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# Characteristics of the Implementation and Computational Demands

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 = taiτai , (10.2) T2 = 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| + [, T2]| HF〉 = 0  , (10.4) Ωμ2 = 〈μ2| + [, T2] + [[, T2], T2]| HF〉 = 0  , (10.5)

where again

= exp(- T1)exp(T1),

and μ1 and μ2 are, respectively, the sets of all singly and doubly excited determinants. For MP3 the energy is computed from the first-order amplitudes ( tiajb(1) ) as

 EMP3, tot = EHF + EMP2 + EMP3 (10.6) = 〈HF| + [, T2(1)]| HF〉 + tμ2(1)〈μ2|[, T(1)2]| HF〉 (10.7)

with = - . To evaluate the fourth-order energy one needs in addition to the first-order also the second-order amplitudes, which are obtained from the solution of the equations

 〈μ1|[, T1(2)] + [, T(1)2]| HF〉 = 0 (10.8) 〈μ2|[, T2(2)] + [, T(1)2]| HF〉 = 0 (10.9) 〈μ3|[, T3(2)] + [, T(1)2]| HF〉 = 0 (10.10)

From these the fourth-order energy correction is computed as:

 EMP4 = tμ2(1)〈μ2|[, T(2)1 + T(2)2 + T(2)3] + [[, T(1)2], T(1)2]| HF〉 . (10.11)

Eqs. (10.5) and (10.7) - (10.11) are computational much more complex and demanding than the corresponding doubles equations for the CC2 model. If 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 (6) . If for the same molecule the number of one-electron basis functions N is increased the costs scale with (4) . (For RI-MP2 and RI-CC2 the costs scale with the system size as (5) and with the number of basis functions as (3) .) The computational costs for an MP3 calculations are about the same as for one CCSD iteration. For MP4 the computational costs are comparable to those for two CCSD iteration plus the costs for the perturbation triples correction (see below).

Subsections

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