N2 DMC energy is better then exact

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N2 DMC energy is better then exact

Postby Vladimir_Konjkov » Fri Mar 09, 2018 12:12 pm

Hello CASINO developers

I'm using the QChem program to get trial WFN

Qchem has “valence” active space methods VOD which allows you to get WFN and T2-amplitudes to construct CASINO input
https://www.q-chem.com/qchem-website/ma ... ctive.html

When I create VOD single determinant WFN for N2 molecule in cc-pVQZ basis I've got very strange energy:
-109.541614957491 +/- 0.001054789189 while the exact value -109.5421 from PHYSICAL REVIEW E 86, 036703 (2012) Framework for constructing generic Jastrow correlation factors

VMC energy has ordinary value -109.468(3) ; var = 3.40(9)

Correlation length is also as usual
N_corr: 15.010449449672 +/- 2.159204234545

I thought that the DMC is variational method and the lower the energy the better however, this result raises suspicions.
How can I make a check to find an error in the calculations?
All input files and results in the attachment.

Best Vladimir.
Attachments
tmax_2_1024_1__SD.tgz
N2 input
(1010.36 KiB) Downloaded 235 times
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Re: N2 DMC energy is better then exact

Postby Ryan_Hunt » Fri Mar 09, 2018 12:33 pm

Vladimir,

I'm not sure what you are asking here. You're ground state energy is: -109.542(1) a.u., whilst the exact value (from Ref. 38 of the quoted PRE article) is -109.5421 a.u. These two are consistent with each other (at the level of statistical significance you are working), and your VMC energy is slightly higher (-109.468(3) a.u.), which is to be expected.

If you do go ahead and reduce your DMC error bar to investigate further, be aware that population control bias and time step bias may muddy your comparison to the exact energy (in principle you should extrapolate data from two or more DMC simulations to remove these biases - see the paper here https://link.aps.org/doi/10.1103/PhysRevE.83.066706 for some best practices). Your attached input files relate to a DMC calculation at /one/ time step and at /one/ population size, therefore you shouldn't necessarily expect to reproduce the exact energy very well.

Regards,
Ryan.
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Re: N2 DMC energy is better then exact

Postby Katharina Doblhoff » Fri Mar 09, 2018 12:45 pm

Also, you can have a fixed node error. If you are not working with the full wave function that gave the reference result in the first place (and I think that the value you cite was obtained by CCSDT), DMC may well also give an energy that is higher than the value from the "standard" quantum chemical approach.
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Re: N2 DMC energy is better then exact

Postby Vladimir_Konjkov » Fri Mar 09, 2018 1:14 pm

Ryan_Hunt wrote:Vladimir,

I'm not sure what you are asking here. You're ground state energy is: -109.542(1) a.u., whilst the exact value (from Ref. 38 of the quoted PRE article) is -109.5421 a.u. These two are consistent with each other (at the level of statistical significance you are working), and your VMC energy is slightly higher (-109.468(3) a.u.), which is to be expected.

If you do go ahead and reduce your DMC error bar to investigate further, be aware that population control bias and time step bias may muddy your comparison to the exact energy (in principle you should extrapolate data from two or more DMC simulations to remove these biases - see the paper here https://link.aps.org/doi/10.1103/PhysRevE.83.066706 for some best practices). Your attached input files relate to a DMC calculation at /one/ time step and at /one/ population size, therefore you shouldn't necessarily expect to reproduce the exact energy very well.

Regards,
Ryan.


Hello Rian.

Initially, I was conducting calculations with 5 determinant WFN obtained from QChem VOD and of course I did calculations for different time steps and populations:
dtdmc: 0.0034; configurations = 1024; E = -109.545857117528 +/- 0.000705444506
dtdmc: 0.0011; configurations = 1024; E = -109.547862314555 +/- 0.000929138775
dtdmc: 0.0034; configurations = 3092; E = -109.546467643406 +/- 0.000787726772
From this we see that even linear extrapolation to zero timestep and infinity population leads to unreasonable, better then exact results.

I can perform there calculations for single determinant but while the best ever SD DMC is -109.5060(7), my SD DMC results is -109.541(1) and I don't think that time step and population biases are so great in my case, to disregard this difference.
Time step and population bias for 5 determinant WFN is order of 0.01 Ha, so I expect they should be comparable in the SD case.

Best Vladimir.
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Re: N2 DMC energy is better then exact

Postby Vladimir_Konjkov » Fri Mar 09, 2018 1:28 pm

Katharina Doblhoff wrote:Also, you can have a fixed node error. If you are not working with the full wave function that gave the reference result in the first place (and I think that the value you cite was obtained by CCSDT), DMC may well also give an energy that is higher than the value from the "standard" quantum chemical approach.


Hello Katharina

-109.5421 Ha energy for N2 is obtained from https://www.tandfonline.com/doi/abs/10. ... 2331339323 pretends to be exact value for non-relativistic electronic energy with mHa accuracy.

And yet my result seems suspicious to me, although I can immediately ask for a Nobel Prize for the best in the world one determinant approximation of the electronic state of a nitrogen molecule. :lol:

Best Vladimir.
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Re: N2 DMC energy is better then exact

Postby Ryan_Hunt » Fri Mar 09, 2018 1:52 pm

Vladimir,

Thanks for your reply. I'm having a hard time chasing up what exactly has gone into the "exact energy" quoted in the earlier referenced PRE. For example, can you find where the experimental atomization energy of N_2 used to form the electronic ground state energy actually came from? (I can't :s).

Cheers,
Ryan.
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Re: N2 DMC energy is better then exact

Postby Pablo_Lopez_Rios » Fri Mar 09, 2018 2:08 pm

Hi Ryan,

If I am not mistaken the energy quoted in the PRE is just the entry corresponding to the row labelled N2 and the column labeled E_e in Table 2 of https://doi.org/10.1080/00268970512331339323 .

Best,
Pablo
Hey there! I am using CASINO.
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Re: N2 DMC energy is better then exact

Postby Ryan_Hunt » Fri Mar 09, 2018 2:16 pm

Pablo,

That's right - however, this value was pieced together from various other places. Some portions of it (vibrational zero point energy, scalar relativistic corrections) are from different theoretical calculations (which presumably have some uncertainties) and one portion of it (the atomisation energy) is taken from experiment. I can't find which experiment. I can track down the other contributions, but not the experimental value for the atomisation energy.

Cheers,
Ryan.
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Re: N2 DMC energy is better then exact

Postby Ryan_Hunt » Fri Mar 09, 2018 2:19 pm

(all this to say, in this case, I think we might also want to scrutinise the "exact" result as well as we do the DMC result)
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Re: N2 DMC energy is better then exact

Postby Vladimir_Konjkov » Fri Mar 09, 2018 2:45 pm

Ryan_Hunt wrote:Vladimir,

Thanks for your reply. I'm having a hard time chasing up what exactly has gone into the "exact energy" quoted in the earlier referenced PRE. For example, can you find where the experimental atomization energy of N_2 used to form the electronic ground state energy actually came from? (I can't :s).

Cheers,
Ryan.


Ryan

Experimental atomization energy for N2 is taken from https://doi.org/10.1063/1.460205
where it approximated by G1 and G2 theory
G1 = 224.8; G2 = 223.8; Exp = 225.1 difference is order of 0.001 Ha, it's a chemical accuracy.

So taking into account Exp value 225.1 , ZPVE correction 0.0048 Ha and scalar relativistic correction -0.0002 Ha and energy of free N atom -54.5892 Ha
We get
-54.5892 * 2 - 225.1 / 627.509 - 0.0048 - 0.0002 = -109.542(1) with a chemical accuracy as error of Experimental atomization energy propagates through all our equation.

Best, Vladimir
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