TABLE IV.

Generation of two-electron RI integrals required for constructing CIS(D′) hole and particle density matrices. Batching of integrals and reordering of tensors have been omitted for clarity.

Step	Operation	Prefactor	Scaling
1	Calculate integrals $\left(\mathrm{\Upsilon }|\alpha \beta \right)$		$\mathcal{O}\left(N_{\mathrm{\Gamma }}{N}_{\alpha }^2\right)$
2	$\left(\mathrm{\Upsilon }|A^{\prime }\beta \right)\leftarrow {\sum }_{\alpha }{C}_{\alpha A^{\prime }}\left(\mathrm{\Upsilon }|\alpha \beta \right)$	$V^{\prime }$	$\mathcal{O}\left(N_{\mathrm{\Gamma }}{N}_{\alpha }^2\right)$
	$\left(\mathrm{\Upsilon }|\overline{A^{\prime }}\beta \right)\leftarrow {\sum }_{\alpha }{\overline{\mathrm{\Lambda }}}_{\alpha A^{\prime }}^{\text{CIS}}\left(\mathrm{\Upsilon }|\alpha \beta \right)$	$V^{\prime }$
	$\left(\mathrm{\Upsilon }|\alpha I^{\prime }\right)\leftarrow {\sum }_{\beta }{C}_{\beta I^{\prime }}\left(\mathrm{\Upsilon }|\alpha \beta \right)$	$O^{\prime }$
	$\left(\mathrm{\Upsilon }|\alpha \overline{I^{\prime }}\right)\leftarrow {\sum }_{\beta }{\overline{\mathrm{\Lambda }}}_{\beta I^{\prime }}^{\text{CIS}}\left(\mathrm{\Upsilon }|\alpha \beta \right)$	$O^{\prime }$
3	$\left(\mathrm{\Upsilon }|A^{\prime }i\right)\leftarrow {\sum }_{\beta }{C}_{\beta i}\left(\mathrm{\Upsilon }|A^{\prime }\beta \right)$	$V^{\prime }$	$\mathcal{O}\left(N_{\mathrm{\Gamma }}{N}_{\alpha }O\right)$
	$\left(\mathrm{\Upsilon }|A^{\prime }ī\right)\leftarrow {\sum }_{\beta }{\overline{\mathrm{\Lambda }}}_{\beta i}^{\text{CIS}}\left(\mathrm{\Upsilon }|A^{\prime }\beta \right)$
	$\left(\mathrm{\Upsilon }|\overline{A^{\prime }}i\right)\leftarrow {\sum }_{\beta }{C}_{\beta i}\left(\mathrm{\Upsilon }|\overline{A^{\prime }}\beta \right)$
4	$\left(\mathrm{\Upsilon }|a{I}^{\prime }\right)\leftarrow {\sum }_{\alpha }{C}_{\alpha a}\left(\mathrm{\Upsilon }|\alpha I^{\prime }\right)$	$O^{\prime }$	$\mathcal{O}\left(N_{\mathrm{\Gamma }}{N}_{\alpha }V\right)$
	$\left(\mathrm{\Upsilon }|āI^{\prime }\right)\leftarrow {\sum }_{\alpha }{\overline{\mathrm{\Lambda }}}_{\alpha a}^{\text{CIS}}\left(\mathrm{\Upsilon }|\alpha I^{\prime }\right)$
	$\left(\mathrm{\Upsilon }|a\overline{I^{\prime }}\right)\leftarrow {\sum }_{\alpha }{C}_{\alpha a}\left(\mathrm{\Upsilon }|\alpha \overline{I^{\prime }}\right)$
5	$I_{A^{\prime }i}^{\mathrm{\Gamma }}\leftarrow {\sum }_{\mathrm{\Upsilon }}{\left(\mathrm{\Gamma }|\mathrm{\Upsilon }\right)}^{-1∕2}\left(\mathrm{\Upsilon }|A^{\prime }i\right)$	$V^{\prime }$	$\mathcal{O}\left(N_{\mathrm{\Gamma }}^{2}O\right)$
	$I_{A^{\prime }ī}^{\mathrm{\Gamma }}\leftarrow {\sum }_{\mathrm{\Upsilon }}{\left(\mathrm{\Gamma }|\mathrm{\Upsilon }\right)}^{-1∕2}\left(\mathrm{\Upsilon }|A^{\prime }ī\right)$
	$I_{\overline{A^{\prime }}i}^{\mathrm{\Gamma }}\leftarrow {\sum }_{\mathrm{\Upsilon }}{\left(\mathrm{\Gamma }|\mathrm{\Upsilon }\right)}^{-1∕2}\left(\mathrm{\Upsilon }|\overline{A^{\prime }}i\right)$
6	$I_{a{I}^{\prime }}^{\mathrm{\Gamma }}\leftarrow {\sum }_{\mathrm{\Upsilon }}{\left(\mathrm{\Gamma }|\mathrm{\Upsilon }\right)}^{-1∕2}\left(\mathrm{\Upsilon }|a{I}^{\prime }\right)$	$O^{\prime }$	$\mathcal{O}\left(N_{\mathrm{\Gamma }}^{2}V\right)$
	$I_{āI^{\prime }}^{\mathrm{\Gamma }}\leftarrow {\sum }_{\mathrm{\Upsilon }}{\left(\mathrm{\Gamma }|\mathrm{\Upsilon }\right)}^{-1∕2}\left(\mathrm{\Upsilon }|āI^{\prime }\right)$
	$I_{a\overline{I^{\prime }}}^{\mathrm{\Gamma }}\leftarrow {\sum }_{\mathrm{\Upsilon }}{\left(\mathrm{\Gamma }|\mathrm{\Upsilon }\right)}^{-1∕2}\left(\mathrm{\Upsilon }|a\overline{I^{\prime }}\right)$