Exploring the full catalytic cycle of rhodium(i)–BINAP-catalysed isomerisation of allylic amines: a graph theory approach for path optimisation

The reaction mechanism of the cationic rhodium(i)–BINAP complex catalysed isomerisation of allylic amines was explored using the artificial force induced reaction method with the global reaction route mapping strategy.


Introduction
Noyori et al. reported the BINAP-metal-catalysed hydrogenation reaction of enamides with excellent enantioselectivity; we have previously examined the mechanisms of this reaction using the quantum mechanics/molecular mechanics (QM/MM) method. [1][2][3][4][5][6][7] Otsuka et al. reported a highly regioselective and enantioselective 1,3-hydrogen shi reaction of allylic amines using a cationic rhodium(I)-BINAP catalyst (Fig. 1). [8][9][10][11][12][13] The asymmetric 1,3-hydrogen shi of diethylgeranylamine forms a corresponding citronellal (R,E)-enamine, which generates a stereogenic carbon centre from a prochiral compound. This reaction has been used in the stereoselective synthesis of Lmenthol. 14 In addition, the isomerisation includes a C-H bond cleavage step. Thus, knowledge of the mechanism of this reaction would be highly meaningful in the context of metalcatalysed C-H functionalisation chemistry. 15 Three reaction mechanisms have previously been reported based on experimental and computational studies (Fig. 2). [16][17][18] All of these mechanisms start from the bis-solvent coordinate Rh(I)-BINAP complex I, and the bis-substrate coordinate complex II is formed through ligand substitution. The difference in the three mechanisms lies in the C(1)-H s-bond cleavage step. The rst mechanism, proposed by Noyori, is a dissociative mechanism and includes the formation of the transient iminium-Rh(I)-hydride complex IV through b-hydride elimination from the mono-amine coordinate complex III. 16 Complex IV is converted to the h 3 -enamine coordinate complex V by the transfer of a hydride from the rhodium to C(3). Ligand substitution from complex V forms complex II or complex III through the bis-amine coordinate Rh(I) complex VI. However, ab initio molecular orbital studies showed that the associative mechanism is more advantageous than Noyori's dissociative mechanism. 17 A second mechanism, Takaya and Noyori's mechanism, was then proposed. They suggested that the C(1)-H oxidative addition occurs directly from complex II without liberation of an NR 3 molecule. This mechanism involves a Rh(I)/ Rh(III) two-electron-redox process through a characteristic distorted-octahedral Rh(III) hydride complex VII. Complex VI is formed by reductive elimination from complex VII. Later, Fig. 1 The asymmetric 1,3-hydrogen shift of diethylgeranylamine.
Ujaque and Espinet proposed a more reasonable allylmetal mechanism based on density functional theory (DFT) studies. 18 This mechanism begins with intramolecular isomerisation of k 1 -(N)-coordinate complex II to h 2 -(C]C)-coordinate complex VIII, which then undergoes C(1)-H s-bond cleavage to form distorted-octahedral h 3 -allyl complex IX. Aer a conformational exchange from IX to X, the reaction concludes with reductive elimination to form complex VI.
The three mechanisms were proposed based on assumptions of the intermediates and transition states (TSs). To eliminate the need for this assumption to allow exploration of the reaction paths, we used a reaction path search method called articial force induced reaction (AFIR), which can algorithmically predict reaction paths and TSs on reaction paths. To analyse a complicated reaction path network obtained using the AFIR method, Prim's algorithm was employed. Although kinetic approaches such as rate constant matrix contraction 19,20 and kinetic Monte Carlo [21][22][23] have been used in evaluating quantities such as overall rate constants and branching ratios, graph theory approaches including Prim's algorithm would be useful to visualise an overview of highly multistep reaction paths. Furthermore, our own N-layered integrated molecular orbital + molecular mechanics (ONIOM) method was employed to reduce the computational cost, which enabled us to use the structure of the Rh(I)-BINAP complex without replacing the functional groups. In the studies by Noyori et al. 17 and Ujaque and Espinet et al., 18 two PH 3 molecules were used as a model of the BINAP   ligand for the same purpose; as a result of this simplication, the associative mechanism appeared to be the most viable. However, our results, without simplication of the Rh(I)-BINAP complex, demonstrated the advantages of the dissociative mechanism over the associative mechanism.
2 Computational methods

Outline of the computational procedure
An outline of our computational procedure is shown in Fig. 3. We employed the AFIR method coded in the global reaction route mapping (GRRM) programme to explore the reaction path network. [24][25][26][27][28][29][30][31] There are two modes for the AFIR method.
In the multi-component mode (MC-AFIR), the force is applied between two or more reactant molecules to induce a reaction between the reactants. In the single-component mode (SC-AFIR), fragments are dened automatically in an entire system (molecule or complex) and the force is applied between the fragments. In this study, the MC-AFIR and SC-AFIR were used, respectively, in the coordination state sampling and in the path sampling within the coordination complexes. The graph consists of connections between nodes and edges. In the present study, a node and an edge represent a local minimum (LM) and a TS, respectively. Prim's algorithm was employed to construct a minimum spanning tree (MST) from the graph. Examination of the algorithm procedure demonstrates that the MST shows energetically reasonable paths. Finally, some LMs and TSs were reoptimised and analysed at higher computational levels for more detailed study.

Coordination state search with the MC-AFIR method
The models that we considered are shown in Fig. 4. The full cationic Rh(I)-BINAP complex XI was used as a catalyst. To simplify the models of the substrates, N,N,3-trimethylbut-2en-1-amine XII and trimethylamine XIII were used as the substrates. To examine the enantiotopic face, two methyl groups on the allylic group were indicated as R t (trans) and R c (cis); the priority of R t was established to be higher than that of R c . The ONIOM-type hybrid QM/MM model was employed to reduce the computational cost. [32][33][34][35][36][37][38][39] The QM and MM regions in the ONIOM method were dened as given in Fig. 4. The MM region was treated at the universal force eld (UFF) level, while the QM region was treated at the B3LYP level in conjunction with the LANL2DZ basis set for the rhodium atom and the 6-31G basis set for other atoms (BS1). [40][41][42][43] The initial geometries of the fragments were optimised, respectively, at the B3LYP/BS1 level. Electrostatic potential (ESP) atomic charges for the MM region were calculated for each fragment at the same level as the geometry optimisation of the fragments; the ESP atomic charges were xed while the AFIR search was performed. We considered the dissociative mechanism and the associative mechanism separately. In the dissociative mechanism, the cationic Rh(I)-BINAP complex X and the allylic amine XI were included in the computational model, and the articial forces were applied between the fragments (X and XI). In the associative mechanism, the trimethylamine XII was appended to the computational model of the dissociative mechanism, and articial forces were applied between three fragments (X, XI and XII). The AFIR search was initiated from randomly generated initial orientations and directions of the fragments; this type of AFIR search is called MC-AFIR. The g value for the MC-AFIR search was set to 300 kJ mol À1 . The path sampling in the MC-AFIR search was terminated when the last ten new AFIR paths resulted in AFIR paths found earlier. All approximate LMs and TSs obtained with the MC-AFIR search were fully optimised without articial forces at the ONIOM level.  5 Graph G of the reaction path network and extraction procedures for determining the rational path. The TS between LM (i) and LM (j) is denoted as TS i-j . Unselected and selected vertices are coloured grey and orange, respectively. All edges denote the reaction path network, and the thick orange edges denote MST.

Reaction path search with the SC-AFIR method
Aer the MC-AFIR search was performed, we applied an SC-AFIR search, which is suitable for nding stepwise paths. The computational level for the SC-AFIR search was the same as for the MC-AFIR search. The initial structures for the SC-AFIR search were selected from the results of the MC-AFIR search; each selected LM was found to be the precursor of the C(1)-H bond cleavage step. All approximate LMs and TSs obtained with SC-AFIR were fully optimised without articial force at the ONIOM level. Aer the MC-AFIR and SC-AFIR searches, the connections between the LMs and TSs were veried by intrinsic reaction coordinate (IRC) analysis from each TS. 44 During the search, the articial force is applied, and paths are somewhat deviated from actual LMs, TSs and IRCs. We emphasise that the LMs, TSs and IRCs nally obtained and discussed below are "actual" local minima, rst-order saddle points, and steepest descent paths in the mass-weighted coordinate, respectively, on the adiabatic potential energy surface. Both AFIR searches were performed with a developmental version of the GRRM programme; the energies, gradients and Hessians were computed with Gaussian 09 (Rev. D01). 31,45 Technical details of the MC-AFIR and SC-AFIR methods and their recent applications to organic reactions can be found in recent review papers. 46,47

Representation of the reaction network and extraction of the reaction path
Aer the comprehensive search for reaction paths is complete, the remaining problems are to understand the entire reaction network and to select probable paths from the greatly extended reaction network. We showed the reaction network in the form of graph G (Fig. 5). A vertex and an edge represent a LM and a TS, respectively. A number of paths from an arbitrary vertex to another vertex are combinations of the edges. If the numbers of vertices and edges increase, nding probable paths becomes much more difficult because of the exponential growth of the number of combinations. Therefore, we applied Prim's algorithm to solve the combination problem. 48 The algorithm accepts a graph G and outputs a tree T by repeating a selection of an edge with a smaller weight. If the energy of the TS is used as the weight of the edge, the algorithm is applicable to obtain a tree T of reaction paths. Because there is a unique path from a starting point to another point in this tree T, the discussion of the reaction paths becomes easy to understand. The concrete procedure is as follows. 1: select an arbitrary vertex from the graph G, which becomes a starting point of the reaction (e.g. vertex 1 is selected in Fig. 5(b)). 2: select a vertex which has not been selected previously and which is connected to the selected vertex. If there is more than one possible edge, the most stable  one is selected (e.g. three edges connect to vertex 1 in Fig. 5(c). The most stable edge is 1-2; thus, it will be selected). 3: select a vertex which has not been selected previously and which is connected to the vertex in the set of selected vertices (Fig. 5(d)). If there is more than one possible edge, the most stable one is selected. 4: perform step 3 as many times as possible. Finally, by this procedure, we can draw a tree T in which the edge with the highest weight is excluded (Fig. 5(e)). This algorithm is known to provide an MST which is uniquely formed from a network. Thus, any vertex can be selected as the starting point. In the present study, the rst vertex was chosen from a vertex with a maximum degree (maximum number of edges connected to the vertex). The numbers of the vertices were arranged in the order of the vertex chosen by Prim's algorithm. Using this order, the smaller number is given to a vertex which passes through energetically lower TSs from the starting vertex. The Gephi programme was employed to illustrate the graphs. 49 The Gibbs free energies at the ONIOM(B3LYP/BS1:UFF) level (25 C) were used for the analysis of the reaction paths.
Here, the notations of the LM and the TS are explained. The number of the LM is decided according to Prim's algorithm, and the LM itself is represented by this number (e.g. i, j, k in Fig. 6). The TS which connects i and j is written as TS i-j . Some reaction steps were discussed together. For example, in Fig. 6, if a reaction path from i to k is discussed, both LMs and a TS with the highest energy (TS j-k ) in the path are focused on. To indicate both ends clearly, notation such as TS j-k (i / k) is introduced.

Full DFT optimisation and single point energy calculations
Some LMs and TSs which were selected from the results of the MC-AFIR and the SC-AFIR reaction path searches were reoptimised without the ONIOM model. The dispersion-corrected B3LYP + D3 functional in conjunction with the Stuttgart/ Dresden effective core potentials, the associated SDD basis set for a rhodium atom, and the 6-31G(d) basis sets for the other atoms (BS2) were employed for the entire system. 50-52 Aer geometry optimisation, IRC calculations were performed to conrm that the TS is connected to the correct LMs. 44 Single point energy calculations were performed with the same functional as that used for the full DFT geometry optimisation. The BS2 basis sets were replaced with a triple zeta valence polarisation basis set, def2-TZVP (BS3). 53,54 A polarised continuum model using the integral equation formalism

Energy decomposition analysis
Energy decomposition analysis (EDA) was performed for the key TSs of the stereochemistry-determining step. 59 The MC-AFIR coordination searches for the dissociative mechanism provided four types with two diastereomers in each type (Fig. 8); the searches for the associative mechanism provided one type with two diastereomers (Fig. 9).
In the dissociative mechanism, the chelate complex MC1 is the most energetically stable LM. The C]C double bond of the substrate in MC1 coordinates to the Rh(I) centre from the siface. Another chelate complex, MC2 (DG ¼ 15.3 kJ mol À1 ), whose C]C double bond is coordinated from the re-face, is an energetically higher LM than MC1 (DG ¼ 0.0 kJ mol À1 ). Next, the stable LMs are classied into N-coordinate types (MC3 and MC4); these have a Rh(I)-N coordinated bond and a Rh(I)/H-C(1) interaction. The coordination mode between the Rh(I) Fig. 12 The Gibbs free energy profile along the path to 54 (type A and B) and the molecular structures on the path, calculated at the ONIOM(B3LYP/BS1:UFF) level (25 C). The energies are given relative to 1 in kJ mol À1 . centre and the H-C(1) s-bond forms a side-on geometry, and the H-C bond distance (1.16Å) is longer than the general H-C bond distance (1.10Å), showing a typical agostic interaction. 61 The energy difference between the complexes MC3 and MC4 is very small (DG ¼ 1.3 kJ mol À1 ). The next type is a C]Ccoordinated type. C]C-coordination from the si-face and reface directions forms the C]C-coordinate Rh(I) complexes MC5 and MC6, respectively. Both complexes also have agostic interactions. The p-coordination bond of C(2)]C(3) is inclined toward the C(2) atom because the agostic interaction attracts the Rh(I) atom to the H-C(1) s-bond. There is a large energy difference between MC5 and MC6 (DG ¼ 38.7 kJ mol À1 ), which will be discussed in Section 3.3.4. The last type of complex has a Rh(I)/H-C(1) interaction and a vacant coordination position on the Rh(I) atom (MC7 and MC8). The rhodium atom of MC7 interacts with the pro-S-hydrogen atom, while that of MC8 interacts with the pro-R-hydrogen atom. The Rh(I)/H-C(1) of both complexes is linear. Thus, this interaction is classied as an anagostic interaction. 61 The four agostic complexes (MC3, MC4, MC5 and MC6) were selected as the initial structures for the SC-AFIR reaction path searches of the dissociative mechanism.
In the associative mechanism, the results of the MC-AFIR search include only N,H-coordinate complexes MC9 and MC10 (Fig. 9); these have a Rh(I)-N coordination bond and a Rh(I)/H-C(1) interaction. The Rh(I)/H-C(1) interaction is classied as an anagostic interaction, similar to MC7 and MC8. No bisamine coordinate Rh(I)-BINAP complex like complex II, which was identied using SC-AFIR, was found.
Further results for the coordination state of the bis-substrate coordinate Rh(I)-BINAP complex will be discussed in Section 3.4. Both of the N,H-coordinate complexes (MC9 and MC10) were selected as initial structures for the SC-AFIR reaction path searches of the associative mechanism.

Reaction networks (results of the SC-AFIR method)
The MC-AFIR and SC-AFIR reaction path searches for dissociative and associative mechanisms gave two reaction path networks, which are shown as graphs G D and G A in Fig. 10. The subscripts 'D' and 'A' denote dissociative and associative mechanisms, respectively. The former consists of 361 LMs and 344 TSs, while the latter consists of 257 LMs and 257 TSs.
To discuss the networks, all LMs and TSs are classied into the following three classes: C1: a 'reactant' complex, which has two C(1)-H bonds in the substrate (green in Fig. 10); C2: a 'hydride' complex, which is classied as neither a reactant nor a product complex (yellow in Fig. 10); C3: a 'product' complex, which has one C(3)-H bond in the substrate (red in Fig. 10). The longest C-H bond distance was set as 1.3Å.

Dissociative mechanism (G D )
According to the present rules, the chelate complex 1 is the starting point in the dissociative mechanism (Fig. 11). A nitrogen atom and a C]C double bond from the re-face direction are coordinated to a Rh(I) atom in complex 1. All of the paths can be classied into six types (A-F in Fig. 11) by their combinations of intermediates. The paths from complex 1 to the product (C3) are summarised in Table 1. 3.3.1 Path to 54 (the most favourable reaction path). A path to 54 (type A) is the most favourable reaction path and forms an  (E,R)-enamine; this stereochemistry is consistent with the experimental results. The path to 54 starts from the chelate complex 1; complex 1 is converted to the C]C-coordinate complex 46, which experiences an agostic interaction between the Rh(I) atom and the H S -C(1) bond. The energy prole is shown in Fig. 12. The Rh-bound H atom (H S ) migrates to the Rh(I) atom as a hydride with the assistance of an electron supplied from the N atom; this results in the formation of an iminium ion structure (47). Importantly, the C(1)]N double bond of the iminium ion moiety is not bound to the Rh(I) atom, in contrast to the h 2 -coordination in the mechanism proposed by Noyori et al. (IV and VII in Fig. 2). Then, a conformational change through rotation around the p-coordination axis forms another C]C-coordination Rh(I) hydride complex 53, which is the rate limiting step of path A to 54 (DG ¼ 27.3 kJ mol À1 ). Finally, the (E,R)-enamine coordinate complex 54 is formed through hydride migration.
One path to 54 (type B) is consistent with the mechanism of Noyori et al. In this path, the Rh(I)-bound iminium ion IV in Fig. 2 is formed through hydride migration from the Ncoordinate Rh(I) complex III. The path to 54 (type B) forms the Rh(I)-bound iminium ion 22 through hydride migration from the N-coordinate Rh(I) complex 8. Then, intramolecular coordination switching occurs (TS   (22 / 53)), when the iminium ion of the substrate is eliminated from the Rh(I) centre and the C]C double bond of the substrate coordinates to the Rh(I) centre to form the C]C-coordinate Rh(I) hydride complex 53. Because the distance between the iminium ion and the Rh(I) centre is longer than that of the other intermediates, the energy of TS 23-49 (22 / 53) is higher (DG ¼ 79.1 kJ mol À1 ) than that of the other LMs and TSs. As a result, the type B path is disadvantageous compared with the type A path.
3.3.2 Path to 92, 119 and 224 (types C, E and F; formation of (Z)-enamine). A path to 92 (type C) forms the (Z,R)-enamine. As a result of the reaction path search, an early number was assigned to the product of the path of type C. However, the (Z)enamine was not obtained in the experiment. These contradictory results are explained by the thermodynamic stabilities of the products (Fig. 13). Because the transformation from an allylic amine to a (Z)-enamine is an endergonic process (DG ¼ 4.3 kJ mol À1 ), the type C path can be ruled out. Both the path to 119 (type E) and the path to 224 (type F), which form (Z)enamines, can also be ruled out for the same reason.
3.3.3 Path to 96 (formation of (E,S)-enamine). Paths to 96, 99 and 108 give the (E,S)-enamine, which is the opposite enantiomer of 54 ((E,R)-enamine). The energy prole is shown in Fig. 14. These paths share the path from 1 to hydride complex 93 and then branch off to each destination. Because the differences mainly arise from a small conformational change in the BINAP ligand and because the paths are similar, the paths to 96 are discussed as a representative. There are three paths (A, B and D) to 96. The path to 96 (type A) is similar to the path to 54 (type A) except for the unstable TS  for the change of coordination mode (1 / 107; DG ¼ 84.6 kJ mol À1 ). The path to 96 (type B) is almost the same as the path to 54 (type B). The path of type D passes an s-cis-iminium ion coordinate Rh(I) complex 116 (DG ¼ 48.6 kJ mol À1 ), which is much higher in   energy compared with the s-trans isomer 102 (DG ¼ 13.0 kJ mol À1 ); thus, the type D path is not preferred.

R/S selection.
The type A path is the preferred path to 54; it affords the (E,R)-enamine (the major product in the experiment) and is the most reasonable path. The path to 96 (type A) forms the (E,S)-enamine, which is a minor product in the experiment, and passes through comparably unstable LMs and TSs. The difference between the two type A paths determines the R/S selectivity of the product; this difference between the path from 1 to 46 (Fig. 12) and the path from 1 to 107 (Fig. 14) is remarkable. Both paths (type A) were compared at higher computational levels (Fig. 15).
The difference between the paths (type A) can be observed from the chelate complex. The chelate complex 33 (DG ¼ 10.0 kJ mol À1 ), in which the C]C double bond is coordinated to Rh(I) from the re-face, is higher in energy than another chelate complex, 4 (DG ¼ À13.3 kJ mol À1 ), in which the C]C double bond is coordinated from the opposite face. The reactions on both paths A (1 / 46 and 1 / 107) proceed while maintaining the Rh(I)/C]C bond and the energy difference. Accordingly, the R/S selectivity of the product depends on the coordination state of the enantiotopic face of the allyl group.
N-C(1) bond rotations (45 / 46 and 113 / 107) occur on the type A path; this is a signicantly high energy process. The rotation connects a C]C-coordinate Rh(I) complex (45 or 113) with the synclinal (sc) conformation of the dihedral angle H-C(1)-N-lp (lone pair) to a complex with an antiperiplanar (ap) conformation (46 or 107). The sc conformation must form immediately aer the Rh(I)/N bond cleavage. In contrast, the ap conformation is advantageous from the viewpoint of the stereoelectronic effects between the non-bonding orbital of the nitrogen (n N ) and the C(1)-H antibonding orbital ðs * Cð1ÞÀH Þ. The orbital overlap can be conrmed by NBO analysis, and the strength of the donor-acceptor interaction between the n N and the s * Cð1ÞÀH orbitals can be conrmed by the second-order perturbation energy E(2) (Fig. 16). As the donor-acceptor interaction increases in strength (45: E(2) ¼ 3.2 kJ mol À1 / 46: E(2) ¼ 45.1 kJ mol À1 ), the C(1)-H bond length increases slightly (45: 1.14Å / 46: 1.18Å). In other words, the interaction weakens the C(1)-H bond. This factor is important for C(1)-H activation.
The N-C(1) bond rotation is a more energy-demanding step in the paths from 1 to 46 and from 1 to 107 (Fig. 15). Comparing the respective TSs of the N-C(1) bond rotation, the BINAP ligand at TS [45][46] (the si-face) maintains C 2 -symmetry very well, while that for TS 113-107 (the re-face) is largely distorted, especially for two of the pseudo-equatorial phenyl groups (Fig. 17). This is because the dialkylamino group and the methyl group on the C(3) atom in TS 113-107 (re-face) occupy the upper le space (second quadrant) and the lower right (fourth quadrant) (Fig. 17), which was originally occupied by the pseudoequatorial phenyl groups of the BINAP ligand, to avoid steric repulsion between the substrate and the phenyl group; therefore, the BINAP ligand is largely distorted. We performed EDA to investigate the energy difference between TS 45-46 and TS 113-107 ( Table 2). The energy difference (DDE ¼ 19.1 kJ mol À1 ) between these TSs is consistent with the deformation energy difference (DDEF ¼ 16.9 kJ mol À1 ) and the interaction energy difference (DINT ¼ 2.2 kJ mol À1 ). The contributions of the DDE mainly originate from the DEF (88.9%), and the contribution of INT is small (11.1%). Furthermore, the difference in deformation energy in the allylamine moiety (DDEF A ¼ À0.1 kJ mol À1 ) is negligible, and that of the BINAP-Rh(I) moiety (DDEF B ¼ 17.0 kJ mol À1 ) occupies the majority of the DDEF.

Associative mechanism
While some product complexes (red vertex in Fig. 10) are present in the network of the dissociative mechanism, there is no product complex in the network of the associative mechanism. To conrm the coordination state of all of the LMs, the Rh(I)-N (allylic amine) distance and the distance between the  Rh(I) atom and the centre of the C(2)]C(3) bond are summarised in the scatter plot shown in Fig. 18, in which the LMs are divided into two groups by the Rh(I)-N (TMA) distance. Some C]C-coordinate Rh(I) complexes have no TMA coordination (orange plot in Fig. 18); however, no C]C-coordinate Rh(I) complex is formed with TMA coordination (blue plot in Fig. 18). This result is different from the allylic mechanism of Ujaque and Espinet et al. 18 (to nd a C]C-coordinate Rh(I) complex, 647 approximate TSs were searched with the MC-AFIR and SC-AFIR methods for the associative mechanism). In the previous study by Ujaque and Espinet et al., the models that they used to conclude that the associative mechanism and allylic mechanism are favourable were too small (e.g. they used two PH 3 molecules as a model of the BINAP ligand). However, BINAP is a bulky ligand; thus, the coordination of the comparably bulky C]C bond to the Rh(I) centre is impossible in the associative mechanism. There are eight paths of C(1)-H bond cleavage in the graph G A (231 / 400, 241 / 369, 279 / 376, 291 / 359, 315 / 384, 374 / 369, 379 / 384 and 402 / 376). The most preferred path among the eight paths starts from the Ncoordinate Rh(I) complex 291 (Fig. 19). Aer 291 is formed, a hydride on C(1) is abstracted by the Rh(I) centre, and the complex is separated into the N-hydride-coordinate Rh(I) and iminium ion fragments 359 (Fig. 19). The eight C(1)-H cleavage paths shown above proceed similarly. The hydride complex 368 has the largest coordination number (ve-coordinate Rh(III) complex) in the entire network, which is formed from 359 through Rh(I)-C(1) bond formation. 368 has a square pyramidal geometry, as shown in Fig. 20.
In all of the hydride complexes except for complex 368, the substrate is located far from the Rh(I) centre (see the ESI, Section 2.1 †). Because the vacant position of complex 368 is blocked by a phenyl group of the BINAP ligand, the formation of the six-coordinate distorted-octahedral complex proposed by Takaya and Noyori et al. 17 and Ujaque and Espinet et al. 18 appears to be impossible. Furthermore, the relative free energy of complex 368 (DG ¼ 116.1 kJ mol À1 ) is energetically unstable. Accordingly, the associative mechanisms of Takaya and Noyori et al. and Ujaque and Espinet et al. are unfavorable.

Conclusions
We explored the reaction paths of the cationic Rh(I)-BINAPcatalysed isomerisation of allylic amines by employing the MC-AFIR and SC-AFIR methods combined with the ONIOM method; furthermore, we determined the most rational path from the complicated reaction network by employing Prim's algorithm. We conclude that the associative mechanism is impossible because there is no path leading to the product complex in the associative mechanism; also, the intermediates in the associative mechanism are unstable compared with those in the dissociative mechanism. In the dissociative mechanism, a path to 54 (type A) is the most rational path; it affords the (E,R)-enamine, which is consistent with the experimental results. Some paths afford the (Z)-enamine. However, the path to the (Z)-enamine is an endergonic process; therefore, the (Z)enamine is not formed. The path to 96 leads to the opposite enantiomer of 54. The C]C-coordinate Rh(I) complexes on the path to 54 (e.g. 45: DG ¼ 20.0 kJ mol À1 ) are comparably stable intermediates. In contrast, those on the path to 96 (e.g. 113: DG ¼ 43.9 kJ mol À1 ) are higher in energy. The difference arises from the steric repulsion between a pseudo-equatorial phenyl group and a substrate; the phenyl group on the dialkylamino group side is especially distorted. The path to 96 is not preferred because of the instability of the C]C-coordinated intermediates. The catalytic cycle is summarised in Fig. 21. The cycle starts from the chelate complex 1 or from the N-coordinate complex 5 or 8. Complex 1 is converted to a C]C-coordinate complex 46, which has an agostic interaction between the Rh(I) atom and the H-C(1) bond. In 46, the C]C double bond is coordinated to the Rh(I) atom from the si-face direction. Then, the hydride ion on the C(1) atom migrates to the Rh(I) atom and the hydride complex 47 is formed, in which the lone pair of the nitrogen atom assists hydride migration. Complex 53 is a conformational isomer of 47. Finally, hydride migration from the Rh(I) atom to the C(3) atom forms the product complex 54 ((E,R)-enamine). Aer the formation of 54, the catalytic cycle moves to the next cycle through substitution.
The aim of the present study is to demonstrate the possibility of determining a reaction mechanism without assumption of TSs. We achieved this using the MC-AFIR and SC-AFIR methods. Fig. 21 Catalytic cycle of the cationic Rh(I)-BINAP-catalysed isomerisation of allylic amines, as determined by a comprehensive reaction path search without assumption of TSs using the MC-AFIR and SC-AFIR methods and the selection of a rational reaction path using Prim's algorithm. The Gibbs free energies are denoted with and without parentheses, calculated at the B3LYP + D3(PCM)/BS3// B3LYP + D3/BS2 level (60 C). The energies are given relative to 1 in kJ mol À1 . Furthermore, the E/Z and S/R selectivity of the product could be explained. In addition, we demonstrated the usefulness of Prim's algorithm to nd a rational combination of elementary reaction steps. We hope the present study demonstrates the successful application of the AFIR method and Prim's algorithm.