Integral-direct implementation and resolution-of-the-identity approximation:

The computationally most demanding (in terms floating point operations) steps of a CCSD calculation are related to two kinds of terms. One of the most costly steps is the contraction

$$\sum_{{cd}}^{}$$ (10.16)

where a, b, c, and d are virtual orbitals. For small molecules with large basis sets or basis sets with diffuse functions, where integral screening is not effective, it is time-determing step and can most efficiently be evaluated with a minimal operation count of ${\tfrac{{1}}{{2}}}$O²V² (where O and V are number of, respectively occupied and virtual orbitals), if the 4-index integrals (ac| bd ) in the MO are precalculated and stored on file before the iterative solution of the coupled-cluster equation, 10.4 and 10.5. For larger systems, however, the storage and I/O of the integrals (ac| bd ) leads to bottlenecks. An alternatively, this contribution can be evaluated in an integral-direct was as

$$\sum_{{cd}}^{} \sum_{{\kappa\lambda}}^{} \sum_{{\mu \nu}}^{}$$ (10.17)

which, depending on the implementation and system, has formally a 2-3 times larger operation count, but allows to avoid the storage and I/O bottlenecks by processing the 4-index integrals on-the-fly without storing them. Furthermore, integral screening techniques can be applied to reduce the operation count for large systems to asymptotic scaling with $\cal {O}$($\cal {N}$⁴).

In TURBOMOLE only the latter algorithm is presently implemented. (For small systems other codes will therefore be faster.)

The other class of expensive contributions are so-called ring terms (in some publications denoted as C and D terms) which involve contractions of the doubles amplitudes t_aibj with several 4-index MO integrals with two occupied and two virtual indeces, partially evaluated with T₁-dependent MO coefficients. For these terms the implementation in TURBOMOLE employs the resolution-of-the-identity (or density-fitting) approximation (with the cbas auxiliary basis set) to reduce the overhead from integral transformation steps. Due this approximation CCSD energies obtained with TURBOMOLe will deviate from those obtained with other coupled-cluster programs by a small RI error. This error is usually in the same order or smaller the RI error for a RI-MP2 calculation for the same system and basis sets.

The RI approximation is also used to evaluate the 4-index integrals in the MO basis needed for the perturbative triples corrections.