Band designation in Chapter 5

Fermi Resonance(2 vib levels)

3N-63N-5 (Linear)

Normal Modes

1 ev = 1.60 X 10-12 erg

= 23 kcal/mole

= 12,000 K

= 8,000 cm-1

= 12,000 Å (1.2 μm)

= 2.4 X 1014 hz

= 12398 Å

= 1239.9 nm

100 cm-1 = 144 K

= 100 μm

= 3000 GHz

Energy Scale (order of magnitude)

Electronic transition E ~ e2/a

e = electronic charge a = Bohr radius

E ~ 10 eV ~ 105 cm-1

Vibrational transition v =

k = “spring constant” ≈ D/a2 D = dissociation energy

D ~ eV a ~ A k ~ 105 dynes cm-1

v ~ 103 cm-1

Rotational transition J = angular momentum

= mωa2 = nh K.E. = mω2a ~

E ~ 10 cm-1

• Thermal energy kT ~ mv2 (1 K ≈ 0.7 cm-1)

T = 300 K E ≈ 200 cm-1∴ Energy hierarchyElectronic > Vibrational > Thermal > Rotation

H = H1+ H2 (t)

(3.1)

(stationary states)

(3.9)

(3.10)

Rotational transitionVibrational transition

ro-vibronic transition (electronic)

SPECTROSCOPY OVERVIEW

upper state denoted by single apostrophe lower “ “ “ double “

v’ J’

v’’ J’’

ELECTRONIC SPECTROSCOPY

(visible, ultraviolet)

Born - Oppenheimer approximation

Motion of nuclei show pared to motion of electrons

∴ can separate Hamiltonian into nuclear and electronic terms

Calculate electronic potential surface as a function of internuclear coordinate

Qualitatively, different electronic states have different electron cloud geometries electric dipole allowed transitions

State designations

Linear molecules g spin multiplicity = 25 + 1

S = ∑si = 0 g=1 singlet

total electronic orbital angular momentum ½ 2 doublet

0 ∑ state 1 3 triplet

1 π

2 Δ

Nonlinear molecules

Z = species of symmetry point group to which molecule belongs

Different electron orbital configurations may behave differently with respect to symmetry operations of point group

Individual electronic states characterized by another letter

X – ground state

A, B, C, … - excited states of same spin multiplicity as ground state

a, b, c, … - excited states with different spin multiplicity

4g

Electric dipole selection rulesLinear molecules Δ = 0, ±1 (randomly oriented molecules) + - in Σ – Σ transitions (notation reflects symmetry of electronic wave functions with respect to *reflection in plane containing internuclear axis) u g (extra designation for homonuclear diatomic molecules indicating eve or odd number of odd orbitals) reflected at 0 ΔS = 0 (spin-orbit coupling weak) weaker magnetic dipole, electric quadruple transitions have different rulesNonlinear molecules symmetry of product of upper and lower states has to be of same species as transition operator ΔS = 0 (spin-orbit coupling weak )*

abso

rptio

n

fluor

esce

nce

inte

rsys

tem

cr

ossi

ng

nonr

adia

tive

deca

y

phosphorescence

S = 0S = 1

S = 0 S = 1or

(1927)

Franck-Cendon factor

Hönl-London factor

Fig 3.8 anharmonicity

relative intensity of hot band no fundamental reflects population of initial states

overtone weaker than fundamental, reflects degree of anharmonicity

combination bands can be very strong if cause large change in dipole moment

Isotopic Bands: 13CO 2, C18OO, HDO

Terminology Vibrational transitions

+

2

1

0

1

0

2

Combination band

v1 v2v1 Normal modes

hot band

fundamental

Overtone (due to anharmonicity)

Rotation – Vibration bands (p. 90)

emission

absorption

J’ = J” + 1J’ = J”J’ = J” - 1

v’

v”J”

13Transition v’ , J’ – v’’ , J’’ upper lower

J’ = J’’ + 1 R – branch

J’ = J’’ Q – branch

J’ = J’’ – 1 P – branch

P RQ

Δv = ± 1 ΔJ = 0 (some cases) ±1Δv = arbitrary for different e states

I2

v1 = 213.2 cm-1

T = 300 K (= 208.7 cm-1)

vibrational population

Rotational population

N2O doubling to rotationcoupling of degenerate vibration

Spectral line shape

• Natural: arises from uncertainty principle due to finite lifetime of excited state ΔvΔT ∼ 1 ΔT = lifetimeImbuntant for resonance scattering of atomic lines

e. g. H (Lyα) 0 (1304 Å)

• Doppler: thermal motion of atoms and moleucles important in mesosphere (~100 km)

• Lorentz: collisions by other molecules. Most importan in bulk of atmosphere

• Voigt: superposition of above 2 (non-interactive)

• Dicke : superposition of Doppler + Lorentz can lead to line “narrowing”!

• Dimer and Pressure-induced by transient state brought together by collision

Oscillator with dumping

Solve for xo

2

Pure sine wave

Finite life time

(3.48)

quantum life timecollision life time

V V + W (3.52) Quantum veresion

Lorentz line width

Natural line width

Doppler Profile

Lurentz Profile

Voigt Profile

(Frequency shift)

(3.79)

(3.48)

4

(3.51)

(collision frequency)

€

FWHM = 2((log2)α D2 +α L

2 )1

2

Doppler + LorentzIn dimensionless units

5

18