BORN-OPPENHEIMER APPROXIMATION

Diagram of Diatomic molecule R = RB −RA

1. Internuclear coordinate, R, a constant value:

2. Schrodinger Equation for electronic motion:

Hel(ri;R)ψ(ri;R) = Eel(R)ψ(ri;R) (1)

where

Hel(ri;R) =∑i

−h2

2m∇2i︸ ︷︷ ︸

K.E. ofelectrons

+ V (ri;R)︸ ︷︷ ︸P.E. withclampednuclei

(2)

3. Assume the total solution has the form:

Ψ(R, ri) = ν(R)ψ(ri;R) (3)

−h2

2µ∇2R +

∑i

−h2

2m∇2i + V (R, ri)

ν(R)ψ(ri;R)

= Eν(R)ψ(ri;R) (4)

4. Kinetic energy operator of the nuclei:

−h2

2µ∇2RΨ(R, ri) = −h

2

2µ∇2Rν(R)ψ(ri;R)

= −h2

2µ

ψ∇2Rν + 2∇Rψ · ∇Rν + ν∇2

Rψ .

−h2

2µ∇2RΨ(R, ri) = −h

2

2µψ(ri;R)∇2

Rν(R)

(5)

5. ’Simplified’ Schrodinger Equation:

ψ−h2

2µ∇2Rν(R)

+

∑i

−h2

2m∇2i + V

ψ︸ ︷︷ ︸

our clampedsolutionsfrom 1 =Eelψ(ri;R)

ν(R) = Eν(R)ψ

Re-arranging (note we can now cancel out theψ(ri;R), as we do not operate on them):

−h2

2µ∇2R + Eel(R)

ν(R) = Eν(R) . (6)

One dim. eq. for nuclear motion, Eel(R) actslike a potential.