6.2 Exchange-Correlation Functionals Available

The following exchange-correlation functionals are available:

For EXX and LHF, see Chapter 18

The XCFun library (Arbitrary-Order Exchange-Correlation Functional Library) by Ulf Ekström and co-workers has been included [55] and some of the functionals implemented there can now be utilized. Among them are the empirically fitted MGGAs M06 and M06-2X from the Truhlar group [56]. XCFun functionals are available for energy, gradient, vib. frequencies and TDDFT excited state energy calculations - with and without RI approximation. For details and the license of XCFun please refer to its web site https://repo.ctcc.no/projects/xcfun/wiki

In detail, the Turbomole own functional library consists of:

Additionally, for all four modules (dscf, grad, ridft, and rdgrad) following hybrid functionals are available (a mixture of Hartree–Fock exchange with DFT exchange-correlation functionals):

The Double-Hydbrid Functional B2-PLYP can be used for single point energy calculations. Note, that one has to run a MP2 calculation after the DFT step to get the correct B2-PLYP energy!

B2-PLYP is a so-called double-hybrid density functional (DHDF) [70] that uses in addition to a non-local exchange contribution (as in conventional hybrid-GGAs) also a non-local perturbation correction for the correlation part. In the following options/restrictions of this method are given:

The non-local perturbation correction to the correlation contribution is given by second-order perturbation theory. The idea is rooted in the ab initio Kohn-Sham perturbation theory (KS-PT2) by Görling and Levy [71,72]. The mixing is described by two empirical parameters ax and ac in the following manner:

EXC (DHDF   ) = (1- ax)EX (GGA  )+ axEX  (HF  )            (6.6)
        +(1 - ac)EC (GGA  )+ acEC (KS  - PT 2),
where EX(GGA) is the energy of a conventional exchange functional and EC(GGA) is the energy of a correlation functional. EX(HF) is the Hartree-Fock exchange of the occupied Kohn-Sham orbitals and EC(KS - PT2) is a Møller-Plesset like perturbation correction term based on the KS orbitals:
                 1 ∑  ∑   (ia|jb)[(ia|jb) - (ib|ja)]
EC(KS  - P T2) = --       ---------------------.
                 2  ia  jb    ei + ej - ea - eb
(6.7)

The method is self-consistent only with respect to the first three terms in Eq. 6.6, i.e., first a SCF using a conventional hybrid-GGA is performed first. Based on these orbitals EC(KS - PT2) is evaluated afterwards and added to the total energy.

For B2-PLYP, B88 exchange [61] and LYP correlation [62] are used with the parameters ax = 0.53 and ac = 0.27. Due to the relatively large Fock-exchange fraction, self-interaction error related problems are alleviated in B2-PLYP while unwanted side effects of this (reduced account of static correlation) are damped or eliminated by the PT2 term.

How to use B2-PLYP:

Or use the b2plypprep script to setp up the calculation.