Equations

Question 1

Draw the conversion chart.

Answer 1

Question 2

State the Born-Oppenheimer approximation.

Answer 2

Ŷ_tot = Ŷ_elŶ_nuc

E_tot = E_el + E_nuc

Ŷ: wavefunction

E_tot: total energy

el: electron
nuc: nucleus

Question 3

State the Boltzmann Law.

Answer 3

At thermal equilibrium, the relative population of the i^th energy level is given by:

n_i/n₀ = g_i/g₀ e^-∆E/k_BT

g: degeneracy

∆E: energy gap

n/n: ratio of the population at n_i compared to n₀

Question 4

State the equation for the energies of hydrogenic orbitals.

Answer 4

E = - (Z²R_X)/n²

Z: nuclear charge

n: principal quantum number

R_X: Rydberg constant for tht atom or ion

Question 5

State the Rydberg constant.

Answer 5

R_X/cm^-1 = (µe⁴)/8h³ce₀²

*Value depends on the mass of the nucleus.

µ: reduced mass

Question 6

State the equation for calculating ionisation energy.

Answer 6

∆E = Z²R_X (1/n₁² - 1/n₂²)

Question 7

State the generic atomic term symbol + annotate.

Answer 7

^2S+1L_J

2S + 1: Spin multiplicity, where S is the total spin quantum number.

L: total orbital angular momentum quantum number.

J: total angular momentum quantum number (how L and S are coupled).

Question 8

What is the atomic term symbol for closed-shell atoms and ions?

Answer 8

¹S₀

e.g. He, Xe, Li⁺, Mg²⁺

Question 9

What is the term symbol for alkali metal atoms?

Answer 9

²S_1/2

i.e. outer shell is Xs¹

Question 10

What is the atomic term symbol for excited alkali metal atoms?

Answer 10

²P_3/2 and ²P_1/2

Question 11

State the selection rules for what we see in a spectrum.

Answer 11

1. ∆J = 0, +/- 1 (not 0 ⇔ 0).
2. ∆S = 0

Question 12

State the Pauli exclusion principle.

Answer 12

No two electrons in an atom/ion can have all quantum numbers the same.

Question 13

What are Hund’s Rules for energy ordering?

When can we apply them?

Answer 13

1. The term with largest S is lowest in energy.
2. For a given S, the term with the largest L is lowest in energy.
3. For a term with several levels:

• If the sub-shell is less than half full, the lowest J level is lowest in energy.
• If the sub-shell is more than half full, the highest J level is lowest in energy.

⇒Only applies to ground state atoms.

Question 14

What is the atomic term symbol for a group state N atom?

Answer 14

⁴S_3/2

Question 15

What is the atomic term symbol for a ground state O atom?

Answer 15

³P₀

Question 16

State the equation for the moment of inertia (I).

Answer 16

I = Σ_im_ir_i²

m_i: mass of the i^th particle

r_i: its perpendicular distance from the axis

Question 17

State the equation for the moment of interia for diatomic molecules.

Answer 17

I = µR²

µ: m₁m₂ ÷ m₁ x m₂

I: rotational equivalent of mass

Question 18

State the eigenvalues for rotational energy levels.

Answer 18

F (J) (=E_J) = BJ (J+1)

J: rotational quantum number

B: rotational constant

Question 19

State the equation for rotational constant B in Hz and cm^-1.

Answer 19

1. B (Hz) = h ÷ 8π²I
2. B (cm^-1) = h ÷ 8π²Ic

⇒ B in cm^-1 is most common, with c being in units of cm^-1

Question 20

What are the rotational selection rules? (3)

Answer 20

1. Heterodiatomics must have a permanent dipole moment to exhibit a pure rotational spectrum.
2. Transitions occur for ∆J = +/- 1
3. Transitions observed at 2B(J+1).

Question 21

State the equation of B used to extract bond lengths.

Answer 21

B = 1 ÷ µR²

Question 22

What are the other points important to rotational spectroscopy? (3)

Answer 22

1. R is isotope independent, is determined by electronic structure.
2. B is isotope dependent - spectra will have different spacings.
3. In the absense of an applied magnetic field, each K levels exhibits (2J+1)-fold degeneracy arising from projection quantum number M_J, levels are split when a magnetic field is applied.

Question 23

What is the equation for the most populated rotational level?

Answer 23

J_max = (-/k_BT ÷ 2B) - 1/2

-/: square root

Question 24

State the equation for rotational terms + why the correction term is required.

Answer 24

F(J) = BJ(J+1) - D[J(J+1)]²

D: centrifugal distortion constant (cm^-1)

The correction term is required because molecules are not rigid rotors, bonds stretch slightly during rotation.

Question 25

State the equation for bond strength from vibrational frequency with units.

Answer 25

w_e² = 4B³ ÷ D

in (cm^-1)²

Question 27

State the B values for H₂, HCl and CO with the standard equation for B.

Answer 27

B = h² ÷ 2I

where h = h ÷ 2π

• H₂: 60.85 cm^-1
• HCl: 10.59 cm^-1
• CO: 1.93 cm^-1

h in equation is the h with the cross stick.

Question 28

State Hooke’s Law.

Answer 28

F = -k_Fx

• k_F: force constant (Nm^-1)
• x: displacement from equilibrium

Question 29

Give the equation for classical harmonic frequency of oscillation.

Answer 29

v_vib = (1 ÷ 2π) x (-/k_F ÷ µ)

where µ = (m₁m₂) ÷ (m₁ + m₂)

Question 30

Give the 2 equations for harmonic vibrational energy levels.

Answer 30

1. w_e = (v_vib ÷ c) = (1 ÷ 2πc) x (-/k_F ÷ µ)
2. G(v) = (v + 1/2) x w_e

v: vibrational quantum number (0, 1, 2…)

w_e: vibrational constant (cm^-1)

k_F: force constant (Nm^-1)

µ: reduced mass

c: speed of light in cm^-1

Question 31

State the generic molecular term symbol.

Answer 31

Question 32

What is the molecular term symbol for N₂ ground state?

Answer 32

¹Σ_g⁺

Question 33

What is the molecular term symbol for NO ground state?

Answer 33

²Π_1/2

Question 34

What is the molecular term symbol for O₂ ground state?

Answer 34

³Σ_g^-

Question 35

State the general principles for electronic state labels (3).

Answer 35

1. Ground electronic state - label X.
2. Higher electronic states with the same spin multiplicity as the ground state are labelled A, B, C…
3. Higher electronic states with a different spin multiplicity are labelled a, b, c…

Question 36

What is the assumption of the Frank-Codon principle?

Answer 36

Electronic transitions take place on such a short timescale that the nuclei remain frozen (R unchanged) during the transition.

Question 37

What does the Frank-Codon principle say about vibrational structure?

Answer 37

The overlap of the vibrational wavefunctions is governed by the nature of the electronic states involved.

Question 38

What is the equation for the dissociation limit in electronic spectroscopy?

Answer 38

G (v_max) = D_e = (w_e²) ÷ (4w_ex_e)

Question 39

What is the equation for the lower state dissociation energy?

Answer 39

D₀” = v₀ + D₀’ - ∆v_atomic

v: v with squiggle over the top