Evaluation of the Fock matrix (both Coulomb, J, and exchange, K, pieces) can be sped up by an approximation known as the resolution-of-the-identity (RI-JK). Essentially, the full complexity in common basis sets required to describe chemical bonding is not necessary to describe the mean-field Coulomb and exchange interactions between electrons. That is, in the left side of
is much less complicated than an individual function pair. The same principle applies to the FTC method in subsection 4.6.5, in which case the slowly varying piece of the electron density is replaced with a plane-wave expansion.
With the RI-JK approximation, the Coulomb interactions of the function pair are fit by a smaller set of atom-centered basis functions. In terms of :
The coefficients must be determined to accurately represent the potential. This is done by performing a least-squared minimization of the difference between and , with differences measured by the Coulomb metric. This requires a matrix inversion over the space of auxiliary basis functions, which may be done rapidly by Cholesky decomposition.
The RI-J can be invoked by either setting RI_J to be true, or (since Q-Chem 5.2) specifying auxiliary basis set for J using AUX_BASIS_J.
The RI method applied to the Fock matrix may be further enhanced by performing local fitting of a density or function pair element. This is the basis of the atomic-RI method (ARI), which has been developed for both Coulomb (J) matrixSodt:2006b and exchange (K) matrix evaluation.Sodt:2008 In ARI, only nearby auxiliary functions are employed to fit the target function. This reduces the asymptotic scaling of the matrix-inversion step as well as that of many intermediate steps in the digestion of RI integrals. Briefly, atom-centered auxiliary functions on nearby atoms are only used if they are within the “outer” radius () of the fitting region. Between and the “inner” radius (), the amplitude of interacting auxiliary functions is smoothed by a function that goes from zero to one and has continuous derivatives. To optimize efficiency, the van der Waals radius of the atom is included in the cutoff so that smaller atoms are dropped from the fitting radius sooner. The values of and are specified as REM variables as described below.