General Physical Chemistry II

Lecture 14

Aleksey Kocherzhenko

October 21, 2014"

A long time ago, in a galaxy far, far away…"

Particle on a sphere"

x

y

z

r

�

✓

Particle of mass constrained to the surface of a sphere of radius"

mr

T = �~22I

@

@r

✓r2

@

@r

◆+

1

sin ✓

@

@✓

✓sin ✓

@

@✓

◆+

1

sin2 ✓

@2

@�2

�

ü  Write down the TISE in spherical coordinates:"

H (✓,�) = E (✓,�)

ü  Express the Hamiltonian in spherical coordinates:"

H = T + V (✓,�)| {z }⌘0

Kinetic energy operatorin spherical coordinates"

�~22I

1

sin ✓

@

@✓

✓sin ✓

@

@✓

◆+

1

sin2 ✓

@2

@�2

�Y (✓,�) = EY (✓,�)

Spherical harmonics"

Eigenvalues and the angular momentum"

x

y

z

r

�

✓

Y mll (✓,�) = (�1)

ml

s(2l + 1) (l �ml)!

4⇡ (l +ml)!Pmll (cos ✓) exp (iml�)

Spherical harmonic: "

E =~2l (l + 1)

2ICorresponding eigenvalue:" E =

J2

2I)

J = ~pl (l + 1)Orbital angular momentum:"

only depends on , not on"

lml

l = 0, 1, 2, ...

Jz = ~ml

Angular momentum around the z-axis (we found this for a particle in a ring):"

ml = 0,±1,±2, ...,±l

Orbital quantum #"

Magnetic quantum #"

Rotational spectroscopy"

The linear rigid rotor "A model for rotation of a diatomic molecule around its center of mass !

)

µ =m1m2

m1 +m2

0

                             

R = r1 + r2

In classical mechanics, a two-body problemis solved by reducing it to a one-body problem!

Reduced mass:!

I = µR2Moment of inertia:!

Ø  We reduced the rigid rotorto a particle on a sphere!!

, where!J = 0, 1, 2, ...Ø  Energy levels:!EJ =~2J (J + 1)

2I

Ø  Projection of the angular momentum: !Lz = ~MJ

L = ~pJ (J + 1)Ø  Angular momentum:!

MJ = 0,±1, ...,±Jwhere!

The rotational constant"

(The units for the rotational constant are Hz)"

The rotational energies,! EJ =~2J (J + 1)

2I

EJ = hBJ (J + 1)can then be rewritten as:!

where! J = 0, 1, 2, ...

The standard notation in spectroscopy is to write the rotational energiesin terms of the rotational constant:!

B =~

4⇡I=

h

8⇡2I

The rotational constant can also be expressed in cm–1:(usually in the range 0.1 – 10 cm–1)!

eB =~

4⇡cI

EJ = hc eBJ (J + 1)Then:!

Nonlinear rigid rotors "Similar to the linear rigid rotor, but for polyatomic molecules the moment of inertia must be defined differently !

The moment of inertia is defined with respect to a certain axis!I =

X

n

mnr2n

Distance from that axis!

Rotational properties of any molecule can be described by three components of I (Ia, Ib, Ic) about three perpendicular axis "

Molecular symmetry"Number of components of I necessary to describe the rotation of the molecule depends on the molecular symmetry"

Component with respect to the molecular axis is zero, the other two components are the same"

All 3 components are the same"

Two components are the same, the third is non-zero"

All 3 components are different"

Table 19.1 in textbook shows moments of inertia for a number of molecules(there is an error for one of the symmetric rotors in both the 5th and 6th editions) "

Symmetric rotor"

EJ,K = hBJ (J + 1) + h (A�B)K2

A =~

4⇡IkB =

~4⇡I?

Two rotational constants:"

K = 0,±1, ...,±JJ = 0, 1, 2, ...

Two quantum numbers:"

The axis of rotation for the molecule is oriented: à perpendicular to the molecular axis, à almost parallel to the molecular axis"K = 0K = ±J

Molecules with three-fold or higher order symmetry axis"

Spherical rotor"

A =~

4⇡IkB =

~4⇡I?

where there were two rotational constants:"

EJ,K = hBJ (J + 1) + h (A�B)K2

For a symmetric rotor we found:"

For a spherical rotor," Ik = I? ) A = B

Thus, the rotational energy only depends on : "J

EJ = hBJ (J + 1)

(This is the same expression that we found for the linear rotor, only the moment of inertia is different)"

J!

Linear non-rigid rotor"Molecules are not really rigid rotors: they distort as they rotate"

Ø  Bond lengths increase à energy levels come closer together"

EJ = hBJ (J + 1)� hDJ2 (J + 1)2Ø  To account for this effect, suppose that"

Centrifugal distortion constant!

D is related to the force constants of bonds:"

Ø  Large when a bond is easily stretched;"

Ø  Small when bond is difficult to stretch."

Allowed and forbidden rotational states"Pauli’s principle: when labels of identical particles are exchanged, "

(B,A) = � (A,B)Ø  for fermions:"

(B,A) = (A,B)Ø  for bosons:"

16O(A) = C =16 O(B)

Example:"

Nuclear spin = 0 à boson!

Ø  Rotate molecule by 180° degrees: the two O atoms are interchanged"Ø  For any molecule rotated by 180° degrees, the wavefunction changes

by a factor of (–1)J, see figure"à For this molecule, only even values of J are allowed"

Microwave spectroscopy"

Transitions between rotational energy levels have wavelengths "

� ⇠ 0.1� 10 cm(in the microwave range of the electromagnetic spectrum)"

Ø  Microwave spectroscopy can be used to determine the molecular geometry"

Ø  For instance, bond lengths can be very accurately determined using this method"

Allowed and forbidden transitions"

µfi =

ZZZ

V

⇤f µ idV

Transitions that can be facilitated by absorbing a photon must have a non-zero transition dipole moment:"

Initial state wavefunction!

Final state wavefunction!

Electric dipole operator!

Spherical redistribution of charge: no associated transition dipole moment, forbidden transition"

Redistribution of charge with non-zero associated transition dipole moment, allowed transition"

(This is also the reason why s à s or p à p transitions in atoms are forbidden, but s à p transitions are allowed)"

Gross and specific selection rules"

Ø  Specifies the general features that a molecule must have in order to have a spectrum of a given kind "

Specific Selection rule:"

Ø  The requirement of the transition moment being non-zero that is expressed in terms of the changes in quantum numbers"

Gross Selection rule:"

Rotational spectrum: molecule must be polar"

Rotational spectrum:"

�J = ±1

�K = 0

Not all transitions between energy levels are allowed"

Why is microwave spectroscopy only possible with polar molecules?"

Ø  If the frequency of the radiation is close to the natural rotational rate of the molecule à the molecule will absorb microwave photons and rotate faster"

Ø  Non-polar molecules (H2, O2, N2, CO2, CH4, CCl4, …) are insensitive to microwaves"

Ø  Polar molecules (e.g., CO) rotate as they try to align themselves with the electric field of the microwaves "

Ø  Since non-polar molecules can’t interact with the electric field, there is no change in their rotation upon exposure to microwave radiation. "

Why do the specific selection rules arise?"When a photon is absorbed or created by a molecule, the angular momentum of the combined system (molecule + photon) must be conserved:"

Ø  When a photon is absorbed, the quantum number J increases by 1, when a photon is emitted, it decreases by one:"

�J = ±1

Ø  No acceleration or deceleration of rotation around this axis:"

�K = 0

Ø  The dipole moment of a polar molecule does not move when a molecule rotates around its symmetry axis"

Rotational transition energies"

Rotational energy levels:"EJ = hBJ (J + 1)

J!

Selection rule:"�E = ±1

�Erot

= EJ+1

� EJ =

= hB [(J + 1) (J + 2)� J (J + 1)] =

= hB�J2 + 3J + 2� J2 � J

�=

= 2hB (J + 1)

⌫J = 2B (J + 1)

Frequency of radiation absorbed in the transition starting from the J

th level:"

Knowing the energy for the transition we can calculate the moment of inertia and therefore the bond length based on the mass !

Lifetime broadening"

The lifetime of excited states is limited by:"

(this follows from solving the time-dependent Schrödinger equation that we are not considering in this class)"

Ø  If the lifetime of a state is finite (all excited states) à its energy cannot be defined exactly:"

�E ⇡ ~⌧ State lifetime  

Lifetime broadening  

Ø  Collisional deactivation (collision of molecules can lead to loss of energy) "

Ø  Spontaneous emission of radiation"

We will look at excited state lifetimes in more detail in a later lecture"

Doppler line broadening"

Doppler line broadening: in gaseous samples molecules move in all directions with velocities distributed according to the Maxwell distribution"

Doppler effect: if source is moving relative to an observer (away or towards with velocity ), than observer sees change in the radiation frequency. "v

⌫receding = ⌫0

✓1� v/c

1 + v/c

◆⇡ ⌫0

1 + s/c

⌫approaching

= ⌫0

✓1 + v/c

1� v/c

◆⇡ ⌫

0

1� s/c

A stationary observer detects the corresponding range of Doppler-shifted frequencies à the absorption (emission) profile arises from all Doppler shifts:"

(Doppler line broadening)  

�� =2�

c

r2RT ln 2

M

Temperature  

Molar mass of molecule  

Gas constant  

Doppler broadening temperature dependence"

(Doppler line broadening)  

�� =2�

c

r2RT ln 2

M

Temperature  

Molar mass of molecule  

Gas constant  

Population of rotational states"For each à rotational states:"J 2J + 1

2J + 1 th energy level is degenerate"J J!

# of degenerate states at J

th level" Boltzmann factor"

Total population of a level:"

PJ / (2J + 1)| {z } exp�hBJ (J + 1)

kBT

�

| {z }

dPJ/dJ=0

Jmax

=

rkB

T

2hB� 1

2

Summary"Ø  The motion of a rigid rotor can be reduced to the motion of a particle on a

sphere: a system for which we have calculated the energy levels earlier"

Ø  Nuclear statistics imposes restrictions on possible molecular rotational states in order to satisfy Pauli’s principle !

Ø  Rotation of an object in 3D can be decomposed into rotation around 3 axes, and is thus characterized by 3 moments of inertia with respect to these axes" "(For highly symmetric molecules, 2 or all 3 of these may be the same.)"

Ø  In spectroscopy, molecular rotational energies are in the microwave range and are usually expressed in terms of the rotational constant:"B = ~/ (4⇡I)

Ø  Microwave spectroscopy allows observing transitions between the rotational states of a molecule if the molecule is polar (gross selection rule)"

Ø  Specific selection rules for rotational transitions are: , (these follow from angular momentum conservation in the photon/molecule system) !

�J = ±1 �K = 0

Ø  Rotational states have finite line width due to broadening!

Ø  Energies of rotationally excited states are comparable to the thermal energy, so multiple rotationally excited states may be populated at room temperature!