Lower Bounds for Coulombic Systems.

Area of Science:

• Quantum Chemistry
• Computational Physics
• Theoretical Chemistry

Background:

• Lower bound theory is not widely used in quantum chemistry due to historically poor quality bounds compared to upper bounds.
• Previous methods, including Temple's theory, were limited to specific basis sets unsuitable for Coulombic systems like atoms and molecules.

Purpose of the Study:

• To develop a novel lower bound theory for quantum chemistry that yields bounds competitive with Ritz upper bounds.
• To address the limitations of previous lower bound methods, particularly for Coulombic systems.

Main Methods:

• Derivation of a simple eigenvalue equation to compute lower bounds.
• Utilizing Ritz eigenvalues and their variances as input, avoiding computation of the full squared Hamiltonian matrix.
• Presentation of a new Cauchy-Schwartz inequality relevant to lower bound theory.

Main Results:

• The derived eigenvalue equation provides lower bounds competitive with Ritz upper bounds for hydrogen and helium atoms.
• Demonstration of the equivalence between Lehmann's methods and the self-consistent lower bound theory within the matrix Hamiltonian framework.
• The new method overcomes the limitations of previous lower bound theories for Coulombic systems.

Conclusions:

• The developed method offers a practical and effective approach to calculating high-quality lower bounds in quantum chemistry.
• This work significantly advances the applicability of lower bound theory to atomic and molecular electronic structure calculations.