Transcript The Born-Oppenheimer

Born-Oppenheimer approximation
H  Tn  Te  Vnn  Vee  Vne
H( R, r )  E( R, r )
Nuclear coordinates
Electronic coordinates
B
A
 He
[Te  Vne  Vee ]e (r; R)  Ee ( R)e (r; R)
( R, r )  n ( R)e (r; R)
Parameter
Born-Oppenheimer approximation (continued)
*
*
*
dr

T


dr

V


dr

 e n  e nn  e [Te  Vne  Vee ]
 E  dr 
*
e
 dr    dr 
*
e
 dr V
*
e nn
2
e
2
  Vnn n  dr e  Vnn n
 dr H     dr H 
*
e
e
n  n
n
*
e
e
e
 Ee n
*
*
dr

T


dr

 e n  e Tn (n e )
 Tn n  dr e  Tn n
2
Born-Oppenheimer approximation (continued)
[Te  Vne  Vee ]e (r; R)  Ee ( R)e (r; R)
*E ] ( R)  E* ( R)
[
T

V

 dr T n  drnne Vnne n  dre [Tne  Vne  Vee ]
*
e n

( R*,
r )  n ( R)e (r; R)
 E  dr
e
BO-approximation
• a particular form of the
wavefunction
• neglect the nuclear
derivatives of the electronic
wavefunction
• neglect the kinetic energy
of the nuclei
QC method
• make a guess for the initial
structure
• determine the electronic
wavefunction
• add the nucleur repulsion
• minimize the total energy
• add nucleur kinetic energy