finite size corrections in 2D

[Dario_Alfe]

What do we mean that in 2D finite size corrections are purely kinetic ? Is it really so, or casino can only deal with dKE in 2D systems ?

[Neil Drummond]

Dear Dario,

Formulae for finite-size corrections in 2D - to both the interaction energy and the kinetic energy - can be found in N.D. Drummond et al., Phys. Rev. B 78, 125106 (2008). The leading-order correction to both the interaction energy and the kinetic energy is positive and scales as N^(-1/4) (i.e., the error in the energy per electron scales as N^(-5/4)).

I suggest removing finite-size errors by extrapolation of twist-averaged energies using the N^(-5/4) scaling.

Best wishes,

Neil.

[Dario_Alfe]

Dear Neil, thanks for this. Casino only computes the KE correction in 2D, does it mean that the formulae for the XC correction are not implemented ?
Using scaling law is difficult if you can only afford a very limited number of different sizes.

Dario

[Neil Drummond]

Dear Dario,

The relevant bit of code predates the finite-size paper by two years, suggesting that the finite-size correction to the exchange-correlation energy wasn't implemented in 2D because the expression for the correction wasn't known at the time. If you have accumulated the static structure factor then you can use Eq. (59) of the finite-size paper to obtain the correction to the exchange-correlation energy. I'll update CASINO to return the XC correction as well. Thanks for drawing attention to the issue.

Note that the corrections don't work as well in 2D as they do in 3D: see the electron-gas results in the finite-size paper. I still think that extrapolation is safer than relying on the corrections to a single system size.

Best wishes,

Neil.

[Dario_Alfe]

Dear Neil,

I see. I do have an accumulated structure factor, but I am not quite sure how to use it in combination to Eq. 59.
Of course, one way to addressing the issue could be to pretend to have a 3D system and use the 3D correction, which indeed seems to work quite well.

I am always a bit wary about extrapolations, especially if I don't have an extensive set of results to check that they are really following the supposed extrapolating function.

Best,

Dario

[Mike Towler]

The relevant bit of code predates the finite-size paper by two years, suggesting that the finite-size correction to the exchange-correlation energy wasn't implemented in 2D because the expression for the correction wasn't known at the time.

That sounds about right. According to the DIARY file my original implementation of finite size corrections to the XC energy from the structure factor was in 2.1.192 from 2006, and I think we just hadn't bothered to do the 2D maths since people were mainly interested in 3D systems (or more likely we just weren't as clever as Neil ). I remember being miffed for about 6 years after I wrote the 2D Ewald summation routines in 1999 that nobody ever did any 2D calculations so it was probably a hangover from that (actually now I come to think about, has anyone every used the 1D 'polymer' routines?).

[Mike Towler]

I'll close this thread with a note that the current beta distribution has now been updated with a couple of relevant finite size correction patches from Neil, which do the following:

--[v2.13.348]---
* Modified the finite-size correction scheme used in CASINO to be more like that
proposed in PRB 78, 125106 (2008). Added a subroutine to evaluate the
constants C_HF, C_3D and C_2D that appear in that paper. The old KE correction
is still evaluated, but is now just for comparing against. The XC correction
in 2D can now be evaluated.
-- Neil Drummond, 2014-05-11

---[v2.13.349]---
* Modified finsize utility to work with the previous patch.
-- Neil Drummond, 2014-05-11

Reduced the verbosity of the calculation of C.