Correlated Flat-Bottom Elastic Network Model for Improved Bond Rearrangement in Reaction Paths

Abstract

This study introduces correlated flat-bottom elastic network model (CFB-ENM), an extension of our recently developed flat-bottom elastic network model (FB-ENM) for generating plausible reaction paths, i.e., collision-free paths preserving nonreactive parts. While FB-ENM improved upon the widely used image-dependent pair potential (IDPP) by addressing unintended structural distortion and bond breaking, it still struggled with regulating the timing of series of bond breaking and formation. CFB-ENM overcomes this limitation by incorporating structure-based correlation terms. These terms impose constraints on pairs of atom pairs, ensuring immediate formation of new bonds after breaking of existing bonds. Using the direct MaxFlux method, we generated paths for 121 reactions involving main group elements and 35 reactions involving transition metals. We found that CFB-ENM significantly improves reaction paths compared to FB-ENM. CFB-ENM paths exhibited lower maximum DFT energies along the paths in most reactions, with nearly half showing significant energy reductions of several tens of kcal/mol. In the few cases where CFB-ENM yielded higher energy paths, most increases were below 10 kcal/mol. We also confirmed that CFB-ENM reduces computational costs in subsequent precise reaction path or transition state searches compared to FB-ENM. An implementation of CFB-ENM based on the Atomic Simulation Environment is available on GitHub for use in computational chemistry research.

1. Introduction

This study extends our previous work on developing a computationally efficient potential energy based on molecular structures for generating plausible reaction paths, termed the flat-bottom elastic network model (FB-ENM).¹ While the previous paper¹ provides detailed background information, a concise overview is presented here.

Understanding chemical reaction processes requires identifying reaction paths and their associated transition states. Computational methods for exploring these aspects are broadly categorized into two types: single-ended methods, which update a single structure, and double-ended methods, which update the entire reaction path connecting a given reactant and product.² This study focuses on the latter category, of which the nudged elastic band (NEB)³⁻⁶ and string methods⁷⁻⁹ are prominent examples.

We have recently developed the direct MaxFlux method, a double-ended transition state searches.¹⁰ This method is derived from the MaxFlux method, which was originally designed to incorporate temperature effects in reaction paths.^11,12 The direct MaxFlux method conversely operates at a near-zero temperature regime to identify transition states. To enhance computational efficiency, we have implemented several numerical techniques in the direct MaxFlux method: basis expansion of the objective function using B-spline functions and adaptive updating of energy evaluation points for the line integral in the objective function. These techniques enable transition state searches with reduced energy evaluations and iterations, resulting in a 50% decrease in total computational cost compared to the NEB method.^13,14 The Python implementation of the direct MaxFlux method is available on GitHub.¹⁵

For reaction path optimization, rapidly generating a plausible initial path is crucial. Such a path, which avoids atomic collisions and preserves the structure of nonreactive parts, can significantly reduce the computational cost of optimization. One approach combines the image-dependent pair potential (IDPP), a computationally efficient potential energy based on molecular structures, with the NEB method to generate such plausible paths.^16,17 Another approach defines a Riemannian metric that renders geodesics as plausible reaction paths.¹⁸ Among them, IDPP has been widely adopted, as implemented, for example, in the Atomic Simulation Environment (ASE).¹⁹

Recently, we proposed FB-ENM, a structure-based potential energy for rapidly generating plausible reaction paths.¹ FB-ENM addresses limitations of IDPP, which is susceptible to energetically unfavorable molecular distortions and bond breaking in nonreactive parts due to its redundant constraints and the functional form of its pair potential. FB-ENM employs a piecewise quadratic function with a flat bottom to impose constraints on interatomic distances. Substantial constraints are applied only to the minimum necessary atom pairs to maintain nonreactive bonds, bond angles, and planar structures. Weak constraints are imposed on other atom pairs to prevent extremely short or long interatomic distances. Consequently, the flat bottom of the total energy in FB-ENM consists of the reactant, product, and other collision-free structures that preserve nonreactive structural elements. The direct MaxFlux method can then determine the shortest path within this flat bottom. A comparison of maximum DFT energies between FB-ENM and IDPP paths for 121 reactions^14,20,21 demonstrated that FB-ENM paths generally yield lower energy paths. Furthermore, using FB-ENM and the direct MaxFlux method, we discovered a sequential rotation mechanism for the rotational isomerization of 9,9’-bianthracene that does not require any bond breaking. In contrast, IDPP generates a high-energy path with unnecessary bond breaking even with a specialized state-of-the-art NEB method.¹⁷

Despite the successful development of structure-based potential energy methods for path generation, such as FB-ENM and IDPP, they still have a common limitation in handling reactions involving chemical bond rearrangements.¹ In these reactions, where some bonds in the reactant break and new bonds form in the product, the transition between bond breaking and formation typically occurs rapidly near the transition state. However, FB-ENM and IDPP lack the ability to control the timing of these bond rearrangements. This limitation arises from their focus on imposing constraints on individual interatomic distances without considering correlations between different atom pairs. Consequently, these methods often produce anomalous reaction paths characterized by an unrealistic lag between bond breaking and formation.

In this study, to enhance the computational efficiency of reaction path/transition state optimization, we introduce a new theoretical concept in structure-based potential energy methods by developing correlated FB-ENM (CFB-ENM). CFB-ENM addresses the limitation above through its novel structure-based correlation terms. Each correlation term imposes a constraint prohibiting the simultaneous breaking of a bond formed only in the reactant and another bond formed only in the product. This theoretically novel approach effectively coordinates bond breaking and formation, eliminating unrealistic lag between these processes in the generated reaction paths. Consequently, CFB-ENM generates energetically more favorable initial paths by providing more chemically accurate representations of bond rearrangements during reactions, thereby reducing computational costs in subsequent precise reaction path/transition state optimization.

This study is organized as follows. In Section 2, we show an example of the limitation of FB-ENM/IDPP and introduce the structure-based correlation terms to overcome this limitation. Section 3 demonstrates that the correlation terms further improve upon FB-ENM. Lastly, Section 4 concludes this study.

2. Methodology

2.1. Notations

The notations in this study align with those in ref (1). An atomic system with n_atoms atoms is represented by a 3n_atoms dimensional vector x, which can also be expressed as a set of three-dimensional vectors r₁, r₂,··· for individual atomic coordinates. Reactant and product structures are denoted by x_r and x_p, respectively. A path connecting these end points is described as x(t), where t ∈ [0, 1].

We introduce the ramp function (or LeRU function) using a superscript plus sign. For a real number a, a⁺ is defined as

1

This notation allows us to alternatively express the FB-ENM pair potential v(r; d^min, d^max), a piecewise quadratic function (eq 5 of ref (1)), as

2

2.2. Common Limitation of FB-ENM and IDPP

As discussed in the Introduction section, during chemical reactions involving bond rearrangement, bond formation often occurs immediately after bond breaking near the transition state. However, FB-ENM and IDPP cannot control the timing of these bond breaking and formation. A simple example is the collinear reaction H₂ + H → H + H₂. We present a qualitative example with DFT calculation (see Section 3.1 for computational details).

Figure 1 illustrates key structures for this reaction. The reactant and product are defined as sufficiently dissociated H₂ and H (Figure 1a and c). The transition state features three closely proximate hydrogen atoms (Figure 1b), where bond breaking and formation occur in quick succession.

Figure 1.

Key structures in the collinear reaction H₂ + H → H + H₂: (a) reactant, (b) transition state, (c) product, and (d) midpoint of FB-ENM/IDPP path. Adjacent numbers indicate DFT energies.

FB-ENM/IDPP paths, however, cannot reproduce such a compact transition state structure. FB-ENM generates the shortest collision-free path, while IDPP uses linear interpolation of interatomic distances. Consequently, both methods yield a path where only the central hydrogen moves, with stationary outer hydrogens. The path midpoint becomes an energetically unstable structure with the central hydrogen dissociated from both outer hydrogens (Figure 1d). Notably, this structure differs from the actual transition state by over 70 kcal/mol, highlighting the significant limitation of FB-ENM/IDPP.

2.3. Correlation Terms

FB-ENM and IDPP fail to control the timing of bond breaking and formation because they only constrain interatomic distances for individual atom pairs, without considering correlations between different pairs. In this subsection, we propose a common form of correlation term that regulates this timing using structural information from a given reactant and product. This correlation term imposes a constraint on a pair of atom pairs, where one pair forms a bond only in the reactant and the other only in the product. This constraint prohibits the simultaneous breaking of these two bonds by a function v_c(r_a, r_b) defined for two interatomic distances r_a and r_b as

3

4

where d^bond_a and d^bond_b are the bond lengths for the atom pairs corresponding to r_a and r_b, respectively. The parameters α^corr₀, α^corr₁, α^corr₂ satisfy the condition 1 ≤ α^corr₀ < α^corr₁ < α^corr₂.

Figure 2 illustrates the regions where ṽ_c and v_c are positive (gray) and where they have a flat bottom with a value of zero (white), with the boundary defined by the hyperbola

5

for r_a > α^corr₀d^bond_a and r_b > α^corr₀d^bond_b. In the gray region, the value of ṽ_c is given by eq 3 without the superscript plus signs. The parameters α^corr₀, α^corr₁, and α^corr₂ play specific roles in defining the hyperbola’s characteristics: The asymptotes of the hyperbola are r_a = α^corr₀d^bond_a and r_b = α^corr₀d^bond_b, and the hyperbola passes through the point (r_a, r_b) = (α^corr₁d^bond_a, α^corr₁d^bond_b). The factorized v_c has a value of unity at (r_a, r_b) = (α^corr₂d^bond_a, α^corr₂d^bond_b).

Figure 2.

Correlation term v_c(r_a, r_b): (gray) positive values, (white) zero. Contour lines at unity intervals.

While v_c is continuous, it is not smooth on the hyperbola expressed by eq 5. For numerical calculations, we introduce smoothing using a small positive parameter ϵ, redefining v_c as

6

where v^orig_c denotes the original, unsmoothed function in eq 4.

In this study, we set (α^corr₀, α^corr₁, α^corr₂, ϵ) = (1.1, 1.5, 1.6, 0.05).

The correlation term v_c(r_a, r_b) defined above exhibits a flat bottom (white region in Figure 2), analogous to FB-ENM proposed in ref (1). Confining the reaction path to this bottom facilitates the mimicking of bond rearrangement. The path is restricted from the region where r_a > α^corr₁d^bond_a and r_b > α^corr₁d^bond_b, preventing simultaneous significant dissociation of atom pairs corresponding to r_a and r_b. Moreover, when r_b is large, r_a becomes nearly α^corr₀d^bond_a, ensuring the bonding of the r_a atom pair, and vice versa for large r_a.

Incorporating v_c into FB-ENM results in a total energy bottom that is the intersection of their respective bottoms. Path optimization using the direct MaxFlux method with this combined energy yields the shortest path through the total energy bottom. This optimal path inherits properties from both FB-ENM and the correlation term: it is collision-free, preserves nonreactive structural elements, and regulates bond rearrangement.

2.4. Selection of Correlated Pairs of Atom Pairs

We now select pairs of atom pairs for applying the correlation term v_c(r_a, r_b). To facilitate this, we define key terms. Following ref (1), we consider an atom pair i, j bonded if

7

where r^cov_i, r^cov_j are the covalent radii²² of atoms i, j, and α^bond is a constant exceeding 1 (set to 1.25 in this study). We define A_r and A_p as sets of atom pairs bonded only in the reactant and product, respectively. A_r,i and A_p,i represent subsets of A_r and A_p containing pairs that include atom i. An atom i is termed a pivot atom if |A_r,i| > 0 and |A_p,i| > 0 (Figure 3b), indicating both bond breaking and formation occur around this atom. In particular, an atom i is termed a single pivot atom if |A_r,i| = 1 and |A_p,i| = 1 (Figure 3c), indicating exactly one bond to be broken and one to be formed around this atom.

Figure 3.

Schematic representation of pairs of atom pairs for correlation terms. Blue dashed lines: bonds present only in the reactant. Orange dotted lines: bonds present only in the product. Black loosely dashed lines: correlated pairs of atom pairs. (a) A nonpivotal pair of atom pairs, (b) multiple pairs of atom pairs mediated by a pivot atom (green), (c) a pair of atom pairs mediated by a single pivot atom (green).

The pairs of atom pairs depicted in Figure 3 exhibit varying degrees of correlation. In Figure 3a, where no pivot atom is present, the correlation is likely weaker due to the spatial separation of the atom pairs. In contrast, cases involving pivot atoms (Figure 3b and c) are expected to show stronger correlations to prevent pivot atom isolation. In configurations with multiple pairs of pairs mediated by a pivot atom (Figure 3b), strong correlation in at least one pair is enough, allowing for weaker correlations among other pairs. The single pivot atom case (Figure 3c) requires a strong correlation in the sole pair of pairs to prevent isolation of the pivot atom. Thus, correlation strength is expected to increase from Figure 3a–c.

To determine the appropriate application range of correlation terms v_c(r_a, r_b) for the pairs of atom pairs shown in Figure 3, we propose three selection patterns. The first pattern applies v_c to all combinations between A_r and A_p, consisting of all types in Figure 3. The sum of correlation terms V_c(x) is expressed as

8

where |r_ξ| and |r_η| represent interatomic distances of each atom pair. The second pattern applies v_c to pairs mediated by pivot atoms, corresponding to Figure 3b and c:

9

The third pattern applies v_c exclusively to pairs mediated by single pivot atoms, as in Figure 3c:

10

The effects of these selection patterns on path optimization with FB-ENM will be examined in the next section.

3. Results and Discussion

3.1. Computational Details

This section compares paths obtained by correlated FB-ENM (CFB-ENM), which refers to FB-ENM with added correlation terms, and the original FB-ENM.¹ We employ the direct MaxFlux method¹⁰ for reaction path optimization, using the same conditions as in ref (1). The initial path is a linear interpolation between reactant and product in Euclidean space, with 20 equidistant energy evaluation points, including end points. These energy evaluation points are not updated and remain fixed throughout the optimization. We iterate the stepwise update of β and additional optimization five times, as proposed in ref (1). Default parameters from our Python implementation¹⁵ are used, except for relaxing the IPOPT^23,24 convergence condition dual_inf_tol to 0.1. CFB-ENM is implemented as an ASE Calculator class,¹⁹ integrated into our Python implementation of the direct MaxFlux method.¹⁵ We also implemented an automated routine for generating paths using CFB-ENM.

To analyze the obtained reaction paths, we calculate realistic potential energies using DFT. In Sections 3.2 and 3.3, we use the B3LYP and B3LYP-D3BJ functionals,²⁵⁻²⁹ respectively, combined with the def2-SVP basis set.^30,31 For reactions involving transition metals (Section 3.4), we employ the ωB97X-D functional³² with the def2-TZVPP basis set.^30,31 All DFT calculations are performed using Gaussian 16.³³

3.2. Collinear Reaction of Triatomic Hydrogen

We analyze the CFB-ENM path for the collinear reaction H₂ + H → H + H₂ presented in Section 2.2. For this reaction, A_r = {(1, 2)} and A_p = {(2, 3)}. As the sole correlation term is mediated by a single pivot atom, all three selection patterns from Section 2.4 yield identical results.

Figure 4a maps FB-ENM and CFB-ENM paths onto (r₁₂, r₂₃), where the indices correspond to the numbers in Figure 1a. The FB-ENM path forms a straight line between reactant and product, indicating atom 2 moves between fixed atoms 1 and 3. The CFB-ENM path, despite slight winding due to the finite basis in the direct MaxFlux method, approximates the shortest path within the bottom of the correlation term, passing near the transition state.

Figure 4.

Comparison of FB-ENM (blue) and CFB-ENM (orange) paths for the collinear reaction H₂ + H → H + H₂. (a) Path mapping onto (r₁₂, r₂₃). R, TS, and P denote reactant, transition state, and product, respectively. (b) DFT energy at each evaluation point, with the dotted line indicating TS energy.

Figure 4b depicts DFT energies along these paths. Since the FB-ENM path traverses a region where large r₁₂ and r₂₃ values isolate atom 2, it exhibits high maximum energy. Conversely, the CFB-ENM path passes near the transition state where r₁₂ and r₂₃ are moderately small, resulting in a significantly lower maximum energy. These results demonstrate that the CFB-ENM path effectively mimics the bond rearrangement.

3.3. ZBA121

We now employ the reaction set from ref (14). This set was originally created by Zimmerman²⁰ and Birkholz and Schlegel,²¹ then reconstructed by Ásgeirsson et al.¹⁴ In the present study, we refer to it as ZBA121, named after the first authors. The set comprises 121 reactions involving 3 to 52 atoms of H, B, C, N, O, F, Mg, P, S, Cl, and Br. We calculated reaction paths using FB-ENM and CFB-ENM for all reactions in this set.

We focus on reaction 34, a hydrogen exchange between ammonia and formaldehyde (Figure 5), which is the worst case for FB-ENM compared to IDPP.¹ We designate the exchanged hydrogens as H₁ (N-bonded in reactant ammonia) and H₂ (C-bonded in reactant formaldehyde). N–H₁ and C–H₂ bonds exist only in the reactant, while N–H₂ and C–H₁ bonds form only in the product. Consequently, four CFB-ENM correlation terms are possible. As all involve single pivot atoms in this case, the three selection patterns proposed in Section 2.4 yield identical results.

Figure 5.

Reaction 34 in set ZBA121, exchange of two hydrogen atoms between ammonia and formaldehyde. Exchanged hydrogen atoms in yellow and orange. Structures: (a) reactant, (b) product, (c) FB-ENM path snapshot with the highest DFT energy, (d) CFB-ENM path snapshot with the highest DFT energy, (e, f) transition states from two distinct reaction paths.

Figure 6a–d maps FB-ENM and CFB-ENM paths onto the four interatomic distance pairs for the correlation terms. The FB-ENM path traverses a state where all N–H₁, N–H₂, C–H₁, and C–H₂ distances simultaneously become large, indicating isolation of the exchanged hydrogens, as noted in ref (1). Consequently, the maximum DFT energy along the FB-ENM path significantly exceeds the two transition state energies (Figure 6e). The IDPP path exhibits similar behavior, also resulting in high DFT energies (see ref (1) for a detailed comparison between FB-ENM and IDPP paths).

Figure 6.

Comparison of FB-ENM (blue) and CFB-ENM (orange) paths for reaction 34 in set ZBA121. (a–d) Path mappings onto (r_CH2, r_CH1), (r_NH1, r_CH1), (r_CH2, r_NH2), and (r_NH1, r_NH2). R and P denote reactant and product, respectively. (e) DFT energy at each evaluation point, with the dotted lines indicating TS energies. Green curve represents IDPP path from ref (1).

In contrast, since the correlation terms in the CFB-ENM path inhibit simultaneous increase of interatomic distances (Figure 6a–d), the CFB-ENM path passes approximately midway between the two transition states, where two hydrogens remain in close proximity to either ammonia or formaldehyde. This leads to a substantially lower maximum DFT energy compared to FB-ENM or IDPP paths (Figure 6e). These results demonstrate that CFB-ENM successfully improves upon FB-ENM even in the worst case for FB-ENM.

Next, we analyze the statistical aspects of ZBA121. We calculated DFT energies along FB-ENM and CFB-ENM paths for all 121 reactions and determined the difference in their maximum energies (Tables S1–S3). For CFB-ENM, we initially employed the correlation terms from eq 10, which impose minimal constraints on pairs of atom pairs mediated by single pivot atoms. Figure 7 illustrates the distribution of these differences, excluding 19 reactions without single pivot atoms and 24 reactions where FB-ENM and CFB-ENM paths coincide due to complete inclusion of the FB-ENM path within the flat-bottom of CFB-ENM. This leaves 78 reactions for analysis.

Figure 7.

Distribution of maximum DFT energy differences between FB-ENM and CFB-ENM (eq 10) paths for ZBA121 reactions, excluding cases where paths coincide. Blue: all reactions; orange: reactions forming quasi-four-membered rings (Q4MRs); green: reactions forming Q4MRs consisting of C=CCH.

Results indicate that CFB-ENM paths exhibit lower energy than FB-ENM in 64 of 78 reactions (82%), with an average energy difference of −30.0 kcal/mol, signifying substantial improvement. In the 14 reactions (18%) where CFB-ENM paths have higher energy, 11 show differences within only 10 kcal/mol, and 3 are below 25 kcal/mol. These findings suggest that CFB-ENM paths with single pivot atom-mediated correlation terms significantly improve upon FB-ENM paths.

Next, we increase the correlation terms in CFB-ENM. Figure 8a illustrates the distribution of maximum DFT energy differences between CFB-ENM paths using pivot atom-mediated terms (eq 9) and single pivot atom-mediated terms (eq 10). Figure 8b shows the distribution between paths using all combination terms (eq 8) and single pivot atom terms. Excluding reactions with coinciding paths, the distributions cover 10 and 31 reactions, respectively. See Tables S1–S3 for the individual differences.

Figure 8.

Distribution of maximum DFT energy differences between CFB-ENM paths using various selection patterns for ZBA121 reactions, excluding cases where paths coincide. (a) Difference between patterns from eqs 9 and 10. (b) Difference between patterns from eqs 8 and 10.

These distributions reveal no clear trend in energy differences from additional correlation terms. Figure 8b notably demonstrates that correlating spatially distant bond changes can sometimes significantly increase energy differences. This suggests that imposing correlations on uncorrelated pairs of atom pairs may increase energetic instability. These findings indicate that single pivot atom-mediated correlation terms are the primary contributors to reaction path improvement.

We investigate reactions where CFB-ENM paths with single pivot atom-mediated correlation terms (eq 10) exhibit higher energy than FB-ENM paths. The reactions with the largest energy differences, highlighted in green in Figure 7, share a common feature: hydrogen transfer between the ends of three carbon atoms linked by one single and one double bond (Figure 9). In these cases, CFB-ENM correlation terms force the path to pass through a structure resembling a four-membered ring (Figure 9, center), which we term a quasi-four-membered ring (Q4MR). Such structures typically lead to significant bond angle strain, making them energetically unfavorable. Consequently, CFB-ENM paths involving Q4MRs likely have higher energy than FB-ENM paths. This view is further supported by the prevalence of reactions with non-C=CCH Q4MRs in the upper range of the energy difference distribution (Figure 7 orange). The particularly large energy differences for C=CCH Q4MRs are likely due to π-conjugation making the carbon chain more rigid. Based on these findings, we have added an option to our CFB-ENM implementation to exclude Q4MR-generating correlation terms.¹⁵

Figure 9.

Schematic of CFB-ENM path for propylene isomerization. A quasi-four-membered ring (Q4MR) forms in the middle of the path, consisting of the transferred hydrogen and three carbon atoms.

Finally, we compared the computational costs of DFT-based transition state searches using CFB-ENM or FB-ENM paths as initial paths. Among the reactions shown in Figure 7, we analyzed 67 cases, excluding those where Q4MR occurred. For each reaction, we performed transition state searches using the direct MaxFlux method with both types of initial paths. We set 5 movable energy evaluation points, with other parameters following refs (10,15). Eleven reactions were excluded from the comparison as they converged to different structures (RMSD > 0.5 Å) depending on the initial paths.

Figure 10 presents the computational cost comparison for the remaining 56 reactions. The left column shows scatter plots of two differences: the highest energy along the initial paths (from Figure 7) versus the number of iterations required for convergence. The differences in iteration counts are presented in two ways: as fractions relative to the FB-ENM cases (upper panels) and as raw numbers (lower panels).

Figure 10.

Computational cost comparison between transition state searches using CFB-ENM and FB-ENM paths as initial paths. Left column: Scatter plots of the difference in the highest energy along initial paths versus the difference in the number of iterations required for convergence. Solid lines represent least-squares regression lines. Right column: Distributions of the iteration count differences. In both columns, upper and lower panels show the differences as fractions relative to FB-ENM cases and as raw numbers, respectively.

The scatter plots reveal a positive correlation between the difference in the highest energy of initial paths and the computational cost for convergence, as shown by the least-squares regression lines (solid lines). Some cases show increased computational cost with CFB-ENM despite its lower initial path energy. This is unavoidable since the direct MaxFlux method’s convergence depends on multiple factors beyond the initial path energy, such as updates of movable energy evaluation points and various constraints on the variables. More importantly, since CFB-ENM paths generally exhibit lower energy profiles than FB-ENM paths (Figure 7), the positive correlation leads to distributions that significantly skew toward reduced iterations (Figure 10 right column). These results indicate that using CFB-ENM paths as initial paths is likely to reduce the computational costs of transition state searches.

3.4. MOBH35

While ZBA121 mainly consists of small molecules containing only main group elements, we now compare the performance of CFB-ENM and FB-ENM across a broader range of elements and molecular sizes. Here, we use a reaction set called MOBH35,³⁴ which consists of 35 reactions involving transition metal complexes. This reaction set includes complexes with diverse transition metals such as Sc, Ti, Mn, Fe, Nb, Mo, Ru, Rh, Pd, Ta, W, Re, Os, Ir, and Pt. Additionally, these systems are relatively large, with the number of atoms reaching up to 92.

As in previous examples, we generated reaction paths using both CFB-ENM and FB-ENM and compared the maximum DFT energies calculated along these paths. To increase the number of comparison samples, we applied CFB-ENM correlation terms to all possible combinations (eq 8). Different paths were obtained between CFB-ENM and FB-ENM for 13 reactions. Figure 11 shows the distribution of differences in maximum DFT energies between the two paths. The distribution is similar to that of ZBA121 (Figure 7). Specifically, while there are some cases where CFB-ENM shows higher energies, in most cases, it yields lower energies. Furthermore, while the energy differences in the former cases are all around a few kcal/mol, in the latter cases, nearly half exhibit significant differences of several tens of kcal/mol. These results suggest that CFB-ENM improves upon FB-ENM not only for simple reactions like those in ZBA121 but also for larger molecules composed of diverse elements.

Figure 11.

Distribution of maximum DFT energy differences between FB-ENM and CFB-ENM (eq 8) paths for MOBH35 reactions, excluding cases where paths coincide.

3.5. Limitations

While CFB-ENM generally produces energetically better paths than FB-ENM, as shown in previous examples, here we examine its limitations. As described in Section 2, CFB-ENM synchronizes bond breaking and formation events. However, applying correlation terms to weakly correlated atom pairs can lead to problematic results, particularly in long-distance bond rearrangements that naturally occur in multiple steps.

Consider the example of proton transfer in protonated p-aminobenzoic acid mediated by a single ammonia.³⁵ In this reaction, ammonia collision causes tautomerization from amide-protonated to carboxyl-protonated p-aminobenzoic acid (Figure 12a and c). When generating a path with CFB-ENM, we add a correlation term between the proton-amide and proton-carboxyl bonds. This forces the proton, nitrogen, and oxygen atoms to come close together along the path, resulting in an energetically unfavorable, bent p-aminobenzoic acid structure (Figure 12b).

Figure 12.

Long-distance bond rearrangements in CFB-ENM paths: (a–c) Proton transfer in p-aminobenzoic acid mediated by ammonia, showing (a) amide-protonated reactant, (b) CFB-ENM path midpoint, and (c) carboxyl-protonated product; (d–f) Hydrogen shift in 1,3-pentadiene, showing (d) reaction scheme, (e) FB-ENM path midpoint, and (f) CFB-ENM path midpoint.

Although CFB-ENM has some limitations, it can sometimes handle long-distance bond rearrangements effectively. Take the intramolecular hydrogen shift in all-trans 1,3-pentadiene (Figure 12d). Both FB-ENM and IDPP generate similar, energetically unfavorable paths where the hydrogen completely isolates from one terminal carbon and moves to the other while the carbon chain remains static (Figure 12e). In contrast, CFB-ENM with appropriate parameters brings the hydrogen and both terminal carbons into proximity, proceeding through an all-cis-like carbon chain (Figure 12f). This matches the known [1,5] sigmatropic rearrangement mechanism, demonstrating that CFB-ENM can produce more realistic paths in some cases.

These examples show that the success of CFB-ENM in handling long-distance bond rearrangements depends on the specific reaction system. The structural changes required to satisfy correlation terms may be either favorable or unfavorable, necessitating careful consideration when applying CFB-ENM to such cases.

4. Conclusions

We recently developed the flat-bottom elastic network model (FB-ENM),¹ a computationally efficient potential energy based on molecular structure, to generate plausible reaction paths between a given reactant and product. This study proposes correlated FB-ENM (CFB-ENM) as an extension of FB-ENM. FB-ENM addresses limitations of the widely used image-dependent pair potential (IDPP),¹⁶ such as structural distortion and vulnerability to bond breaking. However, structure-based approaches like FB-ENM and IDPP cannot regulate the timing of bond breaking/formation, leading to problems such as anomalous atom isolation in rearrangement reactions.¹

CFB-ENM addresses this problem by adding structure-based correlation terms to FB-ENM, controlling bond breaking/formation timing. Each of these correlation terms imposes a constraint on a pair of atom pairs, one forming a bond only in the reactant and the other only in the product. This constraint restricts simultaneous lengthening of interatomic distances, i.e., simultaneous bond breaking. This ensures that when one bond breaks, the other forms immediately.

The region unrestricted by these correlation terms is represented as a flat bottom, similar to FB-ENM. Thus, the bottom of the total energy, combining FB-ENM and correlation terms, becomes the intersection of FB-ENM’s flat bottom and that of the correlation terms. Optimizing the path under this potential energy using our recently developed direct MaxFlux method^10,15 yields the shortest path inheriting properties from both FB-ENM and correlation terms.

We proposed three selection patterns for correlation terms (eqs 8–10) and evaluated their performance using various reaction sets. For 121 reactions involving main group elements, paths generated using the minimal constraint pattern (eq 10) showed significant improvement over FB-ENM paths, with 82% of reactions exhibiting lower maximum DFT energies by an average of 30.0 kcal/mol. In cases where CFB-ENM produced higher energy paths, most increases were below 10 kcal/mol and often involved quasi-four-membered ring (Q4MR) formation. For larger systems, we tested 35 reactions involving transition metal complexes and found similar improvements, demonstrating CFB-ENM’s applicability across diverse molecular systems.

Our analysis of long-distance bond rearrangements revealed both strengths and limitations of CFB-ENM. While the method can successfully handle some cases by identifying energetically favorable paths (e.g., [1,5] sigmatropic rearrangement in pentadiene), it may generate unrealistic paths in reactions that naturally proceed through multiple steps. The success of CFB-ENM in such cases depends on whether the structural changes required to satisfy correlation terms are energetically favorable.

Furthermore, we demonstrated that CFB-ENM paths can significantly reduce computational costs in subsequent transition state searches. Using the direct MaxFlux method, we found a positive correlation between the reduction in maximum path energy and the decrease in required iterations for convergence. This improvement in computational efficiency was observed across a wide range of reactions where CFB-ENM generated lower-energy initial paths.

Several directions for future development of CFB-ENM can be considered. The parameters determining the CFB-ENM potential shape have not been thoroughly optimized in this study, and their optimization, possibly including element dependence, is desirable. The selection criteria for pairs of atom pairs could also be improved. For example, considering that long-distance bond rearrangements often lack strong correlations, an option to exclude correlation terms when atoms are separated beyond a certain distance could be beneficial. It is important to note that FB-ENM and CFB-ENM are fundamentally designed to generate approximate reaction paths between given reactants and products. While refinement of these methods is valuable, expanding their practical applications is equally important.

In summary, CFB-ENM represents a significant theoretical advancement in structure-based potential energy methods. While FB-ENM improved upon IDPP by preventing unnecessary structural distortion and bond breaking, CFB-ENM introduces the novel concept of correlation terms that effectively coordinate bond rearrangements. The method generates collision-free paths that preserve nonreactive structural elements while regulating bond breaking/formation timing, providing an efficient approach for reaction path exploration despite some limitations in specific cases.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.4c01549.

• Additional information about the individual results for 121 reactions in set ZBA121 and 35 reactions in set MOBH35 (PDF)

The authors declare no competing financial interest.