Density matrix formulation of the nuclear-electronic orbital approach with explicit electron-proton correlation

Area of Science:

• Quantum Chemistry
• Computational Chemistry
• Theoretical Chemistry

Background:

• Accurate treatment of electron-proton correlation is crucial for understanding chemical phenomena.
• Existing methods often struggle with the complexity of mixed nuclear-electronic systems.
• The development of efficient computational approaches is needed.

Purpose of the Study:

• To present the density matrix formulation of the nuclear-electronic orbital explicitly correlated Hartree-Fock (NEO-XCHF) approach.
• To extend the NEO-XCHF method for multielectron, multiproton systems and general wave functions.
• To derive expressions for densities, total energy, and Fock matrices.

Main Methods:

• Density matrix formulation of NEO-XCHF.
• Inclusion of explicit dependence on nuclear-electronic distances using Gaussian-type geminal functions.
• Extension to open-shell and multireference wave functions.
• Derivation of general expressions for one- and two-particle densities and total energy.
• Hartree-Fock-Roothaan equations solved iteratively.

Main Results:

• General expressions for densities and energy derived.
• Specific expressions for closed-shell systems obtained.
• Iterative solution to self-consistency achieved.
• Density matrix representation facilitates approximate methods and DFT functional development.

Conclusions:

• The NEO-XCHF approach provides a computationally practical framework for electron-proton correlation.
• The density matrix formulation enables development of approximate methods and DFT functionals.
• This method consistently treats electron-electron and electron-proton correlation.