The intermediates \( \chi \) are defined as $$ \begin{equation} \langle b| \chi |c \rangle =\langle b|f|c\rangle - \frac{1}{2} \sum_{kld} \langle bd|t|kl\rangle \langle kl|v|cd\rangle \tag{15} \end{equation} $$ $$ \begin{equation} \langle k| \chi |j\rangle = \langle k|f|j\rangle + \frac{1}{2} \sum_{cdl} \langle kl|v|cd\rangle \langle cd|t|jl\rangle \tag{16} \end{equation} $$ $$ \begin{equation} \langle kl| \chi |ij\rangle = \langle kl|v|ij\rangle + \frac{1}{2} \sum_{cd} \langle kl|v|cd\rangle \langle cd|t|ij\rangle \tag{17} \end{equation} $$ $$ \begin{equation} \langle kb| \chi |cj\rangle = \langle kb|v|cj\rangle + \frac{1}{2} \sum_{dl} \langle kl|v|cd\rangle \langle db|t|lj\rangle \tag{18} \end{equation} $$ $$ \begin{equation} \langle ab| \chi |cd\rangle = \langle ab|v|cd\rangle \tag{19} \end{equation} $$
With the introduction of the above intermediates, the CCD equations scale now as \( \mathcal{O}(n_{h}^{2}n_{p}^{4}) \).