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