Convergence patterns and rates in two-state perturbation expansions.

Area of Science:

• Quantum Chemistry
• Computational Chemistry
• Theoretical Physics

Background:

• The Møller-Plesset expansion is a common method for calculating electron correlation energy.
• A simple two-state model has previously rationalized the convergence of these expansions.
• Existing models were limited to specific perturbation forms.

Purpose of the Study:

• To generalize the two-state model for analyzing nondegenerate perturbation expansions.
• To investigate the convergence of arbitrary two-state perturbation expansions.
• To identify characteristics governing perturbation series convergence.

Main Methods:

• Detailed analysis of analytic expressions for energy corrections in a general two-state model.
• Examination of correction series for various perturbation types (symmetric and asymmetric).
• Classification of convergence patterns based on archetype, rate, period, and sign pattern.

Main Results:

• Identified seven convergence archetypes: zigzag, interspersed zigzag, triadic, ripples, geometric, zigzag-geometric, and convex-geometric.
• Demonstrated that symmetric perturbations exhibit periodic recurrence in energy corrections, unlike asymmetric ones.
• Established relations between perturbation form and energy correction characteristics.

Conclusions:

• The generalized two-state model provides a comprehensive framework for understanding perturbation series convergence.
• The identified characteristics (archetype, rate, period, sign pattern) offer insights into slow or divergent series.
• This model aids in diagnosing and understanding convergence issues in Møller-Plesset and other perturbation expansions.