Efficient and robust optimization of nuclear and electronic orbitals within the nuclear Hartree product representation

The nuclear-electronic orbital (NEO) approach treats electrons and specified nuclei quantum mechanically on the same level, providing a straightforward framework for including nuclear quantum effects beyond the standard Born-Oppenheimer approximation. Both density functional theory and wave function methods have been developed within the NEO framework. Despite its success, some NEO applications are limited by the intrinsically difficult self-consistent field (SCF) convergence of the Hartree-Fock or Kohn-Sham electronic and nuclear orbitals. Herein, we demonstrate that using the nuclear Hartree product representation improves the efficiency and stability of the NEO-SCF optimizationprocedure compared to the nuclear Slater determinant representation. Both representations lead to the same NEO-SCF energy because the proton-proton exchange energy is typically much smaller than the SCF energy convergence threshold. Faster and more robust NEO-SCF convergence is observed using this strategy for a test set of 92 molecules containing multiple quantum protons. We determine that the improved convergence behavior stems from the absence of nuclear self-Coulomb and self-exchange terms in the Hartree product representation. Furthermore, we show that both electronic and nuclear orbitals can be optimized simultaneously through extensions of the geometric direct minimization and trust-radius augmented Hessian algorithms. These algorithms can be useful for converging challenging systems, as demonstrated by calculations on the enzyme ribonucleotide reductase and the UO2(OH)4 complex.