Exploring Chromium (vi) Dioxodihalides Chemistry: Is Density Functional Theory the Most Suitable Tool?

A comparative systematic study of the CrO 2 F 2 compound has been performed using different conventional ab initio methodologies and density functional procedures. Two points have been analyzed: first, the accuracy of results yielded by each method under study, and second, the computational cost required to reach such results. Weighing up both aspects, density functional theory has been found to be more appropriate than the Hartree–Fock ͑HF͒ and the analyzed post-HF methods. Hence, the structural characterization and spectroscopic elucidation of the full CrO 2 X 2 series ͑XϭF,Cl,Br,I͒ has been done at this level of theory. Emphasis has been given to the unknown CrO 2 I 2 species, and specially to the UV/visible spectra of all four compounds. Furthermore, a topological analysis in terms of charge density distributions has revealed why the valence shell electron pair repulsion model fails in predicting the molecular shape of such CrO 2 X 2 complexes.


I. INTRODUCTION
Chromium ͑VI͒ dioxodihalides form a class of versatile oxidizing agents able to deliver oxygen atoms to a great deal of organic groups.These oxotransition metal complexes of chromium in higher oxidation states have been taken as functional chemical models for cytochrome P-450. 1 In particular, the electronic structure of chromium ͑VI͒ dioxodichloride has been studied in order to elucidate how this kind of systems can mimic mixed-function oxygenases of biological importance and in what fashion the oxygen ligands participate in electrophilic reactions. 2Results suggested that the reactivity of CrO 2 Cl 2 involve charge-transfer interactions, concluding that a consistent description of the electrophilic oxygen is obtained based on covalent rather than electrostatic interactions with the substrate.Very recently, Ziegler and Li 3 studied theoretically the methanol oxidation by CrO 2 Cl 2 and the reaction enthalpies involved in the activation of C-H and O-H bonds.They suggested that O-H addition to the Cr-O bond is an important step of such catalytic process.While CrO 2 Cl 2 has attracted most interest of both experimental and theoretical chemists, [4][5][6][7][8] the other dioxodihalides of the same chromium family have received much less attention.Unlike CrO 2 F 2 and CrO 2 Cl 2 the third member of this series is difficult to prepare: 9 even below room temperature, CrO 2 Br 2 is thermally unstable, most of its physico-chemical properties being still unknown.However, it was found to exist as a monomeric species, as revealed by its molecular weight in CCl 4 . 10Finally, CrO 2 I 2 has not been synthesized yet.
2][13][14][15][16][17][18][19][20][21] Thus, whereas the application of mechano-quantum methodology on organic systems has reached a stage not only interpretative but also predictive, 22 transition-metal compounds are more difficult to be described due to four main reasons.First, the quantity of atoms taking part cover practically the whole periodic table.Second, the variety of bonds which are made cannot necessarily be considered as covalent bonds like in organic compounds.Third, for transition-metal systems there is usually more than one possible hybridization scheme.Finally, bonds between metal and nonmetal atoms are difficult to be theoretically treated, due to the many orbitals of the metal having an active role.In general, the energy of these systems suffers from nonsystematic errors, and much more accurate calculations than for organic compounds are required.As far as chromium ͑VI͒ dioxodihalides are concerned, apart from the aforementioned studies on CrO 2 Cl 2 compound, only a theoretical description of the chromium ͑VI͒ dioxodifluoride has been given by Deeth, 23 including optimized geometries and vibrational frequencies.To our knowledge, calculations on the two heavier members of the family ͑XϭBr,I͒ are still missing.
tool in determining the structures and energetics of transition-metal complexes, 29,30 little work exists for UV/vis spectra calculations.0][41] Regarding heavier systems, perhaps the most comprehensive work is that of Sosa et al. 42 who have studied structures and frequencies for a number of first and second-row transition-metal complexes at the local density approximation ͑LDA͒ level.In this approach, HF and LDA results were compared, the latter being in better agreement with experiment.However, similar works on transition metals comparing HF, post-HF, and nonlocal DFT methods are still lacking.
Despite several calculations for CrO 2 F 2 and CrO 2 Cl 2 complexes, 23,42 no comparative studies exist for the complete CrO 2 X 2 series ͑XϭF,Cl,Br,I͒.Thus, the main aim of the present paper is to assess the geometrical, vibrational, and electronic properties of the family of species CrO 2 X 2 .In particular, theoretical studies on the electronic spectra of these compounds are reported for the first time and compared to experimental UV/visible spectra when possible.Special emphasis is placed on CrO 2 Br 2 and CrO 2 I 2 species.
Another purpose of this article consists of searching the optimum theoretical and computational conditions to perform such analyses.Thus, before pursuing our main goal, we have carried out a systematic study on the CrO 2 F 2 compound in order to find the most suitable methodology to be employed, calculating several properties at different levels of theory, and further comparing not only the quality of results but also the computational effort required for each tested methodology.With regards to DFT calculations, several functionals and packages have been used.An interesting goal pursued when comparing different DFT methodologies is to discover the disadvantages and benefits of different available density functionals, in this particular case, for transitionmetal systems.These kind of studies can also help to understand how the calculated properties can be improved with the inclusion of nonlocal corrections.
Finally, our studies are also prompted by the continued interest in a comprehension of why the valence shell electron pair repulsion ͑VSEPR͒ predictions for transition-metal complexes fail in some cases. 43In the present case, a ЄOCrO bond angle is expected to be larger than a ЄXCrX angle ͑Xϭhalogen͒, according to the VSEPR principle stating that double formal bonds require a greater proportion of the coordination sphere around a central atom than single bonds do.Interestingly, our DFT results reproduce the observed ЄOCrOϽЄXCrX order, whereas both the VSEPR model and some HF calculations are in contradiction to experimental evidence.

II. COMPUTATIONAL DETAILS
HF and post-HF ab initio calculations have been carried out with the system of programs GAUSSIAN 92,44 the only exception being complete active space self-consistent field ͑CASSCF͒ calculations, which have been performed by means of the GAMESS program. 45FT calculations have been done with the programs DMol, 46 ADF, 47 and GAUSSIAN 92.Both functionals of the density ͑local͒ and the density gradient ͑nonlocal͒ have been used.On the first category, calculations have been performed within LDA 48 in the parametrization of Vosko, Wilk, and Nusair ͑VWN͒ 49 or, alternatively, in the Hedin-Lundqvist/ Janak-Morruzi-Williams ͑JMW͒ 50 local correlation functional.In the more sophisticated nonlocal density approximation ͑NLDA͒, pure Becke's ͑B͒ 51 or hybrid Becke's 3 parameters' ͑B3P͒ 52 nonlocal correction for the exchange were added, as well as Lee-Yang-Parr ͑LYP͒ 53 or Perdew's ͑P͒ 54 inhomogeneous gradient corrections for correlation.An accurate integration grid has been chosen everywhere.
As regards to the particular features of each program, DMol employs numerical functions for the atomic basis sets, and the various integrals arising from the expression for the energy equation are then evaluated over a grid.The DMol calculations have been done with a double numerical basis set augmented by polarization functions ͑DNP͒.For comparison with traditional molecular orbital methods, DNP can be considered in terms of size as a polarized double-basis set.However, this basis set is of significantly higher quality than a normal molecular orbital double-basis set, because exact numerical solutions for the atom are used. 46Harmonic vibrational frequencies have been evaluated by finite differences of analytic gradients.
6][57][58][59] For the sake of clarity, we have labeled the Gaussian basis sets used in the first part of this paper as shown in Table I. Second derivatives of the energy have been computed analytically.
In the ADF program, the implementation is centered around an optimized numerical integration scheme, extensive use of point group symmetry being made.Basis functions are Slater-type orbitals ͑STOs͒.The Coulomb potential is evaluated through a fitting of the charge density with Slater-type exponential functions centered on the atoms ͑fit functions͒. 60hus, a triple-basis set has been used for the 3s, 3p, 3d, 4s, and 4p orbitals of chromium.For fluorine (2s,2p), chlorine (3s,3p), bromine (3d,4s,4p), and oxygen (2s,2p), doublebasis sets augmented by an extra polarization function have been employed. 61A similar basis set for iodine was not available, so a triple-STO basis extended with a polarization function has been used instead for this halogen.It is unlikely that the choice of a better basis set for iodine could affect the validity of results when comparing trends along the series.Electrons in inner shells have been treated within the frozen core approximation. 62Harmonic vibrational frequencies have been calculated by applying numerical differentiation to the energy gradients.
Spin-restricted calculations have been performed for the ground states and for the lower-lying excited states.The vertical excited-state energies have been estimated through the configuration interaction singles theory ͑CIS͒ 63 for HF calculations, and through the sum method developed by Ziegler et al. 29,64 for DFT calculations.
Electronic analyses of the Laplacian of the density and location of bond critical points ͑BCPs͒ through topological analyses have been done using the program ELECTRA 65 developed in our laboratory.

III. RESULTS AND DISCUSSION
This section is divided in two subsections.It begins by considering a previous systematic study of the CrO 2 F 2 compound in order to elucidate which methodology must be used in terms of computational effort required and quality of results obtained.Once the optimum theoretical conditions are established in Sec.III A, we give place to the proper investigation of the full family of CrO 2 X 2 compounds in Sec.III B, which is actually the main objective of this paper.

A. A systematic study of the CrO 2 F 2 complex
Schematically, results are presented in the following order: First, calculated structural parameters, harmonic frequencies, and excitation energies are compared to available experimental data, the accuracy yielded by each methodology under study being also analyzed.Second, a comparison is made in terms of CPU time required for evaluating the test compound at different levels of theory.Finally, conclusions drawn from both previous points lead to a decision about the computationally most efficient choice.

Molecular structures
The CrO 2 F 2 molecule, which belongs to the C 2v point group, has the atoms located at roughly tetrahedral positions.Many investigations 66,67 have revealed and confirmed a pseudotetrahedral C 2v structure for this molecule, which is not exclusive of CrO 2 F 2 , but is also extensible to the full CrO 2 X 2 series.Hence, only four geometrical parameters are needed to characterize the present compound.Table II gathers the observed geometrical parameters for CrO 2 F 2 , along with those computed at the HF level.All calculated distances ͑which are not mean distances of the vibrational ground state, but equilibrium distances͒ are systematically shorter than the observed bond lengths.Most basis sets yield bond lengths with a deviation у0.05 Å relative to experimental data.The fact that the rather poor basis F gives accidentally a low deviation of 0.015 Å for Cr-O and 0.043 Å for Cr-F is not so interesting ͑basis G gives already a 0.10 Å devia-tion͒.It is important that the HF limit result is ϳ0.10 Å off.
Regarding bond angles, nearly all basis sets can reproduce the experimental relation ЄFCrFϾЄOCrO, the only exception being basis sets A and G.However, it is surprising that the best description corresponds to basis sets C and D, which use pseudopotentials for Cr.Apart from these two basis sets, other all-electron basis sets underestimate the ЄFCrF angle by more than 2.0°, and overestimate the ЄOCrO angle.The comparison of results from different basis sets allows us to conclude with fair confidence that systematic underestimation of the Cr-O and Cr-F distances is due to intrinsic errors of the HF method and not to the use of unbalanced or too small basis sets.Due to computational limitations, subsequent calculations at post-HF level have been carried out through use of the medium-sized basis set G.
The effect of including electron correlation by means of post-HF methodologies can be seen from Table III.Metalligand distances are slightly improved at the MP2 level, but they are now larger than experimental values as a result of correlation effects being overestimated by the MP2 method. 41,68Neither ЄFCrF nor ЄOCrO bond angles are properly described, the deviation being larger than that in Table II.Moreover, the quality of results provided by the CISD calculation only slightly overcomes the quality of HF results.
A more complex method has been used to improve the  geometrical description of CrO 2 F 2 .In particular, the CASSCF values reported in Table III correspond to a calculation with an 11-orbital active space ͓basically the six occupied 2p atomic orbitals ͑AO͒ of the two oxygen atoms and five unoccupied 3d AO of Cr͔ and 12 valence electrons, summing up a total of 15416 configurations.Such an active space yields an accurate description of bond lengths, the error lying below 0.01 Å.However, the computed bond angles remarkably deviate from the observed data, especially as regards to ЄFCrF.This is probably due to the absence of the 2p AO from fluorine ligands when accounting for the make-up of the active space.Such orbitals are more stabilized than the 2p AO of oxygen.To get a more balanced active space one should include the 2p AO from O and F; however, these orbitals cannot be included simultaneously unless a huge active space beyond 11 orbitals is considered, which goes well beyond our currently available computational resources.
An alternative way of considering electron correlation effects is given by the use of DFT.Table IV gathers the optimized geometries for each DFT method and program tested.All basis sets considered are of similar quality.There is a good agreement between observed and calculated geometries.Bond angles and lengths computed by means of the ADF and DMol programs are slightly better than those yielded by GAUSSIAN 92.This is probably due to the kind of basis sets employed.It must be remarked that for GAUSSIAN 92 we have not made use of basis sets especially developed and optimized for DFT calculations.Some time ago, it was found 69 that the shape of valence orbitals for atoms such as Cr differ considerably between HF and LDA results.More recently, 70 the use of LDA-optimized basis sets was recommended for the study of chemical energetics as well as geometries.However, HF-optimized basis sets are quite used in DFT calculations, partly because of the experience accumulated from HF calculations.
Chromium-ligand bond distances reported in Table IV are no longer consistently underestimated as in Table II, DFT errors ranging from negative to positive values.Changing from LDA to GGA functionals leads to a lengthening of all bonds by 0.02 Å.Thus, the addition of gradient terms corrects the roughly 10% LDA underestimation of Pauli's repulsions 71 and, hence, corrects rather short distances between the first-row transition metal and the ligand predicted by LDA. 42An alternative approach to the use of pure functionals refers to the use of functionals which include a mixture of HF exact exchange with DFT exchange correlation functional.For this system, the quality of the geometrical parameters yielded by a hybrid functional ͑B3P͒ is slightly inferior to that obtained from an also nonlocal yet pure functional ͑BP͒.
Comparison of Tables III and IV shows an improvement in DFT results for bond angles, which are now reproduced to within ϳ1°.Interestingly, they are quite insensitive to the choice of the LDA or GGA method. 23In all cases, the ЄOCrO angle is predicted to be smaller than the ЄFCrF angle, in excellent agreement with the experimental results reported by Garner and Mather. 66s a whole, we have found that the molecular structure of the CrO 2 F 2 compound is not correctly described at the HF level.Post-HF methods based on the reference HF wave function do not systematically improve the HF results and can also yield important errors.In this compound, DFT methods offer more accurate geometries.

Harmonic vibrational frequencies
A second stage of this subsection, focuses on a comparative analysis from a vibrational-spectroscopic point of view.For CrO 2 F 2 , there are nine normal modes of vibration; four of stretching ͑͒, two of bending ͑␦͒, two of rocking ͑͒, and one of torsion ͑͒.Symmetry labels corresponding to each normal mode, together with vibrational frequencies and absorption intensities, are collected in Table V.
The average error for the harmonic frequencies computed at the HF level exceeds 23%.Using basis set B, the error is lower than 13.4%, whereas using basis sets A, G, or J the error exceeds 26.7%.However, the absorption intensities corresponding to bending modes predicted by basis set B are slightly too high.All-electron basis sets G and J have the advantage of predicting frequencies which are systematically above those experimentally observed, and furthermore the error is always of the same magnitude.Therefore, applying a correction factor of 0.85-0.95 to harmonic vibrational frequencies 72 it is possible to reproduce experimentally observed fundamentals.The global error for spectroscopic results obtained at a post-HF level is larger than expected.For instance, the MP2 calculation clearly underestimates low frequencies and overestimates high frequencies, yielding an error ϳ37.6%.Such a deviation can be understood keeping in mind that the HF wave function, taken as a reference for the MP2 calculation, already exhibits some deficiencies.On the other hand, har-monic vibrational frequencies computed at CISD level ͑19.7% error͒ are slightly better than HF frequencies computed with the same basis set.This notwithstanding, none of the HF or post-HF methods used manage to yield the correct increasing ordering of the vibrational frequencies.
Examining DFT frequencies from Table VI, all errors fall below 14.2% and, on average, do not exceed 9%.Vibrational frequencies and absorption intensities fit the experimental pattern with better accuracy than previously seen for non-DFT procedures.This is not surprising at all from the structural parameters shown in Table IV, where optimized geometries already reproduced experimental data quite accurately.In general, the most important trends pointed out when analyzing results of Table IV can be seen again in Table VI.Thus, a change of the functional leads to the same conclusions as those mentioned in the previous section.For   instance, the LDA approach exaggerates the bond strength because it underestimates Pauli's repulsions, leading to LDA frequencies higher than nonlocal frequencies, and also higher than the experimentally observed.

Method/ basis set
Among DFT calculations of Table VI, the minimum error ͑2.3%͒ corresponds to a calculation which combines ͑1͒ a pure nonlocal functional and ͑2͒ a TZϩDZP quality basis set ͑the program being ADF͒.In principle, such a good agreement could be attributed to a cancellation of errors or might be considered fortuitous, because the experimentally observed fundamentals are inherently anharmonic, whereas the present calculations correspond to harmonic vibrations.Indeed, without simplifying theoretical calculations through the harmonic approximation, computed frequencies would have been slightly different.Actually, the addition of anharmonic corrections usually has the effect of decreasing the calculated frequencies.The magnitude of this reduction is quite variable, e.g., for CO, it is 27 cm Ϫ1 ͑ CO harm ϭ2170 cm Ϫ1 , CO ϭ2143 cm Ϫ1 ͒. 73 Therefore, should we add anharmonic corrections to calculated frequencies, they would probably become even closer to experimental values. 74

Electronic spectra
An interesting point which has not received much attention in the literature concerns the relative stability of the low-lying electronic states for CrO 2 F 2 .The lowest energy transitions are assigned as (b 1 ,a 2 ,b 2 )→(a 1 ,a 2 ), yielding six singlet and six triplet excited states (A 1 ,A 2 ,2B 1 ,2B 2 ).Excited energies corresponding to vertical transitions for the CrO 2 F 2 species have been computed at the HF, CASSCF͑12,11͒, and DFT levels of theory.We have found that all three methodologies yield that the first singly excited state has B 2 symmetry, which is in agreement with experimental investigations.The lowest singly excited state of CrO 2 F 2 was assigned from rotational analysis as 1 B 2 ͑Ref.75͒ ͑accounting for a change of axes from the rotational sys-tem͒.The small separation between this state and nearby states deriving from the five other promotions can be taken as a good criterion for testing the accuracy of a given methodology.HF predicts a second singly excited state of 1 A 2 symmetry, whereas both correlated CASSCF and DFT energies agree in yielding a 1 B 1 state.Taking into account a previous study of CrO 2 Cl 2 using the symmetry adapted cluster-configuration interaction method, 76 the latter symmetry is likely more reliable than the former.
Another comparison between different levels and conditions of calculation can be made from Table VII, where the energetic difference between the ground state and the first excited state computed by each methodology is analyzed.The experimental gap is reported to be 2.6 eV; 77 our post-HF calculations yield a reasonable energy of 2.74 eV, whereas HF yields 4.5 eV, clearly overestimating the gap, and DFT yields an intermediate energy of 3.23 eV.HF results remarkably differ from experiment.DFT energies fall within the expected error range for this kind of determination ͑about Ϯ0.5 eV͒.Noticeably, better results are obtained when nonlocal functionals are used.Foresman et al. 78 reported that the effect of adding polarization functions decreases the accuracy of adiabatic and vertical transition energies while increasing the accuracy of excited state potential energy surfaces.Results from Table VII show that, here, omission of a polarization function changes the excitation energy only marginally.The best predictions are yielded by CASSCF.
The general improvement observed when changing from HF to CASSCF is mainly due to the favorable composition of the active space.As mentioned above, such a composition causes a correct structural description of the ЄOCrO angle ͑but not the ЄFCrF angle͒.It also accounts for the good agreement found with the experimental energy of the HOMO-LUMO transition, because in our CASSCF͑12,11͒ calculation the chosen outer orbitals of chromium and oxygen are precisely those directly related to this transition.Moreover, the two lowest CASSCF energies ͑2.74 and 2.90 eV͒ are accurate enough to suggest that the experimentally reported band I of CrO 2 F 2 electronic spectrum, which was found to be approximately centered at 2.6 eV, 77 can be actually assigned to the six transitions derived from the parent 1t 1 →2e transition in CrO 4 2Ϫ ͑vide infra͒.Our DFT calculations locate this band at a slightly higher energy ͑3.23 and 3.35 eV͒.The deviation is attributed to the tendency of DFT to yield electronic transition energies above experimental data. 29

Computational cost
So far, only the quality of results has been analyzed.Above discussions have dealt with the accuracy yielded by each method, highlighting DFT and CASSCF as the better candidates to perform our analysis of CrO 2 X 2 species.Another important aspect refers to the computational effort spent on reaching such accuracy.
Inclusion of electron correlation effects in computational chemistry is expensive.One of the most remarkable features of DFT is said to be its low cost in relation to post-HF methods. 29We have compared the CPU times required to evaluate the energy and gradient for the particular case of CrO 2 F 2 by means of different methodologies using the 88function basis set G. All calculations have been done under the same conditions, identical basis set and direct SCF ͑i.e., recalculating the integrals in each SCF cycle͒, irrespective of the program.Tests have been run on an IBM Risc/6000-355 computer.Table VIII gathers the relative times needed to make a single-point calculation.We have focused our attention to the calculation of energy plus gradients because they are essential to optimize geometries, and further to determine vibrational frequencies.For the sake of clarity, times have been scaled to time spent at HF level.The reported values do not correspond to a unique SCF cycle, but to the total time spent to reach self-consistency.
From non-DFT results, it can be deduced that the time increases gradually and proportionally to the complexity of the calculation.Looking at the first column, DFT methods are about 6 times more expensive than HF, and in turn the CI/CASSCF are 15-20 times more expensive than DFT.From the percentages shown, the time spent on calculating the gradient is large for MP2, medium for SCF and CI, and small for the other methods.The percentage of time needed to compute the energy with respect to energy plus gradients through the CASSCF method is the largest because this method optimizes at the same time the MO and CI coefficients.As seen in the first column, the MP2 calculations take as much time as DFT does.Noteworthy, the comparison between local and nonlocal DFT results leads to the conclusion that including gradient corrections to the functionals makes the calculations slightly more expensive, but the improvement of the quality of results largely compensates for this small decrease on the computational speed as a result of adding nonlocal corrections.On the contrary, the CASSCF ͑12,11͒ calculations become too expensive but do not yield the quality offered by DFT for the compound studied here, at least regarding geometries.By increasing the active space one could finally reach the desired accuracy, but then the CPU time would hugely increase as well.Therefore, DFT is an efficient methodology for the CrO 2 F 2 compound, in addition to being accurate enough.
Calculations presented in the next part of this article have been performed through nonlocal DFT.In particular, among the more reliable conditions of calculation, the BP functional and the TZϩDZP quality basis set as implemented in the ADF program have been selected.

B. The family of compounds CrO 2 X 2 (X‫؍‬F,Cl,Br,I)
The goal of this section is the proper characterization ͑both structural and spectroscopic͒ of the full CrO 2 X 2 series of molecules.We begin with a detailed examination of molecular geometries and vibrational frequencies.After that, the UV/vis spectra are analyzed.For some of these chromium compounds, there is still a lack of experimental data, and hence our corresponding theoretical results are useful as predictions.Finally, attention is focused to the rationalization of the reported geometrical parameters for CrO 2 X 2 , in contrast to SO 2 F 2 , by means of Bader analyses based on the charge density.

Molecular structures
As seen in Table IX, the agreement between calculated and observed parameters ͑available just for XϭF 67 and XϭCl 4 ͒ is satisfactory for both bond angles and bond lengths, the only exception being the calculated ЄClCrCl angle, which deviates by 2.7°.On the whole, the average error for bond distances of this two species is inferior to 0.008 Å, and for bond angles is smaller than 3°.Regarding XϭBr,I, despite the lack on experimental references, it is found that the Cr-X distances follow the expected trend of increasing with an increase on the halogen atomic volume.About Cr-O distances, changing the halogen from F to I has a very minor effect.

Harmonic vibrational frequencies
As far as experimental vibrational spectroscopic data are concerned, they have been reported only for the two lightest members. 9,79Several assignments have also been made for CrO 2 Br 2 , 9 reporting frequencies corresponding to the higher energetic region of the IR spectrum.No experimental data for CrO 2 I 2 are available yet.
Results from Table X can be analyzed by columns or by rows.First, it is useful to consider metal-ligand bonds as diatomic molecules, described by the harmonic oscillator model.Under this consideration, two parameters are responsible for the behavior of the frequencies along the halogen series: the mass and the force constant.It is found that the heavier the halogen, the lower the vibrational frequency.For instance, the frequencies related to the normal mode ͑MX 2 ͒ B 2 gradually decrease from F to I.This is precisely the expected behavior according to the halogen masses.For the same reason, it is not surprising that the symmetric and an- tisymmetric Cr-O stretching vibrations ͑columns 8 and 9, respectively͒ appear at frequencies higher than those of the two analogous Cr-X vibrations ͑columns 6 and 7͒, because all halogen atoms are heavier than oxygen atom.When in a given vibration the halogen atom plays an unimportant role, then the force constant becomes quite determining.In such cases, the ordering of frequencies is less clear than before.Thus, for vibration ͑MO 2 ͒ A 1 , the theoretical frequency of CrO 2 F 2 should be higher than for CrO 2 Cl 2 , yet it is not.This disagreement can be attributed to a failure of the structural parameters ͑Table IX͒, where the computed distance Cr-O is already larger in CrO 2 F 2 than in CrO 2 Cl 2 , in disagreement with experiment.
If the analysis is now performed by rows, the mean deviations for CrO 2 F 2 and for CrO 2 Cl 2 are 10 and 13 cm Ϫ1 , respectively.In both cases, it leads to an average error of Ͻ3%.However, the sign of errors is not constant, so it is difficult to predict whether the experimental values for XϭBr,I will appear above or below our computed frequencies.Examining Table X, only a hint is provided: for higher frequencies the calculated values exceed those experimentally observed, whereas for low frequencies, calculations underestimate experimental values.
As far as absorption intensities are concerned, there is a good agreement between theory and available experimental data. 9,79The derivative of the dipole moment with respect to the normal coordinate decreases when going from F to I, due to the decreasing charge separation in the Cr-X bond along the series.Thus, for a given Cr-X stretching frequency, the variation of the intensity along the series fits the expected change in the dipole moment with vibration.In CrO 2 I 2 , only the three highest vibrational frequencies are estimated to be very strong.On the basis of the known compounds, our theoretical predictions for CrO 2 I 2 ͑and partially for CrO 2 Br 2 ͒ spectra are reasonable approximations, awaiting for future experimental confirmation.

Electronic spectra
Since the CrO 2 F 2 system possesses an approximately tetrahedral configuration, there is a certain similarity between spectroscopic results obtained for this species and those of the isoelectronic molecules CrO 4 2Ϫ or MnO 4 Ϫ , belonging to T d point group.Thus, one can study the electronic spectra of CrO 2 X 2 taking as reference CrO 4 2Ϫ and then perturbing the electronic structure by changing one or more of the oxygen ligands into halogen atoms ͑CrO 4 2Ϫ →CrO 3 X Ϫ →CrO 2 X 2 ͒. 7 However, there is some difficulty in establishing a direct correspondence between the MOs of these molecules 76 because the reduction of the symmetry (T d →C 3v →C 2v ) causes a mixture of the orbitals.
Figure 1 depicts a diagram of the energetic levels for the highest occupied orbitals and the lowest unoccupied orbitals for the ground state of CrO 4 2Ϫ (T d ), CrO 3 F Ϫ (C 3v ), and CrO 2 F 2 (C 2v ).On the basis of CrO 4 2Ϫ , the highest filled orbitals of CrO 2 F 2 derive from the correlation t 1 (T d )→a 2 ϩb 1 ϩb 2 (C 2v ), whereas the two lowest empty orbitals have symmetries a 1 and a 2 with the former being lower lying, and follow the correlation e(T d )→a 1 ϩa 2 (C 2v ).The lowest experimental energy transitions are 1t 1 →2e for CrO 4 2Ϫ ͑3.3 eV͒, 80 1a 2 →9e for CrO 3 F Ϫ ͑2.8 eV͒, 81 and 7b 2 →14a 1 for CrO 2 F 2 ͑2.6 eV͒. 77able XI shows the distributions of electronic charge for the ground-state frontier orbitals of CrO 2 F 2 .As far as the numeration of the orbitals is concerned, it is important to note that, for instance, the orbital we have labeled 5b 2 corresponds to the orbital reported as 7b 2 in the literature. 7This TABLE X. Vibrational frequencies ͑in cm Ϫ1 ͒ and absorption intensities a in parentheses ͑in km/mol͒ corresponding to the nine normal modes of CrO 2 X 2 compounds ͑XϭF, Cl, Br, I͒, together with their labels of symmetry.is due to the omission of core orbitals in our numeration.All indices are, then, shifted.

Species
The highest occupied orbitals derive from the correlation 1t 1 (T d )→2a 2 ϩ5b 1 ϩ5b 2 (C 2v ), having a relative energy ordering 5b 2 Ͼ2a 2 Ͼ5b 1 and only slight chromium character.Among these, the 5b 2 has the greatest oxygen amplitude and the smallest halogen amplitude, so that the highest occupied orbital is still mainly an oxygen lone pair.The two lowest unfilled orbitals belong to the species of symmetry a 1 and a 2 , with the former being lower lying, and are mainly of chromium 3d character.This follows again from the correlation e(T d )→a 1 ϩa 2 (C 2v ).The 9a 1 orbital has greater chromium and fluorine character and smaller oxygen character than the 3a 2 orbital.Our relative contributions to each orbital shown in Table XI are in good agreement with the calculations previously reported by Miller, Tinti, and Case 7 using the X␣-scattered wave method.
From an experimental point of view, most of the electronic spectroscopy information is focused on CrO 2 Cl 2 , and secondarily, on CrO 2 F 2 .͑The latter has partially been dealt with in Sec.III A 3͒.In the same way, the three orbitals below the dashed line arise from the 1t 1 (T d )→2a 1 ϩ5b 1 ϩ5b 2 (C 2v ) correlation, but unlike in CrO 2 F 2 , they do not longer have a predominant character of oxygen; the halogen contribution increases now from Cl to I, exceeding the oxygen contribution.
The six excitations derived from the parent 1t 1 →2e transition have been collected in Table XIII.Among these excitations, the most interesting transition corresponds to that involving a lower energy, namely, the 5b 2 →9a 1 for CrO 2 Cl 2 , the 7b 2 →12a 1 for CrO 2 Br 2 , and the 4a 2 →12a 1 for CrO 2 I 2 .Noticeably, for XϭF,Cl,I such a transition takes place between HOMO and LUMO orbitals, as expected, but not for XϭBr, where the excitation of an electron from the 7b 2 orbital leads to a slightly lower energy ͑2.42 eV͒ than exciting an electron from the HOMO 4a 2 orbital ͑2.50 eV͒.
Analyzing the calculated HOMO-LUMO transitions along the series, it is found that they gradually decrease from F to I ͑3.23Ͼ2.69Ͼ2.50Ͼ1.92͒,which is correlated to the established order in the electrochemical series.The heavier the halogen atom, the less separated are the frontier orbitals in the molecule ͑Table XII͒.This is in agreement with the principle of maximum hardness 82 which states that stability of chemical systems increases with larger HOMO-LUMO differences; soft molecules have a small energy gap, and hard molecules, a large gap.Therefore, our spectroscopic results confirm how unstable the CrO 2 Br 2 compound is, and predict an even more difficult synthesis of CrO 2 I 2 .
The most detailed study on the experimental electronic spectrum of CrO 2 Cl 2 was reported for the species in the gas phase, 77 revealing a lowest excitation energy of 2.4 eV.Our computed gap for the HOMO-LUMO transition in the   83 Charge transfer bands in the visible region of the spectrum are possible when ligands have nonshared electron pairs of high energy ͑like the oxygen or the halogen atoms here͒ and the metal has low empty orbitals.Although we have not computed their oscillator strengths, the studied excitation energies correspond to dipole-allowed transitions, so the calculated excitation energies for CrO 2 X 2 show that these species absorb in the spectral region ranging 450-650 nm, and suggest that they are likely to be colored compounds in the gas phase ͑from an orange CrO 2 F 2 to purple-reddish CrO 2 Br 2 and CrO 2 I 2 ͒.

Rationalization of bond angles by means of Bader analyses
Unlike SO 2 F 2 , the structure of CrO 2 X 2 compounds does not agree with the VSEPR theoretical predictions.The expected trend for bond angles on the basis of this model is ЄXCrXϽЄOCrO.However, both experiment and calculations disagree with VSEPR and predict an opposite behavior for the four chromium ͑VI͒ dioxodihalides ͑see Table IX͒.In order to clarify this disagreement, a topological analysis of the electronic distribution and a study of charges has been performed for CrO 2 F 2 and SO 2 F 2 at the DFT level using basis set G. For the latter system, the ЄFSF and ЄOSO bond angles are 94.9°and 125.0°, respectively, so the bond angle ordering is ЄFSFϽЄOSO, as predicted by the VSEPR model.
Electronic analyses of the density 84,85 have become a widespread technique to analyze interactions between atoms in molecules.This kind of analysis is based on the study of the topological properties of electron density distributions and its derivatives ͑gradient vector and Laplacian͒.The socalled bond critical points ͑BCPs͒ exhibit a minimum value of electron density in a path connecting two nuclei but have an electron density maximum in a direction orthogonal to such a path.The BCPs of CrO 2 F 2 and SO 2 F 2 are collected in Table XIV, together with the Laplacian of the electron den-sity ٌ͑ 2 BCP ͒, and the ratio of the perpendicular contractions of the density to its parallel expansion ͉͑ 1 ͉/ 3 ͒, which also provide important information on the nature of the chemical interaction between atoms. 84Thus, a representation of ٌ 2 BCP exhibits spherical nodes in an atom ͑values of the radius for which ٌ 2 BCP ϭ0͒, their number being related to the atomic shell structure.Negative values denote regions where electron density is locally concentrated while positive values involve regions where electron density is depleted.
The nature of the bonds in CrO 2 F 2 and SO 2 F 2 can be discussed from Table XIV.As far as Cr-O and S-O bonds are concerned, the kind of interaction is basically the same: a closed-shell interaction.Thus, in both cases ٌ 2 BCP Ͼ1, and ͉ 1 ͉/ 3 Ͻ0.25.Values are quite similar irrespective of the central atom being Cr or S. On the other hand, when comparing Cr-F and S-F bonds, different types of interaction are found.The former can be classified as an ionic bond, while the latter exhibits the characteristics of covalent interactions.Two effects lead to this latter conclusion: First, the S-F bond has a negative value of the Laplacian at the BCP, indicating that there is a considerable amount of electron density at the BCP between S and F, whereas ٌ 2 BCP for Cr-F is positive.Second, ͉ 1 ͉/ 3 for S-F is just slightly lower than one, whereas for Cr-F it is about four times smaller, showing that the chemical interactions are intermediate for S-F, yet closed-shell for Cr-F.
Mulliken charge distribution analyses of both compounds confirm the ionic nature of the Cr-F bond.The charge separations in CrO 2 F 2 are especially outstanding ͑Cr ϩ2.20, O Ϫ0.59, F Ϫ0.51͒, whereas in SO 2 F 2 become less pronounced ͑S ϩ1.17, O Ϫ0.37, F Ϫ0.22͒, leading to the conclusion that the repulsions between nonshared electron pairs of ligands when the central atom is a transition metal are basic to determine the stereochemistry of CrO 2 F 2 ; on the contrary, they are not taken into account by the VSEPR theory.Our reasons agree with the assumptions made by Garner and Mather 66 from a study of charges.Consequently, it is suggested that the unexpectedly large ЄFCrF bond angle is partially caused by strong electrostatic repulsions between fluorine atoms, whose high concentrated density comes from a poorer ability of chromium d orbitals to accommodate it, and from the influence of having a less electronegative Cr than S as the central atom.

IV. CONCLUSIONS
Chromium ͑VI͒ dioxodihalides have been theoretically characterized from both the structural and spectroscopic ͑IR and UV/vis͒ points of view.Computed geometries, vibrational frequencies and excitation energies agree fairly well with experiment for CrO 2 F 2 and CrO 2 Cl 2 , whose previous investigations have been taken as reference.Thus, computed bond lengths and angles reproduce observed data within 0.008 Å and 3°, respectively; harmonic IR frequencies deviate by an average error of only 3%, and excitation energies corresponding to the visible spectrum fall within the expected margin of error for such a kind of determination.However, the most interesting conclusions concern the unknown CrO 2 Br 2 and CrO 2 I 2 species, for which predictions are made.Thus, for instance, the Cr-I bond distance is expected to be about 2.5 Å, and the computed lowest excitation energy of CrO 2 Br 2 ͑2.42 eV͒ and CrO 2 I 2 ͑1.92 eV͒ suggests that the latter compound will be more reactive ͑less stable͒ than the former.
We have also performed a comparative systematic study on the CrO 2 F 2 compound, using different methodologies, in order to select the most appropriate computational way of carrying such an investigation.After a first stage, CASSCF͑12,11͒ and DFT procedures have revealed themselves as the most suitable methods, according to the accuracy of results.A second aspect to be treated has concerned computational effort and requirements.Thus, it has been analyzed which efficiency each level of theory could offer.A compromise between both analyses has recommended use of DFT, since it showed to be accurate enough and simultaneously much more cost effective for studying properties of a medium-sized molecule like CrO 2 F 2 , and, by analogy, suitable for the full CrO 2 X 2 series.
Finally, as an interesting outset of the structural calculations, we have also investigated why the VSEPR theory is unable to reproduce correctly the molecular geometries of these species.The contradiction between the VSEPR model, computational predictions and experimental evidence has been elucidated on the basis of electronic analyses, comparing CrO 2 F 2 and SO 2 F 2 molecules.Values of the density at the bond critical points, together with information brought about by the Laplacian of the density, allow to conclude that the failure of the VSEPR theory in CrO 2 X 2 is mainly due to the ionic nature of the Cr-F bond in CrO 2 F 2 , as compared to the intermediate character of the S-F bond in SO 2 F 2 , resulting in relevant coulombic repulsions between fluoride ligands.

FIG. 1 .
FIG. 1. Energy level diagram for the higher occupied and lower unoccupied orbitals in the ground states of CrO 4 2Ϫ (T d ), CrO 3 X Ϫ (C 3v ), and CrO 2 X 2 (C 2v ).The three highest filled orbitals of b 2 , a 2 , and b 1 symmetry in CrO 2 X 2 have the order shown for XϭF,Cl but they follow the ordering a 2 Ͼb 1 Ͼb 2 for XϭBr,I.

TABLE I .
Basis sets used for calculations on CrO 2 F 2 using the GAUSSIAN 92 program.

TABLE II .
Geometrical parameters for CrO 2 F 2 calculated at ab initio HF level.Bond distances in Å and angles in degrees.

TABLE III .
Geometrical parameters for CrO 2 F 2 calculated at post-HF level with basis set labeled G. Bond distances in Å and angles in degrees.

TABLE IV .
Geometrical parameters for CrO 2 F 2 computed at DFT level.Bond distances in Å and angles in degrees.
a From Ref. 67.

TABLE V .
Vibrational frequencies ͑in cm Ϫ1 ͒ and absorption intensities a in parentheses ͑in km/mol͒ corresponding to the nine normal modes of CrO 2 F 2 , computed at ab initio level.

TABLE VI .
Vibrational frequencies ͑in cm Ϫ1 ͒ and absorption intensities a in parentheses ͑in km/mol͒ corresponding to the nine normal modes of CrO 2 F 2 computed at DFT level.

TABLE VII .
Comparison of the calculated and experimental energies ͑in eV͒ for the lowest energy transition in CrO 2 F 2 .
a From Ref.77.

TABLE VIII .
Comparison of CPU times a for CrO 2 F 2 system computed through different methods.The time spent at HF level is taken as reference.

TABLE IX .
Geometrical parameters for CrO 2 X 2 compounds.Bond distances in Å and bond angles in degrees.
Table XII gathers the orbital energies for the ground-state of CrO 2 F 2 , CrO 2 Cl 2 , CrO 2 Br 2 , and CrO 2 I 2 .The two orbitals above the dashed line of the table derive from the e(T d →a 1 ϩa 2 (C 2v ) correlation, as in CrO 2 F 2 .

TABLE XI .
Electronic charge distributions ͑in a.u.͒ for the ground-state orbitals of CrO 2 F 2 that take part in the studied excitations.

TABLE XII .
Ground-state orbital energies ͑in a.u.͒ for CrO 2 X 2 ͑XϭF, Cl, Br, I͒.CrO 2 Cl 2 overestimates the observed value by 0.3 eV.Regarding CrO 2 F 2 , the deviation error sign is also positive.Thus, the predicted values of 2.50 and 1.92 eV for CrO 2 Br 2 and CrO 2 I 2 , respectively, must be taken with caution; experimental evidence will probably reveal, when available, slightly lower energies.CrO 2 F 2 is reported to be a volatile violet-red solid, subliming at 29.6 °C to give an orange vapor.

TABLE XIII .
Excitation energies ͑in eV͒ for the six lower transitions of CrO 2 X 2 ͑XϭF, Cl, Br, I͒.Experimental value in Ref. 77 is 2.4 eV.TABLE XIV.Electronic analyses of the density a for CrO 2 F 2 and SO 2 F 2 .BCP is the electron density at the BCP ͑in a.u.͒ and ٌ 2 is the Laplacian of the electron density at the BCP ͑in a.u.͒.The value of the ratio between the perpendicular and the parallel curvatures ͉͑ 1 ͉/ 3 ͒ b is also given.
b a The density function was obtained at BP/3-21G* level using the GAUSSIAN 92 program.b 1 and 3 are eigenvalues of the Hessian matrix at the BCP.