13.3.1 Overview

The Coulomb Attenuated Schrödinger Equation (CASE) approximation ²³ Adamson R. D., Dombroski J. P., Gill P. M. W.
Chem. Phys. Lett.
(1996), 254, pp. 329. Link follows from the KWIK algorithm ³¹² Dombroski J. P., Taylor S. W., Gill P. M. W.
J. Phys. Chem.
(1996), 100, pp. 6272. Link in which the Coulomb operator is separated into two pieces using the error function, Eq. (5.12). Whereas in Section 5.6 this partition of the Coulomb operator was used to incorporate long-range Hartree-Fock exchange into DFT, within the CASE approximation it is used to attenuate all occurrences of the Coulomb operator in Eq. (4.2), by neglecting the long-range portion of the identity in Eq. (5.12). The parameter $\omega$ in Eq. (5.12) is used to tune the level of attenuation. Although the total energies from Coulomb attenuated calculations are significantly different from non-attenuated energies, it is found that relative energies, correlation energies and, in particular, wave functions, are not, provided a reasonable value of $\omega$ is chosen.

By virtue of the exponential decay of the attenuated operator, ERIs can be neglected on a proximity basis yielding a rigorous $𝒪(N)$ algorithm for single point energies. CASE may also be applied in geometry optimizations and frequency calculations.