Section 3 Ml l R t ti lS t Section 3 Ml l R t ti lS t Molecular R ot ationalSpectroscopy Lectures 4‐6 Molecular R ot ationalSpectroscopy Lectures 4‐6 Quantum theory of atoms / molecules Previously: Quantum Mechanics of atoms / molecules Molecular Rotations and Spectroscopy Mechanics Molecular Rotations and Spectroscopy Diatomic Molecules (revision) Beyond diatomics Beyond rigid rotors: Complications of nuclear spin statistics
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Section 3 M l l R t ti l S t
Section 3 M l l R t ti l S tMolecular Rotational Spectroscopy
Lectures 4‐6Molecular Rotational Spectroscopy
Lectures 4‐6
Quantum theoryof atoms / molecules
Previously: Quantum Mechanicsof atoms / molecules
Molecular Rotations and Spectroscopy
Mechanics
Molecular Rotations and SpectroscopyDiatomic Molecules (revision)Beyond diatomicsBeyond rigid rotors: y gComplications of nuclear spin statistics
3.1 The spectroscopist as sleuth3.1 The spectroscopist as sleuthIt is now possible to following nuclearIt is now possible to following nuclear motion in real time (femtochemistry):
Photodissociation of ICN
Ahmed Zewail, N b l P i 1999
Photodissociation of ICN→ Very “classical” pictures
(balls on surfaces)
Most spectroscopy however is performed in the “frequency domain” i e as a
Nobel Prize 1999
Can be used in many guises:i T d t i l l t t d ti
Most spectroscopy, however, is performed in the frequency domain i.e., as a function of frequency or wavelength.
i. To determine molecular structures and properties ii. To determine chemical composition (e.g., extraterrestrial)iii. To determine properties (Temperature, velocities, etc.)
3.1.1 Determining Chemical Composition3.1.1 Determining Chemical Composition
IR (vibrational) spectroscopyIR (vibrational) spectroscopy useful for determining functional groups in synthetic chemistry.y
But spectroscopy is also useful beyond the lab:beyond the lab:
The Orion NebulaThe Orion Nebula
Molecular identification in space: Emission SpectraMolecular identification in space: Emission Spectra
>130 molecules / ions have been identified in interstellar space by their rotational>130 molecules / ions have been identified in interstellar space by their rotational emission spectra (rf‐astronomy)
Use a star behind the cloud as light source for direct absorption
← λ
uvvisibleInfra‐red visibleInfra red
1st identified in 1921, >300 lines observed throughout the galaxy.
After 89 years, not one line has been conclusively assigned!
At least that was true until Jan 2011.....Group leader, an ex‐Balliol / PTCLDPhil student!
C=C=CH
HH
Molecular l
Molecular lEnergy LevelsEnergy Levels
i.e., typically ΔEel >> ΔEvib >> ΔErot
Different electronic states (electronic arrangements)(electronic arrangements)
λΔ ≈
≈E 2 x 104 – 105 cm‐1
500 – 100 nm
102 – 5 x 103 cm‐1
100 μm – 2 μm
3 – 300 GHz (0.1 – 10 cm‐1)
Transitions at λVis – UV
00 μ μinfrared
10 cm – 1 mmmicrowave
Molecular Rotational Energy Levels and SpectroscopyMolecular Rotational Energy Levels and Spectroscopy3.2 The moment of inertia3.2 The moment of inertia
Definition: The moment of inertia I of a system about an axis passing through the centre of mass is given by; ∑= iirmI 2 where mi is the mass of the ith particle and∑
iiirmI
e.g.,: For a diatomic molecule:
i pri is its perpendicular distance from the axis
2222
211
2 RrmrmrmIi
ii μ=+== ∑g ,
m1 m2
R)( 21
21
mmmmμ
+=
Where the reduced mass,
≈ μR
I is the rotational equivalent of mass. F b d t ti b t i ith l l it JFor a body rotating about an axis with angular velocity ω;
The angular momentum, J = I ω (c.f. p = mv)
J
μ RThe rotational kinetic energy, E = J2/2I = ½ I ω2
(c.f. E = p2/2m = ½mv2)
μ R
ω
3.3 Quantum Rotation: The Diatomic Rigid Rotor3.3 Quantum Rotation: The Diatomic Rigid Rotor
The rigid rotor represents the quantum mechanical “particle on a sphere” problem:
Rotational energy is purely kinetic energy:
2 2 22 2
2 2
2 12 2
H Er r r r
ψ ψ ψμ μ
⎛ ⎞∂ ∂= − ∇ = − + + Λ =⎜ ⎟∂ ∂⎝ ⎠
22
22H E
rψ ψ ψ
μ= − Λ =Which, for constant r, simplifies to
The solutions resemble those of the particle on a ring with cyclic boundary conditions ψ(φ+2π)=ψ(φ) and are called spherical harmonics ( ) ( ) ( )Y θ φ θ φ= Θ Φψ(φ+2π)=ψ(φ) and are called spherical harmonics
The energy eigenvalues (the bit we will be interested in) are given by
( ) ( ) ( )l l llm lm mY ,θ φ θ φ= Θ Φ
( )2
12JJmE I
+= J J
The energy eigenvalues (the bit we will be interested in) are given by
with J = 0, 1, 2, 3....and m = 0 ±1 ±2 ±Jand mJ = 0, ±1, ±2,..... ±J
3.3 Quantum Rotation: The Diatomic Rigid Rotor3.3 Quantum Rotation: The Diatomic Rigid Rotor J530B
Eigenvalues: ( )1 E B= +J J J
( )2
JouleB in
J is the rotational quantum number = 0, 1, 2, 3,...B is the rotational constant given by
420B
( ) Joule 2
B inI
=
These are usually given as wavenumbers or
312B
These are usually given as wavenumbers or rotational terms:
( )1EF Bhc
= = +JJ J J 0 0
1
22B6B
B( )hc
2 2 28 8h hBcI c Rπ π μ
= =H2 60.85 cm‐1
CO 1.93 cm‐1
HCl 10 59 cm‐1
B
8 8cI c Rπ π μ HCl 10.59 cm 1
n.b., There is no zero point energy associated with rotation, i.e., EJ=0= 0p gy J 0 Rotational energy levels get more widely space with increasing JSo how fast do molecules rotate?
3.3 Quantum Rotation: The Diatomic Rigid Rotor3.3 Quantum Rotation: The Diatomic Rigid Rotor
11BI
∝ provides the spectral link to molecular geometric structure
F diatomics it i it li it1 1B ∝ → extract bond lengths (strictly ⟨1/R2⟩)For diatomics it is quite explicit: 2BI Rμ
∝ = → extract bond lengths (strictly ⟨1/R2⟩)
Isotope Effects:Isotope Effects: R is isotope independent (the electronic problem) but clearly the reduced mass does change and so e.g.,
37 36B u u uμ35 37
37 35
37 36 1 001538 35
H Cl H Cl
H Cl H Cl
B u.u u. .B u u.u
μ
μ= = =For H35Cl and H37Cl:
MJ
12
12
Degeneracy of Rotational LevelsDegeneracy of Rotational Levels
In the absence of external fields each J level exhibits (2J+1)-fold degeneracy arising from the projection quantum number MJ:
0
1
‐1
0
1
‐1the projection quantum number MJ: ‐2 ‐2e.g., J = 2
3.4 Populations of J levels3.4 Populations of J levels
A G S l ti R lA. Gross Selection Rule: For a molecule to exhibit a pure rotational spectrum it must posses a permanent dipole moment.
420B(otherwise the photon has no means of interacting –“nothing to grab hold of”)
312BB. Specific Selection Rule: ΔJ = ± 1
0 01
2
2B
6B(Conservation of angular momentum → max. ΔJ = ± 1But need to change parity (see rotational wavefunctions))
( )( ) ( )1
1 2 1v F F
B B+
= −
= + + − +J J
J J J J
Transitions observed at:
Equally spaced ( )( ) ( )( )
1 2 1 2 1
B BB
+ + +
= +
J J J JJ
q y plines, separation 2B
n.b., all lab‐based rotational spectroscopy is performed in absorption due to very slow spontaneous decay rates (A coefficient ∝ ν3 and ν small)
3.7 e.g., Rotational Spectrum of CO3.7 e.g., Rotational Spectrum of CO
Rotational spectrum of CO (300K)
( )( ) ( )1
1 2 1v F F
B B+
= −
= + + − +J J
J J J J( )( ) ( )( ) 2 1B= +J
Transitions in the microwave region:1‐100 cm‐1 (λ = 1 cm – 100 μm)
Lines spaced by 2B
Spectral Profile governed by population of lower p g y p plevels and J dependence of the transition strength.
3.8 General Classification of Molecules3.8 General Classification of Molecules
Recall Definition: The moment of inertia I of a system about an axis passing
Back to the moment of inertia
through the centre of mass is given by; ∑=i
iirmI 2
We classify polyatomic molecules on the basis of their moments of inertia about three mutually perpendicular axes through the centre of mass (principal axes).
Call these axes a, b, c and (and thus Ia, Ib, Ic) such that Ic=Imax and
cba III ≤≤
3.9 General Classification of Molecules3.9 General Classification of MoleculesTd Oh Ih
I. Spherical tops: cba III ==
Zero dipole moment ∴ no pure rotational absorption spectrum
II. Symmetric tops: (two identical Ii)
a) Prolate tops:cba III =<
cba III <=b) Oblate tops:
III. Asymmetric tops: cba III <<
d d f d l1 1B
3.10 Rotational terms3.10 Rotational terms
For diatomics we defined a rotational constant2B
I Rμ∝ =
In general we require three such rotational constants:
hB~hA~hC~b
bcIhB 28π
=acI
hA 28π=
ccIC 28π
=as wavenumbers:
≥ ≥≥ ≥
H2O molecule
‐1
‐1
27.9 cm14.5 cm
AB
==
‐1
14.5 cm9.3 cm
BC = But, we can no longer relate
these constants explicitly to p yindividual bond lengths within the molecule.
KBAJJBF KJ −++= )~~()1(~ 2,
3.11 Prolate tops3.11 Prolate tops
JKJ
±±±==
……
,2,1,0,3,2,1,0
Levels labelled JKa
J is the total angular momentum or rotational quantum number and
Ka the projection quantum numbera p j q(for projection on the unique, a axis).
)J(JJ 1 KJJK JK
)J(JJ 1+= KJa =
J = 2
J
JK=0 KK=-1K=-2
66 6
6
J
JK=0 KK=-1K=-2
6666 66
66
J = 2
2
K +2K=+1
JK
66
K +2K=+1
JK
6666
20
2221
K=+2K=+1JK K=+2K=+1JKn.b., Each level has 2J+1 degeneracy (arising from MJ)
In addition, each level K > 0 has extra two‐fold degeneracy (±K)
Levels labelled JKcJ is the total angular momentum
or rotational quantum number andKc the projection quantum number
(for projection on the unique, c axis).
Oblate tops are typically flat “discus” – like molecules (e.g., benzene)
n.b., Each level has 2J+1 degeneracy (arising from MJ)In addition each level K > 0 has extra two‐fold degeneracy (±K)In addition, each level K > 0 has extra two fold degeneracy (±K)
3.13 Don’t confuse various projections3.13 Don’t confuse various projections
( )= +J J J
K refers to a projection on a body‐fixed axis
(in this case, for a prolate top, the a axis)( , p p, )MJ refers to a projection on
a space‐fixed axis
3.14 Energy levels for Symmetric tops3.14 Energy levels for Symmetric tops
21 K)B~C~()J(JB~F
Oblate top terms
21 K)B~A~()J(JB~F ++
Prolate top terms
21 K)BC()J(JBF K,J −++=<0
1 K)BA()J(JBF K,J −++=>0
234
3
4J
K=0 K=1 K=2
23
211
0
2
K=0 K=1 K=20
1
K 0K 0 K 1 K 2
K ‐ stacks K ‐ stacks
For a given J, energy increases with K For a given J, energy decreases with K
b
3.15 Linear Molecules (C∞v, D∞h)3.15 Linear Molecules (C∞v, D∞h)
∞== A~I hence0 a
Special, limiting case of prolate top:
∞A,Ia hence 0c
Only K = 0 exists, so
( )( )1 0 1 2 3F B , , , ....= =J
J J + J
b
3.16 Linear Molecules (C∞v, D∞h)3.16 Linear Molecules (C∞v, D∞h)
Alas, for the vast majority of molecules there is no simple general analytical form for the rotational levels. Some molecules are described as “near prolate” and “near oblate” tops. In general, terms can be derived by matrix diagonalisation.oblate tops. In general, terms can be derived by matrix diagonalisation.
3.23 Effects of External Fields3.23 Effects of External Fields
We know that every rotational level, J, comprises 2J+1 states due to space quantization:
MJe.g., J = 2MJ = -2, -1, 0, 1, 2
1
2
1
2MJ 2, 1, 0, 1, 2 MJ
0
‐1
0
‐11
‐2
1
‐2
In the absence of external fields these states are all degenerate. However, external E(or B) fields can lift this degeneracy:(or B) fields can lift this degeneracy:
E
3.23 Electric Fields: The Linear Stark Effect3.23 Electric Fields: The Linear Stark Effect
An electric dipole μ interacts with an applied field resulting in an interaction energy
μ μ θ=μ
E
θμ cosμ μ θ= θθμ cos
Effect on energy levels: Symmetric topsEffect on energy levels: Symmetric tops
Consider J level of CH3F:3
μ points along the C3 axis with J at some angle to it:
But we know this angle from the ratio of JA and J (and hence J, K): cos =
Kαthe ratio of JA and J (and hence J, K):)1(
cos+JJ
α
As J precesses, the component of μperpendicular to J cancels leaving only μJ:perpendicular to J cancels leaving only μJ:
cos ==Kμαμμ
)1(cos
+==
JJJ μαμμ
So, an external field making an angle β to J yields an i iinteraction energy
βμ cosEJ−)1(
cos+
=JJ
M Jβand)(
cos),,( −= EMKJE JJStark βμ
.)1()1()1( +
−=
++−=
JJEKM
JJM
EJJK JJ μ
μ
n.b., Estark ∝ μ, E, K, MJ but Estark ∝ |J|‐2
− EKM Jμ3.24 Electric Fields: Symmetric Tops3.24 Electric Fields: Symmetric Tops
Example: .)1(
),,(+
=JJEKMMKJE J
JStarkμ
Consider the transition 21 ← 11 in a symmetric top:
For J = 2, K = 1, MJ = ±2, ±1, 0
Consider the transition 21 ← 11 in a symmetric top:
0,,2 EEEStarkμμ
±±= ,6
,6Stark
In J 1 K 1 M ±1 0
0EES kμ
±=
In J = 1, K = 1, MJ = ±1, 0
0,2
EStark ±=
Eff t th t f t i t
3.25 Electric Fields: The Effects on the Spectrum3.25 Electric Fields: The Effects on the Spectrum
Effect on the spectrum of a symmetric rotor:
Remember the selection rule ΔMJ = 0.
No field
Useful for determining absolute values of J in complex spectra.
N l ( t t ) F i d lt l i i l
3.26 Effects of nuclear spins on rotational energy levels3.26 Effects of nuclear spins on rotational energy levels
Nucleons (protons, neutrons) are Fermions and as a result nuclei possess spin angular momentum I with corresponding quantum number I.
If the mass number is even: I is integral such nuclei are BosonsIf the mass number is even: I is integral such nuclei are Bosonsodd: I is half‐integral such nuclei are Fermions
This can affect rotational energy Levels in two ways:
1) The nuclear spin gives rise to a magnetic moment which can interact1) The nuclear spin gives rise to a magnetic moment which can interact with external magnetic fields (the basis of NMR) and internal magnetic fields to give nuclear hyperfine structure: small splittings in the spectrum
2) They can determine whether or not rotational levels in symmetric molecules actually exist as a result of nuclear spin statistics
Recall the Pauli Principle:
“Any acceptable wavefunction must be anti‐symmetric with respect to the exchange of two identical fermions and totally symmetric with
respect to the exchange of identical bosons”
3.27 Nuclear Spin statistics in H2: ortho‐ and para‐ hydrogen3.27 Nuclear Spin statistics in H2: ortho‐ and para‐ hydrogen
W h f h l i i h l f i hi h b
ψtot = ψelψvibψrotψns
We have to account for the nuclear spin in the total wavefunction which becomes:
Consider ψns [c.f.electron spins in Section 2.16]
In H each nucleus is a fermion with s = ½ and thereforem = ±½ (or α β)In H2 each nucleus is a fermion with s = ½ and therefore ms = ±½ (or α, β)
Four combinations are possible: α(1)α(2), β(1)β(2), α(1)β(2), β(1)α(2)
But we are interested in symmetry with respect to exchange so take linear combinations of latter two and we achieve:
( ) ( )( ) ( )1 21 2
α αβ β Symmetric to exchange( ) ( )
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
1 1 2 2 1 21 1 2 2 1
α β α β
β β
⎡ ⎤+⎣ ⎦⎡ ⎤
Symmetric to exchange[3 of]
( ) ( ) ( ) ( )1 2 2 12
α β α β⎡ ⎤−⎣ ⎦ Anti‐symmetric to exchange[1 of]
3.27 Nuclear Spin statistics in H2: ortho‐ and para‐ hydrogen3.27 Nuclear Spin statistics in H2: ortho‐ and para‐ hydrogen
ψtot = ψelψvibψrotψns
ψtot must be anti‐symmetric with respect to the permutation of H nuclei: 12P ψ ψ=−ψtot
In the ground state of H2 :ψel (1Σg
+) is symmetric with respect nuclear interchange and
12
ψel ( g ) y p gAll v = 0 vibrational levels are symmetric
So, for acceptable ψtot we need anti‐symmetric ψrotψns productsSo, for acceptable ψtot we need anti symmetric ψrotψns products
ψrot: permutation equivalent to a c2 rotation
even J levels are symmetricodd J levels are anti‐symmetric
( )2 1rot rotcψ ψ= −J
All even J levels correspond with Anti‐Symm ψ of which there is oneAll even J levels correspond with Anti Symm ψns of which there is oneAll odd J levels correspond with Symm ψns of which there are three
3.27 Nuclear Spin statistics in H2: ortho‐ and para‐ hydrogen3.27 Nuclear Spin statistics in H2: ortho‐ and para‐ hydrogen
The extra statistical weighting of ortho‐H2 (odd Jlevels) over para‐H2 (even J) means intensity alternations of 3:1 in rotationally resolved spectra of H2.
Of course H2 doesn’t exhibit a pure rotational absorption spectrum but this effect is clear in the
o‐H2 and p‐H2 are essentially different forms of H2: They do not interconvert except in the presence of a high spin catalyst.
o‐H2 has a J = 1 ground state and thus exhibits zero‐point rotational motion = 2B
3.28 Nuclear Spin Statistics: General3.28 Nuclear Spin Statistics: GeneralConsider symmetry of ψel ψ ib individually. ψel is usually symmetric but beware O2Consider symmetry of ψel ψvib individually. ψel is usually symmetric but beware O2
ground state (3Σg–) which is anti‐symmetric.
In general, the statistical weighting of . of SYM 1no Iψ +nuclear spin functions is given by:
. of SYM 1. of ANTI‐SYM
ns
ns
no Ino I
ψψ
+=
Example 1: 14N214N is a Boson (I = 1), ∴ ψtot SYMψel is SYM (1Σg
+), hence:
ψtot = ψelψvibψrotψns
SYMAS
S S S SAS
Weight:
21
2:1 intensity alternationψel ( g ), AS AS 1
Example 2: 16O2 ψtot = ψelψvibψrotψns Weight: even J