4.6.8 CASE Approximation

The Coulomb Attenuated Schrödinger Equation (CASE) approximation⁸ follows from the KWIK algorithm²³² 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.