BORN-OPPENHEIMER APPROXIMATION
Diagram of Diatomic molecule R = RB − RA
1. Internuclear coordinate, R, a constant value: 2. Schr¨odinger Equation for electronic motion: ˆ el(ri; R)ψ(ri; R) = Eel(R)ψ(ri; R) (1) H where 2 −¯ h ˆ el(ri; R) = H ∇2i + V (r{zi; R)} | i 2m | {z } P.E. with K.E. of electrons clamped nuclei 3. Assume the total solution has the form: X
Ψ(R, ri) = ν(R)ψ(ri; R)
h ¯2
−¯ h2
(2)
(3)
X 2 2 − ∇R + ∇i + V (R, ri) ν(R)ψ(ri; R) i 2m 2µ
= Eν(R)ψ(ri; R) (4)
4. Kinetic energy operator of the nuclei: −¯ h2 2 h ¯2 2 ∇RΨ(R, ri) = − ∇Rν(R)ψ(ri; R) 2µ 2µ h ¯ 2 =− ψ∇2Rν + 2∇Rψ · ∇Rν + ν∇2Rψ . 2µ −¯ h2 2 h ¯2 ∇RΨ(R, ri) = − ψ(ri; R)∇2Rν(R) 2µ 2µ (5) 5. ’Simplified’ Schr¨odinger Equation:
ψ −
h ¯2 2µ
2 ∇Rν(R) +
X
−¯ h2
i 2m
2 ∇i + V ψ ν(R) = Eν(R)ψ
|
{z
}
our clamped solutions from 1 = Eelψ(ri; R) Re-arranging (note we can now cancel out the ψ(ri; R), as we do not operate on them):
−
h ¯2 2µ
2 ∇R + Eel(R) ν(R) = Eν(R) .
(6)
One dim. eq. for nuclear motion, Eel(R) acts like a potential.