FERENC BILLES STRUCTURAL CHEMISTRY

FERENC BILLES

STRUCTURAL CHEMISTRY

Chapter 1.

INTERACTIONS OF ATOMS AND MOLECULES

WITH PARTICLES AND EXTERNAL FIELDS

experimental method

qualitative information quantitative information

structure elucidation identification quantitative determination

theoretical research preparative research analysis model industry, agriculture

Elucidation of the molecular structure

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:  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 () characterizes the effectivity of the collision. If N particles impact into a surface of the

target with  particle density, the number of the produced reactions (collisions, absorptions, etc.) will be

s=.N

If the particle stream (particle/cross-section unit) is , and there are n particles on the target surface,

s=.n.

Unit of collision cross-section is called barn, 1 barn=10^-28 m².

The impulse (p) of a photon (velocity v=c) is

the impulse of a particle with velocity v < c is p=m.v

c

.

p _ h 

Elastic scattering





 





2 2

1

1

E  E h   h 

     

Inelastic scattering





 E

 E

E

Induced scattering (coherent)







.

h 

 E

 2 h 

 E

Partially inelastic scattering





) (

2 2

* 1

1

      

 E E



 E

 

 E

( 

 

)

Induced partially inelastic scattering (coherent)







a

b

b



 

 E

 2 

 E

( 

 

)

Spontaneous scattering



E ₁ ^*  E ₂   ₂

Inelastic scattering with particle change





a

b

Fig. 1.8

 _{1 a}  E ₁   _{2 b}  E ^* ₂

in general:

from photon to electron: h I m v

I E E

a e e



  2 



I: ionization energy, 

:photon frequency,

m

: electron mass, v

: electron velocity

Interactions with electric field

charge system of charges Q

position vectors r

a charge at the point P position vector R

permittivity 

potential: U Q

i i

 ₄ ¹ _  _{R r} 

Expanding into series around P, second member:

p   ^{Q i i} r

i

dipole moment

Cl Cl

C

H C

H H

C H

C

Cl Cl

H

C

H C

Cl Cl

I II III

Next term: quadrupole moment,

characterizes asymmetricity of charge distribution

example:

External electric field acts: F  QE

Total dipole moment: ^{p p} _o ^{E 1E }

2

2 ...

^ polarizability tensor^,  hyperpolarizability.

distorsion polarization

E acts on p: torque T  p E orientation polarization

P

p





i

V

Vector of polarization V: volume

P   

E





Dielectric induction vector characterizes the surface charge density

D   E

Strong field: D and E not collinear: ^{D  }_o^{E  P  }_o





e

r





    1 

0

Relative permittivity (dielectric constant)

Molar polarization: characterizes the polarization state of the substance (Clausius and Mosotti): P M

M r

r

 





 

1 M: molecular mass, : density 2



 



 

 

 

kT 3

p 3

N M

2 P 1

2

o A r

r

M 





 More detailed: 

N_A: 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  Q v 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: m_s, electron angular moment: spin (s)

e: electron charge, m_e: electron mass

Correspondence principle of quantum mechanics: quantities of classical physics are substituted by operators that act on wavefunctions.

For electrons:

e

B

m

e 2

 



 2

 h



h is the Planck constant,

ˆ s

m

s   m

e

(h-bar) mˆ s

sˆ 2

 

B 



Electron on an atomic orbit:

angular moment l, magnetic moment: m.

The right side of this equation differs from the similar equation for the electron in the factor 2 for electron spin:

The corresponding quantum chemical expression is:

m l 2 m

e

e





m

s   m

e

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):

M

B

L ˆ ˆ

g    

m ˆ lˆ   



Moments for a nucleus

Nuclear angular moment: I.

Magnetic moment M_I, 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!

g_N is the Landé factor of the nucleus, _is the nuclear magneton, m_p is the proton mass:

I N

I ˆ M ˆ g

  

m

2 e 

N





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 m

e 2 2

4m r Δ  e

The resulted diamagnetic moment is

m l

T p

E

B     

     

Precession of the magnetic moment

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

B , 

m

The direction of B (external field) is per definitionem the z axis. The angular velocity  and B are collinear.

For electrons

B B

ω   

h

g μ

_e ^and _N are the magnetogyric ratios for electrons and nuclei, respectively.

For nuclei

ω  B  

B

h

g

μ

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:

the system absorbs the wave.

The relaxation process of the magnetic moment is observable.

 Theoretical basis of

NMR (nuclear magnetic resonance), and ESR (electron spin resonance) methods.







 m d

M ^ 

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,





i

m

M V

M is proportional (in the case of weak fields) to the magnetic field strength H

H M  



₀: permeability of vacuum, (1.25664x10^-6 VsA^-1m^-1), _m: magnetic susceptibility,

H B  

The magnetic field strength is determined by B and not by H:

magnetic permeability weak field, linear:

_r is the relative permeability

m

r



 

 _{  } 1

Stronger field: B and H are not parallel, 0 _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.

Curie's law: ^m A

T B

  A>0 and B are constants At temperture T: ferromagnetism 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 E_j to higher level E_i:









 E  E

 E

 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:

   

t

0

2

K

t exp iω t dt

a 1







j



i

t: time, t_p: time of process (absorption), and

dτ ψ

Kˆ ψ

K

^ 

is the operator of perturbation,

K ˆ

Potential energy operator: multiplication with potential energy (U).

 pE



 U K

p: change in the dipole moment during the perturbation.

The expression for K_ij

K

^ E  

 p 

d 

The integral in this equation is called transition moment of the process:









p

d





P

P² is the transition probability.

2. The effect of population

According to Boltzmann's distribution law



 



 



 



 



 kT

exp hν kT

E exp E

N

N _{i j}

j i

N number of atoms (population) in the energy level (i or j).

The process is drived by (N_j-N_i)/N_j.

Frequency dependence of populations at 298K

/Hz N_i/N_j 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.

M1 M2

detector light source

beamsplitter

computer plotter

 x = v t interference

*

*

control of M1

2. Fourier Transform spectrometer

Interferograms

One-beam spectra

Double-beam spectra

Incident light (rates)

1. reflects on the sample surface, reflectivity  2. absorbs by the sample, absorptivity 

3. transmits the sample, transmittivity 

 1



  



A

spectrum

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.

L

inewidth

The natural linewidth is determined by Heisenberg's uncertainty law:

Energy uncertainty: E=h., Time uncertainty: t=,

average lifetime of excited state The natural linewidth:

 

2

 1

2π

δE.δt  h

Doppler effect

An atom or a molecule nears to the detector with velocity v and emits light with frequency ₀ (wavelength ).

The observed frequency increases by v/.

If the particle moves away from the detector, the frequency decreases by v/.

c v





 

Since ^=c/_o (c is the velocity of light in vacuum)

The velocity distribution in a gas follows Boltzmann's law, the spectral line gets a well-defined profile.

Line broadening

2kT ) exp( mv

I I

2

o





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.

Effect of nuclear spin

The spectrum

The intensity of experimental spectra is measured as transmittance of the sample (often

%): I⁰

T  I

or as absorbance ^lg

 

I lg I

A ^o   



 



 

I is the transmitted light intensity, I_o is the incident one.

The intensity of the reflected light

is measured as reflectance:

^lg   ^r

I lg I R

r

o

   

 



 

I_r is the intensity of the reflected light, r is called reflectivity.

The independent variable of the spectra is either frequency, or wavenumber

c

ν~  ν

₀ is the nominal frequency of the band , FWHH is the full width at half hight.

Characteristic data of a band