Chapter 5: Molecules

Question 1

List three types of quantised motion in a diatomic molecule.

Answer 1

In a diatomic molecule the (i) rotational, (ii) vibrational and (iii) electronic motions are all quantised.

Question 2

Under what conditions can the Born-Oppenheimer approximation be employed in solving the time-independent Schr¨odinger equation for a molecule?

Answer 2

The Born-Oppenheimer approximation can be employed in solving the timeindependent Schr¨odinger equation for a molecule when the motion of the electron(s) is much faster than the motion of the nuclei. This is generally the case in low energy electronic states when the nuclei are much heavier than the electrons.

Question 3

Give a general expression for the form of the total wavefunction of a diatomic molecule under the Born-Oppenheimer approximation.

Answer 3

Question 4

Give an expression for the Hamiltonian of the molecular hydrogen cation H+ 2 .

Answer 4

Question 5

Write the wavefunctions for the two lowest-lying electronic states of H+ 2 in terms of linear combinations of atomic orbitals.

Answer 5

Question 6

What is meant by the term overlap integral when referring to the electronic structure of H+ 2 ?

To what values does the overlap integral tend as R → ∞, and as R → 0?

Answer 6

The overlap integral in H+ 2 is the integral of the product of the 1s electronic orbital centred on proton A and the 1s electronic orbital centred on proton B, and its value depends on the internuclear distance R. As R → ∞ the value of the overlap integral tends to zero. As R → 0 the value of the overlap integral tends to one.

Question 7

How does the symmetry of the electronic wavefunctions affect the stability of the H+ 2 cation?

Answer 7

Question 8

What do the functions describing the energy dependence of the electronic states of H+ 2 represent within the Born-Oppenheimer approximation?

Answer 8

Within the Born-Oppenheimer approximation the functions describing the energy dependence of the electronic states of H+ 2 represent the potentials in which the nuclei move.

Question 9

Give an expression for the Hamiltonian of the neutral H2 molecule in atomic units

Answer 9

Question 10

Give expressions for the electronic wavefunctions of the two lowest energy states of the H2 molecule that account for the spins of the electrons.

Answer 10

Question 11

What effect do the overlap, Coulomb and exchange integrals have on the stability of the molecule?

Answer 11

Question 12

What do the functions describing the energy dependence of the electronic states of H+ 2 represent within the Born-Oppenheimer approximation?

Answer 12

Within the Born-Oppenheimer approximation the functions describing the energy dependence of the electronic states of H+ 2 represent the potentials in which the nuclei move.

Question 13

How does the symmetry of the electronic wavefunctions affect the stability of the H+ 2 cation?

Answer 13

In the case of the symmetric electronic wavefunction, ψ+, for any internuclear distance R, there is always a non-zero probability of finding the electron at every position between the two protons. For some internuclear distances this non-zero electron charge density between the two protons neutralises the Coulomb repulsion between them and leads to the formation of a stable bound molecular state. In the case of the antisymmetric electronic wavefunction, ψ−, for any internuclear distance R, the probability of finding the electron at the mid-point between the two protons is always zero. Consequently, the Coulomb repulsion between them cannot be completely neutralised and a stable bound molecular state cannot be formed. This antisymmetric electronic state is repulsive at all internuclear distances.

Question 14

Give an expression for the Hamiltonian of the neutral H2 molecule in atomic units

Answer 14

Question 15

Answer 15

Question 16

What are the three main types of transitions that can occur between the quantum states in a diatomic molecule, and in what regions of the electromagnetic spectrum do these transitions lie?

Answer 16

The three main types of transition that can occur between the quantum states in a diatomic molecules are: - Pure rotational transitions ⇒ Microwave region - Vibration-rotation transitions ⇒ Infrared region - Electronic transitions ⇒ Visible/ultraviolet regions

Question 17

Answer 17

Question 18

What is the difference between De and D0 when referring to the dissociation energy of a diatomic molecule?

Answer 18

De is the equilibrium dissociation energy, the energy difference between the minimum of the Born-Oppenheimer potential energy curve and the energy of the system as the internuclear distance tends to infinity. D0 is the dissociation energy from the ground vibrational state with v = 0 in the Born-Oppenheimer potential. It accounts for the zero-point vibrational energy of the molecule and is therefore slightly smaller than De.

Question 19

For what range of parameters can the Born-Oppenheimer potential energy curve of a diatomic molecule be treated using the harmonic approximation?

Answer 19

The Born-Oppenheimer potential energy curve of a diatomic molecule can be treated using the harmonic approximation for small displacements from the equilibrium internuclear distance and for low-lying vibrational states

Question 20

What property must a diatomic molecule possess for it to undergo pure rotational transitions?

Answer 20

Question 21

In the pure rotational spectrum of NaCl why are the spectral lines equally spaced?

Answer 21

Question 22

Derive Spin orbit equation