A Low-Order Permutationally\nInvariant Polynomial Approach\nto Learning Potential Energy Surfaces Using the Bond-Order Charge-Density\nMatrix: Application to C<sub><i>n</i></sub> Clusters for <i>n</i> = 3–10, 20

A representation\nfor learning potential energy surfaces (PESs)\nin terms of permutationally invariant polynomials (PIPs) using the\nHartree–Fock expression for electronic energy is proposed.\nOur approach is based on the one-electron core Hamiltonian weighted\nby the configuration-dependent elements of the bond-order charge density\nmatrix (CDM). While the previously reported model used an <i>s</i>-function Gaussian basis for the CDM, the present formulation\nis expanded with <i>p</i>-functions, which are crucial for\ndescribing chemical bonding. Detailed results are demonstrated on\nlinear and cyclic C<sub><i>n</i></sub> clusters (<i>n</i> = 3–10) trained on extensive B3LYP/aug-cc-pVTZ\ndata. The described method facilitates PES learning by reducing the\nroot 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\nthe conventional PIP approach. This is equivalent to using CDM and\nan <i>sp</i> basis with a PIP of order <i>M</i> to achieve the same RMSE as with the conventional method with a\nPIP of order <i>M</i> + 2. Implications for large-scale\nproblems are discussed using the case of the PES of the C<sub>20</sub> fullerene in full permutational symmetry.