Molecular Structures, Acid−Base Properties, and Formation of Group 6

ARTICLE pubs.acs.org/JPCC

Molecular Structures, Acid
Base Properties, and Formation of Group 6 Transition Metal Hydroxides Shenggang Li, Courtney L. Guenther, Matthew S. Kelley, and David A. Dixon* Chemistry Department, The University of Alabama, Shelby Hall, Box 870336, Tuscaloosa, Alabama 35487-0336, United States

bS Supporting Information ABSTRACT: Density functional theory (DFT) and coupled cluster theory (CCSD(T)) were used to study the group 6 metal (M = Cr, Mo, W) hydroxides: MO3
m(OH)2m (m = 1
3), M2O6
m(OH)2m (m = 1
5), M3O9
m(OH)2m (m = 1, 2), and M4O11(OH)2. The calculations were done up to the complete basis set (CBS) limit for the CCSD(T) method. Molecular structures of many low-energy conformers/isomers were located. Brønsted acidities in the gas phase and pKa values in aqueous solution were predicted for MO3
m(OH)2m (m = 1
3) and MnO3n
1(OH)2 (n = 2
4). In addition, Brønsted basicities and Lewis acidities (fluoride affinities) were predicted for MO3
m(OH)2m (m = 1
3) as well as the metal oxide clusters MnO3n (n = 1
3). The metal hydroxides were predicted to be strong Brønsted acids and weak to modest Brønsted bases and Lewis acids. The pKa values can have values as negative as
31. Potential energy surfaces for the hydrolysis of the MnO3n (n = 1
4) clusters were calculated. Heats of formation of the metal hydroxides were predicted from the calculated reaction energies, and the agreement with the limited available experimental data is good. The first hydrolysis step leading to the formation of MnO3n
1(OH)2 was predicted to be exothermic, with the exothermicity becoming less negative as n increases and essentially converged at n = 3. Reaction rate constants for the hydrogen transfer steps were calculated using transition state theory and RRKM theory. Further hydrolysis of MnO3n
1(OH)2 tends to be endothermic especially for M = Cr. Fifty-five DFT exchange-correlation functionals were benchmarked for the calculations of the reaction energies, complexation energies, and reaction barriers by comparing to our CCSD(T) results. Overall, the DFT results for the potential energy surfaces are semiquantitatively correct, but no single functional works for all processes and all three metals. Among the functionals benchmarked, the wB97, wB97X, B1B95, B97-1, mPW1LYP, and X3LYP functionals have the best performance. Linear correlations between the calculated reaction barrier and reaction energy for hydrogen transfer reactions to the terminal dO atom and to the bridge O atom were found to be quite different, indicating their different reaction properties. The calculated Lewis acidity (fluoride affinity) was found to best correlate with the calculated adsorption energy, the dissociative adsorption energy, and the reaction barrier for hydrogen transfer reactions to the terminal dO atom.

’ INTRODUCTION Transition metal oxides (TMOs) and their associated compounds have received wide attention due to their rich acid/ base and redox chemistries and their ubiquitous applications in industry as catalysts and catalyst supports.1 Transition metal hydroxides (e.g., M(OH)n) and oxyhydroxides (e.g., M(dO)m(OH)n
2m) are of great importance due to their presence in many TMO-catalyzed processes (we will use the term hydroxides for both classes of compounds hereafter) and their acid
base chemistries.1c TMO clusters and related compounds have been used as models of actual catalysts and in some cases, such as the polyoxometallates, are the actual catalysts.2 The TMO and hydroxide clusters described herein are smaller than most of the polyoxometallates and can also be considered as models of these larger structures. Besides their acid/base and redox chemistries, many TMOs are semiconductors with a sizable band gap (MoO3, 3.2 eV; WO3, 2.6 eV)3 and have been extensively studied for their photocatalytic activities for H2O splitting and r 2011 American Chemical Society

CO2 conversion chemistries and other applications for the efficient utilization of the solar energy especially with TiO2 (band gap: 3.1 eV) and its mixed oxides.4 In addition, metal oxides can also be used as solid state gas sensors.5 Catalytic acid
base and redox chemistries account for over 90% of practical applications of metal oxides as catalysts. A fundamental understanding of TMO-catalyzed chemical transformations is needed so that new catalysts can be systematically developed from first principles, and thus, the acid
base properties are important in our understanding of TMO-catalyzed reactions. A key property of TMO catalysts is the acid
base character of the active site, and a key reaction to generate acidic sites is the hydrolysis of TMO species. The gas-phase molecular acidity, a well-studied property from both theory and experiment, Received: November 18, 2010 Revised: February 24, 2011 Published: March 31, 2011 8072

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Figure 1. Molecular structures of the metal hydroxides MO3
m(OH)2m (M = Cr, Mo, W; m = 1
3). Calculated metal oxygen bond lengths in Å at the B3LYP/aD level and relative energies at 0 K in kcal/mol at the CCSD(T)/CBS//B3LYP/aD level. M = Cr (Black), Mo (Red), and W (Blue).

is given by the free energy of the reaction AH f A
þ Hþ. The gas-phase basicity is defined as the negative of the free energy of the reaction B þ Hþ f BHþ. The temperature for these definitions is usually taken as 298 K. For example, an excellent correlation between the calculated gas-phase acidities at the density functional theory (DFT) level and the solution-phase acidity constants for a number of first row transition metal and inorganic oxyacids has been found.6 In addition, we have recently corrected the very strong gas-phase acidity scale on the basis of high level ab initio calculations at the G3MP2 and CCSD(T)/ CBS (complete basis set) levels.7 In that study, we also used a self-consistent reaction field (SCRF) approach8 with the conductor-like screening model (COSMO)9 to predict the pKa values of very strong acids in aqueous solution. In the current study, we use a similar approach to calculate the pKa values of the metal hydroxides in aqueous solution. We have recently developed a quantitative Lewis acidity scale based on calculated fluoride affinities (FAs) where the FA is defined as the negative of the enthalpy of the reaction A þ F
f AF
.10 Due to its high basicity and small size, the fluoride anion Lewis base reacts with essentially all Lewis acids. This scale has enabled us to predict the Lewis acidities of a wide range of inorganic acids, including the very strong Lewis acids, and organic molecules. A number of experimental and computational studies have been performed on the structures, properties, and reactions of the group 6 transition metal hydroxides. MO2(OH)2 and related molecules were produced and identified by infrared spectroscopy in an argon matrix by reacting laser-ablated metal atoms with H2O2.11 Cr was found to favor the formation of lower oxidation state compounds, whereas Mo and W favor higher oxidation state products. Reactions of group 5 (V, Nb, Ta) and 6 (W) transition metal oxide and hydroxide anions with H2O, HCl, and O2 were studied in a fast-flow reactor.12 In contrast to the Nb and Ta species, the tungsten oxide anions that were studied displayed no reactivity toward H2O. Ion
molecule reactions of CrxOyHz
with H2O and O2 were probed using a quadrupole ion trap

secondary ion mass spectrometer.13 It was found that some of the anions are also unreactive toward H2O or O2 under their experimental conditions. Very recently, Jarrold and co-workers have studied the reactions of small tungsten and mixed molybdenum
tungsten oxide oxygen-deficient cluster anions with H2O, D2O, and/or CO2 with mass spectrometry in a fast-flow reactor.14 It was found that H2 was released upon the reaction of a metal oxide cluster with H2O. Molecular structures and thermodynamic properties of small group 6 transition metal oxide and hydroxide clusters have been studied by DFT and ab initio methods.15 Hydrolysis of the M
O
M bridging oxygen in transition metal oxide cluster (M = Sc, Ti, V, Cr, Mn) was studied with the DFT method.16 The formation of gaseous CrO2(OH)2 was proposed to explain the observed erosion of the protective chromimum oxide scale on high-temperature alloys at high temperatures and humidities. Extensive studies have shown that one must take care in the calculation of the geometries and vibrational frequencies of transition metal compounds with ab initio molecular orbital methods. In general, one needs to use some type of correlation treatment, often beyond the second-order Møller
Plesset perturbation theory (MP2) level. The DFT method, especially with gradient-corrected exchange-correlation functionals, has been shown to yield good predictions of the structures and frequencies of transition metal complexes at a fraction of the computational cost of more expensive post-Hartree
Fock (HF) alternatives.17 The B3LYP18,19 hybrid exchange-correlation functional provides reasonable structures and energies especially for M = Mo and W.20 The high atomic numbers for the second and especially the third row transition metals imply that relativistic effects must be properly included to attain even semiquantitative accuracy.21,22 For most of the properties of interest in the proposed work, the core electrons are chemically inert so one can eliminate the core contributions from direct consideration by using pseudopotentials (PPs) or effective core potentials (ECPs).23
25 We have shown that one can get excellent results for the structures and 8073

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Figure 2. Molecular structures of the metal hydroxides M2O6
m(OH)2m (M = Cr, Mo, W; m = 1
5). Selected metal oxygen bond lengths in Å at the B3LYP/aD level and relative energies at 0 K in kcal/mol at the CCSD(T)/aT//B3LYP/aD level (at the CCSD(T)/aD//B3LYP/aD level for M2O(OH)10). M = Cr (Black), Mo (Red), and W (Blue).

frequencies of a broad range of transition metal complexes with ECPs and DFT when comparison is made to experiment.26 A

range of exchanged-correlation functionals for the DFT method have been extensively benchmarked for various applications,27 8074

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Figure 3. Molecular structures of the metal hydroxides MnO3n
m(OH)2m (M = Cr, Mo, W; n = 3, 4; m = 1, 2). Selected metal oxygen bond lengths in Å at the B3LYP/aD level and relative energies at 0 K in kcal/mol at the CCSD(T)/aD//B3LYP/aD level (at the CCSD(T)/aT//B3LYP/aD level for M3O8(OH)2). M = Cr (Black), Mo (Red), and W (Blue).

which can help one choose the appropriate functional for the problem of interest. We have recently studied the molecular structures, acid
base properties, electron affinities, electron excitation energies, clustering energies, and heats of formation of groups 6 and 4 TMO clusters of the form (MO3)n (M = Cr, Mo, W; n = 1
6) and (MO2)n (M = Ti, Zr, Hf; n = 1
4), as well as the partially reduced TMO cluster M3O8 (M = Cr, W).20a
g,28 In this study, we extend our prior work to the studies of the molecular structures, vertical electron excitation energies, acid
base properties, and heats of formations for the molecular clusters MO2(OH)2, MO(OH)4, M(OH)6, M2O5(OH)2, M2O4(OH)4, M2O3(OH)6, M2O2(OH)8, M2O(OH)10, M3O8(OH)2, M3O7(OH)4, and M4O11(OH)2 for M = Cr, Mo, and W (the group 6 transition metals). In addition, we describe the potential energy surfaces for the addition of H2O to the transition metal oxide clusters. The calculations have been done by using DFT and correlated ab initio molecular orbital theory methods. We also benchmarked a large number of DFT exchange-correlation functionals for the calculations of potential energy surfaces by comparing the DFT results with our more accurate ab initio results.

’ COMPUTATIONAL METHODS Equilibrium geometries and harmonic vibrational frequencies were calculated at the DFT level with the B3LYP18,19 exchangecorrelation functional. Geometry optimizations for the local minima were carried out with the Berny algorithm.29 For transition state optimizations, the synchoronous transit-guided

quasi-Newton (STQN) method was usually employed.30 Alternatively, for initial geometries sufficiently close to the transition states, the Berny algorithm was used instead for the transition state geometry optimizations. These geometry optimizations were done in the redundant internal coordinates.31,32 The above geometry optimizations and frequency calculations were performed with the aug-cc-pVDZ basis set for H and O33 and the aug-cc-pVDZ-PP basis set for Cr, Mo, and W based on small core ECPs.20e,34 This combination of basis sets will be abbreviated as aD hereafter. Electronic energies at the DFT level with the B3LYP functional at the above optimized geometries were also calculated with the aug-cc-pVTZ basis set for H and O and the aug-ccpVTZ-PP basis set for Cr, Mo, and W. This combination of basis sets will be abbreviated as aT. For selected reaction pathways, the electronic energies at the DFT level were also calculated with an additional 54 exchange-correlation functionals. These functionals can be catorgized into (1) local spin density approximation (LSDA): SVWN5;35,36 (2) generalized gradient approximations (GGAs): BLYP,37,19 BP86,37,38 BRxP86,39,38 BPW91,37,40 BB95,37,41 PW91,40 mPWPW91,42,40 mPWLYP,42,19 G96LYP,43,19 PBE,44 mPWPBE,38,44 OLYP,45,19 TPSS,46 PBEhPBE,47,44 PKZB,48 VSXC,49 Handy’s family of functionals HCTH/93, HCTH/147, HCTH/407,50 τ-HCTH,51 M06-L,52 and the BP86, PBE, PW91, and TPSS functionals with the long-range correction of Hirao and co-workers53 (LC-BP86, LC-PBE, LC-PW91, LC-TPSS), and (3) hybrid GGAs: B3P86,18,38 B3PW91,18,40 B1B95,41 B1LYP,54 mPW1PW91,42 B98,55 B97-1,50a B97-2,56 PBE0 (also known as PBE1PBE),57 HSE1PBE, HSE03, HSE06,58 PBEh1PBE,47,44 8075

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O3LYP,59 TPSSh,46 τ-HCTHhyb,51 BMK,60 X3LYP,61 and the long-range corrected functionals CAM-B3LYP,62 LC-wPBE,63 wB97, and wB97X.64 Electronic energies at the B3LYP optimized geometries were also calculated at the CCSD(T) level65
68 with the aD and aT basis sets. In addition, core
valence correlation corrections for the 1s2 electrons on O and (n
1)s2(n
1)p6 electrons on Cr (n = 3), Mo (n = 4), and W (n = 5) were calculated at the CCSD(T) level with the cc-pVDZ basis set for H, the cc-pwCVDZ basis set for O,69 and the cc-pwCVDZ-PP basis set for Cr, Mo, and W;34 this combination of basis sets will be abbreviated as wCVDZ. For the molecules with one metal atom, the CCSD(T) calculations were also preformed with the aQ basis set (aug-ccpVQZ for H and O and aug-cc-pVQZ-PP for Cr, Mo, W). The CCSD(T) energies were extrapolated to the complete basis set (CBS) limit using a mixed Gaussian/exponential formula.70 The cardinal numbers for the aD, aT, and aQ basis sets are 2, 3, and 4, respectively. Our recent studies on the group 6 TMO clusters have shown that the effect of the choice of the cardinal numbers in this extrapolation scheme is fairly small.20a In addition, the core
valence corrections including a relativistic correction were also calculated at the CCSD(T) level with the second-order Douglas
Kroll
Hess Hamiltonian71
73 and the aug-ccpwCVTZ-DK basis set (the aug-cc-pVTZ-DK basis set for H).74,75 This combination of basis sets will be abbreviated as awCVTZ-DK. For H and O, the diffuse and/or tight functions from the aug-cc-pVTZ basis set for H and the aug-cc-pwCVTZ basis set for O were used to augment the cc-pVTZ-DK basis set. For W, additional high angular momentum functions of the form 2f2g1h were added to treat the correlation effect of the 4f14 electrons, as the 4f orbitals on W lie very close in energy to the 2s orbital on O. For comparison, the core
valence corrections were also calculated at the CCSD(T)-DK/wCVTZ-DK, CCSD(T)/ awCVTZ, CCSD(T)/wCVTZ, and CCSD(T)/awCVDZ levels. The definition of these basis sets are similar to that for the above awCVTZ-DK and wCVDZ basis sets. Our previous studies20e on the thermochemistry of TMO clusters have shown that the core
valence corrections calculated at the CCSD(T)/awCVTZ level can be substantially different from those calculated at the CCSD(T)-DK/awCVTZ-DK level especially for calculating the total atomization energies (TAEs), and the core
valence corrections calculated at the CCSD(T)/wCVDZ level are usually closer to those calculated at the CCSD(T)-DK/awCVTZ-DK level than those calculated at the CCSD(T)/awCVDZ level for calculating the normalized clustering energies. Scalar relativistic corrections (ΔERel) were calculated from eq 1 following our previous work,20e where ΔEawCVTZ-DK and ΔEawCVTZ are the valence electronic energy contributions to a given property calculated at the CCSD(T)-DK/awCVTZ-DK and CCSD(T)/ awCVTZ levels, respectively. ΔERel ¼ ΔEawCVTZ-DK
ΔEawCVTZ

ð1Þ

The ΔERel correction can be considered to be a sum of two terms.20e One of them is the scalar relativistic correction calculated from the expectation value of the mass-velocity and Darwin (MVD) operator from the Breit
Pauli Hamiltonian76 for the CISD/aT wave function. The other term is the pseudopotential correction, which was defined by eq 2 and accounts for the error introduced by using the ECP.20e ΔEPP, corr ¼ ΔEawCVTZ-DK
ðΔEawCVTZ þ ΔEMVD Þ

ð2Þ

The above approach follows ours and others’ work on the accurate prediction of the heats of formation and other properties for a wide range of compounds.77 Vertical excitation energies to the lowest few singlet and triplet states from the singlet ground state equilibrium geometries for the low-lying neutral structures were calculated with the timedependent (TD) DFT method78 at the B3LYP/aD level. The asymptotic correction for this functional was employed,78f,g but this correction is not expected to significantly affect the excitation energies for the low-lying excited states.20 Vertical excitation energies from the singlet ground state to the lowest few singlet excited states for selected clusters were also calculated with the equation-of-motion (EOM) approach at the EOM-CCSD level79 with the aD basis set. The reaction rate constant at temperature T at the high pressure limit from the thermodynamic formulation of classical transition state theory (TST)80 is given by eq 3. ! ! kB T ΔSq
ΔH q exp k¥ ðTÞ ¼ σ exp ð3Þ h R RT In the above equation, σ is the symmetry number;81 kB is the Boltzmann constant; h is the Planck constant; R is the ideal gas constant; and ΔSq and ΔHq are the entropy and enthalpy changes from the reactants to the transition state at temperature T. The activation energy (Ea) from the Arrhenius expression80 is related to ΔHq in eq 3 by Ea = ΔHq þ (1
Δn)RT, where Δn is the change in the number of particles from the reactants to the transition state, 0 for unimolecular reactions and
1 for bimolecular reactions. For unimolecular reactions, the reaction rate constants can also be calculated using RRKM theory.80 The strong collision model82 can be applied to simplify the RRKM calculations. For reactions involving hydrogen transfer, the tunneling effect can become important especially at low temperatures.83 The tunneling coefficient κ(T) can be calculated by the Wigner formula,84 as shown in eq 4. kðTÞ ¼ 1 þ

  1 hν 2 24 kB T

ð4Þ

In the above equations, ν is the absolute value of the imaginary frequency. The DFT calculations were performed with the Gaussian 09 program package.85 For the pure DFT methods, the density fitting approximation was employed to speed up the calculations.86,87 The density fitting sets were automatically generated from the atomic orbital primitives.85 The CCSD(T) and EOM-CCSD calculations were carried out with the MOLPRO 2009.1 program package.88 The TD-DFT calculations were carried out with the NWChem 5.1 program package.89,90 The calculations were performed on the local Xeon and Opteron based Penguin Computing clusters, the Xeon based Dell Linux cluster at the University of Alabama, the Opeteron and Xeon based Dense Memory Cluster (DMC) and Itanium 2 based SGI Altix systems at the Alabama Supercomputer Center, and the Opteron based HP Linux cluster at the Molecular Science Computing Facility at Pacific Northwest National Laboratory. Molecular visualization was done using the AGUI graphics program from the AMPAC program package.91 Reaction rate constant calculations based on TST and RRKM thoeries were done with the KHIMERA program.92 8076

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Table 1. Relative Energies (kcal/mol) at 0 K for the Different Conformers/Isomers of the Metal Hydroxides Calculated at the CCSD(T)/aT//B3LYP/aD (CCSD(T)/CBS///B3LYP/aD for the Monometal Compounds and CCSD(T)/aD//B3LYP/aD for M2O(OH)10, M3O7(OH)4, and M4O11(OH)2) and B3LYP/aT//B3LYP/aD Levels CCSD(T)

B3LYP

M = Cr

M = Cr

1a

0.0

0.0

1b

0.4

0.5

1c

0.9

1d M(OH)6

1e 1f

M2O5(OH)2

2a

0.0

0.0

2b

27.5

29.2

molecule MO2(OH)2 MO(OH)4

M2O4(OH)4

structure

CCSD(T) M = Mo

B3LYP

CCSD(T)

B3LYP

M = Mo

M=W

M=W

0.0

0.0

0.0

0.0

0.3

0.4

0.3

0.3

0.2

0.0

0.0

0.0

0.0

0.0

0.0

0.3

0.4

0.2

0.2

0.0 2.0

0.0 0.6

0.0 0.3

0.0
0.1

0.0 0.0

0.0
0.3

0.0

0.0

0.0

0.0

13.7

17.0

9.0

13.9

2c

-

-

2.9

0.2

0.0

0.0

2d

2.5

9.2

0.0

0.0

3.2

4.7

2e

0.0

0.0

3.1


1.8

5.5

2.7

2f

43.6

44.3

30.7

28.8

28.5

29.2

2g 2h

-

-

0.0 8.2

0.0 11.8

0.0 7.0

0.0 10.7

2i

-

-

7.9

12.0

7.8

11.7

2j

-

-

16.8

15.8

8.2

9.3

2l

-

-

0.0

0.0

0.0

0.0

2m

0.0

0.0

24.9

17.2

17.9

12.1

3a

26.8

27.8

5.9

7.9

0.0

0.0

3b

0.0

0.0

0.0

0.0

3.1


0.4

M3O7(OH)4

3c 3d

-

-

0.0 12.4

0.0 16.8

0.0 1.7

0.0 8.9

M4O11(OH)2

3e

23.1

23.6

4.9

3.3

0.0

0.0

3f

0.0

0.0

0.0

0.0

0.4

3.9

M2O3(OH)6

M2O(OH)10 M3O8(OH)2

a

a

See Figures 1
3 for the structures of the conformers/isomers.

’ RESULTS AND DISCUSSION Equilibrium Geometries. Optimized molecular structures including selected metal oxygen bond lengths at the B3LYP/ aD level for the metal hydroxides are shown in Figure 1 for the compounds with a single metal atom, in Figure 2 for the compounds with two metal atoms, and in Figure 3 for the compounds with three and four metal atoms. The different isomers and conformers of these hydroxides are labeled following the order of their appearance in these figures. The metal and oxygen atoms in some of these figures are labeled with upper- and lowercase Greek letters, respectively. All of the discussions below about geometries, frequencies, and atomic charges93 refer to results calculated at the B3LYP/aD level of theory unless noted. Although no accurate experimental geometries exist for the hydroxides, we have recently shown for MO2 (M = Ti, Zr, Hf) that the B3LYP/ aD bond lengths and angles are within 0.02 Å and 1 of the experimental values, respectively, similar in accuracy to the CCSD(T)/aT optimized structures. 20g For MO3 (M = Cr, Mo, W) and M2O6, we have also shown that the B3LYP/aD bond lengths and angles are within 0.02 Å and 1 from the CCSD(T)/aT results, respectively.20a Thus, we expect the B3LYP/aD geometries for these hydroxides to have a similar accuracy. For molecules with more than one conformer/isomer with significant geometry differences, the relative energies calculated at the CCSD(T)/aT//B3LYP/aD level (CCSD(T)/CBS//

B3LYP/aD for the monometal compounds, CCSD(T)/aD// B3LYP/aD for M2O2(OH)8, M2O(OH)10, M3O7(OH)4, and M4O11(OH)2) are also given in these figures, with more details given in Table 1 and the Supporting Information. All discussions hereafter about relative energies, acid
base properties, reaction barriers, and reaction energies refer to those calculated at the above CCSD(T) levels unless noted. Single Metal Hydroxides. The simplest hydroxide is MO2(OH)2 (1a). Another conformer (1b) which is 0.3
0.4 kcal/mol higher in energy was located. The metal
oxygen bond lengths calculated for these two conformers are essentially the same, and their geometry difference lies in the relative orientation of the
OH groups. In both conformers, the metal atom is bonded with two terminal oxygen atoms (labeled dO hereafter) and two hydroxyl groups (labeled
OH hereafter). The fact that these two conformers lie very close in energy indicates that any interaction between the two
OH groups must be very weak, consistent with the large separation between the
OH groups. For example, for M = Cr, the H atom on one
OH group is ∼3.36 and 3.41 Å away from the O atom on the other
OH group for conformers 1a and 1b, respectively, making any hydrogen bonding between the
OH groups very weak. In addition, the distance between the H atom on the
OH group to its closest dO atom is also similar for these two conformers, ∼2.81 and 2.80 Å, respectively. The MdO bond length is about 0.01
0.02 Å shorter than that in MO3.20a The M
OH bond 8077

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0.20 Å longer than that of the MdO bond and is within 0.01 Å of that of the bridge M
O bond in M3O9 (Cr, 1.765 Å; Mo, 1.902 Å; W, 1.905 Å) and M4O12 (Cr, 1.757 Å; Mo, 1.895 Å; W, 1.896 Å).20b,c,e We have previously predicted the average MdO bond energy in MO3 to be 111, 137, and 153 kcal/mol for M = Cr, Mo, and W, respectively.20a,e Assuming the average MdO bond energy in M2O6 is the same as that in MO3, the average M
O bond energy in M2O6 was calculated to be 79, 97, and 108 kcal/mol.20a,e,94 Using a similar approach, the average M
O bond energies in M3O9 and M4O12 are calculated to be 84, 104, and 117 kcal/mol for M3O9 and 86, 106, and 119 kcal/mol for M4O12. On the basis of the calculated M
O bond lengths as shown above, we expect the average M
OH bond energy in MO2(OH)2 to be approximately 85, 105, and 120 kcal/mol for M = Cr, Mo, and W, respectively. Assuming the average MdO bond energy in MO2(OH)2 is the same as that in MO3, the average M
OH bond energies in MO2(OH)2 are calculated to be 89, 109, and 124 kcal/mol from the theoretical heat of formation of MO2(OH)2 obtained in this study as discussed in the following section and the experimental heats of formation of the M and O atoms95 and the OH radical.77k This is consistent with our estimation based on the average M
O bond energy in MnO3n (n = 2
4). Molecular orbital analysis for MO3 and MO2(OH)2 suggests significant contributions from the metal
oxygen d
pπ bonding to the MdO bonds but substantially smaller contributions to the M
OH bonds. The W
O
H bond angles in WO2(OH)2 (1a) are predicted to be ∼125, consistent with the significant ionic character of the M
O bond. An increase in the ionic character increases the X
O
Y bond angle as found for Li
O
Li (180), which is completely ionic.96 The O
M
O
H dihedral angles are largely determined by the hydrogen bonding. For example, the O
W
O
H dihedral angles are calculated to be
12.6,
132.1, and 107.2 for WO2(OH)2 (1a). The appearance of a small dihedral angle of about
13 indicates the importance of the hydrogen bonding over steric effects. The metal oxygen bond lengths are very similar for M = Mo and W, which are much longer than that for M = Cr, consistent with our previous results on group 6 TMO clusters.28,20a
20c,20e,20f Replacing a dO atom in MO2(OH)2 with two
OH groups gives rise to MO(OH)4. There are significantly more conformations for MO(OH)4 than for MO2(OH)2, as we have found up to 10 conformors for MO(OH)4. At the CCSD(T)/aT//B3LYP/ aD level, the energies of these conformers relative to the ground state conformer are up to 7 kcal/mol higher. Two important conformers, 1c and 1d, are shown in Figure 1, with the former predicted to be the ground state structure for M = Mo and W and the latter for M = Cr. Four additional conformers for M = Cr and two additional conformers for M = Mo and W were calculated to lie within 1 kcal/mol of the ground state conformer. Details of these structures and other conformers are given in the Supporting Information. All of these conformers are best considered as distorted square pyramidal structures, with the lowest-energy conformers (1c and 1d) having the dO atom in the axial position for M = Mo and W as shown in Figure 1. For M = Cr, two of the lowest-energy conformers have the
OH group in the axial position. In the ground state conformer of WO(OH)4 (1c), the OdW
OH angles are calculated to be between 102 and 108. For conformer 1c, the metal oxygen bonds split into three groups: the MdO bond, two “regular” M
OH bonds (labeled R and β), and two elongated M
OH bonds (labeled γ and δ).

ARTICLE

The MdO bond is about 0.02 Å shorter than that in MO2(OH)2. The “regular” M
OH bonds have bond lengths similar to those in MO2(OH)2, whereas the elongated M
OH bonds are longer by ∼0.08 Å for M = Cr and by ∼0.05 Å for M = Mo and W. Using an approach similar to the one described above, the average M
OH bond energies in MO(OH)4 are calculated to be 69, 91, and 105 kcal/mol, which are lower than those in MO2(OH)2 by ∼20 kcal/mol. Similar observations are made for conformer 1c for M = Cr and Mo. For M = W, conformer 1d has C2 symmetry. Hydrogen bonding appears to play a role in determining the relative orientations of the
OH groups and their distances from the metal atom. For example, for conformer 1c for WO(OH)4, the hydrogen bond distances Hβ 3 3 3 Oγ, Hγ 3 3 3 Oδ, and Hδ 3 3 3 OR are 2.37, 2.29, and 2.76 Å, respectively, and the atoms HR, Oβ, and the dO atom are not involved in hydrogen bonding. The
O(H) atom from the regular M
O bonds (OR and Oβ) are at best only involved in very weak hydrogen bonds. Replacing the final dO atom in MO(OH)4 with two additional
OH groups gives rise to M(OH)6. Similar to the case of MO(OH)4, we found up to 8 conformers for M(OH)6 with their energies differing by up to 6 kcal/mol at the CCSD(T)/aT// B3LYP/aD level. The two lowest-energy conformers (1e and 1f) are shown in Figure 1, which are very close in energy for M = Mo and W. However, unlike the case of MO(OH)4, no additional conformers lie within 1 kcal/mol from the ground state conformer. Details of the other conformers are given in the Supporting Information. All these conformers can be considered as pseudo-octahedral. The ground state conformer of M(OH)6 (1e) is significantly distorted from the ideal octahedron with only C3 symmetry. The M
OH bond distances split into two groups. Three of them have bond lengths similar to those in MO2(OH)2 with HO
M
OH bond angles of ∼94. The other three are longer by ∼0.12 Å for M = Cr and by ∼0.08 Å for M = Mo and W, with corresponding smaller HO
M
OH bond angles of ∼84. The average M
OH bond energies in M(OH)6 are calculated to be 61, 83, and 97 kcal/mol, which are lower than those in MO2(OH)2 by 25
30 kcal/mol. The distortion predicted for M(OH)6 is typical for octahedral compounds with stereoactive lone pairs97 such as XeF6 and IF6
as well as the structures of W(CH3)6 and Mo(CH3)6.98 The distortion away from octahedral symmetry due to the absence of electrons in the d shell98d in M(OH)6 is enhanced by hydrogen bonding between the ligands. Similar to the ground state conformer of MO(OH)4, in the ground state conformer of M(OH)6 the
O(H) atom from the regular M
OH bond does not form hydrogen bonds, whereas that from the elongated M
OH bond forms two hydrogen bonds. These hydrogen bonds are similar to those predicted for MO(OH)4, for example, 2.33 and 2.46 Å for W(OH)6. In contrast, the MF6 compounds for M = Mo and W are predicted to have Oh symmetry,20i,j in agreement with experimental neutron99 and X-ray100 diffraction crystal structures and electron diffraction101 for WF6 and an X-ray diffraction crystal structure of MoF6 as well as an EXAFS study102 of MoF6. The distortions in M(OH)6 due to hydrogen bonding are larger than the Jahn
Teller distortions in MF6 and MF6
with transition metals to the right of the Group 6 metals.20i,j Similar geometric distortions are present in conformer 1f, although the
O(H) atom from two of three regular M
OH bonds also forms a weaker hydrogen bond. Dimetal Hydroxides. Two unique isomers for M2O5(OH)2 are shown: the chain structure (2a) and the ring structure (2b). The chain structure was predicted to be lower in energy than the 8078

dx.doi.org/10.1021/jp111031x |J. Phys. Chem. C 2011, 115, 8072–8103

The Journal of Physical Chemistry C ring structure for all three metals, although the energy difference decreases substantially from M = Cr (∼28 kcal/mol) to M = Mo and W (9
14 kcal/mol). The chain structure is stabilized by the release of the bond strain in the four-membered ring (
M
O
M
O
). A more symmetric chain conformer with C2 symmetry was also located (Supporting Information), which is essentially isoenergetic to the asymmetric chain structure 2a. Similarly, a slightly different ring conformer was also located (Supporting Information), which lies up to 3 kcal/mol higher in energy than the ring structure 2b. In the chain structure, the bridge M
O and M
OH bond lengths are essentially the same as those of the M
OH bonds in MO2(OH)2 for a given metal. The M
O
M bond angle varies from ∼130 for M = Cr to ∼150 for M = Mo and W. This is due to the increasing ionic bonding character from M = Cr to M = Mo and W, as indicated by the increasing atomic charges discussed in a following section. In comparison, the X
O
X bond angle is 104.5 for X = H, which is essentially covalent, and 180 for X = Li,96 which is essentially ionic. In the chain structure, each
OH group forms a hydrogen bond with a neighboring dO atom. In the ring structure, one
OH group forms a hydrogen bond with one bridge
O
atom, and the other forms no hydrogen bond. For this bridge
O
atom, the bridge M
O bond for the M without any
OH group is shorter than that in M2O6 by ∼0.08 Å for M = Cr and ∼0.04 Å for M = Mo and W,20a whereas the other is much longer and is ∼2.0 Å for all three metals. Similar observations can be made for the other bridge
O
atom for M = Mo and W, albeit with a smaller difference in the M
O bond lengths; for M = Cr, the two M
O bonds are comparable in bond length. Thus, for the ring structure, one metal center has two MdO bonds and two somewhat “regular” bridge M
O bonds, whereas the other metal center has one MdO bond, three more or less regular bridge M
O or M
OH bonds, and one very long bridge M
O bond as indicated by the bond distances. For M2O4(OH)4, four unique isomers (2c to 2f) were located for M = Mo and W. Three of these isomers retain the fourmembered ring of (
M
O
M
O
). For M = Cr, geometry optimization starting from isomer 2c optimizes to a hydrogen bonded complex of two CrO2(OH)2 molecules, which is 23.9 kcal/mol lower in energy than the ground state isomer of Cr2O4(OH)4 (2e) at the B3LYP/aD level. For M = Mo and W, this type of hydrogen-bonded complex is 6.4 and 4.8 kcal/mol lower in energy than the ground state isomer of M2O4(OH)4 (2d for M = Mo and 2c for M = W) at the B3LYP/aD level. Thus, for all three metals, M2O4(OH)4 is metastable with respect to dissociation into a complex of two MO2(OH)2 molecules. Furthermore, four additional conformers for isomers 2c and 2d were located for M = W and two for M = Mo. Their structures and energetics are given in the Supporting Information. Isomers 2c
2e for M = Mo and W and isomers 2d and 2e for M = Cr lie fairly close in energy as shown in Figure 2 and Table 1, although the ground state isomer for these three metals differs from one another. The ground state of M2O4(OH)4 was predicted to be isomer 2e for M = Cr, isomer 2d for M = Mo, and isomer 2c for M = W. This is consistent with the increasing energy difference between the chain and ring isomers of M2O5(OH)2 from M = Cr to Mo to W. Isomer 2c has Ci symmetry, and the different types of O atoms (dO,
O
,
OH) are equally distributed between the two metal centers. Two of the four bridge M
O bonds have bond lengths similar to those in the chain isomer of M2O5(OH)2 (2a), whereas the other two are longer by 0.10
0.15 Å. Isomer 2d is derived from isomer 2c by replacing a bridge
O
atom

ARTICLE

with a bridge
OH
group. For M = Cr, isomer 2d has a Cr(dO)2(OH) moiety with the Cr forming long bonds to the
O
and
OH
bridges. The other Cr atom is in a Cr(dO)(OH)2 moiety with a regular bridge Cr
O bond and a longer bridge Cr
OH bond. For M = Mo and M = W, the bridging
O
atom is more equally shared between the two M atoms, but the bridging
OH
group is less equally shared. If the orientation of the
OH
bridge is changed so that its H atom can form a hydrogen bond to the dO atom in the M(dO)2OH moiety, one gets isomer 2e, which has an M(dO)2OH moiety with a shorter bridge M
O bond and an M(dO)(OH)3 moiety with a longer bridge M
O bond. Isomer 2e is closely related to the chain isomer 2a of M2O5(OH)2. However, for all three metals, the two bridge M
O bond lengths in isomer 2e are significantly different, whereas those in the chain isomer of M2O5(OH)2 (2a) are essentially the same. For isomer 2e, one of the bridge M
O bonds is shorter than those in the chain isomer of M2O5(OH)2 by 0.05
0.09 Å, whereas the other one is much longer (by up to 0.35 Å for M = Cr) and has a bond length of ∼2.02 Å for all three metals with very small variations of only 0.01 Å. For isomer 2d, the difference between the distances from the bridge
OH
group to the two metal centers increases from M = Cr to Mo and W, although both are longer than the typical M
OH bond lengths by at least 0.14 Å. Isomer 2f with all four
OH groups located on one metal center was predicted to be substantially higher in energy than the other structures by 25
30 kcal/mol for M = Mo and W and ∼40 kcal/mol for M = Cr. Four unique isomers (2g
2j) were located for M2O3(OH)6 for M = Mo and W. For M = Cr, geometry optimizations starting from these isomers led to hydrogen-bonded complexes consisting of two CrO2(OH)2 and one H2O molecules. Four additional conformers of isomers 2g and 2j were also located and are given as Supporting Information. Isomer 2g was predicted to be the most stable for M = Mo and W, which has a chain structure with three
OH groups on each metal center. Similar to isomer 2e of M2O4(OH)4, isomer 2g of M2O3(OH)6 has two bridge M
O bonds of significantly different lengths and two hydrogen bonds between two pairs of
OH groups. In isomer 2g, one bridge M
O bond is much longer than the other one, whose bond length is between those of “typical” bridge M
O and terminal MdO bonds. The moiety which has OH groups donating the lone pairs has the shorter bridge M
O bond. The other three isomers all retain the four-membered (
M
O
M
O
) ring, although two of them have a bridge
OH group. These three isomers lie within 10 kcal/mol of the ground state structure except for isomer 2j for M = Mo. The two isomers for M2O3(OH)6 with a bridge
OH group lie very close in energy, but they differ in the distribution of the other
OH groups. In both cases, the metal bridge
OH bond distances are much longer than the typical M
OH bond lengths, similar to that for isomer 2d of M2O4(OH)4. Only one unique isomer (2k) was located for M2O2(OH)8 for M = Mo and W, which has four
OH groups on each metal center. For M = Cr, geometry optimization starting from this structure results in a hydrogen-bonded complex consisting of two CrO2(OH)2 and two H2O molecules. Isomer 2k has the maximum number of
OH groups without breaking the fourmembered (
M
O
M
O
) ring. For M2O(OH)10, two distinct conformers (2l and 2m) were located for M = Mo and W, with more conformers given in the Supporting Information, most of which lie within a few kilocalories per mole of the ground state conformer (2l). Both 8079

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The Journal of Physical Chemistry C

ARTICLE

Table 2. Atomic Charges on the Metal and Oxygen for the MO3
n(OH)2n (M = Cr, Mo, W; n = 0
3) Clusters from the Natural Population Analysis at the B3LYP/aD Levela,b molecule

M


OH or
O
or
OH


dO

CrO3

C3v

1.02


0.34 (x3)

-

CrO2(OH)2 CrO(OH)4

1a 1d

0.76 0.71


0.21 (x2)
0.13


0.17 (x2)
0.01 (R),
0.14 (β),
0.23 (γ),
0.20 (δ)

Cr(OH)6

1e

0.55

-

0.02 (R, x3),
0.20 (β, x3)

MoO3

C3v

1.64


0.55 (x3)

-

MoO2(OH)2

1a

1.58


0.47 (x2)


0.32 (x2)

MoO(OH)4

1c

1.57


0.40


0.26 (R),
0.24 (β),
0.31 (γ),
0.36 (δ)

Mo(OH)6

1e

1.47

-


0.17 (R, x3),
0.32 (β, x3)

WO3

C3v

1.81


0.60 (x3)

-

WO2(OH)2 WO(OH)4

1a 1c

1.80 1.81


0.56 (x2)
0.50


0.34 (x2)
0.29 (R),
0.30 (β),
0.35 (γ),
0.37 (δ)

W(OH)6

1e

1.72

-


0.23 (R, x3),
0.35 (β, x3)

Cr2O6

D2h

0.79 (x2)


0.19 (x4)


0.42 (x2)

Cr2O5(OH)2

2a

0.76, 0.77


0.18,
0.19,
0.20,
0.22


0.47 (R),
0.14 (β, x2)

2b

0.69 (A), 0.76 (B)


0.04 (A),
0.23 (B, x2)


0.38 (R),
0.44 (β),
0.01 (γ),
0.11 (δ)

2d

0.73 (A), 0.67 (B)


0.10 (A),
0.16 (B, x2)


0.36 (R),
0.22 (β),
0.13 (γ),
0.06 (δ),
0.21 (ε)

2e

0.72 (A), 0.75 (B)


0.05 (A),
0.23 (B),
0.36 (ζ)


0.46 (R),
0.12 (β),
0.02 (γ),
0.04 (δ),
0.19 (ε)

2f D2h

0.47 (A), 0.77 (B) 1.59 (x2)


0.23 (x2)
0.45 (x4)


0.36 (R, x2),
0.07 (β, x2),
0.05 (γ, x2)
0.70 (x2)

Cr2O4(OH)4

Mo2O6 Mo2O5(OH)2 Mo2O4(OH)4

W2O6 W2O5(OH)2 W2O4(OH)4

a

structure

2a

1.59 (x2)


0.45 (x2),
0.46 (x2)


0.76 (R),
0.30 (β, x2)

2b

1.53 (A), 1.58 (B)


0.34 (A),
0.48 (B, x2)


0.66 (R),
0.69 (β),
0.20 (γ),
0.25 (δ)

2c

1.56 (x2)


0.39 (x2)


0.66 (R, x2),
0.28 (β, x2),
0.23 (γ, x2)

2d

1.58 (A), 1.51 (B)


0.37 (A),
0.42 (B),
0.45 (B)


0.65 (R),
0.36 (β),
0.26 (γ),
0.24 (δ),
0.33 (ε)

2e

1.57 (A), 1.58 (B)


0.36 (A),
0.47 (B),
0.57 (ζ)


0.72 (R),
0.28 (β),
0.21 (γ),
0.22 (δ),
0.32 (ε)

2f

1.42 (A), 1.59 (B)


0.49 (x2)


0.63 (R, x2),
0.24 (β, x2),
0.14 (γ, x2)

D2h 2a

1.84 (x2) 1.82 (x2)


0.54 (x4)
0.54 (x2),
0.55 (x2)


0.75 (x2)
0.82 (R),
0.32 (β, x2)

2b

1.77 (A), 1.81 (B)


0.44 (A),
0.57 (B, x2)


0.73 (R),
0.75 (β),
0.24 (γ),
0.28 (δ)

2c

1.80 (x2)


0.49 (x2)


0.74 (R, x2),
0.31 (β, x2),
0.26 (γ, x2)

2d

1.82 (A), 1.74 (B)


0.47 (A),
0.53 (B),
0.55 (B)


0.73 (R),
0.38 (β),
0.29 (γ),
0.27 (δ),
0.35 (ε)

2e

1.82 (A), 1.82 (B)


0.46 (A),
0.57 (B),
0.65 (ζ)


0.78 (R),
0.32 (β),
0.25 (γ),
0.28 (δ),
0.34 (ε)

2f

1.68 (A), 1.81 (B)


0.59 (x2)


0.70 (R, x2),
0.26 (β, x2),
0.19 (γ, x2)

Atomic charges on H are summed with the O. b See Figures 1 and 2 for the molecular structure and the labels of O in MO(OH)4 and M(OH)6.

conformers have five
OH groups on each metal center. They differ in that no hydrogen bond is formed between the two sets of
OH groups for conformer 2m, which makes it more symmetric than the other conformer. The asymmetric conformer 2l was predicted to be more stable than the symmetric conformer 2m for M = Mo and W by ∼18 and 25 kcal/mol. For M = Cr, geometry optimization starting from the asymmetric conformer decomposes into a hydrogen-bonded complex consisting of two CrO2(OH)2 and three H2O molecules. For isomer 2e of M2O4(OH)4, isomer 2g of M2O3(OH)6, and conformer 2l of M2O(OH)10, one of the bridge M
O bonds is much shorter than the other one. For instance, for M = W, the two bridge M
O bonds are 1.85 and 2.02 Å for M2O4(OH)4, 1.83 and 2.03 Å for M2O3(OH)6, and 1.79 and 2.12 Å for M2O(OH)10. The shorter bridge M
O bond in M2O(OH)10 for M = Mo and W is only 0.06
0.08 Å longer than that of the MdO bond in MO2(OH)2. Again, the moiety which has the OH groups donating the lone pairs has the shorter M
O bond. Tri- and Tetrametal Hydroxides. Two unique isomers for M3O8(OH)2 are shown in Figure 3, the ring isomer 3a and the

chain isomer 3b. The chain isomer was predicted to be substantially lower in energy than the ring isomer for M = Cr. For M = Mo, although the chain isomer is still lower in energy, the ring isomer is only ∼6 kcal/mol higher in energy. For M = W, the ring isomer was predicted to be more stable than the chain isomer by ∼3 kcal/mol. The trend in the energy difference between the chain and ring isomers for M3O8(OH)2 is similar to that for M2O5(OH)2 in that it substantially decreases from M = Cr to Mo to W. As in the case of M3O9,28 the ring isomer of M3O8(OH)2 features a six-membered puckered ring for M = Cr and an essentially planar ring for M = Mo and W. The ring isomer for M = Cr also differs from those for M = Mo and W in that both
OH groups form hydrogen bonds with the terminal dO atoms on the neighboring metal atoms due to the ability of the ring to readily pucker. The chain isomer of M3O8(OH)2 has an eightmembered ring resulting from the hydrogen bonding between an OH group on one terminal metal center and an OH group (for M = Cr) or an dO atom (for M = Mo and W) from the other terminal metal center. For M = Mo and W, six of the atoms in this eight-membered ring are nearly in the same plane. The OH 3 3 3 O 8080

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The Journal of Physical Chemistry C hydrogen bond length in the chain isomer is 2.01, 2.01, and 1.97 Å for M = Cr, Mo, and W respectively, similar to the value of 1.93 Å in (H2O)2.103 The effect of the hydrogen bond on the M
OH bond lengths is not large. For the ring isomer, the metal center with M
OH bonds has one MdO bond, two somewhat regular M
OH bonds, and two elongated bridge M
O bonds. The longest bridge M
O bond in the ring isomer of M3O8(OH)2 is ∼1.91 Å for M = Cr and ∼1.96 Å for M = Mo and W, longer than that in the ring isomer of M2O5(OH)2. Two unique isomers (3c and 3d) were located for M3O7(OH)4 for M = Mo and W, as shown in Figure 3. Isomer 3c has one OH group on one terminal metal center and three OH groups on the other terminal metal center. Isomer 3d has all four OH groups on one metal center, maintaining the six-membered ring. For both M = Mo and W, the four M
OH bonds on this metal center have the same bond length of ∼1.86 Å, and the two M
O bonds have the same bond length of ∼2.05 Å. This is different from the case of M(OH)6 with three regular and three elongated M
OH bonds. The chain isomer 3c was predicted to be lower in energy for both metals, although for M = W, the ring isomer 3d is only ∼2 kcal/mol higher in energy. Both structures feature hydrogen bonding between the
OH group and a neighboring terminal dO atom. The ring isomer (3e) and the chain isomer (3f) of M4O11(OH)2 are shown in Figure 3. The similarity between the isomers for M3O8(OH)2 and M4O11(OH)2 lies in that the chain isomer is more stable for M = Cr and Mo, and the ring isomer is more stable for M = W. For the chain isomer, the hydrogen bonding is between the
OH groups for M = Cr and between the
OH group and the dO atom for M = Mo and W. The ring isomer of M4O11(OH)2 differs from that of M3O8(OH)2 in that for all three metals hydrogen bonds are formed between the
OH groups and dO atoms on neighboring metal centers, and the structure is substantially puckered. The geometry of the chain isomer of M4O11(OH)2 is quite flexible with a number of lowenergy (floppy) modes and thus is very difficult to optimize. For M = Mo and W, the chain isomer exhibits a long coordination bond between a terminal dO atom on one metal center to another metal center, with an M
O distance of 3.45 and 2.67 Å, respectively. For M = W, this bond distance is similar to the bond distance of 2.63 Å between the internal O atom and the metal center on the C4 axis with a terminal MdO bond in the inverted cage structure of M6O18.28 The bond length for this MdO bond is longer by 2.28 In general, they shift toward lower frequencies as the number of
OH groups increases with the exception from MO(OH)4 to M(OH)6 for M = Mo and W. Vertical Excitation Energies. Table 4 presents the calculated vertical excitation energies for the lowest singlet and triplet excited states with the EOM-CCSD and TD-DFT methods for the metal hydroxides as well as the metal oxides. For (CrO3)n (n = 1
5), we have recently shown that the lowest TD-DFT triplet vertical excitation energy calculated at the B3LYP/aD level is ∼0.5 eV higher for n = 1
3, ∼0.4 eV higher for n = 4, and ∼0.3 eV higher for n = 5 than the experimentally estimated adiabatic value.20c Thus, we expect our calculated TD-DFT vertical excitation energies for these hydroxides to be reasonably accurate, within about 0.5 eV. The EOM-CCSD vertical excitation energies are also consistent with the above experimental data. As there are no high-quality experimental vertical excitation energies with which to directly compare, it is not yet possible to determine which theory gives more accurate excitation energies for these compounds. The singlet excitation energies calculated with the TD-DFT method are higher than those calculated with the EOM-CCSD method by up to 0.5 eV for M = Cr. For M = Mo and W, the TD-DFT results are usually lower than the EOMCCSD results by up to 0.4 eV. The triplet excitation energies calculated with the TD-DFT method are substantially lower than the singlet excitation energies calculated with the same method by up to ∼1 eV, as we have noted previously.28 For the (MO3)n ring clusters, the lowest singlet and triplet excitation energies increase as n increases up to n = 4. The calculated excitation energies for n = 3 are very close to those for 8083

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The Journal of Physical Chemistry C

ARTICLE

Table 5. Gas-Phase Acidities (ΔG298K, kcal/mol) and Aqueous-Phase pKa Values for the Metal Hydroxides Calculated at the CCSD(T)/aT//B3LYP/aD Level (CCSD(T)/CBS//B3LYP/aD for the Monometal Compounds and CCSD(T)/aD//B3LYP/aD for M4O11(OH)2)a ΔG298K molecule

b

structure

M = Cr

ΔG298K

pKa M = Cr

M = Mo

pKa M = Mo

ΔG298K

pKa

M=W

M=W
1.5

MO2(OH)2

1a

308.6


4.3

309.8


1.3

309.6

MO(OH)4

1cc

294.0


12.6

300.7


5.6

305.1

0.3

M(OH)6

1e

291.5


12.3

296.5


8.7

303.1


4.6

M2O5(OH)2

2a

290.6


7.1

295.2

1.6

294.2

2.1

M3O8(OH)2

2bd 3a

261.5 265.5


27.5
25.2

283.5 270.7


11.0
12.9

284.5 273.8


9.2
10.4

3b

281.2


10.3

281.2


5.2

278.7


6.3

3ed

247.7


31.0

264.8


14.1

267.8


12.9

3f

274.5


11.2

277.3


5.3

277.5


4.5

M4O11(OH)2

The experimental pKa value of HNO3 is
1.4. b See Figures 1
3 for the molecular structure. c 1d for M = Cr. d For M = Cr, no ring structure was obtained for the anion. a

n = 4, indicating that these properties rapidly converge as we have previously predicted.28 The singlet excitation energies calculated at the EOM-CCSD level for n = 4 are ∼2.8, 4.1, and 4.7 eV for M = Cr, Mo, and W. The experimentally determined band gaps for bulk MO3 for M = Mo and W vary substantially depending on the method of measurement, and those compiled by Payne3 are 3.2 and 2.6 eV for M = Mo and W, respectively, which are substantially smaller than our converged EOM-CCSD values by 0.9 and 2.1 eV. For the MO3
m(OH)2m (m = 1
3) clusters, the calculated lowest singlet and triplet excitation energies are higher than those for MO3 except for the triplet excitation energies for M = Cr for m = 2 and 3. There is a substantial increase in the excitation energy from MO3 to MO2(OH)2 by 1.7, 2.4, and 3.3 eV calculated at the EOM-CCSD level, and the excitation energy then decreases considerably upon further
OH substitutions by 0.9
1.1 eV for M = Cr, ∼0.7 eV for M = Mo, and 0.3
0.7 eV for M = W. The excitation energies for the different conformations are usually similar with differences up to 0.3 eV. The first excitation (band gap) is due to the HOMO
LUMO transition. Figure 4 presents the highest occupied and lowest unoccupied molecular orbitals (HOMO and LUMO) for the MO3
n(OH)2n (n = 1
3) clusters. The HOMOs and LUMOs of the metal oxide clusters have been discussed elsewhere.20b,c,28 The HOMOs of these clusters are dominated by O 2p orbitals, whereas the LUMOs are dominated by metal d orbitals. For CrO2(OH)2, the HOMO has nearly equal contribution from the dO and
OH groups, whereas for MoO2(OH)2 and WO2(OH)2, the HOMO has little contribution from the
OH groups. The contribution for the dO atoms can be considered as a 2pπ type orbital and that from the OH group to be a 2p lone pair. For MO(OH)4 and M(OH)6, the HOMO consists of mostly the 2p orbitals from the elongated
OH groups. The LUMOs of these hydroxides are of metal d character similar to those for MO3. Thus, if an dO atom is present, the excitation is essentially from an dO 2pπ orbital to a vacant metal d orbital consistent with the excess negative charge on the dO atom and the formal þ6 character of the metal. When the OH groups dominate the coordination to the metal, the excitation is an O 2p lone pair to an empty metal d orbital. These would be expected to be higher in energy as found (Table 4). The result that the excitation energy increases with increasing coordination number

is consistent with the usual expectation that the band gap converges to the bulk from above the band gap rather than below it in terms of the size of the nanoparticle. However, if a dO atom is present, the excitation can be substantially lower than the bulk band gap as found for the MO2 (M = Ti, Zr, Hf) nanoclusters.20d,g These results are also consistent with the changes in the atomic charges. From MO3 to MO(OH)4, the positive charge on the M atom is reduced, which leads to destabilized metal d orbitals. At the same time, the negative charge on the dO atom is reduced, which leads to stabilized O 2pπ orbitals. This results in larger HOMO
LUMO gaps and thus higher excitation energies. For the MnO3n
1(OH)2 (n = 2
4) clusters, the chain isomers have higher excitation energies than the ring isomer for a given n by up to 1.3 eV. For n = 2, the chain isomer has much higher excitation energies than M2O6, whereas for n = 3 and 4, the chain isomer has comparable excitation energies to MnO3n. The ring isomer usually has lower excitation energies than MnO3n for n = 2
4. Acid
Base Properties. Calculated Brønsted acidities and basicities and Lewis acidities are listed in Tables 5
7. The Br€onsted basicities and Lewis acidities for the (MO3)n (n = 1
3) clusters calculated at the same levels of theory are also listed in Tables 6 and 7. Additional information is given as Supporting Information. Due to the presence of multiple lowlying conformations, these acid
base properties were calculated from the lowest-energy structures unless otherwise noted. The pKa values shown in Table 5 ae absolute values calculated from their relative acidities to nitric acid7 using the experimental pKa of nitric acid (
1.4).106 Brønsted Acidities. The Brønsted acidities of MO2(OH)2 for M = Cr, Mo, and W are comparable, ∼310 kcal/mol (ΔG298K). These are very strong gas-phase acids, and they are slightly weaker than H2SO4 (ΔG298K = 304.0 kcal/mol).107 In aqueous solution, the acidity of MO2(OH)2 was predicted to be higher for M = Cr (pKa
4.3) than for M = Mo and W (pKa
1.3 and
1.5). They are weaker acids than H2SO4, the pKa value of which was predicted to be
7.0 with a similar approach.7 The pKa value of MO2(OH)2 for M = Mo and W is comparable to the experimental value of
1.4 for HNO3.106 MO(OH)4 is a stronger gasphase Brønsted acid than MO2(OH)2, and the difference in their Brønsted acidities decreases from ∼15 kcal/mol for M = Cr to 8084

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Table 6. Brønsted Basicities (ΔG298K, kcal/mol) for the Metal Hydroxides Calculated at the CCSD(T)/aT//B3LYP/aD (CCSD(T)/CBS//B3LYP/aD for the Monometal Compounds) and B3LYP/aT//B3LYP/aD Levelsa CCSD(T) b

molecule

site

B3LYP

CCSD(T)

B3LYP

CCSD(T)

B3LYP

M = Cr

M = Cr

M = Mo

M = Mo

M=W

M=W

MO3

dO

166.1

165.0

179.0

179.0

179.2

179.8

MO2(OH)2

dO

167.9

172.0

179.3

182.8

184.4

187.4

MO(OH)4


OH dO

173.9 162.6

168.6 168.7

170.8 177.3

167.3 182.9

162.7 183.4

160.9 188.9


OHc

219.6

221.2

202.3

201.2

198.2

198.2

M(OH)6


OHc

223.6

223.9

209.1

209.4

206.8

208.3

M2O6

dO

162.9

163.9

174.0

174.4

174.7

175.3


O


159.4

155.6

162.7

160.4

152.7

151.6

dO

167.7

167.9

176.3

176.5

175.1

175.7


O


165.5

159.7

160.8

157.3

146.4

144.3

M3O9 a

Also listed are the Brønsted basicities for the metal oxides. b Ground state structure assumed. See Figure 1. c For M = Cr, H2O formed from the protonation binds the cluster via a hydrogen bond.

Table 7. Fluoride Affinities (ΔH0K, kcal/mol) for the Metal Hydroxides Calculated at the CCSD(T)/aT//B3LYP/aD (CCSD(T)/CBS//B3LYP/aD for the Monometal Compounds) and B3LYP/aT//B3LYP/aD Levelsa CCSD(T) b

molecule

site

MO3

a

B3LYP

CCSD(T)

B3LYP

CCSD(T)

B3LYP

M = Cr

M = Cr

M = Mo

M = Mo

M=W

M=W

121.5

121.9

133.8

133.2

148.2

145.1

MO2(OH)2

56.1

49.6

72.6

66.7

78.3

71.3

MO(OH)4

53.6

50.3

69.4

62.5

73.7

66.1

M(OH)6

34.1

26.4

58.5

51.6

60.2

51.5

123.8 81.3

119.7 73.7

M2O6

terminal bridge

90.2 -

87.3 -

107.9 71.8

104.9 65.4

M3O9

terminal

89.1

84.9

105.3

101.6

117.9

113.2

bridge

76.4

68.9

108.1

97.7

119.4

107.7

Also listed are the fluoride affinities for the metal oxides. b Ground state structure assumed. See Figure 1.

∼10 kcal/mol for M = Mo to ∼5 kcal/mol for M = W. In aqueous solution, MO(OH)4 was predicted to be a stronger acid than MO2(OH)2 for M = Cr and Mo but a weaker acid for M = W. CrO(OH)4 was predicted to be a stronger acid than H2SO4 in both the gas phase and aqueous solution. M(OH)6 is a slightly stronger acid than MO(OH)4 in both the gas phase and aqueous solution, with the difference in their gas-phase Brønsted acidities being less than 5 kcal/mol. For M2O5(OH)2, we calculated the Brønsted acidities for both the chain and ring isomers. The neutral chain to anionic chain represents the acidity of the lowest-energy neutral. For the ring isomer of Cr2O5(OH)2, deprotonation causes the ring to open forming the chain isomer of Cr2O6(OH)
, so its Brønsted acidity is calculated relative to the chain isomer of the anion. Thus, it is calculated to be a much stronger acid than the other M2O5(OH)2 species due to the large difference in the energies of the parent neutral isomers (2a and 2b in Figure 2). For M = Mo and W, deprotonation of the ring isomer leads to an anionic ring isomer. The ring isomer of M2O5(OH)2 is a stronger gas-phase acid than the chain conformer, and the difference in their gasphase Brønsted acidities is ∼30 kcal/mol for M = Cr and ∼10 kcal/mol for M = Mo and W. This is partially due to the fact that the metal center with the
OH groups in the ring conformer resembles MO(OH)4, whereas that in the chain

conformer resembles MO2(OH)2. Nevertheless, the chain isomer of M2O5(OH)2 is a much stronger gas-phase acid than MO2(OH)2 as well as MO(OH)4. For M = Cr and Mo, it is comparable in acidity to M(OH)6, whereas for M = W, it is a stronger acid than M(OH)6 by ∼10 kcal/mol. In aqueous solution, the pKa value of the ring isomer of M2O5(OH)2 is 10
20 pKa units more negative than its chain isomer. The pKa value of the chain isomer is between those of MO2(OH)2 and MO(OH)4/M(OH)6 for M = Cr but higher by ∼3 than MO2(OH)2 for M = Mo and W. In fact, the chain isomer of M2O5(OH)2 for M = Mo and W is calculated to be the weakest acid in aqueous solution among the acids in Table 5. In terms of the Brønsted acidities, M3O8(OH)2 and M4O11(OH)2 are similar to M2O5(OH)2 in that the ring isomer is always a stronger acid than the chain isomer. The strength of the these acids in the gas phase roughly follows the order of MO2(OH)2 < MO(OH)4 e M(OH)6 e M2O5(OH)2 (chain) < M2O5(OH)2 (ring) < M3O8(OH)2 (chain) e M4O11(OH)2 (chain) < M3O8(OH)2 (ring) < M4O11(OH)2 (ring) for a given metal and W e Mo < Cr for a given form of acid. In aqueous solution, however, the strengths of these acids roughly follow a different order of M2O5(OH)2 (chain) < MO2(OH)2 < MO(OH)4 e M3O8(OH)2 (chain) e M4O11(OH)2 (chain) e M(OH)6 < M2O5(OH)2 (ring) < M3O8(OH)2 (ring) < 8085

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The Journal of Physical Chemistry C M4O11(OH)2 (ring) for a given metal. The calculated first gasphase acidities of these polynuclear hydroxides are comparable to those calculated for M6O19H2 of 266.7 and 273.4 kcal/mol for M = Mo and W at the B3LYP level (0 K values)28 and for PW12O40H3 of 256
258 kcal/mol at the BP86 level.108 As the cluster increases in size, it tends to become a stronger Brønsted acid. This can be partially attributed to the increasing charge delocalization effect in the conjugate base. For example, in the gas phase, the Brønsted acidity of Cr2O5(OH)2 (chain) is ∼18 kcal/mol lower than that of CrO2(OH)2. The calculated atomic charges (Table 2) on the M, dO, and
OH atoms/ groups are similar for these two acids. In their conjugate bases, the
OH group in CrO3(OH)
, which carries a charge of
0.35e, is replaced by the (
O
)Cr(dO)2(OH) moiety in Cr2O6(OH)
, which carries a charge of
0.49e. The additional charge delocalization in Cr2O6(OH)
helps to stabilize the anion, thus making Cr2O5(OH)2 (chain) a stronger acid than CrO2(OH)2. The pKa values for the Cr compounds show that they are predicted to be extremely strong acids in aqueous solution with effective pKa’s as large as
31 for the ring structure 3e. The more stable open-chain structure 3f has a value closer to
11. For a given structure, the acidities tend to decrease with increasing atomic number of the metal atom. We have previously predicted the pKa’s using the same approach for the well-known very strong acids (CF3SO2)3CH (
17.4), CF3SO3H (
12.5), and FSO3H (
11.4).7 The pKa’s can be in this range for many of the Cr compounds and for the larger Mo and W clusters. Using the same approach, the pKa’s for Mo6O19H2 and W6O19H2 are predicted to be
13.3 and
16.0, respectively. Thus the larger clusters for all of these species will be very strong acids with acidities comparable to or greater than that of H2SO4 under most conditions. Brønsted Basicities. For the (MO3)n (n = 1
3) clusters, the terminal dO atoms are more basic than the bridge
O
atoms in the same molecule, and the difference in their Brønsted basicities is 1, the formation of MnO3n
2(OH)4 can become favorable. Table 8 also presents the clustering energies of MO2(OH)2 and MO(OH)4, given for reactions (15) and (16). At 298 K, reaction (15) (2MO2(OH)2 f M2O4(OH)4) is endothermic by ∼14 kcal/mol for M = Cr and exothermic by about
13 and
26 kcal/mol for M = Mo and W, respectively. The free energy change for this reaction is very endothermic for M = Cr, slightly exothermic for M = Mo, and quite exothermic for M = W, indicating the formation of W2O4(OH)4 is quite favorable, although as noted above the hydrogen-bonded complex of two 2MO2(OH)2 is more stable than M2O4(OH)4). Reaction (16) is endothermic by ∼9 kcal/mol for M = Mo and exothermic by about
11 kcal/mol for M = W. However, the free energy change for both M = Mo and W for reaction (16) is endothermic,

indicating that the formation of M2O2(OH)8 is unfavorable for both metals. For M = Cr, M2O2(OH)8 exists as a hydrogenbonded complex of MO2(OH)2 and H2O molecules. Heats of Formation. Table 9 collects the heats of formation calculated at 298.15 K for the group 6 metal oxide clusters and their hydroxides. The heats of formation for the metal oxide clusters were previously calculated from the theoretical total atomization energies and experimental heats of formation of the atoms95 for the monomers and dimers and from the theoretical normalized clustering energies and monomer heats of formation for the trimers and tetramers.20e Those for the hydroxide clusters are calculated from the reaction energies in Table 8 using the theoretical heat of formation of H2O77k and the metal oxide clusters.20e The error bars associated with these calculated heats of formation due to the uncertainty in the heat of formation of the atoms are (1.0n kcal/mol for M = Cr, (0.9n kcal/mol for M = Mo, and (1.5n kcal/mol for M = W, where n is the number metal atoms in the molecule.95 We have previously shown that our calculated heats of formation for most of these metal oxide clusters are within the experimental error bars, although the experimental error bars are often quite large. For MO2(OH)2, our calculated heats of formation at 298 K of
204.2 ( 0.9 kcal/mol and
219.5 ( 1.5 kcal/mol are in good agreement with the experimental values of
203.4 ( 1.0 kcal/mol and
216.5 ( 1.0 kcal/mol for M = Mo and W, respectively.95 Thus, our calculated heats of formation for these compounds should be close to “chemical accuracy ((1 kcal/mol)” especially for the smaller clusters. For M2O5(OH)2, M2O3(OH)6, and M2O(OH)10, we calculated their heats of formation using the reaction enthalpies for reactions (4a), (6), and (8) or reactions (12
14). These two approaches give slightly different heats of formation by up to ∼1 kcal/mol as shown by the numbers in the paratheses in Table 9. Hydrolysis Potential Energy Surfaces. The potential energy surfaces for the hydrolysis reactions are shown in Figures 5
8. 8088

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Figure 5. Potential energy surfaces for the reaction MO3 þ 3H2O f M(OH)6 (M = Cr, Mo, W) calculated at the CCSD(T)/CBS//B3LYP/aD level. Relative energies at 0 K from the reactants in kcal/mol and selected bond lengths in Å in italic. M = Cr (Black), Mo (Red), W (Green).

Molecular structures of the reactant complexes and transition states with selected bond lengths are shown in these figures, with more detailed information given as Supporting Information. Entropy and enthalpy changes from the reactant complexes to the transition states, imaginary frequencies, reaction rate constants (good to about 1 decimal place), and tunneling coefficients for selected hydrolysis reactions are listed in Table 10. For the RRKM calculations, the temperature range is 273
773 K, and the pressure range is 76
1520 Torr. N2 is used as the carrier gas. The calculated reaction rate constants from the TST and RRKM theories agree with each other very well at 298 K. MO3 þ H2O f MO2(OH)2. We have previously28 shown that the group 6 metal oxide MO3 should be very strong Lewis acids but fairly weak Brønsted bases (also see Tables 6 and 7). The Lewis acidities of MO3 were predicted to be 120
150 kcal/mol for these metals (Table 7). Thus, they should form very energetically stable Lewis acid complexes with H2O. As shown in Figure 5, the complexation energies between MO3 and H2O are quite high, 35
45 kcal/mol. The O3M
OH2 bond distance (Figure 5a) is ∼0.3 Å longer than the M
OH bond distance in MO2(OH)2. Complexation leads to only small changes in the MdO and O
H bond distances. Natural population analysis shows a substantial amount of charge transfer of 0.15e to 0.2e from H2O to MO3. MO2(OH)2 is formed by proton transfer from the coordinated H2O to the dO atom in the initial Lewis acid complex. Reaction barriers from the reactant complexes (Figure 5 and Table 10) were calculated to range from 15
25 kcal/mol. The energies of these transition states are still well below the energies of the separated reactants by 15
25 kcal/mol. The imaginary frequencies for the transition states (Table 10) are substantial,

1630
1760 cm
1, suggesting a role for tunneling. Compared to that in the Lewis acid complex, the O3M
OH2 bond in the transition state (Figure 5b) is shortened by ∼0.1 Å, and the MdO bond to which the proton is being transferred lengthens by ∼0.1 Å. The O
H bond in the H2O for the proton being transferred is elongated by ∼0.3 Å. The distance between this H atom and the dO atom is 1.26 Å for M = Cr, 1.31 Å for M = Mo, and 1.36 Å for M = W, which is longer than the elongated O
H bond in the dissociating H2O for M = Mo and W. This is to be expected as the transition state for an exothermic reaction is usually an early transition state, and the higher the exothermicity, the earlier the transition state.111 The changes in the O
H bond lengths are assisted by bending the H2O toward the dO atom. The OdM
OH2 bond angles are 102
105 in the Lewis acid complex and are reduced to 75
82 in the transition states. The natural population analysis shows little change in the charge on the transferred hydrogen, which remains constant at about þ0.55e. The reaction barrier from the Lewis acid complex decreases from M = Cr to Mo to W, as the reaction becomes more exothermic. The natural population analysis shows a significant increase of the charge on the metal atom from M = Cr to Mo and W by 1.22e, 2.03e, and 2.33e, respectively. Calculated reaction rate constants (Table 10) increase substantially from M = Cr to Mo to W, as expected based on the calculated decrease in the reaction barriers. The tunneling factor at 298 K for this reaction is estimated to be 3.5 to 4 for these three metals due to the high imaginary frequencies of the transition states. MO2(OH)2 þ H2O f MO(OH)4. MO2(OH)2 is a much poorer Lewis acid than MO3. The fluoride affinities of MO2(OH)2 range from 50 to 75 kcal/mol (Table 7), which are lower than those of 8089

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Figure 6. Potential energy surfaces for the reaction M2O6 þ 2H2O f M2O4(OH)4 (M = Cr, Mo, W) calculated at the CCSD(T)/aT//B3LYP/aD level. Relative energies at 0 K from the reactants in kcal/mol. M = Cr (Black), Mo (Red), W (Green).

Figure 7. Potential energy surfaces for the reaction M3O9 þ H2O f M3O8(OH)2 (M = Cr, Mo, W) calculated at the CCSD(T)/aT//B3LYP/aD level and for the reaction M3O8(OH)2 þ H2O f M3O7(OH)4 (M = Mo, W) calculated at the CCSD(T)/aD//B3LYP/aD level. Relative energies at 0 K from the reactants in kcal/mol. M = Cr (Black), Mo (Red), W (Green).

MO3 by 70
75 kcal/mol for a given metal. Thus, its interaction with H2O is much weaker than that for MO3. Several Lewis acid

complexes by MO2(OH)2 and H2O can be formed for M = Mo and W, and one of them is shown in Figure 5c, which has a 8090

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Figure 8. Potential energy surfaces for the reaction M4O12 þ H2O f M4O11(OH)2 (M = Cr, Mo, W) calculated at the CCSD(T)/aD// B3LYP/aD level. Relative energies at 0 K from the reactants in kcal/mol. M = Cr (Black), Mo (Red), W (Green).

complexation energy of ∼5 kcal/mol which is much smaller than that for MO3 of >35 kcal/mol. No Lewis acid complex was predicted for M = Cr, and only a hydrogen-bonded complex is formed. More Lewis acid complexes for MO2(OH)2 and H2O with M = Mo and W are given as Supporting Information. Except for the hydrogen-bonded complex, hydrogen transfer from H2O to the dO atom can occur. Thus, there are multiple pathways connecting the various conformations of the Lewis acid complex between MO2(OH)2 and H2O to the various conformations of MO(OH)4, as indicated by the shaded areas on the potential energy surfaces in Figure 5. The reaction barriers for these different pathways may vary by a few kilocalories/mole. For example, the reaction barriers at 298 K (ΔHq) for the formation of the two lowest-energy structures of MO(OH)4 (1c and 1d) are calculated to be 36.2 and 33.0 kcal/mol for M = Cr, 24.4 and 22.5 kcal/mol for M = Mo, and 19.1 and 18.1 kcal/mol for M = W, respectively. Thus, the reaction rate constants for different pathways will exhibit some variation, and any measured rate constant would represent an average over the various values for the different conformers. MO(OH)4 þ H2O f M(OH)6. The reaction of MO(OH)4 with H2O to form M(OH)6 is similar to that of MO2(OH)2 with H2O. On the basis of our calculated fluoride affinities (Table 7), M = Cr MO(OH)4 is a much weaker Lewis acid than MO2(OH)2, whereas for M = Mo and W they are comparable. No Lewis acid complex was predicted for M = Cr, and only a hydrogen-bonded complex is formed. For M = Mo and W, we located several complexes, one of which is shown in Figure 5e, and the others are given as Supporting Information. A different kind of complex from that shown in Figure 5e other than the hydrogen-bonded complex can be formed, where the dO atom in MO(OH)4 and the coordinated H2O molecule occupy the opposite positions in a pseudo-octahedral structure. Transfer of hydrogen from H2O to the dO atom in MO(OH)4 can only occur in a complex where they are adjacent. For the lowestenergy pathway leading to the formation of the lowest-energy conformation of M(OH)6, the reaction barriers at 298 K (ΔHq) are 36.8, 28.5, and 23.8 kcal/mol for M = Cr, Mo, and W, respectively.

ARTICLE

MnO3n þ H2O f MnO3n
1(OH)2. Figures 6
8 present the potential energy surfaces for the reactions of MnO3n with H2O. These reactions occur in a similar fashion as those for MO3 by first forming a Lewis acid complex, followed by hydrogen transfer from H2O to the dO atom to form the ring structure or to the (M)
O(H) atom to form the chain structure. The complexation energies between the MnO3n (n = 2
4) clusters and water are much smaller than those for MO3, and they further decrease substantially from n = 2 to n = 3 by ∼10 kcal/mol. From n = 3 to n = 4 there is little change in the complexation energy. Although these reactions are all exothermic, their exothermicities are much smaller than those for MO3, and they are further reduced from n = 2 to n = 3. Again there is little change in the reaction enthalpy from n = 3 to n = 4. The reaction barriers leading to the ring isomers are ∼32 kcal/mol for M = Cr, ∼25 kcal/mol for M = Mo, and ∼20 kcal/mol for M = W, and they change little from n = 2 to n = 4. The variations in the reaction barriers leading to the chain isomers are within a few kilocalories/mole from n = 2 to n = 4. The reaction barriers for the reactions of MO3 and H2O are between the two different reaction barriers for the large clusters. Except for M = W for n = 3 and 4, the formation of the chain structure is thermodynamically more favorable. However, the reaction barrier calculated for the formation of the chain structure is substantially smaller than that for the formation of the ring structure (Table 10), so the formation of the chain structure is also kinetically more favorable for n = 2. In the case of M = W for n = 3 and 4, the formation of the chain structure is favored by kinetics, whereas the formation of the ring structure is favored by thermodynamics; thus, it should be possible to control the outcome of these reactions by choosing the reaction conditions. The calculated reaction rate constants are much higher for the reaction leading to the formation of the chain isomer than that leading to the ring isomer by 5 to 20 orders of magnitude. In addition, the calculated reaction rate constants increase substantially from M = Cr to Mo to W for the formation of the ring isomer, whereas those for the formation of the chain isomer are usually of similar magnitude. Table 10 also presents the calculated reaction rate constants for the dehydration reactions. The dehydration of MO2(OH)2 has a large reaction barrier (ΔHq) of 35
55 kcal/mol and increases from M = Cr to Mo to W. The dehydration of the two isomers of M2O5(OH)2 have similar reaction barriers, and they are much smaller than those for MO2(OH)2, 12
16 kcal/mol for M = Cr, 18
19 kcal/mol for M = Mo, and 22
26 kcal/mol for M = W. The dehydration of M3O8(OH)2 has even smaller reaction barriers than those for M2O5(OH)2, and the reaction barriers for the hydrolysis and dehydration are rather close especially for M = W and for the chain isomer. MnO3n
1(OH)2 þ H2O f MnO3n
2(OH)4. Figures 6 and 7 also present the potential energy surfaces for the further hydrolysis reactions of the larger clusters. Those for the reactions M2O4(OH)4 þ H2O f M2O3(OH)6 for M = Mo and W are given as Supporting Information. These reactions occur in a similar fashion as described above. Adsorption and Dissociation Energies. Table 11 presents the calculated adsorption (physisorption) and dissociative adsorption energies at 298 K at the CCSD(T) level. The adsorption energy is the enthalpy change from the reactant to the Lewis acid
base complex, and the dissociative adsorption energy is the enthalpy change for the hydrolysis reaction as given in Table 8. Compared to the larger metal oxide clusters and the hydroxides, MO3 has much more negative adsorption and dissociative 8091

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Table 10. Imaginary Frequencies (ωimg, cm
1) for the Transition States Calculated at the B3LYP/aD Level, Tunneling Coefficients Calculated from Equation 4, Entropy (ΔSq, cal/mol/K) and Enthalpy (ΔHq, kcal/mol) Changes from the Reactant Complexes to the Transtion States, Rate Constants from RRKM Theory (kuni, s
1) Calculated at 298.15 K (760 Torr for kuni), and Approximate Expressions for the Rate Constants from the RRKM Theory Calculated at the CCSD(T)/CBS//B3LYP/aD Level for Reaction (1) and at the CCSD(T)/aT//B3LYP/aD Level for Reactions (4) and (9)a reactionb (1) CrO3 3 3 3 H2O f CrO2(OH)2 CrO2(OH)2 f CrO3 3 3 3 H2O (1) MoO3 3 3 3 H2O f MoO2(OH)2 MoO2(OH)2 f MoO3 3 3 3 H2O (1) WO3 3 3 3 H2O f WO2(OH)2 WO2(OH)2 f WO3 3 3 3 H2O (4a) Cr2O6 3 3 3 H2O f Cr2O5(OH)2 (2a) Cr2O5(OH)2 (2a) f Cr2O6 3 3 3 H2O (4b) Cr2O6 3 3 3 H2O f Cr2O5(OH)2 (2b) Cr2O5(OH)2 (2b) f Cr2O6 3 3 3 H2O (4a) Mo2O6 3 3 3 H2O f Mo2O5(OH)2 (2a) Mo2O5(OH)2 (2a) f Mo2O6 3 3 3 H2O (4b) Mo2O6 3 3 3 H2O f Mo2O5(OH)2 (2b) Mo2O5(OH)2 (2b) f Mo2O6 3 3 3 H2O (4a) W2O6 3 3 3 H2O f W2O5(OH)2 (2a) W2O5(OH)2 (2a) f W2O6 3 3 3 H2O (4b) W2O6 3 3 3 H2O f W2O5(OH)2 (2b) W2O5(OH)2 (2b) f W2O6 3 3 3 H2O (9a) Cr3O9 3 3 3 H2O f Cr3O8(OH)2 (3a) Cr3O8(OH)2 (3a) f Cr3O9 3 3 3 H2O

κ

ΔSq

ΔHq

kunic

1762.1

4.0


5.8

24.1

3.0  10
6

1665.5 1627.7

3.7 3.6

135.3

1.0

kuni(T,P)d
16

(1.3  1012)P0.155 exp(
24.6/RT)

0.0
6.6

38.9 17.6

1.2  10 2.8  10
2

(7.9  1011)P0.155 exp(
38.4/RT) (2.5  1011)P0.69 exp(
17.9/RT)


1.1

46.4

1.7  10
22

(2.0  1012)P0.069 exp(
46.6/RT)


6.6

16.3

0.26

(2.0  1011)P0.089 exp(
16.5/RT)


1.2


28

54.8

1.1  10

(1.3  1012)P0.089 exp(
55.0/RT)


3.0

8.2

1.2  10

(2.5  1010)P0.354 exp(
7.1/RT)


9.5

15.9

0.10

(8.7  108)P0.354 exp(
14.8/RT)

6


12

2.9


3.8

31.9

4.0  10

(2.5  1012)P0.001 exp(
32.4/RT)

1080.3

2.1


3.5
1.7

12.5 10.3

8.2  102 8.2  104

(2.5  1012)P0.001 exp(
13.0/RT) (7.9  1010)P0.316 exp(
9.2/RT)

1544.9

3.3

1395.7

1078.6 1595.4 1436.0

(9b) Cr3O9 3 3 3 H2O f Cr3O8(OH)2 (3b) Cr3O8(OH)2 (3b) f Cr3O9 3 3 3 H2O

160.6

(9a) Mo3O9 3 3 3 H2O f Mo3O8(OH)2 (3a) Mo3O8(OH)2 (3a) f Mo3O9 3 3 3 H2O

1492.4

(9b) Mo3O9 3 3 3 H2O f Mo3O8(OH)2 (3b) Mo3O8(OH)2 (3b) f Mo3O9 3 3 3 H2O (9a) W3O9 3 3 3 H2O f W3O8(OH)2 (3a)

1010.2

W3O8(OH)2 (3a) f W3O9 3 3 3 H2O (9b) W3O9 3 3 3 H2O f W3O8(OH)2 (3b) a

ωimg


13.7

17.9

5.1  10
4

(1.4  108)P0.316 exp(
16.6/RT)


3.9

24.5

9.4  10
7

(2.0  1012)P0.009 exp(
25.0/RT)


4.5

19.0

7.6  10
3

(1.3  1012)P0.009 exp(
19.4/RT)


1.4

9.1

6.4  105

(4.0  1010)P0.374 exp(
7.7/RT)


12.8

22.7

2.1  10
7

(9.1  107)P0.374 exp(
21.1/RT)

3.5


3.5

20.9

5.3  10
4

(2.0  1012)P0.028 exp(
21.4/RT)

3.0


4.2
7.4

26.1 32.1

5.3  10
8 4.7  10
13

(1.3  108)P0.028 exp(
26.4/RT) (4.0  1011)exp(
32.6/RT)


2.3

10.5

4.2  104

(5.0  1012)exp(
11.1/RT)

12.0

2

6.2  10

(2.5  1011)P0.097 exp(
12.0/RT)


2

(4.0  1010)P0.097 exp(
16.8/RT)


8

2.1

1.0


5.3
9.3

1540.2 1021.5

3.1

16.7

3.2  10


5.0

25.7

8.2  10

(1.3  1012)P0.001 exp(
26.2/RT)


3.9

15.7

2.9

(2.0  1012)P0.001 exp(
16.1/RT)

12.9

5.3  10

(6.3  1011)P0.148 exp(
12.8/RT)

2.0


2.9

3.3


7.9
5.2

8.5 21.1

6.9  10 1.4  10
4

(5.0  1010)P0.148 exp(
8.4/RT) (1.0  1012)P0.007 exp(
21.7/RT)


4.7

21.9

5.1  10
5

(1.0  1012)P0.007 exp(
22.3/RT)

13.5

2

1.9  10

(7.9  1011)P0.133 exp(
13.5/RT)

10.9

3

1.1  10

(5.0  1010)P0.133 exp(
10.8/RT)

2.0

W3O8(OH)2 (3b) f W3O9 3 3 3 H2O


2.7
8.0

2 4

Symmetry numbers are not included in the calculated rate constants. The symmetry number for the reverse reaction of reaction (1) is 2, and those for the other reactions are 1. The temperature range is 273
773 K for both the TST and RRKM calculations. For the RRKM calculations, the pressure range is 76
1520 Torr, and the carrier gas is N2. The strong collision model is used for reaction (1) for M = Cr. b See Figures 1
3 and 5
7 for the molecular structures. Ground state structures assumed unless specified. c k¥ from TST may differ by up to ∼10% from kuni. d In the rate constant expression, A(Pξ)exp(
ΔE/RT), P is in Torr, ΔE is in kcal/mol, and and T is in K.

adsorption energies, which is consistent with its large Lewis acidity as discussed in the previous section. Replacing one or two dO in MO3 by two
OH groups results in much less negative adsorption energies (from
1 to
13 kcal/mol) and positive dissociative adsorption energies (up to 40 kcal/mol) except for WO2(OH)2, which can have a slightly negative dissociative adsorption energy of up to
6 kcal/mol. Thus, except for M = W, the formation of the OdM(OH)4 site is an endothermic process. For the lowest-energy structures of the larger metal oxide clusters, only the M(dO)2(
O
)2 site is available. Although this site is similar that in MO2(OH)2 in the number of dO and
O bonds, the adsorption and dissociative adsorption energies are very different. For the MnO3n clusters, the adsorption energy becomes less negative by ∼10 kcal/mol from n = 2 to n = 3 and 4, and it is essentially converged at n = 3 to
4 kcal/mol for M = Cr,


12 kcal/mol for M = Mo, and
16 kcal/mol for M = W. Similarly, the dissociative adsorption energy also becomes less negative (more positive for M = Cr for hydrogen transfer to the terminal oxygen) by 10
15 kcal/mol for M = Cr, 10
20 kcal/ mol for M = Mo, and 15
30 kcal/mol for M = W. The dissociative adsorption energy for hydrogen transfer to the terminal oxygen is also essentially converged at n = 3 to
18 kcal/mol for M = Cr,
4 kcal/mol for M = Mo, and
18 kcal/mol for M = W. For hydrogen transfer to the bridge oxygen, the dissociative adsorption energy is essentially converged for M = Mo to
8 kcal/mol, whereas for M = Cr and W, the reaction of water with larger clusters needs to be further studied to test the convergence of the dissociative adsorption energy. The fast convergence of the adsorption and dissociative adsorption energies for the group 6 metal oxide clusters is not surprising, as other properties such as the geometry parameters, vertical excitation 8092

dx.doi.org/10.1021/jp111031x |J. Phys. Chem. C 2011, 115, 8072–8103

The Journal of Physical Chemistry C

ARTICLE

Table 11. Adsorption and Dissociation Energies at 298 K (ΔHad and ΔHdis, kcal/mol) Calculated at the CCSD(T)/aT//B3LYP/ aD Level.a

b

molecule

site

ΔHad

ΔHdis

ΔHad

ΔHdis

ΔHad

ΔHdis

M = Cr

M = Cr

M = Mo

M = Mo

M=W

M=W

MO3

M(=O)3


39.8

MO2(OH)2

M(=O)2(
OH)2

(
8.0)c


54.6

MO(OH)4 M2O6

M(=O)(
OH)4 M(=O)2(
O
)2

(
8.2)c
12.7

M3O9

M(=O)2(
O
)2


4.1


8.8 (b), 17.5 (t)


13.0

M4O12

M(=O)2(
O
)2


4.3


4.1 (b), 18.4 (t)


12.2

28.2 to 33.3 33.6 to 39.7
20.5 (b), 6.7 (t)


38.3


67.1


5.4 to
10.1
1.1 to
11.4
21.1


43.5

5.5 to 10.4


6.4 to
11.4


82.1
5.6 to 1.1


2.8 to
12.6
27.6

2.1 to 7.0
41.2 (b),
32.9 (t)


8.6 (b),
3.1 (t)


17.3


14.6 (b),
18.0 (t)


8.1 (b),
3.7 (t)


15.8


17.4 (b),
18.0 (t)

12.4 to 19.3
28.7 (b),
15.6 (t)

a

At the CCSD(T)/CBS//B3LYP/aD level for the mononuclear compounds and at the CCSD(T)/aD//B3LYP/aD level for M3O8(OH)2 and M4O11(OH)2. ΔHad = E(adsorption product)
E(A)
E(H2O); ΔHad = E(dissociation product)
E(A)
E(H2O), where A is the metal oxide or metal hydroxide. ΔHad equals the reaction energy in Table 8. b Ground state structure assumed. See Figure 1. c Forms a hydrogen-bonded complex.

energies, and electron attachment energies also reach convergence rather quickly at n = 3 or 4.20c,28 It is worthwhile to compare the adsorption properties of the group 6 transition metal oxide clusters to those of the titanium dioxide clusters.112 The experimental physisorption energies for H2O on the TiO2 surfaces were measured to be
14 to
24 kcal/mol.113,114 The experimental dissociative adsorption energies for H2O on the TiO2 surfaces have been measured to be
56,115
42,116
118 and
18 to
23 kcal/mol.119 Both the adsorption and dissociative adsorption energies depend strongly on the active site as shown in our computational studies on the interaction of H2O with the TiO2 nanoclusters.112 Although hydrolysis of the group 6 transition metal oxide clusters and the TiO2 nanoclusters follows a similar path, i.e., the formation of the Lewis acid
base complex followed by hydrogen transfer from H2O to the dO or
O
atom, there is a substantial difference between them. While the adsorption energy of MO3 is predicted to more negative by 8
13 kcal/mol than that of TiO2, those of the larger MO3 clusters are predicted to be less negative by ∼16, 8, and 2 kcal/mol for M = Cr, Mo, and W for the dimers and by up to 29, 21, and 18 kcal/mol for the trimers and tetramers for the respective metals. The adsorption energy for the product of the first hydrolysis step for the monomer is also quite different. The adsorption energy of MO2(OH)2 is predicted to be only up to
13 kcal/mol, whereas that of TiO(OH)2 is predicted to be about
29 kcal/mol, similar to that of TiO2. Similar comparisons can be made for the dissociative adsorption energies. Accurate Potentional Energy Surface Prediction. The accurate prediction of the potential energy surface (PES) requires the accurate prediction of the reaction energy and the reaction barrier, which determine the reaction equilibrium and reaction rate, respectively. Here, we examine the basis set and other requirements for obtaining high-quality PESs with our composite approaches based on the CCSD(T) method.77 Table 12 presents the valence electronic energy contributions, core
valence correlation corrections, and scalar relativistic corrections to the reaction energies, reacton barriers, and complexation energies for reactions (1)
(3) as defined in Table 8. The reaction barrier is the energy difference between the transition state and the reactant complex, except for reaction (2) for M = Cr and reaction (3) for all three metals. The complexation energy is the energy difference between the separated reactants and the reactant complex. The energy contributions to the relative energies of the stationary points on the PES are given as Supporting Information.

Valence Electronic Energy. Compared to the valence electronic energy contributions calculated at the CCSD(T)/CBS level, those calculated at the CCSD(T)/aD level underestimate the exothermicity (or overestimate the endothermicity) for the reaction energy, overestimate the reaction barrier, and underestimate the complexation energy, except for reaction (3) where the opposite effect was observed. The difference between the valence electronic energy contributions calculated at the CCSD(T)/aD and CCSD(T)/CBS levels is up to ∼3 kcal/mol for the reaction energy and ∼1 kcal/mol for the reaction barrier and complexation energy. The basis set effect is larger for M = W than for M = Cr and Mo. The valence electronic energy contributions calculated at the CCSD(T)/aT level are usually within ∼0.3 kcal/mol from those calculated at the CCSD(T)/CBS level. For reaction (2) for M = Cr and reaction (3) for M = Cr and Mo, the difference between the CCSD(T)/aT and CCSD(T)/CBS valence electronic energy contributions is up to ∼1 kcal/mol. Thus, the CCSD(T)/aT valence electronic energies are usually sufficiently close to those at the CCSD(T)/CBS level, and even the CCSD(T)/ad energies are semiquantitative. Core
Valence Correlation Correction. The core
valence correlation corrections calculated at the CCSD(T)-DK/awCVTZ-DK level are up to ∼1 kcal/mol for the reaction energies and complexation energies and usually less than 0.5 kcal/mol for the reaction barriers. The core
valence correlation corrections to the reaction energies and complexation energies are larger for reaction (1) than for reactions (2) and (3). The core
valence correlation corrections calculated at the CCSD(T)-DK/wCVTZ-DK level are within 0.4 kcal/mol from those calculated at the CCSD(T)-DK/awCVTZDK level. Those calculated at the CCSD(T)/awCVTZ level can differ from the CCSD(T)-DK/awCVTZ-DK results by 0.8 kcal/ mol, whereas those calculated at the CCSD(T)/wCVTZ level are usually in slightly better agreement with the CCSD(T)-DK/ awCVTZ-DK results. Similar conclusions can be reached when comparing the core
valence corrections calculated at the CCSD(T)/awCVDZ and CCSD(T)/wCVDZ levels to the CCSD(T)DK/awCVTZ-DK results, with those calculated at the CCSD(T)/ wCVDZ level being superior to those at the CCSD(T)/awCVDZ level. The core
valence corrections calculated at the CCSD(T)/ wCVDZ level differ from the CCSD(T)-DK/awCVTZ-DK results by up to 0.7 kcal/mol for reaction (1) and up to 1.5 kcal/mol for reactions (2) and (3), although their differences for the reaction barriers and complexation energies are usually much smaller than those for the reaction energies. Thus, for calculating the PES for the larger clusters, inclusion of the core
valence corrections at the 8093

dx.doi.org/10.1021/jp111031x |J. Phys. Chem. C 2011, 115, 8072–8103

The Journal of Physical Chemistry C

ARTICLE

Table 12. Electronic Energy Contributions to Reaction Energies (RE), Reaction Barriers (RB), and Complexation Energies (CE) in kcal/mol for Reactions (1)
(3)a

reaction

b

ΔEVal

ΔEVal

ΔEVal

ΔEVal

aD

aT

aQ

CBS

ΔECV wCVDZ

ΔECV awCVDZ

ΔECV

ΔECV

ΔECV

wCVTZ

awCVTZ

wCVTZ-DK

ΔECV

ΔERel

ΔEMVD

þ0.43

awCVTZ-DK

(1) M = Cr RE


52.74


54.54


54.62


54.61


1.74


2.30


1.68


1.94


1.03


1.25

þ0.09

RB

28.67

27.78

27.90

28.01


0.44


0.58


0.28


0.26


0.13


0.11


0.11

0.00

CE

40.19

40.89

40.89

40.87

þ1.06

þ1.56

þ1.23

þ1.45

þ0.94

þ1.14


0.15


0.30

RE


65.11


66.71


66.78


66.78


1.70


2.37


1.39


1.70


0.91


1.16

þ0.01

þ0.21

RB

21.88

20.99

21.09

21.19


0.31


0.45


0.10


0.10


0.01

0.00


0.10


0.01

CE

38.86

39.27

39.22

39.17

þ1.00

þ1.52

þ1.05

þ1.30

þ0.81

þ1.01


0.08


0.16

RE


79.41


81.87


82.11


82.19


0.60


1.39


0.47


0.80


0.46


0.83

þ0.15

þ0.21

RB

19.64

18.95

19.08

19.19


0.11


0.20

þ0.10

þ0.08

þ0.35

þ0.35

þ0.16


0.02

CE

43.47

44.31

44.32

44.29

þ0.36

þ1.01

þ0.47

þ0.71

þ0.77

þ1.06


0.07


0.16

(1) M = Mo

(1) M = W

(2a) M = Cr RE

28.90

27.74

28.05

28.30


1.21


1.65


0.72


0.82


0.36


0.45


0.07

þ0.24

RB

38.38

37.33

37.79

38.14


0.91


1.32


0.45


0.55


0.21


0.29


0.08

þ0.15

(2b) M = Cr RE

27.57

26.43

26.82

27.12


1.07


1.51


0.51


0.61


0.07


0.17


0.08

þ0.25

RB

35.14

34.30

34.70

35.00


0.89


1.26


0.45


0.54


0.22


0.30


0.09

þ0.15

(2a) M = Mo RE

5.78

5.03

5.07

5.12


1.84


2.34


1.03


1.23


0.53


0.67


0.21

þ0.17

RB

28.93

28.01

28.13

28.24


0.93


1.12


0.57


0.59


0.41


0.43


0.07

þ0.02

CE

7.44

7.11

6.95

6.85

þ0.53

þ0.86

þ0.36

þ0.50

þ0.16

þ0.26

þ0.11


0.10

(2b) M = Mo RE

6.22

5.56

5.64

5.72


1.91


2.36


1.05


1.24


0.51


0.64


0.17

þ0.17

RB

26.76

25.85

25.99

26.11


1.06


1.32


0.62


0.69


0.43


0.48


0.06

þ0.03

CE

7.84

7.52

7.36

7.27

þ0.36

þ0.58

þ0.30

þ0.40

þ0.15

þ0.21

þ0.12


0.10

RE


5.33


6.50


6.70


6.79


1.20


1.76


0.29


0.51

þ0.44

þ0.22


0.11

þ0.16

RB

23.93

22.80

22.83

22.89


0.57


0.71


0.26


0.29


0.06


0.09


0.19

þ0.02

CE

8.90

8.48

8.41

8.37

þ0.51

þ0.88

þ0.14

þ0.29


0.17

0.00


0.05


0.09

RE


5.06


6.33


6.46


6.51


1.21


1.74


0.28


0.50

þ0.48

þ0.25


0.03

þ0.16

RB

22.59

21.47

21.56

21.65


0.64


0.78


0.25


0.30


0.01


0.06


0.15

þ0.02

CE

10.03

9.67

9.63

9.61

þ0.43

þ0.74

þ0.16

þ0.30


0.10

þ0.04


0.12


0.10

(2a) M = W

(2b) M = W

(3) M = Cr RE

32.17

31.55

32.11

32.51


1.38


1.79


0.59


0.67


0.13


0.22


0.12

þ0.24

RB

38.56

38.37

39.02

39.46


0.98


1.39


0.37


0.47


0.11


0.19


0.09

þ0.16

(3) M = Mo RE

10.76

11.06

11.41

11.64


2.10


2.77


1.06


1.19


0.70


0.80


0.05

þ0.17

RB

23.35

23.68

24.21

24.55


1.59


2.20


0.89


1.02


0.67


0.76


0.02

þ0.13

(3) M = W RE

0.13

0.52

0.53

0.52


1.38


1.87


0.41


0.59

þ0.30

þ0.08


0.01

þ0.16

RB

17.30

17.79

18.07

18.24


1.20


1.64


0.49


0.65


0.01


0.19

þ0.01

þ0.13

a

The reaction barrier is defined as the energy difference between the transition state and the reactant complex. b See Table 8 and Figures 1 and 5 for the molecular structures of the reactants, reactant complexes, transition states, and products.

CCSD(T)/wCVDZ level is preferable to those at the CCSD(T)/ awCVDZ level. However, the core
valence corrections calculated

at the CCSD(T)/wCVDZ level are likely to improve the quality of the reaction barriers and complexation energies more than the 8094

dx.doi.org/10.1021/jp111031x |J. Phys. Chem. C 2011, 115, 8072–8103

The Journal of Physical Chemistry C

ARTICLE

Table 13. Absolute and Average Deviations (kcal/mol) from the CCSD(T)/CBS//B3LYP/aD Results for the Reaction Energies (RE), Reaction Barriers (RB), and Complexation Energies (CE) for Reaction (1) Calculated with the DFT Exchange-Correlation Functionals and the aT Basis Set at the B3LYP/aD Geometries absolute deviations functional

RE(Cr)

RE(Mo)

RE(W)

RB(Cr)

RB(Mo)

average

RB(W)

CE(Cr)

CE(Mo)

CE(W)

RE

RB

CE

All

B1B95


2.3


1.9


0.2


5.0


3.8


3.7


3.4


2.4


3.1

1.5

4.2

2.9

2.9

B1LYP


3.2


2.3


0.2


3.2


1.5


1.4


2.8


1.9


2.9

1.9

2.1

2.5

2.2

B3LYP


3.5


2.3

0.2


4.1


2.3


2.2


2.7


1.7


3.0

2.0

2.9

2.5

2.4

B3P86


5.3


3.5


1.3


6.3


4.6


4.4


1.5


1.1


2.1

3.4

5.1

1.6

3.3

B3PW91 B97-1


3.0
3.4


1.3
2.0

0.8 0.2


5.9
5.0


4.2
3.2


4.0
3.1


3.5
2.8


3.0
2.0


4.0
3.1

1.7 1.9

4.7 3.8

3.5 2.6

3.3 2.7

B97-2


0.5

0.5

2.2


4.6


2.8


2.8


5.4


4.3


5.0

1.1

3.4

4.9

3.1

B98


3.2


2.1

0.0


4.9


3.1


3.0


3.0


2.1


3.1

1.8

3.7

2.7

2.7

BB95

0.3

4.2

8.0


8.7


6.3


6.1


6.4


6.7


8.9

4.2

7.0

7.3

6.2

BLYP


0.8

3.4

7.4


6.5


3.9


3.6


5.6


5.9


8.4

3.9

4.6

6.6

5.0

BMK


8.7


11.9


9.5


4.5


2.6


2.3

1.4

4.7

3.7

10.0

3.1

3.2

5.5

BP86


2.8

1.5

5.2


9.0


6.5


6.2


4.2


4.7


6.8

3.2

7.2

5.2

5.2

BPW91 BRxP86


0.7
7.9

3.6
3.0

7.1 0.7


8.6
6.1


6.1
3.7


5.8
3.3


6.2 0.0


6.6
1.3


8.6
3.4

3.8 3.9

6.9 4.4

7.1 1.6

5.9 3.3

CAM-B3LYP


6.1


6.8


5.9


3.7


2.7


2.6

0.2

1.7

1.6

6.3

3.0

1.2

3.5

1.3

5.2

8.9


7.0


4.4


4.0


7.9


8.0


10.2

5.1

5.1

8.7

6.3

G96LYP HCTH/147

3.3

6.4

9.5


6.4


3.9


3.8


8.9


8.4


10.3

6.4

4.7

9.2

6.8

HCTH/407

5.0

8.0

10.9


5.7


3.2


3.2


10.1


9.4


11.4

7.9

4.0

10.3

7.4 8.9

HCTH/93

7.1

9.9

12.9


6.0


3.4


3.4


12.4


11.5


13.4

10.0

4.3

12.4

HSE1PBE


5.9


4.1


2.2


5.6


4.0


3.9


0.8


0.5


1.3

4.1

4.5

0.9

3.1

HSE03 HSE06


6.0
5.8


4.3
4.1


2.4
2.2


5.5
5.6


3.9
4.0


3.8
3.9


0.7
0.9


0.4
0.5


1.2
1.3

4.2 4.1

4.4 4.5

0.8 0.9

3.1 3.1

LC-BP86


12.2


16.0


16.8


6.4


6.7


6.5

5.6

8.0

9.5

15.0

6.5

7.7

9.7

LC-PBE


10.1


13.8


14.7


6.4


6.7


6.5

3.5

6.0

7.4

12.9

6.5

5.6

8.3

LC-PW91


10.2


14.0


15.0


6.0


6.4


6.2

3.7

6.2

7.7

13.1

6.2

5.8

8.4

LC-TPSS


9.3


13.4


14.5


5.4


5.7


5.6

2.9

5.6

7.2

12.4

5.6

5.2

7.7

LC-wPBE


3.5


6.4


7.1


4.4


4.1


4.2


2.1

0.6

1.9

5.7

4.2

1.5

3.8

M06


5.1


2.3


1.3


3.4


2.1


1.7


1.1


1.6


1.8

2.9

2.4

1.5

2.3

M06-2X


5.1


7.5


7.7


3.1


2.2


2.5

1.8

3.7

4.2

6.8

2.6

3.2

4.2

M06-HF


6.5


14.8


15.5


5.8


4.6


5.0

4.2

9.5

10.5

12.3

5.1

8.0

8.5

M06-L


5.0

0.0

1.1


3.8


1.9


1.7


2.0


3.4


3.6

2.0

2.4

3.0

2.5

mPW1LYP


4.5


3.4


1.4


3.3


1.7


1.6


1.5


0.7


1.8

3.1

2.2

1.3

2.2

mPW1PBE


4.4


3.0


1.3


5.8


4.3


4.2


2.3


1.7


2.4

2.9

4.8

2.1

3.3

mPW1PW91 mPW3PBE


4.4
4.3


3.0
2.4


1.4
0.3


5.5
6.2


4.0
4.6


3.9
4.4


2.2
2.3


1.6
2.0


2.3
2.9

2.9 2.3

4.5 5.1

2.0 2.4

3.1 3.3

mPWLYP


2.5

1.9

5.9


6.6


4.1


3.8


3.9


4.4


7.0

3.4

4.8

5.1

4.5

mPWPBE


2.4

2.1

5.6


9.1


6.6


6.3


4.6


5.1


7.2

3.4

7.3

5.6

5.4

mPWPW91

5.3


2.4

2.0

5.6


8.8


6.3


6.0


4.5


5.1


7.1

3.3

7.0

5.6

O3LYP

4.2

6.2

8.7


5.4


3.3


3.4


9.8


8.8


10.3

6.3

4.0

9.6

6.7

OLYP

7.1

10.4

13.7


6.8


4.2


4.2


12.6


12.0


14.2

10.4

5.1

13.0

9.5

PBE


3.6

1.0

4.8


9.2


6.8


6.5


3.3


4.0


6.2

3.1

7.5

4.5

5.0

PBE0 PBEh1PBE


5.3
5.8


3.8
4.2


2.0
2.4


5.9
5.6


4.4
4.0


4.3
3.9


1.3
0.9


0.8
0.5


1.6
1.2

3.7 4.1

4.9 4.5

1.3 0.9

3.3 3.1

PBEhPBE


4.3

0.5

4.3


8.7


6.2


5.9


2.7


3.5


5.7

3.0

7.0

4.0

4.6

PKZB

3.9

7.3

10.6


6.9


4.5


4.5


9.3


9.1


11.1

7.3

5.3

9.8

7.5

PW91


4.4

0.2

3.9


9.1


6.7


6.4


2.6


3.4


5.5

2.8

7.4

3.8

4.7


16.4


10.8


6.7


13.6


11.7


11.0

7.6

5.4

3.2

11.3

12.1

5.4

9.6

SVWN5 τ-HCTH τ-HCTHhyb TPSS

2.4

5.0

7.8


7.3


5.1


4.7


8.3


7.5


9.1

5.0

5.7

8.3

6.3


3.7
3.4


1.9 0.2

0.7 3.3


6.0
8.0


4.1
5.9


3.8
5.6


2.9
3.6


2.2
3.9


3.5
5.5

2.1 2.3

4.6 6.5

2.8 4.3

3.2 4.4

8095

dx.doi.org/10.1021/jp111031x |J. Phys. Chem. C 2011, 115, 8072–8103

The Journal of Physical Chemistry C

ARTICLE

Table 13. Continued absolute deviations functional TPSSh VSXC

RE(Cr)

RE(Mo)


4.0
6.1


1.5
0.2

average

RE(W)

RB(Cr)

RB(Mo)

RB(W)

CE(Cr)

CE(Mo)

CE(W)

RE

RB

CE

All

0.9 2.5


6.7
2.8


5.0
1.1


4.8
1.1


2.9 1.1


2.7
1.4


3.7
3.0

2.1 2.9

5.5 1.7

3.1 1.8

3.6 2.1

wB97


4.7


7.9


8.8


1.7


1.0


1.1

0.0

3.2

4.6

7.1

1.2

2.6

3.7

wB97X


5.3


7.0


7.5


2.2


1.2


1.3

0.1

2.4

3.4

6.6

1.6

2.0

3.4

X3LYP


4.3


2.8


0.6


4.1


2.3


2.2


1.8


1.3


2.4

2.6

2.9

1.8

2.4

reaction energies. Inclusion of the core
valence corrections calculated at the CCSD(T)-DK/wCVTZ-DK or CCSD(T)/wCVTZ level is preferable if the calculations can be performed. Scalar Relativsitic Corrections. The scalar relativistic corrections calculated using eq 1 are very small (e0.2 kcal/mol) for the reaction energies, reaction barriers, and complexation energies. The scalar relativistic corrections calculated as the expectation values of the mass-velocity and Darwin (MVD) operators are also small but can differ from those calculated by eq 1 by 0.4 kcal/ mol, and their difference is the pseudopotential correction given by eq 2. Thus, inclusion of the MVD corrections alone for the scalar relativistic corrections does not necessarily improve the results, although the effect is expected to be very small. DFT Performance. Tables 13 and 14 present the benchmark results for calculating the reaction energies, reaction barriers, and complexations energies for reactions (1)
(4) and (9) with the various DFT exchange-correlation functionals widely available in the current generation of the computational chemistry programs. Absolute deviations for these energies from the CCSD(T)/CBS or CCSD(T)/aT results are given for reaction (1) in Table 13, and those for the other reactions are given as Supporting Information. Table 14 lists the average deviations over the metal (Cr, Mo, W) and the type of energy (reaction energy, reaction barrier, and complexation energy). MO3 þ H2O f MO2(OH)2. For the calculated reaction energies for reaction (1) for M = W, a number of DFT exchangecorrelation functionals give results very close to the CCSD(T)/ CBS results, and nearly all of them are hybrid functionals. The reaction energies calculated using the B1B95, B1LYP, B3LYP, B97-1, B98, and mPW3PBE functionals are within 0.3 kcal/mol of the CCSD(T)/CBS results. The reaction energies calculated using the B3PW91, BRxP86(pure GGA), τ-HCTH(pure GGA), TPSSh, and X3LYP functionals are within 1 kcal/mol of the CCSD(T)/CBS results, and those from the B3P86, M06, M06L(pure GGA), mPW1LYP, mPW1PBE, mPW1PW91, and PBE0 functionals are within 2 kcal/mol of the CCSD(T)/CBS results. For the same reaction energy for M = Mo, pure functionals give better results than hybrid functionals. The reaction energies calculated using the M06-L, PW91, TPSS, and VSXC functionals are within 0.2 kcal/mol of the CCSD(T)/CBS results, using the B97-2, PBE, and PBEhPBE functionals are within 1 kcal/mol of the CCSD(T)/CBS results, and using the B1B95, B3PW91, B971, BP86, mPWLYP, mPWPW91, τ-HCTHhyb, and TPSSh functionals are within 2 kcal/mol of the CCSD(T)/CBS results. For M = Cr, fewer functionals give good results. The BB95 functional gives the best result with an error of 0.3 kcal/mol. The B97-2, BLYP, and BPW91 functionals also give good results with errors less than 1 kcal/mol. The G96LYP functional has an error of 1.3 kcal/mol. All of these except B97-2 are pure functionals. The fact that pure functionals usually perform better for the

energetics for the first row transition metal compounds and hybrid functionals perform better for the third row transition metal compounds has been previously attributed to the larger multireference characters of the first row transition metal compounds.20a For the compounds involved in reaction (1), the T1 diagnostic calculated at the CCSD(T)/aT level is ∼0.04 for M = Cr, ∼0.03 for M = Mo, and ∼0.025 for M = W. For the average deviation of the reaction energies for reaction (1) for these three metals, the B97-2 functional has the best performance with an average error of ∼1 kcal/mol and a maximum error of ∼2 kcal/mol. For the complexation energies for all three metals, the HSE1PBE, HSE03, HSE06, and PBEh1PBE functionals have average errors of less than 1 kcal/mol, and the CAM-B3LYP, LCwPBE, M06, mPW1LYP, and PBE0 functionals have average errors of e1.5 kcal/mol. The B3P86, BRxP86, VSXC, and X3LYP functionals also predict reasonable complexation energies for M = Cr and Mo. All of the long-range corrected functionals predict larger complexation energies than the CCSD(T)/ CBS method. The DFT reaction barriers generally agree less well with the CCSD(T)/CBS results than do the reaction energies. Most functionals underestimate both the reaction barrier and the complexation energy. For all three metals, the wB97 functional has the best performance with an average error of ∼1 kcal/mol. For M = Mo and W, the reaction barriers calculated by the B1LYP, M06, M06-L, mPW1LYP, VSXC, wB97, and wB97X functionals are within 2 kcal/mol of the CCSD(T)/CBS results. For M = Cr, only the wB97 functional has an error of less than 2 kcal/mol as compared to the CCSD(T)/CBS results. For reaction (1) for all three metals, no exchange-correlation functional is capable of predicting the reaction energies, reaction barriers, and complexation energies with an average error of less than 2 kcal/mol. The B1LYP, B3LYP, M06, M06-L, mPW1LYP, VSXC, and X3LYP have average errors of e2.5 kcal/mol. The B1LYP functional has the best overall results for reaction (1) with a maximum error of slightly over 3 kcal/mol. The B3LYP and X3LYP functionals have maximum errors of slightly over 4 kcal/mol. Average Deviations for All Reactions. Absolute deviations for reactions (2)
(4) and (9) are given as Supporting Information. Similar conclusions can be drawn for these reactions, and we discuss the average deviations for all these reactions as given in Table 14. We have averaged the absolute deviations over both the metal and the type of energy, as well as the global average over all reactions. For the average deviations of the reaction energies, the B1B95 and B3PW91 functionals have the best performance with average errors of