. 2017 May 16;146(19):194102. doi: 10.1063/1.4983277

TABLE III.

Working equations and their scaling for the evaluation of CIS(D) density matrices.

Step	Scaling
1 Construct CIS contribution to density matrices
${\bar{D}}_{i j}^{CIS (D)} \leftarrow \sum_{a} c_{i}^{a} c_{j}^{a}$	$n_{occ}^{2} n_{virt}$
$D_{a b}^{CIS (D)} \leftarrow \sum_{i} c_{i}^{a} c_{i}^{b}$	$n_{occ} n_{virt}^{2}$
2 Compute intermediate $Y_{a i}^{Q}$
$Y_{a i}^{Q} \leftarrow \sum_{c} J_{a c}^{Q} c_{i}^{c}$	$n_{occ} n_{virt}^{2} n_{aux}$
$Y_{a i}^{Q} \leftarrow - \sum_{j} J_{i j}^{Q} c_{j}^{a}$	$n_{occ}^{2} n_{virt} n_{aux}$
3 Compute intermediate $V_{i j}^{a b}$
$V_{i j}^{a b} = \sum_{Q} Y_{a i}^{Q} J_{b j}^{Q}$	$n_{occ}^{2} n_{virt}^{2} n_{aux}$
4 Calculate $c_{i j}^{a b}$ coefficients
$c_{i j}^{a b} = (V_{i j}^{a b} + V_{j i}^{b a}) / (D_{i j}^{a b} + ω_{CIS})$	$n_{occ}^{2} n_{virt}^{2}$
5 Compute intermediate $X_{i j}^{a b}$
$X_{i j}^{a b} = 2 c_{i j}^{a b} - c_{i j}^{b a}$	$n_{occ}^{2} n_{virt}^{2}$
6 Compute density matrices
${\bar{D}}_{i j}^{CIS (D)} \leftarrow \sum_{k a b} c_{i k}^{a b} X_{j k}^{a b}$	$n_{occ}^{3} n_{virt}^{2}$
$D_{a b}^{CIS (D)} \leftarrow \sum_{i j c} c_{i j}^{c a} X_{i j}^{c b}$	$n_{occ}^{2} n_{virt}^{3}$