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Abstract
Multi-reference effects can be covered by density functional theory (DFT) either implicitly via the exchange-correlation functional or explicitly via the form of the Kohn-Sham wave function. With the help of the exchange hole it is shown that the self-interaction error of the exchange functional will mimic long-range electron correlation effects if restricted Kohn-Sham theory is used. Functionals based on Slater or Becke exchange have a relatively large self-interaction error and, therefore, lead to a relatively large implicit coverage of long-range correlation, which, because of the possibility of doublecounting of electron correlation, has to be considered when using these functionals in connection with two- or multi-configurational descriptions based on ensemble DFT methods such as REKS (spin-Restricted Ensemble-referenced KS-DFT). Arguments are given that a REKS description of a multireference problem avoids a double-counting of long-range correlation effects, in particular as in this situation the self-interaction error of the exchange functional simulates more short- rather than longrange correlation effects. There is, however, no guarantee that the short-range effects are not doublecounted, namely once via the exchange and once via the correlation functional. Therefore, one should use hybrid functionals such as B3LYP in connection with multi-reference DFT methods because for hybrid functionals the self-interaction error and by this the implicit coverage of long(short)-range correlation effects is reduced due to the admixture of exact exchange. This rule applies also to broken-symmetry UDFT, which performs better with hybrid rather than GGA functionals. A way of avoiding the implicit coverage of multi-reference effects is given by the combination of wave function theory and DFT methods. The advantages and disadvantages of CAS-DFT are discussed and it is shown that an effective reduction of a double-counting of correlation effects is possible within this method.
Highlights
It is generally assumed that standard Kohn-Sham density functional theory (DFT) [1,2,3,4] carried out with approximate exchange-correlation functionals as they are currently in use covers beside exchange correlation effects just short-range Coulomb correlation
One calculates the correlation energy of the homogeneous electron gas (HEG) with the density ρ in a way that is comparable to the CASSCF description: Excitations are allowed from the occupied orbitals only into virtual orbitals up to a certain energy limit, which is adjusted in the way that the active space has just the reference density ρref
It is a reflection of the fact that electron correlation is covered by DFT implicitly via the form of the exchange-correlation functional and any explicit additional coverage of correlation effects is difficult to adjust in a way that any double-counting correlation is avoided
Summary
It is generally assumed that standard Kohn-Sham density functional theory (DFT) [1,2,3,4] carried out with approximate exchange-correlation functionals (approximate DFT) as they are currently in use covers beside exchange correlation effects just short-range Coulomb correlation (dynamic electron correlation). For the explicit account of multi-reference effects, DFT offers today two major routes, namely a) the extension of standard DFT to ensemble DFT [27,28,29,30] and b) the merging of DFT with techniques used in WFT The latter possibility includes methods such as two-configurational DFT (GVB-DFT) [31] and multi-configurational DFT such as CAS-DFT. The objectives of this work are threefold: a) First, we will clarify what types of correlation effects are introduced by approximate exchange functionals under different conditions, i.e. RDFT, UDFT or BS-UDFT. In this connection, difference densities and representations of the exchange hole will be used to support the discussion.
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