Revisiting the polytopal rearrangements in penta-coordinate d7-metallocomplexes: modified Berry pseudorotation, octahedral switch, and butterfly isomerization

New rearrangement mechanism prototypes are introduced to account for the non-ideal coordination and fluxional behavior of five-coordinate metalo-complexes.

A host of hypothetical pathways has been proposed over the past decades for different metal and ligand combinations. 2-11,13a,14,16-43 For the isomerization in TBP complexes, Muetterties and co-workers have outlined six general mechanisms following topological considerations based on nuclear magnetic resonance (NMR) data. 26 Of these, the Berry pseudorotation (BPR; see Fig. 1)the perhaps most prominent intramolecular exchange mechanism in coordination chemistry 34appears to be the only one that could be conrmed based on its ability to explain the experimentally observed stereochemical non-rigidity of penta-coordinate compounds. 3,6,19,24,26,[36][37][38] The classical BPR mechanism proceeds through a regular SP transition state (TS) of C 4v symmetry featuring a large apical-M-basal (droop) angle q (Fig. 1). 14,[20][21][22] In equilibrium SP structures, q is typically ranging from 105 to 125 . 22 The TS BPR connects two chemically equivalent TBP stereoisomers of D 3h symmetry, in which the axial and equatorial ligands as switched. The characteristic stereomutation coordinate shown in Fig. 1 indicates that two basal ligands (1 and 2) are moving towards the trans-position pivotal ligand 5, while the other two (3 and 4) are simultaneously moving away from it. 26b, 34 Another famous mechanism postulated by Muetterties is the tetrahedral jump (TJ) 3,8,43 (initially called tetrahedral tunneling 8 ) for quasi-tetrahedral HML 4 and H 2 ML 4 complexes, in which "hydrogen atoms have been considered to be at tetrahedral facial positions in the ground state". 43 To explain their NMR data, the authors proposed a concerted process, in which "a [ligand-metal-ligand] angle increases with concomitant transverse of a hydrogen atom to this affected tetrahedral edge, and then the [ligand-metal-ligand] angle decreases as the H atom goes to a new tetrahedral face" 43 (see Fig. 2).
It is worth stressing that the TJ mechanismdespite being the subject of an extensive discussionhas never been conrmed by either experiment or calculations. 8,32,59b Except for the BPR, the mechanisms outlined in ref. 26 also have never been observed, and the predicted barriers are too high to be accessible under realistic conditions. 24 A combination of two of these mechanisms constitutes another famous polytopal process, i.e., the turnstile rotation (TSR). 39,40 It involves an internal rotation of a ligand pair relative to the other three ligands. The TSR interchanges one axial and two equatorial ligands in a motion that resembles that of a turnstile gate. However, the TSR mechanism was dismissed too, in a series of electronic structure studies 24,41,42 in favor of the BPR. It should be noted, though, that a revised TSR version involving a threefold cyclic permutation (120 rotational motion vs. 60 in the original TSR mechanism 39 ) was suggested by Lammertsma et al. 24 A reverse BPR, in which an SP complex rearranges through a TBP TS, has been postulated in early studies as well. 14,44,45 More recently, the term "reverse BPR" has been used for pathways with edge-bridged (or edge-capped 31 ) tetrahedral distortions involving a bending motion of the axial ligands in the opposite direction to the one shown in Fig. 1. 27 Distortions at different points of the reverse BPR coordinate have been mapped in ref. 31 for a set of experimental structures to complement the data provided by Muetterties for regular BPRs. 46 In addition, a mini-review by Gusev and Berke 11 has stressed the importance of two principal types of ligand motion migratory and replacementto distinguish the dynamic behavior of metal polyhydrides MH n L m (with n $ 3). The authors have primarily used this classication to rationalize the uxional mechanisms in six-and seven-coordinated complexes, while its applicability to ve-coordinate systems remains less clear.
The paper at hand addresses the question, which (if any) of the polytopal rearrangement mechanisms suggested in the past are relevant for penta-coordinate (formally) d 7 -systems. Our work offers rst-principles mechanistic ndings from a careful and comprehensive analysis of the potential energy surfaces (PESs) of a suitable set of coordination compounds. We focus  on the iron tetracarbonyl hydride radical HFec(CO) 4 as a simple and clean prototype of this class of systems, and then expand the scope of our discussion to a number of isolobal analogues.
HFec(CO) 4 was rst observed by Krusic et al. in the mid-1980s using electron paramagnetic resonance (EPR) spectroscopy. [47][48][49] Nagorski and Mirbach proposed to utilize this radical as a precursor for the coordinatively unsaturated catalyst HFec(CO) 3 in the hydrogenation of 1-octane. 50 Our work was motivated by the observation of several elementary reactions involving polytopal rearrangements during a study of radical pathways in the photochemical and thermal hydrogenation of ethylene by Fe(CO) 5 -derived catalysts. (The study of the corresponding molecular pathways was reported in ref. 15.) We found that these rearrangements play an intricate role in the switching between alternative catalytic pathways. A full account of our ndings for the [HFec(CO) 4 /HFec(CO) 3 ]-catalyzed hydrogenation of ethylene will be reported elsewhere.
In Section 2 of this paper we will provide a short overview of the computational methodology used in current work. Section 3 describes our investigations of the stationary points found on the HFec(CO) 4 PES. We will analyze the resulting isomers, polytopal rearrangement mechanisms, and overall PES features before putting them into the context of prior work on this topic. We will also rationalize our ndings based on electronic structure considerations, subsequently compare the HFec(CO) 4 prototype with other coordination compounds, and discuss the transferability of our ndings. We will summarize the insights gained from this work in Section 4.

Methods and computational details
The study at hand is primarily based on Kohn-Sham density functional theory (DFT) 52,53 using generalized gradient approximation (GGA) 54 and hybrid functionals. 55 In spite of a number of well-known deciencies and limitations for more complex electronic structure situations, DFT is widely used as a baseline tool in computational coordination chemistry. 56 Its performance has been extensively studied for various properties and systems, including cases in the same problem domain as the work presented here. 15,36,51,[57][58][59]64 For instance, the barrier height of the iconic BPR rearrangement in Fe(CO) 5 was computed by Harris et al. at the BP86 GGA level to be 2.1 kcal mol À1 , 36 which Fig. 3 Equilibrium geometries of the HFec(CO) 4 isomers (left), calculated at UB3LYP/BS-I and UCCSD/BS-III (data in parentheses) level of theory and isosurface plots of the corresponding singly occupied molecular orbitals (SOMOs) at ROB3LYP level (right). The SOMOs underscore that the unpaired electron has primarily metal-d z 2 character (partially hybridized with ligand p z -orbitals) in both isomers, and that it is localized along the (a) cisand (b) trans-directions to the corresponding Fe-H bonds, respectively. A Mulliken population analysis places 92% of the unpaired electron density on the Fe atom in the qSP and 82% in the EBT structure.
is in remarkably good agreement to the experimental activation energy of 1.6 AE 0.3 kcal mol À1 obtained via temperaturedependent two-dimensional infrared spectroscopy. 36 While no experimental data is available for the geometries of the HFec(CO) 4 isomers to benchmark the employed geometry optimization approach, it is known that DFT successfully predicts the geometries of the precursor H 2 Fe(CO) 4 (ref. 58 and 59a) and various open-shell derivatives of the HFec(CO) 4 radical (a review can be found in ref. 15).
In addition to the DFT study, we also provide higher-level ab initio wavefunction theory results to support key ndings of our work. These include second-order Møller-Plesset perturbation theory (MP2), 60 and coupled cluster theory with singles, doubles (CCSD), and perturbative triples amplitudes (CCSD(T)). 61 The latter is considered the "gold standard" of computational quantum chemistry (see, e.g., ref. 15,[62][63][64]. Carbon monoxide is a strong-eld ligand in the spectrochemical series, and the metal in the HFec(CO) 4 complex is thus in oxidation state +1 with a low-spin doublet (S ¼ 1/2) electron conguration. Our analysis of the corresponding HFec(CO) 4 PES is based on the following computational protocol: we performed a full geometry optimization using the BP86 functional 65 (which is known as an economic path to accurate geometries in coordination compounds 56 ) in conjunction with the LanL2TZ(f) effective core potential (ECP) 66 for iron and the augmented double-z Dunning basis aug-cc-pVDZ 67 for all other elements. We abbreviate this custom compound basis set as BS-I. To assess the quality of these baseline-level results, we reoptimized the geometries using the hybrid functional B3LYP and for selected cases the MP2 and CCSD level. The DFT results are in good agreement with those from CCSD, while MP2 shows larger discrepancies. MP2 is wellknown to underestimate the metal carbonyl (M-CO) bond distances for the rst-row transition metal complexes. 25,68,69 For one of the isomers in this study, we obtained the following Fe-H bond distances, which conrm this issue: MP2: 1.35Å; DFT: 1.52Å; CCSD: 1.56Å (cf. Fig. 3).
We also tested for basis set dependence by employing the allelectron augmented triple-z Pople basis 6-311+G(2d,p) (abbreviated BS-II), which is known to yield good geometries when combined with B3LYP. 70 The differences were very minor, which demonstrates that BS-I is an adequate choice.
We performed frequency calculations at the same DFT levels to conrm the nature of each stationary point and to obtain thermodynamic corrections to the electronic energies (including zero-point energy corrections) and Gibbs free energies. Our study also employed internal reaction coordinate (IRC) calculations, for which the end point geometries were fully optimized to ensure connectivity between proper minima.
To obtain the best possible accuracy in the energetics, we performed high-level CCSD(T) single-point calculations on the stationary points obtained from B3LYP. The coupled cluster calculations employed the before-mentioned ECP for Fe and the standard double-z 6-31G(d,p) 71 basis for the remaining elements. We abbreviate this basis set as BS-III. We also performed allelectron calculations with the BS-II basis set. To assess the degree of multireference character of a given system (which may limit the validity of CCSD(T) results), we performed T1-amplitude checks.
All calculations were carried out with the Gaussian 03 program package, 72 utilizing the unrestricted, open-shell framework to account for the radical nature of the coordination compounds at hand. A restricted open-shell approach was used to create the radical single occupied molecular orbital (SOMO) plot in Fig. 3 and the orbital diagram in Fig. 4. We employed the ne grid option throughout, and an even ner grid (225 974 points) in selected cases to conrm the validity of small imaginary frequencies in Berry-type TSs. Excess spin densities were computed using the Mulliken population analysis, 73 and we performed tests to assess the extent of basis-set dependence. The analysis results presented below are based on the largest basis set employed in this work, i.e., BS-I.
The generally good agreement between the results obtained from different methods (see Table 1)with the exception of MP2underscores their robustness and applicability, and thus supports the choices we made in designing our computational protocols. The distinct differences between the MP2 results and those of all other methods are readily explained based on the before-mentioned failure of MP2 to correctly describe the M-CO bonds in the systems at hand. Since the origin of the discrepancies is clear we will largely omit discussing the awed MP2 results in Section 3. A comprehensive summary of all identied structures is provided as ESI (Table S1 †).

Isomers of HFec(CO) 4
Based on the EPR data of Krusic and co-workers as well as an extended Hückel theory analysis, HFec(CO) 4 has previously been assigned an SP equilibrium geometry of C 4v symmetry with the hydrogen located in the apical position. [47][48][49] No evidence has been reported to support a TBP structure with an axial H-ligand.
Our rst-principles characterization of HFec(CO) 4 and its PES are summarised in Table 1. None of the presented results (at any level of theory) support the postulated SP equilibrium structure, nor do they support the other ideal penta-coordination polyhedron, i.e., a TBP isomer. Rather than being a minimum on the PES, we have identied the SP structure to be a low-lying saddle point. The true minimum is a quasi-SP (qSP) structure of C s symmetry with a basal hydrogen (see Fig. 3a), which is in accord with the geometries reported for a number of HM(CO) 4 complexes. 23 In addition to the qSP equilibrium structure, an edge-bridged tetrahedron (EBT) of C 2v symmetry (see Fig. 3b) emerged as another relevant isomer. EBT structures are well-known and constitute a separate class of penta-coordinate complexes. 19,27,31 The distorted, low-symmetry qSP equilibrium geometry is about 10 kcal mol À1 more stable than the EBT isomer. Our analysis of the spin density distribution (Fig. 3) shows some differences in the directionality and localization of the metalcentered radical electron between the qSP and EBT isomers: both are more pronounced for the former, which may be attributed to a less hindered Fe-center and which may indicate a relatively larger reactivity. 15 HFec(CO) 4 can be expected to exhibit a high propensity for dimerization, 48 which has been conrmed experimentally by the formation of the bimetallic H 2 Fe 2 (CO) 8 dimer. Its geometry represents two qSP structures linked to each other via radical centers, [47][48][49] which is consistent with the above consideration.
There are several ways in which we can rationalize the computed qSP and EBT isomer structures of HFec(CO) 4 . One way is to employ Hoffmann's isolobal concept, 4 according to which HFec(CO) 4 can be represented as an isolobal analogue of the d 7 -fragment of an ML 6 complex, such as Mn(CO) 5 4 precursor and remove an axial H, then this strong perturbation is expected to shi one of the metal-ligand s-antibonding orbitals of the degenerate e g set to lower energies, thus creating a low-lying acceptor orbital in the emerging qSP radical. The DFT results shown in Fig. 4 illustrate this notion. It is worth noting that d 6 HMn(CO) 4 also adopts a qSP structure, however, in this case the H occupies a basal position. In contrast to HFec(CO) 4 , HMn(CO) 4 also features a less stable SP isomer with the H in apical position. 23 As an alternative precursor for HFec(CO) 4 , we can consider the diamagnetic [HFe(CO) 4 ] À with d 8 electron conguration. The anion exhibits a TBP-type structure, 74,75 for which we can assign an idealized D 3h symmetry. If we remove one electron to furnish HFec(CO) 4 , this would leave one unpaired electron in a degenerate e 0 -orbital. 20,21 The resulting Jahn-Teller distortion 19,76,77 would li the degeneracy of this set of orbitals and lead to the formation of the qSP and EBT structures observed for HFec(CO) 4 . This distortion should occur along a reverse-BPR pathway, i.e., upon upward bending of the two axial ligands 3 and 4 in the direction of the pivotal ligand 5 as shown in Fig. 1. Ward et al. have reported such an EBT distortion in d 0 MD 2 L 3type compounds containing two equatorial p-donor ligands D. 27 Another pathway to distort the ideal D 3h symmetry and li the degeneracy of the e 0 -orbitals may proceed through the formation of distorted-TBP (dTBP) structures, which exhibit an acute angle a in the equatorial plane. 28 Our calculations identied such a dTBP-type geometry, albeit for the transition structures (see Fig. 6 and 7).
Despite all this evidence, Jahn-Teller-type distortions alone do not suffice to explain why HFec(CO) 4 adopts an EBT structure for the second isomer. This becomes evident when we consider the electronically analogous complexes of the heavier homologues Ru and Os, which adopt SP rather than EBT geometries (see Table 2). We will revisit and discuss this issue in more detail in Section 3.6.
A number of higher symmetry structures, including the aforementioned SP, emerge as saddle points on the PES. They connect the (stereo-)isomers and thus represent TSs of the corresponding polytopal rearrangements. We have identied two main mechanisms for uxional rearrangement, i.e., a modied BPR (mBPR) anddue to its conceptual similarity to Muetterties' TJan octahedral switch (OS). For the interconversion between the non-equivalent qSP and EBT structures we discovered a buttery isomerization (BFI) mechanism. It is worth noting thatconsistent with our discussion in Section 1 and ref. 24, 41 and 42no TS for a TSR mechanism could be found on the HFec(CO) 4 PES.

Modied Berry pseudorotation
As indicated in the previous section, our calculations show that the symmetric SP structure is not a minimum but a saddle point on the HFec(CO) 4 PES. Closer inspection reveals that it interconverts two EBT stereoisomers, and the corresponding trajectory points to a BPR-type pathway (i.e., two equatorial ligands are simultaneously moving up, another two down, and vice versa; see Fig. 5).
However, unlike for the classical BPR mechanism (Fig. 1), no high-symmetry TBP structures could be identied to represent the end-points of the inversion. An IRC analysis of the TS yielded the two EBT stereoisomers as the true end-point structures instead. The DFT barrier heights (DE, which include zero-point corrections) for this mBPR mechanism range from 2.0 to 3.1 kcal mol À1 (see Table 1). The corresponding Gibbs free energies (DG) are somewhat higher (by about 1 kcal mol À1 ) due to the entropy decrease in the high-symmetry TS mBPR . (This effect is not seen for the other mechanisms described below, for which the values for DE and DG are nearly identical.) The computed barriers are close to the experimental activation energy of 1.6 AE 0.3 kcal mol À1 for the classical BPR in Fe(CO) 5 , 36 which suggests that our results are reasonable. Our single-point all electron CCSD(T) calculations predict a similar value for DE (3.1 kcal mol À1 ), whereas the ECP-based calculations yielded a somewhat higher barrier of 7.0 kcal mol À1 .

Octahedral switch
In addition to the TS mBPR , we have located two saddle points on the HFec(CO) 4 PES that each connect a pair of chemically identical qSP stereoisomers. Both pathways of this octahedral switch mechanism describe a single ligand migration to a vacant position of an "octahedron" via a TS of C 2v symmetry. The initial qSP structure is considered the idealized, non-vacant part of this octahedron. We distinguish the cases where (i) H migrates from a basal position of the qSP to the vacant position of the octahedron via an EBT TS (this case is denoted as OSH; see Fig. 6), and (ii) a bulkier CO migrates to the same vacant octahedral position via a TS of dTBP geometry (this case is denoted as OSL; see Fig. 7). Unlike the TJ mechanism described in Fig. 2, in which the ligand migration occurs between faces of the tetrahedron, the OS mechanism involves ligand migration between (open) vertices of the "octahedron", resulting in the apical-basal ligand exchange (switch).
It is worth noting that the OSH mechanism has some resemblance to the pivoting mechanism suggested by Albright et al. for penta-coordinate d 6 -complexes, 19 except that the latter includes a pivoting of two ligands, whereas in the former, only one ligand is moving. In the OSL, two other equatorial ligands of the dTBP TS are also moving concurrently in opposite directions to maintain zero net angular momentum in a manner to support the OS and the respective stereoisomer formation. This phenomenon resembles that described by Berke, Gusev, and co-workers for Ir(H) 2 X(P t Bu 2 Ph) pentacoordinate systems, where X is a halogen. 11,29,78 However, in contrast to the OSL mechanism, the X-ligand in ref. 29 does not change its position. For the OSH mechanism we found even closer parallels to early work in the eld. While Burdett et al. proposed the inverse Berry twist (SP / [TBP] ‡ / SP 0 ) involving singlet excited states to explain the photochemical behavior of d 6 M(CO) 5 complexes, 14 they also postulated an alternative mechanism involving a C 2v TS. However, based on extended Hückel theory calculations and the resulting d-orbital stabilization energies, it was concluded at the time that repulsive interactions would make such a TS unfavorable compared to the TBP TS for d 7 and d 8 complexes, and that it could only be relevant for d 6 low-spin complexes. The OS mechanism is also somewhat reminiscent of the uxionality of "clusters" that feature a vacant site on a polyhedral surface 79-83 following the simple bond-stretch isomerism as postulated by Stohrer and Hoffmann. 84 Such an isomerization occurs, e.g., between two identical SP intermediates of the (C 5 H 5 ) + cation via a C 2v TS, which was predicted at the MINDO/3, 85 MP2, 86 and DFT 82,83 levels. Saillard and co-workers have calculated this "migration" for the isoelectronic P 5 + and Sb 5 + as well as B-substituted clusters. 82,83 Even though the bond-breaking processes in these "hollow" clusters require only relatively low activation energies (30 kcal mol À1 at MP2/DZP 86 ), they are still signicantly higher than those for OS rearrangements in metallocomplexes. The DFT-level DE for the OSH mechanism range from 2.7 to 2.9 kcal mol À1 and for the OSL from 3.9 to 4.2 kcal mol À1 (the corresponding DG are only very slightly higher, i.e., by about 0.2 kcal mol À1 ; see Table 1), i.e., the OS barriers for the light H-atom is lower by about 1 kcal mol À1 than the one for the heavier CO.
The MP2 results favor the OSL mechanism with a barrier of 6.3 kcal mol À1 over the OSH mechanism with 8.5 kcal mol À1 . The ECP-based CCSD(T)/BS-III results are essentially degenerate with OSL narrowly winning out (5.2 vs. 5.3 kcal mol À1 ), whereas our highest level results from the all-electron CCSD(T)/BS-II calculations conrm the DFT predictions favoring the H-shi (3.7 vs. 6.1 kcal mol À1 ). Considering the small energetic differences and the method inherent margins of error, we cannot make a conclusive statement concerning the order of these two rearrangement pathways.

Buttery isomerization
The mBPR and OS mechanisms described so far facilitate uxional rearrangements that connect chemically indistinguishable species, i.e., qSP to qSP 0 and EBT to EBT 0 , respectively. There also has to be a mechanism that interconverts the qSP isomer into the EBT and vice versa. This mechanism should intuitively include the migration of H or CO-ligands, as well as deformation of the acute angle a. An examination of the two distinct HFec(CO) 4 isomers (see Fig. 3) suggests that the isomerization could simply proceed through a small shi of the H position. However, a direct H-migration scan (see Fig. S1, ESI †) reveals that such a pathway does not end up at the EBT isomer, but rather the TS OSH , which has a similar EBT geometry. The only difference is the acute angle a. On the other hand, a variation of a provides a simple picture of consecutive transformations, and the results of such an a-scan are presented in Fig. 8.
The rst maximum of the energy prole is an approximate representation of the TS for the qSP to EBT isomerization. It features a distorted EBT (dEBT) geometry of C s symmetry (with asymmetrical wing angles being 125 and 101 , respectively) and a barrier height of about 11 kcal mol À1 . As expected, the a angle has an intermediate value of 134 between that of the two isomers (99 and 153 for qSP and EBT, respectively). The shallow minimum (well-depth of about 1-2 kcal mol À1 ) and small reverse barrier indicate that the EBT is kinetically unstable. Since the EBT is also the higher-energy (i.e., thermodynamically unstable) isomer, it is expected that it will have a transient character. A further increase of a leads to a regular SP structure, which corresponds to the TS mBPR that interconverts EBT stereoisomers as discussed in Section 3.2. The actual TS given in Table 1 was determined following the a-scan by full gradient norm optimization. Since this TS combines the beforementioned H-migration with a buttery-type motion of the COligands, we call this the buttery isomerization (BFI) mechanism. The DE for the BFI mechanism range from 8.3 to 9.5 kcal mol À1 at the DFT level (DG are 0.2 kcal mol À1 higher in each case), and between 10.2 and 11.1 kcal mol À1 at the MP2 and CCSD(T) levels (see Table 1). The BFI is thus energetically more demanding than the uxional rearrangements as they involve more signicant molecular deformations. Fig. 9 shows a schematic representation of the HFec(CO) 4 PES with all the stationary points introduced in the previous subsections. The qSP minimum is the stable isomer of HFec(CO) 4 , but it can undergo uxional stereoisomerization, either via one of the OS pathways or a BFI / mBPR / BFI 0 sequence with metastable EBT stereoisomers as intermediates.

The HFec(CO) 4 PES
Since the consecutive rearrangement results in a relatively high overall barrier (13.8 kcal mol À1 at all electron CCSD(T) level), the OS will be the dominant process at low temperatures. It is worth stressing that the different mechanisms lead to the rearrangement of different ligands. In fact, the polytopal rearrangements can be traced through the changes in the apical positions of the ligands (the corresponding ligand number is encircled for clarity in Fig. 9) as well as the relative orientation of ligands 2 and 4 (highlighted in pink).
The OS stereoisomerization is shown on the le side of the PES diagram. It features a swap in the apical and basal ligand positions, while the corresponding ligands 2 and 4 remain cisoriented to each other and the angular orientation of the Hatom with respect to 2 and 4 does not change. In the processes on the right side, the H-atom is the pivoting ligand, and 2 and 4 rearrange from cis-to trans-orientation. The latter can only be achieved through the energetically less favorable BFI / mBPR / BFI 0 sequence, which is only accessible at more elevated temperatures. Only a traversal of the full sequence (facilitated by the deformation of a) inverts the 1-3 trans-and 1-4 cis-orientation and replaces the 2-H trans-with the 3-H transorientation. It would be an interesting (albeit challenging) task for experimentalists to discriminate these steps to prove the theoretical conclusions drawn from this study.

Comparison with other d 7 HMc(CO) 4 systems
The isomers (i.e., qSP and EBT) and mechanisms (i.e., mBPR, OSH, OSL, and BFI) introduced in Sections 3.1-3.4 for the prototype system HFec(CO) 4 may offer a more general picture of polytopal arrangements in ve-coordinate systems, at least for d 7 -complexes. We have thus expanded our DFT study of the HFec(CO) 4 radical to an initial set of related coordination compounds. Systems that are electronically homologous to HFec(CO) 4 but that feature different central metal atoms are an obvious rst choice for a comparative study. In the following, we will provide such a comparison and discuss the transferability of our ndings. The results of this work are summarized in Table 2.
All 3d 7 -systems studied here, i.e., [HCo(CO) 4 ]c + , HFec(CO) 4 , and [HMn(CO) 4 ]c À , feature qSP isomers as the global minima on their respective PESs and they represent the end-point structures for OS-rearrangements (see Fig. 9). Each example also features a metastable EBT isomer between 8.5 and 10.7 kcal mol À1 above the qSP structure. EBT stereoisomers interconvert via the proposed mBPR mechanism in all cases. Finally, we also nd the BFI mechanism as the mode of isomerization between the qSP and EBT isomers in each of our 3d 7 examples. Our discussion in Sections 3.1-3.5 is thus fully applicable for the other 3d 7 -systems considered here as well. While the three 3d 7compounds have identical electron congurations, they differ in their nuclear charges. The barrier heights for the polytopal rearrangements correlate strongly with the nuclear charge, i.e., they are highest for the Co and lowest for the Mn case. This is consistent with our expectation for electrostatically induced rigidity in these complexes (i.e., the electrons in the bonding orbitals are subject to a stronger electrostatic attraction from the nucleusexacerbated by the decreasing atomic radii from Mn to Fe to Cowhich makes the ligands less mobile and susceptible to rearrangements).
The two remaining d 7 radicals in the iron group (i.e., HRuc(CO) 4 and HOsc(CO) 4 ) also feature qSP structures, but instead of the additional EBT isomers, they adopt symmetric SP structures with H in the apical position. EBT isomers and the corresponding mBPR stereoisomerization pathways found in the 3d 7 -systems could not be identied in the 4d 7 and 5d 7 cases. The SP structures are marked by relatively large apical-M-basal angles of 91.2 and 91.8 for Ru and Os, respectively. The qSP structures are found to be slightly higher in energy (i.e., less stable) than the SP isomers by 0.7 and 1.7 kcal mol À1 , respectively, i.e., for the Ru and Os radical complexes the qSP structures are not the global minima on the PES. (However, considering the small energetic differences and inherent margins of error of DFT, this is not a conclusive nding.) The isomerization from qSP to SP proceeds through a BFI-type mechanism.
The preference of the SP over the EBT structure may be traced back to the differences in the covalent radii of the metal centers (Ru: 1.37Å; Os: 1.38Å; Fe: 1.24Å (ref. 87)). Classical interligand interaction theory 10,18 based on the partitioning of the interligand interactions into 6-exp (one-variable Kitaigorodsky-type) and Coulomb potentials makes it possible to distinguish the contributions of size and polarity of ligand spheres, respectively, to the stabilization of a polyhedral structure. Qualitatively, the increase in the covalent radius of the central metal (assuming xed valence-shell conguration) results in longer M-L coordination bonds (i.e., a larger coordination sphere) and consequently in an increased interligand spacing. Such an increased spacing (which would be expected for the Ru and Os complexes relative to the Fe prototype) will alleviate steric repulsion between the ligands and will thus make coordination polyhedra with smaller baseline ligand separation (such as SP in comparison to EBT) more competitive. 10 We point out that all isomer energy differences and all barriers heights of the d 7 systems shown in Table 2 are smaller than 11 kcal mol À1 and thus energetically very close. The presence of such shallow PESs and closely related stationary points underscores the challenges of studying the nature, mechanistic intricacies, and impact of polytopal rearrangements.

Conclusions
Using an array of DFT and ab initio approaches, we have demonstrated that the penta-coordinate d 7 complex HFec(CO) 4 has two thermally accessible isomersa qSP equilibrium structure and a metastable EBTand that the ideal highsymmetry SP geometry that was advocated in earlier work is in fact not a stable structure but a TS. We can rationalize the lower symmetry qSP and EBT structures as being derived from isolobal, higher symmetry precursors that become subject to Jahn-Teller-type distortions upon transition to the d 7 radical, which would otherwise feature a degenerate orbital occupied by a single electron.
Both isomers are subject to uxional rearrangements (i.e., stereoisomerization) as well as interconversion (i.e., isomerization) during which they traverse different polyhedral forms. We have analysed these intramolecular polytopal rearrangement processes for the HFec(CO) 4 prototype system and could identify three new mechanisms that facilitate them: mBPR, OS, and BFI. We could also show that the host of other mechanisms that have been proposed for this class of compounds in the past are ultimately not applicable. Our study thus lls a gap in the mechanistic understanding of rearrangement processes for systems involving distorted equilibrium structures. We have characterized and rationalized each mechanism in detail and shown that they are valid for other 3d 7 -systems as well. The d 7complexes of the higher rows exhibit a somewhat different situation with an SP equilibrium geometry and a metastable qSP isomer. The discrepancy may be attributed to the weaker interligand interactions found in systems with larger coordination spheres.
The barrier heights for all mechanisms are on the order of 10 kcal mol À1 or smaller, and the barrier differences are within a few kcal mol À1 . We note that some of these differences are basically within the margin of error of the applied computational methods, so that we cannot claim a denite order in these cases. The shallow nature of the PESs of the systems at hand indicates that experimental studies on the prevalent mechanisms will be challenging.
In upcoming publications, we will investigate the impact of the nature of the ligand (in particular its electronic structure and propensity for different types of coordinative bonds) on these newly proposed polytopal rearrangement mechanisms, and we will also expand the scope of our investigation to pentacoordinated complexes with d 6 and d 8 electron conguration.