4.6 Large Molecules and Linear Scaling Methods

4.6.8 CASE Approximation

The Coulomb Attenuated Schrödinger Equation (CASE) approximation25 follows from the KWIK algorithm237 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 ω 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 ω 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.

OMEGA
       Controls the degree of attenuation of the Coulomb operator.
TYPE:
       INTEGER
DEFAULT:
       No default
OPTIONS:
       n Corresponding to ω=n/1000, in units of bohr-1
RECOMMENDATION:
       None

INTEGRAL_2E_OPR
       Determines the two-electron operator.
TYPE:
       INTEGER
DEFAULT:
       -2 Coulomb Operator.
OPTIONS:
       -1 Apply the CASE approximation. -2 Coulomb Operator.
RECOMMENDATION:
       Use the default unless the CASE operator is desired.