The ECP matrix elements are arguably the most difficult one-electron integrals
in existence. Indeed, using current methods, the time taken to compute the ECP
integrals can exceed the time taken to compute the far more numerous electron
repulsion integrals. Q-Chem 5.0 implements a state-of-the-art ECP implementation
J. Phys. Chem. A
(2018), 122, pp. 3066. based on efficient recursion relations and upper bounds. This method relies on a restricted radial potential , where the radial power is only ever zero, i.e. . Whilst true for some ECPs, such as the Stuttgart-Bonn sets, many other ECPs have radial potentials containing and terms. To overcome this challenge, we fit these ECP radial potentials using only terms. Each and term is expanded as a sum of three terms, each with independent contraction coefficient and Gaussian exponent . The Gaussian exponents are given by a predetermined recipe and the contraction coefficients are computed in a least squares fitting procedure. The errors introduced by the ECP fitting are insignificant and of the same order as those introduced by numerical integration present in other ECP methods. For the built-in ECPs, fitted variants of each are now provided in the $QCAUX directory, e.g., fit-LANL2DZ. For user-defined ECPs with or terms, Q-Chem will perform a fit at run time with the additional rem keyword ECP_FIT = TRUE.