Molecular Solubility, Hydration Shells, and Inverse Spectral Tomography via 3D Lone-Pair Fiedler Vector Optimization

Abstract

Predicting aqueous solubility (log S), co-crystallization, and solvation dynamics remains one of the formidable bottlenecks in drug discovery, formulation engineering, and physical chemistry. Conventional approaches rely either on computationally expensive quantum mechanical methods (DFT/MD) or black-box machine learning models (e.g., Graph Neural Networks / QSAR) that lack physical interpretability. Here, we present a novel, explainable Spectral Graph Theory framework that predicts multi-solvent solubility and solvation structures at O(N log N) computational complexity. By integrating 3D water lone-pair tetrahedral geometry (109.5°) into a normalized graph Laplacian L_norm, our model calculates the Fiedler value (λ2)—the second smallest eigenvalue measuring algebraic connectivity—to quantify hydration shell ordering and micro-solvation free energy. Furthermore, by coupling Fiedler phase complementarity (vi · vj < 0) with Kabsch Singular Value Decomposition (SVD), optimal dimer/solvent orientations are determined analytically without grid searching. We introduce the Fiedler Crystallization Anomaly Index (FCAI) to disentangle micro-hydration attraction from solid-state crystal lattice packing penalties. Finally, we formulate Inverse Spectral Tomography, demonstrating that an unknown solute’s functional groups and molecular architecture can be reconstructed within seconds from an 8-solvent solubility fingerprint vector (y_exp ∈ R^8). Validated across benchmark pharmaceuticals (Carbamazepine, Nicotinamide, D-Mannitol) and diverse organic solvents, this framework establishes a unified graph-spectral theory bridging molecular structure, solvent percolation, and thermodynamic solubility. Keywords: Molecular Solubility, Spectral Graph Theory, Fiedler Value, Water Lone-Pair Geometry, Kabsch SVD Alignment, Inverse Spectral Tomography, Drug Discovery.