A Low-Order Permutationally Invariant Polynomial Approach to Learning Potential Energy Surfaces Using the Bond-Order Charge-Density Matrix: Application to C_<i>n</i> Clusters for <i>n</i> = 3–10, 20

A representation for learning potential energy surfaces (PESs) in terms of permutationally invariant polynomials (PIPs) using the Hartree-Fock expression for electronic energy is proposed. Our approach is based on the one-electron core Hamiltonian weighted by the configuration-dependent elements of the bond-order charge density matrix (CDM). While the previously reported model used an <i>s</i>-function Gaussian basis for the CDM, the present formulation is expanded with <i>p</i>-functions, which are crucial for describing chemical bonding. Detailed results are demonstrated on linear and cyclic C_<i>n</i> clusters (<i>n</i> = 3-10) trained on extensive B3LYP/aug-cc-pVTZ data. The described method facilitates PES learning by reducing the root mean squared error (RMSE) by a factor of 5 relative to the <i>s</i>-function formulation and by a factor of 20 relative to the conventional PIP approach. This is equivalent to using CDM and an <i>sp</i> basis with a PIP of order <i>M</i> to achieve the same RMSE as with the conventional method with a PIP of order <i>M</i> + 2. Implications for large-scale problems are discussed using the case of the PES of the C₂₀ fullerene in full permutational symmetry.