Application of the String Method to Determine the Ambimodal Nature of Transition States in Bifurcation Reactions

ABSTRACT

Recent quantum chemical studies have revealed that a wide range of organic reactions exhibit post‐transition‐state bifurcation (PTSB), in which a single transition state (TS) leads directly to two or more distinct products along barrierless downhill pathways. Such TSs are commonly referred to as ambimodal TSs. In contrast, conventional intrinsic reaction coordinate (IRC) calculations typically connect the TS to only one of the possible products and therefore fail to identify alternative non‐IRC products. To assess the ambimodal character of TSs in PTSB reactions, we analyzed relaxed pathways connecting a TS directly to a non‐IRC product using the double‐ended string method. Application of this approach to a total of 33 reaction systems, including pericyclic reactions (among them two trifurcation systems) and non‐pericyclic reactions, enabled unambiguous classification of TS‐product connections as either a static downhill connection or a static small‐barrier connection, depending on whether an energy maximum exists below the TS. The proposed method provides a practical and robust tool for the static topological analysis of PTSB reactions and serves as an effective complement to on‐the‐fly dynamical trajectory simulations.

The double‐ended string method evaluates the ambimodal nature of transition states by connecting them directly to non‐IRC products. The resulting pathways unambiguously distinguish static downhill connections from static small‐barrier connections on potential energy surfaces, serving as a practical and robust tool for the static topological analysis of PTSB reactions.

1. Introduction

The concept of the transition state (TS) has long been central to our understanding of chemical reaction mechanisms, dynamics, and kinetics [1, 2]. In quantum chemical calculations, a TS is typically defined as a first‐order saddle point on a multidimensional potential energy surface (PES). At such a point, the mass‐weighted Hessian matrix possesses a single negative eigenvalue, and the corresponding eigenvector, referred to as the reaction coordinate, points toward the reactant and product regions. Within this framework, the intrinsic reaction coordinate (IRC), defined as the steepest‐descent path connecting the TS to the adjacent minima, has become the standard representation of a reaction pathway on the PES [3, 4]. Recent advances in automated reaction‐path search methods, such as those implemented in the GRRM program [5, 6, 7, 8, 9], have made it increasingly routine to identify stationary points and their connecting IRCs, enabling the construction of reaction networks for complex chemical systems. However, the conventional one‐TS‐to‐one‐product paradigm breaks down when a single TS leads directly to two or more distinct products without the intervention of an intermediate or an additional TS. This phenomenon is known as post‐transition‐state bifurcation (PTSB) [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. In such cases, the TS is often described as an ambimodal TS because it can evolve toward multiple product channels. PTSB has been reported in a wide variety of chemical systems, including pericyclic reactions, radical addition reactions involving bond rotation, and enzyme‐catalyzed cycloadditions [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. Nevertheless, reaction networks constructed solely from IRC calculations inherently follow only the steepest‐descent pathway toward a single product valley. As a result, bifurcating pathways that are not represented by IRCs are systematically excluded from conventional reaction networks [8, 9], leading to an incomplete description of the underlying reaction landscape. Therefore, the development of methodologies capable of reliably identifying PTSB is essential.

To identify PTSB behavior, Tantillo and coworkers previously demonstrated that the product reached by an IRC can switch upon slight changes in the initial structure used for the IRC calculation, the substituents, or the choice of DFT functional, and that such switching is often accompanied by a characteristic shoulder in the energy profile [21, 22]. This phenomenon provides one indication of the presence of PTSB. PTSB originates from a change in the topology of the PES from a valley to a ridge [23, 24, 25, 26, 27, 28, 29]. Two related concepts are commonly used to describe this bifurcative feature: the valley‐ridge inflection (VRI) point, which represents the fundamental topological origin of the bifurcation, and the valley‐ridge transition (VRT) point, which is defined by a less stringent criterion (see Section S1.9). For practical purposes, the VRT is typically evaluated along the IRC and is defined as the point at which an eigenvalue of the Hessian projected onto the subspace orthogonal to the reaction path changes sign from positive to negative [4, 23]. Consequently, projected harmonic vibrational analysis along the IRC has been proposed as a means of detecting PTSB [30, 31, 32, 33, 34]. However, because the VRI and VRT coincide along the IRC only in symmetric systems, the true VRI generally does not lie on the IRC [35]. Moreover, a system may exhibit genuine bifurcation without displaying any VRT point along the IRC, despite clearly undergoing PTSB [36]. In addition, this vibrational‐analysis approach requires the repeated evaluation of Hessian matrices along the reaction path, which can be computationally demanding. More recently, Soler et al. proposed a dimensionality‐reduced optimization (DRO) approach. In this method, a geometry optimization is initiated from a structure located just beyond the TS on a dimensionality‐reduced PES, where the coordinates leading toward known products are constrained. When the optimization is performed in the full 3 N − 6‐dimensional space, where N is the number of atoms in the system, it naturally converges to the product already connected to the TS through the IRC. In contrast, when the optimization is carried out on a constrained (3 N − 6 − X)‐dimensional PES (X > 0), convergence to a different product is regarded as strong evidence for the ambimodal character of the TS [37].

Despite the availability of the static methods described above for detecting PTSB, on‐the‐fly ab initio molecular dynamics (AIMD) trajectory simulations remain the gold standard in PTSB research [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. Unlike static approaches, AIMD explicitly incorporates dynamical effects, including finite‐temperature motion, zero‐point energy, recrossing, entropic contributions, and dynamical matching, thereby enabling quantitative prediction of product branching ratios. AIMD simulations can also uncover unexpected reaction pathways without requiring prior knowledge of all possible product structures. Nevertheless, trajectory‐based analyses are not without limitations. Their results can depend on the details of the simulation protocol, such as the sampling of initial conditions and the distribution of quantized zero‐point vibrational energy among the normal modes. Moreover, the reliable characterization of extremely minor reaction channels often requires hundreds or even thousands of trajectories, making such calculations computationally demanding.

In the present work, we therefore set aside dynamical effects and instead propose a strategy based on the double‐ended string method [38, 39, 40, 41, 42, 43, 44, 45, 46] that focuses on the static topographical features of ambimodal TSs. The string method, which relies solely on gradient information, is capable of locating the minimum‐energy pathway connecting any two specified structures on the PES. Here, we apply it directly between a TS of interest and a candidate non‐IRC product. If the resulting string pathway is entirely downhill in energy, it provides static evidence that the non‐IRC product is accessible from the TS without an intervening barrier. Such a pathway establishes a static downhill connection between the TS and the non‐IRC product, thereby recovering a topological link that is missed by the IRC and yielding a more complete reaction network. We argue that the existence of this static downhill connection demonstrates the ambimodal nature of the TS and provides strong evidence for the possibility of PTSB. In contrast to AIMD simulations, the present approach is robust in that it is independent of the choice of temperature, initial momenta, and sampling protocol. Precisely because it neglects dynamical effects, however, it cannot provide quantitative information on product branching ratios. The scope of the present work is therefore intentionally restricted to the identification and characterization of static potential‐energy pathways showing bifurcation behavior. Accordingly, we present the string method not as a substitute for trajectory‐based simulations, but as a practical and complementary approach for detecting and characterizing ambimodal TSs associated with PTSB.

2. Computational Methods

A schematic PES for a typical PTSB reaction is shown in Figure 1 as a function of two hypothetical molecular coordinates. As illustrated in the figure, a typical PTSB system consists of three equilibrium structures on the PES: Reactant, Product 1, and Product 2. Here, TS₁ is assumed to be located between Reactant and Product 1, and the corresponding IRC pathway is represented by a solid line. Product 1 and Product 2 are interconnected through a second transition state, TS₂, and the corresponding IRC pathway is shown by a dotted line. The VRI point is expected to lie in the vicinity of the bifurcation region. As mentioned in the Introduction, it is important to note that automated reaction‐path search algorithms implemented in programs such as the GRRM code [5, 6, 7] can readily identify these three equilibrium structures and determine the two IRC pathways in a fully automated manner. However, the ambimodal nature of TS₁ cannot be inferred solely from IRC calculations. Establishing whether TS₁ is ambimodal generally requires additional analyses, such as on‐the‐fly AIMD simulations initiated from structures in the vicinity of TS₁. Therefore, it is interesting to employ the string method [38, 39, 40, 41, 42, 43, 44, 45, 46] to investigate the energetic characteristics of the minimum‐energy pathway connecting TS₁ and Product 2, which is not directly connected to TS₁ through the IRC.

FIGURE 1.

Schematic two‐dimensional contour map of the PES of a typical PTSB reaction plotted as a function of two hypothetical molecular coordinates. The reaction system consists of three equilibrium structures: Reactant, Product 1, and Product 2. Here, TS₁ is assumed to be located between Reactant and Product 1, and the corresponding IRC pathway is represented by a solid line. Product 1 and Product 2 are interconnected through a second transition state, TS₂, and the corresponding IRC pathway is shown by a dotted line. The string optimization is applied to the path (a dashed line) between TS₁ and Product 2, which is not directly connected to TS₁ along the IRC.

In the string method, the calculation is initiated from a series of structural images that are typically distributed uniformly along a linear interpolation between the two endpoint structures. During the optimization, forces are iteratively applied to relax the images toward the minimum‐energy pathway, while the images are simultaneously redistributed to maintain approximately uniform spacing between neighboring images throughout the optimization process. It is worth noting that the path‐optimization procedure does not require Hessian evaluations, so that the cost of a string calculation is governed entirely by the number of gradient (force) evaluations. In the present study, we employ the string method implemented in the open‐source PIMD code (version 2.7.0) developed by Shiga [47]. The string path optimization, including both the evolution and the reparameterization step, was performed throughout in mass‐weighted Cartesian coordinates, the same coordinate system in which the IRC is conventionally defined. Numerical details of the string method we employed in this work are also described in Section S1 (Figures S1 and S2).

We applied the string method to a total of 33 reaction systems (from entries 1 to 33) summarized in Figure 2 (see also Tables 1, 2, 3). The first 12 systems (from entries 1 to 12) correspond to an intramolecular cycloaddition between diene and triene [15, 34] connected by a tether consisting of a two‐atom chain. This reaction may proceed through an ambimodal TS to yield either a (6 + 4) Diels‐Alder cycloadduct or a (4 + 2) cycloadduct. In our previous study [34], subtle modifications to the tether could alter the potential energy landscape but did not significantly disturb the PTSB behavior [48]. The next 15 systems (entries 13–27) correspond to an intramolecular cycloaddition between cyclobutadiene and butadiene [48], in which these two reacting fragments are also connected by a tether composed of either a three‐atom or four‐atom chain. This reaction has been previously shown to exhibit a PTSB character, yielding either a six‐membered (4 + 2) Diels‐Alder product or a four‐membered (2 + 2) cycloadduct from a single TS depending on the tether [34, 48]. For entries 1–27, Figure 2 also shows the three key C‐C internuclear distances (R ₁, R ₂, and R ₃) that characterize the bifurcation dynamics, where the C‐C(R ₁) bond is immediately formed after passing through the TS, followed by formation of either the C–C(R ₂) or C–C(R ₃) bond during the subsequent nuclear dynamics.

FIGURE 2.

Reaction systems investigated in this work. The TS₁ (see Figure 1) and product structures are shown for each system. Entries 1–12 correspond to intramolecular cycloaddition reactions between diene and triene moieties connected by different tethers. Entries 13–27 correspond to intramolecular cycloaddition reactions of tethered butadiene systems connected by various linkers. Hydrogen atoms are omitted in the definition of tethers. Entry 28 represents the bimolecular cycloaddition reaction between tropone and dimethylfulvene. Entry 29 represents the bimolecular tripericyclic reaction between tropone and cycloheptatriene. Entry 30 represents the bimolecular tripericyclic reaction between 8,8‐disubstituted heptafulvene and 6,6‐dimethylfulvene. Entries 31 and 32 represent the radical additions of methyl and fluoromethyl radicals to methylallene, respectively. Entry 33 represents the asynchronous nitrene insertion into a C–C $\mathrm{\sigma }$‐bond.

TABLE 1.

Summary of the results of IRC and string calculations for entries 1 to 28.

Entry	Tether a	Product 1 (IRC)	Product 2 (non‐IRC)	String result b , c (TS1–Product 2)	VRT
Intramolecular diene‐triene cycloaddition leading to either (6 + 4) or (4 + 2) product
Reaction 1	C–C	(6 + 4)	(4 + 2)	Downhill	Yes
Reaction 2	C–N	(6 + 4)	(4 + 2)	Downhill	Yes
Reaction 3	C=C	(6 + 4)	(4 + 2)	Downhill	Yes
Reaction 4	N–C	(6 + 4)	(4 + 2)	Downhill	Yes
Reaction 5	N–N	(4 + 2)	(6 + 4)	Downhill	Yes
Reaction 6	N–O	(4 + 2)	(6 + 4)	Downhill	Yes
Reaction 7	O–C	(6 + 4)	(4 + 2)	Downhill	Yes
Reaction 8	O–N	(4 + 2)	(6 + 4)	Downhill	Yes
Reaction 9	O–O	(4 + 2)	(6 + 4)	Downhill	Yes
Reaction 10	C=N	(6 + 4)	(4 + 2)	Downhill + min	No
Reaction 11	N=C	(6 + 4)	(4 + 2)	Downhill + min	No
Reaction 12	N=N	(6 + 4)	(4 + 2)	Downhill + min	No
Intramolecular cyclobutadiene‐butadiene cycloaddition leading to either (4 + 2) or (2 + 2) product
Reaction 13	C–C–C	(2 + 2)	(4 + 2)	Downhill + min	Yes
Reaction 14	C=N–C	(2 + 2)	(4 + 2)	Downhill + min	Yes
Reaction 15	C–N=C	(2 + 2)	(4 + 2)	Downhill + min	Yes
Reaction 16	C=C–N	(2 + 2)	(4 + 2)	Downhill + min	Yes
Reaction 17	N–C=N	(2 + 2)	(4 + 2)	Downhill + min	Yes
Reaction 18	C–C–C–C	(4 + 2)	(2 + 2)	Downhill + min	Yes
Reaction 19	C–O–C–C	(4 + 2)	(2 + 2)	Downhill + min	Yes
Reaction 20	C–C–O–C	(4 + 2)	(2 + 2)	Downhill + min	Yes
Reaction 21	C–O–O–C	(2 + 2)	(4 + 2)	Downhill + min	Yes
Reaction 22	C–C=N	(2 + 2)	(4 + 2)	Downhill + min	No
Reaction 23	C–N–C	(2 + 2)	(4 + 2)	Downhill + min	No
Reaction 24	C–O–C	(2 + 2)	(4 + 2)	Downhill + min	No
Reaction 25	C–O–O	(2 + 2)	(4 + 2)	Downhill + min	No
Reaction 26	N=C–C	(2 + 2)	(4 + 2)	Downhill + min	No
Reaction 27	N–N–N	(2 + 2)	(4 + 2)	Downhill + min	No
Bimolecular cycloaddition between tropone and dimethylfulvene
Reaction 28		(6 + 4)	(6 + 4)′	Small barrier	No

^a Hydrogen atoms are omitted in the tether definition.

^b “Downhill + min” means the potential energy profile contains a pseudo‐minimum along the optimized string path.

^c “Small barrier” means the potential energy profile contains a pseudo‐minimum and a low energy barrier (above the TS₁ energy level) along the optimized string path.

TABLE 2.

Summary of the results of IRC and string calculations for entries 29 and 30.

Entry	Product 1 (IRC)	Product 2 (non‐IRC)	String result a , b (TS1–Product 2)	Product 3 (non‐IRC)	String result a , b (TS1–Product 3)	VRT
Bimolecular tripericyclic reaction leading to three different cyclic products
Reaction 29	(6 + 4)	(6 + 4)′	Downhill	(8 + 2)	Small barrier	Yes
Reaction 30	(4 + 6)	(6 + 4)	Downhill + min	(8 + 2)	Downhill min	No

^a “Small barrier” means the potential energy profile contains a pseudo‐minimum and a low energy barrier (above the TS₁ energy level) along the optimized string path.

^b “Downhill + min” means the potential energy profile contains a pseudo‐minimum along the optimized string path.

TABLE 3.

Summary of the results of the string calculations for the non‐pericyclic systems, entries 31–33.

Entry	Reaction	Product 1 (IRC)	Product 2 (non‐IRC)	String result a	VRT
Reaction 31	CH3 radical addition to methylallene	Z‐allylic radical	E‐allylic radical	Downhill + min	Yes
Reaction 32	CH2F radical addition to methylallene	Z‐allylic radical	E‐allylic radical	Downhill + min	Yes
Reaction 33	Asynchronous nitrene insertion into a C–C σ‐bond	[4.2.1] adduct	[3.2.2] adduct	Downhill + min	Yes

^a “Downhill + min” means the potential energy profile contains a pseudo‐minimum along the optimized string path.

The next system (entry 28) is a bimolecular cycloaddition reaction between tropone and dimethylfulvene, which has been previously shown to lead either to one of the two different (6 + 4) cycloadduct products via the ambimodal TS [49]. This system is chosen because the previous theoretical study by Soler et al. [37] using the DRO approach suggests that this system does not show the ambimodal nature of the TS. Thus, it is of particular interest to investigate this reaction system using the present string‐method approach and compare the results with the findings of Soler et al. [37]. The next two systems (entries 29 and 30) correspond to tripericyclic reactions that were previously shown, through AIMD simulations, to yield three distinct cycloaddition products [50, 51]. More specifically, for entry 29, two different (6 + 4) cycloadducts and one (8 + 2) cycloadduct were found to be formed. In contrast, entry 30 was shown to produce (4 + 6), (6 + 4), and (8 + 2) cycloadducts. Figure 2 illustrates the four key internuclear distances (R ₁, R ₂, R ₃, and R ₄) that characterize these two tripericyclic reactions. Among these distances, formation of the R ₁ bond occurs immediately after passage through the TS, followed by the formation of the R ₂, R ₃, or R ₄ bond, ultimately leading to different cycloadduct products. It is worth noting that these two tripericyclic reactions were also studied by Soler et al. [37]. For entry 29, they found that the reaction pathways leading from the TS to the two (6 + 4) cycloadducts exhibit downhill character, whereas a small barrier was suggested to exist along the pathway connecting the TS to the (8 + 2) product. For entry 30, the IRC originating from the TS was found to lead only to the (4 + 6) cycloadduct, while the pathways connecting the TS to the (6 + 4) and (8 + 2) products were not purely downhill. Their finding implies that, from a static perspective, the corresponding TS may not be regarded as a truly ambimodal TS. Instead, Soler et al. suggested the presence of small barriers along the pathways leading to the (6 + 4) and (8 + 2) products and argued that the TS should be classified as a dynamical ambimodal TS, since these barriers can be overcome by additional energy [37]. Thus, it is of particular interest to reexamine these reaction systems using the present string‐method calculations and assess the validity of the conclusions reached by Soler et al. [37].

Entries 1–30 are all pericyclic cycloadditions, chosen so that a large, internally consistent family of closely related systems could be scanned systematically. The string method itself, however, contains nothing specific to pericyclic reactivity: it requires only a TS structure, a product not connected to that TS by the IRC, and gradients on the PES. To demonstrate the applicability of the string method to different types of PTSB reaction directly, we have added three non‐pericyclic PTSB systems, entries 31–33. Entries 31 and 32 are the additions of a methyl radical and a fluoromethyl radical to methylallene, respectively [20]. In both cases, passage through the TS leads to both the Z‐ and E‐allylic radicals through a sequential process: completion of C–C bond formation is followed by torsion around the allylic single bond [20]. Unlike the pericyclic reactions described above, the reaction pathways here are analyzed by focusing on two key geometrical parameters: the distance R between the allene and the approaching radical carbon, and a specific dihedral angle ϕ that characterizes the torsion around the allylic bond after radical addition (Figure 2). In this coordinate definition, upon radical addition, ϕ ≈ 0° and ϕ ≈ 180° correspond to the Z‐ and E‐allylic radicals, respectively. Entry 33 is the asynchronous insertion of a nitrene into a C–C σ‐bond, in which the ambimodal TS leads to two bicyclic frameworks [52]. None of these three involves a pericyclic reaction, and they span both open‐shell doublet and closed‐shell neutral systems.

In the present computational study, for entries 1–27, we used the machine‐learning potentials called AIQM2 [53] to obtain potential energy values and the derivatives at a given structure on the PES. Details for the AIQM2 method are also provided in our previous paper [34]. Regarding entries 28–30, the computational levels were selected as follows, and all DFT calculations were performed using the Gaussian 09 program package [54]. For entries 28 and 30, calculations were performed at exactly the same DFT levels as the original AIMD studies by Houk and co‐workers (B3LYP‐D3/6‐31G(d) for entry 28 [49] and ωB97X‐D/6‐31G(d) for entry 30 [51]), enabling a direct comparison with their dynamical results. For entry 29, although we reduced the basis set size to ωB97X‐D/6‐31G(d) due to computational cost constraints compared to the ωB97X‐D/def2‐TZVP level used by Houk and co‐workers [50], this modification is not expected to alter the essential features of the reaction pathways. For the three non‐pericyclic systems (entries 31 to 33), the level of theory was in each case chosen to match that of the original dynamics study. Entries 31 and 32 are open‐shell doublets treated at the (U)B3LYP‐D3(BJ)/def2‐SVP level with the SMD continuum model for 1,4‐dioxane and an ultrafine integration grid, following ref. [20]. Entry 33 is a neutral closed‐shell reaction and is treated at B3LYP/6‐31+G(d,p) with an ultrafine grid, following ref. [52].

We have also checked if each reaction system has a VRT point on the IRC pathway by evaluating projected vibrational harmonic frequencies along the IRC [33]. For discussions on the computational costs and coordinate dependence of the string method, see Sections S1.7 and S1.8 (Table S1).

3. Results and Discussion

Figure 3a shows a representative example of a double‐ended string optimization connecting TS₁ and Product 2, which is not directly connected to TS₁ through the IRC pathway, for Reaction 1 (an intramolecular diene–triene cycloaddition linked by the C–C tether). The string paths are projected onto the (R ₂, R ₃) coordinate plane. The string optimization, consisting of 101 images, was initiated from a linearly interpolated path connecting the two endpoint structures. Several intermediate stages of the string optimization, together with the final optimized pathway, are displayed in the figure. For comparison, the two IRC pathways passing through TS₁ and TS₂ are also shown, and the VRT point on the IRC pathway connecting TS₁ and Product 1 is indicated. The optimized string pathway exhibits the expected behavior for an ambimodal TS. Specifically, it follows the IRC pathway connecting TS₁ and Product 1 up to the vicinity of the VRT region, after which it gradually deviates toward the Product 2 channel and ultimately reaches Product 2 without encountering an additional barrier. The corresponding potential energy profile along the final optimized string path is presented in Figure 3b as a function of the reaction‐path length (in units of (amu)^1/2Å). The resulting energy profile displays a continuous downhill trend from TS₁ to Product 2, indicating that the pathway connecting these two endpoint structures is barrierless. TS₁ and Product 2 are therefore joined by a static downhill connection. Since the IRC pathway from TS₁ to Product 1 is likewise a barrierless downhill pathway, the PES topology of this reaction clearly shows that the TS₁ structure is smoothly connected to the two products via energetically downhill paths.

FIGURE 3.

A typical example of double‐ended string optimization connecting TS₁ and Product 2 for Reaction 1 (intramolecular cycloaddition reaction between diene and triene with the C–C tether). In panel (a), the string path consisting of 101 images is projected onto the (R ₂, R ₃) coordinate plane. The string optimization was initiated from a linearly interpolated path connecting these two endpoint structures. Several intermediate stages of the string optimization, together with the final optimized pathway, are presented. Two IRC pathways passing through TS₁ and TS₂ and the VRT point are also plotted. In panel (b), the potential energy profile along the final optimized string path is shown as a function of the reaction path length (measured from the energy level of TS₁ in units of (amu)^1/2Å). The potential energy profile along the IRC through TS₁ is also shown.

Figure 4 presents another example of a string optimization, which exhibits behavior somewhat different from that shown in Figure 3. Figure 4a displays the optimized string pathway projected onto the (R ₂, R ₃) coordinate plane for Reaction 10 (an intramolecular diene–triene cycloaddition connected by a C=N tether). Notice that this system does not have a VRT point on the IRC pathways passing through TS₁. In the region where R ₂ > ~2.2 Å, the optimized pathway closely follows the IRC passing through TS₁. Beyond this region, however, the pathway bends markedly toward Product 2. The corresponding potential energy profile along the optimized string pathway is shown in Figure 4b. Interestingly, a local minimum feature is observed at x ~ 2.5 (amu)^1/2Å, which is the reaction path length measured from the TS₁ structure in mass‐weighted coordinates. We note that this minimum occurs approximately in the region where the pathway exhibits pronounced curvature in Figure 4a. To examine whether this minimum corresponds to a genuine intermediate on the PES, we carried out a conventional geometry optimization starting from the structure corresponding to the local minimum on the potential energy profile along the optimized string path. We found that the optimization converged directly to Product 1, indicating that no stable intermediate exists in this region of the PES. Because the optimized string path connects TS₁ to Product 2 in an overall downhill manner without passing through any energy level higher than TS₁, Reaction 10 likewise possesses a static downhill connection to the non‐IRC product. The additional sub‐TS₁ barrier along this pathway suggests that Product 2 is less directly accessible than in Reaction 1, but this should not be directly interpreted as a quantitative estimate of the branching fraction, because trajectories in general depart from the string and need not encounter that barrier.

FIGURE 4.

Same as Figure 3 but for Reaction 10 (intramolecular cycloaddition reaction between diene and triene with the C=N tether).

The results of the string optimizations for Reactions 1–27 are summarized in Table 1, together with information on the presence or absence of a VRT point along the corresponding IRC pathway. The final optimized string pathways for Reactions 1–12 are shown in Figure 5, and the corresponding potential energy profiles are presented in Figure 6. Similarly, the optimized string pathways for Reactions 13–27 are displayed in Figure 7, while the corresponding potential energy profiles are shown in Figure 8.

FIGURE 5.

Optimized string paths between TS₁ and Product 2 (non‐IRC product) for Reaction 1–12 (intramolecular cycloaddition reaction between diene and triene by a tether chain) plotted on the (R ₂, R ₃) coordinate plane. Two IRC pathways passing through TS₁ and TS₂ are also plotted. The VRT points are also shown (if VRT occurs).

FIGURE 6.

Potential energy profiles along the optimized string paths for Reaction 1 to 12 (intramolecular cycloaddition reaction between diene and triene by a tether chain) plotted as a function of reaction path length defined using mass‐weighted coordinates. The potential energy profile along the IRC through TS₁ is also shown. Energy is measured from the energy level of TS₁.

FIGURE 7.

Optimized string paths between TS₁ and Product 2 (non‐IRC product) for Reaction 13–27 (intramolecular cycloaddition reaction between cyclobutadiene and butadiene connected by a tether chain) plotted on the (R ₂, R ₃) coordinate plane. Two IRC pathways passing through TS₁ and TS₂ are also plotted. The VRT points are also shown (if VRT occurs).

FIGURE 8.

Potential energy profiles along the optimized string paths for Reaction 13 to 27 (intramolecular cycloaddition reaction between cyclobutadiene and butadiene connected by a tether chain) plotted as a function of reaction path length defined using mass‐weighted coordinates. The potential energy profile along the IRC through TS₁ is also shown. Energy is measured from the energy level of TS₁.

Inspection of the results presented in Figures 5, 6, 7, 8 shows that a static downhill connection between TS₁ and Product 2 is present in every case (entries 1–27), since the overall potential energy profile connecting TS₁ and Product 2 is downhill throughout. At the same time, many systems exhibit pseudo‐minima along the optimized string pathways, although we verified that geometry optimizations starting from the structures at these pseudo‐minima always arrived at the Product 1 structures without exception (the XYZ coordinates of these pseudo‐minima are provided in the Supporting Information). For a representative C=N system (entry 10), a 2D relaxed‐scan calculation was performed to map out the overall energy landscape, confirming that the observed pseudo‐minima are apparent topological features rather than genuine reaction intermediates (see Section S2 and Figure S3). The barrier heights associated with these features, measured relative to the corresponding pseudo‐minimum, span a relatively wide range of energies. It is reasonable to expect that reactions with zero or very small barriers along the optimized string pathway will exhibit pronounced bifurcation behavior, since both product channels are readily accessible from the vicinity of TS₁. In contrast, when the sub‐TS₁ barrier arising from such a pseudo‐minimum is appreciable, access to the non‐IRC channel is expected to be less direct from a dynamical viewpoint. Another important observation is that the presence of a VRT point on the IRC pathway is not a prerequisite for a static downhill connection to the non‐IRC product. In other words, the system can exhibit static downhill character regardless of the presence or absence of a VRT point. This finding is consistent with the recent theoretical study by Taketsugu and co‐workers [36].

Figure 9 and Table 1 summarize the string optimization results for Reaction 28, a bimolecular [6 + 4] cycloaddition between tropone and dimethylfulvene [49], computed using $21$ images. As shown in Figure 9a, the optimized pathway runs nearly parallel to the IRC passing through TS₂. Along this path, the energy profile descends into a shallow pseudo‐minimum before crossing a small barrier of 0.27 kcal/mol above the TS₁ level (Figure 9b). TS₁ and Product 2 are therefore joined by a static small‐barrier connection rather than a static downhill connection. This energetic feature provides a clear rationale for why the DRO approach by Soler et al. [37] failed to locate Product 2, as optimizations initiated from TS₁ in reduced coordinate space would become trapped in the vicinity of the pseudo‐minimum. Furthermore, the presence of this small barrier qualitatively agrees with the minor branching fraction (94.0:6.0) reported in the AIMD study by Houk and co‐workers [49] and supports Soler et al.'s argument that the TS is not strictly ambimodal from a static PES perspective. Although a 0.27 kcal/mol barrier is readily accessible at room temperature and string profiles alone cannot quantitatively determine branching fractions, the present observation demonstrates that the string optimization method can provide a reliable means of establishing whether a static downhill connection to a non‐IRC product exists, and of distinguishing that from a static small‐barrier connection.

FIGURE 9.

Results of the string optimization for the bimolecular cycloaddition reaction between tropone and dimethylfulvene (Reaction 28; see Figure 2 for the coordinate definitions). The string optimization, consisting of 21 images, was initiated from a linearly interpolated path connecting the two endpoint structures, TS₁ and Product 2. (a) Final optimized string pathway and IRC paths through TS₁ and TS₂ projected onto the (R ₂, R ₃) coordinate plane. (b) Corresponding potential energy profile along the optimized string pathway plotted as a function of the reaction‐path length, where energy is measured from the energy level of TS₁. The potential energy profile along the IRC through TS₁ is also shown.

Next, we examine the results of the string optimizations for Reactions 29 and 30, which have previously been studied as representative trifurcation reactions yielding three distinct products from a single TS through on‐the‐fly AIMD simulations [50, 51]. The results of the string calculations for Reactions 29 and 30 are summarized in Figure 10 and in Table 2. As shown in Figure 10b for Reaction 29, the potential energy profile along the optimized string pathway leading to Product 2 exhibits a completely downhill character. In contrast, the pathway leading to Product 3 initially descends from TS₁ but subsequently encounters a barrier whose energy lies above that of TS₁. This result indicates that TS₁ is joined to Product 1 and Product 2 by static downhill connections, whereas TS₁ and Product 3 are joined by a static small‐barrier connection. This behavior is consistent with the findings of Soler et al. [37] obtained using the DRO approach. Furthermore, Houk and co‐workers reported a branching fraction of only 0.8% for Product 3, while the corresponding branching fractions for Products 1 and 2 were calculated to be 63.6% and 35.7%, respectively [50]. These results are also in qualitative agreement with the present string‐optimization analysis.

FIGURE 10.

Optimized string pathways connecting TS₁ to Product 2 and TS₁ to Product 3 for Reaction 29 (bimolecular tripericyclic reaction between tropone and cycloheptatriene) and Reaction 30 (bimolecular tripericyclic reaction between 8,8‐disubstituted heptafulvene and 6,6‐dimethylfulvene). The string optimizations were performed using 21 images initialized from linearly interpolated paths between the respective endpoint structures. (a) Optimized string pathways for Reaction 29 projected onto the three‐dimensional (R ₂, R ₃, R ₄) coordinate space (see Figure 2 for the definition of the reaction coordinates). The IRC pathway is also shown. (b) Corresponding potential energy profiles along the optimized string pathways for Reaction 29 plotted as functions of the reaction‐path length, where energy is measured from the energy level of TS₁. The potential energy profile along the IRC through TS₁ is also shown. (c) Same as panel (a) but for Reaction 30. (d) Same as panel (b) but for Reaction 30.

For Reaction 30, the optimized string pathways leading to both Product 2 and Product 3 exhibit static downhill connections, as shown in Figure 10d. This finding is consistent with the previous on‐the‐fly AIMD study [51], which yielded a branching ratio of 87:3:3, but differs from the conclusions reached by Soler et al. [37], who reported that no static downhill connection could be located for this system. This discrepancy can be qualitatively understood from the pseudo‐minima along both optimized string pathways, similar to those observed for the tethered systems discussed above. Similar to Reaction 28, a DRO starting near TS₁ can be trapped near a pseudo‐minimum rather than converging to the final product, which likely explains why the DRO method failed to locate the downhill connection that the full‐dimensional string calculation reveals. Although the present pseudo‐minima differ in detail from the structures identified by Soler et al., owing to the different dimensionality of the two coordinate spaces, this comparison highlights a general limitation of the DRO approach and underscores the usefulness of the full‐dimensional string method. Although Products 2 and 3 are statically accessible from TS₁, the presence of pseudo‐minima along their string profiles indicates a less direct descent than toward Product 1. This feature qualitatively aligns with the minor branching fractions from the AIMD calculations reported by Houk and co‐workers (only 3% and 3% for Products 2 and 3, respectively [51]), though string profiles alone cannot quantitatively determine branching ratios.

Finally, we turn to the three non‐pericyclic systems, entries 31 to 33, which were added to establish that the string approach is not restricted to pericyclic reactivity. The results are collected in Table 3 and Figures 11 and 12. In all three cases the optimized string connecting TS₁ to the non‐IRC product is downhill throughout, so that a static downhill connection is present in every case. For entries 31 and 32, representing the radical additions of $·{\mathrm{CH}}_3$ and $·{\mathrm{CH}}_2\mathrm{F}$ to methylallene, Hong and co‐workers reported trajectory branching ratios of 129:6 and 93:20 favoring the Z‐allylic radical [20]. In these systems, the competing products are the Z‐ and E‐allylic radicals, which differ not by bond formation or cleavage but by torsion around the allylic single bond. In both cases, the string optimization connecting TS₁ to the non‐IRC E‐product produced a downhill energy profile (Figure 11a,b for entry 31 and Figure 11c,d for entry 32). This result demonstrates that the string method is readily applicable to torsional bifurcations as well as bond‐forming ones. Entry 33 connects TS₁ to the bicyclic product that is not reached by the IRC, for which Tantillo and co‐workers reported a branching fraction of 3 out of 75 productive trajectories [52]; the downhill string profile (Figure 12) is consistent with that channel being open, while the small branching fraction is a dynamical consequence that cannot be obtained from the present static energy profile. In summary, the present results demonstrate that string method can also be applied to non‐pericyclic reaction systems showing PTSB behavior.

FIGURE 11.

Results of the string optimization for the radical additions to methylallene: Reaction 31 (methyl radical addition) and Reaction 32 (fluoromethyl radical addition). (a) Optimized string pathway connecting TS₁ to Product 2 and IRC path for Reaction 31 projected onto the two‐dimensional space defined by the radical–carbon distance R and the specific dihedral angle ϕ (where ϕ ≈ 0° and ϕ ≈ 180° correspond to the Z‐ and E‐allylic radicals, respectively; see Figure 2). (b) Potential energy profile along the optimized string pathway plotted as a function of the reaction‐path length for Reaction 31, where the energy is measured from the energy level of TS₁. The potential energy profile along the IRC through TS₁ leading to Product 1 is also shown for comparison. (c) Same as panel (a) but for Reaction 32. (d) Same as panel (b) but for Reaction 32.

FIGURE 12.

Results of the string optimization for the asynchronous nitrene insertion into a C–C σ‐bond (Reaction 33; see Figure 2 for the coordinate definitions). The string optimization, consisting of 21 images, was initiated from a linearly interpolated path connecting the two endpoint structures, TS₁ and Product 2. (a) Final optimized string pathway and IRC path through TS₁ projected onto the reaction coordinate space. (b) Corresponding potential energy profile along the optimized string pathway plotted as a function of the reaction‐path length, where energy is measured from the energy level of TS₁. The potential energy profile along the IRC through TS₁ leading to Product 1 is also shown.

3.1. Additional Remarks on the String Approach

In this section we briefly discuss the limitations of the string calculation. The string path is a purely static object, determined by the shape of the PES alone, and the following limitations are intrinsic. (i) Both endpoints must be supplied, so only product channels already envisaged can be tested, whereas trajectories can and do reveal unanticipated products. (ii) Trajectories predict branching ratios whereas the string method does not; even if a purely downhill path exists on the PES, dynamic effects may prevent trajectories from accessing that product valley, meaning the method can identify a static connection to a product that is not dynamically formed. (iii) Trajectories can account for the entropy of competing pathways, and a product that is uphill on the PES can dominate for entropic reasons, as seen in the ammonium ylide rearrangements of Singleton and co‐workers [55]; such a channel would be judged minor by any purely static criterion, meaning that string profiles cannot reliably indicate the expected magnitude of branching fractions. (iv) Quasiclassical trajectories include zero‐point vibrational energy; the string path does not. (v) Recrossing is frequent on bifurcating surfaces and is captured by trajectories but not by the string. (vi) Trajectories can account for temperature and the ensemble of energies crossing the TS, which a static barrier height cannot capture. (vii) Much like the IRC, the string pathway neglects atomic momenta, and real trajectories may depart from it; thus, a barrier along this particular path serves at best as a qualitative flag.

It is worth mentioning that the above‐mentioned points can also affect the choice of terminology. Following ref. [37], one might be tempted to classify a TS as “statically” or “dynamically” ambimodal depending on whether a barrier above the TS₁ energy is present along the reaction path. We refrain from using these terms, however, because “ambimodal” inherently carries an established dynamical meaning—namely, that trajectories initiated at the TS yield multiple distinct products—whereas the string method evaluates purely static properties of the PES. We instead state that a TS and a non‐IRC product are joined by a static downhill connection when the optimized string contains no point higher in energy than the TS, and by a static small‐barrier connection when the highest point exceeds it by a small amount. These descriptors strictly target only the topographical features of the calculated pathway and avoid referencing dynamic branching ratios, thereby enabling unambiguous verification through static calculations. On the other hand, the static string approach offers clear advantages. Trajectory outcomes inherently depend on dynamic simulation parameters and can suffer from dynamic artifacts, such as sensitivity to initial momentum sampling, temperature choices, and zero‐point energy leakage [15]. Furthermore, statistical sampling limits carry the inherent risk of omitting minor reaction channels. In contrast, string calculations are free from these dynamic constraints; they deterministically characterize the intrinsic static PES topography and evaluate the existence of a downhill pathway to a specified product at a computational cost unaffected by branching bias. Moreover, unlike multidimensional grid‐based PES scans (such as 2D relaxed scans), which suffer from the arbitrariness of defining collective variables and prohibitive computational scaling, the string method yields an unambiguously determined pathway at relatively low computational cost. A practical and robust workflow is therefore to first identify plausible product minima using chemical intuition or automated global search methods such as GRRM, and subsequently employ string calculations to systematically screen the static accessibility of each candidate.

Because the string pathway is optimized to minimize perpendicular forces, projected frequencies—and hence zero‐point energies and vibrational entropies—could in principle be evaluated along well‐behaved regions of the pathway, as previously demonstrated for bifurcating systems within variational transition state theory by Gonzalez‐Lafont, Moreno, and Lluch [56]. Extending the present approach to such free‐energy profiles represents the most promising route toward quantitative static predictions. However, when quantitatively evaluating and predicting product branching ratios, caution is required regarding the sole use of the current string method, which traces only static pathways on the PES. Its primary role remains that of a secondary, complementary tool alongside trajectory simulations, which are essential whenever a quantitative determination of branching ratios is required.

4. Conclusions

We demonstrated a double‐ended string method optimization scheme connecting TSs directly to non‐IRC products to evaluate the ambimodal nature of TSs across 33 reaction systems. The benchmark encompasses 30 pericyclic reactions (including tripericyclic systems exhibiting trifurcation) and 3 non‐pericyclic pathways spanning radical additions and nitrene insertions, establishing the broad applicability of the scheme to diverse bifurcating topologies.

By directly optimizing pathways between designated endpoints, this approach provides a practical means to verify whether a static downhill connection to a non‐IRC product exists on the PES, bypassing the dynamical parameter sensitivity inherent to trajectory simulations (such as thermal sampling, initial momentum distributions, and zero‐point energy partitioning). Unlike grid‐based scans, it avoids the arbitrariness of selecting collective variables; unlike DRO approaches, it overcomes trapping at pseudo‐minima along the reaction path. Importantly, the method unambiguously recovers the missing topological edges between ambimodal TSs and alternative product channels that are inherently overlooked by standard IRC calculations, without risking the omission of minor channels due to trajectory sampling limits. Although the string alone cannot predict quantitative branching ratios governed by atomic momenta and entropic effects, it serves as a robust static‐topological tool that reliably complements on‐the‐fly AIMD simulations. Future extensions to free energy calculations and automated reaction network exploration will further broaden its utility toward a comprehensive understanding of complex chemical reactions.

Author Contributions

Keita Yamaki: data curation, methodology, writing – original draft, visualization, investigation. Hayato Yamada: data curation, investigation, visualization. Daichi Okamoto: data curation, investigation, visualization. Shoto Nakagawa: investigation, visualization, data curation. Toshiyuki Takayanagi: conceptualization, funding acquisition, investigation, writing – review and editing, project administration, supervision.

Funding

This work was supported by the Japan Society for the Promotion of Science, Grant‐in‐Aid for Scientific Research (grant No. JP25K08573).

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Figure S1: Dependence of the optimized string pathway from TS₁ to Product 2 on the increment parameter Δt (0.01 ≤ Δt ≤ 4.0), with the number of images fixed at 101.

Figure S2: Dependence of the optimized string pathway from TS₁ to Product 2 on the number of images ($N_{\text{images}}$), with the increment parameter Δt fixed at 1.0.

Table S1: Number of images, convergence iterations, and total gradient evaluations for each reaction system.

Figure S3: Two‐dimensional potential energy surface (PES) obtained from 2D relaxed‐scan calculations and reaction pathways for the $\mathrm{C}[chemistry\; double\; bond\; solid\; lines]\mathrm{N}$ system (Reaction 10).

Data Availability Statement

The data that supports the findings of this study are available in the Supporting Information of this article.