Vibrational Spectroscopy
• Observe transitions between vibrational levels
of molecules
– Changes in rotation as well; Ro-vibrational spectra
• Observed as:
– one photon absorption & emission (mostly in midIR spectral region)
– As inelastic scattering of light (Raman)
– As nonlinear 4-wave mixing (CARS)
• Universally applicable.
– All polyatomic and heteronuclear diatomics have
at least one IR allowed fundamental transition
– All molecules have a Raman vibrational spectra.
• Vibrational spectroscopy can be applied to
liquids, solids, and surfaces as well as in gas
phase.
• There is a high degree of qualitative chemical
information in a vibrational spectrum
– Different types of chemical bonds lead to IR
absorption bands in predicable spectral intervals.
– Bernath has a table of such IR frequencies
• These are really wavenumbers not frequencies
– In “finger print region” the normal modes are
quite mixed but can complex spectra can be used
for identification
• Modern electron structure calculations can
predict vibrational spectra with remarkable
accuracy for most molecules
– B3LYP DFT is widely used; after empirical scaling
errors of a few % are typical.
• IR spectroscopy of rare gas matrices have given
the first spectroscopic evidence for many
unstable and highly reactive species
• IR spectra provides window into Intramolecular
dynamics – how energy deposited in one mode
can flow to other modes.
– These “IVR” processes are fundamental to most
chemical reactions.
• Once vibrational spectra are assigned, they
can be used to selectively detect molecules
– IR absorption typically stronger an MW absorption
• For OCS strongest IR integrated line strength ~2000x
larger than for strongest rotational transition at room T.
– FTIR’s are widely used for chemical process control
• IR spectroscopy can be used to probe with
~5mm micron spatial resolution
– Raman & CARS microscopy can be extended down
to ~200 nm resolution
• IR spectroscopy can follow population changes
into the fsec time domain
– Time vs. Freq. resolution trade-off.
Theory Foundations
• Vibrational spectroscopy based upon BornOppenheimer approximation
– Vibrational states found by solving SE for nuclear
motion on potential surface generated by
electronic motion.
• Almost all IR absorption and emission bands
arise from electric dipole transition moments
• Raman transitions arise from vibrational
dependence of molecular polarizability
How do we solve the vibrational SE?
• For diatomic molecules, vibrational energies and
wavefunctions can be solved essentially exactly at
trivial computational cost by numerical
integration
• For small polyatomic molecules (3-5 atoms),
vibrational states can be found by variational
calculations in a large basis set.
• For most larger molecules, vibrational motion
treated by (quasi degenerate) perturbation
theory calculations
– Double perturbation theory: Anharmonicity &
vibration-rotation interactions.
Diatomic Molecules
• Wavefunctions are produce of spherical
harmonic function (rotational wavefunction)
times a radial function c(r). Radial SE:
d 2c
2m
J(J 1)
2   2 E  U(r) 
dr
h
r2
• With c(0)=0. Probability density |c(r)|2

Harmonic Oscillator – Rigid Rotor
• Expand
2
3
4
k2
k3
k4
U(r)  U 0  r  re   r  re   r  re   .....
2
6
24
– Keep only harmonic (k2) term.
J(J 1)
J(J 1)
• Replace


r
r
• Ignore fact that r does not go to minus infinity
 now solve this problem by elementary
• We can
quantum mechanics
 1 
•
˜v   hcB J(J 1)
E U hc
2
v,J
˜


0
1
2c
2
e
k2
m
2 
e
h
Be  2 2
8 cmre
Show Dunham Expansion &
Morse Oscillator worksheets
Vibrational Spectroscopy II:
Polyatomic molecules
Key reference sources:
E.B. Wilson, J.C. Decius, and P.C. Cross, Molecular
Vibrations: Theory of Infrared and Raman
Vibrational Spectra (Dover)
G. Herzberg, Molecular Spectra and Molecular
Structure: II Infrared and Raman Spectra of
Polyatomic Molecules
• Starting point is Harmonic theory of vibrations
– Even when we want higher accuracy, we almost
always start from the Harmonic theory and do
either perturbation theory or variational calcs.
• We will start with analysis of purely vibrational
motion and then go on to ro-vibrational
spectroscopy
Harmonic expansion
• Let us have N atom molecule with equilibrium
positions of atoms xie, yie , zie (i = 0….N-1) and
instantaneous positions xi, yi , zi.
• Define mass weighted coordinates by:
q3i  mi x i  x ie  q3i1  mi y i  y ie  q3i2  mi zi  zie 
• In terms of these coordinates the Kinetic
Energy is given by:
1
2
T
qÝ

2
i
i
Potential Energy
1
1
1
r
)
V (q)  Ve   f i, j qi q j   f i,(3)j,k qi q j qk 
f i,(4j,k,l
qi q j qk ql K

2 i, j
6 i, j,k
24 i, j,k,l
f i, j
  2V 
 
q q 
 
 i j e
1
mi m j
  2V 

x x 

 i j e
There is no term linear in q’s because we expanded around the equilibrium
position. InHarmonic theory, we truncate after quadratic terms. f is a
symmetric matrix with positive definite eigenvalues (or else the re position
would be a saddle point instead of an equilibrium positions. Thus we can
write the eigenvalues as wi2 with orthonormal eigenvectors lji. Define normal
modes of vibration Qi by:
Qi   l ji q j
q j   l j,i Qi
j
i
Classical Solutions
• Qi (t) = Ki cos(  i t + fi) so wi is the angular
velocity of i’th normal mode. Divide by 2c to
get wavenumber.
q j (t)   l j,i Qi (t)   l j,i K i cosw i t  f i 
• Define dipole derivatives
i

i
r 
r  
r 




q
m
m
m
r
j


mi    

l j,i
   



Qi  j q j Qi  j q j 
• To first order in q (harmonic approximation for
r
r
r
 dipole moment)
m( t)  me   mi Qi (t)
i
• Classically, an oscillating dipole moment
radiates at its frequency of oscillation. Thus,
we can expect a normal mode will produce
emission (and thus by the Einstein relations,
absorption also) when at least one component
of mi is nonzero.
• Polarizability derivative
a 
i
a 

 
 q 
l ji
j 
j 
,   x, y, or z
– Normal mode will produce Raman scattering with
wavenumber shift wi/2c if at least one

component
of ai is nonzero
Zero frequency modes
• For a nonlinear (linear) molecule, there will
always be 6 (5) normal mode vectors l that
have zero eigenvalues. 3 of these will
correspond to the x,y, and z translation of the
entire molecule, which can have zero restoring
force. 3(2) correspond to infinitesimal
rotation of the molecule, which also has zero
restoring force.
Symmetry Analysis
• Each normal mode is part of irreducible
representation of the Point Group of the molecule’s
equilibrium structure.
• Form a 3N dim. Rep. from the x,y,z displacement
coordinates of each the N atoms
– Reduce this into the irreps.
– Subtract irreps for x,y, and z (for center of mass motion)
and Rx, Ry, and Rz (for rotations)
– The remaining irreps give the distribution of the normal
modes among symmetry classes.
– The number of totally symmetric modes equals the
number of unique structural parameters of equilibrium
structure.
Symmetry (cont.)
• Normal modes which have the symmetry of x,y, or z
are IR symmetry allowed.
• The fundamental transition will have components of
the dipole matrix element that have same symmetry
as the Cartesian components.
• Normal modes which have symmetry of x2, y2, or z2
have polarized Raman scattering fundamentals
• Those with symmetry of xy, yz, or xz have
depolarized Raman scattering fundamentals.
Quantum Solutions
• Let Pi be momentum conjugate to normal
mode Qi.
h
H   Pi2  w i2Qi2 
2 i

1 
E   hw i n i  

2 
i
  
i
HO
ni
(Qi )
IR & Raman Selection Rules
• If we approximate:
m me   mi Qi
r
r
r
– (double harmonic approximation) i
– Qi have nonzero transition matrix elements between
states with ni = ±1 while all other nj =0.
(Fundamentals)

• Same ni = ±1 selection rule applies to Raman
Spectroscopy if we truncate polarization expansion
at linear terms.
• Exclusion Rule: For molecules with center of
inversion, IR allowed fundamentals are all of u
symmetry, while all Raman fundaments are g.
Matrix Elements
  w /h
n | Q | n 1 
n | Q2 | n  2 
n 1
2
n 1n  2
n | P | n 1  ih

n | Q | n 1 
2
n 1
2
n
2
n | Q2 | n 
2n 1
2
n | P | n 1  ih
n | Q2 | n  2 
nn 1
2
n
2
In general Qm has matrix elements with n = -m, -m+2….+m. Similarly
for Pm. Q can be written as sum of raising and lowering operators. P as
i times the difference.
Overtones and Combination bands
• Spectra show transitions from ground state to
states with multiple quanta in one mode
(overtones) or multiple modes (combination
bands) principally for two reasons
– The dipole (and polarizability) are not simply
linear functions of the normal modes, but have
higher order terms in their expansions. The
corresponding orders in the expansions allow
transitions with the same order changes in
vibrational quantum number
Overtones and Combinations (cont).
• The potential has higher order terms beyond
harmonic terms.
– For stretching vibrations, the potential is more like Morse
Oscillator which has rapidly decreasing but nonzero matrix
elements for all n.
– Off-diagonal terms such as Qa2Qb, will mix the state with nb
= 1 with the one with na = 2. This allows the state with na =
2 to “borrow” or “steal” intensity from the nb = 1
fundamental.
• This is known as Fermi resonance when third order term is
involved and is quite common.
Internal coordinates
• Potential energy surface much easier to parameterize in
terms of bond lengths and bond angles rather than
atomic Cartesian coordinates.
• We can use internal coordinates for vibrational problem.
– In general, these internal coordinates are curvilinear and give
very complex kinetic energy operators.
– More common to define rectilinear internal coordinates that
have atoms move in straight lines.
– Minimum energy to bend is an arc that is somewhere between
rectilinear motion and constant bond length bend.
– These internal coordinates are not “orthogonal” which
introduces come complications.
Internal coordinates (cont)
• Commonly used internal coordinates are:
– bond lengths: rij = |ri – rj|
– bond angles qijk equal to the angle between unit
vectors j->I and j->k
– and torsional angles tijkl equal to the angle
between planes formed by atoms i,j,k and that
formed by atoms j,k,l
– Out of plan distance h, equal to the distance of an
atom i from the plan formed by atoms j,k,l
• For each internal coordinate, St, define a 3N
vector Bt which has components equal to the
derivatives of St with respect to the Cartesian
coordinates of the atoms. The set of column
vectors Bt make a matrix B.
– Rectilinear coordinates defined by St = Si Bti xi
– Explicit expressions for B elements given in
Wilson, Decius, and Cross
• Define G matrix by
3N 1
Gtu 

i0
1
Bt,i Bu,i
mi
• Kinetic energy:
dSu 
Pt   G t,u
dt
u
1
r 1 dSt
1
1 r tp
dSu
1
T   Pt  Gt,u  Pu  P  G P  
 G t,u 
2 t,u
2
2 t,u dt
dt u
• Define F matrix elements by
  2V 
Ft,u  


S

S
 t u e
• The square of the vibrational angular frequencies
are the eigenvalues of the matrix G.F.
– Note that even though G and F are symmetric matrices,
their product is not (since they do not commute)
– Eigenvalues are still real and positive definite or stable
minimum energy point.
– Define matrix, L, of right handed eigenvectors by
•
•
•
•
G.F.λ = L. , where  is diagonal matrix of eigenvalues
L normalized by Lt G-1 L = I (identity matrix), Lt F L = 
This matrix is not orthogonal L-1 = Lt G-1.
S = L.Q, Q = L-1.S, Ps = (L-1) t .PQ PQ = L t .PS
– L matrix is real which means all atoms reach end
points of vibration at same time (though for
asymmetric modes, some could be at outer
turning points when others at inner turning pts.
Redundant Coordinates
• It is often useful to define more internal coordinates than
there are normal modes because of angle constraints.
– For planer molecule with 3 atoms bound to common atom (for
example H2CO, the sum of the three bond angles = 2.
– For 4 bonds from common atom (often C), there are six bond angles,
but only 5 are independent.
– For ring compounds, the sum of the internal bond angles are  (N-2).
• We could use constraint to express one angle in terms of the
others, but that will kill the symmetry.
• If we ignore the constraints, but put in physically realizable
force constants, we will get zero eigenvalue for normal mode
that is perpendicular to constraint condition.
Symmetry Coordinates
• We can do a linear transformation of the internal
coordinates to produce a set of symmetry
coordinates that transform as irreducible reps of the
point group.
– We can use symmetry projection operators to generate
these from the internal coordinates.
– Each symmetry coordinate will be linear combinations of
symmetrically equivalent internal coordinates (i.e. those
that transform into one another)
• If we transform the F and G matrices by the matrix
that transforms to symmetry coordinates, they will
each block diagonalize into symmetry blocks. Same
for FG product.
Isotopic Substitution
• The shift in vibrational spectrum with isotopic
substitution is often used to assign character
of normal modes.
p
2
i
– By first order perturbation theory: dE   2m
dm
m
– Thus the shift gives us the kinetic energy of the
substituted atom in each normal mode.
k
i
i
i

k
• Redlich-Teller Product rule: (WDC sect. 8.5)
w 'k  I'A I'B I'C M' 3N 1 mi
 w   I I I M  m'
 A B C 
k
i
k 1
i0
3N 6
Vibrational Angular Momentum
• Define Vibrational Angular Momentum
operators (Pa):
N
ast  
a l
l
s,i t,i
i1  ,  x,y,z
 xyz   yzx   zxy  1  xzy   zyx   yxz  1 a  0 otherwise
r r
(a  b )a  a a b
 ,
3N 6

Pa  ih  stQs
Qt
s,t
a
Effective Moment of Inertia


a a
I'aa  Iaa  s,ut.u QsQt

s,t  u
 a  
I'a  Ia  s,ut.u QsQt

s,t  u
m  I'
1

a  x, y,z
a    x, y,z
Wilson –Howard Hamiltonian (1936)
1
1
H  m1/ 4  (J  P ) m , m 1/ 2 (J  P ) m1/ 4  m1/ 4  Psm 1/ 2 Psm1/ 4  V (Q)
2
2
 ,
s
m  det( m )
• This is so complicated that it is almost
impossible to use without further
simplification!
Watson Hamiltonian
Mol. Phys. 15, 479-90 (1968)
• James Watson showed he could dramatically simplify
the Wilson-Howard Hamiltonian
1
h2
2
H   m , J  P  J  P   2  V (Q)  U(Q)
2  ,
2 t Qt

U 

1
m
8 a aa
The U term (which arises from K.E.) depends only on Q’s and
thus is a like a mass dependent kinetic energy term.

The Watson Hamiltonian has been used to do essentially exact
variational calculations of ro-vibronic energy levels of
polyatomic molecules.
Parallel Vibrational Band of a Symmetric top molecule
Parallel transitions (DK = 0)
 (K)   0  A' B'  A"B"K
sub
0
Beff  B  DJK K 2
2
Aeff  A DK K 2
The P,R branches follow polynomial in m expression as for linear molecule
but with |m| > K.
Honl-London
Factors
DJ  1 AKJ
DJ  0
AKJ
(J 1) 2  K 2

(J 1)(2J 1)
K2

J(J 1)
DJ  1 AKJ
J2  K2

J(2J 1)
Perpendicular Vibrational Band of
Symmetric Top
 0sub (K)   0  A'(1  2)  B'  2A'(1  )  B'K  A' B''  A"B"K 2
Honl-London Factors
DJ  1

DJ  0
DJ  1
AKJ
(J  2  K )J 1  K 

(J 1)(2J 1)
AKJ 
J 1 mK J mK 
AKJ 
J(J 1)
J 1 mK J mK 
and RR branches (K = J) are strong,
RP and PR (K = -J) weak
PP

J(2J 1)