CHAPTER 12 Molecular Spectroscopy 1: Rotational and Vibrational Spectra
I. General Features of Spectroscopy.
A. Overview.
1. Bohr frequency rule:
hc Ephoton = hν = = hcν˜ = Eupper −Elower λ
1 ν˜ = -1 Also remember λ (usually in cm ) € 2. Types of spectroscopy (three broad types):
-emission
€
-absorption
-scattering
Rayleigh Stokes Anti-Stokes
CHAPTER 12 1
3. The electromagnetic spectrum:
Region λ Processes gamma <10 pm nuclear state changes x-ray 10 pm – 10 nm inner shell e- transition UV 10 nm – 400 nm electronic trans in valence shell VIS 400 – 750 nm electronic trans in valence shell IR 750 nm – 1 mm Vibrational state changes microwave 1 mm – 10 cm Rotational state changes radio 10 cm – 10 km NMR - spin state changes
4. Classical picture of light absorption – since light is oscillating electric and magnetic field, interaction with matter depends on interaction with fluctuating dipole moments, either permanent or temporary.
Discuss vibrational, rotational, electronic
5. Quantum picture: Einstein identified three contributions to transitions between states:
a. stimulated absorption
b. stimulated emission
Stimulated absorption transition probability w is proportional to the energy density ρ of radiation through a proportionality constant B
w=B ρ
The total rate of absorption W depends on how many molecules N in the light path, so:
W = w N = N B ρ
CHAPTER 12 2 Now what is B?
Consider plane-polarized light (along z-axis), can cause transition if the Q.M. transition dipole moment is non-zero.
µ = ψ *µˆ ψdτ ij ∫ j z i
Where µˆ z = z-component of electric dipole operator of atom or molecule
Example: H-atom € €
- e r
+ ˆ er µ z = −
Coefficient of absorption B (intrinsic ability to absorb light)
2 µij B = 6 2€ Einstein B-coefficient of stimulated absorption εo
if 0, transition i j is allowed µ ij ≠ →
µ = 0 forbidden € ij € 6. Selection rules – QM-derived rules, based on the evaluated transition dipole moments which say what transitions are allowed or forbidden. €
Examples: H-atom selection rules. sà p allowed Δml = 0, ±1 Δl = ±1 Δn = anything
2s
1s sà s forbidden
CHAPTER 12 3 Molecular vibration – a vibrational mode must result in the oscillation of a dipole moment in order to appear in vibrational spectrum.
Example: HCl has 1 vibr mode. IR active. O2 has 1 vibr mode. IR inactive.
A molecule has: 3N–6 vibr modes, if non-linear 3N-5 vibr modes, if linear
CO2 3 * 3 – 5 = 4 vibr modes
Vibration state changes allowed:
Δv = ±1
Since selection rules are usually derived from approximate ψ’s and without relativity effects, some “forbidden” transitions actually can occur with a small probability.
Example: Vibration Δv = ±2 (1st overtone) Δv = ±3 (2nd overtone)
Molecular rotations – molecule must possess a permanent dipole moment to absorb in a pure rotation spectrum.
7. Stimulated Emission.
Stimulated emission transition probability w’ is given by
w’=B’ ρ
and the total rate of stimulated emission W’ depends on number of molecules N’ in the excited state as:
W’ = w’ N’ = N’ B’ ρ
Einstein was able to show that B=B’
For example, if you had one molecule in the lower state and one molecule in the upper state, an incident photon would have EQUAL probability of stimulating an emission as stimulating absorption.
Implication: continuous bombardment of a sample with high intensity resonant light can only induce up to but not beyond a 50/50 population of lower and upper state.
CHAPTER 12 4 Wnet = W - W’
= NBρ - N’B’ρ
= NBρ - N’Bρ
= (N - N’)Bρ
Actually there is one more event happening that negligibly affects the above equation, and that is spontaneous emission.
8. Spontaneous emission: Upper to lower state transition occurring even in absence of radiation.
So the downward rate would be modified to:
W’ = N’(A + Bρ)
Where A = coefficient of spontaneous emission:
$ 8πhν3 ' A = & 3 )B % c (
9. Two principal factors determine intensity of absorption or emission:
€ a) inherent probability of transition.
2 B as seen before ∝ µij
b) population of atoms or molecules in initial state.
€ −Ei /kBT Population of state i N i ∝gie
Ei = energy of state i gi = degeneracy of state i -23 -1 kB = 1.38 x 10 J K Boltzmann constant T = K temperature € 10. Emission intensities can be increased by promoting excited state populations above their room T values by:
-flame excitation -electrical discharge
CHAPTER 12 5
11. Emission spectroscopy (spectroscope) – H-atom Experiment.
-emitted light generated in e- discharge tube -passed through narrow slit -dispersed: quartz prism; diffraction grating -detected by: photography; photomultiplier tube (pmt)
12. Absorption Spectroscopy (spectrophotometer) – Dye Experiment.
a) need light source in appropriate frequency range. -klystron for microwaves -Nernst filament for IR (a ceramic containing rare earth oxides) -synchrotron – wide range -common incandescent bulb – visible
b) grating or prism to select out monochrome light passed through cell.
c) single beam instrument – sample cell and blank cell are placed alternately in the beam. Spectra are subtracted later.
d) double beam – beam is split and passed simultaneously through sample and blank.
e) detectors – photomultiplier tube (PMT); photo plate; thermocouple (IR); crystal diode (microwaves).
13. Intensity Definition.
Itot = total intensity = intensity of light of all λ passing through a unit area per unit time.
I(λ) = intensity per unit wavelength interval
I(λ)dλ = intensity of radiation per area per time in wavelength interval λ to λ+dλ
CHAPTER 12 6 14. The Beer-Lambert Law: dependence of intensity on concentration of substance in sample.
loss of intensity in dx interval = -dI = κ(λ) c I dx
where κ(λ) is intrinsic ability of material to absorb light at λ, c is concentration, and dx is thickness
−dI = κ(λ)cdx I
I(b) b dI − = κ(λ)cdx ∫ I ∫ Io o
I(b) −ln = κ(λ)cb Io I +ln o = κ(λ)cb I(b) I 2.303log o = κ(λ)cb I(b)
Define absorbance as
Io Aλ ≡ log absorbance at λ (or optical density) € I(b)
Aλ = ελbc Beer Lambert Law
€ κ(λ) ε = = extinction coeff or molar absorption coeff λ 2.303 CHAPTER 12 7
€ 1 molar absorptivity units = mol cm ( L)
b = path length in cm
c = concentration in€ mol/L (molarity)
I(λ) T(λ) = transmittance = I λ ( )o
I(λ) %T(λ) =% transmittance = x100% I λ € ( )o
Integrated absorption coefficient €
A = ∫ ε(ν˜ )dν˜ band
15. Raman spectroscopy.
€Look at scattering of high intensity visible energy photons (from laser) from the sample.
Sample may absorb incident photon and immediately re-emit at slightly different ν. Δε matches vibrational transition in the molecule.
Raman has advantage that some vibrational modes that are IR inactive may be Raman active. (see lab manual)
CHAPTER 12 8 B. Linewidths - peaks have a finite width in frequency or wavelength.
1. Doppler broadening.
Gas phase Doppler effect
Actual freq molecule will absorb if moving veloc v away:
ν ν " = o 1 v + c
Moving toward:
ν € ν " = o 1 v − c
Measure width of line at half-height
1/2 2ν $ 2k T ' € δν = o & B ln2) where m is mass and T is temp c % m (
T δν ∝ increases with Temp and decreases with mass € m
2. Lifetime broadening.
€ Let τ = average lifetime of being in excited state.
Uncertainty principle δE ≈ τ
shorten the τ, the greater the blurring δE of the energy of the excited state €
5.3cm−1 δν˜ ≈ τ(ps)
Processes responsible for finite lifetimes τ of the excited state
a. collisional deactivation (work at low P to minimize effect) € b. spontaneous emission (can’t change this) τ↓ ν↑ CHAPTER 12 9 II. Pure Rotational Spectra. (microwave region)
A. Energy levels depend on symmetries, moments of inertia.
atoms 2 2 I = mixi where x is distance of atom i from rot axis ∑ i i
Molecules have 3 moments of inertia: € € Ia < Ib < Ic (principal axis)
2 2 2 Ja Jb Jc Classical rotational energy E = + + 2Ia 2Ib 2Ic
Jα = ang mom about α axis
€ J2 J J 1 2 Convert to QM each → ( + )
B. Four types of Rigid Rotors.
1. Spherical €rotors, 3 equal moments of inertia (CH4)
2. Symmetric rotors, 2 equal moments of inertia, one different (NH3)
3. Linear rotors, one moment of inertia is zero (CO2, HCl)
4. Asymmetric rotors, 3 different moments of inertia (H2O)
Table 12.1 Moments of inertia
HCl
HCN
CO2
CHAPTER 12 10
CH3Cl
NH3
SF4Cl2
CH4
SF6
CHAPTER 12 11 C. Linear rotor.
Rotation about internuclear axis has zero moment of inertia, no energy
Other two rotations have equal moment of inertia = I 2 EJ = J(J+1) 2I
ΔJ = + 1 Selection rule (also molecule must be polar to have pure rot spectrum)
€ ΔmJ = 0, + 1
degeneracy of rot level J = 2J + 1 = gj
For rotational absorption transitions:
J-1 à J
Derive ΔEJ defined as EJ - EJ-1
Set hν = ΔEJ
Solve for ν˜ wavenumber
€
Result:
ν˜ (J← J−1) = 2BJ
where
B ≡ the rotation constant € 4πcI CHAPTER 12 12
€ D. Pure Rotational Spectrum.
“Pure” meaning only rotation state transitions, no vibration state changes. Pure rotational transitions are in microwave region of spectrum.
At right is pure rot spectrum of a linear molecule.
Wavenumbers of transitions are equally spaced at 2B, 4B, 6B, 8B, etc.
spacing between them = 2B
Typical B falls in range:
0.1 to 10 cm-1, so in microwave
Peak heights vary according to equilibrium population of the energy levels and their degeneracy
Population of level J
−EJ /kBT NJ ∝gJe
g J = 2J+1
€ E. Centrifugal distortion effect.
We have treated rotors as rigid. In reality, they can flex, and the magnitude of their flex increases as rotational energy increases.
3 ν˜ (J← J−1) = 2BJ− 4DJJ
CHAPTER 12 13 € III. Vibrational Spectra of Diatomics.
A. Molecular vibrations.
1. Here we are concerned with motions of nuclei in a potential well which, to a first approximation and for small amplitude vibrations, is parabolic.
1 2 V(x) = kx where x ≡ R −Re 2
R=instantaneous internuclear separation
Re=equilibrium internuclear separation € (bond length)
k = force constant of the bond (curvature of parabola)
2. We have previously solved the Schrod Eq for this system
Energy levels are evenly spaced, given by:
E ( 1) 0,1,2,3... v = υ + 2 ν υ = where :
1 1 $k '2 ν = & ) µ=reduced mass 2π %µ (
3. Vibration terms G(v) of a molecule:
These are energy levels but expressed in wavenumbers (cm-1)
G( ) ( 1) υ = υ + 2 ν where 1 1 $k '2 ν = & ) 2πc %µ ( E G(υ) = v hc
CHAPTER 12 14 4. Selection Rules for vibration of molecules.
a. The electric dipole moment of the molecule must change when the atoms vibrate in order to absorb in the infrared (be infrared active).
For diatomics this means they must possess a dipole moment. HCl vs N2
For polyatomics, there will be several modes of vibration, some of which are IR active and some which are not, depending on whether the motion produces a change in the dipole.
A polyatomic need not have a permanent dipole moment to have some modes which are IR active.
b. For purely harmonic vibrations, the vibration state can only change by one.
Δ υ = ±1
This means that there should be only one absorption line in the pure IR spectrum, where € Transition wavenumber
= G(υ + 1) − G(υ) = ν
5. At room temperature the equilibrium distribution of energy in molecules has almost all molecules in the ground vibrational state, so the fundamental transition.
v=0 to v=1
is the only transition contributing to the spectrum. ν = (1 ← 0) = G(1) − G(0)
6. Real molecules show anharmonic vibrations, especially in higher vibr levels. Do The true potential energy curve is not De parabolic, and the levels are thus not evenly spaced but becomes shortened at higher energy:
CHAPTER 12 15 Potential energy curve can be approx. by so-called Morse potential:
2 V(R) hcD 1 e−a(R−Re ) = e{ − }
An anharmonicity correction term can be derived for the vibrational energy level equation since the Schrodinger Eq can be solved in the Morse potential. (See text)
€ Anharmonicity also loosens up the selection rule such that weak overtones appear in the spectrum, in which level changes of + 2 are possible.
B. Vibration-rotation spectra.
When heteronuclear diatomics (e.g., HCl) are studied in the gas phase, their IR spectra will show rotational state changes accompanying the vibrational transition v=0 à 1.
This is called rotational fine structure.
The rotation of the molecule will change by ΔJ = + 1 while vibration is changing v=0 to 1.
CHAPTER 12 16
Equation generating this spectrum:
ν˜ R−branch = ν˜ + 2B(J+1) J = 0,1,2...
˜ ˜ 2BJ J 1 2 3 ν P−branch = ν − = , , ...
Derive by using a combined vibration-rotation term S(v,J), which is the energy when molecule is in vibrational state v and rotational state J.
€ Let’s do this without anharmonicity effects in the vibration, and without centrifugal distortion of the rotor.
S(υ,J) = G(υ) + F(J)
or S (υ,J) = (υ +1/2)ν˜ + BJ(J+1)
R-branch ΔJ = +1
€ ν˜ R−branch = S(υ +1,J+1) −S(υ,J) or
˜ ˜ 2B J 1 ν R−branch = ν + ( + )
where ν˜ is the wavenumber of pure vibrational trans
P€- branch ΔJ = -1
€ ν˜ P−branch = S(υ +1,J−1) −S(υ,J) or
˜ ˜ 2BJ ν P−branch = ν −
Corrections for real diatomics:
vibrational anharmonicity xe €centrifugal stretching De (very small) *vibr-rot coupling αe
CHAPTER 12 17 IV. Vibrational Spectra of Polyatomics.
One bond vibr is coupled to another – leads to complicated motion.
Simplified by resolving nuclear vibrations into “normal modes” of vibration.
Only normal modes of motion resulting in change in dipole moment can absorb from electromagnetic field (infrared active).
IF N atoms in molecule:
3N-6 normal modes of vibration (non-linear molecules)
3N-5 (linear)
H2O CO2
IR spectra useful in compound identification.
CHAPTER 12 18 Greenhouse Gases
CHAPTER 12 19 V. Raman Scattering Spectra.
A. Overview.
1. Means of studying spectroscopically inactive vibr and rotational transitions.
2. Based on inelastic scattering of photons and molecules.
3. In elastic collision – no energy transfer.
4. In inelastic collision – molecule changes quantum states.
ν scattered ≠ νincident
5. Mechanism of light scattering:
€ -oscill field induces an oscillating dipole moment in molecule (even if it has no permanent dipole). (induced dipole strength ~ α, polarizability)
-oscill dipole emits electromagnetic radiation, predominantly of same frequency. (Rayleigh scattering)
-molecule may also emit light of slightly lower frequency (Stokes) or higher frequency (anti-Stokes).
h ν + E# # = hν # + E#
where h ν is incident light, E " " is molec energy in initial state, h ν " is scattered light, and E " is final state. € E " −E" " = h(ν − ν ") = hΔν € € € where E " −E" " is determined€ by quantum levels and h Δν is called “Raman shift” (Δν independent of ν) € B. Selection rules for Raman: € €
1. rotational Raman ΔJ = ±2
2. vibrational mode is Raman active if polarizability of molecule changes during vibration.
Δν = ±1 (Stokes and anti-Stokes)
3. If molecule possess a center of inversion, then an IR active vibration will be Raman inactive, and vice versa.
CHAPTER 12 20 C. Examples.
1. CO2 linear molecule with center of inversion.
IR inactive (no dipole fluct) but Raman active (polarizability changes).
IR active, Raman inactive
IR active, Raman inactive
IR active, Raman inactive
2. Tetrahedral molecule like Methane or carbon tet (no center of inversion)
3N-6 = 3x5-6 = 9 modes all modes Raman active
1 of these. 3 of these. IR inact. IR active.
2 of these. 3 of these. IR inactive. IR active.
CHAPTER 12 21
NOTES:
CHAPTER 12 22