Efficient Evaluation of Two-Center Gaussian Integrals in Periodic Systems

Area of Science:

• Computational Chemistry
• Quantum Chemistry
• Solid-State Physics

Background:

• Calculating periodic integrals is crucial for understanding condensed matter systems.
• Existing methods can be computationally intensive, especially for complex systems.

Purpose of the Study:

• To develop an efficient algorithm for calculating periodic integrals over Gaussian basis functions.
• To improve the computational speed of electronic structure calculations for periodic systems.

Main Methods:

• Utilized Poisson's summation formula to partition lattice summations between real and reciprocal space.
• Employed an efficient McMurchie-Davidson recurrence relation for real-space summation.
• Derived and implemented expressions for reciprocal-space summation, optimizing term reuse.

Main Results:

• Achieved exponentially fast convergence for both real and reciprocal space sums.
• Demonstrated efficient calculation of two-center Gaussian integrals (overlap, kinetic, Coulomb).
• The algorithm's performance is only a factor of 5-15 slower than molecular integrals, highlighting its efficiency.

Conclusions:

• The developed method provides a significant computational speedup for periodic integral calculations.
• The algorithm is particularly efficient for highly contracted basis functions with large exponents.
• This work lays the groundwork for efficient three-center Coulomb integral calculations in periodic systems.