Riemannian Trust Region Method for Minimization of the Fourth Central Moment for Localized Molecular Orbitals

Area of Science:

• Quantum Chemistry
• Computational Chemistry
• Theoretical Chemistry

Background:

• Localized molecular orbitals (MOs) are crucial for understanding chemical bonding and enhancing quantum correlation treatments beyond mean-field approximations.
• While occupied MOs are readily localized, generating orthonormal localized virtual MOs presents significant computational challenges.
• Orthonormal MOs are essential for efficient group theoretical methods in multireference configuration interaction (MRCISD) and perturbation theories.

Purpose of the Study:

• To develop a robust and efficient computational method for generating orthonormal localized virtual molecular orbitals.
• To overcome the limitations of standard optimization algorithms in localizing virtual and partially occupied MOs using fourth moment cost functions.
• To provide a method that enhances both qualitative understanding and quantitative accuracy in molecular electronic structure calculations.

Main Methods:

• Adoption of the powers of the fourth moment cost function.
• Application of a trust region algorithm on an orthonormal Riemannian manifold, incorporating approximate retractions.
• Coupling Riemannian trust region outer iterations with truncated Conjugate Gradient inner loops to avoid computationally expensive linear equation or eigenvalue solutions.

Main Results:

• Successfully generated orthonormal localized virtual MOs, overcoming issues with negative Hessian eigenvalues common in standard methods.
• Demonstrated the algorithm's efficacy on model systems like H10 and chemically relevant molecules such as cyclobutadiene (c-C4H4) and the propargyl radical (C3H3).
• Validated the method for occupied, virtual, and active space orbitals within the multiconfiguration self-consistent field (MCSCF) theory.

Conclusions:

• The developed Riemannian trust region algorithm provides an effective solution for localizing virtual molecular orbitals.
• This method enhances the utility of localized MOs in advanced quantum chemical calculations and chemical bonding analysis.
• The algorithm's efficiency and applicability across different orbital spaces offer significant advantages for computational chemistry research.