MetalâMetal Bonding in Trinuclear, Mixed-Valence - ACS Publications

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Metal−Metal Bonding in Trinuclear, Mixed-Valence [Ti3X12]4− (X = F, Cl, Br, I) Face-Shared Complexes Jinasena W. Hewage,† Germán Cavigliasso,‡ and Robert Stranger*,‡ †

Department of Chemistry, University of Ruhuna, Matara, Sri Lanka Research School of Chemistry, Australian National University, Canberra, Australian Capital Territory 0200, Australia

‡

S Supporting Information *

ABSTRACT: Metal−metal bonding in structurally characterized In4Ti3Br12, comprising linear, mixed-valence d1d2d1 face-shared [Ti3Br12]4− units with a Ti−Ti separation of 3.087 Å and strong antiferromagnetic coupling (Θ = −1216 K), has been investigated using density functional theory. The antiferromagnetic configuration, in which the single d electron on each terminal TiIII (d1) metal center is aligned antiparallel to the two electrons occupying the central TiII (d2) metal site, is shown to best agree with the reported structural and magnetic data and is consistent with an S = 0 ground state in which two of the four metal-based electrons are involved in a twoelectron, three-center σ bond between the Ti atoms (formal Ti−Ti bond order of ∼0.5). However, the unpaired spin densities on the Ti sites indicate that while the metal−metal σ interaction is strong, the electrons are not fully paired off and consequently dominate the ground state antiferromagnetic coupling. The same overall partially delocalized bonding regime is predicted for the other three halide [Ti3X12]4− (X = F, Cl, I) systems with the metal−metal bonding becoming weaker as the halide group is descended. The possibility of bond-stretch isomerism was also examined where one isomer has a symmetric structure with identical Ti−Ti bonds while the other is unsymmetric with one short and one long Ti−Ti bond. Although calculations indicate that the latter form is more stable, the barrier to interconversion between equivalent unsymmetric forms, where the short Ti−Ti bond is on one side of the trinuclear unit or the other, is relatively small such that at room temperature only the averaged (symmetric) structure is likely to be observed.

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INTRODUCTION Trinuclear, transition metal complexes of the type [M3X12]z−, where X is a halide atom, can exist in two forms (Scheme 1)a

comprises face-shared bioctahedral metal centers where adjacent metals are bridged by three halide atoms. The linear face-shared [M3X12]z− complexes belong to a more general class of linear trinuclear metal structures in which the metal−metal interaction can vary from weak antiferromagnetic coupling through to multiple metal−metal bonding. These systems have attracted significant interest in recent years due to the possibility of exhibiting the phenomenon of “bondstretch isomerism”4,5 where a molecule can exist in two distinct structural forms which have the same spin state but differ only in the length of one or more bonds. For this to be possible, the ground state potential energy surface of the isolated molecule must be characterized by the presence of separate minima corresponding to the two isomeric forms and with a sufficient energy barrier between the minima to prevent interconversion. While the existence of true bond-stretch isomers is a rare occurrence, some of the better known examples involve polynuclear transition metal complexes, including the dinuclear system [(Cp*RuCl)2(μ-Cl)2]6,7 and trinuclear species of the type [M3(dpa)L2] or [M3(dpa)LL′], where M = Co or Cr, dpa is the dipyridilamide anion, and L, L′ are either halogens or

Scheme 1

triangular metal cluster, exemplified by the classic [Re3Cl12]3− structure, and a linear metal arrangement for which the [Ru3Cl12]4− and [Mo3I12]4− structures are representative examples.1−3 The triangular [Re3Cl12]3− structure type possesses ideal D3h symmetry where adjacent metal atoms are directly bonded to one another and also share a common bridging halide ligand. The linear [Ru3Cl12]4− structure type, on the other hand, possesses D3d symmetry in the ideal case and © XXXX American Chemical Society

Received: June 25, 2015

A

DOI: 10.1021/acs.inorgchem.5b01435 Inorg. Chem. XXXX, XXX, XXX−XXX

Article

Inorganic Chemistry molecular anions.8−13 A number of linear trinuclear [M3X12]z− systems have also been reported including complexes with M = Ru, Mo, and Ti and X = Cl, Br, and I.2,3,14−16 These systems exhibit significant variation in their structural and magnetic properties and, consequently, the extent of metal−metal interaction. The mixed-valence [Ru3Cl12]4− system, which comprises a central RuII and terminal RuIII ions, has been structurally characterized with a Ru−Ru distance of 2.805 Å.2 On the basis of SCF-Xα-SW calculations,17 this distance is consistent with direct metal−metal bonding and a Ru−Ru bond order of 1/2. The [Mo3X12]3− system is experimentally known for X = Cl, Br, and I,3,14 but only the iodide complex has been structurally characterized in the form of (PPh4)3[Mo3I12].3 For this compound, all three Mo centers are in the +3 oxidation state corresponding to a d3 metal configuration. Crystallographically, the [Mo3I12]3− anion has centrosymmetric (Ci) symmetry, but its structure is close to ideal D3d molecular symmetry with a Mo−Mo distance of 3.258 Å. The reported room-temperature magnetic moment of 3.17 μ B for (PPh4)3[Mo3I12]3 was substantially reduced from that expected for weakly coupled MoIII centers but was reasonably close to the value predicted for three unpaired electrons and a S = 3/2 ground state. Compared with the reported room-temperature magnetic moments 3,14 of 2.02 and 2.60 μ B for the corresponding Cl and Br complexes, a strong dependence on the halide is evident, with the antiferromagnetic coupling progressively weakening as the halide group is descended. On the basis of the structural and magnetic data it was proposed that the [Mo3I12]3− anion possessed a metal−metal σ bond of order 1/2 with the remaining seven metal-based electrons occupying essentially nonbonding orbitals. Recently, we investigated the electronic structure and metal− metal bonding in the linear trinuclear [Mo3X12]3− (X = F, Cl, Br, I) system18 using density functional theory (DFT) in combination with the broken-symmetry approach of Noodleman and co-workers.19,20 In the past we used a similar approach to investigate a diverse series of related dinuclear face-shared [M2X9]z− and edge-shared [M2X10]z− systems and showed that this methodology was capable of describing the entire range of metal−metal interactions from weak antiferromagnetic coupling through to multiple metal−metal bonding.21−35 In our study of the trinuclear [Mo3X12]3− system, we also examined the possibility of bond-stretch isomerism and, consequently, the existence of both symmetric (D3d) and unsymmetric (C3v) forms of these complexes as shown in Scheme 2. In the symmetric form, the metal−metal distances are identical, whereas the unsymmetric structure has one short and one long metal−metal distance. From this work, we were able to show that while the metal− metal interaction in the crystallographically characterized iodide complex was qualitatively consistent with a two-electron threecenter σ bond between the metal atoms, it was evident that the

electrons in this bond were not fully paired off and so still contributed to the strong antiferromagnetic coupling observed for this complex. The unsymmetric form of the complex, which has a short Mo−Mo triple bond and a long bond where the Mo centers are essentially decoupled, was found to be the most stable structure. However, the energetic barrier between the two unsymmetric forms, where the short metal−metal bond is on one side of the trinuclear unit or the other, was relatively small such that interconversion between the two would ensure that the averaged structure at room temperature was symmetric. The same outcome was predicted for the analogous [Mo 3 Cl 12 ] 3− and [Mo 3 Br 12 ] 3− complexes but not for [Mo3F12]3−. In the latter system, the energetic barrier for interconversion was substantially higher, and consequently, the unsymmetric form of the structure was predicted to persist, even at room temperature. Despite the crystallographic evidence for the symmetric form of [Mo3I12]3−, a vibrational frequency analysis of the [Mo3X12]3− (X = F, Cl, Br, I) system revealed that in each case the symmetric configuration was in fact a transition state structure and consequently cannot correspond to a true minimum on the ground state potential energy surface. Therefore, irrespective of whether the symmetric or unsymmetric form is observed at room temperature, the [Mo3X12]3− system cannot be considered as an example of true bond-stretch isomerism. In the current study, we apply the same approach to investigate the electronic structure and metal−metal interactions in the [Ti3X12]4− (X = F, Cl, Br, I) system with the focus on the crystallographically characterized [Ti3Br12]4− species.16 Unlike the d3d3d3 coupled [Mo3X12]3− system comprising only MoIII centers, the [Ti3X12]4− series represents a mixed-valence d1d2d1 system in which the two terminal metal sites are occupied by TiIII ions while the central metal is TiII. The structure of the related face-shared [Ti3Cl12]3− complex has also been reported,15 but in this system all Ti atoms are in the +3 oxidation state. Although the structure of this complex, in the form of the (PPh4)3Ti3Cl12 salt, is asymmetric, the Ti−Ti distances are similar, 3.098 and 3.185 Å. The reported roomtemperature magnetic moment of 2.07 μB is significantly reduced from that expected for three weakly coupled d1 centers, suggesting strong antiferromagnetic coupling between the TiIII ions. For the mixed-valence [Ti3X12]4− system, both In4Ti3Br12 and K4Ti3Br12 have been structurally characterized with Ti−Ti distances of 3.087 and 3.20 Å, respectively.16,36 In both cases, the molecular structure of the [Ti3Br12]4− anion is symmetric, and for In4Ti3Br12, ideal D3d symmetry is observed. From the temperature dependence of magnetic susceptibility, a large Weiss constant of Θ = −1216 K was obtained for In4Ti3Br12,16 indicating strong antiferromagnetic coupling between the metal centers. Analogous to the [Mo3X12]3− system, the possibility of bond-stretch isomers for the [Ti3X12]4− (X = F, Cl, Br, I) series is also explored by generating the potential energy curves for the various spin states of the symmetric (D3d) and unsymmetric (C3v) forms of these complexes.

Scheme 2

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CALCULATION DETAILS

All density functional calculations reported in this article were carried out with the Amsterdam Density Functional (ADF) package.37−39 Functionals based on the generalized-gradient-approximation (GGA) involving the exchange and correlation expressions due to Perdew, Burke, and Ernzerhof40 (labeled PBE) were utilized. Basis sets of triple-ζ quality and two (TZ2P) polarization functions were employed.41 Relativistic effects were included by means of the zeroB


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Inorganic Chemistry order regular approximation (ZORA),42−44 and the conductor-like screening model (COSMO) was used for the treatment of solvation effects.45 In earlier studies,18,46 we have shown that this methodology gives very good agreement with key structural parameters sensitive to the metal−metal interaction across a number of dinuclear and trinuclear complexes. However, we note that the small energy differences associated with the ground state spin ladder levels calculated in this work will, to some extent, be dependent on the choice of density functional. Default convergence parameters were used for self-consistent-field (SCF) procedures (Becke grid of normal quality) and geometry optimizations (gradient tolerance of 0.001).47,48 To establish the nature of the stationary points, frequency calculations were carried out using analytical second derivatives.49−51 Plots of the molecular orbitals were generated with the MOLEKEL program52 using data in MOLDEN format53,54 derived from the ADF TAPE21 files.

For the central TiII (d2) and terminal TiIII (d1) sites, the occupations of the t2g orbitals in C3v local symmetry are [(a1)1(e)1] and [(a1)1], respectively. Here it is assumed that the metal−metal interaction is sufficient to stabilize the a1 level relative to the e level such that for the TiIII sites, only the a1 level is occupied. The strong antiferromagnetic coupling (Θ = −1216 K) reported for In4Ti3Br12 is consistent with this orbital occupation.16 In this coupling regime, the [d1d2d1] configuration for the trimer corresponds to the [(a1)1(e)0 × (a1)1(e)1 × (a1)1(e)0] coupling mode which gives rise to S = 0, 1, and 2 spin states. These spin states can be described in the localized (C3v) limit by the single electron configurations shown on the left in Figure 2 but in the delocalized limit correlate with the

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RESULTS AND DISCUSSION From the crystallographically characterized compounds In4Ti3Br12 and K4Ti3Br12,16,36 the molecular structure of the symmetric form of the trinuclear [Ti3Br12]4− face-shared complex shown in Scheme 2 has ideal or approximate D3d symmetry with Ti−Ti distances of 3.087 and 3.20 Å, respectively, for the In and K salts. The Ti centers occupy distorted octahedral sites with C3v local symmetry for which the metal-based t2g orbitals involved in metal−metal interactions transform as a1 and e symmetry and participate in σ and δπ overlap, respectively, with adjacent metal centers. Analogous to their dinuclear [M2X9]n− counterparts,21,55 the δπ label indicates mixed (2/3δ + 1/3π) character due to the local trigonal symmetry. In the delocalized limit, the overlap of the metalbased t2g orbitals result in a1g, eg, eu, a2u, eg, and a1g molecular orbitals. The relative energies and orbital plots for these levels are shown in Figure 1 for the optimized D3d structure of [Ti3Br12]4−.

Figure 2. Correlation between the C3v and the D3d molecular orbital descriptions of the symmetric (D3d) form of [Ti3Br12]4−.

Figure 1. Calculated energy levels and orbital plots for the metal-based molecular orbitals in the symmetric (D3d) form of [Ti3Br12]4−.

The lowest lying a1g level is σ bonding between all three metal centers, while the highest lying a1g level is the σ* antibonding counterpart. The lowest and highest lying eg levels are weakly δπ bonding and antibonding, respectively, between all three metal centers. The remaining eu and a2u molecular orbitals comprise t2g orbitals only on the terminal metal sites and, consequently, are nonbonding between the metal centers, analogous to those found in [Mo3I12]3− and [Ru3Cl12]4−.17,18

following D3d molecular-orbital configurations which are also depicted on the right in Figure 2 BS (MS = 0) [(a1g)2 (eg )1(e u)0 (a 2u)1(eg )0 (a1g)0 ] S = 1 [(a1g)2 (eg )1(e u)0 (a 2u)1(eg )0 (a1g)0 ] S = 2 [(a1g)1(eg )1(e u)0 (a 2u)1(eg )0 (a1g)1] C


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Inorganic Chemistry In the localized limit, the S = 2 state corresponds to all metalbased electrons being aligned parallel, corresponding to the weakly coupled limit. The broken-symmetry (BS) state with Ms = 0 shown in Figure 2 represents the fully antiferromagnetic configuration where the electrons on terminal metal centers are aligned antiparallel to those on the central metal. The other broken-symmetry state with Ms = 1 corresponds to the electrons on the central metal being aligned parallel to one terminal metal site but antiparallel to the other. In the delocalized limit it corresponds to an S = 1 spin state where two electrons are paired up in a metal−metal σ bond while the remaining electrons are aligned parallel in the eg and a2u orbitals. The connection between the localized (C3v) singleion and delocalized (D3d) molecular-orbital descriptions of these states shown in Figure 2, is that when the antiferromagnetic coupling within a subset (σ or δπ) of electrons is weak, the associated ferromagnetic state, where the weakly coupled electrons are aligned parallel, lies close in energy to the BS state. Thus, when magnetic coupling between the metal centers is weak, the BS (MS = 0) state will lie close in energy to the ferromagnetic S = 2 state. However, when the magnetic coupling is strong, the BS (MS = 0) state will lie close in energy to the S = 1 state where two of the metal-based electrons are paired off in a metal−metal σ bond. In addition to the states described above, we must also consider another delocalized spin triplet state corresponding to the orbital configuration [(a1g)2(eg)2(eu)0(a2u)0(eg)0(a1g)0] shown in Figure 2. For this configuration, two metal-based electrons are again paired off in a metal−metal σ bond with the remaining two electrons aligned parallel in the lowest lying eg orbital which in principal is δπ bonding on the basis of the MO plots shown in Figure 1. Unlike the S = 1 and 2 states, this state does not correlate with any local C3v configuration based on the [(a1)1(e)0 × (a1)1(e)1 × (a1)1(e)0] coupling mode and, therefore, to distinguish it from the other spin triplet state is designated as S = 1′. Symmetric Structures. The optimized metal−metal distances, total bonding energies, and metal spin densities (derived from a Mulliken population analysis) for the BS (MS = 0), S = 1, S = 2, and S = 1′ states of the symmetric D3d structures of [Ti3X12]4− (X = F, Cl, Br, I) are given in Table 1, while the key structural parameters derived from the BS calculation are compared with those obtained experimentally for the crystallographically characterized [Ti3Br12]4− species in Table 2. The optimized structures of the BS and S = 1 states have almost identical energy, within 0.05 eV, whereas the S = 2 state lies nearly 0.4 eV to higher energy as a result of the metal−metal σ bond being broken. The optimized Ti−Ti distances of 3.08 and 3.04 Å for the BS and S = 1 states, respectively, are in very good agreement with the experimental value of 3.09 Å for the In+ salt of [Ti3Br12]4−. The calculated Ti−Ti distance for the fully ferromagnetic S = 2 state has increased to 3.49 Å, consistent with the fact that for this configuration, all metal-based electrons are weakly coupled. This structural comparison suggests that at least some of the metal-based electrons in [Ti3Br12]4− are undergoing metal− metal bonding interactions. From the data in Table 1, the optimized energy for the delocalized S = 1′ state is actually lower than the BS state by around 0.06 eV and, consequently, represents the minimum energy structure for the symmetric (D3d) form of [Ti3Br12]4−. However, since the optimized Ti−Ti distance of 2.97 Å for the S = 1′ state is 0.12 Å shorter than the experimental value, it

Table 1. Calculated Metal−Metal Distance (Ti−Ti in Angstroms), Total Bonding Energy (EB in electronvolts), and Spin Densities (ρ) for the BS (MS = 0), S = 1, S = 2, and S = 1′ States of the Symmetric (D3d) Forms of [Ti3X12]4− halide (X) F

state BS S= S= S= BS S= S= S= BS S= S= S= BS S= S= S=

Cl

Br

I

1 2 1′ 1 2 1′ 1 2 1′ 1 2 1′

ρ(Ti1,3), ρ(Ti2), ρtot

Ti−Ti

EB

2.571 2.541 2.943 2.488 2.920 2.893 3.326 2.839 3.081 3.043 3.493 2.974 3.266 3.237 3.720 3.136

−104.8433 −104.8687 −104.2804 −104.9561 −81.6474 −81.6974 −81.1548 −81.7058 −74.4379 −74.4844 −74.0626 −74.5036 −65.0247 −65.0499 −64.8656 −65.0866

−0.59, 0.85, 1.14, 0.60, −0.62, 1.01, 1.30, 0.74, −0.71, 1.12, 1.34, 0.81, −0.82, 1.28, 1.44, 0.92,

1.18, 2.36 0.30, 2.00 1.92, 4.20 0.76, 1.96 1.22, 2.46 0.03, 2.05 1.59, 4.19 0.53, 2.01 1.39, 2.81 −0.16, 2.40 1.60, 4.28 0.44, 2.06 1.58, 3.22 −0.35, 2.91 1.68, 4.56 0.36, 2.20

Table 2. Comparison of Calculated and Observed Structural Parameters for the Symmetric (D3d) Forms of [Ti3Br12]4− parameter c

Ti−Ti Ti−Br (bridging) Ti−Br (terminal) Br−Ti−Br (bridging) Br−Ti−Br (terminal)

calculationa

experimentb

3.081 Å 2.651−2.654 Å 2.602 Å 89−90° 92°

3.086 Å 2.602−2.624 Å 2.546 Å 88−92° 90°

a Values based on the BS (Ms = 0) state. bExperimental data for In4Ti3Br12.16 cTi−Ti distance is 3.20 Å for K4Ti3Br12.36

would appear that the BS state is a better representation of the [Ti3Br12]4− species in the solid state where crystallographic packing effects are likely to affect the metal−metal separation. Indeed, for the analogous K+ salt, the Ti−Ti has increased to 3.20 Å.36 We will address this issue further in the section on magnetic properties and show that once spin projection is taken into account, the energy of the antiferromagnetic S = 0 state is predicted to be the true ground state. The other calculated structural parameters listed in Table 2 are also in quite good agreement with the crystallographic data. In previous work,21−35 largely focused on face-shared [M2X9]z− and edge-shared [M2X10]z‑ dinuclear systems, we have shown that a satisfactory description of the entire range of metal−metal interactions can be achieved with an approach based on the analysis of the BS potential energy curve in terms of the curves for the associated spin states. The latter states result when one or more subsets of metal-based electrons are involved in metal−metal bonding while the remaining electrons are weakly coupled. This methodology has been successfully applied to both even and odd electron systems and more recently to d3d3d3 face-shared [M3X12]3− (M = Mo, W; X = F, Cl, Br, I) trinuclear systems.18 The same approach can be used to analyze the potential energy curves of the BS (MS = 0) and associated S = 1 and 2 states for the symmetric D3d structures of [Ti3X12]4− as shown for the bromide species in Figure 3 and for the remaining halide complexes in Figure S1 (see Supporting Information). These curves were obtained by systematically varying the metal−metal D


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Inorganic Chemistry

As yet, we have not considered the delocalized S = 1′ spin triplet state. The potential energy curve for this state is compared with the BS state in Figure 4 for [Ti3Br12]4− and for

Figure 3. Potential energy curves for the BS (MS = 0) and associated S = 1, 2 states of the symmetric (D3d) form of [Ti3Br12]4−.

distance while allowing all other structural parameters to fully optimize. At long Ti−Ti distances greater than 3.5 Å, the BS curve lies close to the S = 2 state, indicating that for these metal−metal separations all metal-based electrons are weakly coupled. For Ti−Ti distances less than 3.5 Å, the S = 2 state diverges to higher energy from the BS state, suggesting that in the latter the σ metal-based electrons are no longer weakly coupled but increasingly paired up in a metal−metal σ bond. Interestingly, the BS curve is observed to be convergent with the S = 1 associated state for Ti−Ti separations from 2.0 to 4.0 Å. This outcome can be rationalized on the basis of the molecular-orbital plots shown in Figure 2, where it can be seen that the only difference between the orbital configurations for the BS and S = 1 states is the flipping of the electron in the a2u level which is metal−metal nonbonding. To obtain some measure of the metal−metal δπ bonding interaction, the potential energy curve for the S = 1 associated state is compared with the curve generated for the alternative S = 1 state in Figure S2 (Supporting Information) arising from the [(a1g)2(eg)0(eu)1(a2u)1(eg)0(a1g)0] configuration where the electron originally occupying the δπ metal−metal bonding eg orbital is moved to the nonbonding eu level. This configuration maintains the metal−metal σ bond but has no δπ bonding. The two S = 1 curves are observed to diverge for Ti−Ti distances less than 3.1 Å, indicating that for Ti−Ti distances above this, the δπ interaction is negligible. In the case of the experimental [Ti3Br12]4− structure, the observed Ti−Ti distance of 3.086 Å would suggest that δπ metal−metal bonding is essentially absent. On the basis of the data in Table 1 and the potential energy curves shown in Figure S1, a similar conclusion is reached for the other halide systems [Ti3X12]3− (X = F, Cl, I). In all cases, the S = 2 curve lies above the BS curve except at long Ti−Ti distances where the two curves converge. The point of convergence progressively increases as the halide group is descended, corresponding to Ti−Ti distances greater than 3.0, 3.4, 3.5, and 3.7 Å, respectively, for X = F, Cl, Br, and I. Analogous to the Br system, the potential energy curves for the BS and S = 1 states are essentially superimposed over the entire range of metal−metal distances from 2.0 to 4.0 Å, consistent with the orbital picture described above. The minimum in the BS curve also shifts to longer Ti−Ti distances, corresponding to 2.57, 2.92, 3.08, and 3.27 Å for X = F, Cl, Br, and I, respectively.

Figure 4. Potential energy curves for the BS (Ms = 0) and S = 1′ states of the symmetric (D3d) form of [Ti3Br12]4−.

the remaining halide systems in Figure S3 (Supporting Information). In the case of [Ti3Br12]4−, it is apparent that the two curves converge for Ti−Ti distances greater than 3.0 Å but at shorter distances the S = 1′ curve lies to lower energy and thus becomes the ground state for the symmetric D3d structures. It is worthwhile commenting on why the delocalized S = 1′ state lies lower in energy than the BS state (and also the associated S = 1 state) at shorter metal−metal separations. From Figure 2 it is apparent that while both states have two electrons paired up in the a1g metal−metal σ bonding orbital, only one electron occupies the eg metal−metal δπ bonding orbital for the BS state, leading to a formal bond order of 0.75 (1/2σ + 1/4δπ) between adjacent Ti atoms, whereas two electrons occupy this orbital for the S = 1′ state, resulting in a bond order of 1.0 (1/2σ + 1/2δπ). At Ti−Ti distances below 3.1 Å, the direct overlap between the metal-based δπ orbitals becomes relevant, and so the greater bond order for the S = 1′ state leads to its additional stabilization over the BS state. Essentially the same trend is observed from the potential energy curves shown in Figure S3 (Supporting Information) for the other halide systems (X = F, Cl, I). In each case, the BS and S = 1′ states are superimposed at relatively large Ti−Ti distances, but at Ti−Ti separations below 2.6 (X = F), 2.9 (X = Cl), and 3.4 Å (X = I), the S = 1′ state is stabilized relative to the BS state and, analogous to [Ti3Br12]4−, corresponds to the energy minimum in all halide members for the symmetric (D3d) structures (see Table 1). The extent of delocalization arising from the metal−metal bonding interactions is best gauged from examining the unpaired spin densities on the metal centers shown in Table 1. For [Ti3Br12]4−, the total unpaired spin density (ρtot), summed over all three metal centers, is 2.81 for the BS state, whereas the corresponding values for the S = 1 and 2 associated states are 2.40 and 4.28, respectively. The unpaired spin density for the BS state lies between the values for the S = 1 and 2 states but clearly much closer to the former. This implies that in the BS state, the metal-based electrons are not completely paired off in a metal−metal σ bond, and consequently, the effective bond order will be slightly less than 0.5. The small E


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Inorganic Chemistry

short and long metal−metal distances. For each halide complex, it is evident that the unsymmetric BS curve is energetically lower for all metal−metal separations. The barrier in the unsymmetric (C3v) BS curve coincides with the minimum in the symmetric (D3d) BS curve, corresponding to a structure where both metal−metal distances are equal. In all cases, the energetic barrier between the two minima of the unsymmetric BS curve is small, less than 10 kJ mol−1. However, the barrier height increases as the halide group is descended from approximately 1 kJ mol−1 for the fluoride complex to 9 kJ mol−1 for the iodide. Interestingly, while the barrier heights in the related [Mo3X12]3− series ranged between 6 and 29 kJ mol−1, the reverse trend was observed where the barrier was greatest for the fluoride complex and smallest for the iodide.18 The metal−metal distances and energies of the minima in the symmetric (D3d) and unsymmetric (C3v) BS curves for all four complexes are given in Table 3. The short metal−metal

reduction in the bond order is consistent with the 0.04 Å increase in the metal−metal distance for the BS relative to the S = 1 state. Comparing the unpaired spin densities of 2.36 (F), 2.46 (Cl), 2.81 (Br), and 3.22 (I) in the BS state across the halide members, it is evident that as the group is descended, the metal-based σ electrons become progressively decoupled, reflected in the increasing value of the metal−metal distance in the BS state from 2.57 Å for the F complex to 3.27 Å for the iodide. The unpaired spin densities for the delocalized S = 1′ state are actually lower than the BS state for all halide members. Again, this is due to the S = 1′ state having two electrons occupying the metal−metal bonding δπ level compared to only one for the BS state. The enhanced metal−metal δπ interaction results in shorter Ti−Ti distances for the S = 1′ state and stronger coupling between the metal-based σ electrons. Unsymmetric Structures. In principle, [Ti3X12]4− complexes can possess C3v molecular symmetry where one of the metal−metal bond lengths is longer than the other (Scheme 2). This possibility is apparent from the electron configuration of the associated S = 1 state shown in Figure 2 where a nonsymmetric coupling arrangement occurs between the central Ti center and the two terminal Ti sites. Since the electron occupying the a1 single-ion orbital on one terminal site is aligned antiparallel to the a1 electron on the central metal, they can participate in a metal−metal σ bond, resulting in a shorter metal−metal distance. In contrast, the a1 electron on the other terminal site is parallel to the central metal, and therefore, only weak ferromagnetic coupling is possible leading to a longer metal−metal distance. To address the possibility that the unsymmetric forms of [Ti3X12]4− may correspond to the more stable structure, the potential energy curves for both the symmetric (D3d) and the unsymmetric (C3v) BS states are compared in Figure 5 for

Table 3. Comparison of Calculated Metal−Metal Distance (Ti−Ti in Angstroms) and Total Bonding Energy (EB in electronvolts) for the BS (MS = 0) States of the Symmetric (D3d) and Unsymmetric (C3v) Forms of [Ti3X12]4− halide (X)

symmetry

F

D3d C3v D3d C3v D3d C3v D3d C3v

Cl Br I

Ti−Ti 2.571 2.434, 2.920 2.708, 3.081 2.802, 3.266 2.872,

2.741 3.200 3.390 3.716

EB −104.8433 −104.8572 −81.6474 −81.6606 −74.4379 −74.4663 −65.0247 −65.1130

distance in the unsymmetric forms ranges from 2.43 Å for the fluoride complex to 2.87 Å for the iodide and is clearly smaller than the analogous long metal−metal distances which vary from 2.74 to 3.72 Å. The energies and molecular-orbital plots of the metal-based orbitals for the unsymmetric (C3v) form of [Ti3Br12]4− are shown in Figure 6. A qualitative description of the bonding in this complex corresponds to a formal bond order of 1.5 (σ + 0.5δπ) between the two Ti atoms with the shortest metal− metal separation, the remaining electron being mostly localized

Figure 5. Potential energy curves for the BS (MS = 0) states of the symmetric (D3d) and unsymmetric (C3v) forms of [Ti3Br12]4−.

[Ti3Br12]4− and in Figure S4 (Supporting Information) for all four halide systems. The unsymmetric BS curves were generated by lowering the molecular symmetry to C3v and systematically varying one of the metal−metal distances while allowing all other structural parameters, including the second metal−metal distance, to fully optimize. In general, the curves for the unsymmetric BS states for all four halide systems exhibit a shallow double-minima structure representing the two possible configurations with alternating

Figure 6. Orbital energy levels and spatial plots for the unsymmetric (C3v) BS (MS = 0) state of [Ti3Br12]4−. F


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Inorganic Chemistry

[Ti3Br12]4−, the unpaired spin densities of 1.34 (terminal Ti) and 1.60 (central Ti) for the ferromagnetic S = 2 state indicate that when the magnetic coupling is weak (calculated Ti−Ti distance of 3.49 Å), electron density is transferred from the central to the terminal metal centers. As a result, the spin densities are not truly representative of a d1d2d1 coupled system but instead are closer to a configuration in which all three metals have similar unpaired spin densities. The total unpaired spin density of 2.40 for the S = 1 state compared to 4.28 for S = 2 is more or less consistent with the pairing of approximately two electrons in a metal−metal bond. This is certainly the case for the delocalized S = 1′ state where the calculated spin density is 2.06. However, while the S = 1′ state is calculated to lie lower in energy than the BS state for the symmetric structures, its orbital occupancy cannot account for the strong antiferromagnetic behavior observed for the Br complex. For this state, two electrons are delocalized in the a1g metal−metal σ bond with the remaining two electrons occupying the lowest eg level (see Figure 2). The degeneracy of this level ensures that the two unpaired electrons are aligned parallel in orthogonal orbitals and consequently are not able to participate in antiferromagnetic coupling. The BS state, which is calculated to lie approximately 6 kJ mol−1 above the S = 1′ state, more closely reflects the magnetic coupling in the ground state of [Ti3Br12]4−. The total unpaired spin density of 2.81 for the BS state suggests that the electrons involved in the metal−metal σ bond are not fully paired off and consequently still contribute to the magnetic coupling. The same conclusion is reached for the F, Cl, and I complexes as the total unpaired spin density in the BS state is at least 0.3 electrons greater than for the corresponding S = 1 state where two electrons are fully paired off in a metal−metal σ bond. The increase in the total unpaired spin density from 2.36 (X = F) to 3.22 (X = I) for the BS state as the halide group is descended reflects the progressive weakening in the metal−metal bonding as the metal separation increases from 2.57 Å for the fluoride complex to 3.27 Å for the iodide. The spin densities obtained for the BS state for the Br complex suggest that electron density is lost from all three metal centers. Furthermore, the ρ values of 0.71 (terminal Ti) and 1.39 (central Ti) are in a ratio of 1:2:1, suggesting that a model which assumes all four electrons participate in the magnetic coupling is more realistic than one in which two of the magnetic coupled electrons are paired off in a metal−metal bond. In this coupling model where S1 = S3 = 1/2 and S2 = 1, total spin states with S = 0, 1, and 2 states are possible for the trinuclear system. In the analysis to follow, we make use of the broken-symmetry approach combined with spin projection to determine the energies of the low-lying spin states in the experimentally characterized [Ti3Br12]4− complex and compare its behavior with that for the analogous complexes with X = F, Cl, and I. For trinuclear complexes, and polynuclear systems in general, the broken-symmetry approach can give rise to more than one antiferromagnetic state by locating the spin-up and spin-down electrons on different metal centers (see Figure 2). In the case of the [Ti3X12]4− system, two BS states are possible. A symmetric BS state with MS = 0 results when the magnetic electrons on the central metal are aligned antiparallel to those on the terminal metal centers. This BS state retains the full D3d electronic symmetry of the complex. The other BS state with MS = 1 arises when the electrons on the central metal are aligned parallel to one terminal site but antiparallel to the other.

on the a1 single-ion orbital of the other terminal Ti atom. Not unexpectedly, the energetic difference between the symmetric and the unsymmetric forms of [Ti3X12]4− is sensitive to the identity of the halide with the unsymmetric structure becoming increasingly stabilized relative to the symmetric form as the halide changes from F to I. On the basis of the definition by Parkin,56 the symmetric (D3d) and unsymmetric (C3v) forms of the [Ti3X12]4− systems can be considered as examples of bond-stretch isomerism as they correspond to “molecules that differ only in the length of one or more bonds”. However, a further consideration, as noted by Rohmer and Bénard,4 is that the identity of the two structures must be established as separate minima. Analogous to our earlier study on [Mo3X12]3−,18 vibrational frequency analysis confirms that the unsymmetric forms of [Ti3X12]4− are indeed minima but the symmetric structures correspond to transition states which connect the two possible configurations of the unsymmetric species where the short metal−metal bond alternates from one side of the molecule to the other. The failure to establish a minimum structure for the symmetric forms of [Ti3X12]4− implies that this system cannot give rise to bond-stretch isomers. While the calculations indicate that the unsymmetric form of [Ti3Br12]4− is more stable, the symmetric structure observed crystallographically for In4Ti3Br12 can be straightforwardly rationalized on the basis that the calculated barrier to interconversion is very small, around 3 kJ mol−1. In fact, the real barrier will be even smaller as the true antiferromagnetic S = 0 ground state for the symmetric structure will lie lower in energy than the BS state. Consequently, it is likely that the experimental structure simply corresponds to the symmetric average of the two equivalent unsymmetric configurations. Indeed, the reported crystal structure of In4Ti3Br12 exhibits a thermal ellipsoid on the central Ti ion which is elongated toward the terminal Ti atoms. While this elongation could be the result of crystallographic disorder, on the basis of our calculations, it is more likely a reflection of rapid interconversion between the unsymmetric forms. Since the calculated barrier to interconversion is at most only 9 kJ mol−1 for all [Ti3X12]4− species (see Table 3), the transition between the two equivalent unsymmetric forms is likely to be quite facile at room temperature. As a result, none of these systems are expected to exist in the unsymmetric form. This outcome differs from that found for the related [Mo3X12]3− system where the calculated barrier to interconversion for [Mo3F12]3− was large enough to trap the structure in the unsymmetric form.18 Magnetic Properties. The magnetic susceptibility measurements on In4Ti3Br12 between 108 and 298 K revealed that the compound was strongly antiferromagnetic with a large Weiss constant of Θ = −1216 K.16 At 300 K, the average magnetic moment per Ti center was measured at approximately 1.58 μB, which is significantly lower than the spin-only value of 2.16 μB expected on the basis of two TiIII and one TiII centers. However, the room-temperature magnetic moment is quite close to 1.63 μB per Ti atom, which is the value predicted if two of the four metal-based electrons are paired off in a twoelectron three-center bond while the remaining two electrons are weakly coupled. The unpaired spin densities given in Table 1, which were obtained from a Mulliken population analysis of the BS, S = 1, S = 2, and S = 1′ states, shed further light on the magnetic coupling in the [Ti3X12]4− (X = F, Cl, Br, I) system. For G


Article

Inorganic Chemistry

where S′ = S1 + S3 and S = S′ + S2. Therefore, for S1 = S3 = 1/2 and S2 = 1, the values that S′ can take are 0 and 1, while the total spin S has values of S = 0, 1, and 2. It should be noted that in applying a Heisenberg formalism, we have ignored the orbital degeneracy of the ground state spin manifold. However, the large antiferromagnetic coupling is driven by the σ bonding interaction between the metal centers which only involves electrons in the nondegenerate a1 orbital on each Ti atom. In essence, the single electron in the degenerate e level on the central TiII atom will contribute a constant ferromagnetic term to each of the spin states, but this contribution is likely to be small given the negligible overlap between the δπ orbitals on the Ti centers and is therefore neglected. Using expression 5 above and eqs 2−4 derived earlier for Jij, one can show that the true antiferromagnetic S = 0 state will lie −2J12 lower in energy than the BS state. Since the magnetic exchange coupling constant, J12, is calculated to be quite large (between 320 and 1140 cm−1), this ensures that the S = 0 state will lie lower than the delocalized S = 1′ state and thus corresponds to the true ground state for the [Ti3X12]4− system. In the case of [Ti3Br12]4− for which −J12 is 757 cm−1, the S = 0 ground state is predicted to lie nearly 1000 cm−1 below the S = 1′ state. This energy difference implies that at room temperatures the S = 1′ state is essentially unpopulated, and consequently, the ground state magnetic behavior for [Ti3Br12]4− will be dominated by the S = 0 state, consistent with the strong antiferromagnetic behavior observed for this system.

Because of the asymmetric coupling arrangement, this BS state possesses C3v electronic symmetry. In general, the BS state of lowest energy will reflect more closely the spin coupling arrangement existing in the ground state. For spin-coupled polynuclear metal complexes, the energies of the broken-symmetry states are given by H=

∑ −2Jij

Sî ·Sĵ (1)

ij

where the expectation values ⟨Ŝi·Ŝj⟩ correspond to +SiSj and −SiSj for parallel and antiparallel alignment of spins on metal centers Mi and Mj, respectively, and Jij are the magnetic exchange coupling constants. Applying eq 1 to a symmetric linear trinuclear cluster with S1 = S3 = 1/2 and S2 = 1 leads to the following expressions for the relative energies of the highspin S = 2, BS (MS = 0), and BS (MS = 1) states E[S = 2] = −2J12 − J13 /2

(2)

E[BS(MS = 0)] = 2J12 − J13 /2

(3)

E[BS(MS = 1)] = J13 /2

(4)

where J12 parametrizes the exchange coupling between terminal and central metal sites and J13 the coupling between the two terminal sites. Unfortunately, evaluation of the J13 values was precluded on the basis that it was not possible to obtain energies for the BS (MS = 1) state, due to SCF convergence issues. However, analogous to the [Mo3X12]3− series, the J13 parameters are expected to be quite small with the antiferromagnetic coupling dominated by the J12 values. From the above expressions and the calculated energies for the S = 2 and BS (MS = 0) states in D3d molecular symmetry given in Table 1, the values of J12 for all halide members can be determined. For [Ti3Br12]4−, a value of J12 = −757 cm−1 is obtained, indicative of strong antiferromagnetic coupling and consistent with the large antiferromagnetic Weiss constant of Θ = −1216 K reported for this system.16 This value of J12 is also of a similar magnitude to the value of J = −430 cm−1 and J = −380 cm−1 reported for In+ and Cs+ salts of the related dinuclear Ti(III) complex [Ti2Br9]3−.57,58 The trend in calculated −J12 values of 1135, 993, 757, and 321 cm−1 for the [Ti3X12]4− series with X = F, Cl, Br, and I, respectively, reflects that found for the unpaired spin densities on the metal centers. Specifically, as the halide group is descended, the −J12 values indicate a progressive reduction in the metal−metal interaction which in turn is reflected in the increase in metal−metal separation from approximately 2.57 (X = F) to 3.27 Å (X = I) seen in Table 1. Finally, while the delocalized spin triplet S′ = 1 state is calculated to lie lower in energy than the antiferromagnetic BS state, the true antiferromagnetic S = 0 state may be even lower in energy. Since the latter state is multideterminant, its energy cannot be calculated directly using DFT methods which are, by nature, single-determinant based. Instead, this possibility can be addressed by determining the relative energies of the low-lying, exchange-coupled spin states from the following expression derived from the Heisenberg spin Hamiltonian for a symmetric trinuclear system18

■

CONCLUSION The metal−metal interaction in structurally characterized In4Ti3Br12, comprising linear, face-shared [Ti3Br12]4− units with terminal TiIII (d1) and central TiII (d2) sites, has been investigated using broken-symmetry density functional theory and compared with the related halide (X = F, Cl, I) members. In the absence of spin projection, which is necessary to determine the energy of the multideterminant, antiferromagnetic S = 0 state, the lowest energy structure associated with all four mixed-valence d1d2d1 trinuclear [Ti3X12]4− species corresponds to the delocalized, symmetric (D3d) spin-triplet configuration in which two of the four metal-based electrons are paired up in a two-electron, three-center σ bond between the Ti atoms, corresponding to a formal Ti−Ti bond order of 0.5. The remaining two electrons are aligned parallel in the degenerate metal−metal δπ bonding level. The strong antiferromagnetic coupling observed for the Br complex, however, is not consistent with this delocalized bonding picture, which necessarily enforces ferromagnetic coupling between the δπ electrons. The calculated metal−metal distance for this spin-triplet state is also at least 0.12 Å shorter than observed experimentally. The conflict can be resolved on the basis that not only does the antiferromagnetic brokensymmetry configuration, in which the single d electron on each of the terminal metal centers is aligned antiparallel to the two electrons occupying the central metal site, lie very close in energy but also the calculated Ti−Ti distance of 3.08 Å for this state is in excellent agreement with the crystallographically determined value of 3.09 Å. Furthermore, on the basis of spin projection, the antiferromagnetic S = 0 state is predicted to lie −2J12 (nearly 1000 cm−1) lower in energy than the BS state and, consequently, will correspond to the true ground state for [Ti3Br12]4−, indeed all [Ti3X12]4−.

E(S′, S) = −J12 [S(S + 1)] + J12 [S′(S′ + 1)] −J13[S′(S′ + 1)]

(5) H


Article

Inorganic Chemistry

(3) Fettinger, J. C.; Gordon, J. C.; Mattamana, S. P.; O’Connor, C. J.; Poli, R.; Salem, G. Inorg. Chem. 1996, 35, 7404−7412. (4) Rohmer, M.-M.; Bénard, M. Chem. Soc. Rev. 2001, 30, 340−354. (5) Pantazis, D. A.; McGrady, J. E. J. Am. Chem. Soc. 2006, 128, 4128−4135. (6) McGrady, J. E. Angew. Chem., Int. Ed. 2000, 39, 3077−3079. (7) Feng, Q.; Rauchfuss, T. B.; Wilson, S. J. Am. Chem. Soc. 1995, 117, 4702−4703. (8) Clérac, R.; Cotton, F. A.; Daniels, L. M.; Dunbar, K. R.; Murillo, C. A.; Pascual, I. Inorg. Chem. 2000, 39, 748−751. (9) Clérac, R.; Cotton, F. A.; Dunbar, K. R.; Lu, T. B.; Murillo, C. A.; Wang, X. J. Am. Chem. Soc. 2000, 122, 2272−2278. (10) Clérac, R.; Cotton, F. A.; Daniels, L. M.; Dunbar, K. R.; Kirshbaum, K.; Murillo, C. A.; Pinketon, A. A.; Schultz, A. J.; Wang, X. J. Am. Chem. Soc. 2000, 122, 6226−6236. (11) Clérac, R.; Cotton, F. A.; Daniels, L. M.; Dunbar, K. R.; Murillo, C. A.; Wang, X. Inorg. Chem. 2001, 40, 1256−1264. (12) Rohmer, M. M.; Strich, A.; Bénard, M.; Malrieu, J. P. J. Am. Chem. Soc. 2001, 123, 9126−9134. (13) Berry, J. F.; Cotton, F. A.; Lu, T.; Murillo, C. A.; Roberts, B. K.; Wang, X. J. Am. Chem. Soc. 2004, 126, 7082−7096. (14) Delphin, W. H.; Wentworth, R. A. D.; Matson, M. S. Inorg. Chem. 1974, 13, 2552−2555. (15) Chen, L.; Cotton, F. A. Polyhedron 1998, 17, 3727−3734. (16) Schroeder, M.; Hartwig, S.; Krämer, K. W.; Decurtins, S.; Hillebrecht, H. Inorg. Chem. 2012, 51, 8385−8393. (17) Bursten, B. E.; Cotton, F. A.; Fang, A. Inorg. Chem. 1983, 22, 2127−2133. (18) Cavigliasso, G.; Stranger, R. Inorg. Chem. 2008, 47, 3072−3083. (19) Noodleman, L.; Norman, J. G. J. Chem. Phys. 1979, 70, 4903− 4906. (20) Noodleman, L. J. Chem. Phys. 1981, 74, 5737−5743. (21) Lovell, T.; McGrady, J. E.; Stranger, R.; Macgregor, S. A. Inorg. Chem. 1996, 35, 3079−3080. (22) McGrady, J. E.; Stranger, R.; Lovell, T. J. Phys. Chem. A 1997, 101, 6265−6272. (23) Stranger, R.; McGrady, J. E.; Lovell, T. Inorg. Chem. 1998, 37, 6795−6806. (24) Lovell, T.; Stranger, R.; McGrady, J. E. Inorg. Chem. 2001, 40, 39−43. (25) Stranger, R.; Turner, A.; Delfs, C. D. Inorg. Chem. 2001, 40, 4093−4100. (26) Petrie, S.; Stranger, R. Polyhedron 2002, 21, 1163−1175. (27) Stranger, R.; Lovell, T.; McGrady, J. E. Polyhedron 2002, 21, 1969−1977. (28) Cavigliasso, G.; Stranger, R. Inorg. Chem. 2003, 42, 5252−5258. (29) Cavigliasso, G.; Stranger, R. Inorg. Chem. 2004, 43, 2368−2378. (30) Cavigliasso, G.; Comba, P.; Stranger, R. Inorg. Chem. 2004, 43, 6734−6744. (31) Cavigliasso, G.; Stranger, R. Inorg. Chem. 2005, 44, 5081−5091. (32) McGrady, J. E.; Stranger, R.; Lovell, T. Inorg. Chem. 1998, 37, 3802−3808. (33) Stranger, R.; Lovell, T.; McGrady, J. E. Inorg. Chem. 1999, 38, 5510−5518. (34) Cavigliasso, G.; Lovell, T.; Stranger, R. J. Chem. Soc., Dalton Trans. 2006, 2017−2025. (35) Cavigliasso, G.; Yu, C.-Y.; Stranger, R. Polyhedron 2007, 26, 2942−2948. (36) Jongen, L.; Meyer, G. Z. Anorg. Allg. Chem. 2004, 630, 1732. (37) ADF2013; SCM, Theoretical Chemistry, Vrije Universiteit: Amsterdam, The Netherlands, 2013; http://www.scm.com. (38) Fonseca Guerra, C.; Snijders, J. G.; te Velde, G.; Baerends, E. J. Theor. Chem. Acc. 1998, 99, 391−403. (39) te Velde, G.; Bickelhaupt, F. M.; van Gisbergen, S. J. A.; Fonseca Guerra, C.; Baerends, E. J.; Snijders, J. G.; Ziegler, T. J. Comput. Chem. 2001, 22, 931−967. (40) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865−3868.

The unpaired spin density [ρtot = 2.81] on the terminal and central Ti sites in [Ti3Br12]4− for the BS state indicate that while the metal−metal interaction is strong, the two electrons are not fully paired off in a metal−metal σ bond and consequently still contribute to the antiferromagnetic coupling in this complex. The calculated magnetic exchange coupling constant of −J12 = 757 cm−1 is consistent with this partially localized bonding picture and also the large reported antiferromagnetic Weiss constant of Θ = −1216 K. On the basis of calculations, the same overall bonding scenario applies to the other three halide systems [Ti3X12]4− (X = F, Cl, I). In the case of [Ti3F12]4− and [Ti3Cl12]4−, both the lower, unpaired spin densities [ρtot = 2.36 (F); ρtot = 2.46 (Cl)], shorter metal− metal distances [2.57 Å (F); 2.92 Å (Cl)], and larger calculated exchange coupling constants [−J12 = 1135 cm−1 (F); −J12 = 993 cm−1 (Cl)] relative to the Br complex indicate that the metal−metal interaction is stronger in these systems. For [Ti3I12]4−, the increase in the metal−metal distance to 3.27 Å is indicative of a substantial reduction in the metal−metal interaction, and this is reflected in the higher unpaired spin density of ρtot = 3.22 and significantly smaller exchange coupling constant of −J12 = 321 cm−1. The possibility of so-called bond-stretch isomerism was also examined. In principle, for the [Ti3X12]4− motif, one isomer corresponds to the symmetric (D3d) structure in which both Ti−Ti bonds are identical, while the other isomer has an unsymmetric (C3v) structure with one short and one long metal−metal bond. The calculations indicate that the unsymmetric form is more stable and that the symmetric structure is simply a transition state connecting two energetically equivalent unsymmetric forms where the short metal− metal bond alternates between one side of the trinuclear unit and the other. However, the barrier to interconversion between the unsymmetric forms is relatively small, less than 9 kJ mol−1, such that at room temperature only the averaged (symmetric) structure is likely to be observed.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.5b01435. Additional potential energy curve data for the brokensymmetry (BS), S = 1, S = 2, and S = 1′ states of [Ti3X12]4− (X = F, Cl, I) systems (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the Australian National University and the National Computational Infrastructure (NCI) for access to supercomputing facilities.

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REFERENCES

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