A Low-Order Permutationally Invariant Polynomial Approach to Learning Potential Energy Surfaces Using the Bond-Order Charge-Density Matrix: Application to C n Clusters for n = 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 s -function Gaussian basis for the CDM, the present formulation is expanded with p -functions, which are crucial for describing chemical bonding. Detailed results are demonstrated on linear and cyclic C n clusters ( n = 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 s -function formulation and by a factor of 20 relative to the conventional PIP approach. This is equivalent to using CDM and an sp basis with a PIP of order M to achieve the same RMSE as with the conventional method with a PIP of order M + 2. Implications for large-scale problems are discussed using the case of the PES of the C 20 fullerene in full permutational symmetry.