Solutions of the Schrödinger
equation for the ground helium by
finite element method
Jiahua Guo
Introduction
•
•
•
•
Schrödinger equation
Problems with traditional methods
Helium atom system
Challenge of solving He with FEM
Governing equations
In Cartesian coordinates, the spin-independent, nonrelativistic Schrödinger
equation for the two electrons in the helium atom is:
 1
1
2 2 1
H   12   22    
2
r1 r2 r12 
 2
In spherical coordinates:
H  E
1  1   2   1   2   L12 L22  2 2
1
 r1
  2
 r2
  2  2    
H   2
2  r1 r1  r1  r2 r2  r2  r1
r2  r1 r2 r12
L is the angular momentum operator and can be written as:
L12  ( L2ix  L2iy  L2iz )  
1
 
 
 sin 12

sin 12 12 
12 
Thus the Hamiltonian operator can be finally written as:
1 1   2   1   2    1
1 1
 
  2 2
1
 r1
  2
 r2
   2  2 
 sin  12
   
H   2
2  r1 r1  r1  r2 r2  r2   r1
 12  r1 r2 r12
r2  sin  12  12 
Boundary conditions & Formulation
• Boundary conditions
r1 , r2  (0, rc ]
 0
12 [0,  ]
When out of the boundary
• Formulation
Coefficient form of eigenvalue PDE in FEMlab:
   (cu  u)  u  au  d u


 0.5 





c    0.5


1 1
1 
 2  2  

 
2  r1
r2  


1
   
 r1
a

1
r2

ctg 12
2
 1
1 
 2  2  
r2  
 r1
2 2
1
 
r1 r2
r12  r22  2r1r2 cos12
Solution
E = -2.7285 hartree
= -74.22eV
(Experimental result:
Eexp = -78.98eV
The slice scheme of the helium wave function
with the lowest eigenvalue
Validation
• Energy levels for hydrogen atom
Energy
level
Energy value based on
Bohr model (hartree)
Energy value calculated by
FemLab (hartree)
Error
n=1
-0.5000
-0.5014
0.28%
n=2
-0.1250
-0.1247
0.24%
• Atomic orbits
1s
2s
2px
2py
2pz
Conclusion
• Schrödinger equation can be simplified by
decreasing some variables, making it an
equation with fewer dimensions.
• FEMlab is a good tool when trying to find out
the eigenvalues of energy of some threedimensional systems (e.g. hydrogen and
helium atoms). However, it can’t deal with a
complicated many-body Schrödinger equation.
Thank you!