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The segmented contracted basis sets are usually handled well by the QM programs than generally contracted basis sets, as the primitives are not repeated. The correlation consistent basis sets (cc-PVnZ) are all generally contracted, and optimized for electron correlation calculations such as CCSD.

Would it be possible to make segmented contracted basis sets that performed well for CCSD and other correlation methods? I have been told by my lecturer that in correlated calculations, the HF calculation is the fast step, so there is no real benefit to using segmented contraction. However, I am still curious as to whether it is possible, and if not then why?

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Yes, it is perfectly possible. As I've discussed here, it is possible to convert generally contracted basis sets into (somewhat) segmented sets without any formal loss of accuracy; next, one would discard functions with tiny coefficients and reoptimize the segmented exponents and contraction coefficients to end up with a segmented basis set.

However, as you (or your lecturer) said, segmented basis sets are much less beneficial for correlated calculations than SCF theory, since whereas SCF is usually dominated by the cost of forming the Fock matrix (i.e. calculating the two-electron integrals), in correlated calculations the cost of forming the integrals is often small compared to the post-HF calculation itself. For instance, the traditional AO $\to$ MO integral transform has a $\mathcal{O}(N^5)$ computational cost, while CCSD has an iterative $\mathcal{O}(N^6)$ computational cost.

Also note that the Karlsruhe def2 sets employ segmented contractions, and the sets with two sets of polarization functions (def2-TZVPP and def2-QZVPP) are aimed for post-HF calculations, so you should be able to find out the speed difference yourself by comparing calculations in the def2 sets to corresponding ones with the cc sets.

Addendum: I should also mention the Sapporo basis sets, which are likewise segmented and optimized for post-Hartree-Fock calculations.

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    $\begingroup$ Because of the bounty that was spent here, I sent the question to Susi, along with a couple other people who do work in this area. Wim Klopper, leader of the TURBOMOLE software that works very well with def2 basis sets, has replied with "the answer to the question by Susi Lehtola is perfect. In particular, the segmented Karlsruhe basis set with “PP” are meant for correlated calculations." $\endgroup$ – Nike Dattani Jan 4 at 20:32

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