27 Molecular Physics
In this section, we will study the binding and excitation of simple molecules. Atoms bind into
molecules by sharing electrons, thus reducing the kinetic energy. Molecules can be excited in three
ways.
- Excitation of electrons to higher states.E∼4 eV
- Vibrational modes (Harmonic Oscillator). Nuclei move slowly in background of electrons.E∼
0.1 eV - Rotational modes (L=n ̄h). Entire molecule rotates.E∼0.001 eV
Why don’t atoms have rotational states?
The atomic state already accounts for electrons angular momentum around the nucleus.
About which axes can a molecule rotate?
Do you think identical atoms will make a difference?
This material is covered inGasiorowicz Chapter 20,and inCohen-Tannoudji et al. Com-
plementsCVI,EV II,CXI.
27.1 The H+ 2 Ion
The simplest molecule we can work with is the H+ 2 ion. It has two nuclei (A and B) sharing one
electron (1).
H 0 =
p^2 e
2 m−
e^2
r 1 A−
e^2
r 1 B+
e^2
RABRABis the distance between the two nuclei.
The lowest energy wavefunction can be thought of as a (anti)symmetric linear combination of an
electron in the ground state near nucleus A and the ground state near nucleus B
ψ±(
~r,R~)
=C±(R) [ψA±ψB]whereψA=
√
1
πa^30 e−r 1 A/a (^0) is g.s. around nucleus A.ψAandψBare not orthogonal; there is overlap.
We must compute the normalization constant to estimate the energy.
1
C±^2
=〈ψA±ψB|ψA±ψB〉= 2± 2 〈ψA|ψB〉≡ 2 ± 2 S(R)
where
S(R)≡〈ψA|ψB〉=
(
1 +
R
a 0+
R^2
3 a^20)
e−R/a^0These calculations are “straightforward but tedious” (Gasiorowicz).