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Characteristics of the Implementation and Computational Demands

In CCSD the ground-state energy is (as for CC2) evaluated as

ECC = 〈HF| H| CC〉 = 〈HF| H exp(T)| HF〉  , (10.1)

where the cluster operator T = T1 + T2 consist of linear combination of single and double excitations:

T1 = $\displaystyle \sum_{{ai}}^{}$taiτai , (10.2)
T2 = $\displaystyle {\frac{{1}}{{2}}}$$\displaystyle \sum_{{aibj}}^{}$taibjτaibj . (10.3)

In difference to CC2, the cluster amplitudes tai and taibj are determined from equations which contain no further approximations apart from the restriction of T to single and double excitations:

Ωμ1 = 〈μ1|$\displaystyle \hat{{\tilde{H}}}$ + [$\displaystyle \hat{{\tilde{H}}}$, T2]| HF〉 = 0  , (10.4)
Ωμ2 = 〈μ2|$\displaystyle \hat{{\tilde{H}}}$ + [$\displaystyle \hat{{\tilde{H}}}$, T2] + [[$\displaystyle \hat{{H}}$, T2], T2]| HF〉 = 0  , (10.5)

where again

$\displaystyle \hat{{\tilde{H}}}$ = exp(- T1)$\displaystyle \hat{{H}}$exp(T1),

and μ1 and μ2 are, respectively, the sets of all singly and doubly excited determinants. Eq. (10.5) is computational much more complex and demanding than the corresponding doubles equations for the CC2 model. If $ \cal {N}$ is a measure for the system size (e.g. the number of atoms), the computational costs (in terms of floating point operations) for CCSD calculations scale as $ \cal {O}$($ \cal {N}$6). If for the same molecule the number of one-electron basis functions N is increased the costs scale with $ \cal {O}$($ \cal {N}$4). (For RI-MP2 and RI-CC2 the costs scale with the system size as $ \cal {O}$($ \cal {N}$5) and with the number of basis functions as $ \cal {O}$($ \cal {N}$3).)



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