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We recently introduced [J. Chem. Phys. 152 2020, 204103] thenuclear-electronic all-particle density matrix renormalization group method(NEAP-DMRG) to solve the molecular Schr\"{o}dinger equation, based on astochastically optimized orbital basis, without invoking the Born-Oppenheimerapproximation. In this work, we combine the DMRG with nuclear-electronicHartree-Fock (NEHF-DMRG), treating nuclei and electrons on the same footing.Inter- and intra-species correlations are described within the DMRG withouttruncating the excitation degree of the full configuration interaction wavefunction. We extend the concept of orbital entanglement and mutual informationto nuclear-electronic wave functions and demonstrate that they are reliablemetrics to detect strong correlation effects. We apply NEHF-DMRG to theHeHHe$^+$ molecular ion, to obtain accurate proton densities, ground-statetotal energies, and vibrational transition frequencies by comparison withstate-of-the-art data obtained with grid-based approaches and modernconfiguration interaction methods. For HCN, we improve on the accuracy of thelatter approaches with respect to both ground-state absolute energy and protondensity which is a major challenge for multi-reference nuclear-electronicstate-of-the-art methods.

Quantum Proton Effects from Density Matrix Renormalization Group Calculations