Note that \( n_u\gg A \) in general. In textbooks, one reads that CCD (and CCSD) cost only \( A^2n_u^4 \). Our most expensive diagrams, however are \( A^4n_u^4 \). What is going on?
To understand this puzzle, let us consider the last diagram of Figure xx . We break up the computation into two steps, computing first the intermediate
$$ \begin{align} \chi_{ij}^{kl}\equiv {1\over 2} \sum_{cd} \langle kl\vert V\vert cd\rangle t_{ij}^{cd} \tag{28} \end{align} $$at a cost of \( A^4n_u^2 \), and then
$$ \begin{align} {1\over 2} \sum_{kl} \chi_{ij}^{kl} t_{kl}^{ab} \tag{29} \end{align} $$at a cost of \( A^4n_u^2 \). This is affordable. The price to pay is the storage of the intermediate \( \chi_{ij}^{kl} \), i.e. we traded memory for computational cycles. This trick is known as factorization.