4.6 Large Molecules and Linear Scaling Methods 4.6.8 CASE Approximation 4.7 Dual-Basis Self-Consistent Field Calculations

4.6.9 occ-RI-K Exchange Algorithm

The occupied orbital RI-K (occ-RI-K) algorithm⁶²⁰ 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,

$\displaystyle E_{K}$	$\displaystyle=$	$\displaystyle-\sum_{\mu\nu}P_{\mu\nu}K_{\mu\nu}$	(4.50)
	$\displaystyle=$	$\displaystyle-\sum_{\mu\nu}c_{\mu i}K_{\mu\nu}c_{\nu i}$
	$\displaystyle=$	$\displaystyle-\sum_{i}K_{ii}$

and

\frac{\partial E_{K}}{\partial\Delta_{ai}}=2K_{ai}

(4.51)

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

U=e^{(\Delta-\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.

\displaystyle\frac{\mbox{\#~{}of~{}RI-K~{}quartic~{}operations}}{\mbox{\#~{}of% ~{}occ-RI-K~{}quartic~{}operations}}=\frac{oNX^{2}+oN^{2}X}{o^{2}X^{2}+o^{2}NX% +o^{2}NX}=\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

4.6.9.1 occ-RI-K for exchange energy gradient evaluation

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:

	$\displaystyle E_{K}^{x}$	$\displaystyle=$	$\displaystyle(ij\|ij)^{x}$		(4.54)
		$\displaystyle=$	$\displaystyle\sum_{\mu\nu P}\sum_{ij}c_{\mu i}c_{\nu j}C_{ij}^{P}(\mu\nu\|P)^{x% }-\sum_{RS}\sum_{ij}C_{ij}^{R}C_{ij}^{S}(R\|S)^{x}.$		(4.54)

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:

\displaystyle C_{\mu\nu}^{P}=\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"

Example 4.19 Q-Chem input for a energy and gradient calculations with occ-RI-K method.

$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

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4.6.9 occ-RI-K Exchange Algorithm

4.6.9.1 occ-RI-K for exchange energy gradient evaluation