The rij tests.

The general form of the integrals tested is .

  The analytical evaluation of this integral is given in analytical4.doc .htm.  The main code is in rijtest.wpj orijtest.zip and includes setup.for, random.for, hfuns.for, xnfind.for, and xnefin.for.  The code hfuns is the code used for determining the analytical answers.

The codes leading to orijtest.zip were found under \wsteve\Gauss\testing>testing.mdp  which is a power station ide that also contains most of the code discussed in Fmonte.doc .htm.   The e-e probablility of selection can be made very high.  For N=6 an e-e or pele of 0.9 produces an epsilon connection for the integrals tested.  Value from 0.01 through 0.7 appear to work well. 

            The main code contains the various parameters that are changed in the tests.           The initial testing on the normalization integral yields.

Method

electrons #

α

β

Analytical

configurations

Numerical estimate

Residual

BAR

3

1

1

0.503362

1048576

0.494202 +- 0.0064

-1.4

M 0.7

3

1

1

0.5033618

1048576

0.494123 +-0.006235

-1.5

M 0.7

2

1

1

2.7732

1048576

2.801653 +-0.01323

+2.2

M 0.7

3

1

4

1.105 E-2

1048576

1.098 E-2 +-0.0049 E-2

-1.4

 

The final testing involves calculating the normalization integral, the integral times the sum of 1/rI, times the sum 1/ri2, times the double sum of 1/rij, and finally times the double sum of 1/rij2.  These tests were run by selecting the first electron with respect to the nucleus, and the next with respect to the nucleus (30% probability) and also with respect to one of the electrons already selected (70% probability).

     NUP=6

     beta=4

     alpha=1

     CALL RSEED(123,451)

     CALL SETUP(NGUIDE,GCON)

C ** zero all sums

     DO IMC=1,NMC

C PICK FIRST ELECTRON BIASED AS RANDOM

       CALL XNFIND(X,WXT,1)

       DO I=2,NUP

C PICK ELECTRONS 2 TO NUP WITH RESPECT TO NUCLEUS AND ELECTRONS

         CALL XNEFIN(X,WNXT,I-1,I)

         wxt=wxt*wnxt

       ENDDO


 

In each table space the first number is the Monte-Carlo estimate of the integral,

The second is the Monte-Carlo error estimate.  The third is the analytical value of the integral.  The fourth is Res = (Monte-Carlo  Analytical)/(Monte Carlo error estimate).

NMC=    1048576

N    α   β

∫ Gaussian

∫G∑1/rI

∫G∑1/rI2

∫G∑1/rIJ

∫G∑1/rIJ2

2     1    4

0.4448+- 0.0016

0.44236  R  1.46

1.3765+- 0.0039

1.3720  R   1.13

3.3453+- 0.0071
 3.3423 R   0.43

 1.4585+-0.0035
 1.4553 R  0.93

    7.5235+-0.0103
    7.5201 R  0.32

 

6     1    4

6.1953E-9+-  0.0952
6.2788E-9 R 0.88

9.77 E-8+-  0.14
9.919E-8 R1.03

 4.065E-7 +- 0.055
 4.102E-7 R  0.66

5.197E-7+-  0.074

 5.260E-7 R 0.85

4.569E-6 +-  0.051

 4.615E-6  R 0.90

 

6  1 -0.03

842224+-   121893

847225 R   -0.04

4862637+-593633

4732944  R  0.22

7232194 +- 590436

 6922013  R 0.53

 8209386  +- 1041249

 8111858  R 0.09

8192845 +- 727167
8133365  R  0.08

 

6 1 -0.03

congenAtom

850762+-  7093

847225  R 0.50

4758031 +- 32287

4732945 R 0.78

6956772+-   30507

6922014  R 1.14

8155839  +-   60505

8111860 R 0.73

8164505+-   54555

8133367 R 0.57

 

The last line above is fom congen, the newest version of the point selector described in ..\BiasedSelectionMonteCarloAtom.docx .htm

            It is tested by rijtest2.wpj rijtest2.zip