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Chapter 5
CHEMICAL
CHARACTERIZATION(I)
Fundamentals of Quantum
Chemistry and Spectroscopy
1
5-1. Introduction to Chemical Characterization
A. Chemical Characterization
 Chemical composition：
- based on elements： elemental composition
- based on compounds： phase composition
- based on functional group , e,g.,-OH or –NH
adsorbed or bonded on solid surfaces.

Energy state
-Band gap
-Ionization potential
-Binding energy
2
B. Cartogories of Methods for Chemical Characterization
 Wet chemical techniques (classical)
Spectroscopic techniques
-Optical spectroscopy
-Electron spectroscopy
-Ionic spectroscopy
Probing
C. Classical wet chemical techniques
the material is fused by heating in the presence of a flux,
dissolved in acid (or basic), and then analyzed using a standard
precipitation, titration, or colorimetric technique.
Example :
(1)Chemical analysis of minerals.
(2)Chemical analysis of solid waste from wafer cutting.
3
5-2. Spectroscopic Techniques：fundamentals
photons, electrons, ions
probing
Spectroscopic
signals
Electronic and optical
properties of materials
Samples
Interaction : changes of
energy states (transition)
 Type of particles:
-independent atoms or molecules；
-ions, atoms or molecules in solids；
-electrons in atoms or molecules；
-electrons in solids.
Energy states (electronic
configulation or band
structure) ：types of
motion and particle
(Quantum Chemistry)
 Type of motion of particles：
- Translation - Rotation - Vibration - Electronic
mode
#11
#15
#8
#86
4
5-2-2. Energy States and Orbitals
Classical mechanics:
(Newton’s Law of Motion
: F=ma)
Quantum mechanics
Macroscopic particles
(solutions)
Microscopic particles
Schrodinger Equation
(solutions)
Motion or behavior
of macroscopic particles
but fail to describe the
behavior of microscopic particles
E:
Energy states
Energy levels
Energy gap
(electronic configulation or
band structure)
ψ: probailities of finding the particles
orbitals
(the probability of finding the particle
in the space d = ψ* ψd )
5
time-independent Schrödinger Equations (

2
steady state in Chem.E.)
/ 2m1   2 / x12   2 / y12   2 / z12 
 
2
/ 2mn    / x    / y    / z
2
2
n
2
2
n
2
2
n
  E   E
(form 2)
P
•E≡total energy＝ Ek ( kinetic energy)＋ Ep ( potential energy)
• EP  f (t ) and EP  f (coordinate) only
•
( e.g., Ep＝1/2kx2, or Ep＝-kq1q2/r)
=h/2π
•m1: mass of particle1, m2: mass of particle2, mn: mass of particle n
•x1, y1, z1: coordinates (position) of particle1
•xn, yn, zn: coordinates (position) of particle n
#15
#16
66
●time-independent Schrödinger Equations
(form 1)
 : wave function (of the system composed of n particles)
 is not a function of t (a function of coordinates only)
(  is used to calculate the probability of finding the particle)
(form 2)
(form 3)
Schro.
Eg.
2
2
2



2 : Laplacian operator, 2  2  2  2
x
y
z
spec
74
7
A. TRANSLATION
(A-1) The model：one particle in a one-dimentional box
one particle in a two-dimensional box
one particle in a three-dimentional box
2D
(A-2) One Particle in a One-Dimensional Box
Ep = 0 , 0≦ x ≦a
 , x＜ 0 and x＞ a
F11.11
F2.11
boundary conditions
No real system has a Ep like this, but it can be used as a crude
model for dealing with Pi electrons in conjugated molecules and free
electrons in metallic particles.
time-independent Schrödinger Equations

2
/ 2m1   2 / x12   2 / y12   2 / z12 
 
2
/ 2mn   2 / xn2   2 / yn2   2 / zn2   EP  E
- h2 /2m ˙ d2 ψ /dx2 ＋Ep ψ =E ψ
- h2 /2m ˙ d2 ψ /dx2 =Eψ
(h)
(Ep=0)
8
(a) The solutions
  n / a  2mE /
2mE /
2
 n 2 2 / a 2 ,
2
2
2
 4h/4π
En= n2 h2/ 8ma2
n ( Quantum Number)
En
2
2
n= 1,2,3,…
En (h2/ 8ma2 )
4
16 h2 / 8ma2
16
3
9 h2 / 8ma2
9
2 (Excited States)
4 h2 / 8ma2
4
h2 / 8ma2
1
1 (ground State)
(11.149)
ψn
E
2
h
( 2 /a) 2 sin(4π/a) 7
8ma 2
( 2 /a) 2 sin(3π/a)
h2
5
2
8
ma
2
( 2 /a) sin(2π/a)
h2
3
( 2 /a) 2 sin(π/a)
8ma 2
9
(b) Energy gap

E n 1 n  n  1  n 2
E  h 
hc

2
 2n  1

h2
 2n  1
8ma 2
h2
h2
8ma 2
8ma 2
m or/and a  E  and  ()
m or/and a  E  and  ()
m 
(macroscopic particle)
or a  E  0
＊a ：free particle  energy is not quantized (but
contineous) no energy gap
10
(c) practical applications

En 1n  n  1  n
2
2

h2
8ma 2
 2n  1
h2
8ma 2
(C-1) describing
the translation of an atom or a molecule
hc
E in
 ha
 container
 of the atom or the molecule
m: mass
a: dimension of the container
Energy gap
radio wave region
(λ>1m)
No practical applications for spectroscopic measurement
(C-2) A crude model for estimating the energy gap of a free
electron in a nano sized metallic particle (“If being considered as
electronic: would be a more complicated in solving the
Schrodinger.)
m: mass of the atom or the molecule
a: dimension of the container
nano particles : a ↓
ΔE 
11
(
semiconductor or ceramic nanoparticles : a ↓
Eg )
(B) Rolation
QC
H2： H
H
O2 ： O
O
(B-1) The model： a rigid rotator
reduced mass,
μ=
moment of inertia,
m1υm2
I = μr2
ｍ1m2
ｍ1+m2
12
(B-2) The Schrodinger equation
Ĥ µ ψ µ= E µ ψ µ
(F)
Ĥ = (-ħ2 /2m) V2 + Ep
(F-R)
for a rigid rotator: r = constant , Ep = 0
for spherical coordinates

V2 =

 
 
 
/ (r2 r) ( r2 /  r ) θ, ø + [1 / (r2 sinθ) ( / θ[(sin θ / θ) ]θ, ø
+ (1 / r2 sin2θ) ( 2 / ф 2 ) r , θ
Classical Treatment
(1) energy of a rigid rotator
T = (½) I ω 2
(11.214)(a)
ω : angular velocity
I: moment of inertia, i.e., I = µr2 (b)
(2) angular momentum
L=Iω
T = L2 /(2 I)
(11.213) (c)
13
(B-3) The soluions
Energy levels
Microwave region
(λ>mm)
Energy gap
Microwave spectroscpic
Measurement (not widely used)
The solution to Eq. (F-R) , like the solution to Hermite’s equation ,
is obtained by the power series method. When one does solve Eq. (F-R),
the energy levels of rotational motion is obtained as：
Eι ＝( h2 / 2I ) J（J+1）
(g)
J= 0, 1, 2,… (rotational quantum number)
Once again, we obtain a set of discrete energy levels.
-
The rigid rotator is a model for a rotating diatomic molecule
-
Equation (g) gives the allowed energies of a rigid rotator.
transitions from only adjacent states are allowed △ J=  1 selection rule.
-
△E = EJ+1 －EJ ＝( h2/ 2I) [（ J +1）（ J +2）－ J （ J +1）]
= h2/ I （J+1） ＝ h2/ 4π2 I（J+1）
(selection rule：the rule for the allowed transition of energy state)
14
ΔE =
h2
( J+1)
4π2 I
= 2B ( J+1)
(
h2
B=
8π2 I
J+1
)
J
ΔE 1
0=
2B
ΔE 2
1=
4B
15
(C) Vibration
qc
(c-1) The model：one-dimentional harmonic oscillator
N2： N
N
O2 ： O
O
16
actual potential
energy of
vibration of
diatomic molecule
potential
energy of a
HarmonicOscillator
(x)
(No significant difference at low vibrational energy, i.e., at low
vibrational quantum number or not at high temperature)
17
(c-2) Quantum Mechanical Treatment
from total energy (H) operator (Ĥ) transformation ( or from time-independent
Schrödinger Eq. )
Ĥ μψμ = Eμ ψμ
(F)
Ĥ μ = － (ħ2/ 2μ) ( 2 / x2 ) + (1 /2 ) kx 2
Eμ = Ek + Ep = Px2 / 2μ + (1 /2 ) kx
2
(11.156)
(11.153, 154)
e.q. (F) omitting subscript
[(-ħ2/ 2μ) ( 2 / x2 ) + (k /2) x 2 ] = E μ
(11.157)
e.q. (11.15) → k = 4 π2 ν02 m
Ep = (1 /2) kx 2 = 2 π2 mν02 x 2
e.q. (11.157) →
d 2 ψ / dx 2 + ( 8 π 2 μ / h 2 ) ( E －2 π 2 μ ν02 x 2 ) μ = 0
set λ = 8 π 2 μ E / h 2
and α2 = ( 8 π 2 μ / h 2 ) ( 2 π 2 μ ν02 )
(G)
(H)
(I)
(J)
(K)
18
(c-3) Energy levels
Energy level
Energy gap
IR region
(λ:μm-mm)
vibrational spectroscopic measurement
λ= 8π2μ E / h2
α2 = (8π2μ/ h2) ( 2π2μ  0)
 0=1/2π(k/m) ½
(J)
(K)
The vibrational energy:
En = ( n+ ½ ) h  0, n= 0,1,2….. (vibrational quantum number)
(X)
n=0, En=E0 ≡ zero point energy : E0=½ h  0
Even in its lowest state the system has energy greater than that which it
would have if it were at rest in it equilibrium position.
△ En+1↔n =nr0
The frequency emitted or absorbed by a transition between adjacent
energy levels is equal to the classical vibration frequency ν0.
19
1
En  ( n  )h 0
2
n = 0, 1, 2, ……..
E  En  En"
Selection rule:
n =  1
E  E n1  E n  h 0
(approximately equal energy gap between two adjacent status)
20
The Harmrnic Oscillator Accounts for the Infrared
Spectrum of a Diatomic Molecule
1
En  ( n  )h 0
2
n = 0, 1, 2, ……..
(x)
E  En  En"
Selection rule ： n =  1
 E  En1  En  h 0
 0   obs
The fundamental vibrational frequency for HC1,
 obs  2.90  10 cm
3
1
(IR)
21
(D) Electronic
Unit of motion：electrons (in a atoms, a molecule or in a
solid)
● Simple system：
F19-1
Hydrogen-like Atoms
• Spherical polar coordinates. r, θ and ψ (polar and
azimuthal angles)
●
r
x2  y2  z2
 Ze2
V  V (r) 
40 r

2
2
2 ( r, , )  V ( r ) ( r, , )  E ( r,  ,  )
(7-12)
where
2
1


1


1





2  2  r 2   2
 sin    2 2
r r  r  r sin   
  r sin   2
(7-13)
22
(D-1) Energy States
F19-6
Energy level
Energy gas
UV, visible,
X-ray
(λ<700nm)
widely used for spectroscopic measurement
back
23
D. ENGERY BAND STRUCTRUES OF SOLIDS
D-1. Electronic Configuration and Energy States
of Single Atoms
◎ Shells: designated by integers (principle
quantum number,I.e., 1,2,3, etc), subshells: by
letters (s,p,d, and f)
QC1
◎ Each of s, p, d, and f subshells: one, three, five,
and seven states
◎ Two electrons of opposite spin per state:
Pauli exclusion principle
T2.1 F2.2 F2.4
F2.3
T2.2
Line Spectra
24
26
D-2. Electronic Configuration and Energy States
of Solid Materials : band structure
◎ As the atoms come within close proximity of one
another, electrons are acted upon, or perturbed, by
the electrons and nuclei of adjacent atoms. Each
distinct atomic state may split into a series of closely
spaced electron states: electron energy band.
◎ At the equilibrium spacing , band formation may not
occur for the electron subshells nearest the nucleus.
◎ The number of states within each band will equal
the total of all states contributed by the N atoms.
For example, an s band will consist of N states,
and a p band of 3N states. Each energy state may
accommodate two electrons, which must have
518.4 F18.2 F18.3 27
oppositely directed spins.
◎ For Solid：
◎ The electrical properties are a consequence of its
electron band structure: the arrangement of the
outermost electron bands and the way in which
they are filled with electrons.( Eletrical conduction
occurs only when there are available
positions(empty states or holes) for electrons to
move.)
◎ Four different types of band structures are possible
at 0 k:
• The first : partially filled(valence)band: conductor
(Eg=0, i.e., no band gap), e.g., copper, one 4s
electron(per atom), only half the available electorn
positions within this 4S band are filled.
28
• The second: overlap of an empty (conduction)
band and a filled (valence) band: conductor
(Eg=0, i.e., no band gap), e.g.,Mg, the 3s and
3p bands overlap.
• The final two: valence band (completely filled )
is separated from conduction band (empty) by a
energy band gap. For materials that the band
gap is relatively wide (Eg>3eV,Figure 18.4c):
insulations; for Eg=0.02-3eV: semiconductors
(Figure 18.4d)
◎ The energy corresponding to the highest filled
state at 0 k is called the Fermi energy Ef .
F18.4
f2.12
t2.7
t2.6
E g Nano
29
E. Schematics of Energy States and Typical Spectra
(E-1) Rotation：for diatomic and polyatomic molecules
only；not for solid
30
32
34
35
(E-3) Electronic (－vibration and－rotation)
(E-3-1) atoms or molecules
36
One particle in a three dimensional Box
3-D
Ep = 0 , 0≦x≦a , 0≦y≦b , 0≦z≦c
∞ , x＜0 and x＞a , y＜0 and y＞b , z＜0 and z＞c
ψ = 0 outside the box
ψ = ? Inside the box
We only concern with the stationary states
Eq(b) =>
-h2/ 2m ( 2 ψ/ x2 + 2ψ / y2 + 2ψ / z2 )=Eψ
Using the method of separating variables
∵ E can be divided into Ex+ Ey + Ez
assume, ψ= X(x) Y(y) Z (z)
-h2/ 2m [Y(y) Z (z)X〞(x)+ X(x) Z (z) )Y〞(y) + X(x) Y(y) Z 〞(z)]
= E X(x) Y(y) Z (z)
dividing both sides by X(x) Y(y) Z (z)
-h2/ 2m ˙ X〞(x) / X(x) － h2/ 2m ˙ Y〞(y) / Y(y) － h2/ 2m ˙ Z 〞(z) / Z (z) = E
39
ψ = X(x) Y(y) Z (z) = (8/ abc )1/2 sin nx πx /a • sin ny πy/b • sin nzπz / c
E = Ex + Ey +Ez = h2/ 8m ( nx2/ a2 + ny 2/ b2 + nz2/ c2 )
↑
↘
(11.150)
↖
Y-direction
Z-direction
Kinetic
every associated motion
in the X-direction
Degeneracy
if a= b=c in Eq. (11.150)
E= h 2 /8ma 2 ( nx2 + ny 2 + nz2 )
nx , ny, nz (quantum number)
( 2,2,1 )( 2,1,2) (1,2,2)
(2,1,1) (1,2,1) (1,1,2)
(1,1,1 )( lowest energy
zero point energy)
(11.151)
ψ
En(energy level)
ψ221 , ψ212 , ψ122
9 h 2 /8ma 2
ψ211 , ψ121 , ψ112
6 h 2 /8ma 2
ψ111 (lowest energy state
3 h 2 /8ma 2
i.e. ground state)
‧One set of quantam number(e.g.,(2,2,1))
one quantum
‧More than one quantum state having the same energy level： Degeneracy
‧Degree of Degeneracy
energy level
40
X-ray fluorescence: x-rays are used to stimulate
secondary characteristic x-radiation; quantitative
analyses with an atomic number greater than that of
sodium
Spec
31
B-3. Infrared Spectroscopy
(Diffuse Reflectance Infrared Spectroscopy, DRIFT)
absorption of IR due to vibrations, determine
molecular structure and trace molecular anion
impurities, especially to detect and identify the
species absorbed on the surface of solid particles.
41
Examples:
Spec
 hydrated crystals
 chemically modified carbon nano tube (CNT) to obtain
strong bonding with ploymers.
 AlN/Polymer composites: chemically modified AlN particles
to bond strongly with epoxy.
AlN－OH
4.1
23
3.6
 Dye-sensitized solar cell：TiO2 surface structure to be
absorbed by dye motecules.
TiO2 －OH + R－COOH
TiO2 －OOC－R
42
frequencies ranging from 1012to1014Hz(3-300 μm
wavelength) infrared (IR) regions.
In the infrared experiment, the intensity of a beam of
infrared radiation is measured before and after it interacts
with the sample as a function of light frequency. A plot of
relative intensity versus frequency is the “infrared
spectrum.” A familiar term “FTIR” refers to Fourier
Transform Infrared Spectroscopy.
(d) Raman spectroscopy
Spec
An indirect coupling of high-frequency radiation (such as
visible light) with vibrations of chemical bonds
43
translation
time-independent
Schrodinger Eg.
Ep
(potential
energy)
rotation
vibration
electronic
78
energy band
band
energy level (state)
energy
gap
single atoms
sharp
discret
band
gap
solids
79