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Year 2 - Chem > Atomic & Rotational Spec > Flashcards

Atomic & Rotational Spec Flashcards

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1
Q

How is an EM wave described?

A

Transverse wave of perpendicular, sinusoidally oscillating electric and magnetic fields

E = E0 sin(kx - ωt + φ)

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2
Q

What is cvac?

A

299,792,458 ms-1

Work out as c2vac = 1 / μ0ε0, and μ0 given in exam

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3
Q

What is the linear momentum of a photon?

A

ρ = E/c = hν/c = h/λ

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4
Q

What is the angular momentum of a photon?

A

Quantum number: jph = 1

Magnitude: |jph| = Sqrt(jph (jph +1)) = hbar * Sqrt(2)

As bosons - integer spin

Helicity of +/- 1 only - not 0 as projects AM on direction of travel, gives left or right circually polarised light

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5
Q

What is wavelength and frequency dependent on?

A

Wavelength - refractive index of medium

Frequency - independent of the medium

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6
Q

What is the wavenumber?

A

vbar = 1 / λvac

Units: cm-1

E = hc* vbar

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7
Q

What is a common units mistake with wavenumbers?

A

1 cm-1 = 100 m-1

VERY COMMON MISTAKE

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8
Q

What is the hamiltonian of a molecular system?

A

H^tot = H^e + H^n = T^e + V^ee + V^ne + T^n + V^nn

T^ is KE operator

V^ is PE operator between different particles

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9
Q

What is the Born-Oppernheimer Approx?

A

φtot = φel(q,Q)φn(Q)

Etot = Eel+Enuc

Where q is electron coordinates, Q is nuclear coordinates

Can be done due to difference in mass between e- and nuclei, so can separate motion

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10
Q

What are the transition required for the different forms of spectroscopy?

A

From largest ΔE:
Electronic - different electronic states (arrangement, or MOs/AOs), 500-100 nm so UV-Vis

Vibrational - different vib states of one elec state, 100 nm -2 μm, infrared

Rotational - different rot states of one vib state, 10 cm - 1mm, microwave

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11
Q

What is population of energy level i in Botlzmann law?

A

ni = (N/q) * gi * exp(-Ei/kT)

where q is molecular partition function

gi is the degen of ith level

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12
Q

What is the formula for molecular partition function?

A

q = Σi gi * exp(-Ei/kT)

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13
Q

What are the three standard interactions of light and matter?

A

Stimulated absorption - M + hn -> M*

Stimulated emmision - M* + hn -> M + 2hn

Spontaneous emmision - M* -> M + hn

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14
Q

What occurs in stimulated absorption and what is its rate?

A

Photon lost and system absorbs its energy, must have exact energy difference between E1 (lower) and E2 (higher)

Rate of absorption: dn1/dt = -B12 * ρ(E21)*n1

Where B12 is Einstein coefficient, and ρ(E21) is radiation enerergy density

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15
Q

What is the radiation energy density?

A

ρ(E) = (8πhv3/c3)(1/exp(E/kT)-1)

energy of radiation field in m-3

Exy is when energy between x and y energy level

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16
Q

What occurs in stimulated emmision?

A

Photon hits excited e-, additional photon created with same frequency, polarization, direction and phase of original

e- relaxes to lower e- state

dn2/dt = -B21*ρ(E21)n2

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17
Q

What occurs in spontaneous emmision?

A

Photon created and e- relaxes from “E2 ->E1”

dn2 /dt = -A*n2

Where A is einstein coefficient for spontaneous emmision

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18
Q

What occurs to Einstein coefficients at eqm?

A

M -> M* and M* -> M

dn1/dt = 0 so B12ρ(E21)n1 = A21n2 + B21ρ(E21)n2

Simplifies to give g1B12 = g2B21 and A21 = (8πhv3/c3)*B21

A α v3B so only one independent coefficient, decay occurs fastest

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19
Q

What are allowed transitions for electronic spectroscopy?

A

E0 =! 0 as need a photon

Em0 - Ej0 = +/- hω as must conserve energy (photon equal to energy difference)

Transition dipole moment, R21 =! 0

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20
Q

What is the transition dipole moment, R21 ?

A

R21 = 2|μ^|ψ1>

where ψ2 is final, and ψ1 initial

and TDM operator μ^ = Σi qiri^ where q is charge on particle and r^ is position vector

μ^ operates on ψ1 to give new state, TDM therefore represents transition amplitude of ending up in final state ψ2 which is determined by overlap integral of ψ2 with the transformed μ^ψ1

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21
Q

What are the selection rules and transitions for H-Atom in atomic spec?

A

Δn unrestricted

Δl = +/- 1

Δml = 0,+/- 1

Transitions at wavenumber vbar = ΔE/hc = Z2Ry(1/n12 - 1/n22)

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22
Q

How is the Δl = +/- 1 selection rule for H-atom spec dervied?

A

Photon as AM of | lphoton| = Sqrt2 * hbar

Total AM must be conserved in emssion/absorption process: lF = li + Sqrt2*hbar

But lF is quantised to 0, sqrt2*hbar, sqrt6*hbar, etc.

Vectorially, max and min when Δl = +/- 1

Δl =! 0 for a different reason

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23
Q

What is the magnitude of l for a H-atom?

A

Quantised

projection on z-axis lz = ml * hbar

l | = hbar * Sqrt(l*(l+1)) = 0, Sqrt2 * hbar, Sqrt6 * hbar, etc

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24
Q

Why is Δl =! 0 for a H-atom spectra?

A

Non-zero TDM so integrand must be totally symmetric under symmetry of group

μ^ is an odd operator so ψF and ψi must have opposite parities as (-1)l is the parity of an AO

Therefore Δl =! 0 for symmetry reasons

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25
Q

Where does the selection rule of Δml = 0, +/- 1 for a H-atom spectra arise from?

A

Due to the helicity of the photon being σ = +/- 1

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26
Q

What is the problem with the schrodinger equation for atoms other than H?

A

Electron repulsion Σi =!j Vij term makes it insolvable

As in general need 3N spatial and N spin coordinates

Solved using orbital approx

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27
Q

What is the orbital approx for a many e- system?

A

Assume ψspace is product of n, one-e- wavefunctions/orbitals

Each orbital has [(-hbar2/2me) * ∇i2 + ViN + average(Σi=!j Vij)]*φ(ri) = E*φ(ri)

average of sum means that e- i experiences mean field of all other electrons

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28
Q

What are the problems with the orbital approx?

A

ψspace should be a linear combination of orbitals with define symmetry wrt e- permutation

Is not so doesn’t satisfy Pauli’s principle

Also neglects spin and correlation

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29
Q

How does the energy of a ns orbital compare in a alkali metal to a H-atom?

A

Energy level now depends on n and l

for alkali metals the ns orbital experiences more attraction as higher Z and penetrating so lower in E

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30
Q

What are valence excitations in atomic spec?

A

Core excitations at higher energies

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31
Q

What factors can be used to account for spectrum of non-H atoms?

A

Zeff - change in the Z

Quantumm defect - changes the n quantum number

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32
Q

What are the different series present in emission atomic spec?

A

Sharp: ns -> np

Principal: np -> ns

Diffuse: nd -> np

Fundamental: nf -> nd

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33
Q

What are the selection rules for non-H atomic spectra?

A

Δn is unrestricted

Δl = +/- 1

Δml = 0, +/- 1

Δj = 0, +/- 1

ΔS = 0

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34
Q

What causes spectral fine spectra in atomic spec?

A

Spin orbit coupling - increases as atomic number does

Orbital AM l couples to spin AM s to give a total AM j

j = s+ l

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35
Q

What are the allowed values of j in atomic spec?

A

Range given by Clebsch-Gordon series

j = l+s, l+s-1, …, | l-s|

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36
Q

How is the spin-orbit coupling constant calculated in atomic spec?

A

l * s = 1/2 *(j2 - l2 - s2) from j = l + s

Sub in eigenvalues: l * s = hbar2/2 * [j(j+1) - l(l+1) - s(s+1)]

E of a given level: E = 1/2 hcA * [j(j+1) - l(l+1) - s(s+1)]

where A is spin-orbit coupling constant proportional to Z4/n3l3

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37
Q

What is the fine spectra which arises from spin-orbit coupling?

A

line at specific l and s splits to two lines for different j values

lines at j values split to 2j+ 1 due to different mj values ( when in absence of external fields)

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38
Q

Where does the Δj = 0,+/-1 selection rule in atomic spec arise from?

A

Conservation of angular momentum

NOTE: cannot go from 0 to 0 as absorption of photon gives angular momentum

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39
Q

What is russel-saunders coupling?

A

AM L and total spin AM S arise from additions of possible li or si from Clebsch-Gordon

Couple to give total AM J = L+S, L+S-1, …, |L-S|

Coupling is the different values of J which can be seen

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40
Q

What are fermions and bosons wrt spin?

A

Fermions have 1/2 integer spin

Bosons have integer spin

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41
Q

What is the Pauli principle?

A

Wavefn must be anti-symmetric wrt exchange of two identical fermions and symmetric wrt exhange of bosons

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42
Q

What are the spin combinations for two e-?

A

α(1)α(2), β(1)β(2) - both are symmetric

α(1)β(2), α(2)β(1) - no defined symmetry so must take linear combination

1/sqrt2[α(1)β(2) +/- α(2)β(1)] where + is symm and - is anti

First two and + linear combination form a triplet which are symm

Final - linear combination is a singlet

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43
Q

What is the singlet spin state?

A

One e- spin up and the other down

Cancellation to give S = 0, S = 0, Ms = 0, a single arrangement

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44
Q

What is the triplet spin state?

A

Three arrangements with S = 1

Ms = 0 - one up one down spin

Ms = +1 - two up spin

Ms = -1 - two down spin

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45
Q

How are triplet and singlet states paired with spin functions?

A

ψ = (1/sqrt2) [φ1s(1)φ2s(2) +/- φ1s(2)φ2s(1)]

This linear combination is to give symmetry, + is symm and - is anti

Triplet has symm spin wavefn so must have ψ-

Singlet has antisymm spin wavefn so must have ψ+

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46
Q

Explain why e- with wavefn ψ+/- have different chances of being found at the same place

A

P^(ψ-(1,2)) = -ψ-(2,1)

when r1 = r2 then ψ-(1,1) = -ψ-(1,1) so ψ-(1,1) = 0 and 2e- cannot exist at same place

however for ψ+ there is a max

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47
Q

What is the Fermi hole and heap?

A

e- in the triplet have a 0 prob of being in same location - seen as a dip in graph

e- in the singlet have a maximum in graph when in same location, so more repulsion

Means triplet is lower in energy overall

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48
Q

What does the ΔS = 0 selection rule mean for Grotian diagrams?

A

Singlet to triplet or vice versa transition forbidden

As every triplet state is lower than corresponding singlet

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49
Q

What does: configuration, terms, levels, and states refer to?

A

Configuration: number of e- in each orbital

Terms: different L & S config (atomic symbols in SL), from spin correlation

Levels: different J levels of terms, from magnetic coupling of total spin and total orbital AM

States: all 2J+1 possibilites of the mj

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50
Q

What must you note when working out the possible term symbols?

A

Pauli Exclusion principle - cannot have identical quantum numbers (q.n.)

E.g. 3D for Carbon has L=2 and ML = 2 compenent, meaning ml1 = ml2 = 1 and S = 1 so Ms=1 and so ms1 = ms2 = 1/2, same l and n so has same q.n.

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51
Q

What is a microstate table?

A

Table of every combination of ms and ml

Can eliminate all terms of different Σml values

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52
Q

What are the lowest energies for different terms in Russell-Saunders coupling?

A

Term with largest S is lowest in energy

For given S the term with largest L is lowest in energy

When several levels: if subshell less than half full then lowest J level is lowest in energy but if more than half full then highest J level

Assumes spin correlation >> orbital am coupling >> spin-orbit coupling

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53
Q

When is LS (Russell-Saunders) coupling relevant?

A

Only for ground states of atoms

Valid for period 1-2 and some TM

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54
Q

When does LS coupling occur and when does j-j coupling occur?

A

At large Z spin-orbit coupling is strong then j-j coupling

As spin prefers to couple with its own orbital a.m. to give total j

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55
Q

What is the notation for j-j coupling?

A

ji - total a.m. of individual e- orbital a.m. and spin a.m. couple

J - sum of ji

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56
Q

What is the normal Zeeman effect and when is it applicable?

A

Applicable when S=0, so is a singlet

External field B lifts degeneracy of ML components which leads to splitting in the spectrum

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57
Q

What is the interaction energy in the classic Zeeman effect?

A

Classically: E = -m*B = -γeLzB

QM: E= -γeMLB*hbar = μBMLB

where γe = -e/2m and μB (bohr magneton) = -γe*hbar

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58
Q

What is the selection rule involved in the normal zeeman splitting?

A

ΔML = 0, +/- 1

Single signal therefore splits

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59
Q

What is the anomalous zeeman effect?

A

When atom of S =! 0 is subjected to an external field

Causes splitting into 2J+1 levels, and the energy of which is dependent on J, L and S

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60
Q

What is the energy splitting in the anomalous zeeman effect?

A

E = -γe(L + 2S).B

Only need to consider projection of L & S on B

L.B = (L.J**)(**B**.**J)/J2

E = -γegJB**.**J = -γegJMJB*hbar

So proportional to MJ and B, not degen splitting so numerous peaks for each

ΔMJ

where Lande factor gJ (J,L,S) = 1 + [J(J+1) + S(S+1) - L(L+1)]/2J(J+1)

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61
Q

What is the range for transitions in rotational spec?

A

10cm to 1mm wavelength

In the microwave

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62
Q

What is the moment of inertia in a molecule?

A

I = Σi miri2 where mi is mass of ith particle and ri is perpendicular distance from axis

Rotational equivalent of mass

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63
Q

What is the angular momentum and rotational KE of a molecule?

A

J = I*ω

E = J2/2I = 0.5*I*ω2

(where I is inertia)

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64
Q

What is the diatomic rigid rotor equivalent to in QM?

A

Rigid rotor represents particle on a sphere

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65
Q

What is the rotational energy in a particle on a sphere?

A

Purely kinetic energy

H^ψ = [-(hbar)2/2μ]*∇2ψ = Eψ

Where ∇2 = δ2/δr2 + (2/r)(δ/δr) + (1/r2)*Λ2 in spherical polar coordinates

Cyclic boundary conditions that ψ(φ+2π) = ψ(φ)

Gives Spherical harmonics as the solutions

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66
Q

What are the energy eigenvalues of the particle on a sphere?

A

EJ,Mj = j^2/2I = J(J+1)*hbar2/2I = BJ(J+1)

with rotational q.n. J = 0,1,2,3…

rotational a.m. projection q.n. mJ = 0, +/- 1, +/- 2… +/- J

B is rotational constant in joules B = hbar2/2I

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67
Q

What are rotational spec values given in?

A

Given as wavenumbers or rotational terms

B~ = B/hc = h/8π2cI = h/ h/8π2cμR2

No zero point energy associated with rotation - get more widely spaced with increasing J

As B~ α 1/μR2 for diatomics relate this to bond lengths <1/R2>

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68
Q

What is rotational spec dependence on isotope?

A

Reduced mass does change so B~ ratio changes

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69
Q

What is the degeneracy present in rotational levels when no external field?

A

Each J level has 2J+1 degenerate states

Arises from projection quantum number MJ

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70
Q

What is the population of J levels in rotational spec?

A

From Boltzmann distribution, ni α gj*exp(-Ej/kT)

gj = 2J+1 and EJ = hcB~J(J+1)

Most populated when dni/dJ = 0

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71
Q

What is the most populated J-level in rotational spec?

A

Energy relative to ZPE, NJ/N0 = (2J+1)*exp(-E/kT)

dni/dJ = 0

dni/dJ = [2-(2J+1)2*hcB~/kT} * exp[-hcB~J(J+1)/kT] = 0

Jmax = Sqrt[kT/2hcB~] - 1/2

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72
Q

How does the most populated J-level in rotational spec change with T and spacing of the levels?

A

As temp increases the more are in a higher most populated rot J-level

Higher B, and hence spacing, then lower the most populated rot J-level

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73
Q

What is the general solution of a rotational wavefunction?

A

ψml (φ) = Exp(imlφ)/Sqrt[2π]

Where ml = +/- Sqrt[2EI]/hbar

Cyclic boundary conditions restrict to ml = 0, +/- 1, +/- 2, +/- 3

Real part plotted shows symmetry, odd & even J levels have opposite parity

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74
Q

What is the parity of different rotational wavefunctions?

A

Parity - symmetry wrt inversion

Parity = (-1)J where J is the J level occupied

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75
Q

What is a gross selection rule?

A

Properties required to do a form of spectroscopy

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76
Q

What is the gross selection rule for rotational spectra?

A

Must posses a permanent dipole moment

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77
Q

What is the specific selection rule of rotational spec?

A

ΔJ = +/- 1 : Due to conservation of a.m. as need to change parity

ΔK = 0 : required for non-linear or polyatomic

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78
Q

What is the energy in rotational spec given in terms of FJ?

A

FJ = EJ / hc = B~J(J+1)

where B~ = B/hc

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79
Q

At what energies are transitions observed in rotational spec?

A

v~ = FJ+1 - FJ = B~(J+1)(J+2) - B~J(J+1)

v~ = 2B~(J+1)

Equally spaced lines with separation of 2B

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80
Q

How do the gaps between J levels change as J increases? (rotational spec)

A

As J increases the gap between levels increases due to J2 dependence of rotational eigenvalues

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81
Q

What info does rotational spec give you?

A

Info on geometry

For diatomic molecules it gives bond length

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82
Q

What is the profile of rotational spec determined by?

A

Population of lower levels and J dependence of transition strength

See that population of many levels as kBT large wrt rotation energy

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83
Q

How are polyatomic molecules defined in rot spec?

A

Moments of inertia about three mutually perpendicular axes through centre of mass

Called axis a,b and c

so Ia < Ib < Ic with Ic - max

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84
Q

What is a spherical top in rotational spec?

A

Molecule with zero dipole moment so doesn’t appear in spectra

Ia = Ib = Ic

E.g. Td, Oh

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85
Q

What are the classes of moleucles in rot spec?

A

Spherical tops - identical Ii

Symmetrical tops - two identical Ii

Asymmetrical tops - all unique Ii

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86
Q

What are the symmetrical tops in rot spec?

A

Prolate tops - long and thin molecules with Ia b = Ic

Oblate tops - discus type molecules with Ia = Ib < Ic

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87
Q

What are the rotational terms for diatomics?

A

Constant for each moment of inertia for the three axes

A~ > B~ > C~ as Ic is the largest

Cannot relate to bond lengths individually as includes angles and lengths

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88
Q

What is the Ka qn?

A

Projection quantum number for projection on the unique a axis

Where a is body-fixed axis in prolate, and c in oblate

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89
Q

How are prolate tops labelled in rot spec?

A

JKa - J is total a.m. and Ka is projection q.n.

Each level has 2J+1 degen (from MJ) and if K>0 has two fold degen of +/-K

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90
Q

What are the permitted values of J and K in prolate tops in rot spec?

A

FJ,K = B~J(J+1) + (A~-B~)K2

So J = 0, 1, 2, 3…. and K = 0, +/- 1, +/- 2,…, +/- J

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91
Q

What is the magnitude of J and projection of J (Ja) in a prolate top (rot spec)?

A

|J| = Sqrt[J(J+1)] * hbar

Ja = K * hbar , is what extent it rotates wrt principle axis

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92
Q

What does the relation of K and J mean in rot spec of a prolate?

A

When K≈J then rot a.m. is nearly parallel to principle axis

K = 0 then rot a.m. is perpendicular to principle axis

93
Q

What is the notation for levels in rot spec of an oblate top?

A

JKc where J is total a.m. and Kc is projection q.n. (on unique axis, c)

Each level has 2J+1 degen (from MJ) and when K>0 two-fold extra degen (+/- K)

94
Q

What are the J and K values in oblate top rotational spec?

A

FJ,K = B~J(J+1) + (C~-B~)K2

C~ <= B~ so term is -ve

95
Q

What is the relation of K and MJ in rotational spec?

A

NONE, K =! MJ
K refers to projection on a body-fixed axis

MJ is a projection on a space-fixed axis

96
Q

What are the energy levels for symmetric tops in rot spec?

A

Prolate: (A~-B~)K2 > 0 so for a given J the energy increases with K

Oblate: (C~-B~)K2 < 0 so for a given J the energy decreases with K

97
Q

What is the case of rotational spec for linear molecules?

A

Ia = 0 and so A~ = ∞

As only K = 0

FJ = B~J(J+1) and J = 0,1,2,3…

98
Q

What is the case of rotational spec for a spherical top?

A

A~=B~=C~

FJ = B~J(J+1) where J = 0,1,2,3…

Degen = (2J+1)2

No pure rotational spec as no dipole moment

99
Q

What is the spectra for prolate and oblate tops compared to linear?

A

Wihtin rigid rotor approx they are the same

Have equally spaced lines with separation = 2B~

So no info on unique axis

100
Q

What is centrifugal distortion and the problem for rot spec?

A

Bonds stretch during rotation

So inertia and rot constants change with J

Treat as perturbation to rigid rotor as terms not large in molecules at room T

101
Q

What is the rotational term including centrifugal distortion?

A

F(J) = B~J(J+1) - D~J(J+1)2

Where D~ is centrifugal distortion constant in cm-1

D~ = 4B~3/ωe2 and ωe is vibrational frequency (stronger bonds cause lesser distortion)

102
Q

Where are transitions when including centrifugal distortion?

A

Values are smaller than expected - more prominent as J increases

Transitions occur at v~(J) = F(J+1) - F(J) = 2B~(J+1) - 4D~(J+1)3

103
Q

How can you plot v~(J) when there is centrifugal distortion?

A

v~(J) = 2B~(J+1) - 4D~(J+1)3

Plot (J+1)2 on x-axis and v~(J)/(J+1)

Gives intercept = 2B~ and gradient = -4D~

104
Q

Where do transitions occur in symmetric tops with centrifugual distortion?

A

v~ = F(J+1, K) - F(J,K)

v~ = 2(B~-D~JKK2)(J+1) - 4D~J(J+1)3

105
Q

What is the degeneracy of rotational levels?

A

Each J levels has 2J+1 which are degenerate due to space quantization

When there is a field (E or B) the degenercy is lifted

106
Q

What is the linear stark effect?

A

Electric dipole μ interacts with an applied field, results in an interacting energy

μ.E = -μEcosθ

Effects energy levels of symmetrical tops

107
Q

How does the stark effect influence energy of symmetric tops in rot spec?

A

μJ = μ*cosα = μ*K/Sqrt[J(J+1)] where μJ is projection along J

External field making an angle β to J which leads an interaction energy

Estark = -μJ*E*cosβ = -μEKMJ/[J(J+1)]

where E is electric field, and MJ means there are different values

108
Q

What is the stark energy proportional to?

A

Estark α |J|-2​

109
Q

How does the stark effect cause splitting in symmetric tops?

A

As depends on MJ it causes distinct energy levels

Has larger splitting when J is smaller

110
Q

What is seen due to the Stark effect for a symmetric rotor?

A

Selection rule: ΔMJ = 0 which will cause splitting

E.g.

So if from J=1 to 2 there is one peak, when field applied will get 3 separate peaks μE/3 in energy apart for MJ 1, 0, -1 of the J=1

Equal in intensity and only 3 as can only do three transitions from J=1 to 2 to keep selection rule

111
Q

What is the spin angular momentum and quantum number I?

A

I is the a.m. and I is the quantum number

112
Q

How does spin quantum number change with mass number? (for rot spec)

A

Mass number Even: I is integral (0,1,2,3..) nuclei are bosons

Mass number Odd: I is 1/2 integral (1/2, 3/2, …) nuclei are fermions

113
Q

How does spin angular momentum affect energy levels? (in rot spec)

A

Nuclear spin causes magnetic moment which interacts with external magnetic fields & internal magnetic field (gives small splittings)

Determines whether rotlevels in symm molecules actually exist - due to nuclear spin stats

114
Q

What nuclear spin statistics must be considered in rot spectroscopy?

A

ψtot = ψelψvibψrotψns

ψtot if boson is symm, if fermion is antisymm

Consider ground state of molecule, and then work out symmetry of each part, ψeland ψvib are constant

Gives several combination, find weighting of them which is shown in spectra

115
Q

How is the stat weighting of nuclear spin functions?

A

of sym ψns / # of antisym ψns = (I + 1) / I

See the ratio of this or lack of some in the spectra

116
Q

What is the stark energy for a symmetric top?

A

E = -(μEKMJ)/[J(J+1)]

define for J and K then will have splitting to MJ levels

117
Q

How does the stark energy for splitting between MJ change with J?

A

Larger splitting between levels when J small

118
Q

What is the effect of an electric field on rot spectroscopy?

A

Introduces selection rule ΔMJ = 0

Leads to splitting in spectrum to number of MJ levels in the lower state

119
Q

Is ψel symmetric or antisymmetric?

A

Find term symbol for the molecule (e.g. H2 has 1Σg+)

Usually symmetric but need to check, O2 is the normal exception

120
Q

Is ψvib symmetric or anti?

A

All v=0 vib levels are symmetric

121
Q

Is ψrot symmetric or anti?

A

Even J are symmetric, odd J levels are anti

from (-1)J

122
Q

Is ψns symmetric or anti?

A

Nuclear spin

Even mass no then I is an integer and a boson (symmetric)

Odd - I is a half-integer and so fermion (antisymm)

123
Q

Is ψtotal symmetric or anti?

A

Even mass - boson, must be symm
Odd mass - fermion, must be anti

124
Q

How do you do a nuclear spin stat for a question?

A

Find mass number - this determines ψtotal and ψns

Find symmetry of elec state to give ψel

Then find combinations to give correct ψtotal, remember that NS must be correct for the compound!

125
Q

What is vibrational spec?

A

Infrared

126
Q

What is Hookes law and the PE of a compound from it?

A

F = -kFx

PE: V(x) = -∫F dx = (kFx2)/2

127
Q

What is the eigenvalues of a harmonic oscillator?

A

Gv = (v+1/2)ωe

where v is vib qn and ωe is vibrational const

128
Q

What is the equation for ωe?

A

ωe = νvib/c~ = (1/2πc~) Sqrt[kF/μ]

where :
νvib is the classical freq of oscillation
kF is the force const
μ is the reduced mass

129
Q

What is the energy of a harmonic oscillator?

A

E = hc~G~(v+1/2)

E = hν(v+1/2)
where ν is freq

E = hbarωe(v+1/2)

130
Q

What is the ZPE of a harmonic oscillator?

A

E = (1/2)hν
where ν is freq

131
Q

What is the spacing of energy levels in a harmonic oscillator?

A

Equally spaced non-degen energy levels

132
Q

What do prob densities in the harmonic oscillator change at different v qns?

A

As q increases the prob density becomes more classical
Penetration into classically forbidden regions

133
Q

What is the hamiltonian for a harmonic oscillator?

A

H^ = -(1/2)(d2/dq2) + (1/2)q2

where q is vib coord, is 0 at eqm bond length

134
Q

What is the lowest energy eigenfunction of HO?

A

ψ0 = exp(-q2/2)

135
Q

What is the gross selection rule of IR (vib spec)?

A

Dipole moment that changes during vibration
TDM =! 1

136
Q

What is the specific selection rule of IR (vib spec)?

A

Δv = +/- 1
where v is rot qn

137
Q

How are the selection rules of vib spec dervied?

A

R21 =! 0

R21 = (dμ/dq) ∫ψ*qψ dq

(dμ/dq) =! 0, dipole changes - gross selec rules

∫ψ*qψ dq =! 0, leads to Δv=+/- 1 which is specific selec rule

138
Q

What are transitions at with simple HO?

A

v~ = ωe

One line in spectrum

139
Q

How does the potential in HO relate to dissociation limit?

A

As bond stretches the bond gets weaker so slope decreases to limit

140
Q

What is a the morse potential?

A

V(R) = De[ 1-exp(-β(R-Re))]2

Gives more accurate than HO

141
Q

What are the energy eigenvalues in the morse potential?

A

G~ = (v+1/2)ωe - (v+1/2)2ωexe

where v=0,1,2,… vmax
where xe is the anharmonicity constant

142
Q

What is xe, the harmonicity const?

A

xe = ωe/4De

where De is the dissociation energy

143
Q

What is the experimental dissociation energy (D0) in vib?

A

D0 = De - ZPE

where De is dissociation energy to 0

144
Q

What is the specific selection rule of morse oscillator?

A

Δv = +/- 1, (+/-2, +/-3, etc. are weaker)

145
Q

What transition types are present in morse oscillator?

A

Fundamental (0->1)
1st overtone (0->2)
1st hot band (1->2)

146
Q

What can occur when a vib transition occurs?

A

Rot transition also occurs

Etot = Evib + Erot

147
Q

What are the selection rules for vib and rot transitions?

A

Δv = +/- 1, (+/- 2, +/- 3, etc weaker)

ΔJ = +/- 1

148
Q

What is the R-branch and P-branch?

A

For rot vib spec:
when ΔJ = +1, transitions give rise to R-branch with higher wavenumbers

when ΔJ = -1, transitions give rise to P-branch with lower wavenumbers

149
Q

What is the vibrational and rotational terms energy terms?

A

S~ = (v+1/2)ωe - (v+1/2)2ωexe + B~J(J+1)

where only last term is from rot

150
Q

What is the Q-branch in rotation vibration spectra?

A

Signal when ΔJ=0

Absent unless additional angular momentum (from elec a.m.) or degen bending mode present

151
Q

What is the spacing in the R and P branch?

A

Spacing in the branch = 2B~

152
Q

What is the spacing between R and P branch?

A

Central gap = 4B~

153
Q

What is the point of origin (ω0) between the R and P branch in rot vib spec?

A

ω0 = ωe - 2ωexe

154
Q

What is the vib dependence of rotational const?

A

B~ α 1/μR2

Treat as <1/R2>, which decreases with v

155
Q

How does the rotational const at different v relate?

A

B~0 > B~1 > B~v>1

B~v = B~e - α(v+1/2)

156
Q

How can you determine the rotational const (B~) ?

A

Use transitions with common upper/lower levels to determine rot const for each level

157
Q

What is a bandhead in vib spec?

A

At high J the wavenumber separation of adj transitions can change sign

Causes transitions to bunch up

158
Q

What is a normal mode?

A

Independent, harmonic vibrations of polyatomic molecules

These vibrations must: leave centre of mass unmoved, involve all atoms moving in phase, and transform as an irrep of molecular point group

159
Q

What is coherent motion?

A

Means all atoms moving in phase and going back to original position at the same time

160
Q

How many dof does a polyatomic molecule have?

A

N isolated atoms have 3N dof

161
Q

How many trasnslations do linear and non-linear polyatomic molecules have?

A

Linear molecule = 3
Non-linear = 3

162
Q

How many rotations do linear and non-linear polyatomic molecules have?

A

Linear = 2
Non-linear = 3

163
Q

How many vibrations do linear and non-linear polyatomic molecules have?

A

Linear = 3N-5
Non-linear = 3N-6

164
Q

How can you find if a normal mode is an irrep of point group?

A

Perform symm operations on stretches
1 if unchanged and -1 if swapped

165
Q

What are the irreps of normal modes of CO2?

A
166
Q

What is the symmetry of the gs vibrational wavefn?

A

ψ0 transforms as totaly symmetric irrep of relevant point group

167
Q

What is the symmetry of the v=1 excited vib level?

A

ψ1 has the same symmetry as the normal mode
This is because has same symm as normal coord q (which is a linear function)

168
Q

What is the symmetry of vib levels 2 and 3?

A

ψ2 transforms as totally symm IR, as proportional goes via q2

ψ3 transforms as same symm of normal mode, as proportional to q+q3

when degen modes then use direct product tables

169
Q

What is a selection rule for polyatomic molecules which come from a normal mode being harmonic?

A

New one: Δvi = +/- 1

Dipole moment must change with vib (standard one)

170
Q

What is the TDM of a polyatomic molec?

A

TDM separates across x,y, and z
Each is scalar, and at least 1 of them must transform as totally symm irrep

Check if direct prod transform as totally symmetric, with v’ being final

171
Q

What does μx, y, or z transform as?

A

Transforms as x,y,z from character table

172
Q

What is a parallel or perpendicular mode?

A

Parallel - if transition moment of a mode is parallel to symmentry axis

Perp - same as above but perpendicular

173
Q

What is required for a fundamental transition to be allowed and what kind of mode is this?

A

Symm of normal mode (and therefore v=1) is the same as x,y,z
Due to the starting one being totally symm

174
Q

What is a combination band?

A

Simultaneous excitation of both symm and asymmetric stretches

e.g. v1=0, v3=0 -> v1=1, v3=1
Direct prod of how v1 and v3 gives how they transform overall

175
Q

What are present in parallel and perpendicular bands?

A

Parallel (Σ-Σ) - ΔJ=+/- 1, so P & R branches

Perpendicular (Σ-Π) - ΔJ=0,+/- 1 (Q,P, and R branches)

176
Q

Why does a Q branch occur in vib spec?

A

Occurs due to additional angular momentum from degen bending mode

177
Q

What is the Born-Oppenheimer Aprrox?

A

ψtotal(r,Q,θ) = ψel(r) ψvib(r) ψ(θ)

Assumption: single ψel for a given config for a given elec config as separable to nuc coord

178
Q

How does the BO approx effect the PE?

A

1 value for PE at specific separation so e- can arrange in lowest E config quickly

179
Q

What term symbols are used for diatomics?

A

Classified wrt angular momentum around internuclear axis, λ

λ analogous to mI in atoms

180
Q

What is the term symbol for two pz orbitals combining?

A

mI = 0, combine to give

σ,σ* (λ=0)

where σ means cylindrical symm about the internuclear axis

181
Q

What is the term symbol of a diatomic with px and y combined?

A

Each has mI = +/- 1, combine to give

π,π* (λ = +/- 1)
where π means antisymmetry about internuclear axis

182
Q

What is overall angular momentum and how is it calculated?

A

Λ = Σ λi
λ is for each e-

Λ =0 gives Σ
Λ =+/- 1 gives Π
Λ =+/- 2 gives Δ

183
Q

What is the format of the molecular term symbol?

A

2S+1ΛΩ

where
Λ = total angular momentum
Ω = |Λ + Σ|
and Σ = projection of S on internuclear axis

184
Q

When can you give u/g notation for a diatomic?

A

Homonulcear

u - antisymm
g - symm

185
Q

What are the direct products between u and g states?

A

g x g = g

u x u = g

g x u = u x g = u

186
Q

When can you assign +/- in the molec orbitals?

A

For sigma terms

It is with respect to reflection in a plane containing the internuclear axis

187
Q

When there is a singlet state, what does it mean about the symmetry?

A

Singlet: ψspin is anti, so ψspace must be symm

Therefore g,+ states

Triplet must be paired with g,- states

188
Q

What things come from applying the BO approx to the TDM for elec spec?

A

BO Approx allows for separation: ψtot = ψel(r) ψvib(R)

TDM therefore required elec and vib part to not equal to 0

189
Q

What are the selection rules in electronic spec?

A

ΔΛ = 0, +/- 1
ΔS = 0
ΔΣ = 0
when they exist:
g <-> u
+ <-> +, and - <-> -

190
Q

What does the Franck-Condon factor come from?

A

Square of vib overlap integral (from square of TDM)

Deals with vibrational changes when transition occurs

191
Q

What is the assumption of the Franck-Condon principle?

A

R unchanged as:
Elec transitions take place on such a short timescale that the nuclei reamin frozen

192
Q

What is the Franck-Condon principle?

A

Probability of a transition between elec states is governed by square of overlap integral of two vib wavefunctions

No selection rule governing allowed vib changes which accompany the elec transition

193
Q

What determines overlap of vib wavefunctions in different elec states?

A

Short progression: when similar bonding state (e.g. bond to bond) then 0->0 more likely

Long progression: then different bonding (e.g. bond to anti) then 0->excited states

194
Q

How do find dissociation energies from the morse oscillator?

A

Do dGv/d(v+1/2) = ωe - 2ωexe (v+1/2)

at the dissociation limit (v+1/2)max ->0

gives: G~max = De = ωe2/(4ωexe)

195
Q

What is the Birge-Sponer Extrapolation?

A

Plot ΔG as a function of (v+1/2)

Area under plot gives dissociation energy

At high v, then true curve falls off and so linear extrapolation performed

196
Q

How is rot structure observed in elec spectra?

A

Only larger changes in rot const observed
So band heads are observed in elec, and can be in either branch

This is because <R2> change can be large depending on bonding character of final and initial state
Large change in Re means B’<<B'' so rot levels more close in upper state

197
Q

When do band heads occur in elec spec?

A

When lines coalesce, dv~/dJ = 0
Can be in either branch (but R more likely)

In R branch: dv~/dJ = (B’ + B’’) + 2(B’-B’’)(J+1)
In P branch: dv~/dJ = -(B’ + B’’) + 2J(B’-B’’)

therefore
(J+1)head = - (B’+B’’)/2(B’-B’’)
Jhead = (B’+B’’)/2(B’-B’’)

198
Q

How can a Q-branch be seen due to elec transition?

A

when ΔΛ

199
Q

How can a Q-branch be seen due to elec transition?

A

When transition has ΔΛ > 0, then additional angular momentum can cause it

Coudl alternatively form vib mode

200
Q

What does photoelectron spec measure?

A

Records ionization energies for removal of e-s
Provides measure of energy of MOs

201
Q

What is the process of photoelectron spec?

A
  1. Excite with fixed λ (above IE)
  2. Measure KE of ejected photo-e-
202
Q

What are the energies invovled with photoelectron spec?

A

overall: hv = I + EM+ + KEe- + KEM+
where I is ionisation energy, and EM+ is the internal energy of the ion

KEe-&raquo_space;> KEM+, due to M being sig heavier so less KE when e- leaves
therefore make assumption

KEe- = hv - I - EM+

203
Q

Why are progressions seen in photoelec spec?

A

Franck-Condon Principle

Removal of strongly bond e- results in substantial red in bonding character
Therefore can occupy various vib energy levels of excited state

204
Q

What does a short or long progression in photoelec?

A

Short - weakly anti/bonding e-

Long - strongly anti/bonding e-

205
Q

What is photon scattering ?

A

Photons scattered by molecules
Weak but still occur

206
Q

What are the two types of photon scattering?

A

Rayleigh scattering - elastic, leaves molecule in same state

Raman scattering - inelastic, leaves molcules in different quantum state (rot or vib)

207
Q

What is the equation for Rayleigh scattering (elastic)?

A

I = I0 [8πNα2/λ4R2] (1+cos2θ)

where N is number of scatterers
α is polarisability, and R is distance between scatterer and observer

208
Q

When does Ralyeigh scattering occur?

A

When the dipole scatterer &laquo_space;λ of light

209
Q

What are the important proprotionality of Rayleigh scattering intensity?

A

I α R-2, inverse square law!

I α λ-4, sig more effective with shorter wavelength of light

210
Q

What is stronger Rayleigh or Raman scattering?

A

Rayleigh

211
Q

What occurs to photons in Raman scattering?

A

Scattered photon has different energy (frequency, wavelength)

212
Q

What are stokes and anti-stokes lines?

A

Result of raman scattering:
Stokes lines - when scattered photon has lost energy
Anti-Stokes - when scattered photon has gained energy

213
Q

How

A

Rayleigh strongest scattering

214
Q

What is the gross selection rule of raman scattering?

A

Raman active requires a molecule to have anisotropic polarisability

where anisotropic = different polarising in one plane compared to another

215
Q

What type of atoms are required for Rayleigh scattering?

A

Polarisability is isotropic - same on each plane

Always is in atoms and some molecules

216
Q

What is the polarisability of a linear molecule?

A

Anisotropic

as the molecule rotates the polarisability presented in E field changes

217
Q

What is the specific selection rule of rotational raman scattering?

A

ΔJ = 0,+/- 2

where 0 is from Rayleigh, + is from stokes , and anti being -ve

2 as is an effective 2-photon process

218
Q

Why does rotational raman occur?

A

When anisotropic molecule rotates the polarisability presented to E field changes

This means induced dipole modulated by rotation
Results in rotational transitions occuring

219
Q

What is the spacing between stokes or anti-stokes lines in rot raman?

A

4B~

220
Q

What is the spacing between the Rayleigh and 1st lines of branches of stokes/anti lines?

A

6B~

221
Q

What does a rot raman spectra look like?

A
222
Q

Why are there intensity differences between odd J and even in rot raman?

A

Due to nuclear spin-stats

223
Q

What must you be careful of when finding B~?

A

Missing lines may be present in spectra

e.g. O2 and CO2 have alternate lines missing (8B~ instead of 2B~ in rot or 4B~ in rot raman)

224
Q

What is teh gross selec rule of vib raman?

A

Polarizability must change during vibration
Therefore: normal mode must transform with same symm as quadratic forms (x2, xy, etc.)

225
Q

What is the specific selection rule for vib raman?

A

Δv = +/- 1

+ve is stokes and -ve is anti

226
Q

Why are anti-stokes rarely observed in vib raman?

A

v>0 are weakly populated

227
Q

What is the rule of mutual exclusion for molecules with centre of inversion?

A

Vib mode may either be Raman or IR active but not both

(only when centre of inversion symm)

228
Q

How can you check if a normal mode is IR or Raman active?

A

IR - transforms as x,y,or z

Raman - transforms as quadratic term (e.g. z2, x2+y2, etc)

Year 2 - Chem (23 decks)

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