FERENC BILLES

STRUCTURAL CHEMISTRY

Chapter 1.

INTERACTIONS OF ATOMS AND MOLECULES WITH PARTICLES AND EXTERNAL FIELDS

experimental method

qualitative information quantitative information

structure elucidation quantitative determinationidentification

theoretical research preparative research analysis

model industry, agriculture

Elucidation of the molecular structure

Types of properties

System Propety Example

atom AA: atomic spectrum

molecule ΔA, MM: vibrational

spectrum

molecular ensamble

ΔA, ΔM, SS: 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

.hp

Elastic scattering

1212211 hhEE

Inelastic scattering

1 1 2 E E*

E1

Induced scattering (coherent)

.

h E h E 1 1 1 22 *

Partially inelastic scattering

)( 1222*11 EE

1 1 2 2 2 1 E E* ( )

Induced partially inelastic scattering (coherent)

a

b

b

1 1 1 1 2 1 12a b b a bE E * ( )

Spontaneous scattering

E E1 2 2*

Inelastic scattering with particle change

a

b

Fig. 1.8

1 1 2 2a bE E *in general:

from photon to electron: h Im v

I E Eae e1

2

2 12 *

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: UQi

ii

1

4 R r

Expanding into series around P, second member:

p rQi ii

dipole moment

Cl

Cl

C

H

C

H H

C

H

C

ClCl

H

CH

C

Cl

Cl

I II III

Next term: quadrupole moment, characterizes asymmetricity of charge distribution

example:

External electric field acts: F EQ

Total dipole moment: p p E E o 1

22 ...

polarizability tensor, hyperpolarizability.

distorsion polarization

E acts on p: torque EpT

orientation polarization

P

p

i

i

VVector of polarization V: volume

P E o e

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: EPED eoo 1

er 1

0

Relative permittivity (dielectric constant)

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

MM

r

r

1

2M: molecular mass, : density

kT3

p

3

NM

2

1P

2

o

A

r

rM

More detailed:

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 v B QB: 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: electron charge, me: electron mass

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

For electrons:

eB m

e

2

is the Bohr magneton,

2

h

h is the Planck constant, s is the spin operator.

sms em

e

(h-bar)

sms2

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:

ml 2me

e

sms em

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):

MLB ˆˆg g is the Landé factor

ml B

Moments for a nucleus

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, is the nuclear magneton, mp is the proton mass:

IN MI ˆˆgN

pm2

eN

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 (pE, E act on p (T=p x E), T is time derivative of angular moment l, this coupled with the magnetic moment m, so

Bme

22

4m

reΔ

The resulted diamagnetic moment is

mlTpEB

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 BBω eh

μg B

e and N are the magnetogyric ratios for electrons and nuclei, respectively.

For nuclei BBω Nh

μg NN

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.

dmM ji *

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

imMV

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

HM m00: permeability of vacuum, (1.25664x10-6 VsA-1m-1), m: magnetic susceptibility,

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

magnetic permeability

weak field, linear:

r is the relative permeability mr

10

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: magnetizationcurve, a hysteresis curve. Its area (curve integral) is proportional to the power of magnetization.

Curie's law: mA

TB 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 Ej

to higher level Ei:

hji EEE

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:

2

ij

t

0

ij2dttiωexptK

1a

p

ji

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

dτψKψK j*iij

is the operator of perturbation, K

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

pEUKp: change in the dipole moment during the perturbation.

The expression for Kij dK *ij ji pE

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

dp ji *P

P2 is the transition probability.

2. The effect of population

According to Boltzmann's distribution law

kT

hνexp

kT

EEexp

N

N ji

j

i

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.

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 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:

Energy uncertainty: E=h., Time uncertainty: t=, average lifetime of excited state

The natural linewidth:

2

1

2π

hδE.δt

Doppler effect (gas phase)

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

v0

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

mvexp(II

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 %): 0I

IT

or as absorbance TlgI

IlgA o

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

The intensity of the reflected light is measured as reflectance:

rlgII

lgRr

o

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

The independent variable of the spectra is either frequency,

or wavenumber c

νν~ or wavelength.

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

Characteristic data of a band