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,

  0.8S + 0.72B88 + 0.2HF + 0.19VWN(V) + 0.81LYP (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)) + 0.25HF + 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)) + 0.1HF + 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:

  $$\bf r \sum_{{ij}}^{} {\frac{{ \phi_i({\bf r}) \phi_j({\bf r}) }}{{ \rho({\bf r}) }}} \left(\vphantom{ - \int d {\bf r'} \frac{\phi_i({\bf r'}) \phi_j(... ...r-r'}\vert } + \langle \phi_i \vert v_x- v_x^{NL} \vert \phi_j \rangle }\right. \int \bf r' {\frac{{\phi_i({\bf r'}) \phi_j({\bf r'})}}{{\vert{\bf r-r'}\vert }}} \left.\vphantom{ - \int d {\bf r'} \frac{\phi_i({\bf r'}) \phi_j(... ...r-r'}\vert } + \langle \phi_i \vert v_x- v_x^{NL} \vert \phi_j \rangle }\right)$$ (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, s₆(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_xE_X(HF) (6.7) + (1 - a_c)E_C(GGA) + a_cE_C(KS - PT2),

where E_X(GGA) is the energy of a conventional exchange functional and E_C(GGA) is the energy of a correlation functional. E_X(HF) is the Hartree-Fock exchange of the occupied Kohn-Sham orbitals and E_C(KS - PT2) is a Møller-Plesset like perturbation correction term based on the KS orbitals:

$${\frac{{1}}{{2}}} \sum_{{ia}}^{} \sum_{{jb}}^{} {\frac{{(ia\vert jb)[(ia\vert jb)-(ib\vert ja)]}}{{e_i+e_j-e_a-e_b}}}$$ (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 E_C(KS - PT2) is evaluated afterwards and added to the total energy.

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.