The wave functions and lowest energies of an He atom
First results and drawings










The mesh of 6-cubes is defined by the Mefisto commands:
Option 61: Mefisto file of the 6-cube (length=20)
This a regular mesh of sub 6-cubes of one 6-cube with 5 EDGES in each of the 6 DIRECTIONS (of type finite differences)
The drawing is done with a projection into R3.
With only 6 EDGES the following error is obtained:
NUMBER of VERTICES = 6**6 = 46 656       (6 edges => 7**6 = 117 649 vertices)
NUMBER of 6-CUBES  = 5**6 = 15 625
 => ERROR in thevvp: NOT ENOUGH MEMORY or SWAP  for MG + KG + Aux
                     115 222 032 DECLARED MEMORY WORDS
                   1 451 343 744 NECESSARY WORDS to store MG + KG + Aux
 => REDUCE the MESH



To construct a finer mesh around the origin and growing with the distance from the origin to reduce the number of finite elements, the 6-cubes are generated by an homothetical mapping of the initial center 6-cube through its 12 5-faces with a number of layers. The mesh of 6-cubes is defined by the Mefisto commands:
Option 62: Mefisto file of the 6-cube mesh construction (length=20)
This a regular mesh of sub 6-cubes of one 6-cube with 50 layers.

But, this time, the number of finite elements is not enough around the origin!


Option 63: Fichier Mefisto of the 6-cube (length=20, 4 layers)
This third mesh is the combination of the 2 previous meshes.
A kernel of 3 finite difference 6-cubes (to obtain more finite elements around the origin) with layers of homothetical 6-cubes through the 5-faces of their boundary (to obtain a progress of the mesh with not too more elements).
For n finite differences of the kernel, there are 
- (n+1)**6 vertices
- (n+1)**6 - (n-1)**6 vertices by layer
-  n**6 + nblayers ( 12 * n**5 ) 6-cubes.

Kernel n=1 => 2**6=       64 vertices  and       665 vertices by layer
Kernel n=2 => 3**6=      729 vertices  and     4 032 vertices by layer
Kernel n=3 => 4**6=    4 096 vertices  and    14 896 vertices by layer
Kernel n=4 => 5**6=   15 625 vertices  and    42 560 vertices by layer
Kernel n=5 => 6**6=   46 656 vertices  and   102 024 vertices by layer
Kernel n=6 => 7**6=  117 649 vertices  and   215 488 vertices by layer
Kernel n=7 => 8**6=  262 144 vertices  and   413 792 vertices by layer
Kernel n=8 => 9**6=  531 441 vertices  and   737 856 vertices by layer
Kernel n=9 =>10**6=1 000 000 vertices  and 1 240 120 vertices by layer  !...

A good accuracy requires a large value of n => a VERY LARGE MEMORY is necessary to solve the problem!

The mesh with n=3 finite differences and 5 layers:


The mesh with 5 layers does not permit the computation of energies, by default of a sufficient main memory.


The computation of energies with the 3 meshes is defined by the same Mefisto commands:
Mefisto file to compute the energies of He atom

With the third mesh, the necessary main memory is
INITIAL SKYLINE MATRIX = 85 335 188 x 8 BYTES
FINAL   SKYLINE MATRIX = 65 103 823 x 8 BYTES = 520 827 404 BYTES

3 SKYLINE MATRICES ARE NECESSARY => 1.6 Gigabytes!

The first 14 computed energies with this last mesh:
                cube        exact       error
ENERGY    1 = -2.816518    -2.90358    < 3.0 %
ENERGY    2 = -2.050233    -2.17522    < 5.7 %
ENERGY    3 = -1.830378    -2.14596    < 15  %
ENERGY    4 = -1.830232    -2.13316    < 15  %
ENERGY    5 = -1.689611
ENERGY    6 = -1.582237
ENERGY    7 = -1.582237
ENERGY    8 = -1.197280
ENERGY    9 = -1.194597
ENERGY   10 = -1.097584
ENERGY   11 = -1.090540
ENERGY   12 = -1.087805
ENERGY   13 = -1.055419
ENERGY   14 = -1.037078

CONCLUSION:
It is evident that the TOO SMALL NUMBER of finite elements of the mesh must be augmented to obtain a better and correct accuracy.
Technically, this augmentation needs a powerful computer with a very large memory and compilers able to manage adresses greater than 2**32. We hope to obtain these ones in the next months.



Some informations on the drawings:














EIGENVALUE 1 = -2.817     COLOR SECTION=EIGENVECTOR1(X,Y,0,0,0)



EIGENVALUE 1 = -2.817     COLOR SECTION=EIGENVECTOR1(X,Y,0,0,0,0)



EIGENVALUE 1 = -2.817     PROFILE Z=EIGENVECTOR1(X,Y,0,0,0,0)



EIGENVALUE 1 = -2.817     PROFILE Z=EIGENVECTOR1(X,Y,0,u=0,0,0)



EIGENVALUE 1 = -2.817     PROFILE Y=EIGENVECTOR1(X,0,Z,0,0,0)



EIGENVALUE 1 = -2.817     PROFILE X=EIGENVECTOR1(0,Y,Z,0,0,0)





EIGENVALUE 2 = -2.050     COLOR SECTION=EIGENVECTOR1(X,Y,0,0,0)



EIGENVALUE 2 = -2.050     COLOR SECTION=EIGENVECTOR1(X,0,Z,0,0,0)



EIGENVALUE 2 = -2.050     PROFILE Z=EIGENVECTOR1(0,Y,Z,0,0,0)



EIGENVALUE 2 = -2.050     PROFILE Z=EIGENVECTOR1(0,Y,Z,0,0,0)



EIGENVALUE 2 = -2.050     PROFILE Y=EIGENVECTOR1(X,0,Z,0,0,0)



EIGENVALUE 2 = -2.050     PROFILE Z=EIGENVECTOR1(0,Y,0,U,0,0)



EIGENVALUE 2 = -2.050     PROFILE Y=EIGENVECTOR1(X,0,0,U,0,0)



EIGENVALUE 2 = -2.050     PROFILE X=EIGENVECTOR1(X,Y,0,0,0,0)





EIGENVALUE 3 = -1.830     COLOR SECTION=EIGENVECTOR1(X,Y,0,0,0)
BR>

EIGENVALUE 3 = -1.830     PROFILE Z=EIGENVECTOR1(X,Y,0,0,0,0)



EIGENVALUE 3 = -1.830     PROFILE Z=EIGENVECTOR1(0,Y,0,U,0,0)



EIGENVALUE 3 = -1.830     PROFILE Y=EIGENVECTOR1(X,0,0,U,0,0)





EIGENVALUE 4 = -1.830     COLOR SECTION=EIGENVECTOR4(X,Y,0,0,0,0)



EIGENVALUE 4 = -1.830     COLOR SECTION=EIGENVECTOR4(X,0,Z,0,0,0) et (X,Y,0,0,0,0)



EIGENVALUE 4 = -1.830     PROFILE Z=EIGENVECTOR4(X,Y,0,0,0,0)



EIGENVALUE 4 = -1.830     PROFILE Y=EIGENVECTOR4(X,0,Z,0,0,0)



EIGENVALUE 4 = -1.830     PROFILE X=EIGENVECTOR4(0,Y,Z,0,0,0)



EIGENVALUE 4 = -1.830     ISOSURFACES of EIGENVECTOR4(X,Y,Z,0,0,0)



EIGENVALUE 4 = -1.830     ISOSURFACES of EIGENVECTOR4(X,Y,0,U,0,0)





EIGENVALUE 10 = -1.098     PROFILE Z=EIGENVECTOR10(X,Y,0,0,0,0)





EIGENVALUE 13 = -1.055     PROFILE Z=EIGENVECTOR13(X,Y,0,U=0,0,0)



EIGENVALUE 13 = -1.055     COLOR SECTION=EIGENVECTOR13(X,Y,0,U=0,0,0)



EIGENVALUE 13 = -1.055     ISOSURFACES of EIGENVECTOR13(X,Y,0,U,0,0)





EIGENVALUE 14 = -1.037     PROFILE Z=EIGENVECTOR14(X,Y,0,0,0,0)



EIGENVALUE 14 = -1.037     PROFILE Z=EIGENVECTOR14(X,Y,0)



EIGENVALUE 14 = -1.037     COLOR SECTION=EIGENVECTOR14(X,Y,0)



EIGENVALUE 14 = -1.037     ISOSURFACES of EIGENVECTOR14(X,Y,Z)


Page written by Alain Perronnet
Last update January 17-th 2007