Comparison of an improved self-consistent lower bound theory with Lehmann's method for low-lying eigenvalues

Area of Science:

• Quantum mechanics
• Computational physics
• Theoretical chemistry

Background:

• Ritz eigenvalues provide only upper bounds for energy levels.
• Calculating lower bounds requires variances, with Weinstein and Temple methods showing slow convergence and poor quality.
• Lehmann's theory optimizes Temple's bounds, and Self-Consistent Lower Bound Theory (SCLBT) further improves upon them.

Purpose of the Study:

• To further improve the Self-Consistent Lower Bound Theory (SCLBT).
• To compare the quality of the improved SCLBT (iSCLBT) with Lehmann's theory.
• To assess the performance of iSCLBT and Lehmann's theory for calculating lower bounds of energy levels.

Main Methods:

• Formulation and implementation of an improved Self-Consistent Lower Bound Theory (iSCLBT).
• Utilizing the Lánczos algorithm for Hamiltonian matrix construction.
• Comparison of iSCLBT and Lehmann's theory using two lattice Hamiltonians.

Main Results:

• The novel iSCLBT demonstrates significant improvement over its previous implementation.
• Both Lehmann's theory and SCLBT variants yield superior lower bounds compared to Weinstein's and Temple's methods.
• iSCLBT and Lehmann's theory show comparable performance in lower bound quality and convergence, with iSCLBT offering more flexible input options.

Conclusions:

• The improved Self-Consistent Lower Bound Theory (iSCLBT) provides a significant advancement in calculating lower bounds for energy levels.
• Both iSCLBT and Lehmann's theory are effective for low-lying excited states, with increasing state inclusion leading to tighter bounds.
• iSCLBT offers advantages over Lehmann's theory, including the potential for improved convergence due to flexible input requirements.