  The recently developed hypercomplex Kohn-Sham (HCKS) theory shows great
potential to overcome the static/strong correlation issue in density functional
theory (DFT), which highlights the necessity of further exploration of the HCKS
theory toward better handling many-electron problem. This work mainly focuses
on approximate functionals in HCKS, seeking to gain more insights into
functional development from the comparison between Kohn-Sham (KS) DFT and HCKS.
Unlike KS-DFT, HCKS can handle different correlation effects by resorting to a
set of auxiliary orbitals with dynamically varying fractional occupations.
These orbitals of hierarchical correlation (HCOs) thus contain distinct
electronic information for better considering the exchange-correlation effect
in HCKS. The test on the triplet-singlet gaps shows that HCKS has much better
performance as compared to KS-DFT in use of the same functionals, and the
systematic errors of semi-local functionals can be effectively reduced by
including appropriate amount of the HCO-dependent Hartree-Fock (HF) exchange.
In contrast, KS-DFT shows large systematic errors, which are hardly reduced by
the functionals tested in this work. Therefore, HCKS creates new channels to
address to the strong correlation issue, and further development of functionals
that depend on HCOs and their occupations is necessary for the treatment of
strongly correlated systems.
