Optimization of pairing orbitals for symmetric e-h bilayer.

[Sharma_Omprakash]

Dear CASINO users,
I am optimizing the pairing trial wave function for symmetric electron-hole bilayer at rs=5 d=1 a.u.. I have used the Gaussian pairing orbital and constrain the parameters so that they correspond to the Gaussian expansion of an exponential (sec. 7.4.8 of manual) in which case Rex is the only optimizable parameter. I have have optimized the Rex in two different methods first by emin and then varmin.
During the the optimization what i faced was as follows.

In case of energy minimization, the optimized energy decreases with opt_cycles and the rex decreases. The energies and Rex values are shown below.

Optimization with emin.

File: ./out
Block averages recomputed for current run.
Corrected for serial correlation using correlation time method.
Energies in au/particle, sample variances in au/particle^2.

VMC #1: E = -0.18769(3) ; var = 1.43(3) (correlation.out.0)
VMC #2: E = -0.18807(4) ; var = 1.68(2) (correlation.out.1)
VMC #3: E = -0.18814(4) ; var = 2.03(2) (correlation.out.2)
VMC #4: E = -0.18822(4) ; var = 2.09(2) (correlation.out.3)
VMC #5: E = -0.18818(5) ; var = 2.54(3) (correlation.out.4)
VMC #6: E = -0.18827(4) ; var = 2.30(2) (correlation.out.5)
VMC #7: E = -0.18829(4) ; var = 2.31(4) (correlation.out.6)
VMC #8: E = -0.18819(5) ; var = 2.38(3) (correlation.out.7)
VMC #9: E = -0.18798(4) ; var = 2.33(2) (correlation.out.8)
VMC #10: E = -0.18823(5) ; var = 2.60(2) (correlation.out.9)
VMC #11: E = -0.18820(5) ; var = 2.72(2) (correlation.out.10)
VMC #12: E = -0.18827(5) ; var = 2.91(2) (correlation.out.11)

grep Rex correlation.out.* ----------------------------------------------

correlation.out.0: 6.3557310791994013 1 ! Rex
correlation.out.1: 5.3320244393576504 1 ! Rex
correlation.out.2: 4.8231850071336186 1 ! Rex
correlation.out.3: 4.7702982867643282 1 ! Rex
correlation.out.4: 4.4891996579918194 1 ! Rex
correlation.out.5: 4.6281052261050277 1 ! Rex
correlation.out.6: 4.6281052261050277 1 ! Rex
correlation.out.7: 4.5830108731240085 1 ! Rex
correlation.out.8: 4.5847864490511760 1 ! Rex
correlation.out.9: 4.2093825037300805 1 ! Rex
correlation.out.10: 4.1290425213591444 1 ! Rex
correlation.out.11: 4.0337631705225609 1 ! Rex

***
On the other hand, if I choose varmin for optimization the optimized energies are slightly large(-0.18753(3)) than the emin (-0.18827(5)) but the values of Rex increase (See the output below).


Optimization with varmin.

File: ./out
Block averages recomputed for current run.
Corrected for serial correlation using correlation time method.
Energies in au/particle, sample variances in au/particle^2.

VMC #1: E = -0.18827(5) ; var = 2.91(2) (correlation.out.0)
VMC #2: E = -0.18511(4) ; var = 1.70(2) (correlation.out.1)
VMC #3: E = -0.18443(3) ; var = 0.81(1) (correlation.out.2)
VMC #4: E = -0.18548(3) ; var = 0.75(1) (correlation.out.3)
VMC #5: E = -0.18508(3) ; var = 0.626(8) (correlation.out.4)
VMC #6: E = -0.18491(3) ; var = 0.68(1) (correlation.out.5)
VMC #1: E = -0.18452(4) ; var = 0.54(2) (correlation.out.0)
VMC #2: E = -0.18391(4) ; var = 0.52(1) (correlation.out.1)
VMC #3: E = -0.18390(4) ; var = 0.52(1) (correlation.out.2)
VMC #4: E = -0.18418(4) ; var = 0.50(1) (correlation.out.3)
VMC #5: E = -0.18430(4) ; var = 0.49(1) (correlation.out.4)
VMC #6: E = -0.18409(4) ; var = 0.50(1) (correlation.out.5)
VMC #7: E = -0.18445(4) ; var = 0.468(8) (correlation.out.6)
VMC #8: E = -0.18459(4) ; var = 0.463(7) (correlation.out.7)
VMC #9: E = -0.18439(4) ; var = 0.454(6) (correlation.out.8)
VMC #10: E = -0.18430(4) ; var = 0.428(5) (correlation.out.9)
VMC #11: E = -0.18480(3) ; var = 0.370(5) (correlation.out.10)
VMC #1: E = -0.18483(4) ; var = 0.373(6) (correlation.out.0)
VMC #2: E = -0.18547(4) ; var = 0.339(5) (correlation.out.1)
VMC #3: E = -0.18585(3) ; var = 0.303(5) (correlation.out.2)
VMC #4: E = -0.18596(3) ; var = 0.297(4) (correlation.out.3)
VMC #5: E = -0.18603(3) ; var = 0.296(7) (correlation.out.4)
VMC #6: E = -0.18702(3) ; var = 0.265(4) (correlation.out.5)
VMC #7: E = -0.18760(3) ; var = 0.247(3) (correlation.out.6)
VMC #8: E = -0.18730(3) ; var = 0.251(3) (correlation.out.7)
VMC #9: E = -0.18744(3) ; var = 0.254(4) (correlation.out.8)
VMC #10: E = -0.18767(3) ; var = 0.248(3) (correlation.out.9)
VMC #11: E = -0.18734(3) ; var = 0.255(3) (correlation.out.10)
VMC #1: E = -0.18742(3) ; var = 0.258(4) (correlation.out.0)
VMC #2: E = -0.18760(3) ; var = 0.249(3) (correlation.out.1)
VMC #3: E = -0.18749(3) ; var = 0.253(3) (correlation.out.2)
VMC #4: E = -0.18751(3) ; var = 0.254(3) (correlation.out.3)
VMC #5: E = -0.18763(3) ; var = 0.251(3) (correlation.out.4)
VMC #6: E = -0.18755(3) ; var = 0.249(3) (correlation.out.5)
VMC #7: E = -0.18759(3) ; var = 0.250(3) (correlation.out.6)
VMC #8: E = -0.18758(3) ; var = 0.254(3) (correlation.out.7)
VMC #9: E = -0.18755(3) ; var = 0.256(6) (correlation.out.8)
VMC #10: E = -0.18753(3) ; var = 0.252(3) (correlation.out.9)
VMC #11: E = -0.18753(3) ; var = 0.250(2) (correlation.out.10)

grep Rex correlation.out.* ---------------------------------------------------

correlation.out.0: 4.0337631705225609 1 ! Rex
correlation.out.1: 5.7285131334489776 1 ! Rex
correlation.out.2: 8.8608508537964958 1 ! Rex
correlation.out.3: 10.088126688522550 1 ! Rex
correlation.out.4: 11.806072364232282 1 ! Rex
correlation.out.5: 12.677022273652353 1 ! Rex
correlation.out.0: 12.677022273652353 1 ! Rex
correlation.out.1: 13.472736135809793 1 ! Rex
correlation.out.2: 14.854384155485882 1 ! Rex
correlation.out.3: 16.683277549782950 1 ! Rex
correlation.out.4: 18.097064504152261 1 ! Rex
correlation.out.5: 18.975776508010462 1 ! Rex
correlation.out.0: 18.975776508010462 1 ! Rex
correlation.out.1: 21.631288089605292 1 ! Rex
correlation.out.2: 21.651508323264316 1 ! Rex
correlation.out.3: 22.921052465899262 1 ! Rex
correlation.out.4: 22.481807040098467 1 ! Rex
correlation.out.5: 23.433834385248467 1 ! Rex
correlation.out.6: 23.697793963844738 1 ! Rex
correlation.out.7: 24.196984784580408 1 ! Rex
correlation.out.8: 24.679199233589529 1 ! Rex
correlation.out.9: 27.418183809728124 1 ! Rex
correlation.out.10: 33.436810805890779 1 ! Rex
correlation.out.0: 33.436810805890779 1 ! Rex
correlation.out.1: 35.042572472044107 1 ! Rex
correlation.out.2: 40.899224225436384 1 ! Rex
correlation.out.3: 42.148064764855022 1 ! Rex
correlation.out.4: 43.397612674552072 1 ! Rex
correlation.out.5: 48.751054456604088 1 ! Rex
correlation.out.6: 54.442442521343445 1 ! Rex
correlation.out.7: 54.477420773760585 1 ! Rex
correlation.out.8: 54.406870466727035 1 ! Rex
correlation.out.9: 56.031027343795550 1 ! Rex
correlation.out.10: 56.431813613262811 1 ! Rex
correlation.out.0: 56.431813613262811 1 ! Rex
correlation.out.1: 57.005053594854544 1 ! Rex
correlation.out.2: 57.119399470648332 1 ! Rex
correlation.out.3: 58.719938006950009 1 ! Rex
correlation.out.4: 59.930875064709952 1 ! Rex
correlation.out.5: 60.369287144632636 1 ! Rex
correlation.out.6: 60.933407185295607 1 ! Rex
correlation.out.7: 61.128374487010518 1 ! Rex
correlation.out.8: 59.999693708323960 1 ! Rex
correlation.out.9: 59.828315565750479 1 ! Rex
correlation.out.10: 60.917331716951942 1 ! Rex

You can see the variance in case of emin varies from 1.43(3) to 2.91(2) whareas in case of varmin it is much lower (as expected) but the value of Rex gradually increases from 4.0 to 60.9.
In short in case of emin, Rex decrease to around 4 while in case of varmin it increases from 4 to 60.
So, Please someone help me to understand this result and selection of appropriate optimization method.

Thanks and Regards,
Rajesh Sharma

[Sharma_Omprakash]

Hi CASINO User,
Please give me some suggestions.

Thanks and regards,
Rajesh Sharma

[Neil Drummond]

Dear Rajesh,

I would always trust energy minimisation rather than variance minimisation when optimising a parameter (such as Rex) that affects the nodal surface of the trial wave function.

Your emin calculation looks OK to me. The energies are bouncing about by rather more than the statistical error bars on the individual VMC calculations, but that is not unexpected since there is noise due to the reoptimisation of the parameters.

Best wishes,

Neil.

[Sharma_Omprakash]

Thank you for your reply.

The energies are bouncing about by rather more than the statistical error bars on the individual VMC calculations, but that is not unexpected since there is noise due to the reoptimisation of the parameters.

Please give me some idea how to avoid the noise due to reoptimization?

Thank you.
Rajesh

[Neil Drummond]

Dear Sharma,

It's inevitable, and is not a problem. If you were to perform repeated VMC calculations with exactly the same wave function then the VMC energies would bounce about by an amount given by the statistical error bars on the VMC energy. If you also perform a stochastic optimisation of the parameters in the wave function, the VMC energies will bounce about by a larger amount, due to the additional uncertainty in the wave function.

If you want the amount of "bouncing" to be smaller in absolute terms then you need to increase the number of configurations written out for use in the optimisation. You only need to do this if you feel that you need greater precision on the energy for your particular application.

Best wishes,

Neil.

[Sharma_Omprakash]

Thank you sir for your kind support.