A Rotationally Invariant Cartesian Ansatz for the Bond‐Order Charge‐Density Matrix Approach to Learning Potential Energy Surfaces

We present a modification of our recently described method for learning potential energy surfaces (PES) via the bond‐order charge‐density matrix (CDM) formalism which makes use of the Hartree‐Fock expression for the electronic energy. The modified approach also employs permutationally invariant polynomials (PIP) as a fitting basis for the individual CDM elements P μν with μ and ν as Cartesian components. The key difference introduced presently is in the representation of P μν as linear combinations of orientation‐independent intrinsic density elements weighted by the bond direction cosines forming a complete basis, in contrast to the direct space‐fixed Cartesian representation employed by us previously. The new approach identically preserves PES's rotational invariance and has the same root‐mean‐square error as the original formulation with ∼2.2 times fewer linear variational parameters. We demonstrate its efficiency on the fitting quality of the PESs of cyclo[10]carbon singlet and linear triplet states in full permutational symmetry space.