[Wien] Meaning of EORB

[tran at theochem.tuwien.ac.at]

> 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).