pcv zoom lecture 2 - eth z · 2020-03-27 · motion electronic vibrational r otational typical...

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