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 testing\rijtest.for which has includes to setup.for, random.for, hfuns.for, xnfind.for, and xnefin.for.  The main code contains the various parameters that are changed in the tests.  The codes setup, random, xnfind and xnefin are the Monte Carlo codes discussed in detail in Fmonte.doc .htm.  The code hfuns is the code used for determining the analytical answers.

            The initial testing on the normalization integral yields.

Method

electrons #

a

b

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    a   b

ò Gaussian

òGå1/rI

òGå1/rI2

òGå1/rIJ

òGå1/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 R–1.03

 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