Arrow diagram theory for non-orthogonal electronic groups: the continued fractions method

Area of Science:

• Quantum Chemistry
• Condensed Matter Physics
• Computational Materials Science

Background:

• Group function theory (GFT) is effective for quantum systems with localized electronic groups.
• The arrow diagram (AD) technique facilitates matrix element calculations for non-orthogonal group functions.
• Extending GFT and AD techniques to large or infinite systems presents normalization challenges.

Purpose of the Study:

• To adapt GFT and AD methods for calculating properties of extended quantum systems.
• To evaluate and improve methods for calculating pre-factors in AD expansions for extended systems.
• To accurately determine the electron density in two-dimensional (2D) Hartree-Fock (HF) models.

Main Methods:

• Utilized group function theory and the arrow diagram (AD) technique.
• Applied a power series expansion method to calculate pre-factors for a 2D HF model.
• Developed and implemented a continued fraction expansion method for pre-factor calculation.

Main Results:

• The power series expansion method showed very slow convergence for the 2D HF model.
• The continued fraction expansion method demonstrated significantly faster convergence to the exact solution.
• The new method was successfully illustrated for calculating electron density in the 2D HF model.

Conclusions:

• Continued fraction expansion offers a more efficient approach for treating extended quantum systems within GFT.
• This method provides a powerful tool for analyzing systems with numerous localized electronic groups.
• The findings pave the way for more accurate computations in materials science and quantum chemistry.