Stability conditions and local minima in multicomponent Hartree-Fock and density functional theory

Area of Science:

• Quantum Chemistry
• Computational Chemistry
• Theoretical Chemistry

Background:

• Multicomponent quantum chemistry enables simultaneous quantum mechanical treatment of electrons and protons.
• Self-consistent-field (SCF) calculations aim to find the global minimum energy state.
• Distinguishing between minima and saddle points is crucial for accurate electronic structure calculations.

Purpose of the Study:

• Derive and present stability conditions for multicomponent Hartree-Fock (HF) and density functional theory (DFT) within the nuclear-electronic orbital (NEO) framework.
• Establish criteria to determine if an SCF solution represents a minimum or a saddle point.
• Analyze potential instabilities in electron-proton systems.

Main Methods:

• Derivation of stability conditions based on the Hessian matrix of SCF solutions.
• Comparison of stability matrices for NEO-HF and NEO-DFT with their time-dependent counterparts (NEO-TDHF and NEO-TDDFT).
• Analysis of internal and external stabilities for theories with varying orbital constraints.

Main Results:

• The Hessian matrix must be positive semi-definite for an SCF solution to be a minimum.
• Stability matrices for NEO-HF and NEO-DFT share structural similarities with NEO-TDHF/NEO-TDDFT matrices.
• Identified three potential instability types: electronic, protonic, and electron-proton vibronic.
• Negative eigenvalues of the stability matrix are necessary, but not sufficient, for imaginary eigenvalues in NEO-TDHF/NEO-TDDFT.
• Demonstrated connections between stationary points in nuclear and orbital spaces using NEO ∆SCF calculations on HCN, HNC, and 2-cyanomalonaldehyde.

Conclusions:

• The derived stability analysis is a valuable tool for characterizing SCF solutions in multicomponent quantum chemistry.
• This method aids in the search for lower-energy solutions in complex electron-proton systems.
• The study provides insights into the nature of stationary points in both conventional and multicomponent electronic structure calculations.