CHM 3411 - Physical Chemistry II
Chapter 13 - Supplementary Material
1. Notation in spectroscopy (general considerations)
Chemists are not always consistent in the notation they use in rotational. vibrational, and electronic
spectroscopy. The following summarizes a consistent presentation of spectroscopy that is commonly used by
spectroscopists.
Energies are usually given in wavenumbers (cm-1), where E(cm-1) = 1/, where  is the wavelength (in cm)
of the photon that is absorbed or admitted when a transition is made between two states a molecule. This is related
to the energy in MKS units (J = Joule) by
E(cm-1) = E(J)/hc
(13.1.1)
where E(cm-1) is energy in wavenumbers and E(J) is energy in Joule. The conversion factor between the two is
1 cm-1 = 1.986 x 10-23 J
(13.1.2)
Often the symbol Ẽ is used when energy is given in wavenumbers. Note that in the material given below we will
always use wavenumbers as our energy unit (which means that rotational and vibrational constants will also be given
in units of wavenumbers).
In labeling states the lower energy state is given a " (double prime) and the higher energy state is given a '
(single prime). This is done for all types of spectroscopy - absorption, emission (fluorescence) and Raman.
2. Pure rotational spectroscopy
Diatomics
In the rigid rotor (RR) approximation the rotational energy of a diatomic molecule is given by the
expression
E = J(J+1) B
(13.2.1)
where B, the rotational constant, is
B = h/82Ic = h/82r2 , I = r2 ,  = mAmB/(mA+mB)
(13.2.2)
In the above expressions I is the moment of inertia, which for a diatomic molecule is equal to r2, where  is the
reduced mass and r is the bond length for the molecule.
For high precision work the rotational energy can be written in terms of a power series expansion in J(J+1),
with
E = J(J+1) Bv - J2(J+1)2 Dv
(13.2.3)
In the above expression Bv takes into account that the rotational constant is different for different vibrational states of
the molecule (because the average bond distance increases as v ncreases), and Dv takes into account the stretching of
the bond as the molecule rotates (due to the centrifugal effect). B v is itself often written as
Bv = Be - (v + 1/2) e
(13.2.4)
since for small values of v the changes in the value of the rotational constant are approximately proportional to v.
Note that Be is the value of the rotational constant (equn 13.2.2) at r = r e (equilibrium bond distance), the bond
distance at the minimum in the molecular potential energy curve for the molecule.
Polyatomics
In discussing rotational energy for polyatomic molecules the molecules are classified by their principal
moments of inertia (Iaa, Ibb, Icc) as follows (note that I must be greater than or equal to zero).
1) Linear molecules (including diatomic molecules)
Iaa = Ibb = 0, Icc  0
2) Spherical top molecules (pure tetrahedral or octahedral)
Iaa = Ibb = Icc  0
3) Symmetric top molecules (molecule has a C3 or higher
proper axis of rotation
4) Asymmetric top molecule
Iaa = Ibb < Icc or Iaa < Ibb = Icc
Iaa < Ibb < Icc
In the rigid rotor approximation the energy levels for a linear polyatomic molecule or a spherical top
molecule are the same as for a diatomic molecule (equn 13.2.1). If the moment of inertia for the molecule is known
then the rotational constant can be found using equn 13.2.2. Note that for a spherical top molecule there are now
three rotational quantum numbers
MJ = 0, 1, 2, ...J
J = 0, 1, 2, ...
K = 0, 1, 2, ...J
(13.2.5)
MJ and K have no effect on the energy, but the presence of a third rotational quantum number does have an effect on
the degeneracy of the rotational energy levels.
For a symmetric top molecules the rotational energy levels are given by the expression
where
E = J(J+1) B + K2 (A - B)
(13.2.6)
A = h/82I║c
(13.2.7)
B = h/82I┴c
In the above expressions I║ is the moment of inertia parallel to the principal axis (and so is the the principal moment
of inertia that is different than the others) and I┴ is te moment of inertia perpendicular to the principal axis (and so
corresponds to the two equal principal moments of inertia). The quantum numbers are the same as for a spherical
rotor (equn 13.2.5) but the degeneracies of the energy levels are in general smaller than those for a spherical rotor
due to the fact that the rotational energy depends on both J and K.
For an asymmetic top molecule there is no general expression for the rotational energy levels. Approximate
values for these levels (in the rigid rotor approximation) can be found using perturbation theory.
Selection rules (pure rotational spectroscopy)
1) Molecule must have a permanent dipole moment. This means that homonuclear diatomic molecules,
some linear polyatomic molecules (such as CO2 and C2H2) and spherical top molecules do not exhibit a pure
rotational spectrum.
2) J = 1
3) MJ = 0, 1
4) K = 0 (for nonlinear molecules)
Note that selection rule on K means that the location of allowed transitions for symmetric top molecules (in the rigid
rotor approximation) is the same as for linear molecules, even though the rotational energy depends on K.
3. Vibrational spectroscopy
Diatomics - Vibrational energy
In the harmonic oscillator (HO) approximation the vibrational energy of a diatomic molecule is given by the
expression
E = (v + 1/2) 
(13.3.1)
where , the vibrational constant, is
 = (1/2c) (k/)1/2
,  = mAmB/(mA+mB)
(13.3.2)
In the above expressions k is the force constant for the molecule and  is the reduced mass of the molecule. Note
that k is related to the curvature of the molecular potential at the equilibrium bond distance, r e, with k = (d2V/dr2)re.
Real molecules deviate from the harmonic oscillator model even at low energies. By analogy to rotation,
the vibrational energy levels for a real diatomic molecule can be written in terms of a power series expansion in (v +
1
/2)
E = (v + 1/2) e - (v + 1/2)2 exe + (v + 1/2)3 eye + ...
(13.3.3)
where e is the value for the vibrational constant at r = re, exe (a single constant, despite how it is written) is the
leading order correction term for the vibrational energy, and eye, ... are higher order correction terms. Real
molecules, unlike the harmonic oscillator, will also have a highest value for vibrational quantum number, usually
written as vmax.
It is interesting to note that equn 13.3.3, cut off after the leading order correction term, gives the correct
expression for the energy levels of a Morse oscillator. A Morse potential has the form
V(r) = De { 1 - exp[-a(r - re)] }2
(13.3.4)
where De is the dissociation energy of the molecule (measured from the minimum in the potential energy curve) and
a is a constant. Solution of the corresponding TISE for this potential gives
EMORSE = (v + 1/2) e - (v + 1/2)2 exe
(13.3.5)
Since a Morse potential is close in appearance to the potential for a real diatomic molecule, equn 13.3.5 is expected
to be a good approximation for the vibrational energy levels of a real diatomic molecule. In this case, the following
relationship, which may be derived for a Morse oscillator, will be approximately correct (it is of course exact for a
Morse potential)
exe = e/De
(13.3.6)
where De is the dissociation energy of the molecule as measured from the minimum of the potential energy curve.
This is closely related to D0, the dissociation energy measured from the v = 0 vibrational energy level. These two
dissociation energies are related by the expression
D0 = De - ( e/2 + exe/4 )
(13.3.7)
Diatomics - Rotational structure and selection rules
If we have sufficient energy to excite a vibrational transition then we can also change the rotational state of
a diatomic molecule. Vibrational spectra will therefore exhibit rotational structure, which can provide additional
information about the molecule.
In the RRHO (rigid rotor-harmonic oscillator) model the combined vibrational + rotational energy of a
diatomic molecule is
E = (v + 1/2)  + J(J+1)B
(13.3.8)
The combined selection rules for an allowed transition are
1) Dipole moment of the molecule changes as the molecule vibrates (which for diatomic molecules means
transitions are allowed for heteronuclear diatomics and forbidden for homonuclear diatomics)
2) v =  1
3) J =  1
For absorption spectroscopy v = +1, while for emission spectroscopy (IR fluorescence) v = -1. Since at room
temperature at equilibrium most diatomic molecules are in the v = 0 vibrational state, the only allowed transition
commonly seen is v" = 0  v' = 1. However, since the selection rule for changes in v is obtained in the harmonic
oscillator approximation, one can occasionally observe weak transitions where v = 2, 3, ... These are called
overtone transitions. Finally, while for most diatomic molecules J =  1, there are a few diatomics where the
selection rule is J =  1, 0. There can also be additional complicating factors for some diatomics, which we will
ignore.
For molecules with initial quantum numbers v" = 0, J", there will be two types of allowed transitions
v" = 0, J"  v' = 1, J' = J"+1
R-branch transitions
v" = 0, J"  v' = 1, J' = J"-1
P-branch transitions
The transitions where J increases by 1 are called R-branch transitions, while those that occur when J decreases by 1
are called P-branch transitons. If there were transitions where J did not change, as is true for some diatomic
molecules, they are called Q-branch transitions. Transitions are labeled by the value of J in the lower energy state,
so, for example, R(3) (for the transition where J increases from 3 to 4), or P(5) (for the transition where J decreases
from 5 to 4). There are R-branch transitions for J" = 0, 1, 2, ..., and P-branch transitions for J" = 1, 2, 3, .... Note
that there is no P(0) transition because the final state required for this (J' = -1) does not exist.
The energy at which a particular allowed vibrational-rotational transition will occur can be found in the
harmonic oscillator approximation using equn 13.3.8
E = E(v',J') - E(v",J")
(13.3.9)
Using equn 13.3.9 and 13.3.8 we can derive the following expressions for the R-branch and P-branch transitions
R-branch
E =  + 2(J"+1) B
(13.3.10)
P-branch
(13.3.11)
E =  - 2J"B
The above predicts two families of transitions, one at energies larger than  (the R-branch transitions) the other at
energies smaller than  (the P-branch transitions). Adjacent peaks within a family of transitions are separated by an
energy equal to 2B, and there is a central gap between the R(0) and P(1) peaks separated by an energy 4B. The
intensity of the observed peaks is determined by a combination of the degeneracy of the initial rotational state and
the Boltzmann factor.
For real molecules, where anharmonicity in the vibrational energy and the change in the rotational constant
with vibration are included, one can use a modified form of 13.3.8 along with equn 13.3.9 to find an improved
expression for the location of R-branch and P-branch transitions. The main effect of including anharmonicity is to
cause the spacing between adjacent peaks in the spectrum to change slightly as J" changes, and to only be
approximately equal to 2B.
Polyatomics
For a polyatomic molecule there will be several different fundamental, or normal mode, vibrations. The
total number is related to the size of the molecule
# normal mode vibrations = 3N - 5 linear molecule
# normal mode vibrations = 3N - 6 nonlinear molecule
where N is the number of atoms per molecule. For small polyatomic molecules it is sometimes possible to write
down the normal mode vibrations by inspection (as done for CO 2 and H2O on page 461 of Atkins). For larger
polyatomic molecules the normal mode vibrations can be classified by their symmetry properties, which can help in
determining the form of each vibration.
In the harmonic oscillator approximation the total vibrational energy for a polyatomic molecule is given by
the expression
E(v1,v2,...vk) = (v1 + 1/2)1 + (v2 + 1/2)2 + ... + (vk + 1/2)k = i=1k (vi + 1/2)i
(13.3.12)
where k is the total number of normal mode vibrations, and vi and i are the vibrational quantum number and
vibrational constant for each normal mode vibration. In the harmonic oscillator approximation v i = 0, 1, 2, ... for
each normal mode vibration.
The selection rules for allowed vibrational transitions are
1) Dipole moment of the molecule changes as the molecule vibrates
2) vi = 0, 1
3) At least one normal mode vibration has a change in its quantum number
For molecules where there are degenerate vibrations (such as in CO2, where the bending vibration has a degeneracy
of 2) the vibrations are usually grouped together and described by a single quantum number. In this case, the
selection rule becomes vi = 0,  1,  2, ..., g, where g is the degeneracy of the vibration.
The types of vibrations observed in absorption polyatomic molecules are classified as follows
1) Fundamental transitions. All vibrational quantum numbers are initially zero. One quantum number
increases by 1, while the other vibrational quantum numbers remain zero. (Example: 0,0,0  0,0,1)
2) Overtone transitions. All vibrational quantum numbers are initially zero. One quantum number
increases by more than 1, while the other vibrational quantum numbers remain zero. (Example: 0,0,0  2,0,0)
3) Combination transition. All vibrational quantum numbers are initially zero. More than one quantum
number changes. (Examples: 0,0,0  0,1,1 ; 0,0,0  2,1,0)
4) Hot band transitions. Transitions where the initial state does not have all of the vibrational quantum
numbers equal to zero. (Example: 0,1,0  0,2,0).
Note that hot band tansitions occur from vibrationally excited initial states. The temperature dependence of a
vibrational transition can be used to identify hot band transitions. Since the fraction of molecules in vibrationally
excited states decreases as temperature decreases, the intensity of a hot band transition should decrease if the sample
is cooled (and increase if the sample is heated). The intensity for the other types of transitions will be relatively
unaffected by changes in temperature.
Generally speaking, allowed fundamental transitions will be very strong, while allowed overtone and
combination transitions will be moderately strong. Forbidden transitions, when they are observed, are usually weak.
For real molecules, where the harmonic oscillator approximation breaks down, the location of overtone and
combination bands will be close to the values predicted by the harmonic oscillator model, but lower in energy due to
the effects of anharmonicity.
The intensity and location of vibrational transitions can often be used to find the vibrational constants for
each normal mode vibration.
Each vibrational transition will have associated rotational structure. However, since this structure is usually
not resolved, we will not discuss rotational structure any further.