J. CHEM. SOC. FARADAY TRANS., 1994, 90(21), 3237-3240 Theoretical Vibrational Energies of [CrO,]’-[MnO,J2-and [Fe0,l2-Robert J. Deeth* and Paul D. Sheen Inorganic Computational Chemistry Group, School of Chemistry, University of Bath, Claverton Down, Bath, UK BA2 7A Y Density functional theory (DFT) calculations of the vibrational energies of the tetrahedral complexes [M0,l2-(M = Cr, Mn, Fe) are reported using both local and non-local functionals. The computed results vary slight with choice of basis set but are less sensitive to the choice of functional. The computed data tend to be slightly too large for the high-energy stretching modes and slightly too small for the low-energy bending/torsion modes. The average deviation between theory and experiment is about 4%, with a maximum average error of only 7% depending on which experimental data are used in the comparison.Terminal metal-oxygen bonds are a common feature of the high oxidation state chemistry of the transition elements.’ They often participate in the important process of 0x0 trans- fer.’ X + AOsXO + A Among the metals of the left half of the transition series a variety of metal-oxo subunits exist, ranging from simple mono-oxo to tetra-oxo species. When two or more terminal oxygen ligands are present, several geometric arrangements become possible and examples of virtually every isomeric configuration can be found, e.g. cis and trans O=M=O units, trigonal planar and trigonal pyramidal MO, units, tetrahedral and ‘see-saw’ MO, units.Among the latter, a range of discrete, tetrahedral tetra-oxo complexes are found for each element of the vanadium through to iron triads. These species represent some of the simplest metal-oxo mol- ecules.’ The richness of metal-oxo chemistry has stimulated a good deal of theoretical interest. Studies in this laboratory have examined (i) the electronic transition energies of a range of d’ complexes of Cr03+ VO” and Moo3+, (ii) the opti- mised structures and vibrational energies of Mo’X, species (M = Cr, Mo; X = F, C1)6 (iii) the optimised bond lengths and electronic structures of [MO,]’-compounds (M = Cr, Mn, Fe).7 In each case, a theoretical model based on density functional theory (DFT) was used and found to give good agreement with experimental data.This contrasts with the results of Hartree-Fock treatments of the M-0 bonding in, for example, tetrahedral oxyanions which are qualitatively and quantitatively po~r.~,~ The present paper extends our earlier DFT treatment of the [MO,]’-species7 to the esti- mation of their vibrational energies. Once again, the DFT approach yields accurate results. Subject to a minor revision of the assignment of the spectrum of [FeO,]’-, the experi- mental bands are reproduced to within 7% or better. Computational Details All DFT calculations were based on the Amsterdam density functional (ADF) program system due to Baerends et al.” with STO basis sets of double-c plus polarisation (DZP) and triple-c plus polarisation (TZP) quality.’ ‘,12 The method has been described in detail el~ewhere.~.’ The uniform electron gas local density appr~ximation’~ (LDA) was used in con- junction with analytical energy gradients’ for all geometry optimisations which were constrained to & symmetry.The LDA correlation energy was computed according to Vosko et UI.’S’~ parametrisation of electron gas data and includes Stoll et d’s’ 7,18 correction for self-interaction. Non-local gradient corrections (GC)to the LDA exchange and corre- lation terms employed the formulations of Becke” and Perdew,20.21 respectively. The lower core shells on the atoms (1s on 0 and up to 2p on the metal atoms) were treated by the frozen-core approximation.22 The total molecular elec- tron density was fitted in each SCF cycle by auxiliary s, p, d, f and g STO functions.23 Harmonic vibrational energies were estimated via finite dif- ference of the analytical first derivatives. The calculations were run without symmetry constraints (i.e.with all nine degrees of freedom), with two additional points used to esti- mate each second derivative. The resulting nine computed vibrational energies were then averaged over symmetry-equivalent components assigned by reference to the com-puted eigenvectors. Notionally equivalent vibrational com-ponents never differed by more than ca. 15 cm-’,which indicates the approximate level of numerical ‘noise’ inherent in this procedure. Our previous DFT study’ on these complexes gave opti- mised M=O distances within 0.02 A of the experimental values.Test calculations using the experimentally observed bond lengths and the DFT-optimised bond lengths gave computed vibrational-energy differences of around 10 cm-’. This error is of the same order as the numerical ‘noise’ described above and hence the averaged experimental bond lengths were used throughout, uiz. 1.658 A,’, 1.659 A’’ and 1.656 for Cr-0, Mn-0 and Fe-0, respectively. The formally d’ [MnO,]’- ion possess a 2E Jahn-Teller active ground state. In principle, a regular & geometry would not correspond to a minimum on the potential-energy surface and would yield an imaginary vibrational frequency. However, there is no experimental evidence for a marked Jahn-Teller di~tortion.~’To circumvent this problem and maintain the high symmetry for comparison with the other two molecules, the unpaired electron was distributed equally between the two components of the 2e partially occupied MO (molecular orbital).The DFT formalism remains consis- tent even with non-integral MO occupation^.'^ This pro- cedure then generates a ’A, state which now corresponds to a (constrained) energy minimum. The fact that this state is not the global minimum does not appear to affect adversely the computed vibrational energies (vide infra). Results and Discussion Accurate second derivatives are required for characterising potential-energy surface features. For example, a minimum gives all real eigenvalues while a saddle point corresponding to a transition state will have exactly one imaginary eigen- value.At present, computation is the onZy way to obtain detailed transition-state geometries. In addition, vibrational energies derived from the second derivatives can be used to compute zero-point energies and, using statistical mechanics, thermodynamic quantities like enthalpy and the entropy.'* A completely theoretical estimation of reaction energetics requires the accurate estimation of the second derivatives and molecular vibrational energies. The calculated and observed data -for the three complexes are collected in Table 1 along with error estimates in terms of percentage differences. Generally, there is a marginal improvement using the larger basis but the difference between local and non-local functionals is slight (uide infra).Considering the order of the vibrational bands; in each case the theoretical assignment, labelled (1) below, yields energies which increase in the order : E < F, < A, < F, The experimental assignment for [CrO,]' -has been reported by Weinstock et aL2' while Grifith and Gonzalez- Vichez have given assignments for the other two complexes on two separate occasion^.^^^^^ Assignment (1) is suggested for both [MnOJ2- and [FeO,]'- in the earlier However, in the later work,3' an alternative assignment, (2), was proposed for [Fe0,I2- based on the polarisation of lines in the Raman spectrum. F, < E < F2< A, This conflicts with the present DFT results.However, given the similarity of the species and the otherwise good numerical agreement, we suggest a revision back to the original assign- ment (1). The most favourable agreement between computed and observed energies uses the earlier data for [Mn0,12- with J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 the more recent band maxima for [Fe0,I2 -reassigned. Under these conditions, the average deviation among the TZP results is between 3 and 5%, with the largest percentage error on any one band being 7.6% (for the lower F, vibration of [MnO4l2- and with the largest energy difference of 55 cm-' for the A, band of [FeOJ2- at the GC TZP level. The largest percentage deviation for any TZP result is 12% corre-sponding to the highest F, band for [FeO,]'- which, at the GC TZP level is 95 cm-higher than experiment, provided one is prepared to accept assignment (2).If assignment (1) is considered but with any of the reported experimental data, the largest error is 7% corresponding to a deviation of 55 cm-' between observed and calculated (GC TZP) A, vibra-tions for [Fe0,]2- The difference between best and worst cases is, therefore, not too large. The best case comparison with respect to the experimental data is illustrated in Fig. 1. At a detailed level, theory tends to overestimate slightly the energies of the highest two bands, which correspond to M-0 stretching vibration^.^' The LDA is noted to lead to overbinding" which could lead to steeper potential wells and hence to computed energies that are too high.However, the non-local GC results, which should correct the overbinding, always give a slightly increased energy relative to the compa- rable LDA TZP data. Evidently, the depth of the potential well is improved using GC functionals but its shape is vir- tually unchanged. Therefore, the GC results for the two high- energy bands, tend to be in slightly worse agreement with the experimental results. In contrast, the LDA calculations for the lower two bands tend to underestimate slightly the experimental values such that the increase found for the GC results brings the latter into slightly better agreement with the experimental. In summary, the LDA results are slightly better than the GC for the two higher-energy bands and slightly worse for the two lower-energy bands.However, it is clear that the effects of Table 1 Observed and calculated vibrational energies (an-'),percentage deviations [A (%)I and the average magnitude of A (%)(I A (%)I) for tetrahedral [MO,]'-complexes; where multiple sets of experimental data are quoted, corresponding sets of A (%) and I A (%) I are also given' vibrational energylcm -' complex method E A (%) F, A(%) A, A(%) F2(%) A(%) lA(%)[ [CrO,l2 - exp.bLDA, DZP LDA, TZP GC, TZP 349 332 323 330 -7.7 -7.5 -5.4 378 356 356 380 -5.8 -5.8 +0.5 846 836 866 870 -1.2 +2.4+2.8 890 894 912 926 +0.5+2.5+4.0 3.8 4.6 3.2 [MnO,]'- exp.'exp.d LDA, DZP LDA, TXP GC, TZP 325 325 307 311 321 -5.5 -5.5 -4.3 -4.3 -1.2 -1.2 332 328 330 336 353 -0.6 +0.6 + 1.2 +2.4 +6.3 -7.6 812 816 856 844 853 +6.4+4.9 +3.9+3.4+5.1 +4.5 820 862 909 876 889 10.9+5.5+6.8+ 1.6+8.9 +3.1 5.6 4.1 4.1 2.9 5.3 4.1 [FeO,]'- exp.dexp.'Vd exp.f LDA, DZP LDA, TZP GC, TZP 320 340 322 292 304 31 1 -8.8 -14.1 -9.3 -5.0 -10.6 -5.6 -2.8 -8.5 -3.4 340 322 340 3 14 328 341 -7.7 -2.5 -7.7 -3.5 + 1.9 -3.5 +0.3 +5.9 +0.3 778 832 790 810 833 845 +4.1 -2.6 +2.5 +7.1 +0.1 +5.4+8.6+1.6+7.0 800 790 832 864 867 885 +8.0 +9.4 3.9+8.4+9.8+4.2 10.6 12.0+6.4 7.2 7.2 5.9 6.0 5.6 4.7 5.6 7.0 4.3 ~~~ See text for explanation of methods and assignments.Ref. 29. 'Ref. 31. Ref. 30. 'Based on published assignment (2). f Based on assign-ment (1). J. CHEM.SOC. FARADAY TRANS., 1994, VOL. 90 1000 r--------- I10001 I 800 800 600 - 600 400 - 400 200 - 200 0 07-800 0 obs.m 600 -. calc.(LDA, DZP) calc.(LDA, TZP) --m400 calc.(GC, TZP) 200 --0 ?--Fig. 1 Graphical comparison of the observed and calculated vibrational energies (cm-')for [MO,]'-complexes listed in Table 1. M: (a) Cr, (b)Mn, (c)Fe. 400380 i" 360 (a' 360t -340 320 300 1 Cr Mn Fe Cr Mn Fe 920 920 900 900 - 880 880 - 860 860 --;(d) 840 840 820 820 800 800 780 780 Cr Mn Fe Fig. 2 Systematic deviations between observed (m) vibrational energies (cm-')and those calculated at the LDA TZP level (0)for vibrational bands (a) E, (b)F,, (c)A, and (6)F, 3240 changing basis set and functional are all relatively small and of a similar order to the numerical ‘noise’ of around 10-15 cm-’. This range can be considered a rough guide to the theoretical error bars and essentially implies that the LDA DZP, LDA TZP and gradient corrected TZP results are fairly comparable.As found for the dioxodihalides of Cr and Mo,~ harmonic vibrational energies from DFT reproduce the experimental data which are intrinsically anharmonic. Evidently, these systems either obey the harmonic oscillator approximation quite well or there is a fortuitous cancellation of errors in the DFT calculations. Either way, the DFT method appears to give consistently reliable results. Assuming assignment (1) throughout, then, one sees that each band tends to decrease in energy in the order [CrOi-> [Mn0,12-> [FeO,]’-.The comparison between theory (LDA TZP) and experiment is displayed in Fig. 2. The only discrepancy seems to be that the lower- energy F, band observed in [MnO,]’- is anomalously low. Otherwise, there appears to be a relatively systematic discrep- ancy between experiment and the DFT results. However, as noted above, a special procedure was adopted for the Mn anion in order to avoid a ground-state orbital degeneracy. It is possible that this procedure is responsible for the apparent anomaly but, given the agreement found for the other three bands, it seems possible that the experimental data may be suspect. The two low-energy bands of [MnO,]’- are report- ed to be only 3-7 cm-’ apart and presumably overlap.It may be that the exact positions of the band maxima are subject to a larger uncertainty. An increase of only 20 cm-l for the lower F, band would establish a consistent trend. However, in the absence of the actual spectra, the reliability of this assertion cannot be judged. Conclusions Density functional theory calculations of the vibrational energies of [CrO,]’-, [MnO,]’-and [FeO,]’-using double and triple-[ STO basis sets give accurate agreement with experimental results subject to a revision of the assign- ment of [FeO4l2- back to the original proposal. The average error for the larger basis sets using both local and non-local functions is only about 4%, with a maximum average devi- ation of about 7% depending on which experimental data one compares with.Theoretical estimates for the two high- energy stretching modes tend to be slightly too small. These errors are of the same order as the numerical noise, which is around 10-15 cm-’. This study, together with the previous work on these complexes,’ demonstrates that DFT provides a very satisfactory description of the potential-energy surface of these molecules in the vicinity of the ground state. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 The authors acknowledge the support of the Engineering and Physical Sciences Research Council for the provision of a stu- dentship (to P. D. S.) and computer hardware (through the Computational Science Initiative and the Science and Materials Computing Committee).References 1 F. A. Cotton and G. 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