4.6.8 occ-RI-K Exchange Algorithm

The occupied orbital RI-K (occ-RI-K) algorithm⁷³⁴ is a new scheme for building the exchange matrix (K) partially in the MO basis using the RI approximation. occ-RI-K typically matches current alternatives in terms of both the accuracy (energetics identical to standard RI-K) and convergence (essentially unchanged relative to conventional methods). On the other hand, this algorithm exhibits significant speedups over conventional integral evaluation (14x) and standard RI-K (3.3x) for a test system, a graphene fragment (C${}_{68}$H${}_{22}$) using cc-pVQZ basis set (4400 basis functions), whereas the speedup increases with the size of the AO basis set. Thus occ-RI-K helps to make larger basis set hybrid DFT calculations more feasible, which is quite desirable for achieving improved accuracy in DFT calculations with modern functionals.

The idea of the occ-RI-K formalism comes from a simple observation that the exchange energy $E_K$ and its gradient can be evaluated from the diagonal elements of the exchange matrix in the occupied-occupied block $K_{ii}$, and occupied-virtual block $K_{ia}$, respectively, rather than the full matrix in the AO representation, $K_{\mu \nu }$. Mathematically,

and

where $\mathrm{\Delta }$ is a skew-symmetric matrix used to parameterize the unitary transformation $U$, which represents the variations of the MO coefficients as follows:

From Eq. 4.52 and 4.53 it is evident that the exchange energy and gradient need just $K_{i\nu }$ rather than $K_{\mu \nu }$.

In regular RI-K one has to compute two quartic terms,¹¹⁹¹ whereas there are three quartic terms for the occ-RI-K algorithm. The speedup of the latter with respect to former can be explained from the following ratio of operations; refer to Ref. 734 for details.

With a conservative approximation of $X\approx 2N$, the speedup is $\frac{3}{4}(N/o)$. The occ-RI-K algorithm also involves some cubic steps which should be negligible in the very large molecule limit. Tests in the Ref. 734 suggest that occ-RI-K for small systems with large basis will gain less speed than a large system with small basis, because the cubic terms will be more dominant for the former than the latter case.

In the course of SCF iteration, the occ-RI-K method does not require us to construct the exact Fock matrix explicitly. Rather, $k_{i\nu }$ contributes to the Fock matrix in the mixed MO and AO representations ($F_{i\nu }$) and yields orbital gradient and DIIS error vectors for converging SCF. On the other hand, since occ-RI-K does not provide exactly the same unoccupied eigenvalues, the diagonalization updates can differ from the conventional SCF procedure. In Ref. 734, occ-RI-K was found to require, on average, the same number of SCF iterations to converge and to yield accurate energies.

The occ-RI-K can be invoked by either setting OCC_RI_K to be true, or (since Q-Chem 5.2) specifying auxiliary basis set for K using AUX_BASIS_K.