4.6.9 occ-RI-K Exchange Algorithm

The occupied orbital RI-K (occ-RI-K) algorithm⁶⁰⁶ 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,

	$E_K$	$=$	$-{\displaystyle \sum _{\mu \nu }}{P}_{\mu \nu }{K}_{\mu \nu }$		(4.49)
		$=$	$-{\displaystyle \sum _{\mu \nu }}{c}_{\mu i}{K}_{\mu \nu }{c}_{\nu i}$
		$=$	$-{\displaystyle \sum _i}{K}_{ii}$

and

$$\frac{\partial E_K}{\partial {\mathrm{\Delta }}_{ai}}=2K_{ai}$$

(4.50)

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:

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

(4.51)

From Eq. 4.49 and 4.50 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,⁹⁷⁶ 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. 606 for details.

${\displaystyle \frac{\text{\# of RI-K quartic operations}}{\text{\# of occ-RI-K quartic operations}}}={\displaystyle \frac{oN{X}^2+o{N}^{2}X}{o^2{X}^2+o^{2}NX+o^{2}NX}}={\displaystyle \frac{N(X+N)}{o(X+2N)}}$

(4.52)

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. 606 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. 606, 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.

$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