I gave up on the wB97X-2(TQZ) functional, which would have an exact decay rate of -1/r, re: my previous question and decided to use DSD-PBEPBE-D3BJ, which is available on vanilla Psi4. I then looked up the decay rate of said functional- pure PBE seems to have exponential decay but there is absolutely no mention of the decay rate of the hybrid PBE0, let alone the double hybrid DSD-PBEPBE-D3BJ.

My question now follows- is there a closed-form, or even an approximate-form, decay rate of DSD-PBEPBE(-D3BJ) in the literature?

  • 4
    $\begingroup$ Decay rate of what quantity? There is no such thing as the "decay rate of a functional". $\endgroup$
    – wzkchem5
    Dec 17, 2022 at 14:53

1 Answer 1


I guess the OP is referring to the asymptotic potential which is -1/r with the optimized effective potential (OEP) but which decays exponentially with density functionals as well as Hartree-Fock exchange in the usual orbital-dependent formulation. I doubt wB97X-2(TQZ) would have -1/r asymptotic form; no reference is given for the claim. In any case, PBE is a semilocal functional and its potential decays exponentially. The usual implementation of PBE0 has the same feature since the 25% fraction of Hartree-Fock exchange does not change the picture. An implementation using the true OEP or approximations thereof like the Krieger-Li-Iafrate (KLI) method would yield an asymptotic potential of -0.25/r for PBE0.

  • $\begingroup$ The -D3BJ part might attenuate it though, by the very definition of what "dispersion correction" must do. $\endgroup$ Dec 23, 2022 at 6:36
  • $\begingroup$ For the wB97X part, the w is an LRC paramater s.t. the potential becomes attenuated to the exact -1/r. $\endgroup$ Dec 23, 2022 at 6:37
  • $\begingroup$ Nope, the omega is the range-separation parameter and wB97X has 100% exact exchange interaction in the long range. However, this still leads to an exponentially decaying potential in the long range, unless you use OEP (or its approximations like KLI) and I do not know any publicly available implementation thereof... $\endgroup$ Dec 23, 2022 at 7:47
  • $\begingroup$ The Q-Chem manual provides a mathematical proof (or sketch thereof) of the -1/r rate for LRC functionals with tuned ω. $\endgroup$ Dec 23, 2022 at 9:23
  • $\begingroup$ A range-separated exchange functional with 100% long-range exchange behaves asymptotically like -1/r; however, this is not the potential that enters the Kohn-Sham equation, which decays exponentially. $\endgroup$ Dec 31, 2022 at 15:40

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