[Wien] Meaning of EORB

[David Tompsett]

Dear Fabien,

Thanks again for the response. A few more questions/comments:

1) I think the expression for exchange is clearer in eq (4) of your PRB 75,
1155108 (2006). This is primarily because the occupation matrices are
written in spin diagonal space. One of the biggest challenges for me in this
field is that most expressions in the modern papers are written in the
rotationally invariant form (in Novak Phys. Stat. Sol. (B) 243, 563 (2006)
rotationally invariant in both spin- and m-space). This also means that the
Gaunt's numbers in the appendix of Czyzyk and Sawatzky PRB 49,14211 (1994)
which are a_mm' and b_mm' become subsumed into just a(m1,m3,m4,m2) in the
rotationally invariant form. So it is quite hard to relate papers to each
other.

2) The Ylvisaker paper is very good at explaining DFT+U, but does not not
really compare to hybrids which is what I'm trying to get a grip on.

3) Essentially from this reading I think that exchange is just the second
term of eq. (2) in Shick? This is the interaction that is included in exact
exchange and Hybrids. The confusing thing is that exact exchange and Hybrids
are referred to as HF (in emails above), but HF itself does include Coulomb
interactions, just not in a way screened by correlations as in DFT+U i.e. HF
includes the bare Hartree term. In exact exchange the direct Coulomb
interaction is ignored, while in hybrids it is left to the local functional.

Cheers,
David.

On Fri, Aug 12, 2011 at 7:19 PM, <tran at theochem.tuwien.ac.at> wrote:

>
> > 1) The relations that Fabian sent to me indicate that EORB includes both
> the
> > interactions and the double counting correction. So Laurence if that is
> > right then EORB is more than just the double counting term.
>
> Yes.
>
> > 2) Interestingly, Novak et al. [Phys. Stat. Sol. (B) 243, 563 (2006)]
> > indicates that for excact exchange both exchange and correlations for
> > E^LDA_xc are subtracted off by the double counting term. This seems in
> > contradiction to the original PBE0 paper (Perdew et al. J Chem Phys 105,
> > 9982(1996)) where only exchange is subtracted off.
>
> Actually, in Novak et al. this is about exact exchange (i.e., HF)
> for correlated electrons (no correlation). This is why E^LDA_xc for the
> correlated electrons is substracted such that only E^HF_x is applied to
> the correlated electrons.
>
> With PBE0, a fraction of E^PBE_x is replaced by E^HF_x and the correlation
> is E^PBE_c.
>
> > 3) Also, it is hard to see difference between the interaction in the two
> > methods because they are very similar in form. If I look at equation (2)
> in
> > both Shick and Novak they look very similar. Is the difference that HF
> only
> > includes direct exchange, while DFT+U includes spin-dependent exchange?
> > By looking at Czyzyk and Sawatzky PRB 49,14211 (1994) relations are given
> in
> > the appendix between the direct U_mm' and indirect J_mm' interactions and
> > the Slater integrals. Does HF just ignore J_mm'?
> >
> > The only other place that they are very different is in the double
> counting
> > terms. But as you say these can matter a lot.
>
> In LDA+U, exchange and Coulomb (correlation) and exchange are improved
> for the correlated electrons, while only exchange in
> Novak et al. is improved (furthermore correlation is zero).
>
> Another difference is that Coulomb and exchange are screened in LDA+U,
> while exchange is unscreened in Novak et al.
>
> I recommend you this paper abot LDA+U:
> Ylvisaker et al., PRB 79, 035103 (2009).
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