Molecular and Vibrational Structure of Tetroxo d0 Metal Complexes in

Article pubs.acs.org/JPCA

Molecular and Vibrational Structure of Tetroxo d0 Metal Complexes in their Excited States. A Study Based on Time-Dependent Density Functional Calculations and Franck−Condon Theory Linta Jose, Michael Seth, and Tom Ziegler* Department of Chemistry, University of Calgary, University Drive 2500, Calgary AB T2N-1N4, Canada S Supporting Information *

ABSTRACT: We have applied time dependent density functional theory to study excited state structures of the tetroxo d0 transition metal complexes MnO4−, TcO4−, RuO4, and OsO4. The excited state geometry optimization was based on a newly implemented scheme [Seth et al. Theor. Chem. Acc. 2011, 129, 331]. The first excited state has a C3v geometry for all investigated complexes and is due to a “charge transfer” transition from the oxygen based HOMO to the metal based LUMO. The second excited state can uniformly be characterized by “charge transfer” from the oxygen HOMO-1 to the metal LUMO with a D2d geometry for TcO4−, RuO4, and OsO4 and two C2v geometries for MnO4−. It is finally found that the third excited state of MnO4− representing the HOMO to metal based LUMO+1 orbital transition has a D2d geometry. On the basis of the calculated excited state structures and vibrational modes, the Franck−Condon method was used to simulate the vibronic structure of the absorption spectra for the tetroxo d0 transition metal complexes. The Franck−Condon scheme seems to reproduce the salient features of the experimental spectra as well as the simulated vibronic structure for MnO4− generated from an alternative scheme [Neugebauer J. J. Phys. Chem. A 2005, 109, 1168] that does not apply the Franck−Condon approximation.

1. INTRODUCTION The electronic structure of tetroxo d0 transition metal complexes has been the subject of numerous studies over the past 70 years. These studies include the recording of low temperature absorption spectra for MnO4−, TcO4−, RuO4 and OsO4,1−7 starting with the investigation of permanganate by Teltow.1,2 In tandem, series of computational investigations based on Hartree−Fock (HF),8−12 post HF,13−17 regular Kohn−Sham density functional theory (ΔSCF),18−20 and time dependent density functional theory (TD-DFT)21−23 are now available in the literature. The many computational studies reveal that a theoretical description of the tetroxo complexes is more difficult and challenging than one might expect based on their modest size and high symmetry.24 Nevertheless, consensus24b has been reached as to the assignment of the electronic spectra for the tetroxo complexes with the possible exception of permanganate. The latter complex is exceptionally difficult to describe,23,24 even in the ground state.24a The objective of the current investigation is to extend the excited state study on tetroxo complexes from a mere characterization of the orbital composition of these states to a systematic description of their potential energy surface. To this end, we optimize the excited state geometries for a number of tetroxo d0 transition metal complexes (MnO4−, TcO4−, RuO4, and OsO4) and compare their distorted structures to the perfect tetrahedral (Td) ground state geometry. We provide as well a full vibrational analysis of the normal modes and the © 2012 American Chemical Society

corresponding frequencies for each of the optimized excited state structures. The vibrational analysis is finally used in conjunction with the Franck−Condon approximation to simulate the vibrational fine structure present in the experimental absorptizon spectra of the investigated complexes. Our simulations are compared to the experimental spectra available in the literature1−7 as well as results based on previous theoretical studies.23 A study of the type proposed here has not been possible until the recent development of TD-DFT based methods that allow for an automated and efficient determination of excited state energy gradients and Hessians.25 However, a recent paper by Neugebauer et al.23 have used an elegant vibronic coupling scheme to simulate the vibrational fine structure in the absorption spectrum of permanganate. We shall compare our findings to the work by Neugebauer et al.23 for MnO4‑ .

2. COMPUTATIONAL DETAILS All the calculations were carried out using the ADF 201026 program. We employed in all cases a valence triple-ζ basis set with two added polarization (TZ2P) functions. The density functional adopted was VWN based on the local density approximation (LDA).27 The introduction of the generalized Received: December 20, 2011 Revised: January 25, 2012 Published: January 26, 2012 1864

dx.doi.org/10.1021/jp212292j | J. Phys. Chem. A 2012, 116, 1864−1876

The Journal of Physical Chemistry A

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gradient approximation (GGA) does not significantly change the calculated excitation energies20−24 neither did hybrid functionals. However, LDA geometries are on the average in somewhat better agreement with the experiment compared to GGA structures as demonstrated in ref 20. All excited state calculations were based on time dependent DFT (TD-DFT).28 The scalar relativistic effects were included through the use of the zeroth order regularized approximation (ZORA).29,30 The overall accuracy parameter (ACINT) for any numerical integration was chosen to be 6. This should ensure that matrix elements calculated by numerical accuracy are correct to 6 digits. The default value for ACINT is 4. The ground state geometries are optimized by standard procedures. Excited state structures are optimized with the excited state geometry optimization method25 recently implemented into the ADF 201026 program. The excited state vibrational frequencies were evaluated by a numerical differentiation of the energy gradients.25 The vibrational fine structure present in the electronic spectra is simulated based on the Franck−Condon approach.31−33 A program (‘fcf ’) utilizing the FC method has been written34−36 and implemented into the Amsterdam Density Functional Package (ADF).26 The ‘fcf ’ package is an auxiliary program, which calculates the Franck−Condon factors from two vibrational mode calculations carried out for two different electronic states of the same molecule.26 This program yields stick spectra which can be replaced with Lorentzian or Gaussian functions to simulate the broadened spectrum. The broadened spectra were simulated by replacing each stick with energy E = Ei and intensity Ii with a Lorentizian functions I(E) = Ii

Figure 1. Molecular orbitals and energy levels for MO4n−.

Ballhausen3 published the first well-resolved experimental spectrum in 1967 based on measurements at liquid hydrogen and helium temperatures in which KMnO4 was dissolved in a sample of KClO4. Their findings have later been confirmed by other experimental investigations. Holt and Ballhausen3 recorded four distinct bands. At lowest energy is a strong band (i) with an origin at 18 072 cm−1 followed by (ii) a completely featureless band between 25 000 and 29 000 cm−1 and (iii) a strong band with maximum at 33 000 cm−1. Finally, a completely featureless band (iv) with maximum intensity at 43 500 cm−1 is observed at higher energy. In our work, the ground state of MnO4− is optimized assuming tetrahedral symmetry. Table 1 lists the calculated vertical excitation

γ (Ei − E)2 + γ2

(1)

Here γ is the parameter specifying the half-width. We used a halfwidth of 60 cm−1 in our simulated spectra unless otherwise stated. Here Ii is evaluated in the fcf- program based on the calculated oscillatory strength and the estimated Franck−Condon factors.

3. RESULTS AND DISCUSSION 3.1.1. Permanganate. Electronic Structure of MnO4− in its Ground and Excited states. Ground State Electronic Structure of MnO4−. In MnO4− and other tetroxo d0 complexes, the nd, (n + 1)s, and (n + 1)p atomic orbitals of the central metal atom and the 2p atomic orbitals of the oxygen ligands are primarily involved in the molecular orbitals that are responsible for the M−O bonding. The 24 valence electrons in the tetrahedral MO4n− (n = 0,1) species of Td symmetry are placed in the 1t2, 1e, 1a, 2t2, and 1t1 levels as shown in Figure 1.20−24 The highest fully occupied level (1t1) is represented by an Mn−O nonbonding molecular orbital localized entirely on the oxygen ligands and exhibiting some O−O antibonding interactions. The 2t2 orbital is also Mn−O nonbonding and is localized on the oxygen ligands with some O−O bonding interactions. The 1a orbital is made of 2pσ on oxygen overlapping in-phase with 4s on the metal. The 1t2 and 1e orbitals are strongly Mn−O bonding combinations between nd and 2p The empty 2e and 3t2 orbitals represent the corresponding antibonding combinations. The 3t2 is placed above 2e, Figure 1, in line with ligand field theory as the antibonding metal−oxygen overlaps are numerically larger in 2e than in 3t2. Electronic Spectrum of MnO4− and Its Assignment. MnO4− has served as a yardstick in the study of the electronic spectra of tetrahedral tetroxo-complexes.19,20,37−43 Holt and

Table 1. Excitation Energies and Oscillator Strengths for the First Three Transitions of MnO4‑ excitation energies (ΔE) and oscillatory strengths ( f)a state

verb ΔEcal

00 c ΔEcal

fcal

ver d ΔEexp

00 e ΔEexp

fexpi

1T2 2T2 3T2

3.11 4.16 5.12

2.81f 3.65g 4.59h

0.0085 0.0012 0.0104

2.34 3.5 4.09

2.23 3.09 3.6

0.03 0.02 0.04

a

Excitation energies in eV. bCalculated vertical excitation energies. Calculated 0−0 excitation energies. dThe experimental vertical excitation energies were taken as the band maxima in ref 3. eThe 0−0 experimental excitation energies were taken as the band onsets in ref 3. fWith respect to optimized C3v structure 1. gWith respect to C2v structure 2. hWith respect to D2d structure 4. iReference 38b. c

energies and oscillator strengths for the first three transitions of MnO4‑ together with the experimental data.3 The agreement between theory and experiment for the vertical excitation energies is only qualitative even after making allowance for some uncertainties introduced by determining the experimental vertical excitation energies from the observed band maxima.23 Nevertheless, our findings are in line with other TDDFT calculations using different functionals.21,22a,b,23 The experimental spectrum exhibits an overlap between the second and third bands, which is not reproduced by our 1865



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0.30 [Td(2T2)], and 0.49 eV [Td(3T2)] in comparison to the ver . Nevertheless, corresponding vertical excitation energies Δcal the optimized excited state Td structures do not represent stable minimum points on the potential energy surface as the three geometries have five imaginary frequencies corresponding to normal stretching (QT2) and bending (QE) modes of T2 and E symmetry, respectively43 (see Figure S1 and Table S1 in the Supporting Information). It is thus clear that MnO4− in the three excited states must distort from Td symmetry along the QT2 or QE normal modes to reach a stable equilibrium point. Geometries of the first three excited states of MnO4‑ were optimized using the excited state optimization routine available25e in ADF 2010. The excited state structures for MnO4‑ are given in Figure 2.

calculations, where the two transition energies are further apart. The energy difference of 0.3 eV observed in the experimental spectrum3 between the two overlapping states is of the same order of magnitude as the general error in TD-DFT calculations on the same or similar systems using LDA, GGA, or hybrid functionals.24b There have been several attempts8−23 to assign the experimental bands to specific orbital transitions. All calculations agree on the assignment of the first band to the HOMO → LUMO (1t1 → 2e) transition. However, there has been some controversy about the assignment of the second (ii) and third (iii) band. In their experimental study, Holt and Ballhausen3 (HB) assigned the third band to the 2t2 →2e transition and the second to the 1t1 → 3t2 excitation. In 1971 this assignment was reversed to (ii) (2t2 → 2e) and (iii) (1t1 → 3t2) by calculations employing the multiple scattering Xα method,18 a forerunner to today’s DFT schemes. There have been several post-HF studies9,13−15 reported in the literature. Symmetry adapted cluster configuration interaction (SAC+CI)13 and similarity transformed equation-of-motion (STEOM)15 calculations supported the HB-assignment whereas Hartree−Fock plus singles and doubles configuration interaction (HF+SDCI)9 and equation-of-motion coupled cluster singles-and-doubles (EOM-CCSD)14 calculations were in agreement with the Xα findings. A series of studies based on DFT (ΔSCF) employing various functionals is also available in the literature.19,20 All of the ΔSCF DFT calculations support the Xα assignment. Finally there has been a recent TD-DFT study by Neugebauer et al.23 In this investigation, the authors simulated the vibronic fine structure of the absorption spectra using TD-DFT in combination with a vibronic coupling method that do not rely on the Franck−Condon approximation as in the present case. Table 2 lists all of the orbital transitions that contribute to the first, second, and third bands in the absorption spectrum of Table 2. Excited States and the Corresponding Contributions from Orbital Transitions for MnO4‑ state

contributing one-electron transitions

1T2 2T2 3T2

1t1 → 2e (88%), 2t2 → 2e (4%) 2t2 → 2e (66%), 1t1 → 3t2 (32%) 1t1 → 3t2 (52%), 2t2 → 2e (20%)

MnO4‑ according to the present calculation. These calculations show in agreement with Neugebauer et al.23 and other TD-DFT calculations21,22 that the second and the third bands can be assigned to a mixture of 2t2 → 2e and 1t1 → 3t2 transitions although the second band has the largest contribution from 2t2 → 2e and the third from 1t1 → 3t2.That the second band primarily is due to 2t2 → 2e and the third to 1t1 → 3t2 is also supported by TD-DFT simulation due to Seth et al.22c,24b of the magnetic circular dichroism (MCD) spectrum of MnO4−. 3.1.2. Geometrical Structure of MnO4− in its Excited states. Geometrical Structures of MnO4− in Its First Three Excited States under Td Constraints. Optimisation of the first three excited states under Td symmetry constraints gives rise to structures with elongated Mn−O bonds of 1.63 [Td(1T2)], 1.64 [Td(2T2)], and 1.65 Å [Td(3T2)], respectively, compared to the ground state with a calculated bond length of 1.60 Å. The energies of the three structures are 2.88 [Td(1T2)], 3.86 [Td(2T2)], and 4.63 eV [Td(3T2)] above the ground state. Thus optimizing the excited state Mn−O bond length within Td constraints leads to an energy lowering of 0.23 [Td(1T2)],

Figure 2. Structures located after distortion of the three first singlet T2 excited states of MnO4‑ .

Geometrical Structures of MnO4− in Its First Three Excited States. Geometry optimizations on the first excited state of MnO4‑ resulted in a C3V structure 1 with an increase in the three equivalent Mn−Ob distances to 1.64 Å compared to the ground state geometry (1.60 Å). On the other hand the unique Mn−Oa distance is only stretched to 1.61 Å. Further, the Oa−Mn−Ob umbrella angle has opened up from 109.4° to 111.4°. The minimum structure 1 is only 56 cm−1 (0.007 eV) more stable than Td(1T2). For the second excited state we were able to locate two structures 2 and 3 of C2v symmetry, Figure 2. The first C2v structure 2 has two elongated Mn−Ob distances of 1.68 Å with the other two (Mn−Oa) being only slightly stretched to 1.61 Å. 1866



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The Oa−Mn−Oa (θaa) and Ob−Mn−Ob (θbb) angles are increased to 111.2° and 116.4°, respectively. On the other hand, structure 3 exhibits Oa−Mn−Oa and Ob−Mn−Ob angles that are compressed to θaa = 105.8° and θbb = 106.8°, respectively, compared to the tetrahedral angle of 109.4°. The corresponding distances are Mn−Oa = 1.62 Å and Mn−Ob = 1.65 Å. We find that 2 and 3 are stabilized by 1695 cm−1 (0.21 eV) and 726 cm−1 (0.09 eV), respectively, compared to Td(2T2). Calculations on the third excited state lead to a structure 4 with D2d symmetry. In this geometry, all of the four Mn−O bond lengths increase to 1.65 Å compared to the ground state bond length of 1.60 Å. However the optimized Mn−O bond lengths in 4 is similar to that found for Td(3T2). Thus, the D2d distortion [Td(3T2)→4] represents primarily an angular change from 109.4° to 112.7°/107.9° along one of the QE bending modes with a resulting stabilization of 323 cm−1 (0.04 ev). We shall now further analyze the calculated distortions from tetrahedral geometry for 1−4 in more details. Origin of the C3v Distortion for the First Excited State of MnO4−. It follows from Table 2 that the first excitation involves the electron transfer from a ligand based O−O antibonding molecular orbital (1t1) to a metal based Mn−O antibonding molecular orbital (2e). The only located stable minimum on the potential energy surface is the C3v structure 1 corresponding to a singlet state of A1 symmetry. The structure 1 can be reached by a distortion from Td along a linear combination of the degenerate (QT2x,QT2y,QT2z) stretching modes with QA1. In the vicinity of Td one such combination is (1/2)√3[3Qt2x − Qt2y − Qt2z − QA1]. The motion along the C3v distortion coordinate splits the 1T2 excited states into A1 and E. The E states under C3v constraints do not have any stable minima. They are however connected via the optimized Td(1T2) geometry to A1 states of C3v symmetry in structures 1′ similar to 1, in which other combinations of the oxygen atoms play the role of Oa and Ob, Figure 3. Here 1′ and 1 are reached by different linear combinations of (QT2x,QT2y,QT2z,QA1).

Figure 4. The intrinsic reaction coordinate (IRC) path from the optimized Td(1T2) structure to 1. The change in energy along the IRC path is plotted against the Mn−Oa distance of 1 defined in Figure 2.

Figure 5 gives a comparison in the energies of different levels in the ground state of Td symmetry and the A1 excited state

Figure 5. Molecular orbital level diagrams for the ground and the excited states of MnO4‑ in their optimized equilibrium structures.

with the C3v equilibrium structure 1. The nonbonding 1a2 and 4e levels in 1 remain more or less unchanged compared to the 1t1 level of the ground state with Td symmetry. However, the antibonding 5e level in 1 decreases in energy compared to the 2e level of the ground state. The stabilization of the 5e orbitals is due to a reduction in their antibonding character as a result of the Mn−Ob stretch. Such a stabilization is favorable as the 5e level holds one electron in 1. It should be noted that the occupation of a single electron in 5e does not lead to any further distortion from C3v as 5ea and 5eb contributes equally to Ψ2A1. We have seen that MnO4‑ in its first excited state exhibits a Jahn−Teller distortion away from Td resulting in a stable 2A1 state with the structure 1 of C 3v geometry. Further, permanganate in its first excited state is “fluxional” in that the interconversion 1 → Td(1T2) → 1′ of Figure 3 has a low barrier of 56 cm−1. The “fluxionality” is driven by the hole in the 1t1 shell of the 1t15 configuration. The “fluxionality is dynamic as the barrier is below the zero-point energy.

Figure 3. Interconversion between two equilibrium structures 1 and 1′ for the first excited state via the optimized Td structure Td(1T2).

We display in Figure 4 the intrinsic reaction path (IRP)26 from the optimized Td structure Td(1T2) for the first excited state to 1. It follows from Figure 4 that the distortion Td(1T2) → 1 results in a stabilization of 0.007 eV or 0.7 kJ/mol. Thus the interconversion 1 → Td(1T2) → 1′ of Figure 3 should have a modest barrier of only 0.7 kJ/mol or 56 cm−1. The 1t1 orbitals of Td symmetry split as 1a2 and 4e in the C3v distortion of the first excited state whereas the 2e orbital of Td symmetry becomes 5e in 1. These orbitals are shown in Figure S2 (see the Supporting Information). In a simple form the wave + function of the A1 state can be expressed as Ψ2A1 = (1/√2)(|Ψ4e a − + Ψ5e | + |Ψ4e Ψ b−|). a b 5e 1867



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Origin of the C2v Distortion for the Second Excited State of MnO4−. The 2t2 → 2e orbital excitation is the primary contributor to the second band in the absorption spectrum of MnO4−. In this case, an electron is transferred from a ligand based O−O bonding molecular orbital (2t2) to a metal-based antibonding Mn−O molecular orbital (2e). The structure 2 of Figure 2 is reached from Td by a Jahn−Teller distortion involving elongations of Mn−Oa and especially Mn−Ob in combination with an increase of the Oa−Mn−Oa (θaa) and Ob− Mn−Ob (θbb) bond angles. The Mn−Ob bond elongation can be accomplished from Td by any of the stretching mode combinations (1/√2)[QA1 − QT2i] (i = x,y,z), Table S1 (see the Supporting Information), whereas the increase of the θaa and θbb bond angles is brought about by one of the QE bending modes. The occupied 2t2 ground state levels of Td symmetry split as 4a1, 2b1, and 2b2 and the virtual 2e ground state levels split as 5a1 and 3a2 in the C2v distortion of the second excited state. The resulting orbitals from the distortion are shown in Figure S3 (see the Supporting Information). In 2 we have that 5a1(2e) is placed below 3a2(2e), Figure 5, as the former becomes less antibonding with an increase of θaa and θbb as well as the Mn−Ob stretch. Thus when 2T2 splits in 2 under the 2T2 distortion into A1[2T2] (4a1 → 5a1), B1[2T2](2b2 → 3a2), and B2(2T2)(2b1 → 3a2), it is A1[2T2] with the approximate wave function Ψ2A1(2T2) = |4a15a1| that is of lowest energy. On the other hand, the two states B1[2T2] and B2[2T2] have as their equilibrium structure 3 in which θaa and θbb by a Jahn−Teller distortion are decreased compared to Td. In this case 3a2(2e) is placed below 5a1(2e). We display in Figure 6 the actual calculated potential surface of the states

also implicated two minimum structures for a distorted 2T2 state. However they did not fully optimize these structures. We shall later see that the existence of two structures strongly will impact the vibronic structure of the second absorption band for MnO4−. Analysis of the D2d Distortion of the Third Excited State of MnO4−. The third band in the absorption spectrum of MnO4‑ involves mainly the 1t1 → 3t2 orbital excitation. In this transition an electron is transferred from a ligand based nonbonding molecular orbital (1t1) to a metal based highly antibonding molecular orbital (3t2), Figure 1. We locate for the third excited state a stable minimum structure 4 of D2d geometry, Figure 2. It can be generated from Td(3T2) by a Jahn−Teller distortion along any of the two bending modes QE, Figure S1 (see the Supporting Information). In the D2d distortion, the states 3T2 split into B2[3T2]and E[3T2]. For B2[3T2] the structure 4 represents a stable minimum. On the other hand, E[3T2] does not have any stable minimum within the D2d domain containing 4. We note that the Td[3T2] geometry with the same Mn−O bond lengths of 1.65 Å as 4 connects different equivalent D2d structures 4 and 4′ via the two QE bending modes. In this way a 3T2 state that becomes part of a E[3T2] in one D2d domain 4 becomes a B2[3T2] state in another domain 4′ of D2d symmetry. We find that the barrier for such a process [4 → Td(3T2) → 4′] only is 323 cm−1 (0.04 ev) or 4 kJ/mol. The 1t1 orbitals split as 1a2 and 3e in the D2d distortion whereas 3t2 becomes 3b2 and 4e. These orbitals are shown in Figure S4 (see the Supporting Information). Figure 5 displays a comparison between the energies of the different levels in the ground state of Td symmetry and the third excited state of D2d symmetry. The ligand based orbitals 1a2 and 3e have nearly the same energy as 1t1. However, the antibonding 3b2 and 4e levels in the D2d distorted excited state decreases in energy compared to 3t2 of the ground state. The lowering in the energy of the dbased orbitals (3b2 and 4e) is due to a reduction in the antibonding character in 3b2 and 4e as a result of the increased Mn−O distances in 4. The Kohn−Sham wave function representing B2[3T2] can be written approximately as (1/√2){|3ea 4eb| + |3eb 4ea|}. It is thus clear that the lowering of 4e will stabilize B2[3T2]. Also the occupation of the 4e level will not lead to any further Jahn−Teller distortion within the D2d domain of 4 since both 4e orbitals are equally occupied although only a total of one electron is in 4e. Futher, the third excited state is “fluxional’ due to the inter-conversion path [4 → Td(3T2) → 4′] with a barrier of only 323 cm−1. This Jahn-Teller “fluxionality” is due to a hole in both the 2t2 and 3t2 levels. 3.1.3. Vibronic Structure of the Absorption Spectrum due MnO4−. Vibrational frequencies were calculated as implemented in ADF.26 Table 3 lists the vibrational frequencies obtained for the ground and excited states of MnO4‑ along with the experimental data.42 The experimental spectrum of MnO4− was recorded3 at liquid hydrogen and helium temperatures using crystals where KMnO4 was dissolved in KClO4. The first experimentally observed absorption corresponding to an allowed transition has its 0−0 origin at 18 072 cm−1 and reveals a progression with ν1 = 768 cm−1 due to the totally symmetric stretch, Figure 7b. The band maximum appears at 18 842 cm−1. Built upon the ν1 progression are several peaks corresponding to multiple quanta of ν3 = 272 cm−1. The simulated spectrum has its onset at 22 600 cm−1 with a progression corresponding to ν1 = 891 cm−1 as well as several peaks representing quanta of ν1 = 315 cm−1. Thus within the experimental resolution the simulated spectrum reproduces

Figure 6. Calculated potential energy surface for A1[2T2] with structure 2 and B1[2T2], B2[2T2] with structure 3 along the totally symmetric bending mode as a function of the Oa−Mn−Oa angle defined in in Figure 2.

A1[2T2], B1[2T2], and B2[2T2] as a function of θaa along the symmetrical bending mode QA1 = (1/√2){Δθaa + Δθbb}. Thus A1[2T2] exhibits a minimum 2 for θaa,θbb > 109.4° whereas B1[2T2] and B2[2T2] have their equilibrium for 3 with θaa,θbb < 109.4°. Both 2 and 3 are related to other structures 2′ and 3′, respectively, via Td[2T2]. We find that 2 → Td[2T2] →2′ has a barrier of 1695 cm−1 (0.21 eV) whereas the barrier for the rearragments 3 → Td[2T2] →3′ is 726 cm−1 (0.09 eV). The structure Td[2T2] can also interconnect A1[2T2] at 2 with B1[2T2], B2[2T2] at 3. The complex Jahn−Teller distortion pattern described here is associated with a hole in the 2e level as well as the 2t2 level. It is interesting that Neugebauer et al.23 1868



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Table 3. Vibrational Frequencies (in cm−1) for MnO4− in Its Ground and Excited States ground state experimentala 348 400 845 930

a

b,c

(E) (T2)d (A1)e (T2)e

calculated 362 410 930 997

b

(E) (T2)d (A1)e (T2)e

1 (C3v) 196 296 315 821 842 891

2 (C2v)

d

(E) (E)d (A1)d (E)e (A1)e (A1)e

244 249 295 303 342 601 716 996

3 (C2v) d

(A1) (A2)d (B1)d (B2)d (A1)d (B2)e (A1)e (A1)e

d

247 (A2) 262 (B1)d 282 (B2)d 349 (A1)d 575 (A1)e 736 (B1)e 877 (A1)e 1032 (B2)e

4 (D2d) 237 280 356 406 673 870

(B2)d (A1)d (E)d (B1)d (E)e (A1)e

Reference 44a. bWagging modes. cFor a description of the normal modes see ref 45. dBending modes. eStretching modes.

structure gives rise to an onset which we name ν0 and ν0′, respectively, Figure 7c . We have in addition two progressions given by ν1 = 996 cm−1 for 2 and ν1′ = 877 cm−1 for 3. We note finally some fine structure built on the symmetrical progression with ν3 = 342 cm−1 (2) and ν3′ = 349 cm−1 (3). A comparison to experiment is somewhat hampered by the fact that the second band from 28 500 cm−1 and up is superimposed on the third band. We have in order to make the comparison to experiment more graphical shifted the 0−0 frequency of the second calculated band to 25 300 cm−1 and the 0−0 transition of the third simulated band to 29 000 cm−1. The simulated spectrum is displayed in Figure 8c and compared to experiment in Figure 8a as well as the simulated spectrum due to Neugebauer et al.23 in Figure 8b. In Figure 8c a similar shift has been performed on the calculated 0−0 transitions to that in Figure 8b. Our spectrum in Figure 8c is seen to be in good qualitative agreement with experiment and the work by Neugebauer et al.23 We should finally mention that the actual potential energy surface along the totally symmetric normal modes were found all to be harmonic. This finding gives us some confidence in our Franck−Condon (FG) approach and the simulated FG spectra presented here. 3.2.1. TcO4−. Electronic Spectrum of TcO4−. Pertechnetate has a tetrahedral geometry in its ground state just like MnO4‑. In 1972, Güdel and Ballhausen5 recorded at 4 K an absorption spectrum of 3 mol % TcO4‑ dissolved in KClO4 crystals. The experimental spectrum exhibits two strong and overlapping bands with onsets at 32 677 and 38 544 cm−1, respectively. The first band arises according to our calculations primarily from the 1t1 → 2e transition and the second mostly as a result of the 2t2 → 2e excitation. Our assignment is identical to that obtained previously by other theoretical studies.16,19,20 The solution spectrum of TcO4‑ has previously been recorded by Mullen et al.4 Our optimized ground state structure affords a Tc−O bond length of 1.71 Å which is very close to the experimental value of 1.72 Å.44a The calculated excitation energies and oscillator strengths are listed in Table 4 along with the experimental data.5 Table 4 also lists all the one-electron transitions that contribute to the first and second band of the TcO4− spectrum. We find a much better agreement between theory and experiment for TcO4− than for MnO4‑. This is the case for both the vertical excitations ΔEver as well as the 0−0 transitions ΔE00. 3.2.2. Geometrical Structures of TcO4− in Its First Two Excited States. Optimisation of the first two excited states

most of the observed features with the exception that the intensity of the ν1 components diminish faster in the calculated spectrum, Figure 7a. Our simulation is based on the Franck−Condon approach that assumes a harmonic potential surface around the equilibrium structure of the two states involved in the transition. We have from calculations confirmed that the real calculated potential along all three totally symmetrical modes are harmonic. Thus the influence of the Jahn−Teller distortion 1 → Td(1T2) → 1′ discussed previously on the vibronic spectrum would enter through an anharmonic coupling of the normal modes. This coupling is neglected here. Neugebauer et al.23 have used a different vibronic coupling theory that do not rely on the Franck−Condon approach. Their simulated spectrum for the first excited state of MnO4− is quite similar to ours. It would thus appear that the interconversion 1 → Td(1T2) → 1′ only is able to have a marginal influence on the vibronic spectrum. This is likely so since the interconversion barrier of 56 cm−1 is much smaller than the zero-point energy. The third experimental band is less resolved. It has an estimated3 onset 3000 cm−1 after the second band at ∼29 000 cm−1 and it clearly reveals a progression of ν1 = 750 cm−1 as well as a band maximum at 33 000 cm−1 corresponding to 4ν1, Figure 7f. Our simulated spectrum has ν1 = 870 cm−1 from the symmetric stretch and exhibits in addition peaks corresponding to a vibrational fine structure with ν3 = 280 cm−1 due to the symmetric bending mode, Figure 7e. As for the first electronic band, the intensity of the ν1 components are seen to diminish faster in the simulated spectrum compared to experiment. We attribute that to a theoretical underestimation of the elongation for the Mn−O bond on excitation. Our calculated spectrum compares well with that obtained by Neugebauer et al.23 However, these authors did not provide explicit information about excited state structures or frequencies. Direct calculations along the two totally symmetric modes ν1,ν3 that primarily contribute to the vibronic structure revealed a harmonic potential surface. We shall assume that the anharmonic perturbation of the vibronic structure due the Jahn−Teller “fluxionality” [4 → Td(3T2) → 4′] is small and not observable at the experimental resolution provided in Figure 7f. The second experimental band shown in Figure 7d starts at 25 000 cm−1 and is nearly featureless although the contours of one or more progressions can be inferred. The simulated spectrum is made complex by the presence of two different structures 2 and 3 and three different states A1[2T2], B1[2T2], and B2[2T2]. The features due to B1[2T2] and B1[2T2] coalesce and will be lumped together in the discussion Thus, each 1869



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Figure 7. Comparison between experimental and calculated spectra for the first three allowed transitions of MnO4‑. (a and b) First allowed transition. (c and d) Second allowed transition. (e and f) Third allowed transition.

within Td symmetry constraints gives rise to structures with elongated Tc−O bonds of 1.76 Å (1T2) and 1.77 Å (2T2) compared to the ground state with 1.72 Å. The two structures [Td(1T2)] and [Td(2T2)] are 4.08 and 4.78 eV above the ground state, respectively. Thus relaxing the Tc−O bond length within Td constraints leads to an energy lowering of 0.48 eV [Td(1T2)] and 0.50 eV [Td(2T2)] in comparison to the ver corresponding vertical excitation energies Δcal The lowering is ‑ considerably larger than for MnO4 due to the stronger antibonding interaction in the 2e orbitals of TcO4‑. The geometries Td(1T2) and Td(2T2) do not represent stationary minimum points on the potential energy surface. The optimized structures yield instead five imaginary frequencies corresponding to the stretching QT2 and bending QE normal modes, as in the case of MnO4‑. Thus TcO4− is in its two lowest

excited states is subject to a Jahn−Teller distortion away from the Td symmetry as indicated in Table 5. Pertechnetate undergoes in its first excited state (1t1 → 2e) a C3v distortion which splits 1T2 into its A1 and E components. The A1 state has a stable minimum C3v structure 5 similar to 1 in which three Tc−Ob bonds are stretched from 1.71 to 1.78 Å whereas the unique Tc−Oa bond is elongated to 1.73 Å and the Oa−Tc−Ob umbrella angle opened up to 112°, Table 5. The minimum C3v structure for E is a transition state. The E components are connected to A1 states in other C3V species via the Td structure as illustrated in Figure 3. It is clear that permanganate and pertechnetate behaves quite similarly in their first excited state. Also, the A1 state of pertechnetate has a wave function similar to that discussed for MnO4‑ in which the two d-based orbitals 5ea and 5eb of Figure 5 are equally occupied and the excited state as a consequence is not prone to a further 1870



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excited state. In the case of pertechnetate the barrier for this rearrangement amounts to 968 cm−1 (0.12 eV). The higher barrier compared to MnO4‑ underlines again the more strongly antibonding character of the 5e orbitals. The second excited state of TcO4‑ with Td symmetry [Td(2T2)] is subject to a D2d Jahn−Teller distortion that lifts the degeneracy of the three 2T2 components into B2[2T2] and E[2T2]. The B2[2T2] state has a stable D2d minimum structure 6 with all four Tc−O bond lengths increased to 1.77 Å compared to the optimized ground state distance of 1.71 Å. Its Oa−Tc−Oa angle that is bisected by the S4 axis is 116°, Table 5. Further, the D2d minimum geometry for E[2T2] is a transition state. We note that Td(2T2) which has the same Tc−O distance as 6 connects different equivalent D2d structures via the two Qe bending modes . In this way a 2T2 component that constitutes a E[2T2] state in one D2d domain 6 becomes a B2[2T2] state in another domain 6′ of D2d symmetry. The barrier for 6 → Td(2T2) → 6′ is calculated to be 1210 cm−1 (0.15 eV). The D2d Jahn−Teller distortion splits the 2t2 level into 2b2, 2ea, and 2eb whereas the degeneracy in 2e is lifted to produce 3a1 and 2b1, Figure 5. The resulting orbitals are depicted in Figure S5 (see the Supporting Information). The Oa−Tc−Oa angle that is bisected by the S4 axis opens up from 109.4° to 116° in going from Td to D2d. The result is a decrease in the M-O antibonding interaction in 3a1 and an increase in 2b1. As a consequence 3a1 is placed below 2b1 for 6, Figure 5. Among the 2t2 ligand orbitals 2b2 is found above 2ea and 2eb. It is thus not surprising that B2[2T2] represented by the 2b2 → 3a1 transition is the 2T2 component in D2d of lowest energy. 3.2.3. Vibronic Structure for the Absorption Spectrum of TcO4−. Figure 9 (bottom left) exhibits the experimental UV absorption spectrum5 recorded at liquid helium temperature using single crystals where TcO4− is dissolved in KClO4. This spectrum displays two somewhat overlapping bands with pronounced vibrational structures. The first band has a 0−0 transition at 32 677 cm−1 and an average progression (ν1) of 795 to 806 cm−1, depending on the polarization of the light source. The progression has at least 6 components and a maximum at 3ν1. The corresponding simulated spectrum in Figure 9 (top left) displays a 0−0 transition at 32 500 cm−1 and a regular progression (ν1) of 881 cm−1 corresponding to the totally symmetric stretching mode, Table 6. The maximum is at 2ν1. For every peak due to ν1 we find in addition a peak with an offset of ν3 = 221 cm−1 due to the totally symmetric bending mode. This feature is not resolved in the experimental spectrum. The additional peaks that arez visible but not assigned in the calculated spectrum correspond to multiples of υ3. Contributions to the theoretical spectrum in Figure 9 from the second totally symmetric stretch at ν2 = 806 cm−1 are too weak to be discernible. The second experimental band has its 0−0 transition around 38,500 cm−1 and a progression in ν1 of 772 to 791 cm−1 with a maximum at 3υ1, Figure 9 (bottom right). The corresponding calculated spectrum places the 0−0 transition at 39 500 cm−1 and display a progression from the totally symmetric stretch of ν1 = 806 cm−1, Figure 9 (bottom left). The maximum is again at 2ν1. Also apparent are contributions from the totally symmetric bending mode ν3 at 238 cm−1, Table 6. In the analysis of the potential energy surface for the two excited states of TcO4− we discovered Jahn−Teller distortion behavior along the paths 5 → Td(1T2) → 5′ and 6 → Td(2T2) → 6′. Since the totally symmetric modes are essential for the vibronic structure, we have explored the actual potential

Figure 8. Second and third band of permanganate. (a) Experiment from ref 3. (b) Simulated spectrum from ref 23. (c) Present work with γ = 180 cm−1.

Table 4. Excitation Energies, Oscillator Strengths, and Contribution from One-Electron Transitions for the First Two Excited States of TcO4‑ a excitation energies and oscillator strengths of TcO4‑ experimentalb

calculated

state

verc ΔEcal

1T2

4.56

2T2

5.28

00 d ΔEcal g

fcal

ver e ΔEexp

00 f ΔEexp

4.02

0.0093

4.32

4.05

0.04

fexp

4.73h

0.0179

5.08

4.79

0.095

i

one electron transitions 1t1 → 2e (78%) 2t2 → 2e (17%) 2t2 → 2e (68%) 1t1 → 3t2 (18%)

a

Excitation energies in eV. bReference 5. cCalculated vertical excitation energies. dCalculated 0−0 excitation energies. eThe experimental vertical excitation energies were taken as the band maxima in ref 5. f The 0−0 experimental excitation energies were taken as the band onsets in ref 5. gWith respect to optimized C3v structure 6. hWith respect to D2d structure 5. iReference 38.

Jahn−Teller distortion away from C3v as a result of a single electron in the 5e shell. In spite of this the C3v structure 5 for TcO4− can undergo a rearrangement 5→ Td(1T2) → 5′ similar to that shown for MnO4‑ in Figure 3. This (Jahn−Teller) rearrangement is a result of the hole in the 1t1 level of the first 1871



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Table 5. Optimized Geometrical Parameters for TcO4‑, RuO4, and OsO4 in the Ground (Gr) and Two First Excited Statesd TcO4‑ Gr

a

RuO4

5b

6c

Gra

7b

OsO4 8

Gra

9b

10c

parameter

Td

C3v

D2d

Td

C3v

D2d

Td

C3v

D2d

M−Oa M−Ob Oa−M−Oa Oa−M−Ob Ob−M−Ob

1.71

1.73f 1.78

1.77

1.68 e

1.70f 1.74

1.73

1.70 e

1.71f 1.76

1.75

115g

e 112 107

115g 115 107

107

107

116g 113 106

106

a

Ground state of Td symmetry. bFirst excited state with C3v geometry. cSecond excited state with D2d symmetry. dDistances in Å and angles in deg. e Tetrahedral angle 109.5°. fUnique bond in C3v geometry. gAngle being bisected by S4 axis in D2d symmetry.

Figure 9. Comparison between experimental and calculated spectra for the first and second allowed transition of TcO4−. (a) First allowed transition calculated. (b) Second allowed transition calculated. (c) First allowed transition experimentally. (d) Second allowed transition experimentally.

Table 6. Vibrational Frequencies (in cm−1) for TcO4‑ in Its Ground and Excited States

discussed above will have any noticeable influence on the vibronic structure of the first two absorption bands of TcO4−, at least not at the experimental resolution provided in Figure 9. We note finally that the intensity of the components in the vibronic progression diminish faster in the simulated spectra with a maximum at 2ν1 than in the experimental spectra with a maximum at 3ν1. We attribute this to an underestimation in the M−O bond elongation of the excited states. We note finally that the second band in TcO4− is much less complicated than in MnO4− since 2T2 here only splits into one state, B2[2T2]. This is in spite of the fact that the excited state in both cases is due to the 2t2 → 2e transition. 3.3.1. RuO4 and OsO4. Electronic Spectrum of RuO4 and OsO4. RuO4 and OsO4 are the only neutral tetrahedral tetroxides for which an absorption spectrum has been recorded. In 1967, Wells et al.6 recorded a high-resolution vapor spectrum of OsO4 and RuO4. They found two prominent bands with maxima at 26 000 and 32 600 cm−1 for RuO4 and at 35 000 and

ground state (Td) exp.d

cal. 318 341 941 942

a

(E) (T2)c,e (T2)b (A1)b

327 350 913 920

5 (C3v) a

(T2) (E)a (A1)b (T2)b

147 209 221 674 806 882

a

(E) (E)a (A1)a (E)b (A1)b (A1)b

6 (D2d) 141 161 177 238 596 699 845

(B1)a (E)a (B2)a (A1)a (B2)b (E)b (A1)b

a

Bending mode. bStretching mode. cWagging modes. dReference 47b. For a description of the normal modes see ref 45.

e

surface along the excites state normal modes from their equilibrium structures and confirmed that the surface is harmonic. It is thus not likely that the Jahn−Teller distortion 1872



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41 500 cm−1 for OsO4. Their findings were confirmed in 1973 by Foster et al.45 in a study that also included higher energy Rydberg transitions. In 1974, Roebber and Wiener7 performed extended Hückel calculations on OsO4 and RuO4. The results of these calculations was an assignment of the first band to the 1t1 → 2e transition and the second band to the 2t2 → 2e excitation for both OsO4 and RuO4. The calculations based on SAC (symmetry adapted cluster) and SAC+CI theories39 confirm the assignment by Roebber and Wiener (RW). There have been several DFT calculations reported in the literature.19,20,40 The HFS-DVM (Hartree−Fock−Slater discrete variational method) calculations by Ziegler et al.19 supported the RW-assignment. They also calculated the oscillator strength of these bands. In 2005, TD-DFT calculations conducted by Menconi et al.24 lead to the conclusion that almost all the excitations observed in the spectra are a mixture of two or more one-electron transitions. However, 1t1 → 2e and 2t2 → 2e are the major contributors to band I and II, respectively. Ground state geometry optimizations were carried out within the tetrahedral geometry. The experimental metal−oxygen distances were reproduced very closely; 1.68 Å for RuO4 and 1.70 Å for OsO4 compared to the experimental metal−oxygen distances of 1.71 Å for RuO444 and 1.71 Å for OsO4.46 The calculated vertical excitation energies and oscillator strengths are listed in Table 7 for RuO4 and in Table 8 for OsO4, respectively, along

Table 8. Excitation Energies, Oscillator Strengths, and Contribution from One-Electron Transitions for the First Two Excited States of OsO4a excitation energies and oscillator strengths of OsO4

a

verc ΔEcal

00 d ΔEcal h

experimentalb

1T2

3.78

3.31

2T2

4.59

4.02i

fcal

ver e ΔEexp

00 f ΔEexp

0.0061

3.22

2.94

0.02

0.0106

3.99

3.78

0.044

b

fexpg

1T2

4.28

2T2

5.04

00 d ΔEcal h

fcal

ver e ΔEexp

00 f ΔEexp

3.45

0.0044

4.34

3.85

0.05

fexp

4.69i

0.0179

5.14

4.83

0.108

g


ver Δcal . The optimized excited state Td structures have five imaginary frequencies corresponding to the stretching QT2 and bending QE normal modes. This suggests that the first two excited states of RuO 4 and OsO 4 within the T d symmetry are not stationary minimum points on the potential energy surface. Thus RuO4 and OsO4 in the first two excited states will Jahn−Teller distort from the Td ground state symmetry. Table 5 lists the excited state structures for both RuO4 and OsO4. The corresponding frequencies for the ground and excited states are given in Table 9 for RuO4 and in 10 for OsO4.

excitation energies and oscillator strengths of RuO4

state

state

verc ΔEcal

a Excitation energies in eV. bReference 48. cCalculated vertical excitation energies. dCalculated 0−0 excitation energies. eThe experimental vertical excitation energies were taken as the band maxima in ref 48. fThe 0−0 experimental excitation energies were taken as the band onsets in ref 48. gReference 38d. hWith respect to optimized C3v structure 1. iWith respect to D2d structure 5.

Table 7. Excitation Energies, Oscillator Strengths and Contribution from One-Electron Transitions for the First Two Excited States of RuO4a calculated

experimentalb

calculated


Table 9. Vibrational Frequencies (in cm−1) for RuO4 in Its Ground and Excited States ground state (Td) exp.a

cal. 328 353 963 991

c

Excitation energies in eV. Reference 45. Calculated vertical excitation energies. dCalculated 0−0 excitation energies. eThe experimental vertical excitation energies were taken as the band maxima in ref 48. fThe 0−0 experimental excitation energies were taken as the band onsets in ref 48. gReference 38d. hWith respect to optimized C3v structure 1. iWith respect to D2d structure 5.

b,c

(E) (T2)d (A1)e (T2)e

328 333 878 921

7 (C3v) b

(T2) (E)b (A1)e (T2)e

159 240 254 731 868 921

b

(E) (A1) (E)b (E)e (A1) (A1)e

8 (D2d) 117 163 221 256 670 748 878

(B1)b (B2)b (E)b (A1)b (B2)e (E)e (A1)e

a

Reference 45. bBending mode. cFor a description of the normal modes see ref 45. dWagging modes. eStretching mode.

with the experimental data.7,45 The two tables also list all the one orbital transitions that are responsible for the first and second band in the RuO4 and OsO4 spectra. 3.3.2. Geometrical Structures of RuO4 and OsO4 in Its First Two Excited States. RuO4 and OsO4 optimized within the Td symmetry constraints for the first two excited states give rise to the structures Td(1T2) and Td(2T2) with M−O distances of 1.73 and 1.74 Å, respectively, for RuO4 compared to 1.76 and 1.77 Å, respectively, for OsO4. The two structures [Td(1T2)] and [Td(2T2)] are 3.42 eV (Ru)/4.13 eV (Os) and 4.16 eV (Ru)/4.84 eV eV(Os) above the ground state, respectively. Thus relaxing the M-O bond length within Td constraints leads to an energy lowering of 0.36 eV (Ru)/ 0.15 eV (Os) for [Td(1T2)] and 0.53 eV (Ru)/0.20 eV (Os) for [Td(2T2)] in comparison to the corresponding vertical excitation energies

Both RuO4 and OsO4 have in their first excited state a C3v structure with three M−Ob bonds elongated to 1.74 Å (Ru)/ 1.76 Å (Os) compared to the ground state distances of 1.68 Å (Ru)/1.70 Å (Os) as well as one shorter bond with the length 1.70 Å (Ru)/1.71 Å (Os). The Oa−M−Ob umbrella angle is given by 115° (Ru)/113° (Os), Table 5. The first band in the absorption spectrum is due to the 1t1 → 2e transition, Tables 7 and 8. Under the C3v distortion of the first excited state, 1T2 splits into A1 and E. The A1 state under C3v constraints gives a stable minimum 7(Ru), 9(Os) whereas the two E components are transition states. The barrier for the 7,9 → Td(1T2) → 7′,9′ transposition of Figure 3 was calculated 1873



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to be 1049 (0.13 eV) and 968 cm−1 (0.12 eV) for RuO4 and OsO4, respectively. The second excited state has a D2d geometry 8 (Ru)/10 (Os) with all the four M−O bond lengths increased by 0.05 Å (Ru)/0.05 Å (Os) compared to the optimized ground state. The T2 excited state splits into B2 and E in the D2d distortion of the second excited state. The B2 state under D2d constraints gives a true minimum that is stabilized by 0.14 eV(Ru)/0.15 eV (Os) compared to [Td(2T2)] whereas the minima for both E components are transition states. Again, a 2T2 component that constitutes a E[2T2] state in one D2d domain 8 (Ru)/10 (Os) becomes a E[2T2] state in another domain 8 (Ru)/10 (Os) of D2d symmetry. The barrier for the Jahn-teller fluxionality 8,10 → Td(2T2) → 8′,10′ is calculated to be 1129 (Ru) and 1741 cm−1 (Os), respectively. 3.3.3. Vibronic Structure for the Absorption Spectrum of RuO4 and OsO4. Table 9 and Table 10 lists the vibrational

frequencies calculated for the ground and the excited states of RuO4 and OsO4, respectively. Figures 10 and 11 compare simulated and experimental spectra48 of RuO4 and OsO4. The first experimental band of both RuO4 and OsO4 has a pronounced vibronic structure. The onsets are at 23 700 (Ru) and 31 000 cm−1 (Os) with progressions of ν1 = 782 (Ru) and 813 cm−1 (Os). The band maxima are found for 3ν1 at 26 000 (Ru) and 35 000 cm−1 (Os). By comparison our modeling affords 0−0 transitions of 26 700 (Ru) and 32 400 cm−1 (Os) with progressions given by ν1 = 921 (Ru) and 968 cm−1 (Os) from the totally symmetric stretch. Also noticeable are peaks corresponding to the symmetric umbrella bending mode with ν3 = 240 (Ru) and 216 cm−1 (Os). We assess the onset of the second experimental band (II) to be 30 500 for RuO4 and 38 500 cm−1 in the case of OsO4. The assessment is hampered by overlap from band I. Further, band II has a clear progression of ν1 = 771 (Ru) and 843 cm−1 (Os). The band maxima are found for 3ν1 at 33 000 (Ru) and 42 000 cm−1 (Os). It is also possible, especially for OsO4, to identify features48 that can be associated with the symmetrical bending mode ν3. The simulated spectra have 0−0 transitions in the range of 32 400 (Ru) and 38 000 cm−1 with visible progressions from the totally symmetrical stretching mode of ν1 = 878 (Ru) and 926 cm−1 (Os). Also present are vibronic bands due to the symmetric bending modes with ν3 = 256 (Ru) and 249 cm−1 (Os). In the ground state the totally symmetric stretching mode ν1 is seen both experimentally and theoretically to be larger for OsO4 than for RuO4. This reflects the observation that M−O bonds are stronger for 5d-elements compared to their 4d congeners as a result of relativistic effects. We note that the totally symmetric stretching modes in the two excited states are found experimentally to be smaller for RuO4 and OsO4. This

Table 10. Vibrational Frequencies (in cm−1) for OsO4 in Its Ground and Excited States ground state (Td) cal. 336 (T2)b,c 339 (E)b 1002 (T2)d 1020 (A1)d

exp.a 327 342 961 968

(T2)b (E)b (A1)d (T2)d

9 (C3v)

10 (D2d)

86 (E)b 216 (A1)b 241 (E)b 743 (E)d 871 (A1)d 958 (A1)d

94 (B1)b 172 (B2)b 233 (E)b 249 (A1)b 704 (B2)d 821 (E)d 926 (A1)d

a

Reference 45. bBending mode. cFor a description of the normal modes see ref 45. dStretching mode.

Figure 10. Comparison between the experimental and calculated spectra for the first and second allowed transition of RuO4. (a) First allowed transition calculated. (b) First allowed transition experimentally. (c) Second allowed transition calculated. (d) Second allowed transition experimentally. 1874



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Figure 11. Comparison between the experimental and calculated spectra for the first and second allowed transition of OsO4. (a) First allowed transition calculated. (b) First allowed transition experimentally. (c) Second allowed transition calculated. (d) Second allowed transition experimentally.

would also be aided by more highly resolved experimental spectra. All excited states exhibit a fluxional behavior in which equivalent structures can be reached via a tetrahedral geometry. The barriers for these rearrangements are in all cases below the zero-point energy and must be described as dynamic Jahn− Teller distortions. It is argued that the influence of the fluxional behavior on the vibronic structure of the absorption spectra will be smaller than the experimental resolution since the potential surface along all the totally symmetric modes were found to be harmonic.

trend is also reproduced in our calculations, Tables 9 and 10. It thus seems that the Os−O bond remains stronger than the Ru−O link, even in the excited states. We note finally that the intensity of the components in the vibronic progression diminish faster in the simulated spectra than in the experimental spectra with a maximum at 3υ1. We attribute that to an underestimation of the M−O bond lengths in the excited states.

■

4. CONCLUDING REMARKS In this work, we have used TD-DFT to study the electronic absorption spectrum of MnO4−, TcO4−, RuO4, and OsO4. We optimized the geometries of the investigated complexes in their excited states using a newly implemented scheme based on TD-DFT. In the excited states, all the investigated complexes exhibit a Jahn−Teller distortion away from the Td ground state symmetry. The first excited state has a C3V geometry for all the investigated complexes. The second excited state has C2v whereas all other investigated complexes have a D2d geometry. The third excited state of MnO4− is of D2d geometry. We find that the first excited state is due to the 1t1 → 2e transition whereas the second band can be assigned to the 2t2 → 2e excitation for all the systems. The third excited state in MnO4− is mostly due to the 1t1 → 3t2 transition. This investigation represents the first computational study on the excited state structures of tetroxo d0 transition metal complexes. We have also simulated the vibrational fine structure present in the electronic absorption spectrum of the tetroxo d0 transition metal complexes MnO4−, TcO4−, RuO4, and OsO4. Our simulations reproduce the salient features including the vibrational progression. More accurate estimates of the excited state energies and structures will be required for a quantitative agreement. A comparison between theory and experiment

ASSOCIATED CONTENT

S Supporting Information *

Figures of orbitals involved in each transition. This material is available free of charge via the Internet at http://pubs.acs.org.

■ ■

AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS This work was supported by NSERC. The computational resources of WESTGRID were used for all calculations. T.Z. thanks the Canadian Government for a Canada Research Chair.

■

REFERENCES

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