Nonlinear optical (NLO) materials are essential for many photonic, telecommunication, and laser technologies, yet discovering better NLO molecules is computationally challenging due to the vast chemical space and competing objectives. We compare evolutionary algorithms for molecular design, targeting four objectives: maximizing the ratio of first-to-second hyperpolarizability $(β/γ)$, optimizing HOMO-LUMO gap and linear polarizability to target ranges, and minimizing energy per atom. We encode molecules as SMILES strings and evaluate their properties using quantum-chemical calculations. We compare NSGA-II, MAP-Elites, MOME, a single-objective $(μ+λ)$ evolutionary algorithm, and simulated annealing. Quality diversity methods maintain archives across a measure space defined by atom and bond count, enabling the discovery of structurally diverse molecules. Our results demonstrate that NSGA-II consistently earns high scores in every objective, leading to high-quality molecules, but MOME does a better job exploring a wide range of possibilities, resulting in higher global hypervolume and MOQD scores. However, each method has strengths and weaknesses, and produced many promising molecules.
Nonlinear optical (NLO) materials can modify the frequency, phase, and/or polarization of light [24,26,31]. Because these properties enable optical communication, optical computing, optical data storage, and optical switching devices, many research groups are actively seeking molecules with specific NLO properties [5,6,30]. Over the past few decades, the development of reliable black-box quantumchemical programs has enabled theoretical chemists to evaluate the properties of NLO molecules solely from their structures.
Despite the commercial importance of NLO materials, the scientific literature contains only general guidelines as to which properties an ideal NLO material should have for a specific application. Most studies focus on a particular property, such as molecular first hyperpolarizability (𝛽) [11,12,18,32,33]. In this paper, we consider an electro-optic modulator [24] as an example NLO device, and identify properties that a molecule must possess to function as an electro-optic modulator (Section 3). We seek molecules that optimize these properties using multiobjective optimization (MOO) [7], quality diversity (QD) [25], multiobjective QD (MOQD) [27], singleobjective evolution, and simulated annealing. Previous work [21] demonstrates that evolutionary algorithms can be used to search for molecules with large hyperpolarizabilities, but we apply a wider variety of approaches to a harder problem: finding molecules that simultaneously satisfy multiple, often conflicting, criteria.
Our results show that MOQD via the MOME algorithm [27] creates the best variety of molecules, covering many diverse niches while also maximizing hypervolume [41], but the multiobjective optimization algorithm NSGA-II [7] uncovers higher scores in each individual objective, providing a different set of trade-offs. Singleobjective methods optimize our main objective of first-to-second hyperpolarizability ratio at the expense of other objectives, but the diversity that is fostered by the QD method MAP-Elites [25] allows it to perform better on a wider range of objectives, even though it is not aware of them. All of these approaches provide an interesting variety of potential NLO materials for further study.
Various search methods have been applied to molecular design. Simulated annealing and evolutionary algorithms have been used to optimize molecular hyperpolarizabilities with semi-empirical quantum chemistry [20,21], representing molecules with SMILES strings [39], which we also use in our experiments (Section 4.1).
Though effective, such optimization techniques generally ignore other aspects of design that impact solution usefulness. In contrast, quality diversity (QD) approaches [4,25] seek diverse collections of artifacts while also maximizing fitness. Graph-based elite patch illumination (GB-EPI [38]) is a QD approach that was applied to small-molecule drug rediscovery benchmarks. GB-EPI evolves graph-based representations instead of SMILES strings, while our experiments combine SMILES with QD.
A separate limitation of standard optimization is its focus on a single objective. Multiobjective Pareto-based optimization addresses this by exploring trade-offs between competing objectives rather than seek diversity in design space, and has also been applied to evolving molecules for small-molecule drug design [10,37].
We apply QD and multiobjective (MO) techniques, along with others, to the problem of discovering molecules for the design of an effective electro-optic modulator.
proportional to hyperpolarizability (𝛽); high 𝛽 enables stronger modulation and smaller devices, while 𝛾 should be small enough to avoid self-phase modulation and optical Kerr effects [31] yet sufficient for intense-field performance [31]. Second, linear polarizability (𝛼) must be balanced: high values provide strong charge transfer (beneficial for 𝛽 and 𝛾), but excessive 𝛼 causes absorption, dispersion, or aggregation. Third, the HOMO-LUMO gap (Δ𝐸) controls electron mobility and optical transparency; small gaps risk visible-light absorption and thermal damage, while large gaps ensure transparency but weaken NLO activity. Fourth, molecules must be thermodynamically stable. These properties are calculable via quantum-chemical programs, which we combine with evolutionary algorithms to search for optimal EO modulators.
SMILES strings representing molecules are evolved using MOO, QD, MOQD, and single-objective evolution, and compared with simulated annealing. The properties of candidate molecules are calculated using the PySCF library [34]. [39]) encodes molecular structures as ASCII strings. We restrict molecules to C, N, O, and H atoms, focusing on organic NLO candidates, following prior work [21]. Atoms are represented by atomic symbols; organic atoms (C, O) omit brackets and use implicit hydrogens based on valence. Bonds are limited to single (implicit -) and double (=) bonds. Branches are represented by parentheses at the attachment point (e.g., C-
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