trial wavefunctions for twistaveraging

General discussion of the Cambridge quantum Monte Carlo code CASINO; how to install and setup; how to use it; what it does; applications.
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Katharina Doblhoff
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Joined: Tue Jun 17, 2014 6:50 am

trial wavefunctions for twistaveraging

Post by Katharina Doblhoff »

Can anybody comment on why it is proposed in the CASINO manual to use scf calculations (including only k-points necessary for k-point unfolding) to generate the trial wavefunction for a particular twist, while the qmcpack manual suggests using non-selfconsistent calculations based on a density that is a priori calculated using a converged k-point grid?

Can one say that using the latter method would bring the wavefunction closer to the "true", infinite wavefunction, thus restricting the nodes to nodes which may be closer to the correct nodes of the infinite system, while the former will bring the wavefunction nodes closer to the "true" nodes of the infinitely repeated supercell and thus the system which is actually described in QMC?

Has anybody ever tested which method actually gives the better results or are there any arguments why one or the other HAS to be better?
Thank you.
Neil Drummond
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Location: Lancaster
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Re: trial wavefunctions for twistaveraging

Post by Neil Drummond »

Dear Katharina,

Within Hartree-Fock theory, using orbitals obtained in a k-point grid finer than the one corresponding to your cell would be suboptimal (the orbitals would not be optimal for the cell being studied). It is obviously unclear whether this continues to apply when you look at orbitals generated by other methods such as DFT or when you include correlation in the wave function.

In general the dependence of the DMC energy on the orbitals is very weak: changing functionals, etc., doesn't change DMC energies by a statistically significant margin.

Using a fine k-point mesh in the orbital-generation calculation could be helpful if there are difficulties getting the HF or DFT code to converge with small numbers of k points.

Best wishes,

Neil.
Katharina Doblhoff
Posts: 84
Joined: Tue Jun 17, 2014 6:50 am

Re: trial wavefunctions for twistaveraging

Post by Katharina Doblhoff »

Hi Neil,

I totally agree that a wavefunction from a converged grid would be suboptimal for the cell. That is what I tried to say with "true nodes of the infinitely repeated supercell and thus the system which is actually described in QMC" in the case of an scf calculation. However, I would not think that this means automatically that the twist averaged or supercell energy resulting from such an scf calculation will be better: Let us assume that DFT is correct. Then the wavefunction at a certain ks has the correct nodes only if a converged k-point grid is used. So inputting a wavefunction from a converged k-point grid will give nodes which represent the true (infinite) system better. Restraining the DMC nodes to these nodes may (or may just as well not) give better or worse energies.

It is of course true that we hope not to depend on the trial wavefunction, but quoting a manual by Luke Shulenberger:
As the goal of the calculation is to produce an extremely accurate DFT wavefunction,
ecutwfc and the k-points should be converged to a very high degree. ... [QMCPACK] will need wavefunctions
at particular k-points to make this [a supercell calculation] work. This process can be automated using setup-qmc.pl ... Instead of a scf calculation, the calculation
will now be nscf so as to avoid changing the charge density from the previously converged
calculation.
So at least there seem to be two doctrines around and I wondered if anybody actually ever performed an actual test.
Neil Drummond
Posts: 113
Joined: Fri May 31, 2013 10:42 am
Location: Lancaster
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Re: trial wavefunctions for twistaveraging

Post by Neil Drummond »

Dear Katharina,

I'm not quite clear whether you are suggesting that the DFT orbitals from the converged k-point mesh will be better for the finite supercell (i.e., will give a lower DMC energy), in which case everyone would agree they should be used, or whether your point is that they will make the energy in the finite supercell closer to the infinite-system energy.

Systematic finite-size errors in the total energy are usually negative. So one can "reduce" finite-size errors by using a poorer trial wave function at small system size. I'm not sure this is a good idea, however. We generally just try to make the trial wave function as good as we can at each system size, as this seems like the best strategy for enabling the elimination of finite-size errors by extrapolation, etc.

Once again, I emphasise that the dependence of the DMC energy on the orbitals is very weak.

Best wishes,

Neil.
Katharina Doblhoff
Posts: 84
Joined: Tue Jun 17, 2014 6:50 am

Re: trial wavefunctions for twistaveraging

Post by Katharina Doblhoff »

Dear Neil,

thank you for your reply. I can totally follow your logic and I do not want to suggest that it should be done differently. All I am saying is that it is apparently sometimes done differently, so I am trying to find a good explanation why. The reason I put forward seems to be the best explanation for me to do so. The argument that this may increase the energy at a certain twist compared to the lowest energy obtainable does not seem 100% fair to me: we actually do not want to describe an infinitely repeated supercell (for which the energy may lie lower), but a real infinite system, which will actually lie higher than the Gamma point energy. However, I do agree that our supercell hamiltonian is not capable to capture this, so you may well be right in saying that it sounds like trying to cancel errors with errors.

All the best,
Katharina
Neil Drummond
Posts: 113
Joined: Fri May 31, 2013 10:42 am
Location: Lancaster
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Re: trial wavefunctions for twistaveraging

Post by Neil Drummond »

Thanks for looking into this - please let us know if you reach any definite conclusions!

Best wishes,

Neil.
Luke_Shulenberger
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Joined: Thu Jun 05, 2014 11:31 am

Re: trial wavefunctions for twistaveraging

Post by Luke_Shulenberger »

I will be interested to see how the results of the calculations go. Although I have not completed formal testing on this point, my thinking is actually a bit different than what is suggested. I'm not certain whether using trial wavefunctions from a converged DFT density will produce the lowest QMC energy for a given size supercell. I was more interested in convergence and finite size corrections.

I have some questions regarding finite size convergence. In the Chiesa et al paper, they argue that the Jastrow factor and structure factor (which give all the necessary information regarding the two body finite size corrections) are insensitive to the size of the supercell considered (in their case, the supercell size only changes the k values at which it can be computed). My thought was that changing the single particle orbitals (and potentially the fixed node error) when moving from one supercell size to the next would break this assumption and change the form of the two body finite size correction for different supercells. While not a problem per se, it seems that this might change the estimation of values at the thermodynamic limit from one cell size to the next.

I have a similar worry about the MPC corrections as they would be using a different periodic density for different cell sizes if an unconverged DFT density were used.

Finally, as a matter of convenience, assuming that the energy difference between using orbitals from a converged and unconverged DFT calculation is minuscule, it would be nice to be able to generate all of your trial wavefunctions for different supercell sizes and boundary conditions in a single DFT calculation rather than having to repeat this step for every individual calculation.
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