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I was trying to write a C++ code following the instruction given in this website. The problem is in the two electron integral file, only 228 entries for 7 molecular orbitals are provided. But my code says it should be 406 (please see below)

int NFI=0; //NFI=number of four electron integrals
for (int i=0; i<NAO; i++){
 for (int j=0; j<=i; j++){
  for (int k=0; k<=i; k++){
   for (int l=0; l<=k; l++){
    int ij=i*(i+1)*0.5+j;
    int kl=k*(k+1)*0.5+l;
     if (kl<=ij){NFI += 1;}
   }
  }
 }
}
printf("%d\n",NFI);

I'm using 4 criteria as can be seen in this code snippet. Can anyone explain me why the number of unique integral is 228, not 406?

As per suggestion, I compared my index with the index provided, here they are

1 1 1 1                   1     1     1     1 
2 1 1 1                   2     1     1     1 
2 1 2 1                   2     2     1     1 
2 2 1 1                   2     1     2     1 
2 2 2 1                   2     2     2     1 
2 2 2 2                   2     2     2     2 
3 1 1 1                   3     3     1     1 
3 1 2 1                   4     4     1     1 
3 1 2 2                   5     5     1     1

Left 4 numbers are generated from my code and right 4 numbers are given in the input file. (3,1,1,1) is missing in the data file provided (I checked the whole file), but it fulfills the criteria of $ij \geq kl$.

Improve this question
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  • 1
    $\begingroup$ You can see all the integrals here sirius.chem.vt.edu/~crawdad/programming/project3/h2o_sto3g/… Print out your indices and compare $\endgroup$
    – Jan Jensen
    Dec 30, 2016 at 10:31
  • 1
    $\begingroup$ I’m not sure whether there’s a need to delete the question. $\endgroup$
    – Jan
    Dec 30, 2016 at 22:56
  • $\begingroup$ OK, I'm deleting the first part. Still I couldn't figure it out. Mathematically also it should be 406 (as 7*8/2=28 and 28*29/2=406) $\endgroup$
    – Osman Mamun
    Dec 30, 2016 at 23:50
  • 2
    $\begingroup$ Have you considered the symmetry of H$_2$O or the NAOs? <sp||ss> = 0 if on the same center. Same goes ofr <gg||gu> (g even under symmetry operator, u odd under symmetry operator, e.g., $C_2$). $\endgroup$
    – Deathbreath
    Jan 18, 2017 at 15:08
  • $\begingroup$ You have a good point, I considered that possibility. As I am a chemical engineering graduate student, I'm not sure how to take symmetry into account. I didn't want to put substantial efforts into that because I'm doing it in my leisure time. Fortunately, I got one complete file that shows lots of zeros. $\endgroup$
    – Osman Mamun
    Jan 18, 2017 at 15:45

1 Answer 1

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Update:: I think that 228 integral was a mistake, I found another project file from Dr. Crawford which listed 406 unique integrals for the same system ($H_2O$ with 7 molecular orbitals). In case if anyone wants to work on that project, they can use this data file.

-----------------------------------------------------------------------
      PSI4: An Open-Source Ab Initio Electronic Structure Package
                          PSI 4.0.0-alpha Driver
T. D. Crawford, C. D. Sherrill, E. F. Valeev, J. T. Fermann, R. A. King,
M. L. Leininger, S. T. Brown, C. L. Janssen, E. T. Seidl, J. P. Kenny,
and W. D. Allen, J. Comput. Chem. 28, 1610-1616 (2007)

                     Additional Contributions by
Francesco Evangelista, Andrew Simmonett, Justin Turney, Jeremiah Wilke
-----------------------------------------------------------------------


Using LocalCommunicator (Number of processes = 1)

Memory level set to 256.000 MB

./plugin_aointegrals.so loaded. Calling plugin plugin_aointegrals.so. Molecular point group: c2v

Geometry (in Angstrom), charge = 0, multiplicity = 1:

   Center              X                  Y                   Z       
------------   -----------------  -----------------  -----------------
       O          0.000000000000     0.000000000000    -0.071151380605
       H          0.000000000000     0.757939245855     0.564612021746
       H          0.000000000000    -0.757939245855     0.564612021746

Number of unique atoms: 2

Atoms equivalency:
   unique atom 0: 0 
   unique atom 1: 1 2 


Nuclear repulsion energy:       8.90770810

## Overlap ##

             1            2            3            4            5            6            7

1    1.0000000    0.2367039    0.0000000    0.0000000   -0.0000000    0.0500137    0.0500137
2    0.2367039    1.0000000    0.0000000    0.0000000   -0.0000000    0.4539953    0.4539953
3    0.0000000    0.0000000    1.0000000    0.0000000    0.0000000    0.0000000    0.0000000
4    0.0000000    0.0000000    0.0000000    1.0000000    0.0000000    0.2927386   -0.2927386
5   -0.0000000   -0.0000000    0.0000000    0.0000000    1.0000000    0.2455507    0.2455507
6    0.0500137    0.4539953    0.0000000    0.2927386    0.2455507    1.0000000    0.2510021
7    0.0500137    0.4539953    0.0000000   -0.2927386    0.2455507    0.2510021    1.0000000

## Kinetic ##

             1            2            3            4            5            6            7

1   29.0031999   -0.1680109    0.0000000    0.0000000   -0.0000000   -0.0045400   -0.0045400
2   -0.1680109    0.8081280    0.0000000    0.0000000   -0.0000000    0.1137475    0.1137475
3    0.0000000    0.0000000    2.5287312    0.0000000    0.0000000    0.0000000    0.0000000
4    0.0000000    0.0000000    0.0000000    2.5287312    0.0000000    0.1993355   -0.1993355
5   -0.0000000   -0.0000000    0.0000000    0.0000000    2.5287312    0.1672036    0.1672036
6   -0.0045400    0.1137475    0.0000000    0.1993355    0.1672036    0.7600319    0.0083249
7   -0.0045400    0.1137475    0.0000000   -0.1993355    0.1672036    0.0083249    0.7600319

## Potential ##

             1            2            3            4            5            6            7

1  -61.6882822   -7.4363118    0.0000000    0.0000000   -0.0186797   -1.6150635   -1.6150635
2   -7.4363118  -10.1101908    0.0000000    0.0000000   -0.2221598   -3.6569582   -3.6569582
3    0.0000000    0.0000000   -9.9595668    0.0000000    0.0000000    0.0000000    0.0000000
4    0.0000000    0.0000000    0.0000000  -10.0957534    0.0000000   -2.0901916    2.0901916
5   -0.0186797   -0.2221598    0.0000000    0.0000000  -10.0553869   -1.8259929   -1.8259929
6   -1.6150635   -3.6569582    0.0000000   -2.0901916   -1.8259929   -5.7165220   -1.5685885
7   -1.6150635   -3.6569582    0.0000000    2.0901916   -1.8259929   -1.5685885   -5.7165220

## One Electron Ints ##

             1            2            3            4            5            6            7

1  -32.6850823   -7.6043227    0.0000000    0.0000000   -0.0186797   -1.6196036   -1.6196036
2   -7.6043227   -9.3020628    0.0000000    0.0000000   -0.2221598   -3.5432106   -3.5432106
3    0.0000000    0.0000000   -7.4308356    0.0000000    0.0000000    0.0000000    0.0000000
4    0.0000000    0.0000000    0.0000000   -7.5670222    0.0000000   -1.8908561    1.8908561
5   -0.0186797   -0.2221598    0.0000000    0.0000000   -7.5266557   -1.6587893   -1.6587893
6   -1.6196036   -3.5432106    0.0000000   -1.8908561   -1.6587893   -4.9564901   -1.5602636
7   -1.6196036   -3.5432106    0.0000000    1.8908561   -1.6587893   -1.5602636   -4.9564901

Two-electron Integrals

( 0  0 |  0  0) =    4.785065404705502
( 1  0 |  0  0) =    0.741380351973408
( 1  1 |  0  0) =    1.118946866342470
( 1  0 |  1  0) =    0.136873385354388
( 1  1 |  1  0) =    0.256633394730974
( 1  1 |  1  1) =    0.817206321526058
( 2  0 |  0  0) =    0.000000000000000
( 3  0 |  0  0) =    0.000000000000000
( 4  0 |  0  0) =   -0.000000000000000
( 2  1 |  0  0) =    0.000000000000000
( 3  1 |  0  0) =    0.000000000000000
( 4  1 |  0  0) =   -0.000000000000000
( 2  0 |  1  0) =    0.000000000000000
( 3  0 |  1  0) =    0.000000000000000
( 4  0 |  1  0) =   -0.000000000000000
( 2  1 |  1  0) =    0.000000000000000
( 3  1 |  1  0) =    0.000000000000000
( 4  1 |  1  0) =   -0.000000000000000
( 2  0 |  1  1) =    0.000000000000000
( 3  0 |  1  1) =    0.000000000000000
( 4  0 |  1  1) =   -0.000000000000000
( 2  1 |  1  1) =    0.000000000000000
( 3  1 |  1  1) =    0.000000000000000
( 4  1 |  1  1) =    0.000000000000000
( 2  2 |  0  0) =    1.115813812152427
( 3  2 |  0  0) =    0.000000000000000
( 3  3 |  0  0) =    1.115813812152427
( 4  2 |  0  0) =    0.000000000000000
( 4  3 |  0  0) =    0.000000000000000
( 4  4 |  0  0) =    1.115813812152427
( 2  0 |  2  0) =    0.024477412258099
( 3  0 |  2  0) =    0.000000000000000
( 3  0 |  3  0) =    0.024477412258099
( 4  0 |  2  0) =    0.000000000000000
( 4  0 |  3  0) =    0.000000000000000
( 4  0 |  4  0) =    0.024477412258099
( 2  2 |  1  0) =    0.256683985810103
( 3  2 |  1  0) =    0.000000000000000
( 3  3 |  1  0) =    0.256683985810103
( 4  2 |  1  0) =    0.000000000000000
( 4  3 |  1  0) =    0.000000000000000
( 4  4 |  1  0) =    0.256683985810103
( 2  1 |  2  0) =    0.037808607416361
( 3  0 |  2  1) =    0.000000000000000
( 4  0 |  2  1) =    0.000000000000000
( 3  1 |  2  0) =    0.000000000000000
( 3  1 |  3  0) =    0.037808607416361
( 4  0 |  3  1) =    0.000000000000000
( 4  1 |  2  0) =    0.000000000000000
( 4  1 |  3  0) =    0.000000000000000
( 4  1 |  4  0) =    0.037808607416361
( 2  2 |  1  1) =    0.817022605320914
( 3  2 |  1  1) =    0.000000000000000
( 3  3 |  1  1) =    0.817022605320914
( 4  2 |  1  1) =    0.000000000000000
( 4  3 |  1  1) =    0.000000000000000
( 4  4 |  1  1) =    0.817022605320914
( 2  1 |  2  1) =    0.180518392104632
( 3  1 |  2  1) =    0.000000000000000
( 3  1 |  3  1) =    0.180518392104632
( 4  1 |  2  1) =    0.000000000000000
( 4  1 |  3  1) =    0.000000000000000
( 4  1 |  4  1) =    0.180518392104632
( 2  2 |  2  0) =    0.000000000000000
( 3  2 |  2  0) =    0.000000000000000
( 3  3 |  2  0) =    0.000000000000000
( 4  2 |  2  0) =    0.000000000000000
( 4  3 |  2  0) =    0.000000000000000
( 4  4 |  2  0) =    0.000000000000000
( 3  0 |  2  2) =    0.000000000000000
( 3  2 |  3  0) =    0.000000000000000
( 3  3 |  3  0) =    0.000000000000000
( 4  2 |  3  0) =    0.000000000000000
( 4  3 |  3  0) =    0.000000000000000
( 4  4 |  3  0) =    0.000000000000000
( 4  0 |  2  2) =   -0.000000000000000
( 4  0 |  3  2) =    0.000000000000000
( 4  0 |  3  3) =   -0.000000000000000
( 4  2 |  4  0) =    0.000000000000000
( 4  3 |  4  0) =    0.000000000000000
( 4  4 |  4  0) =   -0.000000000000000
( 2  2 |  2  1) =    0.000000000000000
( 3  2 |  2  1) =    0.000000000000000
( 3  3 |  2  1) =    0.000000000000000
( 4  2 |  2  1) =    0.000000000000000
( 4  3 |  2  1) =    0.000000000000000
( 4  4 |  2  1) =    0.000000000000000
( 3  1 |  2  2) =    0.000000000000000
( 3  2 |  3  1) =    0.000000000000000
( 3  3 |  3  1) =    0.000000000000000
( 4  2 |  3  1) =    0.000000000000000
( 4  3 |  3  1) =    0.000000000000000
( 4  4 |  3  1) =    0.000000000000000
( 4  1 |  2  2) =    0.000000000000000
( 4  1 |  3  2) =    0.000000000000000
( 4  1 |  3  3) =    0.000000000000000
( 4  2 |  4  1) =    0.000000000000000
( 4  3 |  4  1) =    0.000000000000000
( 4  4 |  4  1) =    0.000000000000000
( 2  2 |  2  2) =    0.880159093375045
( 3  2 |  2  2) =    0.000000000000000
( 3  2 |  3  2) =    0.047444445118384
( 3  3 |  2  2) =    0.785270203138277
( 3  3 |  3  2) =    0.000000000000000
( 3  3 |  3  3) =    0.880159093375045
( 4  2 |  2  2) =    0.000000000000000
( 4  2 |  3  2) =    0.000000000000000
( 4  2 |  3  3) =    0.000000000000000
( 4  2 |  4  2) =    0.047444445118384
( 4  3 |  2  2) =    0.000000000000000
( 4  3 |  3  2) =    0.000000000000000
( 4  3 |  3  3) =    0.000000000000000
( 4  3 |  4  2) =    0.000000000000000
( 4  3 |  4  3) =    0.047444445118384
( 4  4 |  2  2) =    0.785270203138277
( 4  4 |  3  2) =    0.000000000000000
( 4  4 |  3  3) =    0.785270203138277
( 4  4 |  4  2) =    0.000000000000000
( 4  4 |  4  3) =    0.000000000000000
( 4  4 |  4  4) =    0.880159093375045
( 5  0 |  0  0) =    0.159028953813045
( 5  1 |  0  0) =    0.381788700431517
( 5  0 |  1  0) =    0.029107255214247
( 5  1 |  1  0) =    0.088639046604491
( 5  0 |  1  1) =    0.054208207239172
( 5  1 |  1  1) =    0.311911628346485
( 5  2 |  0  0) =    0.000000000000000
( 5  3 |  0  0) =    0.208975170450133
( 5  4 |  0  0) =    0.175289465612119
( 5  0 |  2  0) =    0.000000000000000
( 5  0 |  3  0) =    0.001028596856434
( 5  0 |  4  0) =    0.000862792421254
( 5  2 |  1  0) =    0.000000000000000
( 5  3 |  1  0) =    0.049151047350638
( 5  4 |  1  0) =    0.041228155506758
( 5  1 |  2  0) =    0.000000000000000
( 5  1 |  3  0) =    0.005091750233326
( 5  1 |  4  0) =    0.004270986718219
( 5  0 |  2  1) =    0.000000000000000
( 5  0 |  3  1) =    0.001923027401919
( 5  0 |  4  1) =    0.001613045439387
( 5  2 |  1  1) =    0.000000000000000
( 5  3 |  1  1) =    0.184701361441018
( 5  4 |  1  1) =    0.154928467698285
( 5  1 |  2  1) =    0.000000000000000
( 5  1 |  3  1) =    0.038683534029509
( 5  1 |  4  1) =    0.032447950603007
( 5  2 |  2  0) =    0.009336058122476
( 5  2 |  3  0) =    0.000000000000000
( 5  2 |  4  0) =    0.000000000000000
( 5  3 |  2  0) =    0.000000000000000
( 5  3 |  3  0) =    0.011813922734513
( 5  3 |  4  0) =    0.002078445792233
( 5  4 |  2  0) =    0.000000000000000
( 5  4 |  3  0) =    0.002078445792233
( 5  4 |  4  0) =    0.011079469320272
( 5  0 |  2  2) =    0.054104219459936
( 5  0 |  3  2) =    0.000000000000000
( 5  0 |  3  3) =    0.054291571211162
( 5  0 |  4  2) =    0.000000000000000
( 5  0 |  4  3) =    0.000157151628507
( 5  0 |  4  4) =    0.054236039067604
( 5  2 |  2  1) =    0.054125156914812
( 5  2 |  3  1) =    0.000000000000000
( 5  2 |  4  1) =    0.000000000000000
( 5  3 |  2  1) =    0.000000000000000
( 5  3 |  3  1) =    0.075855681518665
( 5  3 |  4  1) =    0.018227677656995
( 5  4 |  2  1) =    0.000000000000000
( 5  4 |  3  1) =    0.018227677656995
( 5  4 |  4  1) =    0.069414628237947
( 5  1 |  2  2) =    0.306655599347507
( 5  1 |  3  2) =    0.000000000000000
( 5  1 |  3  3) =    0.315738293535417
( 5  1 |  4  2) =    0.000000000000000
( 5  1 |  4  3) =    0.007618611374202
( 5  1 |  4  4) =    0.313046130402530
( 5  2 |  2  2) =    0.000000000000000
( 5  2 |  3  2) =    0.008064485483452
( 5  2 |  3  3) =    0.000000000000000
( 5  2 |  4  2) =    0.006764532589140
( 5  2 |  4  3) =    0.000000000000000
( 5  2 |  4  4) =    0.000000000000000
( 5  3 |  2  2) =    0.176295759644369
( 5  3 |  3  2) =    0.000000000000000
( 5  3 |  3  3) =    0.197638924560966
( 5  3 |  4  2) =    0.000000000000000
( 5  3 |  4  3) =    0.011138225736852
( 5  3 |  4  4) =    0.179964436324773
( 5  4 |  2  2) =    0.147877804962089
( 5  4 |  3  2) =    0.000000000000000
( 5  4 |  3  3) =    0.152251498109802
( 5  4 |  4  2) =    0.000000000000000
( 5  4 |  4  3) =    0.011733162163856
( 5  4 |  4  4) =    0.164484175321776
( 5  5 |  0  0) =    0.517125360977304
( 5  0 |  5  0) =    0.006270522616106
( 5  5 |  1  0) =    0.122013556886528
( 5  1 |  5  0) =    0.019208705902818
( 5  5 |  1  1) =    0.491873027324534
( 5  1 |  5  1) =    0.146642827474200
( 5  5 |  2  0) =    0.000000000000000
( 5  5 |  3  0) =    0.009188483135799
( 5  5 |  4  0) =    0.007707347696814
( 5  2 |  5  0) =    0.000000000000000
( 5  3 |  5  0) =    0.011128525245716
( 5  4 |  5  0) =    0.009334665162233
( 5  5 |  2  1) =    0.000000000000000
( 5  5 |  3  1) =    0.095810960370799
( 5  5 |  4  1) =    0.080366734512074
( 5  2 |  5  1) =    0.000000000000000
( 5  3 |  5  1) =    0.105474819021288
( 5  4 |  5  1) =    0.088472829676139
( 5  5 |  2  2) =    0.471913115835107
( 5  5 |  3  2) =    0.000000000000000
( 5  5 |  3  3) =    0.506704669305872
( 5  5 |  4  2) =    0.000000000000000
( 5  5 |  4  3) =    0.029183336960891
( 5  5 |  4  4) =    0.496392251464638
( 5  2 |  5  2) =    0.021048752318672
( 5  3 |  5  2) =    0.000000000000000
( 5  3 |  5  3) =    0.096048971720199
( 5  4 |  5  2) =    0.000000000000000
( 5  4 |  5  3) =    0.062910576176908
( 5  4 |  5  4) =    0.073818472547383
( 5  5 |  5  0) =    0.026686622886860
( 5  5 |  5  1) =    0.289333860384054
( 5  5 |  5  2) =    0.000000000000000
( 5  5 |  5  3) =    0.237637271677700
( 5  5 |  5  4) =    0.199331385983227
( 5  5 |  5  5) =    0.774605943919898
( 6  0 |  0  0) =    0.159028953813045
( 6  1 |  0  0) =    0.381788700431517
( 6  0 |  1  0) =    0.029107255214247
( 6  1 |  1  0) =    0.088639046604491
( 6  0 |  1  1) =    0.054208207239172
( 6  1 |  1  1) =    0.311911628346485
( 6  2 |  0  0) =    0.000000000000000
( 6  3 |  0  0) =   -0.208975170450133
( 6  4 |  0  0) =    0.175289465612119
( 6  0 |  2  0) =    0.000000000000000
( 6  0 |  3  0) =   -0.001028596856434
( 6  0 |  4  0) =    0.000862792421254
( 6  2 |  1  0) =    0.000000000000000
( 6  3 |  1  0) =   -0.049151047350638
( 6  4 |  1  0) =    0.041228155506758
( 6  1 |  2  0) =    0.000000000000000
( 6  1 |  3  0) =   -0.005091750233326
( 6  1 |  4  0) =    0.004270986718219
( 6  0 |  2  1) =    0.000000000000000
( 6  0 |  3  1) =   -0.001923027401919
( 6  0 |  4  1) =    0.001613045439387
( 6  2 |  1  1) =    0.000000000000000
( 6  3 |  1  1) =   -0.184701361441018
( 6  4 |  1  1) =    0.154928467698285
( 6  1 |  2  1) =    0.000000000000000
( 6  1 |  3  1) =   -0.038683534029509
( 6  1 |  4  1) =    0.032447950603007
( 6  2 |  2  0) =    0.009336058122476
( 6  2 |  3  0) =    0.000000000000000
( 6  2 |  4  0) =    0.000000000000000
( 6  3 |  2  0) =    0.000000000000000
( 6  3 |  3  0) =    0.011813922734513
( 6  3 |  4  0) =   -0.002078445792233
( 6  4 |  2  0) =    0.000000000000000
( 6  4 |  3  0) =   -0.002078445792233
( 6  4 |  4  0) =    0.011079469320272
( 6  0 |  2  2) =    0.054104219459936
( 6  0 |  3  2) =    0.000000000000000
( 6  0 |  3  3) =    0.054291571211162
( 6  0 |  4  2) =    0.000000000000000
( 6  0 |  4  3) =   -0.000157151628507
( 6  0 |  4  4) =    0.054236039067604
( 6  2 |  2  1) =    0.054125156914812
( 6  2 |  3  1) =    0.000000000000000
( 6  2 |  4  1) =    0.000000000000000
( 6  3 |  2  1) =    0.000000000000000
( 6  3 |  3  1) =    0.075855681518665
( 6  3 |  4  1) =   -0.018227677656995
( 6  4 |  2  1) =    0.000000000000000
( 6  4 |  3  1) =   -0.018227677656995
( 6  4 |  4  1) =    0.069414628237947
( 6  1 |  2  2) =    0.306655599347507
( 6  1 |  3  2) =    0.000000000000000
( 6  1 |  3  3) =    0.315738293535417
( 6  1 |  4  2) =    0.000000000000000
( 6  1 |  4  3) =   -0.007618611374202
( 6  1 |  4  4) =    0.313046130402530
( 6  2 |  2  2) =    0.000000000000000
( 6  2 |  3  2) =   -0.008064485483452
( 6  2 |  3  3) =    0.000000000000000
( 6  2 |  4  2) =    0.006764532589140
( 6  2 |  4  3) =    0.000000000000000
( 6  2 |  4  4) =    0.000000000000000
( 6  3 |  2  2) =   -0.176295759644369
( 6  3 |  3  2) =    0.000000000000000
( 6  3 |  3  3) =   -0.197638924560966
( 6  3 |  4  2) =    0.000000000000000
( 6  3 |  4  3) =    0.011138225736852
( 6  3 |  4  4) =   -0.179964436324773
( 6  4 |  2  2) =    0.147877804962089
( 6  4 |  3  2) =    0.000000000000000
( 6  4 |  3  3) =    0.152251498109802
( 6  4 |  4  2) =    0.000000000000000
( 6  4 |  4  3) =   -0.011733162163856
( 6  4 |  4  4) =    0.164484175321776
( 6  5 |  0  0) =    0.154983826095369
( 6  0 |  5  0) =    0.006178562861131
( 6  5 |  1  0) =    0.036324481156051
( 6  1 |  5  0) =    0.018637075964466
( 6  0 |  5  1) =    0.018637075964466
( 6  5 |  1  1) =    0.138899129503383
( 6  1 |  5  1) =    0.120424410513661
( 6  5 |  2  0) =    0.000000000000000
( 6  5 |  3  0) =    0.000000000000000
( 6  5 |  4  0) =    0.002735070642527
( 6  2 |  5  0) =    0.000000000000000
( 6  3 |  5  0) =   -0.009851358272803
( 6  4 |  5  0) =    0.009063088333113
( 6  0 |  5  2) =    0.000000000000000
( 6  0 |  5  3) =    0.009851358272803
( 6  0 |  5  4) =    0.009063088333113
( 6  5 |  2  1) =    0.000000000000000
( 6  5 |  3  1) =    0.000000000000000
( 6  5 |  4  1) =    0.024748716021254
( 6  2 |  5  1) =    0.000000000000000
( 6  3 |  5  1) =   -0.059473510026833
( 6  4 |  5  1) =    0.070735331310826
( 6  1 |  5  2) =    0.000000000000000
( 6  1 |  5  3) =    0.059473510026833
( 6  1 |  5  4) =    0.070735331310826
( 6  5 |  2  2) =    0.135540657848790
( 6  5 |  3  2) =    0.000000000000000
( 6  5 |  3  3) =    0.138097677983239
( 6  5 |  4  2) =    0.000000000000000
( 6  5 |  4  3) =    0.000000000000000
( 6  5 |  4  4) =    0.142908608378693
( 6  2 |  5  2) =    0.016764912164332
( 6  2 |  5  3) =    0.000000000000000
( 6  2 |  5  4) =    0.000000000000000
( 6  3 |  5  2) =    0.000000000000000
( 6  3 |  5  3) =   -0.013036508898613
( 6  3 |  5  4) =   -0.034131389240033
( 6  4 |  5  2) =    0.000000000000000
( 6  4 |  5  3) =    0.034131389240033
( 6  4 |  5  4) =    0.057682230376614
( 6  5 |  5  0) =    0.007800931718220
( 6  0 |  5  5) =    0.025610632874342
( 6  5 |  5  1) =    0.065308066611221
( 6  1 |  5  5) =    0.196819899854417
( 6  5 |  5  2) =    0.000000000000000
( 6  5 |  5  3) =    0.040452941573134
( 6  5 |  5  4) =    0.044287849234219
( 6  2 |  5  5) =    0.000000000000000
( 6  3 |  5  5) =   -0.085489074048849
( 6  4 |  5  5) =    0.126882792323867
( 6  5 |  5  5) =    0.133977082851046
( 6  6 |  0  0) =    0.517125360977304
( 6  0 |  6  0) =    0.006270522616106
( 6  6 |  1  0) =    0.122013556886528
( 6  1 |  6  0) =    0.019208705902818
( 6  6 |  1  1) =    0.491873027324534
( 6  1 |  6  1) =    0.146642827474200
( 6  6 |  2  0) =    0.000000000000000
( 6  6 |  3  0) =   -0.009188483135799
( 6  6 |  4  0) =    0.007707347696814
( 6  2 |  6  0) =    0.000000000000000
( 6  3 |  6  0) =   -0.011128525245716
( 6  4 |  6  0) =    0.009334665162233
( 6  6 |  2  1) =    0.000000000000000
( 6  6 |  3  1) =   -0.095810960370799
( 6  6 |  4  1) =    0.080366734512074
( 6  2 |  6  1) =    0.000000000000000
( 6  3 |  6  1) =   -0.105474819021288
( 6  4 |  6  1) =    0.088472829676139
( 6  6 |  2  2) =    0.471913115835107
( 6  6 |  3  2) =    0.000000000000000
( 6  6 |  3  3) =    0.506704669305872
( 6  6 |  4  2) =    0.000000000000000
( 6  6 |  4  3) =   -0.029183336960891
( 6  6 |  4  4) =    0.496392251464638
( 6  2 |  6  2) =    0.021048752318672
( 6  3 |  6  2) =    0.000000000000000
( 6  3 |  6  3) =    0.096048971720199
( 6  4 |  6  2) =    0.000000000000000
( 6  4 |  6  3) =   -0.062910576176908
( 6  4 |  6  4) =    0.073818472547383
( 6  6 |  5  0) =    0.025610632874342
( 6  5 |  6  0) =    0.007800931718220
( 6  6 |  5  1) =    0.196819899854417
( 6  5 |  6  1) =    0.065308066611221
( 6  6 |  5  2) =    0.000000000000000
( 6  6 |  5  3) =    0.085489074048849
( 6  6 |  5  4) =    0.126882792323867
( 6  5 |  6  2) =    0.000000000000000
( 6  5 |  6  3) =   -0.040452941573134
( 6  5 |  6  4) =    0.044287849234219
( 6  6 |  5  5) =    0.342541413419711
( 6  5 |  6  5) =    0.035634211559360
( 6  6 |  6  0) =    0.026686622886860
( 6  6 |  6  1) =    0.289333860384054
( 6  6 |  6  2) =    0.000000000000000
( 6  6 |  6  3) =   -0.237637271677700
( 6  6 |  6  4) =    0.199331385983227
( 6  6 |  6  5) =    0.133977082851046
( 6  6 |  6  6) =    0.774605943919898

There are 406 unique integrals
Improve this answer
$\endgroup$

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