Atomic and molecular spectroscopy chm323
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Transcript of Atomic and molecular spectroscopy chm323
l (L) = orbital angular momentum quantum number l (L) = Total orbital angular momentum (vector), lz = Projection of l on z-axis (ml) lz = ± l, ± (l -1), ± (l -2)……..0. s (S) = Spin angular momentum quantum number s (S) = Total spin angular momentum (vector) sz (Sz) = Projection of S on z-axis (ml) sz (Sz) = ± s, ± (s-1), ± (s-2)……..0., for integer values of S OR ± s, ± (s-1), ± (s-2)……..1/2, for half-integer values of S j (J) = Total angular momentum quantum number j (J) = Total angular momentum (vector) jz (Jz) = Projection of J on z-axis (ml) jz (Jz) = ± j, ± (j-1), ± (j-2)……0, for integer values of J OR ± j, ± (j-1), ± (j-2)……..1/2, for half-integer values of J
Common notations and its explanations
Fundamentals of Molecular Spectroscopy
CHM 323
Atomic Spectroscopy
l √2 lz = +1ħ
Orbital angular momentum (l)
√2 lz = -1ħ
l
√2 lz = +1ħ
√2 lz = 0
For H-atom (one e-system)
I = I(I+1) ħ ; l = 1, I = √2
s √3/2 sz = +1/2ħ
Spin angular momentum (s)
s
√3/2 sz = +1/2ħ
√3/2 sz = +1/2ħ
s = s(s+1) ħ ; s = ½ , s = √3/2
l √2 lz = +1ħ
s √3/2 sz = +1/2ħ
Orbital angular momentum (l) Spin angular momentum (s)
Orbital and Spin angular momentum (l, s)
l
s
lz = +1ħ
sz = +1/2ħ
√2
√3/2
l
s
lz = +1ħ
sz = +1/2ħ
√2
√3/2
z-axis projection l
s
lz = +1ħ
sz = +1/2ħ √3/2
√2
l
s lz = +1ħ
sz = +1/2ħ √3/2
√2
l
s
j
l = 1 and s = 1/2
j = [j(j+1)]1/2 ħ = √15/2 ħ
l
s lz = +1ħ
sz = +1/2ħ
jz = +3/2 ħ
lz = +1 and sz = +1/2
√3/2ħ
√2 ħ √15/2 ħ
j = Il + sI = I1 + ½I = 3/2
l
s j
Spin-orbit coupling : total electron angular momentum (j)
S-O coupling
2p orbital
l
s √3/2 ħ
√2 ħ
j = [j(j+1)]1/2 ħ = √3/2 ħ
lz = +1ħ
sz = -1/2ħ
l = 1 and s = 1/2 lz = +1 and sz = -1/2
j = Il - sI = I1 - ½I = 1/2
l
s
√3/2ħ j
j jz = +½ħ √3/2ħ
jz = +½ħ S-O coupling
lz = 0 l s √3/2
ħ
√2 ħ
sz = -1/2ħ
j = [j(j+1)]1/2 ħ = √3/2 ħ
s = 1/2 lz = 0 and sz = -1/2
j √3/2 ħ
jz = -½ħ
S-O coupling
s lz = 0 l
√2 ħ
√3/2 ħ sz = +1/2ħ
s = 1/2 lz = 0 and sz = +1/2
j = [j(j+1)]1/2 ħ = √3/2 ħ
j √3/2 ħ jz = +½ħ S-O coupling
l
s +½ħ
√2 ħ
√3/2ħ
lz = -1ħ
l = 1 and s = 1/2 lz = -1 and sz = +1/2
j = [j(j+1)]1/2 ħ = √3/2 ħ
j = Il - sI = I1 - ½I = 1/2
l
s
jz = -½ħ j √3/2ħ
j jz = -½ħ √3/2ħ
S-O coupling
l
s √2 ħ
√3/2ħ lz = -1ħ sz = -1/2ħ
l = 1 and s = 1/2 lz = -1 and sz = -1/2
j = [j(j+1)]1/2 ħ = √15/2 ħ
j = Il + sI = I1 + ½I = 3/2
l s
√15/2ħ jz = -3/2ħ j
√15/2ħ
jz = -3/2ħ j
S-O coupling
lz = +1 and sz = +1/2
lz = +1 and sz = -1/2
lz = 0 and sz = +1/2
lz = 0 and sz = -1/2
lz = -1 and sz = +1/2
lz = -1 and sz = -1/2
Hydrogen atom fine spectrum
From Banwell, page No. 138
Compound doublet
From Banwell, page No.138
Lithium atom fine spectrum From Banwell, page No. 141
Helium atom fine spectrum From Banwell, page No. 147, Fig. 5.11
Compound Triplet
From Banwell, page No. 148, Fig. 5.12
Validation/Application of Atomic Spectroscopy: 1. Zeeman Splitting
m = -e/2m J joules/tesla
m = -eg/2m J joules/tesla
m = -eg/2m J(J+1) ħ J/T
For a point charge of mass m, the magnetic dipole vector is given by,
But electron is not a point charge, so need to include a numerical factor, g, called Landé factor
Substituting value of J;
g = 3/2 + S (S+1) – L(L+1)
2J(J+1) g lies in between 0 and 2
Now, assume that the external magnetic filed is in the z direction, then J assigns the Jz values.
mz = -egh/4pm Jz J/T
Extent of splitting: DE = mz . Bz joules = -egh/4pm Jz Bz joules
DE a Jz . Bz
Jz= ± J, ± (J-1), ± (J-2)……0, for integer values of J OR ± J, ± (jJ1), ± (J-2)……..1/2, for half-integer values of J
-eh/4pm = Bohr Magneton
Example 1 Singlet term transitions
1S0
1P1
MJ or Jz +1 0 -1
wo
w
wo
I I
w
DE is same and depend on Bz
For singlet term transitions, S = 0 and L = J. So g = 1 So splitting is simple and is called “Normal Zeeman effect”
DE = eh/4pm Bz; Jz = 1, g =1
DE = -eh/4pm Bz; Jz = -1, g =1
Bz
MJ or Jz +1
-1
0
DS = 0 DL = ±1 DJ = 0, ± 1 DMJ = 0, ± 1
Example 2 Sodium doublet under magnetic field
Transitions between n=3, l=1 to n=3, l= 0
2S1/2
2P3/2
I
2P1/2
w
I
w Positions of spectra lines in
the absence of applied field
Doublet 10 lines
DS = 0 DL = ±1 DJ = 0, ± 1 DMJ = 0, ± 1
3s1
3p1
MJ or Jz
-1/2
+1/2
+1/2
-1/2
-1/2 +1/2
-3/2
+3/2
Bz Spin orbit coupling
I
w
Coarse spectrum
Fine spectrum Hyperfine
spectrum
DE will be in the ratio 3:1:2 for 2S1/2 : 2P1/2 : 2P3/2 states, since g = 2, 2/3 and 4/3 respectively. Complicated spectrum: “Anomalous Zeeman effect”
Validation/Application of Atomic Spectroscopy: 2. Photoelectron Spectroscopy (PES)
hn = Binding energy + Kinetic Energy
Binding energy = hn - Kinetic Energy
For n = 1, 2, 3, 4, 5, …….etc.
X-ray notations K, L, M, N, O…… etc.
Binding Energy
Kinetic Energy
hn
Principle of PES
K
L
M N
If source is UV, Ultra-violet Photoelectron Spectroscopy (UPES)
If source X-ray, X-ray Photoelectron Spectroscopy (XPES)
PES Example 1: Ar ion
1S22S22P5 1S22S22P63S1
Interpretation of spectrum
Electron from 2P terms show higher B.E.
Ratio of peak area between3s to 3p state is 1:3
Ratio of peak between 2P3/2 and 2P1/2 is 2:1
From Banwell, page No. 155
Summary: Atomic Spcetrosocpy
Electromagnetic radiation – Sources- Wien’s Law ( l = 0.0029/T meters)
HΨ = EΨ; Time independent one dimensional Schrödinger equation HΨ = EΨ; solving for E
E = -RH/n2 Joules (RH = Rydberg constant = 2.18 x 10-18 Joules)
E = -Z2RH/n2 Joules; for any one electron system (H-like system He1+, Li2+ etc.).
DE = RH/h [1/nf2 – 1/ni
2]; for emission spectroscopy
DE = RH/h [1/ni2 – 1/nf
2]; for absorption spectroscopy\
For any one electron system , there will a Z2 term in the numerator
Validation of E Ionization energy Atomic spectroscopy
UV
Visible
NIR
IR
Pfund series : E(n) to E(n=5)
Humphrey series: : E(n) to E(n=6)
And so on…..
HΨ = EΨ; solving for Ψ n, l and m quantum numbers 1s = Ψ100, 2pz = Ψ211 etc… Spin-orbit coupling
Orbital angular momentum, l
Spin angular momentum, s
l (L) = orbital angular momentum quantum number l (L) = Total orbital angular momentum (vector), lz = Projection of l on z-axis (ml) lz = ± l, ± (l -1), ± (l -2)……..0. s (S) = Spin angular momentum quantum number s (S) = Total spin angular momentum (vector) sz (Sz) = Projection of S on z-axis (ml) sz (Sz) = ± s, ± (s-1), ± (s-2)……..0., for integer values of S OR ± s, ± (s-1), ± (s-2)……..1/2, for half-integer values of S j (J) = Total angular momentum quantum number j (J) = Total angular momentum (vector) jz (Jz) = Projection of J on z-axis (ml) jz (Jz) = ± j, ± (j-1), ± (j-2)……0, for integer values of J OR ± j, ± (j-1), ± (j-2)……..1/2, for half-integer values of J
Many electron system L-S coupling (Russell Saunder’s coupling) Z >30, jj coupling
Term symbol (Russell Saunder’s coupling)
2S+1LJ
lower J value, for less than half filled orbitals higher J value, for more than half filled orbitals
Non-equivalent : Examples done in class Sodium, Fluorine, 2s1, 2p1 and 2p1, 3p1
Equivalent electrons: Examples done in class Carbon
3P0
3P1
1D2
1S0
3P2
p2 state (carbon) 15 microstates
http://isites.harvard.edu/fs/docs/icb.topic979814.files/Lecture%2017.pdf
Application/validation of atomic spectroscopy Zeeman effect Photoelectrons pectrosocpy (PES)
Molecular Spectroscopy
Molecular energy levels
Molecular Orbital Theory
Homonuclear diatomic molecules originating from s orbitals
Homonuclear diatomic molecules originating from s and p orbitals
Heteronuclear diatomic molecules
Molecular Term symbols
Selection rules
Hydrogen molecule spectrum
Vibrational coarse structure:
Born-Oppenheimer approximation
Franck-Condon Principle
Dissociation
Rotational fine structure
Predissociation
Fluorescence and Phosphorescence
Spectrophotometer
Beer-Lambert law.
Molecular energy level
Molecular Orbital Theory
Homonuclear diatomic molecules originating from s orbitals
+
In MO theory, the valence electrons are delocalized over the entire molecule, not
confined to individual atom or bonds
MO (Ψ) arise by the interaction of atomic orbitals (Ψ). The atomic orbitals can interact
constructively or destructively to form bonding MO and antibonding MO, respectively.
1s orbital, constructive interference
1sa 1sb
View along the inter-nuclear axis
σ1sg
σ: Cylindrically symmetrical along the bond axis
Bonding molecular orbital
σ
Ψ2 = (Ψ1sa + Ψ1sb)2 = Ψ21sa + Ψ21sb + 2 Ψ1saΨ1sb
Constructive interference term
-
1sa 1sb
View along the inter-nuclear axis
σ*1su
σ: Cylindrically symmetrical along the bond axis
Antibonding molecular orbital
1s orbital, destructive interference
σ
Ψ2 = (Ψ1sa - Ψ1sb)2 = Ψ21sa + Ψ21sb - 2 Ψ1saΨ1sb
Destructive interference term
Arranging the M.O. (Ψ ) in energy diagram
1sa 1sb
σ1sg
(Bonding molecular orbital)
σ*1su
(Antibonding molecular orbital)
Bonding MO: High electron density accumulates between the two nuclei
More stable than individual atoms
Antibonding MO: Less electron density accumulates between the two nuclei
Creates an effect exactly opposite to a bonding –Antibonding, NOT non-bonding
Less stable than individual atoms
1s 1s
σ1sg
σ*1su
Example 1: MO diagram of Hydrogen molecule
Atomic configuration: 1s1
Bond order: ½ (# of bonding MO electrons - # antibonding MO electrons)
For H2 = ½ (2-0) = 1; single bond, diamagnetic
Molecular configuration: (σ1sg)2
Energy
Example 1: MO diagram of He molecule
Atomic configuration: 1s2
Bond order: ½ (# of bonding MO electrons - # antibonding MO electrons)
For He2 = ½ (2-2) = 0; No bond.
Molecular configuration: (σ1sg)2(σ*1su)2
1s 1s
σ1sg
σ*1su
Does He molecule exist????
He-He Not a bond that is seen that often
Weakest chemical bond known
DEd = 0.01 kJ/mol for He2
DEd = 432 kJ/mol for H2
Energy
Example 1: MO diagram of Li2 molecule Atomic configuration: 1s22S1
Bond order: ½ (# of bonding MO electron - # antibonding MO electrons)
For Li2 = ½ (4-2) = 1; single bond, diamagnetic
1s 1s
σ1sg
σ*1su
2s 2s
σ2sg
σ*2su
Molecular configuration: (σ1sg)2(σ1s*u)2 (σ2sg)
2
Energy
DEd = 105 kJ/mol for Li2
Example 1: MO diagram of Be2 molecule
Atomic configuration: 1s22s2
Bond order= ½ (4-4) = 0; No bond
1s 1s
σ1sg
σ*1su
2s 2s
σ2sg
σ*2su
Molecular configuration: (σ1sg)
2(σ*1su)2 (σ2sg)2(σ*2su)2
Energy
DEd = 9 kJ/mol for Be2
Molecular Term Symbol
Comparison of notations
From Banwell, page No.183
Molecular Term Symbol
Where, S is the total spin quantum number Λ is the projection of the orbital angular momentum along the internuclear axis Ω is the projection of the total angular momentum along the internuclear axis u/g is the parity +/− is the reflection symmetry along an arbitrary plane containing the internuclear axis
= I l1 + l2I and Il1 – l2I S = I s1 + s2I and Is1 – s2I Ω= Il1+ S1I and Il1- S1I
Only the projection (orbital, spin and total angular momentum) on z-axis is considered in molecular term symbol assignment
Molecular energy level
Molecular Orbital Theory
Homonuclear diatomic molecules originating from p orbitals
g, Gerade parity
u, ungerade parity
The probability density function can be calculated as did in the s orbital case
pz orbital
The internuclear axis (bond axis) is taken as the z-axis
px orbital (or py)
u, Ungerade parity
g, Gerade parity
Antibonding
Bonding
Orbital parity – gerade (g) and ungerade (u)
Symmetry of orbitals and molecules is of great importance, and we should be able to determine whether orbitals are gerade (g) or ungerade (u) (from German for even or odd).
a a a
b b b
center of
inversion
a = b
not a
center of
inversion
a ≠ b
Parity (g or u) of atomic orbitals:
s-orbital
gerade (g) p-orbital
ungerade (u)
d-orbital
gerade (g)
Parity (g or u) of molecular orbitals:
σ*(1s)u
σ(1s)g
The test for whether an MO is g or u is to find the possible center of inversion
of the MO. If two lines drawn out at 180o to each other from the center, and of
equal distances, strike identical points (a and b), then the orbital is g.
center of
inversion
a = b
a
b
a
b
not a center
of inversion
a ≠ b (sign of wave-
function is
opposite) u
g g
u
Parity (g or u) of molecular orbitals:
π*(2p)g
π(2p)u
The test for whether an MO is g or u is to find the possible center of inversion
of the MO. If two lines drawn out at 180o to each other from the center, and of
equal distances, strike identical points (a and b), then the orbital is g.
center of
Inversion
a = b
not a center of
inversion
a ≠ b
a
b
a
b
Bonding p orbital
antibonding p orbital
g
u
Test 1
σ*pu
π*pg
σ*su
πpu
σsg
σ(1s)
+/- reflection plane symmetery element
“+” “+”
“u”
“g” “g”
“u”
Do a reflection operation in a plane containing the bond axis
p2py M.O p2px M.O
+/- reflection plane assignment
“-” “+”
“u”
“g”
“u”
Do a reflection operation in a plane containing the bond axis
Selection rules
DS = 0 D = 0, ±1 (because DJ = 0, ±1 DΩ = 0, ± 1 S+ ↔ S+ , S-↔ S-
g ↔ u, change of parity during transition is required. So ‘u’ to ‘u’ and ‘g’ to ‘g’ are not allowed.
All selection rules (electronic, vibration and rotational) are based on conservation of angular momentum.
Selection rules are quantum mechanically based on transition moment integral, P = < Ψ*m Ψ>
u x u = g g x u = u x g = u g x g = g
Dipole moment operator, m = mx . my . mz ; = u . u. u = u parity
Molecular Term Symbol Examples
H2 : (σ1sg)2 = 1S+
g
H-2 : (σ1sg)
2 (σ*1su)1 = 2S+u
He2 : (σ1sg)2 (σ*1su)2 = 1S+
g
B2 : (σ1sg)2 (σ*1su)2 (σ2sg)
2 (σ*2su)2 (p2pxu)1 (p2pyu) 1 = 1Dg, 3S-
g and 1S+
u
O2 molecule, bond order = 2
O O2 O
BO = (6-2)/2 = 2
(disregardiing overlap of 2s orbitals)
The ability to predict the number of unpaired electrons in molecules is where MO excels, and Lewis-approach fails.
molecules with unpaired electrons are paramagnetic
Singlet oxygen (1O2)
O O2 O
BO = (6-2)/2 = 2
Singlet Oxygen is an excited state of the ground state triplet 3O2
molecule. It is much more reactive, and will readily attack organic molecules.
The O2 molecule in its excited singlet state which is 25 kcal/mol in energy above the ground triplet state. Irradiation with IR light causes excitation to the singlet state, which can persist for hours because the spin-selection rule inhibits transitions that involve a change of spin state.
Born-Oppenheimer Approximation
ΨTot = ψNuc . ψele
ΨNuc. = Ψvib . Ψrot . Ψtans
ΨNuc. = Ψvib . Ψrot
Since Ψtrans , is negligible
So
Ψtot = Ψvib . Ψrot . Ψele
Mass (nuc) >> mass (ele) V(nuc) << V(ele)
(a) For electron, nucleus is stationary and fixed (b) For nucleus, electron is delocalized
Etotal = Eelectronic + Evibration + Erotation in Joules
εtotal = εelectronic + εvibration + εrotational in cm-1
ΔEtotal = ΔEelectronic + ΔEvibration + ΔErotation
Etotal = Eelectronic + Evibration εtotal = εelectronic + εvibration in cm-1
Vibrational Coarse structure
No restriction in vibronic transition.
Progressions
From McQuarrie and Simon, Fig. 13.8
Absorption spectrum of I2 (g) in the visible region
This spectrum is a n’ progression
From McQuarrie and Simon, Fig. 13.9
The Franck–Condon principle is a rule in spectroscopy and quantum chemistry that explains the intensity of vibronic transitions.
Classical form: Franck-Condon principle states that an electronic transition takes place so rapidly that a vibrating molecule does not change its internuclear distance appreciably during the transition. So the electronic transition occurs vertically in P. E. diagram.
The Franck–Condon principle
Classical form: Vertical transitions
The Franck–Condon principle is a rule in spectroscopy and quantum chemistry that explains the intensity of vibronic transitions.
The Franck–Condon principle
Quantum mechanical form:
The intensity of a vibronic transition is proportional to the square of the
overlap Integral between the two vibrational states wave functions (ψ2) that are
involved in the transition.
ψ=ψe ψN
ψf = ψe,f ψN,f
ψi = ψe,i ψN,i
µ = µe +µN
As ψe,f and ψe,i are orthogonal to each
other so <ψe,f | ψe,i> = 0
Franck Condon factor
Transition moment integral (selection rules)
Quantum mechanical form
<µ> = <ψf |µ|ψi>
= <ψe,f ψN,f |µe+µN| ψe,i ψN,i>
= <ψe,f ψN,f |µe| ψe,i ψN,i>
+ <ψe,f ψN,f |µN| ψe,i ψN,i>
= <ψN,f | ψN,i> <ψe,f |µe| ψe,i>
+<ψe,f | ψe,i> <ψN,f |µN| ψN,i>
= <ψN,f | ψN,i> <ψe,f |µe| ψe,i> + 0
= <ψN,f | ψN,i> <ψe,f |µe| ψe,i>
Transition moment integral <µ> = <ψf |µ|ψi>
= <ψe,f ψN,f |µe+µN| ψe,i ψN,i>
= <ψe,f ψN,f |µe| ψe,i ψN,i>
+ <ψe,f ψN,f |µN| ψe,i ψN,i>
= <ψN,f | ψN,i> <ψe,f |µe| ψe,i>
+<ψe,f | ψe,i> <ψN,f |µN| ψN,i>
= <ψN,f | ψN,i> <ψe,f |µe| ψe,i> + 0
= <ψN,f | ψN,i> <ψe,f |µe| ψe,i>
<µ> = <ψf |µ|ψi>
= <ψe,f ψN,f |µe+µN| ψe,i ψN,i>
= <ψe,f ψN,f |µe| ψe,i ψN,i>
+ <ψe,f ψN,f |µN| ψe,i ψN,i>
= <ψN,f | ψN,i> <ψe,f |µe| ψe,i>
+<ψe,f | ψe,i> <ψN,f |µN| ψN,i>
= <ψN,f | ψN,i> <ψe,f |µe| ψe,i> + 0
= <ψN,f | ψN,i> <ψe,f |µe| ψe,i>
<µ> = <ψf |µ|ψi>
= <ψe,f ψN,f |µe+µN| ψe,i ψN,i>
= <ψe,f ψN,f |µe| ψe,i ψN,i>
+ <ψe,f ψN,f |µN| ψe,i ψN,i>
= <ψN,f | ψN,i> <ψe,f |µe| ψe,i>
+<ψe,f | ψe,i> <ψN,f |µN| ψN,i>
= <ψN,f | ψN,i> <ψe,f |µe| ψe,i> + 0
= <ψN,f | ψN,i> <ψe,f |µe| ψe,i>
<µ> = <ψf |µ|ψi>
= <ψe,f ψN,f |µe+µN| ψe,i ψN,i>
= <ψe,f ψN,f |µe| ψe,i ψN,i>
+ <ψe,f ψN,f |µN| ψe,i ψN,i>
= <ψN,f | ψN,i> <ψe,f |µe| ψe,i>
+<ψe,f | ψe,i> <ψN,f |µN| ψN,i>
= <ψN,f | ψN,i> <ψe,f |µe| ψe,i> + 0
= <ψN,f | ψN,i> <ψe,f |µe| ψe,i>
Quantum mechanical form (in integral terms)
P = ʃ ψN,f . ψN,i dζ ʃ ψe,f µ ψe,i dζ
Franck Condon factor
Transition moment integral (selection rules)
P = ʃ ψ vib,f ψ vib,i dζ ʃ ψe,f µ ψe,i dζ
So the final form is
ΨN = Ψvib . Ψrot . Ψtans
ΨN = Ψvib . Ψrot
ΨN = Ψvib
The Franck–Condon principle is a rule in spectroscopy and quantum chemistry that explains the intensity of vibronic transitions.
The Franck–Condon principle
Quantum mechanical form:
The intensity of a vibronic transition is proportional to the square of the
overlap Integral between the two vibrational states wave functions (ψ2) that are
involved in the transition.
QM form: Max. overlap between ψ2
Case 1: When Re’ = Re”
The Franck–Condon principle Different possibilities
Re = Equilibrium internuclear distance
(0,0) most intense transition
When Re’ = Re”
The Franck–Condon principle Different possibilities
Re = Equilibrium internuclear distance
The Franck–Condon principle
Re = Equilibrium internuclear distance
Intensity profile
(0,0) NOT most intense transition (2,0) most intense transition
Actual intensity profile
Molecule spent most of the time at the edges as vibration energy level increases
Case 2: When Re’ > Re”
N2 molecule: Re” = 1.098 Å. Re’ = 1.22Å & 1.2126 for 1Pg and 3Pg
The Franck–Condon principle
Re = Equilibrium internuclear distance
(0,0) NOT most intense transition
Case 2: When Re’ >> Re”
Intensity profile
I2 molecule: Re” = 2.667 Å. Re’ = 3.025 Å
The Franck–Condon principle
Re = Equilibrium internuclear distance
(0,0) NOT most intense transition (2,0) is the most intense
Case 2: When Re’ < Re”
Intensity profile
Transitions b/n two excited states: Antibonding to bonding or non-bonding
Rotational Fine structure of Electronic-Vibration Transition
Etotal = Eelectronic + Evibration + Erotation in Joules
εtotal = εelectronic + εvibration + εrotational in cm-1
ΔEtotal = ΔEelectronic + ΔEvibration + ΔErotation
εtotal = εelectronic + εvibration + BJ (J+1) ; (J = 0,1, 2, 3 …) B = h/8p2Ic
Dεtotal = D (εelectronic + εvibration) + D {BJ (J+1)} in cm-1
Rotational Selection rule
DJ = 0, ±1; J is the rotational energy level
DJ = ±1; for 1S ↔ 1S transitions DJ = 0, ±1; for 1S ↔ 1D or 1P ↔ 1D transitions
Based on conservation of angular momentum
DJ = +1; R branch DJ = 0; Q branch DJ = -1; P branch
Rotational Fine structure of Electronic-Vibration Transition
Dεtotal = D (εelectronic + εvibration) + D {BJ (J+1)} in cm-1
Dεrotational = B ̍ J ̍(J ̍+1) – B ̎ J ̎(J ̎+1) ----- (1)
Where B’’, B ̍ are Rotational constants in electron ground state and excited state respectively
DJ= +1 J ̍ = J ̎ +1 ------- (2)
DEr = B ̍ (J ̎ +1)(J ̎ +2) – B ̎ J ̎ (J ̎ +1)
Substitute eq 2 in 1
= (J ̎ +1){ B ̍ (J ̎ +2) - B ̎ J ̎ } = (J ̎ +1){ B ̍ J ̎ + 2B ̍ - B ̎ J ̎}
= (J ̎ +1){ B ̍ J ̎ - B ̎ J ̎ + B ̍ + B ̎ + B ̍ - B ̎ } (add and subtracting B ̎)
= (J ̎ +1){ (B ̍- B ̎ )J ̎ + B ̍ - B ̎ + B ̍ + B }
DEr = (B ̍+ B ̎ ) (J ̎ +1) + (B ̍- B ̎ ) (J ̎ +1) 2 -------- (3)
Substitute eq 3 in 1
Total energy DET = (DEe + DEv ) + (B ̍+ B ̎ ) (J ̎ +1) + (B ̍- B ̎ ) (J ̎ +1) 2
R- branch
R- branch
DJ= -1 J ̎ = J ̍+1 ------- (4)
DEr = B ̍ J ̍ (J ̍+1) – B ̎ (J ̍+1)(J ̍ + 2)
Substitute eq 4 in 1
= (J ̍ +1){ B ̍ J ̍ – B ̎ (J ̍ + 2) } = (J ̍ +1){ B ̍ J ̍ – B ̎ J ̍ – 2B ̎ ̍}
= (J ̍ +1){ B ̍ J ̍ – B ̎ J –̍ B ̎ + B ̍ – B ̍ – B ̎ } (add and subtracting B ̍)
= (J ̍ +1){ (B ̍- B ̎ )J ̍ +( B ̍ - B ̎ )- (B ̍ + B ̎ ) }
DEr = (B ̍- B ̎ ) (J ̍ +1)2 - (B ̍+ B ̎ ) (J ̍+1) -------- (5)
Substitute eq 5 in 1
Total energy DET = (DEe + DEv ) - (B ̍+ B ̎ ) (J ̍+1) + (B ̍- B ̎ ) (J ̍ +1)2
P- branch
P- branch
DJ= 0 J ̎ = J ̍ ------- (6)
DEr = B ̍ J ̍ (J ̍+1) – B ̎ J ̍ (J ̍+1)
Substitute eq 6 in 1
= J ̍ (J ̍ +1)( B ̍ - B ̎ ) = (B ̍ - B ̎ ) (J ̍ 2+J ̍ )
DEr = (B ̍ - B ̎ ) J ̍ 2 + (B ̍ - B ̎ ) J ̍ ------- (7)
Substitute eq 7 in 1
Total energy DET = (DEe + DEv )+ (B ̍ - B ̎ ) J ̍ + (B ̍ - B ̎ ) J ̍ 2
Q- branch
Q- branch
Total energy DET = (DEe + DEv ) + (B ̍+ B ̎ ) (J ̎ +1) + (B ̍- B ̎ ) (J ̎ +1) 2 R- branch
B’ < B”
Total energy DET = (DEe + DEv ) - (B ̍+ B ̎ ) (J ̍+1) + (B ̍- B ̎ ) (J ̍ +1)2 P- branch
Total energy DET = (DEe + DEv )+ (B ̍ - B ̎ ) J ̍ + (B ̍ - B ̎ ) J ̍ 2 Q- branch
Overall
Band head
Few example
Dissociation and Pre-dissociation
Molecular abs.
Atomic abs.
Dissociation and Pre-dissociation
Pre-dissociation
Molecule spent most of the time at the edges as vibration energy level increases
Tells the fate of an excited electron…
Jablonski Diagram
S0
v0
vn
vn
v0
vn
v0
vn
v0
v0
S1
S2
Sn
T1
Electronic ground state
Abso
rpti
on
(10
-15 s
)
VC (10-12 s)
ISC (10-12 s)
Fluorescence (10-9 s)
Phosphorescence (10-6 s)
Ener
gy
IC (10-12 s)
Implication of excited state dynamics
1. Photosynthesis
2. Solar energy research
L branch polar glutamate side chain
M branch nonpolar valine side chain
Unidirectional charge separation in Reaction Centre
Electron transfer processes occurs in ns/ps
Spectrophotometers
Absorption spectrophotometer
Diode array spectrophotometer
Monochromator: Grating, prism
Detector: Photographic plate, Photodiode, Photomultiplier tube
(~350 to 800 nm)
(~ 190-350 nm)
Atomic Absorption Spectrophotometer (AAS)
ICP-AES, ICP-OES, ICP-MS
According to the Beer-Lambert law
−𝑑𝐼
𝐼∝ 𝐶 𝑙
−𝑑𝐼
𝐼= 𝑎𝐶 𝑙
− 𝑑𝐼
𝐼= 𝑎𝐶 𝑑𝑙
𝐿
0
𝐼
𝐼𝑜
− ln𝐼
𝐼𝑜
= 𝑎𝐶 𝐿
− log𝐼
𝐼𝑜
= a C L/2.303
−𝑙𝑜𝑔𝐼
𝐼𝑜
= ∈ 𝐶 𝐿
log T = − ∈ 𝐶 𝐿
Where T is known as Transmittance now, Absorbance (A) A = - log T = - (- ∈ 𝐶 𝐿) A = ∈ 𝐶 𝐿
0 L
Io I
Io = Intensity of incident light I = intensity of transmitted light
‘∈’ is called the molar absorptivity or molar extinction coefficient. This is a Parameter defining how strongly a substance absorbs light at a given wavelength per molar concentration