nuclei electrons
Schrödinger equation:
BO - Ansatz
nuclei electrons
Nuclear positions are parameters, not variables.
Purely electronicSchrödinger equation:
Inserting into yields
Purely electronic Schrödinger equation:
e.g. H2+
Eigenfunction(electronic wave function)
Eigenvalue(electronic energy)
This is the nuclear Schrödinger equation in a given electronic state n:
Inserting the solution into yields
Using
and
,
neglected
yields
Nuclear Schrödinger equation:
The C-state of Ne2
Spectroscopy studies the transitions between states associated with the internal motion
Remember PCIII:
Properties of spherical harmonics:
Rigid rotor
Harmonic oscillator
The nuclear Schrödinger equation therefore becomes:
Using a Taylor series expansion around the equilibrium geometry Re:
=0
Better approximations:
The harmonic and anharmonic oscillators
Harmonic oscillator:
Eigenfunctions of Morse oscillator
Eigenfunctions of Morse oscillator
Dissociation energy:
Different types of electronic states:
In the absence of spin-orbit coupling, the projection L of electronic angular momentum L on molecular axis is conserved.
L
L
Term symbol: 2S+1L(g/u)
(+/-)
Origin of the quantum number L
One-electron Schrödinger equation in axially symmetric potential:
Ansatz:
Inserting in SE and multiplying with r2/Y gives:
LCAO and correlation diagrams