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FERENC BILLES
STRUCTURAL CHEMISTRY
Chapter 1.
INTERACTIONS OF ATOMS AND MOLECULES
WITH PARTICLES AND EXTERNAL FIELDS
Elucidation of the molecular structure
experimental method
qualitative information
quantitative information
structure elucidation
identification
theoretical research
preparative research
model
quantitative determination
industry, agriculture
analysis
Types of properties
System
Propety
Example
atom
A
A: atomic
spectrum
molecule
ΔA, M
M: vibrational
spectrum
molecular
ensamble
ΔA, ΔM, S
S: X-ray
diffractogram
The model is only an approach to the reality.
The experiment disturbs the system.
A collision with particles maybe
with electrons, atoms, ions, photons, etc.
An effect with external fields maybe
- effect with external - electric
- magnetic
-electromagnetic fields.
The answer of the system is
- the change of its properties
- or/and emission of one or more particles.
The answer of the particle is
- the change of one or more of its properties.
Non-central collision types:
-elastic, energy change, colliding particle: l remains;
-inelastic, the total energy of the colliding particle
increases the atom or molecule energy;
-partly inelastic.
-coherent, coherence remain during the collison;
-incoherent, coherence ceasing during the collision or
it remains.
Coherence: stationary interference in space and time.
Coherent waves: constant relative phase.
The collision cross-section (s) characterizes the effectivity
of the collision. If N particles impact into a surface of the
target with r particle density, the number of the produced
reactions (collisions, absorptions, etc.) will be
s=s.r.N
If the particle stream (particle/cross-section unit) is F, and
there are n particles on the target surface,
s=s.n.F
Unit of collision cross-section is called barn,
1 barn=10-28 m2.
The impulse (p) of a photon (velocity v=c) is
h.
p
c
the impulse of a particle with velocity v < c is p=m.v
Elastic scattering
  

 1  E1   2  E2
h 1  2   h 1 
Inelastic scattering
E1

*
1  E1  E2
Induced scattering (coherent)
 

.
*
h1  E1
 2 h1  E2
Partially inelastic scattering


1  E1*   2  E2
( 2  1 )
1  E1  2  E*2
( 2  1 )
Induced partially inelastic scattering (coherent)
 b
a
b
1a  1b  E1  21b  E*2
(1a  1b )
Spontaneous scattering

*
E1
 E2   2
Inelastic scattering with particle change
b
a
Fig. 1.8
*
in general: 1a  E1   2b  E2
me v2e
from photon to electron: h1a  I 
I  E*2  E1
2
I: ionization energy, 1a:photon frequency,
me: electron mass, ve: electron velocity
Interactions with electric field
charge system of charges Qi
position vectors ri
a charge at the point P
position vector R
permittivity 
potential:
1
Qi
U

4 i R  ri
Expanding into series around P, second member:
p   Qi ri
i
dipole moment
Next term: quadrupole moment,
characterizes asymmetricity of charge
distribution
example:
I
II
Cl
H
C
Cl
Cl
C
Cl
C
H
III
H
Cl
H
C
C
H
H
C
Cl
External electric field acts: F  QE
distorsion polarization
1 2
Total dipole moment: p  po  E  E ...
2
 polarizability tensor,  hyperpolarizability.
E acts on p: torque T  p  E
orientation polarization
Vector of polarization
P
P  oeE
 pi
i
V: volume
V
o permittivity of vacuum (8.85419 x 10-12 AsV-1m-1),
 e dielectric susceptibility
Dielectric induction vector characterizes the surface charge
density
D  E (weak field)
Strong field: D and E not collinear:
D   o E  P   o 1   e E
Relative permittivity (dielectric constant)

r 
 1 e
0
Molar polarization: characterizes the polarization state of the
r  1 M
substance (Clausius and Mosotti):
PM 
r  2 r
M: molecular mass, r: density
r 1 M NA 
p2 
 

PM 

More detailed:
 r  2 r 3 o 
3kT 
NA: Avogadro constant (6.0214x1023mol-1) first term in parenthesis
average polarizability for distorsion polarization, second term:
orientation polarization,k:Boltzmann constant(1.38066x10-23JK-1)
T: absolute temperature.
Interactions with magnetic field
Lorentz force, external magnetic field (B) acts on
moving (velocity v) charged (Q) particle:
F  Qv  B
B: magnetic induction or magnetic flux density
Elementary magnet is a magnetic dipole, m. B acts on m
T  mB
. Electrons
have magnetic moment from their nature and
from their position (orbit) in the atom or molecule
.
The magnetic moment of the particle is always coupled
with an angular moment.
Electron magnetic moment: ms,
electron angular moment: spin (s)
e
s  m s
me
e: electron charge, me: electron mass
Correspondence principle of quantum mechanics: quantities
of classical physics are substituted by operators that act on
wavefunctions.
For electrons:

 B ŝ   m̂ s
2
e
h
B 
is the Bohr magneton,  
2me
2
h is the Planck constant,
(h-bar)
ŝ is the spin operator.
Electron on an atomic orbit:
angular moment l, magnetic moment: m.
e
l  2 m
me
The right side of this equation differs from the
similar equation for the electron in the factor 2
for electron spin:
e
s  m s
me
The corresponding quantum chemical expression is:
 B l̂   m̂
The total orbital angular moment L (for some electrons) is
coupled to the total orbital magnetic moment M
(L and M vector are sums of individual moments):
ˆ  M
ˆ
gB L
g is the Landé factor
Moments for a nucleus
ˆI
g N  N Iˆ  M
Nuclear angular moment: I.
Magnetic moment MI, it is not zero if the atomic number
is odd (1H) or even with odd mass number (13C).
Pay attention! The sign of the right side is positive!
gN is the Landé factor of the nucleus,
N is the nuclear magneton,
mp is the proton mass:
e
N 
2m p
Diamagnetism
It exists for all molecules independently of other
magnetic effects. It is weak, stronger effects cover it.
Origin: Changing magnetic flux B induces electric
field E, this induces dipole (pE, E act on p
(T=p x E), T is time derivative of angular moment l,
this coupled with the magnetic moment m, so
B  E  p  T  l  m
The resulted diamagnetic moment is
2
2
e r
Δm 
B
4me
Precession of the magnetic moment
B,
According to
Larmor's
theorem the
magnetic
dipoles move in
a field B and
they precess
also around the
direction of B
direction of precession
m
The direction of B (external field) is per definitionem
the z axis. The angular velocity  and B are collinear.
For electrons
For nuclei
μB
ω g
B  eB
h
μN
ω  gN
B NB
h
e and N are the magnetogyric ratios for electrons
and nuclei, respectively.
Another external magnetic field perpendicular to the
first disturbs the stationary state and the magnetic
moments change their directions but continue their
precession.
1. The second field is an electromagnetic wave,
2. its frequency corresponds to the energy difference
of two magnetic levels of the molecule,

3. Magnetic transition moment, not zero: M   i* m  j d
4 the system absorbs the wave.
The relaxation process of the magnetic moment is
observable.
4 Theoretical basis of
NMR (nuclear magnetic resonance), and
ESR (electron spin resonance) methods.
Paramagnetism
The magnetic dipole density of a molecule depends on the
sum of elementary magnetic moments. The vector
of magnetization shows the strength of magnetization,
mi
M 
i V
M is proportional (in the case of weak fields) to the
magnetic field strength H
M  0  m H
0: permeability of vacuum, (1.25664x10-6 VsA-1m-1),
m: magnetic susceptibility,
The magnetic field strength is determined by B and not by H:
weak field, linear:
B  H
: magnetic permeability

 r  1   m
0
r is the relative permeability
Stronger field: B and H are not parallel, r is a tensor.
Very strong field ferromagnetism:
r
m
the substance is
<1
<0
Diamagnetic (Bi)
>1
>0
Paramagnetic(W)
>>1
>>0
Ferromagnetic(Fe)
In the case of ferromagnetic substances: magnetization
curve, a hysteresis curve. Its area (curve integral) is
proportional to the power of magnetization.
A
m   B
T
At temperture T: ferromagnetism 4 paramagnetism
(Curie point)
Hysteresis curve: good magnetic tape, diskette or
pendrive need a magnet with large magnetization area.

Interactions with electromagnetic waves
Wave: disturbance, periodic in time and space,
propagates energy in space and time.
Electromagnetic wave () propagates E perpendicular
to H, both perpendicular to direction of propagation
(transversal wave). E perturbs atom or molecule energy Ej
to higher level Ei:
E  E i  E j  h  
Absorption of photon is possible (inelastic collision).
Light absorption depends on
1. the probability of absorption
2. the relative population of the excited state
3. the average lifetime of the excited state
1. The probability (a) of the process must be larger than zero:
ai  j
1
 2

tp
 K t expiω t dt
ij
2
ij
0
t: time, tp: time of process (absorption), and
K ij   ψi* K̂ψ j dτ
K̂
is the operator of perturbation,
Potential energy operator: multiplication with potential energy (U).
K  U  pE
p: change in the dipole moment during the perturbation.
The expression for Kij
K ij  E  p jd
*
i
The integral in this equation is called transition
moment of the process:
P   i* p j d
P2 is the transition probability.
2. The effect of population
According to Boltzmann's distribution law
 Ei  E j
Ni
 exp 
Nj
kT


hν 

  exp 

 kT 

N number of atoms (population) in the energy level (i or j).
The process is drived by (Nj-Ni)/Nj.
Frequency dependence of populations
at 298K
/Hz
Ni/Nj
108
(1-2)x10-5
1010
0.99
1012
0.85
1013
0.30
1014
10-7
The data follow the exponential law.
3.The average lifetime of the excited state.
This is the average time of existance of a particle
in its excited state.
Long: the saturation of the excited state is easy
Short: its saturation is is difficult.
Type of the excited
state
Average lifetimes (s)
rotational
10-10 – 10-11
vibrational
10-7 – 10-8
electronic (singlet)
10-5 - 10-6
electronic (triplet)
10-2 - 10
The electromagnetic spectrum
Spectrometers used in optical spectroscopy
1.Dispersive spectrometer
Sample: IR after the light source, UV-VIS: after the monchromator
The grating resolves the spectrum. Two beams.The sample beam
(S) is related to the reference beam (R). Half phase S, half phase
R. The electronics balances them and amplifies the signal.
2. Fourier Transform spectrometer
M2
beam
splitter
*
M1
interference
light source
 x= vt
*
detector
computer
control of M1
plotter
Interferograms
One-beam spectra
Double-beam spectra
Incident light (rates)
1. reflects on the sample surface, reflectivity r
2. absorbs by the sample, absorptivity 
3. transmits the sample, transmittivity 
r    1
A spectrum consists of either of lines or bands
A spectral line is the signal of one transition.
A spectral band originates from
- the same transition of several molecules with
somewhat different chemical environment;
- frequencies of several transitions are very close, the
spectrometer cannot resolve the lines.
Linewidth
The natural linewidth is determined
by Heisenberg's uncertainty law:
h
δE.δt 
2π
Energy uncertainty: E=h.,
Time uncertainty:
t=,
average lifetime of excited state
The natural linewidth:
 
1
2
Doppler effect (gas phase)
An atom or a molecule nears to the detector with velocity v
and emits light with frequency 0 (wavelength l).
The observed frequency increases by v/l.
If the particle moves away from the detector,
the frequency decreases by v/l.
Since l=c/o (c is the velocity of light in vacuum)
v
   0
c
Line broadening
The velocity distribution in a gas follows Boltzmann's law,
the spectral line gets a well-defined profile.
mv 2
I  I o exp( 
)
2kT
Effect of nuclear spin
Theoretically the change in the nuclear spin influences
the electronic energy levels of the atom.
.
Practically, however, since this effect is very small its
influence is practically unobservable.
Instrument effect
The measuring instrument influences the line profile, too.
It has a transition function, that modifies the input signal to the
output one.
The result: the instrument broadens the lines and bands.
The spectrum
The intensity of experimental spectra is
I
T
measured as transmittance of the sample (often
I0
%):
I
or as absorbance A  lg  o   lg T 
 I 
I is the transmitted light intensity, Io is the incident one.
The intensity of the reflected light
is measured as reflectance:
 Io 
R  lg    lg r 
 Ir 
Ir is the intensity of the reflected light, r is called reflectivity.
The independent variable of the spectra is either frequency,
or wavenumber
ν
~
ν
c
Characteristic data of a band
0 is the nominal frequency of the band ,
FWHH is the full width at half hight.