Spectral representation of Hartree‐Fock exchange operator

We report a new mathematical result: it is possible to construct a spectral representation of the two particles Coulomb potential in the form of | r ? r ^′|^?1 = ∑_λλg( r ) g_λ( r ^′). We call this formula λ‐decomposition. Two special nontrivial cases of λ‐decomposition are reported together with the numerical analysis of the convergence for one of them. It is shown how λ‐decomposition allows to construct a new fast algorithm for Hartree‐Fock exchange operator calculation, in which the calculation of electron repulsion integrals (ERIs) is completely avoided. The connection between the new method and the resolution of identity and Cholesky decomposition based approaches has been established. Finally, the accuracy of ERIs evaluation within the new approach has been studied numerically. The results demonstrate that it is possible to achieve the accuracy of 10^?10 for the ERIs in wide range of their orbital exponents with relatively small number of terms in λ‐decomposition. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012