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Many-electron atoms
e



The helium atom
ˆ H 
2

r12 r2 Ze

1
2 Ze 2 Ze 2 e2 2   2   4 0 r1 2  4 0 r2 4 0 r12

e2

2

2 1

r1



Orbital approximation Assuming the two electrons move independently
ˆ H approx  
2

2

2 1

2 Ze 2 Ze 2 2 ˆ ˆ   2   H1  H 2 4 0 r1 2  4 0 r2

 approx (r1, 1, 1 ; r2,  2,  2 )   1 (r1, 1, 1 )  2 (r2,  2,  2 )
Eapprox  Z2 Z2   13.60 2  2  n n2   1 

The electron-electron interaction is considered by averaging with the electron density distributions • Ground state configuration of He

1s (1)1s (2) or 1s 2

• Variational treatment of He The ground state energy of the He atom using hydrogen-like wavefunctions The trial wavefunction (1,2)=1s(x1,y1,z1)•1s(x2,y2,z2) = 1s(1)•1s(2)
ˆ ˆ ˆ H  H1  H 2  ˆ Hi   E 
2

e2 4 0 r12  i2  2 xi2  2 yi2  2 zi2

2

 i2

Ze 2 , 4 0 ri

ˆ 1s(1)1s(2)* H 1s(1)1s(2)d 1 d 2  74.8 eV 

Compared to -79.0 eV (Experimental value) -108.8 eV (If the interaction between the two electrons is neglected) Improvement: Change the nuclear charge Z=2 into a variational parameter  The trial wavefunction
3 2

1  (1,2)    e r1 / a0 e r2 / a0   a0    dE e2 27  (2  8 ) 0 d 4 0 a0 

E ( )  ( 

2

27  8

e2 ) 4 0 a0

 

27  1.6875, E (2.6875)  77.5 eV 16

The perturbation method
• The general scheme ˆ ˆ ˆ ˆ For the Hamiltonian operator H, we can always write H  H (0)  H (1) ˆ where H (0) is the Hamiltonian operator for the unperturbed system and is solved ˆ and H (1) is the perturbation The solution is    (0)   (1)   ( 2)  ...
E  E (0)  E (1)  E ( 2)  ...

• The first-order correction to the energy
ˆ E (1)   (0)* H (1) (0) d



E  E (0)  E (1)

• Example: He atom
ˆ ˆ ˆ H  H1  H 2  e2 ˆ ˆ  H (0)  H (1) 4 0 r12
2

ˆ E (1)   (0)* H (1) (0) d  34.0 eV



ˆ ˆ ˆ H ( 0)  H 1  H 2 ...

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Anti. Anti Essays. Retrieved January 16, 2018, from the World Wide Web: http://anties.wressy.com/free-essays/Anti-416175.html