Physically Informed Population Initialization Improves Bonobo Optimizer Searches for Atomic Cluster Global Minima

ABSTRACT

Reliable global optimization of atomic clusters is limited not only by the ruggedness of the potential energy surface but also by the quality of the starting population supplied to the search algorithm. We examine this issue for the Bonobo Optimizer (BO) using a workflow that combines external PyAR‐style population initialization, local Lennard‐Jones relaxation, geometry repair, and structural similarity filtering. The initializer generates physically reasonable and structurally diverse trial clusters before the BO loop begins, while the relaxation and filtering steps map candidates to local minima and reduce redundant population updates. On Lennard‐Jones clusters, the unmodified BO framework becomes unreliable beyond LJ₁₃, and local relaxation with similarity filtering alone remains insufficient for the double‐funnel LJ₃₈ benchmark, reaching the global minimum only six times in 25 runs. Adding the Tabu‐style initializer changes this outcome: LJ₃₈ is solved in all 20 independent runs, and the full LJ₂–LJ₁₀₀ benchmark reaches 943 successes in 1030 runs, corresponding to an overall success rate of 91.5%. The main remaining failures occur for LJ₇₅–LJ₇₇, consistent with the known difficulty of the Marks‐decahedral exceptions. These results identify physically informed population construction as a practical route for improving BO‐based cluster global optimization.

Physically informed, motif‐free population initialization improves Bonobo Optimizer searches for atomic‐cluster global minima. By combining Tabu‐style structure generation with local relaxation and similarity filtering, the workflow improves basin sampling, solves the LJ₃₈ benchmark reliably, and transfers to carbon‐cluster optimization.

1. Introduction

Global optimization is central to computational chemistry because many molecular and cluster properties are governed by the low‐energy regions of a high‐dimensional potential energy surface (PES). As the number of atoms increases, the number of local minima rises rapidly, and the search for the global minimum becomes increasingly sensitive to how candidate structures are generated, relaxed, and retained [1, 2, 3]. This challenge is especially pronounced for atomic clusters, where small structural changes can move the system between competing funnels with distinct motifs and comparable energies.

Several deterministic and stochastic strategies have been developed for global optimization. Simulated annealing, [4, 5, 6, 7] genetic algorithms, [8, 9, 10] differential evolution, [11, 12] ant colony optimization, [13, 14] artificial bee colony optimization, [15, 16, 17] firefly algorithms, [18] particle swarm optimization, [19, 20, 21] random sampling, [22, 23, 24] basin hopping, [25, 26], and metadynamics [27, 28] have all been applied to chemical structure search problems [29, 30]. These methods differ in how they balance exploration and exploitation, but for cluster optimization, a common practical issue remains: the search can become inefficient when the initial population contains high‐energy, poorly packed, or structurally redundant candidates.

The bonobo optimizer (BO), a population‐based stochastic algorithm inspired by bonobo social behavior, was introduced as a flexible optimizer for engineering problems [31, 32]. BO has recently been applied to chemical structure optimization and was shown to perform well for small atomic clusters [33, 34]. However, extending BO from small clusters to medium and larger clusters requires more than increasing the number of iterations. As dimensionality and landscape ruggedness increase, the quality of the initial population and the management of duplicate relaxed structures become critical determinants of whether the population explores useful basins or repeatedly samples the same low‐lying local minima.

Lennard‐Jones (LJ) clusters provide a stringent and well‐characterized benchmark for this question. The LJ potential is simple and inexpensive to evaluate, but LJ_n clusters exhibit rich PES topologies and a rapidly increasing number of local minima with cluster size [25, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50]. Most LJ global minima below 100 atoms are icosahedral, but important exceptions such as LJ₃₈ and LJ₇₅–LJ₇₇ have non‐icosahedral global minima and are associated with difficult funnel‐crossing behavior. These systems therefore test not only whether an algorithm can refine low‐energy structures but also whether it can reach the correct structural basin.

Initial‐structure generation is one of the least neutral choices in any population‐based global optimization workflow. Highly biased initial structures can improve convergence but may encode system‐specific knowledge, while purely random structures can waste computational effort on unphysical or redundant regions of configuration space [25, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49]. The central hypothesis of the present work is that BO can be made more reliable for difficult cluster landscapes by improving the physical quality and diversity of the initial population without imposing known target motifs.

To test this hypothesis, we modify the BO framework by combining external pyar‐tabu population initialization, local relaxation of each trial geometry, and structural similarity control based on Coulomb‐matrix eigenvalues. The initializer generates physically reasonable trial clusters before BO begins, local relaxation maps structures to nearby basin minima, and similarity filtering limits redundant population updates after relaxation.

The modified framework is evaluated first on LJ₂–LJ₁₀₀ clusters, including the double‐funnel LJ₃₈ benchmark and the difficult LJ₇₅–LJ₇₇ Marks‐decahedral exceptions. The same workflow is then compared with the unmodified BO implementation for small carbon clusters and applied to medium‐sized carbon clusters C₂₄, C₃₂, and C₆₀. Through these tests, we assess whether physically informed but motif‐free population construction can improve BO reliability, reduce optimization effort, and transfer from analytic LJ potentials to electronic‐structure‐based cluster searches.

2. Methodology

2.1. BO Framework

The BO is a population‐based stochastic algorithm inspired by bonobo social behavior [31, 32, 33]. In this implementation, each population member is a full cluster geometry represented as a flattened Cartesian coordinate vector. The objective is to minimize the cluster energy over the $3N$ coordinate dimensions of an $N$‐atom system, where $N$ is the atom count, and $P$ is the population size.

The BO update loop operates on a fixed population size $P$ and a set of adaptive control parameters. During the mating phase, an individual is selected for mating, and a temporary subgroup is randomly formed based on the $\mathit{\text{tsgs}}\_\mathit{\text{factor}}$ and the population size $P$. The lowest‐energy structure within this subgroup is identified and compared with the selected individual. If its energy is lower, it is chosen as the mating partner; otherwise, the mating partner is selected randomly from the temporary subgroup. The candidate solution is then updated through one of four mating processes as described in Figure 1. Broadly, the update follows two stochastic paths: either a collective move toward the global best structure or a coordinate‐wise move away from the global best structure toward the bounds or the partner structure. In this work, the proposed vector is clipped to the configured bounds, repaired if the geometry is invalid, locally relaxed, and then compared with the current population. This procedure is carried out sequentially for all individuals in the population during a given generation, and the entire process is repeated iteratively over successive generations until the termination criterion is reached. Figure 2 summarizes the overall scheme used in the modified framework.

FIGURE 1.

Schematic flowchart describing the working of the BO algorithm.

FIGURE 2.

Schematic flowchart describing the modified BO framework.

2.2. Lennard‐Jones Benchmark Potential

The benchmark objective used in the code is the Lennard‐Jones (LJ) potential, because it provides inexpensive energy and gradient evaluations while retaining the rugged, multi‐minimum landscapes characteristic of cluster global optimization. For an LJ_n cluster, the total potential energy is.

$$E_{\mathrm{LJ}}=4\epsilon \sum _{i<j}\left[{\left(\frac{\sigma }{r_{\mathit{ij}}}\right)}^{12}-{\left(\frac{\sigma }{r_{\mathit{ij}}}\right)}^6\right],$$ (1)

where $r_{\mathit{ij}}$ is the distance between atoms $i$ and $j$, $\epsilon$ is the well depth, and $\sigma$ defines the length scale. The code evaluates the LJ potential in reduced units with $\epsilon =\sigma =1$.

2.3. Population Initialization

The initial population is generated by an external pyar‐tabu call in codes/initial_pop.py. In this approach, the initial population of atomic clusters is constructed by sequentially adding atoms (monomers) to a growing fragment (seed) through randomized placement constrained by distance‐based criteria. The command is executed with the seed and monomer XYZ files, the –make‐composite flag, and a target size of num_atoms—1 additional atoms. The –distance‐scaling 1.5 setting enforces physically reasonable interatomic spacing during construction. After pyar‐tabu finishes, the generated XYZ files are read back into memory and used as the initial BO population. This method does not introduce bias but provides an efficient sampling of the configurational space.

The role of pyar‐tabu in the present workflow differs from that in the original PyAR implementations. In the earlier approach, an $n$‐atom cluster was generated recursively by adding one atom at a time [51, 52]. At each intermediate cluster size, the generated structures were locally optimized, screened, and selected before the next atom was added. In the present implementation, pyar‐tabu is used exclusively to generate the initial population for BO. Although the full‐sized $n$‐atom structures are still constructed through sequential atom addition, no local optimization or screening is performed at the intermediate cluster sizes. Once the complete population of full‐sized structures has been generated, the structures are transferred to the BO framework, where local relaxation, similarity checking, and BO‐based structural updates are carried out.

2.4. Geometry Repair and Local Relaxation

Before local optimization, each candidate structure is clipped to the coordinate bounds and repaired with a simple geometry‐fixing routine in codes/make_valid_geom.py. That routine iteratively checks for close contacts using covalent‐radii‐based minimum distances and tests whether the structure is connected using a bond graph defined by a bond‐threshold factor. If the candidate contains disconnected fragments, the smaller components are nudged toward the largest component. If atoms are too close, they are pushed apart along the interatomic vector. The repair loop is attempted up to 100 times.

Each candidate is then locally relaxed with the L‐BFGS‐B optimizer implemented in the codebase. The local objective is the analytic LJ energy, and the gradient is computed analytically and accelerated with numba. The relaxation returns the minimized coordinate vector and its LJ energy. In the current code, this local optimization is applied both to the initial population and to each new BO candidate.

2.5. Structural Similarity Filtering

After relaxation, the candidate is compared against the current population and the best structure using the similarity test in codes/similarity.py. The implemented descriptor is a Coulomb matrix built from atomic numbers and Cartesian distances. For a given structure, an element $M_{\mathit{ij}}^{\text{Coulomb}}$ of the Coulomb matrix is defined as:

$$M_{\mathit{ij}}^{\text{Coulomb}}=\left\{\begin{array}{ll}0.5Z_i^{2.4}& \text{for}i=j\\ \frac{Z_i{Z}_j}{R_{\mathit{ij}}}& \text{for}i\ne j\end{array}\right.$$ (2)

where $Z_i$ and $Z_j$ are the atomic numbers of atoms $i$ and $j$, and $R_{\mathit{ij}}$ is the Euclidean distance between them. The structure is represented by its Coulomb matrix, and the respective eigenvalues are computed. To ensure translational, rotational, and permutational invariance, the eigenvalues of the Coulomb matrix are sorted and compared with the corresponding eigenvalue spectrum of each stored structure. If the Euclidean distance between two spectra is below the similarity threshold, the structures are treated as similar. The Euclidean distance is given by Equation (3):

$$\text{dist}=\sqrt{\sum _{k=1}^n{\left(p_k-q_k\right)}^2}$$ (3)

where $p_k$ and $q_k$ are the $k$‐th eigenvalues of the Coulomb matrices of structures $p$ and $q$, respectively.

This similarity test is used as a diversity filter. In the current implementation, it does not alter the physical LJ energy directly; instead, it influences whether a relaxed candidate is allowed to replace a population member. Similar higher‐energy structures are rejected by the acceptance logic in the BO loop, whereas lower‐energy structures can be accepted normally according to the standard energy‐based selection rule. In addition, dissimilar higher‐energy structures may still be accepted through the stochastic acceptance mechanism controlled by the parameter $p_{\mathit{xgm}}$, which helps preserve population diversity and mitigates premature convergence.

3. Results and Discussion

The results are organized to isolate the contribution of each modification to the BO workflow. We first establish the limitations of the unmodified algorithm on small LJ clusters, then examine the effect of adding local relaxation and structural similarity filtering, and finally evaluate the Tabu‐style initializer on the double‐funnel LJ₃₈ benchmark and on the full LJ₂–LJ₁₀₀ set. The same logic is then used to interpret the carbon‐cluster tests, where the objective is not only successful convergence but also the reduction in optimization effort.

Lennard‐Jones clusters provide a useful benchmark because their energy evaluations are inexpensive while their PESs remain highly multimodal. In the size range considered here, most known global minima are icosahedral, with important non‐icosahedral exceptions at LJ₃₈ and LJ₇₅–LJ₇₇. These exceptions are especially informative because they test whether a search method can move beyond the broad icosahedral funnels that dominate much of the landscape.

The unmodified BO framework successfully located global minimum structures up to LJ₁₃, as reported previously [33]. Its reliability decreased rapidly beyond this range. For LJ₁₄ and LJ₁₅, multiple independent runs showed that convergence to the reference global minimum was infrequent: the global minimum was reached in only 2 out of 10 runs for LJ₁₄ and 1 out of 10 runs for LJ₁₅ (Figure 3). The corresponding energy deviations show that the algorithm was not merely converging slowly; it was often being retained in competing low‐energy basins. These results define the baseline problem addressed by the modified workflow.

FIGURE 3.

Plot of statistical analysis of the performance of unmodified BO on (a) LJ₁₄ and (b) LJ₁₅, respectively.

As a first modification, local optimization and structural similarity checking were integrated into BO. Local optimization maps each trial geometry onto a nearby minimum, so the population evolves through basin representatives rather than unrelaxed high‐energy coordinates. The similarity check reduces repeated replacement by structures that are effectively the same after relaxation.

The local relaxation step in this work was performed using the L‐BFGS optimization algorithm. To determine the most efficient implementation, benchmarking calculations were carried out for Lennard‐Jones clusters ranging from LJ₁₀ to LJ₁₀₀ at intervals of 10 atoms. For each cluster size, 20 initial structures generated were locally relaxed using four different approaches: gradient‐free Python, analytical gradient‐based Python, analytical gradient with NUMBA acceleration, and ASE L‐BFGS. The average wall‐clock time required for the local optimization is presented in Figure 4 as a function of cluster size, $n$. The calculations were performed using a single CPU core on similar computational nodes. In all cases, the computational cost increased with cluster size due to the growing number of pairwise interactions. The gradient‐free implementation showed the slowest performance, whereas inclusion of analytical gradients significantly reduced the optimization time. The NUMBA‐accelerated gradient implementation consistently exhibited the highest efficiency, substantially outperforming both the pure Python and ASE implementations and was therefore adopted throughout this work.

FIGURE 4.

Average wall‐clock time required for local relaxation of structures. All calculations were performed using a single CPU core on similar computational nodes.

Together, these additions improved the baseline workflow and enabled consistent recovery of the icosahedral global minima up to LJ₃₇, with a 100% success rate over 10 consecutive runs for each cluster.

This improvement did not extend automatically to LJ₃₈. LJ₃₈ has a well‐known double‐funnel topology in which the truncated‐octahedral global minimum competes with a broader funnel associated with the second‐lowest minimum [50, 53]. With local optimization and similarity checking included, BO still reached the global minimum only 6 times in 25 independent runs within 1500 iterations. This result is important because it shows that basin‐level relaxation and duplicate control are helpful but may be insufficient when the initial population is poorly positioned with respect to the narrow global‐minimum funnel.

3.1. Generating the Initial Population With the Tabu‐Style Initializer

The failure on LJ₃₈ motivated a closer examination of the initial population. The external Tabu‐style initializer produced a better‐conditioned set of trial geometries than unconstrained random sampling. The initializer is not motif‐directed: it does not encode the truncated‐octahedral structure of LJ₃₈ or any other reference geometry. Its intended role is to supply BO with physically plausible and structurally diverse starting points before stochastic population updates begin.

To assess the impact of initial population quality on optimization performance, we compared two distinct sets of 20 structures each for the LJ₃₈ cluster: one generated using the Tabu‐style method and the other using random sampling. A pairwise cutoff distance of 2 r.u. was employed to define local atomic environments, wherein atoms within this distance from a given atom were considered its neighbors. For each structure in both the population sets, the Lennard‐Jones energy was computed, and the structure with the lowest energy from each set was selected for further analysis. In both selected structures, all interatomic distances within the 2 r.u. neighborhood were evaluated.

Figure 5a shows the resulting interatomic‐distance distributions. The randomly generated structure has a broad distribution, indicating uneven local packing and a mixture of under‐ and over‐separated atom pairs. The Tabu‐style structure has a narrower, more sharply defined distribution centered near physically reasonable nearest‐neighbor distances. Although the mean nearest‐neighbor distance can be slightly lower in the random structure, the broader spread indicates less consistent packing. The same trend is observed for the highest‐energy structures from the two populations (Figure 5b), showing that the difference is a property of the initialization strategy rather than a single favorable example.

FIGURE 5.

Raincloud plots showing the distribution of interatomic distances (in r.u.) for Tabu‐style generated and randomly generated (a) lowest energy and (b) highest energy structures of LJ₃₈.

The same comparison was extended to LJ₂₆, LJ₅₅, LJ₇₅, and LJ₉₅ (Figure 6). The Tabu‐style populations consistently show narrower distance distributions than the random populations, and the difference becomes more pronounced as the cluster size increases. This size dependence is expected: in higher‐dimensional searches, purely random coordinate generation more frequently produces poorly packed or weakly connected configurations, whereas the Tabu‐style initializer returns more structured starting populations.

FIGURE 6.

Raincloud plots showing the distribution of interatomic distances (in r.u) for Tabu‐style generated and randomly generated lowest energy structures of (a) LJ₂₆, (b) LJ₅₅, (c) LJ₇₅, and (d) LJ₉₅.

In addition, Coulomb‐matrix eigenvalue‐based distance distributions were analyzed for LJ₃₈ and LJ₇₅ systems as shown in Figure 7. For each system, 20 structures were generated both randomly and using the Tabu‐based structure generation approach. The pairwise Coulomb‐matrix distances among all generated structures were subsequently evaluated and plotted. In both systems, the structures generated using the Tabu‐based method exhibit significantly broader distance distributions compared to the randomly generated ones, indicating a substantially higher degree of structural diversity. These results demonstrate the effectiveness of the Tabu‐based approach in exploring a wider region of the configurational space and generating more diverse initial candidate structures for global optimization.

FIGURE 7.

Raincloud plots showing the distribution of Coulomb‐matrix eigenvalues based on distances for 20 Tabu‐style generated and randomly generated structures of (a) LJ₃₈ and (b) LJ₇₅.

The energy comparison in Figure 8 further provides evidence of the superiority of the Tabu‐style initialization scheme. For both the LJ₃₈ and LJ₇₅ systems, randomly generated structures exhibit substantially higher and more dispersed LJ energies, whereas the Tabu‐generated structures are confined to a lower and narrower energy range. Thus, the initializer improves both the structural quality, diversity, and the energetic consistency of the starting population.

FIGURE 8.

Energy distribution plots comparing the initial energies of 20 Tabu‐style generated and 20 randomly generated structures for (a) LJ₃₈ and (b) LJ₇₅ clusters. Since the raw Lennard‐Jones energies of all generated structures are positive, lower values of ${\log }_{10}\left(E\right)$ correspond to less repulsive and energetically more favorable initial configurations.

3.2. Impact of the Tabu‐Style Initializer on LJ₃₈ Optimization

Adding the Tabu‐style initializer changed the LJ₃₈ outcome substantially. Over 25 independent runs, the modified workflow located the global minimum in every case. Since the preceding LJ₃₈ test already included local optimization and similarity checking but succeeded only 6 times in 25 runs, this comparison identifies initial‐population quality as the dominant factor behind the observed improvement. The number of steps required for convergence still varies from run to run, which is expected for a stochastic population method, but the endpoint is no longer dominated by trapping in the competing funnel.

The improved performance obtained with the Tabu‐style initializer does not, however, imply that the LJ₃₈ global minimum is already present in the locally relaxed initial population. To examine this possibility, the workflow was terminated immediately after local relaxation of the Tabu‐style generated structures, before any BO structural updates were performed. LJ₆, LJ₁₃, LJ₂₆, LJ₃₈, and LJ₅₅ were selected as representative systems. For each cluster size, five independently generated populations containing 20 structures each were considered, resulting in 100 generated and locally relaxed structures per system. The reference global minimum was recovered in two of the five relaxed initial populations for LJ₆ and in one of the five populations for LJ₁₃. In each successful population, however, only one of the 20 structures relaxed to the global minimum. Thus, only 2 of the 100 structures examined for LJ₆ and 1 of the 100 structures examined for LJ₁₃ relaxed to their respective global minima. In contrast, the global minimum was not found in any of the five populations for LJ₂₆, LJ₃₈, or LJ₅₅. These results indicate that Tabu‐style initialization followed by local relaxation can occasionally recover the global minimum for very small clusters, whose potential energy landscapes contain comparatively few competing minima, but becomes insufficient as the cluster size and landscape complexity increase. In particular, the truncated‐octahedral global minimum of LJ₃₈ was absent from all 100 locally relaxed initial structures. Its consistent recovery by the complete workflow therefore cannot be attributed to its direct inclusion in the starting population. Rather, the Tabu‐style initializer provides a more favorable and diverse distribution of starting configurations across the potential energy landscape, while the subsequent BO updates generate new candidates capable of entering the basin of the global minimum.

We also performed a follow‐up parametric study on key BO parameters: ${\text{tsgs}}_{\text{factor}\_\text{initial}}$ (0.01–0.1), $p_{\mathrm{xgm}\_\text{initial}}$ (0.01–0.1), $\text{scab}$ (1–2), $\text{scsb}$ (1–2), and $\text{rcpp}$ (0.001–0.01), using a fixed population size of $P=20$. Each parameter was varied independently while the others were kept at their mid‐values. The results in Figure S1 show that the modified workflow is much less sensitive to initial parameter choices than the unmodified BO framework [31, 32, 33], with convergence to the LJ₃₈ global minimum observed in all tested cases. The workflow comparisons summarized in Table 1 therefore support a consistent interpretation: Tabu‐style population construction improves the starting basin distribution, while local relaxation and similarity filtering stabilize the subsequent BO search.

TABLE 1.

Summary of the main workflow comparisons used to identify the role of each modification.

Test	Workflow	System(s)	Outcome
Baseline BO	Original BO	LJ14, LJ15	2/10; 1/10
Basin‐level BO	Local relaxation + similarity	LJ2–LJ37	10/10
Double‐funnel test	Local relaxation + similarity	LJ38	6/25
Population‐quality test	Tabu‐style init. added	LJ38	25/25
Similarity ablation	Tabu‐style workflow, no similarity	LJ38	8/10

Note: Success is reported as the number of independent runs that reached the reference global minimum.

3.3. Extension to Larger LJ Clusters (LJ₂ –LJ₁₀₀ )

After the LJ₃₈ benchmark, the modified workflow was applied to LJ_n clusters up to $n=100$. Representative optimized structures are shown in Figure 9, and the aggregate success rates are reported in Table 2. For LJ₂–LJ₇₄, the modified BO recovered the reference global minimum in all 730 independent runs. This result indicates that the combined workflow is reliable not only for small icosahedral clusters but also for substantially larger systems where the number of competing minima is much higher.

FIGURE 9.

Global minimum structures of some Lennard‐Jones clusters with their energy values in reduced units.

TABLE 2.

Aggregated success rates for the modified BO workflow over LJ₂–LJ₁₀₀.

Cluster range	Clusters	Successful runs	Success rate
LJ2–LJ74	73	730/730	100%
LJ75	1	1/10	10%
LJ76	1	2/30	6.67%
LJ77	1	2/30	6.67%
LJ78–LJ100	23	208/230	90.4%
Overall	99	943/1030	91.5%

Note: Success rates were evaluated from 30 independent runs for LJ₇₆ and LJ₇₇ and from 10 independent runs for all other clusters. The full cluster‐by‐cluster data are provided in Table S1.

The effect of the Tabu‐style initializer is size‐dependent. For clusters in the LJ₁₃–LJ₃₈ range, the initializer did not substantially change the outcome for most systems because local relaxation and similarity filtering were already sufficient for rapid convergence, except in the special double‐funnel case of LJ₃₈. For larger clusters, the quality of the starting population becomes more important: poorly conditioned random structures are more likely to relax into redundant or unproductive basins, whereas Tabu‐style structures provide a more useful initial population for BO updates.

Local relaxation and similarity checking steps also remain supporting components of the workflow. In the LJ₃₈ ablation test shown in Figure S2, removing the local relaxation step reduced the success rate from 100% to 0%, highlighting its critical role in guiding candidate structures toward physically meaningful minima. Removing similarity checking, on the other hand, reduced the success rate from 100% to 80% over 10 runs, as shown in Figure S3. This effect is primarily associated with population diversity, where Coulomb‐matrix fingerprints do not introduce new physical information beyond the LJ geometry; instead, they prevent the population from being repeatedly filled with nearly identical relaxed structures.

The main limitation appears in the LJ₇₅–LJ₇₇ region. The reference global minimum was reached in only 1 of 10 independent runs for LJ₇₅ and in 2 of 30 runs for both LJ₇₆ and LJ₇₇. These clusters are known Marks‐decahedral exceptions, so the drop is consistent with the difficulty of crossing from the dominant icosahedral landscape into the narrower non‐icosahedral funnel. Performance recovered for LJ₇₈–LJ₁₀₀, where 208 of 230 runs were successful, as reported in Table S1 and visualized in Figures S4 and S5. The lowest success rate in this range was 30% for LJ₉₈; LJ₈₆, LJ₉₃, LJ₉₅, and LJ₁₀₀ each achieved 80%. Thus, the modified workflow is broadly robust across the benchmark set, but the LJ₇₅–LJ₇₇ failures show that improved initial populations alone do not eliminate all funnel‐crossing limitations.

3.4. Comparison of Unmodified and Modified BO Frameworks

The carbon‐cluster calculations provide a chemically distinct test of the same workflow. The unmodified BO framework had previously been applied to C₃, C₄ (singlet), C₄ (triplet), C₅, C₆ (singlet), and C₆ (triplet), where it could identify the corresponding global minimum structures [33]. In the present work, the modified BO framework was evaluated on the same systems to measure the reduction in search effort. The workflow was also applied to C_n clusters ($n=3$–$12$), where several global minima that were not accessible with the unmodified version were obtained (Figure 10). Energy evaluations and local optimizations for these small carbon clusters were performed with Gaussian 16 at the B3LYP [54, 55, 56]/6–311 + G* [57, 58, 59] level of theory.

FIGURE 10.

Representative carbon‐cluster global‐minimum structures obtained at B3LYP/6‐311 + G*, which were not accessible with the unmodified version.

A direct comparison of convergence behavior is presented in Table 3 and Figure S6. The modified workflow located the global minima in 2, 3, 1, 1, 8, and 6 optimization steps for C₃, C₄ (singlet), C₄ (triplet), C₅, C₆ (singlet), and C₆ (triplet), respectively. The corresponding unmodified BO runs required 96, 319, 215, 288, 365, and 892 steps. The reduction is therefore not marginal: for these systems, the modified workflow reduces the number of optimization steps by factors ranging from 45.6 to 288.

TABLE 3.

Comparison of the number of BO optimization steps required to locate the global minimum structures of small carbon clusters.

System	Unmodified BO	Modified BO	Step reduction
C3	96	2	48.0$\times$
C4 singlet	319	3	106.3$\times$
C4 triplet	215	1	215.0$\times$
C5	288	1	288.0$\times$
C6 singlet	365	8	45.6$\times$
C6 triplet	892	6	148.6$\times$

These results demonstrate that the modified workflow improves optimization efficiency beyond the LJ model. The interpretation is consistent with the LJ benchmarks: BO benefits when the initial population is physically reasonable and structurally diverse before stochastic search begins. Local optimization then maps candidates to chemically meaningful basin minima, while structural similarity filtering limits redundant population updates. Together, these steps reduce wasted evaluations and increase the probability of reaching the relevant low‐energy regions early in the search.

3.5. Applications to Medium‐Sized Carbon Clusters

The modified BO framework was also benchmarked on medium‐sized carbon clusters C₂₄, C₃₂, and C₆₀. For these larger carbon systems, the GFN2‐xTB [60] quantum tight‐binding method was used for energy evaluation and local optimization. The PESs of these clusters were previously studied using Basin Hopping at the GFN2‐xTB level [61]. The global minimum structures obtained with the modified BO workflow match the previously reported structures, as shown in Figure 11. This agreement indicates that the workflow can be transferred from analytic LJ potentials to electronic‐structure‐based cluster optimization, although the higher cost of quantum‐based energy evaluations makes efficient population construction even more important.

FIGURE 11.

Obtained global minimum structures of (a) C₂₄, (b) C₃₂, and (c) C₆₀ at the GFN2‐xTB level.

4. Conclusion

This work examined how the quality and diversity of the initial population affect the performance of the BO for cluster global optimization. Lennard‐Jones clusters up to LJ₁₀₀ were used as the principal benchmark because they combine inexpensive energy evaluations with well‐characterized, highly multimodal PESs. The unmodified BO framework reproduced known minima for very small clusters, but its reliability decreased beyond LJ₁₃, as shown by the low success rates for LJ₁₄ and LJ₁₅.

Introducing local relaxation and structural similarity filtering converted BO into a basin‐level search procedure and substantially improved its reliability, giving 100% success for LJ₂–LJ₃₇ over 10 independent runs per cluster. This modification alone, however, was not sufficient for LJ₃₈, where the global minimum was found only six times in 25 runs. The LJ₃₈ result shows that duplicate control and local relaxation are useful supporting steps, but they do not by themselves solve the population‐quality problem in a double‐funnel landscape.

The PyAR‐inspired Tabu‐style initializer addressed this limitation by generating physically better‐conditioned and structurally diverse starting populations without imposing known target motifs. Interatomic‐distance distributions, Coulomb‐matrix‐based distances, and initial energy profiles showed that the Tabu‐style populations were more compact, diverse, and energetically consistent than randomly generated populations. When this initializer was combined with local relaxation and similarity filtering, BO reached the LJ₃₈ global minimum in all 20 independent runs and showed reduced sensitivity to the tested BO control parameters.

Across the full LJ₂–LJ₁₀₀ benchmark, the modified workflow located the reference global minimum in 943 of 1030 runs, corresponding to an overall success rate of 91.5%. The method was fully reliable for LJ₂–LJ₇₄ and recovered high success rates again for LJ₇₈–LJ₁₀₀, except for LJ₉₈. The main unresolved cases were LJ₇₅–LJ₇₇, where the reference Marks‐decahedral minima were rarely reached in the tested runs. This failure region is consistent with the known difficulty of these non‐icosahedral exceptions and indicates that improved population construction alone does not eliminate all difficult funnel‐crossing problems.

The carbon‐cluster tests further support the transferability of the workflow. For small carbon clusters, the modified framework reduced the number of BO optimization steps by factors of 45.6–288 relative to the unmodified implementation, and it located several structures that were not accessible in the earlier workflow. For medium‐sized carbon clusters, C₂₄, C₃₂, and C₆₀, the GFN2‐xTB searches reproduced previously reported global minimum structures. Overall, these results show that physically informed population construction, basin‐level relaxation, and structural diversity control provide a practical route for strengthening BO‐based cluster global optimization. Future improvements should focus on the remaining funnel‐crossing failures, particularly LJ₇₅–LJ₇₇ and LJ₉₈, where additional strategies such as adaptive restarts, targeted diversity enhancement, or hybrid moves may be required.

Funding

This work was supported by Ministry of Education, India.

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Figure S1: Parametric study on LJ₃₈ after adding the Tabu‐style initializer, local relaxation, and Coulomb‐matrix similarity filtering. Each panel varies one BO control parameter while keeping the remaining parameters fixed at their mid‐values.

Figure S2: Local Optimization ablation test for LJ₃₈. The comparison tests the modified workflow (a) without local optimization and (b) with local optimization.

Figure S3: Similarity‐filtering ablation for LJ₃₈. The comparison tests the modified workflow (a) with and (b) without Coulomb‐matrix similarity filtering after local relaxation, showing the role of duplicate‐basin control in the final BO population.

Figure S4: Percentage success rates for LJ₂–LJ₇₄ over 10 independent runs per cluster.

Figure S5: Percentage success rates for LJ₇₅–LJ₁₀₀ clusters. Success rates were computed from 30 independent runs for LJ₇₆ and LJ₇₇, and from 10 independent runs for all other clusters.

Figure S6: Comparison of the number of steps required to converge to the global minimum structures in the modified and original BO frameworks. For the modified workflow, the actual optimization steps were multiplied by 10 for visual clarity.

Table S1: Cluster‐by‐cluster LJ benchmark results for the modified BO workflow.

Data Availability Statement

The bonobo optimizer codes can be accessed using this GitHub link https://github.com/bhriguchak/modified_bo_framework_codes, and the Supporting Information files (trajectory, 18 output, etc.) can be accessed using the link: https://figshare.com/s/9d37b18c9eab7b7fd139.