Exchange-Correlation Functionals Available

The following exchange-correlation functionals are available:

- LDAs: S-VWN, PWLDA
- GGAs: B-VWN, B-LYP, B-P, PBE
- MGGA: TPSS
- hybrid functionals: BH-LYP, B3-LYP, PBE0, TPSSh
- double-hybrid functional: B2-PLYP (energy calculations only!)

In detail, the functional library consists of:

- The Slater-Dirac exchange functional only (S)
[43,44].
- The 1980 correlation functional (functional V in the paper) of
Vosko, Wilk, and Nusair only (VWN) [45].
- A combination of the Slater-Dirac exchange and Vosko, Wilk,
and Nusair 1980 (functional V) correlation functionals (S-VWN)
[43,44,45].
- The S-VWN functional with VWN functional III in the paper. This
is the same functional form as available in the Gaussian program
[43,44,45].
- A combination of the Slater-Dirac exchange and Perdew-Wang
(1992) correlation functionals
[43,44,46].
- A combination of the Slater-Dirac exchange and Becke's 1988
exchange functionals (B88)
[43,44,47].
- Lee, Yang, and Parr's correlation functional (LYP)
[48].
- The B-LYP exchange-correlation functional (B88 exchange and LYP
correlation functionals)
[43,44,47,48].
- The B-VWN exchange-correlation functional (B88 exchange and VWN
(V) correlation functionals)
[43,44,47,45].
- The B-P86 exchange-correlation functional (B88 exchange, VWN(V)
and

Perdew's 1986 correlation functionals) [43,44,47,45,49]. - The Perdew, Burke, and Ernzerhof (PBE) exchange-correlation
functional
[43,44,46,50].
- The Tao, Perdew, Staroverov, and Scuseria functional (Slater-Dirac, TPSS exchange and
Perdew-Wang (1992) and TPSS correlation functionals)
[43,44,46,51].

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 BH-LYP exchange-correlation functional (Becke's
half-and-half exchange in a combination with the LYP correlation
functional) [43,44,47,48,52].
- The B3-LYP exchange-correlation functional (Becke's
three-parameter

functional) with the form,0.8 *S*+ 0.72*B*88 + 0.2*HF*+ 0.19*VWN*(*V*) + 0.81*LYP*(6.3)

where HF denotes the Hartree-Fock exchange [43,44,47,48,53]. - The B3-LYP exchange-correlation functional with VWN functional
V in the paper. This is the same functional form as available in the
Gaussian program.
- The 1996 hybrid functional of Perdew, Burke, and Ernzerhof,
with the form,
0.75( *S*+*PBE*(*X*)) + 0.25*HF*+*PW*+*PBE*(*C*)(6.4)

where PBE(X) and PBE(C) are the Perdew-Burke-Ernzerhof exchange and correlation functionals and PW is the Perdew-Wang correlation functional [43,44,46,50,54]. - The TPSSH exchange-correlation functional of Staroverov, Scuseria, Tao and Perdew with the
form,
0.9( *S*+*TPSS*(*X*)) + 0.1*HF*+*PW*+*TPSS*(*C*)(6.5)

where HF denotes the Hartree-Fock exchange [43,44,46,51,55]. - The Localized Hartree-Fock method (LHF) to obtain an effective
exact exchange Kohn-Sham potential [56,57] (module
`dscf`only). The LHF potential is:*v*_{x}() =*n*_{s}-*d*+ 〈*φ*_{i}|*v*_{x}-*v*_{x}^{NL}|*φ*_{j}〉(6.6)

where*φ*are Kohn-Sham occupied molecular orbitals,*n*_{s}= 2 for closed-shell systems and*v*_{x}^{NL}is the non-local Hartree-Fock operator.

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

B2-PLYP is a so-called double-hybrid density functional (DHDF)[58] 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. Note the following options/restrictions in the present version of this method:

- single point calculations only (computed with the DSCF and RIMP2/RICC2 modules).
- UKS treatment for open-shell cases.
- can be combined with
*resolution-of-identity*approximation for the SCF step (RI-JK option). - can be combined with the dispersion correction (DFT-D method,
*s*_{6}(B2-PLYP)=0.55).

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[59,60]. The mixing is described by two empirical parameters
*a*_{x} and *a*_{c} in the following manner:

E_{XC}(DHDF) = (1 - a_{x})E_{X}(GGA) + a_{x}E_{X}(HF) |
(6.7) | ||

+ (1 - a_{c})E_{C}(GGA) + a_{c}E_{C}(KS - PT2), |

where

E_{C}(KS - PT2) = . |
(6.8) |

The method is self-consistent only with respect to the first three terms in Eq. 6.7, i.e., first a SCF using a conventional hybrid-GGA is performed first. Based on these orbitals

For B2-PLYP, B88 exchange[47] and LYP correlation[48] are used with the
parameters *a*_{x} = 0.53 and *a*_{c} = 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.

1.2

How to use B2-PLYP:

- during preparation of your input with DEFINE select
`b2-plyp`in the DFT menu. - carry out a DSCF run. Prepare and run a RI-MP2 calculation with either RIMP2 or RICC2 program modules.
- the RI-MP2 program directly prints the B2PLYP energy if this functional has been chosen before
- if you use the RICC2 program the scaled (
*a*_{c}= 0.27) second-order correlation energy. must be added manually to the SCF-energy. - in order to maintain consistency of the PT2 and GGA correlation parts, it is recommend not to apply the frozen-core approximation in the PT2 treatment.