# 4.6.6 Resolution of the Identity Fock Matrix Methods

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, $\rho$ in the left side of

 $\displaystyle(\mu\nu|\rho)=\sum_{\lambda\sigma}(\mu\nu|\lambda\sigma)\mathbf{P% }_{\lambda\sigma}$ (4.47)

is much less complicated than an individual $\lambda\sigma$ 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 $\rho(r)=\lambda\sigma(r)P_{\lambda\sigma}$ are fit by a smaller set of atom-centered basis functions. In terms of $J$:

 $\displaystyle\sum_{\lambda\sigma}\int\textrm{d}^{3}\mathbf{r}_{1}P_{\lambda% \sigma}\lambda\sigma(\mathbf{r}_{1})\frac{1}{|\mathbf{r}_{1}-\mathbf{r}|}% \approx\sum_{K}\int\textrm{d}^{3}\mathbf{r}_{1}P_{K}K(\mathbf{r}_{1})\frac{1}{% |\mathbf{r}_{1}-\mathbf{r}|}$ (4.48)

The coefficients $P_{K}$ must be determined to accurately represent the potential. This is done by performing a least-squared minimization of the difference between $P_{\lambda\sigma}\lambda\sigma(\mathbf{r}_{1})$ and $P_{K}K(\mathbf{r}_{1})$, 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) matrix852 and exchange (K) matrix evaluation.853 In ARI, only nearby auxiliary functions $K(\mathbf{r})$ 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 ($R_{1}$) of the fitting region. Between $R_{1}$ and the “inner” radius ($R_{0}$), 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 $R_{0}$ and $R_{1}$ are specified as REM variables as described below.

RI_J
Toggles the use of the RI algorithm to compute J.
TYPE:
LOGICAL
DEFAULT:
FALSE RI will not be used to compute J.
OPTIONS:
TRUE Turn on RI for J.
RECOMMENDATION:
For large (especially 1D and 2D) molecules the approximation may yield significant improvements in Fock evaluation time when used with ARI.

RI_K
Toggles the use of the RI algorithm to compute K.
TYPE:
LOGICAL
DEFAULT:
FALSE RI will not be used to compute K.
OPTIONS:
TRUE Turn on RI for K.
RECOMMENDATION:
For large (especially 1D and 2D) molecules the approximation may yield significant improvements in Fock evaluation time when used with ARI.

ARI
Toggles the use of the atomic resolution-of-the-identity (ARI) approximation.
TYPE:
LOGICAL
DEFAULT:
FALSE ARI will not be used by default for an RI-JK calculation.
OPTIONS:
TRUE Turn on ARI.
RECOMMENDATION:
For large (especially 1D and 2D) molecules the approximation may yield significant improvements in Fock evaluation time.

ARI_R0
Determines the value of the inner fitting radius (in Ångstroms)
TYPE:
INTEGER
DEFAULT:
4 A value of 4 Å will be added to the atomic van der Waals radius.
OPTIONS:
$n$ User defined radius.
RECOMMENDATION:
For some systems the default value may be too small and the calculation will become unstable.

ARI_R1
Determines the value of the outer fitting radius (in Ångstroms)
TYPE:
INTEGER
DEFAULT:
5 A value of 5 Å will be added to the atomic van der Waals radius.
OPTIONS:
$n$ User defined radius.
RECOMMENDATION:
For some systems the default value may be too small and the calculation will become unstable. This value also determines, in part, the smoothness of the potential energy surface.