Efficient Method for Calculating Effective Core Potential Integrals

Effective core potential (ECP) integrals are among the most difficult one-electron integrals to calculate due to the projection operators. The radial part of these operators may include r(0), r(-1), and r(-2) terms. For the r(0) terms, we exploit a simple analytic expression for the fundamental projected integral to derive new recurrence relations and upper bounds for ECP integrals. For the r(-1) and r(-2) terms, we present a reconstruction method that replaces these terms by a sum of r(0) terms and show that the resulting errors are chemically insignificant for a range of molecular properties. The new algorithm is available in Q-Chem 5.0 and is significantly faster than the ECP implementations in Q-Chem 4.4, GAMESS (US) and Dalton 2016.