densityfitting

Density fitting, also known as the density-fitting approximation or resolution of the identity (RI), is a technique used in quantum chemistry to reduce the computational cost of evaluating electron repulsion integrals. The core idea is to approximate products of basis functions by a linear combination of auxiliary functions drawn from a separate auxiliary basis set. By projecting the four-center two-electron integrals onto this smaller auxiliary space, the most expensive part of electronic structure calculations is turned into operations involving three- or two-center integrals, which are much cheaper to compute and store.

In practice, one introduces an auxiliary basis set {chi_P}. Three-index integrals (mu nu|P) are formed by pairing primary basis functions phi_mu and phi_nu with auxiliary functions chi_P. A metric matrix V_{PQ} is defined from the auxiliary functions, typically V_{PQ} = (chi_P|chi_Q) under the Coulomb operator. The two-electron integral (mu nu|lambda sigma) is then approximated by summing over P and Q: (mu nu|lambda sigma) ≈ sum_{P,Q} (mu nu|P) (V^{-1})_{PQ} (Q|lambda sigma). This reduces computational scaling and memory requirements, benefiting Hartree-Fock, density functional theory, and post-HF methods such as MP2 and CC, particularly for large systems.

Various RI or DF variants exist, such as RI-J (fitting Coulomb energies) and RI-V (fitting two-electron repulsion integrals). The accuracy of density fitting depends on the choice of auxiliary basis and its compatibility with the primary basis; with well-designed sets, errors are small and controllable. Density fitting is widely implemented in quantum chemistry software as a standard acceleration technique. See also resolution of the identity and auxiliary basis sets.

---