Electronic States of Atoms and Molecules

Question 1

Briefly describe the Bohr-Sommerfeld orbit model and the strength and weaknesses of the model.

Answer 1

Electrons can have ellipical orbitals where r is not constant. n is a measure of the long axis of the orbit.

The model gives resonably accurate one electron measurements and gives a physical insight into orbitals. However it cannot describe multiple electron systems.

Question 2

Describe how standing waves can be used to model electrons.

How is this linked to quantum numbers?

Answer 2

Using wave-particle duality of electrons, the energy of electrons can be found using the Schrodinger equation (which terms represent kinetic energy and potential energy).

These eigenfunctions are translated into the shape of molecular orbitals (radial and angular).

Radial part depends on quantum numbers n and l.

Angular part depends on l and m_l.

Question 3

Give the main 5 quantum numbers of an electron.

Answer 3

• Principle - n
• Angular - l
• Magnetic angular - m_l
• Spin - s
• Magnetic spin - m_s

Question 4

Briefly describe the classical and quantum descriptions of angular momentum.

Answer 4

Classical - Angular momentum is perpendicular to the plane of rotation and is equal to the cross product of the position and momentum vectors. l can be broken down into its x, y and z components, with a magnitude from pythagoros. There are no contraints on these values.

Quantum - The momentum and position vectors are replaced with operators and can also be represented by its x, y and z components, or into spherical polar coordinates. The square also represents the magnitude.

Question 5

Define commutation and how this relates to l values.

Answer 5

Two properties commute if they can be specified simaltaneously, [A,B] = AB-BA = 0. The eigenfunctions of l_x, l_y and l_z cannot be solved simaltaneously but they can for one of the components and l².

We typically choose l_z and l².

Question 6

Describe why the hydrogen emission spectrums peaks changes from 1 to 7 when moving from low to high resolution.

Answer 6

The orbital and spin magnetic moments couple together to form different energetic states. These moments product different j states where j is the vector combination of l and s.

Question 7

Describe the process of spin orbit coupling and how it affects the magnetic quantum numbers.

Answer 7

The l and s vectors add together to give j. The l and s vectors now precess about j which means they are no longer well defined, instead m_j is well defined.

Question 8

Describe how to work out the term symbol for a one electron atom.

Answer 8

Question 9

Describe Russell-Saunders coupling for many electron atoms.

Answer 9

1. s and l give negliable coupling.
2. l values strongly couple together to give L, total orbital angular momentum.
3. s values couple to give S, total spin angular momentum.
4. L and S couple to give J, total angular momentum.

All coupling occurs by the Clebsch-Gordan series; J = L+S, L+S-1,…|L-S|.

Question 10

Describe the Pauli principle and how it underpins the function of a HeNe laser.

Answer 10

The Pauli principle states that the total wavefunction of an exchange of electrons must be anti-symmetric. Hence ΔS = 0 during a change of orbital, Δl = ±1.

Electric discharge is used on He to promote the 1¹s electron to 2³s or 2¹s states which are stable due to the Δl = ±1 selection rule.

The energy is transfered to the Ne which has rapid relaxation.

Question 11

What are the selection rules when L and S are good quantum numbers (R-S coupling)?

Answer 11

Δl = ±1, ΔS = 0

The Parity (Laporte) selection rule states that:

ΔL = 0, ±1, but L=0 <-/-> L=0

ΔJ = 0, ±1, but J=0 <-/-> J=0

Question 12

How do you determine the lowest energy configuration of equivalent electrons and find the ground state J value?

Answer 12

Hunds rules: Electrons with the greatest multiplicity are lowest in energy (highest S), with the lowest energy state of these having the largest L value. (MAXIMISE S, THEN MAXIMISE L).

There are 2 cases for the lowest energy J value:

1. Normal multiplet: The lowest J value is lowest energy when the sub-shell is less than half-filled.
2. Inverted multiplet: The highest K value is lowest energy when the sub-shell is more than half-filled.

Question 13

Define the Stark and Zeeman effects and how they affect the M_J levels of a state.

Answer 13

Stark effect - Applying an electric field to a molecule makes the J vector precess about it, making all states with different |M_J| values have different energy.

Zeeman effect - Applying a magnetic field to a molecule makes the J vector precess about it, making all M_J states non-degenerate.

Question 14

For a H₂⁺ molecule, how does the Stark effect change how the orbitals are represented?

Answer 14

The nuclei generate an electric field along the diatomic axis. The Stark effect makes the |m_l| values non-denerate.

This means that l is not a good quantum number but m_l, the component along the diatomic axis, can be well defined and is reffered to as λ.

Hence for d orbitals:

l=2, m_l=λ= ±2, ±1, 0 (Labels δ, π and σ)

Question 15

Describe the coupling effects in many-electron molecules.

Answer 15

The interaction is similar to R-S coupling in atoms:

• Small l values couple together to give L.
• Spins couple together to give S.

However L couples to the electric field (not S) meaning M_L is the new good quantum number.

ML = Λ = 0(Σ), ±1(Π), ±2(Δ), ±3(Φ),…±L

Question 16

How are the values of Λ determined from the electrons in a molecule?

Answer 16

The values are determined from algebraic sum of individual λ values over all electrons:

Λ = Σλ = λ₁ + λ₂ + λ₃…

Where λ are the different |m_l| values.

e.g. O₂ (π_g 2p)²

For π, λ = ±1, Λ = 2, -2, 0

Question 17

Describe the spin coupling situations for spin in multi-electron molecules.

Answer 17

When Λ = 0, S is not coupled and M_S is undefined.

When Λ =/= 0, the orbital motion of electrons creates a magnetic field that couples S. The M_S component is well defined.

M_S = Σ = S, S-1,…-S

S = s₁+s₂, s₁+s₂-1,…|s₁-s₂|

Question 18

How do you work out the term symbol for molecules?

Answer 18

The new quantum numbers, Λ and Σ, couple together to give the total angular momentum, Ω = |Λ| + Σ.

Question 19

What is the term symbol for closed shell molecules?

Answer 19

The totally symmetric representation is Σ⁺_g

Question 20

What shortcut can be used to find the term symbols of molecules?

Answer 20

Combining the symmetry of partially occupied MOs using the data book (pg 84).

Question 21

What are the selection rules for transitions between MO states?

Answer 21

Question 22

How are equivalent electron spin states restricted in molecules?

Answer 22

The total eigen function must be anti-symmetric with respect to electron exchange so symmetric orbital functions (+) mist be paired with anti-symmetric spin states (singlet) and vice versa.