Lennard Jones Potential
The anti-symmetry of parallel spin electrons makes neutral atoms resist
being pushed together.
The slight polarization of the electrons in one atom relative to another gives rise to a slight attraction
between the atoms.
The potential between two neutral atoms can be written as 1
   12   6 
v  r   4        (1.1)
 r 
 r  

Note that the potential minimum is at
 
11
5
 
  
v '  r   4 2 12    6     0
r   r 
 r  
   1 
      .89089872
 r  2
rmin  1.1224620
1/6
vmin
(1.2)
 1  2  1  
 4        
 2   2  
v ''  rmin  
  
 4
r  
7
   13
   
12    6    
 r   
 r
12
6
4 
 
  
  2 12  13    42   
r 
r
 r  
4 
1
1
  2 12  13  42 
4
2
rmin 
12
 2
rmin
(1.3)
So that to a first approximation
1
M.P.Allen and D.J. Tildesley, Computer Simulation of Liquids, Oxford Science Publications, (1987,1989),p 9
1
2
v  r   v  rmin   v ''  rmin  r  rmin 
2
2
(1.4)

 r  rmin  
  1  6 
 

rmin  



On page 22 of Allen and Tildesley is the table
Atom
Source
/k (0K)
(nm)
2
H
Murad and Gubbins
8.6
0.281
He
Maitland, et al3
10.2
0.228
C
Tildesley and Madden4
51.2
0.335
N
Cheung and Powles5
37.3
0.331
O
English and Venables6
61.6
0.295
F
Singer et al7
52.8
0.283
Ne
Maitland et al8
47.0
0.272
S
Tildesley and Madden
183.0
0.352
Cl
Singer et al
173.5
0.335
Ar
Maitland et al
119.8
0.341
Br
Singer et al
257.2
0.354
Kr
Maitland et al
164.0
0.383
While the above is valuable for working with real systems, for today
1
1
 6 (1.5)
12
r
r
The Boltzman constant k = 1.3807  10-23J/0K
1 eV = 1.6022  10-19 J 9
1 Rydberg = 0.5 Hartree = 13.6 eV
aB = 5.29172  10-9 cm
For He
mHe = 4.0026 u
me = 0.000549 u
 He  0.228  109 m  1aB / (5.29172  1011 m)  4.31 aB
v r  
 He  10.2  0 K   1.3807  1023 J /  0 K   1eV / 1.60219  10 19 J   0.5Hartree / 13.6eV
10.2  1.3807  0.5
 102319  0.323  104 Hartree
1.60219  13.6
The Schroedinger equation for Helium atoms 1 and 2 is
12
6

 2 R1 , R2
 R1  R2 
 R1  R2  


 4 He 

  R1 , R2  ( EHartree ) R1 , R2
   He 
2mHe / me
  He  




2





S.Murad and K.E. Gubbins, "Molecular dynamics simulation of methane using a singularity free algorithm". In Computer
modelling of matter, (ed. P. Lykos) ACS Symposium Series Vol. 86, pp. 62-712. American Chemical Society, Washington.
3
G.C. Maitland, et al, Intermolecular forces: their origin and determination, Clarendon Press, Oxford (1981)
4
D.J. Tildesley and P.A. Madden, Mol. Phys. 42, 1137-1156 (1981)
5
Mol. Phys. 30, 921-49 (1975)
6
Proc. R. Soc. Lond. A340, 57-80 (1974)
7
Mol. Phys. 33, 1757-95 (177)
8
G.C. Maitland,M.Rigby,E.B.Smith,W.A. Wakeham, Intermolecular forces: their origin and determination, Clanendon, Oxford
(1981)
9
F. Bueche, Principles of Physics (third edition), McGraw Hill, 1965
Figure 1 h2m.dat is <H2> -1. LJfit was fitted to this by matching the curves at r min = 1.4 using (1.2)
..\wsteve\h2fit LennardJones.zip
The H2 is not spin aligned so there is no statistical repulsion of the type assumed in the Lennard Jones.
The curves are quite different.