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The occupied orbital RI-K (occ-RI-K) algorithm^{620} 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,

${E}_{K}$ | $=$ | $-{\displaystyle \sum _{\mu \nu}}{P}_{\mu \nu}{K}_{\mu \nu}$ | (4.50) | ||

$=$ | $-{\displaystyle \sum _{\mu \nu}}{c}_{\mu i}{K}_{\mu \nu}{c}_{\nu i}$ | ||||

$=$ | $-{\displaystyle \sum _{i}}{K}_{ii}$ |

and

$$\frac{\partial {E}_{K}}{\partial {\mathrm{\Delta}}_{ai}}=2{K}_{ai}$$ | (4.51) |

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:

$$U={e}^{(\mathrm{\Delta}-{\mathrm{\Delta}}^{T})}.$$ | (4.52) |

From Eq. 4.50 and 4.51 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,^{1022}
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. 620 for details.

$\frac{\text{\#ofRI-Kquarticoperations}}{\text{\#ofocc-RI-Kquarticoperations}}}={\displaystyle \frac{oN{X}^{2}+o{N}^{2}X}{{o}^{2}{X}^{2}+{o}^{2}NX+{o}^{2}NX}}={\displaystyle \frac{N(X+N)}{o(X+2N)}$ | (4.53) |

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. 620 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. 620, 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.

OCC_RI_K

Controls the use of the occ-RI-K approximation for constructing the exchange matrix

TYPE:

LOGICAL

DEFAULT:

False
Do not use occ-RI-K.

OPTIONS:

True
Use occ-RI-K.

RECOMMENDATION:

Larger the system, better the performance

A very attractive feature of occ-RI-K framework is that one can compute the exchange energy gradient with respect to nuclear coordinates with the same leading quartic-scaling operations as the energy calculation.

The occ-RI-K formulation yields the following formula for the gradient of exchange energy in global Coulomb-metric RI:

${E}_{K}^{x}$ | $=$ | ${(ij|ij)}^{x}$ | (4.54) | ||

$=$ | $\sum _{\mu \nu P}}{\displaystyle \sum _{ij}}{c}_{\mu i}{c}_{\nu j}{C}_{ij}^{P}{(\mu \nu |P)}^{x}-{\displaystyle \sum _{RS}}{\displaystyle \sum _{ij}}{C}_{ij}^{R}{C}_{ij}^{S}{(R|S)}^{x}.$ |

The superscript $x$ represents the derivative with respect to a nuclear coordinate. Note that the derivatives of the MO coefficients ${c}_{\mu i}$ are not included here, because they are already included in the total energy derivative calculation by Q-Chem via the derivative of the overlap matrix.

In Eq. 4.54, the construction of the density fitting coefficients (${C}_{\mu \nu}^{P}$) has the worst scaling of $\mathcal{O}({M}^{4})$ because it involves MO to AO back transformations:

${C}_{\mu \nu}^{P}={\displaystyle \sum _{ij}}{c}_{\mu i}{c}_{\nu j}{C}_{ij}^{P}$ | (4.55) |

where the operation cost is ${o}^{2}NX+o[\mathrm{NB2}]X$.

RI_K_GRAD

Turn on the nuclear gradient calculations

TYPE:

LOGICAL

DEFAULT:

FALSE
Do not invoke occ-RI-K based gradient

OPTIONS:

TRUE
Use occ-RI-K based gradient

RECOMMENDATION:

Use "RI_J false"

$molecule 0 1 C 0.0000000 0.3057430 5.7138876 C 0.0000000 -0.5442831 4.4256275 C 0.0000000 0.3675825 3.1787857 C 0.0000000 -0.4853210 1.8908707 C 0.0000000 0.4264823 0.6439435 C 0.0000000 -0.4264823 -0.6439435 C 0.0000000 0.4853210 -1.8908707 C 0.0000000 -0.3675825 -3.1787857 C 0.0000000 0.5442831 -4.4256275 C 0.0000000 -0.3057430 -5.7138876 H 0.8973039 0.9418895 5.7433715 H -0.8973039 0.9418895 5.7433715 H -0.8965960 -1.1827891 4.4155828 H 0.8965960 -1.1827891 4.4155828 H 0.8966512 1.0057416 3.1940146 H -0.8966512 1.0057416 3.1940146 H -0.8966512 -1.1234851 1.8761493 H 0.8966512 -1.1234851 1.8761493 H 0.8966507 1.0646443 0.6587590 H -0.8966507 1.0646443 0.6587590 H -0.8966507 -1.0646443 -0.6587590 H 0.8966507 -1.0646443 -0.6587590 H 0.8966512 1.1234851 -1.8761493 H -0.8966512 1.1234851 -1.8761493 H -0.8966512 -1.0057416 -3.1940146 H 0.8966512 -1.0057416 -3.1940146 H 0.8965960 1.1827891 -4.4155828 H -0.8965960 1.1827891 -4.4155828 H 0.8973039 -0.9418895 -5.7433715 H 0.0000000 0.3580832 -6.5913113 H -0.8973039 -0.9418895 -5.7433715 H 0.0000000 -0.3580832 6.5913113 $end $rem JOBTYPE force EXCHANGE HF BASIS cc-pVTZ AUX_BASIS cc-pVTZ-JK OCC_RI_K 1 RI_K_GRAD 1 INCFOCK 0 PURECART 1111 $end