Atomic & Rotational Spec Flashcards
How is an EM wave described?
Transverse wave of perpendicular, sinusoidally oscillating electric and magnetic fields
E = E0 sin(kx - ωt + φ)
What is cvac?
299,792,458 ms-1
Work out as c2vac = 1 / μ0ε0, and μ0 given in exam
What is the linear momentum of a photon?
ρ = E/c = hν/c = h/λ
What is the angular momentum of a photon?
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
What is wavelength and frequency dependent on?
Wavelength - refractive index of medium
Frequency - independent of the medium
What is the wavenumber?
vbar = 1 / λvac
Units: cm-1
E = hc* vbar
What is a common units mistake with wavenumbers?
1 cm-1 = 100 m-1
VERY COMMON MISTAKE
What is the hamiltonian of a molecular system?
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
What is the Born-Oppernheimer Approx?
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
What are the transition required for the different forms of spectroscopy?
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
What is population of energy level i in Botlzmann law?
ni = (N/q) * gi * exp(-Ei/kT)
where q is molecular partition function
gi is the degen of ith level
What is the formula for molecular partition function?
q = Σi gi * exp(-Ei/kT)
What are the three standard interactions of light and matter?
Stimulated absorption - M + hn -> M*
Stimulated emmision - M* + hn -> M + 2hn
Spontaneous emmision - M* -> M + hn
What occurs in stimulated absorption and what is its rate?
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
What is the radiation energy density?
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
What occurs in stimulated emmision?
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
What occurs in spontaneous emmision?
Photon created and e- relaxes from “E2 ->E1”
dn2 /dt = -A*n2
Where A is einstein coefficient for spontaneous emmision
What occurs to Einstein coefficients at eqm?
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
What are allowed transitions for electronic spectroscopy?
E0 =! 0 as need a photon
Em0 - Ej0 = +/- hω as must conserve energy (photon equal to energy difference)
Transition dipole moment, R21 =! 0
What is the transition dipole moment, R21 ?
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
What are the selection rules and transitions for H-Atom in atomic spec?
Δn unrestricted
Δl = +/- 1
Δml = 0,+/- 1
Transitions at wavenumber vbar = ΔE/hc = Z2Ry(1/n12 - 1/n22)
How is the Δl = +/- 1 selection rule for H-atom spec dervied?
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
What is the magnitude of l for a H-atom?
Quantised
projection on z-axis lz = ml * hbar
l | = hbar * Sqrt(l*(l+1)) = 0, Sqrt2 * hbar, Sqrt6 * hbar, etc
Why is Δl =! 0 for a H-atom spectra?
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
Where does the selection rule of Δml = 0, +/- 1 for a H-atom spectra arise from?
Due to the helicity of the photon being σ = +/- 1
What is the problem with the schrodinger equation for atoms other than H?
Electron repulsion Σi =!j Vij term makes it insolvable
As in general need 3N spatial and N spin coordinates
Solved using orbital approx
What is the orbital approx for a many e- system?
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
What are the problems with the orbital approx?
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
How does the energy of a ns orbital compare in a alkali metal to a H-atom?
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
What are valence excitations in atomic spec?
Core excitations at higher energies
What factors can be used to account for spectrum of non-H atoms?
Zeff - change in the Z
Quantumm defect - changes the n quantum number
What are the different series present in emission atomic spec?
Sharp: ns -> np
Principal: np -> ns
Diffuse: nd -> np
Fundamental: nf -> nd
What are the selection rules for non-H atomic spectra?
Δn is unrestricted
Δl = +/- 1
Δml = 0, +/- 1
Δj = 0, +/- 1
ΔS = 0
What causes spectral fine spectra in atomic spec?
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
What are the allowed values of j in atomic spec?
Range given by Clebsch-Gordon series
j = l+s, l+s-1, …, | l-s|
How is the spin-orbit coupling constant calculated in atomic spec?
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
What is the fine spectra which arises from spin-orbit coupling?
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)
Where does the Δj = 0,+/-1 selection rule in atomic spec arise from?
Conservation of angular momentum
NOTE: cannot go from 0 to 0 as absorption of photon gives angular momentum
What is russel-saunders coupling?
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
What are fermions and bosons wrt spin?
Fermions have 1/2 integer spin
Bosons have integer spin
What is the Pauli principle?
Wavefn must be anti-symmetric wrt exchange of two identical fermions and symmetric wrt exhange of bosons
What are the spin combinations for two e-?
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
What is the singlet spin state?
One e- spin up and the other down
Cancellation to give S = 0, S = 0, Ms = 0, a single arrangement
What is the triplet spin state?
Three arrangements with S = 1
Ms = 0 - one up one down spin
Ms = +1 - two up spin
Ms = -1 - two down spin
How are triplet and singlet states paired with spin functions?
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+
Explain why e- with wavefn ψ+/- have different chances of being found at the same place
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
What is the Fermi hole and heap?
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
What does the ΔS = 0 selection rule mean for Grotian diagrams?
Singlet to triplet or vice versa transition forbidden
As every triplet state is lower than corresponding singlet
What does: configuration, terms, levels, and states refer to?
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
What must you note when working out the possible term symbols?
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.
What is a microstate table?
Table of every combination of ms and ml
Can eliminate all terms of different Σml values
What are the lowest energies for different terms in Russell-Saunders coupling?
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
When is LS (Russell-Saunders) coupling relevant?
Only for ground states of atoms
Valid for period 1-2 and some TM
When does LS coupling occur and when does j-j coupling occur?
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
What is the notation for j-j coupling?
ji - total a.m. of individual e- orbital a.m. and spin a.m. couple
J - sum of ji
What is the normal Zeeman effect and when is it applicable?
Applicable when S=0, so is a singlet
External field B lifts degeneracy of ML components which leads to splitting in the spectrum
What is the interaction energy in the classic Zeeman effect?
Classically: E = -m*B = -γeLzB
QM: E= -γeMLB*hbar = μBMLB
where γe = -e/2m and μB (bohr magneton) = -γe*hbar
What is the selection rule involved in the normal zeeman splitting?
ΔML = 0, +/- 1
Single signal therefore splits
What is the anomalous zeeman effect?
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
What is the energy splitting in the anomalous zeeman effect?
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)
What is the range for transitions in rotational spec?
10cm to 1mm wavelength
In the microwave
What is the moment of inertia in a molecule?
I = Σi miri2 where mi is mass of ith particle and ri is perpendicular distance from axis
Rotational equivalent of mass
What is the angular momentum and rotational KE of a molecule?
J = I*ω
E = J2/2I = 0.5*I*ω2
(where I is inertia)
What is the diatomic rigid rotor equivalent to in QM?
Rigid rotor represents particle on a sphere
What is the rotational energy in a particle on a sphere?
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
What are the energy eigenvalues of the particle on a sphere?
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
What are rotational spec values given in?
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>
What is rotational spec dependence on isotope?
Reduced mass does change so B~ ratio changes
What is the degeneracy present in rotational levels when no external field?
Each J level has 2J+1 degenerate states
Arises from projection quantum number MJ
What is the population of J levels in rotational spec?
From Boltzmann distribution, ni α gj*exp(-Ej/kT)
gj = 2J+1 and EJ = hcB~J(J+1)
Most populated when dni/dJ = 0
What is the most populated J-level in rotational spec?
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
How does the most populated J-level in rotational spec change with T and spacing of the levels?
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
What is the general solution of a rotational wavefunction?
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
What is the parity of different rotational wavefunctions?
Parity - symmetry wrt inversion
Parity = (-1)J where J is the J level occupied
What is a gross selection rule?
Properties required to do a form of spectroscopy
What is the gross selection rule for rotational spectra?
Must posses a permanent dipole moment
What is the specific selection rule of rotational spec?
ΔJ = +/- 1 : Due to conservation of a.m. as need to change parity
ΔK = 0 : required for non-linear or polyatomic
What is the energy in rotational spec given in terms of FJ?
FJ = EJ / hc = B~J(J+1)
where B~ = B/hc
At what energies are transitions observed in rotational spec?
v~ = FJ+1 - FJ = B~(J+1)(J+2) - B~J(J+1)
v~ = 2B~(J+1)
Equally spaced lines with separation of 2B
How do the gaps between J levels change as J increases? (rotational spec)
As J increases the gap between levels increases due to J2 dependence of rotational eigenvalues
What info does rotational spec give you?
Info on geometry
For diatomic molecules it gives bond length
What is the profile of rotational spec determined by?
Population of lower levels and J dependence of transition strength
See that population of many levels as kBT large wrt rotation energy
How are polyatomic molecules defined in rot spec?
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
What is a spherical top in rotational spec?
Molecule with zero dipole moment so doesn’t appear in spectra
Ia = Ib = Ic
E.g. Td, Oh
What are the classes of moleucles in rot spec?
Spherical tops - identical Ii
Symmetrical tops - two identical Ii
Asymmetrical tops - all unique Ii
What are the symmetrical tops in rot spec?
Prolate tops - long and thin molecules with Ia b = Ic
Oblate tops - discus type molecules with Ia = Ib < Ic
What are the rotational terms for diatomics?
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
What is the Ka qn?
Projection quantum number for projection on the unique a axis
Where a is body-fixed axis in prolate, and c in oblate
How are prolate tops labelled in rot spec?
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
What are the permitted values of J and K in prolate tops in rot spec?
FJ,K = B~J(J+1) + (A~-B~)K2
So J = 0, 1, 2, 3…. and K = 0, +/- 1, +/- 2,…, +/- J
What is the magnitude of J and projection of J (Ja) in a prolate top (rot spec)?
|J| = Sqrt[J(J+1)] * hbar
Ja = K * hbar , is what extent it rotates wrt principle axis
What does the relation of K and J mean in rot spec of a prolate?
When K≈J then rot a.m. is nearly parallel to principle axis
K = 0 then rot a.m. is perpendicular to principle axis
What is the notation for levels in rot spec of an oblate top?
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)
What are the J and K values in oblate top rotational spec?
FJ,K = B~J(J+1) + (C~-B~)K2
C~ <= B~ so term is -ve
What is the relation of K and MJ in rotational spec?
NONE, K =! MJ
K refers to projection on a body-fixed axis
MJ is a projection on a space-fixed axis
What are the energy levels for symmetric tops in rot spec?
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
What is the case of rotational spec for linear molecules?
Ia = 0 and so A~ = ∞
As only K = 0
FJ = B~J(J+1) and J = 0,1,2,3…
What is the case of rotational spec for a spherical top?
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
What is the spectra for prolate and oblate tops compared to linear?
Wihtin rigid rotor approx they are the same
Have equally spaced lines with separation = 2B~
So no info on unique axis
What is centrifugal distortion and the problem for rot spec?
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
What is the rotational term including centrifugal distortion?
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)
Where are transitions when including centrifugal distortion?
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
How can you plot v~(J) when there is centrifugal distortion?
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~
Where do transitions occur in symmetric tops with centrifugual distortion?
v~ = F(J+1, K) - F(J,K)
v~ = 2(B~-D~JKK2)(J+1) - 4D~J(J+1)3
What is the degeneracy of rotational levels?
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
What is the linear stark effect?
Electric dipole μ interacts with an applied field, results in an interacting energy
μ.E = -μEcosθ
Effects energy levels of symmetrical tops
How does the stark effect influence energy of symmetric tops in rot spec?
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
What is the stark energy proportional to?
Estark α |J|-2
How does the stark effect cause splitting in symmetric tops?
As depends on MJ it causes distinct energy levels
Has larger splitting when J is smaller
What is seen due to the Stark effect for a symmetric rotor?
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
What is the spin angular momentum and quantum number I?
I is the a.m. and I is the quantum number
How does spin quantum number change with mass number? (for rot spec)
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
How does spin angular momentum affect energy levels? (in rot spec)
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
What nuclear spin statistics must be considered in rot spectroscopy?
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
How is the stat weighting of nuclear spin functions?
of sym ψns / # of antisym ψns = (I + 1) / I
See the ratio of this or lack of some in the spectra
What is the stark energy for a symmetric top?
E = -(μEKMJ)/[J(J+1)]
define for J and K then will have splitting to MJ levels
How does the stark energy for splitting between MJ change with J?
Larger splitting between levels when J small
What is the effect of an electric field on rot spectroscopy?
Introduces selection rule ΔMJ = 0
Leads to splitting in spectrum to number of MJ levels in the lower state
Is ψel symmetric or antisymmetric?
Find term symbol for the molecule (e.g. H2 has 1Σg+)
Usually symmetric but need to check, O2 is the normal exception
Is ψvib symmetric or anti?
All v=0 vib levels are symmetric
Is ψrot symmetric or anti?
Even J are symmetric, odd J levels are anti
from (-1)J
Is ψns symmetric or anti?
Nuclear spin
Even mass no then I is an integer and a boson (symmetric)
Odd - I is a half-integer and so fermion (antisymm)
Is ψtotal symmetric or anti?
Even mass - boson, must be symm
Odd mass - fermion, must be anti
How do you do a nuclear spin stat for a question?
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!
What is vibrational spec?
Infrared
What is Hookes law and the PE of a compound from it?
F = -kFx
PE: V(x) = -∫F dx = (kFx2)/2
What is the eigenvalues of a harmonic oscillator?
Gv = (v+1/2)ωe
where v is vib qn and ωe is vibrational const
What is the equation for ωe?
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
What is the energy of a harmonic oscillator?
E = hc~G~(v+1/2)
E = hν(v+1/2)
where ν is freq
E = hbarωe(v+1/2)
What is the ZPE of a harmonic oscillator?
E = (1/2)hν
where ν is freq
What is the spacing of energy levels in a harmonic oscillator?
Equally spaced non-degen energy levels
What do prob densities in the harmonic oscillator change at different v qns?
As q increases the prob density becomes more classical
Penetration into classically forbidden regions
What is the hamiltonian for a harmonic oscillator?
H^ = -(1/2)(d2/dq2) + (1/2)q2
where q is vib coord, is 0 at eqm bond length
What is the lowest energy eigenfunction of HO?
ψ0 = exp(-q2/2)
What is the gross selection rule of IR (vib spec)?
Dipole moment that changes during vibration
TDM =! 1
What is the specific selection rule of IR (vib spec)?
Δv = +/- 1
where v is rot qn
How are the selection rules of vib spec dervied?
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
What are transitions at with simple HO?
v~ = ωe
One line in spectrum
How does the potential in HO relate to dissociation limit?
As bond stretches the bond gets weaker so slope decreases to limit
What is a the morse potential?
V(R) = De[ 1-exp(-β(R-Re))]2
Gives more accurate than HO
What are the energy eigenvalues in the morse potential?
G~ = (v+1/2)ωe - (v+1/2)2ωexe
where v=0,1,2,… vmax
where xe is the anharmonicity constant
What is xe, the harmonicity const?
xe = ωe/4De
where De is the dissociation energy
What is the experimental dissociation energy (D0) in vib?
D0 = De - ZPE
where De is dissociation energy to 0
What is the specific selection rule of morse oscillator?
Δv = +/- 1, (+/-2, +/-3, etc. are weaker)
What transition types are present in morse oscillator?
Fundamental (0->1)
1st overtone (0->2)
1st hot band (1->2)
What can occur when a vib transition occurs?
Rot transition also occurs
Etot = Evib + Erot
What are the selection rules for vib and rot transitions?
Δv = +/- 1, (+/- 2, +/- 3, etc weaker)
ΔJ = +/- 1
What is the R-branch and P-branch?
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
What is the vibrational and rotational terms energy terms?
S~ = (v+1/2)ωe - (v+1/2)2ωexe + B~J(J+1)
where only last term is from rot
What is the Q-branch in rotation vibration spectra?
Signal when ΔJ=0
Absent unless additional angular momentum (from elec a.m.) or degen bending mode present
What is the spacing in the R and P branch?
Spacing in the branch = 2B~
What is the spacing between R and P branch?
Central gap = 4B~
What is the point of origin (ω0) between the R and P branch in rot vib spec?
ω0 = ωe - 2ωexe
What is the vib dependence of rotational const?
B~ α 1/μR2
Treat as <1/R2>, which decreases with v
How does the rotational const at different v relate?
B~0 > B~1 > B~v>1
B~v = B~e - α(v+1/2)
How can you determine the rotational const (B~) ?
Use transitions with common upper/lower levels to determine rot const for each level
What is a bandhead in vib spec?
At high J the wavenumber separation of adj transitions can change sign
Causes transitions to bunch up
What is a normal mode?
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
What is coherent motion?
Means all atoms moving in phase and going back to original position at the same time
How many dof does a polyatomic molecule have?
N isolated atoms have 3N dof
How many trasnslations do linear and non-linear polyatomic molecules have?
Linear molecule = 3
Non-linear = 3
How many rotations do linear and non-linear polyatomic molecules have?
Linear = 2
Non-linear = 3
How many vibrations do linear and non-linear polyatomic molecules have?
Linear = 3N-5
Non-linear = 3N-6
How can you find if a normal mode is an irrep of point group?
Perform symm operations on stretches
1 if unchanged and -1 if swapped
What are the irreps of normal modes of CO2?
What is the symmetry of the gs vibrational wavefn?
ψ0 transforms as totaly symmetric irrep of relevant point group
What is the symmetry of the v=1 excited vib level?
ψ1 has the same symmetry as the normal mode
This is because has same symm as normal coord q (which is a linear function)
What is the symmetry of vib levels 2 and 3?
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
What is a selection rule for polyatomic molecules which come from a normal mode being harmonic?
New one: Δvi = +/- 1
Dipole moment must change with vib (standard one)
What is the TDM of a polyatomic molec?
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
What does μx, y, or z transform as?
Transforms as x,y,z from character table
What is a parallel or perpendicular mode?
Parallel - if transition moment of a mode is parallel to symmentry axis
Perp - same as above but perpendicular
What is required for a fundamental transition to be allowed and what kind of mode is this?
Symm of normal mode (and therefore v=1) is the same as x,y,z
Due to the starting one being totally symm
What is a combination band?
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
What are present in parallel and perpendicular bands?
Parallel (Σ-Σ) - ΔJ=+/- 1, so P & R branches
Perpendicular (Σ-Π) - ΔJ=0,+/- 1 (Q,P, and R branches)
Why does a Q branch occur in vib spec?
Occurs due to additional angular momentum from degen bending mode
What is the Born-Oppenheimer Aprrox?
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
How does the BO approx effect the PE?
1 value for PE at specific separation so e- can arrange in lowest E config quickly
What term symbols are used for diatomics?
Classified wrt angular momentum around internuclear axis, λ
λ analogous to mI in atoms
What is the term symbol for two pz orbitals combining?
mI = 0, combine to give
σ,σ* (λ=0)
where σ means cylindrical symm about the internuclear axis
What is the term symbol of a diatomic with px and y combined?
Each has mI = +/- 1, combine to give
π,π* (λ = +/- 1)
where π means antisymmetry about internuclear axis
What is overall angular momentum and how is it calculated?
Λ = Σ λi
λ is for each e-
Λ =0 gives Σ
Λ =+/- 1 gives Π
Λ =+/- 2 gives Δ
What is the format of the molecular term symbol?
2S+1ΛΩ
where
Λ = total angular momentum
Ω = |Λ + Σ|
and Σ = projection of S on internuclear axis
When can you give u/g notation for a diatomic?
Homonulcear
u - antisymm
g - symm
What are the direct products between u and g states?
g x g = g
u x u = g
g x u = u x g = u
When can you assign +/- in the molec orbitals?
For sigma terms
It is with respect to reflection in a plane containing the internuclear axis
When there is a singlet state, what does it mean about the symmetry?
Singlet: ψspin is anti, so ψspace must be symm
Therefore g,+ states
Triplet must be paired with g,- states
What things come from applying the BO approx to the TDM for elec spec?
BO Approx allows for separation: ψtot = ψel(r) ψvib(R)
TDM therefore required elec and vib part to not equal to 0
What are the selection rules in electronic spec?
ΔΛ = 0, +/- 1
ΔS = 0
ΔΣ = 0
when they exist:
g <-> u
+ <-> +, and - <-> -
What does the Franck-Condon factor come from?
Square of vib overlap integral (from square of TDM)
Deals with vibrational changes when transition occurs
What is the assumption of the Franck-Condon principle?
R unchanged as:
Elec transitions take place on such a short timescale that the nuclei reamin frozen
What is the Franck-Condon principle?
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
What determines overlap of vib wavefunctions in different elec states?
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
How do find dissociation energies from the morse oscillator?
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)
What is the Birge-Sponer Extrapolation?
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
How is rot structure observed in elec spectra?
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
When do band heads occur in elec spec?
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’’)
How can a Q-branch be seen due to elec transition?
when ΔΛ
How can a Q-branch be seen due to elec transition?
When transition has ΔΛ > 0, then additional angular momentum can cause it
Coudl alternatively form vib mode
What does photoelectron spec measure?
Records ionization energies for removal of e-s
Provides measure of energy of MOs
What is the process of photoelectron spec?
- Excite with fixed λ (above IE)
- Measure KE of ejected photo-e-
What are the energies invovled with photoelectron spec?
overall: hv = I + EM+ + KEe- + KEM+
where I is ionisation energy, and EM+ is the internal energy of the ion
KEe-»_space;> KEM+, due to M being sig heavier so less KE when e- leaves
therefore make assumption
KEe- = hv - I - EM+
Why are progressions seen in photoelec spec?
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
What does a short or long progression in photoelec?
Short - weakly anti/bonding e-
Long - strongly anti/bonding e-
What is photon scattering ?
Photons scattered by molecules
Weak but still occur
What are the two types of photon scattering?
Rayleigh scattering - elastic, leaves molecule in same state
Raman scattering - inelastic, leaves molcules in different quantum state (rot or vib)
What is the equation for Rayleigh scattering (elastic)?
I = I0 [8πNα2/λ4R2] (1+cos2θ)
where N is number of scatterers
α is polarisability, and R is distance between scatterer and observer
When does Ralyeigh scattering occur?
When the dipole scatterer «_space;λ of light
What are the important proprotionality of Rayleigh scattering intensity?
I α R-2, inverse square law!
I α λ-4, sig more effective with shorter wavelength of light
What is stronger Rayleigh or Raman scattering?
Rayleigh
What occurs to photons in Raman scattering?
Scattered photon has different energy (frequency, wavelength)
What are stokes and anti-stokes lines?
Result of raman scattering:
Stokes lines - when scattered photon has lost energy
Anti-Stokes - when scattered photon has gained energy
How
Rayleigh strongest scattering
What is the gross selection rule of raman scattering?
Raman active requires a molecule to have anisotropic polarisability
where anisotropic = different polarising in one plane compared to another
What type of atoms are required for Rayleigh scattering?
Polarisability is isotropic - same on each plane
Always is in atoms and some molecules
What is the polarisability of a linear molecule?
Anisotropic
as the molecule rotates the polarisability presented in E field changes
What is the specific selection rule of rotational raman scattering?
ΔJ = 0,+/- 2
where 0 is from Rayleigh, + is from stokes , and anti being -ve
2 as is an effective 2-photon process
Why does rotational raman occur?
When anisotropic molecule rotates the polarisability presented to E field changes
This means induced dipole modulated by rotation
Results in rotational transitions occuring
What is the spacing between stokes or anti-stokes lines in rot raman?
4B~
What is the spacing between the Rayleigh and 1st lines of branches of stokes/anti lines?
6B~
What does a rot raman spectra look like?
Why are there intensity differences between odd J and even in rot raman?
Due to nuclear spin-stats
What must you be careful of when finding B~?
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)
What is teh gross selec rule of vib raman?
Polarizability must change during vibration
Therefore: normal mode must transform with same symm as quadratic forms (x2, xy, etc.)
What is the specific selection rule for vib raman?
Δv = +/- 1
+ve is stokes and -ve is anti
Why are anti-stokes rarely observed in vib raman?
v>0 are weakly populated
What is the rule of mutual exclusion for molecules with centre of inversion?
Vib mode may either be Raman or IR active but not both
(only when centre of inversion symm)
How can you check if a normal mode is IR or Raman active?
IR - transforms as x,y,or z
Raman - transforms as quadratic term (e.g. z2, x2+y2, etc)
