Intracule functional models. V. Recurrence relations for two-electron integrals in position and momentum space

Area of Science:

• Computational Chemistry
• Quantum Mechanics
• Theoretical Physics

Background:

• The Obara-Saika recurrence relation (RR) is fundamental for calculating two-electron integrals in quantum chemistry.
• Efficient computation of multi-dimensional integrals is essential for accurate molecular simulations.

Purpose of the Study:

• To derive novel recurrence relations (RRs) for high-dimensional integrals in phase space, position space, and momentum space.
• To demonstrate the recursive construction of various intracule integrals using these RRs.

Main Methods:

• Adaptation of the Ahlrichs method for deriving the Obara-Saika RR.
• Derivation of an 18-term RR for six-dimensional phase space integrals.
• Derivation of 8-term RRs for three-dimensional position or momentum space integrals.

Main Results:

• An 18-term RR was derived for six-dimensional integrals.
• 8-term RRs were obtained for three-dimensional integrals.
• The 18-term RR simplifies to a 5-term RR for specific intracule integrals in Fourier space.

Conclusions:

• The derived RRs provide an efficient method for calculating complex molecular integrals.
• These RRs facilitate the recursive construction of Position, Momentum, Omega, Dot, and Posmom intracule integrals.
• The study offers a robust framework for advancing quantum mechanical calculations.