentially the same for M ) Ti. The bridge oxygen atom carries more negative natural charge than the terminal oxygen atom by 0.1-0.2e. The metal atom with a terminal oxygen atom in the C3V structure of the dimer carries the most positive charge for M ) Zr and Hf, and the charge distribution is similar for the C2h and C2V structures. The natural charges on the metal atoms are essentially the same for the different structures of the tetramer, ∼1.50e for Ti, ∼2.20e for Zr, and 2.25e for Hf. These values for Ti and Zr are larger than the converged values of Cr and Mo in the (MO3)n clusters by ∼0.3e and 0.2e, respectively, but the values for Hf are very close to the converged values for W.23 Harmonic Frequencies of Neutral Clusters. Harmonic vibrational frequencies of the metal-oxygen stretches calculated at the B3LYP/aD level are presented in Table 4, and those for M ) Ti are also given for comparison. For the monomers, harmonic vibrational frequencies were also calculated at the CCSD(T) level with the aT and awCVTZ basis sets, and those for the symmetric stretch (ν1) are given as Supporting Informa-

tion, as well as the values calculated at the BP86/aD level. Compared to the frequency of ν1 calculated at the CCSD(T)/ awCVTZ level with the core-valence correlation correction included, that calculated at the CCSD(T)/aT level with valence correlation only is smaller by 22 and 27 cm-1 for M ) Ti and Zr, respectively; that calculated at the B3LYP/aD level is larger by 36, 15, and 8 cm-1 for M ) Ti, Zr, and Hf, respectively; and that calculated at the BP86/aD level is smaller by 2, 13, and 20 cm-1 for M ) Ti, Zr, and Hf, respectively. For HfO2, the inclusion of the core-valence correlation at the CCSD(T) level is necessary for the numerical Hessian calculation to give meaningful results. On the basis of the above comparison, the B3LYP/aD method gives better stretching frequencies for the hafnium oxides, the BP86/aD method gives better stretching frequencies for the titanium oxides, and these two DFT methods give similar stretching frequencies for the zirconium oxides. This is not surprising on the basis of our recent benchmark studies on the electronic affinities of group VIB transition-metal oxides, where we found pure functionals to be more suitable for describing the first-row transition-metal compounds due to their larger multireference characters.24 For the monomer, the calculated symmetric MdO stretching frequency at the B3LYP/aD level decreases from M ) Ti to Zr and Hf. The MdO symmetric stretching frequency calculated at the same level of theory for MO3 (1012, 987, and 1001 cm-1 for M ) Cr, Mo, and W) are slightly smaller than that for M ) Ti. The decrease in the harmonic frequency from Ti to Zr and Hf is due to the increase in atomic mass as the MdO bond energy was predicted to increase in that order.28 For the low-energy structures of the (MO2)n (M ) Ti, Zr, Hf; n ) 1-4) clusters, there are always two terminal MdO bonds, except for the C3V structure of the dimer. As shown in

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Li and Dixon

TABLE 4: Harmonic Vibrational Frequencies in cm-1 of the Metal-Oxygen Stretches for the Low-Energy Conformers of the Ground Singlet State for (MO2)n (M ) Ti, Zr, Hf; n ) 1-4) Calculated at the B3LYP/aD Levela

a

molecule

MdO

ZrO2 (C2V) Zr2O4 (C2h) Zr2O4 (C2V) Zr2O4 (C3V) Zr3O6 (Cs) Zr3O6 (C1) Zr3O6 (C2) Zr4O8 (C2V a) Zr4O8 (C2h) Zr4O8 (C2V b) HfO2 (C2V) Hf2O4 (C2h) Hf2O4 (C2V) Hf2O4 (C3V) Hf3O6 (Cs) Hf3O6 (C1) Hf3O6 (C2) Hf4O8 (C2V a) Hf4O8 (C2h) Hf4O8 (C2V b) TiO2 (C2V) Ti2O4 (C2h) Ti2O4 (C2V) Ti2O4 (C3V) Ti3O6 (Cs) Ti3O6 (C1) Ti3O6 (C2) Ti4O8 (C2V a) Ti4O8 (C2h) Ti4O8 (C2V b)

924 (s), 860 (as) 914 (s), 903 (as) 922 (s), 901 (as) 900 915 (s), 896 (as) 921, 899 915 (s), 912 (as) 924 (s), 914 (as) 916 (s), 912 (as) 906 (s), 894 (as) 908 (s), 822 (as) 890 (s), 883 (as) 898 (s), 884 (as) 879 890 (s), 876 (as) 901, 877 892 (s), 891 (as) 901 (s), 894 (as) 891 (s), 888 (as) 885 (s), 876 (as) 1026 (s), 985 (as) 1043 (s), 1022 (as) 1054, 1019 1045 1059 (s), 1031 (as) 1056 (s), 1034 (as) 1046 (s), 1040 (as) 1068 (s), 1054 (as) 1063 (s), 1056 (as) 1048 (s), 1028 (as)

M-O 675 (as + as), 651 (s + s), 594 (s - s), 469 (as - as) 668 (as + as), 645 (s + s), 587 (s - s), 462 (as - as) 786 (s), 667 (as, ×2), 524 (s) 806, 728, 633, 601, 601, 505, 443, 384, 349 815, 750, 686, 631, 506, 500, 461, 375 691, 659, 656, 641, 609, 608, 460, 460 727, 718, 691, 629, 628, 607, 597, 521, 504, 461, 455, 451, 450 718, 698, 669, 621, 605, 592, 549, 532, 529, 497, 473, 424 849, 846, 798, 700, 626, 620, 521, 501, 427, 416, 399, 389 666 (as + as), 653 (s + s), 583 (s - s), 496 (as - as) 657 (as + as), 645 (s + s), 573 (s - s), 487 (as - as) 770 (s), 667(as, ×2), 521(s) 799, 713, 638, 603, 588, 513, 460, 399, 352 810, 747, 679, 646, 510, 481, 461, 388 699, 660, 654, 622, 597, 594, 482, 481 734, 722, 670, 634, 607, 604, 594, 531, 522, 471, 467, 463, 462 727, 712, 637, 609, 601, 588, 553, 539, 532, 519, 483, 435 853, 843, 784, 701, 641, 636, 502, 483, 435, 425, 403, 392 733 (as + as), 733 (s + s), 700 (s - s), 504 (as - as) 725(as + as), 725 (s + s), 694 (s - s), 491 (as - as) 874 (s), 745(as, ×2), 573 (s) 881, 835, 716, 703, 653, 558, 481, 415, 391 891, 820, 784, 695, 623, 569, 516, 395 770, 740, 732, 719, 710, 708, 495, 495 802, 780, 763, 748, 701, 690, 688, 563, 526, 509, 474, 468, 457 801, 757, 750, 749, 682, 678, 630, 578, 561, 527, 486, 433 931, 906, 900, 796, 683, 681, 625, 601, 471, 461, 446, 431

s ) symmetric stretch, as ) asymmetric stretch.

Table 4, the symmetric MdO stretching frequency is always larger than the asymmetric mode. For M ) Zr and Hf, the frequency difference between the MdO symmetric and asymmetric stretches decreases significantly from the monomer (60-90 cm-1) to the larger clusters (less than 20 cm-1), but for M ) Ti, it only slightly decreases. In addition, for M ) Zr and Hf, the MdO symmetric stretching frequency decreases by up to 30 cm-1 from the monomer to the larger clusters, whereas for M ) Ti, it increases by up to 40 cm-1. For the clusters with n > 1, the MdO stretching frequencies range from 1020 to 1070 cm-1 for M ) Ti, 890 to 930 cm-1 for M ) Zr, and 870 to 900 cm-1 for M ) Hf. In contrast to the number of MdO bonds, the number of M-O bonds increases as the cluster size increases. The M-O stretching frequencies are lower than those for MdO due to the lower metal-oxygen bond energy.28 For the C2h and C2V structures of the dimer, four M-O stretching modes arise from the different combinations of the stretches of the four bridge M-O bonds. These vibrational modes are for the M-O-M-O four-member ring, and their frequencies range from 490 to 730 cm-1 for M ) Ti, from 460 to 680 cm-1 for M ) Zr, and from 490 to 670 cm-1 for M ) Hf. For the zirconium oxide clusters, vibrational frequencies between 600 and 700 cm-1 were shown to be present in IR experiments13 and were assigned to Zr-O vibrations in the Zr-O-Zr-O four-member ring, consistent with our calculated harmonic frequencies and IR intensities for the ring vibrations of Zr2O4. The ring modes calculated at ∼670 cm-1 were predicted to have the largest IR intensities followed by those calculated at ∼590 cm-1. For the C3V structure of the dimer, although there are six M-O bonds, there are only four M-O stretching modes. Three of these modes arise from the combinations (a1 + e) of the stretches of the three bridge M-O

bonds on the M center without the terminal oxygen atom. The a1 mode has a substantially higher frequency than the M-O stretches in the C2h and C2V structures of the dimer by 100-140 cm-1, and the frequency of the e mode is in the higher range of the M-O stretching frequencies for the C2h and C2V structures. The fourth M-O stretching mode in the C3V structures of the dimer arises from the totally symmetric combination of the stretches of the other three bridge M-O bonds, and its frequency is in the lower range of the M-O stretching frequencies for the C2h and C2V structures. The M-O stretching modes in the clusters with n > 2 are more complex, but they can be considered as arising from the different M-O stretching modes discussed above for the three structures of the dimer. For example, for the Cs structure of the trimer, the three highest-frequency M-O stretching modes mainly involve M-O stretches on the metal center with no terminal oxygen atom. The highest-frequency M-O stretching mode in this structure is larger than that in the C3V structures of the dimer by up to 30 cm-1. For the trimer and tetramer, the ground-state structure has one more M-O stretching mode than the other two low-lying states, which is consistent with it being the most stable structure. Due to the nature of the different M-O stretching modes, they have a larger frequency span than that for the MdO stretches. They range from 390 to 930 cm-1 for M ) Ti and 350 to 850 cm-1 for M ) Zr and Hf. Equilibrium Geometries of Anionic Clusters. The ground state of the anionic cluster is usually formed by adding an electron to the lowest unoccupied molecular orbital (LUMO) of the neutral ground state. In some cases, for example, for the (C2V a) and C2h structures, the resulting structure further distorts to assume a lower symmetry. The ground state of MO2- (M ) Zr, Hf) was predicted to be the 2A1 state of the C2V structure.

Molecular Structures and Energetics of (ZrO2)n and (HfO2)n The MdO bond lengths in the anion are calculated to be longer than those in the neutral oxide by ∼0.04 Å at the CCSD(T)/aT level. The calculated OdMdO bond angles in the anion are larger than those in the neutral oxide by ∼2° for M ) Zr and ∼4° for M ) Hf. The ground state of M2O4- (M ) Zr, Hf) was predicted to be the 2A1 state of the C3V structure. This is different from M ) Ti, where its ground state was predicted to be the 2A1 state of the C2V structure.27 Similar to the neutral dimer, the C3V structure is stabilized relative to the other two structures from Ti to Zr to Hf. At the CCSD(T)/CBS level, the 2A1 electron configuration of the C2V structure and the 2Ag electron configuration in the C2h structure are higher in energy than the 2A1 state of the C3V structure by ∼3 and 5 kcal/mol for M ) Zr and ∼8 and 9 kcal/ mol for M ) Hf. For M ) Ti, these two structures are lower than the C3V structure by ∼5 and 7 kcal/mol, respectively. For the C2h (C2V) structure, there is an additional electronic state, the 2Bu (2B2) state, which is higher in energy than the 2Ag (2A1) state by 4-7 (7-11) kcal/mol at the B3LYP/aD and BP86/aD levels. We also note that for the anion of the dimer, the C2V structure is lower in energy than the C2h structure by 1-2 kcal/ mol for all three metals, whereas for the neutral cluster, the C2V structure is higher in energy than the C2h structure by 6-8 kcal/ mol. The ground state of M3O6- (M ) Zr, Hf) was predicted to be the 2A′ state of the Cs structure. Similar to M ) Ti,27 optimization for the anion starting from the C1 structure of the neutral cluster gave the above Cs structure. At the CCSD(T)/ aT//B3LYP/aD level, the 2A electron configuration of the C2 structure was predicted to be ∼40 kcal/mol higher in energy than the Cs structure, similar to that for M ) Ti. Thus, the C2 structure is destabilized relative to the Cs structure from the neutral cluster to the anion. In addition, for the C2 structure, there is an additional electronic state, the 2B state, which is higher in energy than the 2A state by ∼1.5 and 2.5 kcal/mol at the B3LYP/aD and BP86/aD levels for M ) Zr and Hf, respectively. The ground state of M4O8- (M ) Zr, Hf) was predicted to be the 2A′ electron configuration of the Cs structure arising from distorting the neutral (C2V a) structure. The 2A1 electron configuration of the (C2V a) structure has one imaginary frequency of 244 and 107 cm-1 for M ) Zr and 175 and 70 cm-1 at the B3LYP/aD and BP86/aD levels, respectively, and lies ∼10 kcal/mol higher in energy than the Cs structure with or without the ZPE correction at the CCSD(T)/aD//B3LYP/aD level. For M ) Ti, this state has one imaginary frequency at the B3LYP/aD level but has none at the BP86/aD level, which was attributed to symmetry breaking with the B3LYP functional.27 Although the 2A′ electron configuration of the Cs structure for Ti4O8- was predicted to be a local minimum at both the B3LYP/aD and BP86/aD levels, it is only slightly lower in energy than the 2A1 electron configuration of the (C2V a) structure by 1, the MdO stretching frequencies range from 930 to 1010 cm-1 for M ) Ti, from 840 to 880 cm-1 for M ) Zr, and from 820 to 860 cm-1 for M ) Hf. The M-O stretching frequencies range from 340 to 860 cm-1 for M ) Ti and from 350 to 800 cm-1 for M ) Zr and Hf. Relative Cluster Energies. For the calculation of the relative energies of the three low-energy confomers of the dimer for M ) Zr and Hf at the CCSD(T) level, the basis set extrapolation effect from aD to CBS is