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

TABLE I.

Working equations and their scaling for the CC2 algorithm.

Step	Scaling
1 Construct intermediates ${\hat{X}}_{a i}^{Q}$ and ${\hat{X}}_{i j}^{Q}$
${\hat{X}}_{a i}^{Q} = \sum_{b} J_{a b}^{Q} t_{i}^{b}$	$n_{occ} n_{virt}^{2} n_{aux}$
${\hat{X}}_{i j}^{Q} = \sum_{a} J_{a i}^{Q} t_{j}^{a}$	$n_{occ}^{2} n_{virt} n_{aux}$
2 Compute the ${\hat{J}}_{j i}^{Q}$ and ${\hat{J}}_{a i}^{Q}$ lists and the contribution of one term in ${\hat{F}}_{a i}$ to $Ω_{a i}$
${\hat{J}}_{i j}^{Q} = {\hat{X}}_{i j}^{Q} + J_{i j}^{Q}$	$n_{occ}^{2} n_{aux}$
$Ω_{a i} \leftarrow - \sum_{Q j} {\hat{X}}_{a j}^{Q} {\hat{J}}_{j i}^{Q}$	$n_{occ}^{2} n_{virt} n_{aux}$
${\hat{J}}_{a i}^{Q} = {\hat{X}}_{a i}^{Q} + J_{a i}^{Q} - \sum_{j} {\hat{J}}_{i j}^{Q} t_{j}^{a}$	$2 n_{occ} n_{virt} n_{aux} + n_{occ}^{2} n_{virt} n_{aux}$
3 Construct intermediate X^Q
$X^{Q} = \sum_{a i} J_{a i}^{Q} t_{i}^{a}$	n_occn_virtn_aux
4 Compute further contributions of ${\hat{F}}_{a i}$ to $Ω_{a i}$
$Ω_{a i} \leftarrow \sum_{Q j} (\sum_{k} {\hat{X}}_{j k}^{Q} t_{k}^{a}) {\hat{J}}_{j i}^{Q}$	$2 n_{occ}^{2} n_{virt} n_{aux}$
$Ω_{a i} \leftarrow \sum_{Q} {\hat{J}}_{a i}^{Q} X^{Q} + t_{i}^{a} (ε_{a} - ε_{i})$	n_occn_virtn_aux + n_occn_virt
5 Calculate ${\hat{F}}_{j b}$
${\hat{F}}_{j b} = \sum_{Q} {\hat{J}}_{b j}^{Q} X^{Q} - \sum_{Q i} {\hat{X}}_{j i}^{Q} J_{b i}^{Q}$	$n_{occ} n_{virt} n_{aux} + n_{occ}^{2} n_{virt} n_{aux}$
6 Construct the $(a i \hat{\|} b j)$ list
$(a i \hat{\|} b j) = \sum_{Q} {\hat{J}}_{a i}^{Q} {\hat{J}}_{b j}^{Q}$	$n_{occ}^{2} n_{virt}^{2} n_{aux}$
7 Compute and contract ${\hat{t}}_{i j}^{a b}$ , calculate the contributions of ${\hat{F}}_{j b}$ to $Ω_{a i}$
${\hat{t}}_{i j}^{a b} = [2 (a i \hat{\|} b j) - (a j \hat{\|} b i)] / D_{i j}^{a b}$	$n_{occ}^{2} n_{virt}^{2}$
${\hat{Y}}_{a i}^{Q} = \sum_{b j} {\hat{t}}_{i j}^{a b} J_{b j}^{Q}$	$n_{occ}^{2} n_{virt}^{2} n_{aux}$
$Ω_{a i} \leftarrow \sum_{b j} {\hat{t}}_{i j}^{a b} {\hat{F}}_{j b}$	$n_{occ}^{2} n_{virt}^{2}$
8 Calculate further contributions of ${\hat{t}}_{i j}^{a b}$ to $Ω_{a i}$
$Ω_{a i} \leftarrow \sum_{Q b} (\sum_{j} J_{b j}^{Q} t_{j}^{a} + J_{b a}^{Q}) {\hat{Y}}_{b i}^{Q} = \sum_{Q b} {\hat{J}}_{b a}^{Q} {\hat{Y}}_{b i}^{Q}$	$2 n_{occ} n_{virt}^{2} n_{aux} + n_{virt}^{2} n_{aux}$
$Ω_{a i} \leftarrow - \sum_{Q j} {\hat{J}}_{j i}^{Q} {\hat{Y}}_{a j}^{Q}$	$n_{occ}^{2} n_{virt} n_{aux}$
9 Calculate CC2 correlation energy
$Δ E_{CC2} = \sum_{Q a i} ({\hat{J}}_{a i}^{Q} + X^{Q} t_{i}^{a} - \sum_{j} {\hat{X}}_{i j}^{Q} t_{j}^{a}) {\hat{Y}}_{a i}^{Q}$	$4 n_{occ} n_{virt} n_{aux} + n_{occ}^{2} n_{virt} n_{aux}$