Chapter 3
STRUCTURE AND PROPERTIES
OF MOLECULES
The molecule is a set of atoms that are in strong chemical
connection to one another building a new substance.
Three types of the strong interactions exist between atoms:
1. Several individual atoms build the system. Each of them
add electrons to the full system. These electrons are
delocalized and move practically without resistance in the
system: the metal bond was built.
2. One of the interacting atoms has low first ionization energy
(e. g. an alkali metal), the second one has high electron affinity
(e.g. a halogen element): easy electron transfer from the first to
the second atom. Ion pair, they attract each other: we have an
ionic bond.
3. Atoms with open valence shells share a part of their
valence electrons, a new electron pair is formed. They
build a chemical bond, the covalent bond. Another possibilty:
an atom has a non-bonded electron pair, the other an electron
pair gap. The electron pair will be common and build a
chemical bond, the dative bond.
The shared electron pair of the molecule moves on
molecular orbitals (MO).
Polar bond: the participition of the electron pair between the
two atoms is unequal.
Delocalized orbital: the bonding electrons are shared under
more than two atoms.
Molecule: finite number of atoms with the exclusion of polymers.
Model: isolated molecule.
Symmetry elements and symmetry
conditions
Object is symmetric if there exist an operation bringing it
in equivalent position. Equivalent: covers the original one.
The operations fulfilling this conditions are symmetry
operations. Symmetry operations belong to symmetry
elements of the object.
Symmery elements are mirror planes,
symmetry centers (inversion),
symmetry axes (girs),
reflection-rotation axes (giroids)
Table of symmetry elements and operations
Symmetry elements of water
z
 yz
C2
y
O
X
H1
H2
 xz
Demonstration of a tetragiroide
S4
Tetragiroide of methane
S4
H
H
C
H
H
Inversion center of trans-hydrogenperoxide
Point groups
Symmetry operations build algebraic groups (G). An algebraic
group is a heap of objects, properties or ideas, characterized as
follows.
- A group operation exists. The group is closed for it, the result is
member of the group.
- a unit element exists (E), X*E=E*X=X
- each X element has its inverse. Y, X*Y=Y*X=E, Y=X-1, X=Y-1
- associativity: (A*B)*C=A*(B*C) A,B,C
G
- conjugate of an element: Y=Z*X*Z-1 and X=Z-1*Y*Z
A set of the group elements that are conjugated
each other builds a class of the group.
X,Y,Z
G
Representations of point groups
The planar water molecule has two  symmetry (mirror)
planes, perpendicular each other. Their crossing axis is a digir,
C2. With a unit element E the build the C2v point group.
There are numbers or matrices those follow this
algorithm. They are the representations of the group. The
possible simplests of them are the irreducible
representations of the group. In spectrocopy they are often
called as symmetry species or simply species.
If the representation is a matrix, the character table contains
the traces of the matrices. The number of representations is
equal to those of the classes.
Representations are generally labelled with G. The used notations:
Indexing of these symbols:
The rows (i) are species, the columns (j) are classes. For
more complicated groups the classes contain more than one
element. The table contains the cij coefficients.
The gir character is +1, the species is A. If it is -1, the
species is B. If xz character is +1, the subscript is 1,
otherwise, if it is -1, the subscript is 2.
Symmetry operations transform atoms in new positions:
Proper operations (Cn, E) can be regarded as rotations:
Improper operations (Sn, , i) are rotations + perpendicular
reflections:
The traces characterize the transformation matrices, they are
independent of the choice of the coordinate system. They are the
characters of the symmetry operation: cj for the jth operation.
p

c j  1  2 cos 2 
n

p  1 ,2 ,..., n  1
The symmetry of the molecules plays important role in the
interpretation of molecular spectra.
The electronic stucture of molecules
Construction of molecular orbitals
The Born-Oppenheimer theory is used: adiabatic
approach. The motion of nuclei are neglected, only the
electrons move. The relativistic effects of the Hamilton
operator are here neglected:
2
2
2
n
n
n
n N
N N Z Z e2
Z e

e
 
2
ˆH  
    


2m e i 1
r
i 1 ji rij
i 1  1 ri
 1  
Z is the atomic number, N is the number of atoms, n is the
number of electrons in the molecule, r is the distance of the
particles.
First term: kinetic energy operator, second: electron-electron
repulsion, third: electron-nucleus attraction, fourth: nucleusnucleus repulsion (costant!). The 2nd-4th terms give the potential
energy operator. D is the nabla operator.
Solution of the Scrödinger equation: exactly only for H 2
Additional approximations (restrictions):
- Molecular wavefunction: production of molecular orbital
functions, depending on the cartesian and spin coordinates;
- Pauli’s principle must be satisfied: (Slater) determinant
wavefunctions are used,
- model of independent particles: each has own orbital (ci)
functions,depending only on their own Cartesian coordinates
(xi).
-Hartree-Fock (HF) method in Roothaan (HFR)
representation : orbital functions expanded into
series using basic functions. Usually atomic orbitals
are applyed in praxis. Linear combinations of atomic
orbitals as molcular orbitals: LCAO-MO.
Solution of the eigenvalue equations for using the
listed approaches:
Self consistent field (SCF) method, it is iterative.
Estimating or assuming values for linear coefficents
for linear combinations, energy is calculated. With
this energy new coeffcients can calculated, with them
one have new energy value, etc., until the deviation
between the energies of two successive steps arrives
the wanted limit.
Shorthand: LCAO-SCF
The symmetry of molecular orbitals
Molecular orbitals have symmetry. The orbital functions maybe
symmetric: their sign does not change under the under effect of
the symmetry operation, cij=+1 (character table);
antisymetric, their sign changes under the effect of the
symmetry operation, cij=-1 (character table);
The symmetry of the molecular orbitals are denoted according
to their symmetry species, but lower case letters are used.
If some belongs to the same species, they are numbered
beginning with them of lowest energy and are used as
coefficients.
z
z
Orbitals of water
molecule.
1 eV = 96.475 kJ mol-1
x
x
1a 1
-557.3 eV
2a
-36.3 eV
1
z
(LCAO-MO calculations)
z
x
Filled ring: + region
x
Empty ring: - region
1b1
-19.3 eV
Possible bond
participations:
3a
1
-15.2 eV
y
1b1 and 2a1
z
x
1b2
-11.9 eV
Localized molecular orbitals
The LCAO-MO results reflects the electronic structure, but
are delocalized. However, they are not suitable for
demonstration of the spatial distribution of the electronic
structure.
The spatial distribution can be introduced with localized
orbitals. The linear combination of the localized orbitals have
symmetries like the localized ones. They are demonstrative,
however, since they are not resuls of quantum chemical
calculations, one cannot speak about their energies.
Localized orbitals of water molecule
(oxygen orbitals), filled: out-of-plane
Examining the localized orbitals the tetrahedral formation
of the four electron pairs around the closed shells of the
oxygen atom is well observable. The binding orbitals are
less localized than the non-binding orbitals. The 1a1
orbital remains unchanged, i.e. localized.
The 1s orbitals of the hydrogen atoms and the 2s, 2px and
2py orbitals of the oxygen atom build the chemical bonds
(2a1 and 1b1). There exist also real delocalized molecular
orbitals, e.g. those of the aromatic rings. Here is difficult to
form localized orbitals.
The covalent bond
The characteristics of the chemical bond
Influences on the formation of the molecular orbitals: (look also
at the Hamilton operator)
- kinetic energy: smaller free space for electrons
higher;
- electron-electron repulsion: increases their distance;
- electron-nucleus attraction: acts on the electron;
- nulceus-nucleus repulsion: important role in the formation of
molecular geometry;
- spin-spin electron interaction: with parallel spin repulsive, with
opposite spin attractive (Pauli principle, Hund rule).
Results for localized elecron pairs:
-Try to avoid one another;
-Try to expanding their possible area,
-Try to come as close to the nucleus as possible
The molecular geometry is the result of the listed
effects.
Formation of the molecular orbital: the electron clouds
of the atoms approach one another.
Hybridization: mixing of the atomic orbitals (y): overlap
integral, measure (grade) of mixing (atoms A and B):
S AB  y y B d
*
A
Mixing of atomic orbitals: chemical bond.
Extreme cases:
- there is not mixing of atomic orbitals, e.g. water 1a1
orbital;
-the participations are equivalent, like H-H bond in
hydrogen molecule, with 1s orbitals.
During the approaching of the atomic orbitals two
levels build, these molecular orbitals:
-the energy of one is lower than those of the atomic
orbials, it is localized between the two atoms, this is
the bonding molecular orbital;
- the energy of the other is increased, it has a nodal
surface, is wide spreaded, this is the antibonding
molecular orbital.
The intoduced model is valid only in case of bonds with
s-s atomic orbitals.
The more atomic orbitals with nearly the same energy
levels are combined in the bond, the greater the
deformation of the original atomic orbitals.
The description of the molecular orbitals is possible only
as the linear combination of several atomic orbitals.
If only elements with atomic number lower than 10 take
part in the molecule, the deformation of the atomic
orbitals is small.
The attractive force between the interatomic electron
clouds and the atomic cores is greater than the repulsive
force between the atomic cores (nucleus and inner
electrons). This is the fundamental reason of the
formation of chemical bonds.
The intramolecular electron affinity of the atoms is
characterized by the electronegativity. Under several
definitions the widely used if that of Mullikan:
X
1
(I  A )
2
Here I is the ionization energy, A is the electronaffinity of
the atom.
The atoms at the first part of the periodic table having
high electronegativity like carbon, nitrogen and oxygen
and can mobilize even two or three electrons to fill their
valence electron shell. The second and third bonds are
weaker than the first one since the interatomic area is
occupied by the electron pair of the first bond
(repulsion). A multiple bond needs atomic orbitals of
appropriate orientation (p or d orbitals) that energy level
is not very high.
The structure of two-atomic molecules
The simplest molecules, suitable for studying the chemical
bond.
Two equivalent atoms: point group:
Infinite gir, 2 operations,
Infinite vertical planes,
Inversion center,
Infinite giroid, 2 operations,
Infinite vertical digirs.
Special labels for species of
diatomic molecules.
, cylindric form
Hetero diatomic molecules have lower symmetry, the
symmetry elements of this group are only the C  gir
and the infinite number of mirror planes, cutting the gir.
Special labels are applied for symmetry species of
diatomic molecules: the Greek letters instead of the
corresponding Latin ones.
The sigma () bond is cylindric, between the two
atoms, maybe s-s. s-p or p-p bond. This is the strongest
bond.
The pi () bond is situated out of the interatomic area,
maybe p-p, p-d or d-d bond, weaker than the sigma
ones.
Look again on
the forms of the
atomic orbitals!
Where are
possible  or 
bonds?
Molecular orbitals
for H2 from 1sA
and 1sB atomic
orbitals. The
antibonding
orbitals are
starred (*).
 g
g 
1
(1s A  1s B )
2(1  S)
 u
 u* 
1
(1s A  1s B )
2(1  S)
Hybridization
A molecular orbital is called hybrid orbital if an atom takes
part in it with more then one orbitals. Measure: participation
of the atomic orbitals in the molecular wavefunction.
Hybridization is possible only in case of  bonds.
The central atom contacts n equivalent atoms or atom
groups. Results: n equivalent orbitals arranging
symmetrically in space and determine the structure.
Example. The ground state of the C atom is 1s22s2p2( 3Po).
If one 2s electron transits to a 2p orbital (according to
Hund's rule) then the electron configuration changes to
1s22sp3 ( 5S2). In space symmetric, equienergetic orbitals,
sp3 hybrids.
These hybrid orbitals are orthogonal to one another
(in algebraic sense), therefore their overalap integrals
are zero. The four 5S2 hybrid molecular wavefunctions
are (atomic orbitals are denoted as c):
Therefore methan has tetrahedral structure. From one
carbon to methan with hybridization
(2 ways):
1. Excitation of C (promotional energy needed),
combined with four hydrogens (energy recovered);
2. C is combined with four H’s, CH4 is in excited state,
energy loss to lower 5S2 state.
In the MO theory the hybridization means the forming
of equivalent orbitals. Beside this sp3 hybrid orbitals
they are formed with ethene (C2H4): sp2 hybrid and
also with ethine (C2H2): sp hybrid.
The substitution demages these hybrid orbitals since
their equivalence disappears.
The hybridization is important in case of complex
compounds of transition elements. Their d orbitals can
form hybrid molecular orbitals with the ligands.
E.g.: spd2 determines a square structure [(PtCl4)2-] ,
sp3d a trigonal bipyramide (PCl5), sp3d2 an octaheder
(SF6) , etc.
Delocalized systems
Organic compounds with conjugated double bonds
are special case of the double bonded molecules.
Beside the first  bonds each second chemical bond
is strengthened though a  bond. However the
electrons of the  bonds spread along the the whole soThe energy levels are
called conjugated system
far over the  levels. The  separation is a good
approach for describing the system.
Restrictions for the simple Hückel method ( levels):
1. for overlap integrals Sij  0 i  j or Sij  1 i  j
2. Hamilton matrix element H ij  y i* Hˆ y j d is denoted as
, i and j on same atom (Coulomb integral);
, i and j on vicinal atoms (resonance integral);
is zero, otherwise.
Both constants have negative sign,  is of higher
absolute value. The eigenvalue equation has the form
H  ES  0
For the ethylene (ethene) molecule (only carbon atoms
are considered):
The results:
 E


 E
 
E1     for bonding  orbital
E 2     for antibonding  orbital
Extended Hückel theory (EHT) for heterocyclic systems:
x=+hx, xy=kxy*, e.g. hN=0.5, kCN=1.
Application of the Hückel theory to benzene: the results of
the eigenvalue equation are ( is assumed as -75 kJ/mol):
E1    2
E2  E3    
E4  E5    
E6    2
Two levels are degenerated. The six  electrons occupy
the lowest E4, E5 and E6 levels. For one carbon atom
E=. Without conjugation is the total energy
6*()=66. With conjugation E2*(2)4*() i.e.
E68. The energy decreased since 2= -150 kJ/mol,
this is the delocalization energy.
The Hückel theory is an acceptable approach for such
cases.
For the point of view of reactivity of molecules two energy
levels are important:
The electron density on the highest occupied molecular
orbital (HOMO) is nearly proportional to the reactivity in
electrophylic reactions.
The electron density on the lowest unoccupied molecular
orbital (LUMO) is nearly proportional to the reactivity in
nucleophylic reactions.
These limit levels play also important role in the
development of the chemical and spectroscopic properties
of the molecule.
The advanced quantum chemical methods result better
approximations, like post-Hartree-Fock and density
functional methods (DFT: density functional theory).
The d orbitals complicate these calculations.
Complex compounds of the transition metals
The d orbitals are important since several transition
metals play role in catalysts and enzymes. Their
description is more complicate than that of the
molecules with atoms below atomic number 10.
Even a simple theory is a good tool in this field.
Bethe's crystal field theory is simple, old, but suitable
also in our days.
The ligands with their negative charges (ion or dipole)
connect the central ion (having positive charge). The bond
is relatively weak. Practically the central ion determines
the molecular structure. The electric field acts on the
crystal field, the spin-orbital interaction and the internal
magnetic field take also part in the Hamilton operator of
the molecule.
The discussion of the nd (n>3) and nf orbitals is
complicate. Our model is the 3d orbital.
Since n=3, the maximal angular quantum number l=2,
magnetic quantum number changes from m=-2 to m=2.
Octahedral complexes with six equvalent ligands
(sp3d2 hybrids) are discussed here. They belong to the
Oh point group.
The angle depending parts of the d orbital functions
determine the symmetry.The 3d x  y and the 3d z2 orbitals
(transforming like 3z2  r 2 ) are symmetric to the xy, xz
and yz mirror planes (3h) and are also symmetrical to
the x, y and z digirs (3C2), therefore they belong to the
symmetry species Eg. The three other d orbitals, dxy, dxz
and dyz are symmetric to the six axis-axis bisectors
(6C2) and to the mirror planes determined by a bisector
and an axis (6d) and to the inversion (i). Therefore
they belong to T2g. (as labels T and F are equivalent).
2
2
Oh karaktertáblázat
The originally five times degenerated energy level splits into
two groups. The ligands connecting to the central ion are
positioned on the coordinate axes. The t2g orbitals (orbitals
are labelled similarly to their symmetry species only small
letters are used) are situated between the coordinate axes,
while the eg orbitals are centred on them. Therefore the
ligands repulse the eg orbitals, so their energy is higher than
that of the t2g ones.
The energy difference between these two orbital groups
depends above all on the electric field generated by the
ligands. The experimentally measured splitting is denoted
by D.
The crystal field theory gives the order of the orbital
energies but only as expressions, their values are not
calculable.
The measure of the splitting in octahedral crystal field is
labelled by 10Dq. Using the experimental data the Dq
becomes calculable (q is the ratio of two matrix elements,
D is a coefficient in the description of the crystal field).
The shift of the band system by the ligands is not taken
into account. Therefore the average energy of the d orbitals
is always 0 D. The splitting is influenced by two effects:
1. The crystal field (metal ion - ligand, d orbital symmetry)
effect.
2. The mutual repulsion of the d electrons.
First effect stronger: strong crystal field,
second effect stronger: weak crystal field.
In a strong crystal field the electrons occupy the energy
levels according to the increasing energy.
Therefore the t2g orbitals are occupied at first, and the
eg ones only later.
The energy of the t2g orbitals is 4Dq lower than
average, while that of the eg orbitals is 6Dq higher than
average.
The t2g orbitals are the bonding ones, the eg's are the
antibonding ones.
d electron configurations in strong octahedral
crystal field
The situation is more complicate in weak crystal fields. The
repulsion of the electrons split the orbitals into several levels.
Sometimes these levels are very close. The electrons occupy the
orbitals according Hund's rule. At first all orbitals are occupied by
one electron. After the occupation of all orbitals in this way the
second electrons join stepwise the first ones with opposite spins.
Octahedral complexes with weak and strong crystal fields differ
in the case of d4, d5, d6 and d7 configurations.
The group spin quantum numbers of these configurations are for
weak crystal fields high, they are high spin states. For strong
crystal fields the group spin quantum number is in these cases
low, they are low spin states. These two types of states are
distinguishable by magnetic measurements.
Comparison of the occupations of energy levels in
weak and strong crystal fields
E
eg
t 2g
d
1
2
d
3
d
4
d
d
5
d
6
7
d
weak crystal field
4
d
5
d
d
6
7
d
strong crystal field
8
d
d
9
10
d
The ligand field theory is the application of the molecular
orbital theory to transition metal complexes. It is very useful if
2
the ligand-metal bond is covalent (e.g. MnO 4 , Fe(CN)6 metal
carbonyls, complexes, etc.). The advances of the method are
the better qualitative description of the molecules and the
quantitative energy values. Most of these kind methods use
semiempirical quantum-chemical models.
Both strong and weak crystal fields are extreme cases. The
real complexes stand between these two models, they crystal
fields are more strong or more weak.
In the case of nd (n>3) and nf orbitals it is necessary to modify
this simple model. The spin-orbital interactions play important
role in these cases.
For comparison: splitting of p2 electron energy levels
under effect of external fields
1S
1S
o
MJ
1D
1D
2
2
0
-2
3P
2
2
3P
p2
H atom level
-2
3P
1
3P
o
electrostatic
interaction
magnetic
interaction
1
-1
0
external
magnetic field
Splitting of d2 electron levels in strong crystal field
1
A
1g
1E
g
3
A
2g
(e g) 2
eg
d2
10 Dq
excitation
of 2 electrons
10 Dq
(e g)(t 2g)
excitation
of 1 electron
10 Dq
t 2g
(t 2g) 2
crystal field
effect
strong field
configurations
1T
1g
1
T2g
3
T
1g
3
T
2g
1
A1g
1
E
g
1
T
2g
3
T
1g
electron-electron
interactions
Splitting of d2 electron levels in weak crystal field
1S
1G
d
2
3P
1D
3F
electron-electron
interaction
1A
1g
1A
1g
1T
1g
1E
g
1T
2g
3T
1g
1
Eg
1T
2g
3A
2g
3T
2g
3T
1g
crystal field
effect
The Jahn-Teller effect is important for transition metal
complexes. If an electron state of a symmetric
polyatomic molecule is degenerated, the nuclei of the
atoms move to come into an asymmetric electron state.
In this way the degenerated state splits. The system will
be stabilized by the combination of the electron orbitals
with vibrational modes. This is not valid for linear
molecules and for spin caused degenerations.
Octahedral complexes (e.g.Fe(CN)62 ) can be distorted
by the Jahn-Teller effect in two forms: into prolate
(stretched) or into oblate (compressed) octahedron,
according to the symmetry of the coupled vibrational
mode (the first case occurs more frequently). The JahnTeller effect is observable also in the electronic spectra
of the transition metal complexes. The spectral bands
split or broaden.
Rotation of molecules
Born and Oppenheimer: the energy of molecules is
may be regarded in first approach as sum of rotational,
vibrational and electron energies. The kinetic energy is
not quantized, and therefore the molecule is studied in a
system fixed to itself. So inertial forces like Coriolis and
centrifugal ones may appear in the system.
Applying a better approach it can be proved, in a good
agreement with the experimental results that these three
types of motions are in interaction. The change in the
vibrational state influences the rotational state, the
change in the electron state influences both the
vibrational and rotational states of the molecule.
Rotational motion of diatomic molecules
The kinetic energy of the rotating bodies is described by
1 2 1 L2
T  E r  I 
2
2 I
I is the moment of inertia, L is the angular moment,  is
the angular velocity. The quantum chemical problem is
calculation of the operator eigenvalues (similar to the
problem of the H atom). Here the eigenvalues of the
angular moment are quantified:
L  J(J  1)  J *
J  0,1,2,3,...
J is the rotational quantum number. The length of the
Lz component is determined by the MJ magnetic
quantum number
Lz  MJ
 J  MJ  J
Using the rigid rotator approach ( the atomic distances
do not change with the change of the rotational energy),
2
Er 
J(J  1)  B' J(J  1)
2I
with
2
B' 
2I
B’ has energy dimension. Since the experimental data
appear in MHz or cm-1 units, the rotational constants
are used in forms B'/h (MHz) or B=B'/hc (cm-1). The
relative positions of the energy levels of a rigid rotator
are shown in next figure. The energy level differences
increase with increasing rotation quantum number. The
energy levels split if an external magnetic or an electric
field acts on the molecule, i.e. the rotational energy
levels are degenerated.
Energy levels and spectral lines of a rigid rotator
J
Er
3
6B
2
4B
~
2B
1
2B
0
According to the definition of the moment of inertia for
the rotational axis of a system of N points
N
I   m i ri2
i 1
m is the mass of the atom, r is its perpendicular distance
from the axis.
The moment of inertia for a diatomic molecule and its
axis crossing the mass center and perpendicular to the
valence line has the form
I  ro2
ro is the distance between the two atoms and

m1 m 2
m1  m 2
is the reduced mass of the molecule.
Rotational spectra of the diatomic molecules
Substituting the eigenfunctions of the rotational states into
the expression of the transition moment
P  y i* Dpy j d
the following selection rules may be derived:
DJ  1
and
M J  1
Supposing constant atomic distances during the
excitation (rigid rotator) the frequencies (as wavenumbers)
of the rotational lines are equidistant (J belongs to the
lower energy state)
E r ,i - E r , j
~

 B(J  1)(J  2)  J(J  1)  2B(J  1)
hc
where
B
h
B
 2
hc 8 cI
The rigid rotator model is a good approach. In the reality,
however, the atomic distances increase with increasing J.
The chemical bonds are elastic, therefore the increasing
centrifugal force stretches the bonds. Result: a greater
moment of inertia, and so a decreasing rotational constant.
For non-rigid (elastic) rotators the distances between the
energy levels decrease with increasing J. Looking the
rotational spectral lines we find their decreasing distance with
the rotational quantum number.
The pure rotational spectra appear in the microwave (MW)
and in the far infrared (FIR) regions. The intensity of the
spectral lines depends on the relative populations of the
energy levels. According to Boltzmann's distribution low:
 E r ,J
N J  N o (2J  1) exp  
 kT



NJ is the number of molecules on the J-th level, 2J+1 is
the degree of degeneration according to the magnetic
quantum number. NJ has a maximum (see the spectra
down).
The rotational spectra can be measured recording
microwave (MW), far infrared (FIR) or Raman (RA) spectra.
Microwave spectra. See the flow chart of the spectrometer.
Excitation: tuneable signal source (SS), this is e.g. a reflex-clystron, or a
Gunn diode. Waves propagate along tubes with squared cross-sections. A
part of the waves crosses the sample (S). Detector: crystal detector (CD).
Its output is proportional to the MW signal intensity. The electronic system
(E) elaborates this signal. Another part of the waves is used for frequency
calibration. They are mixed to the frequency standard (FS) by the
frequency mixer (FM) and the mixed wave is detected by a radio receiver
(RR) that generate the frequency differences. The spectrum will be printed
(P) or presented on the screen of an oscilloscope.
Flow chart of a microwave spectrometer
SS
S
FS
FM
CD
RR
E
P
Far infrared spectroscopy. FT spectrometers are applied.
The optical material is polyethylene, the beam splitter is
polyethylene-terephtalate foil.
The molecule must have a permanent dipole moment, since
otherwise the transition moment is zero. Therefore the
diatomic molecules with two equivalent atoms have not pure
rotational MW or IR spectra. This is the pure rotational IR
spectrum of H35Cl. The H-35Cl distance is calculable form the
line distances.
Raman spectroscopy. Raman spectroscopy is a special
method of the rotational and the vibrational spectroscopy.
This is a scattering spectrum. Spectrum lines are observed
in the direction perpendicular to exciting light (a VIS or NIR
laser beam) beside the original signal
The effect is called Raman scattering, the spectral lines
are lines of the Raman spectrum. The series that appear at
lower frequencies than the that of the exciting beam ( ~ o) are
the Stokes lines, the lines having higher frequencies than ~ o
are the anti-Stokes lines. The intensities of the anti-Stokes
lines are lower than that of the Stokes lines, since the
population of their excited states is smaller. Therefore the
Stokes lines are detected. The Raman shifts, D~ i  ~ o   i
give the frequencies of the rotational lines.
The Raman scattering
Flow chart of a
Raman
spectrometer
sample
~o+-~
i
~

anti-Stokes lines
~
o
Raman scattering
~
o
laser beam
Stokes lines
The Raman lines appear if the polarizability of the
molecule changes during the transition.
Dp  Dα * E
The selection rules are
DJ  2
for equivalent atoms, e.g. H2. This is a difference in
comparison with the MW and IR spectra (the selection
rule is there DJ  1 ). For different atoms
DJ  1,2
Each second line is very weak in the rotational Raman spectrum
of the oxygen molecule, therefore they are not observable in the
spectrum (this is an exclusion). Notice: a line in the middle has
maximal intensity, according to Boltzmann’s distribution low.
O2
The bond length of a diatomic molecule is easily
calculable from the rotational spectrum. According to
equation for the moment of inertia the distance
between the rotational lines is 2B. The distances of the
lines in the H-35Cl spectrum are 20.7 cm-1. Using
27
thementioned equation, I  2.703  10 kg m2.
Therefore the bond length is 129 pm. Similarly, taking
into account the line distance in the Raman spectrum
of oxygen (11.5 cm-1) and it equivalence with 8B, the
bond length in the oxygen molecule is 121 pm.
B
h
B
 2
hc 8 cI
N
I   m i ri2
i 1
Rotational specra of polyatomic mlecules
The calculations of these rotational spectra are carried
out in coordinate systems fixed to the molecule. The
origin is the center of mass, the axes are the principal
axes of the moment of inertia. Those of maximal value
are labelled as C, with minimal one as A, the third is
perpendicular to both is the B.
The rotating moleculas are considered as rotating
tops. According to the relative value of the principal axes
of inertia the can be spherical, symmetric (prolate,
oblate) or asymmetric rotators.
Fo the simplest, spherical rotators the simplest
equation is valid:
E r ,i - E r , j
~

 B(J  1)(J  2)  J(J  1)  2B(J  1)
hc
For symmetric top prolate molecules

Er  hc BJ (J  1)  (A  B)K 2

For symmetric top oblate molecules

Er  hc BJ (J  1)  (C  B)K 2

K is the nutational quantum number, It quantizes the
component of the angular moment to the highest order
symmetry axis of the molecule (e.g. C6 for the benzene
molecule).
Selection rules
for non-linear symmetric top molecules:
DJ  1 DK  0 (IR)
DJ  1, 2 DK  0 (RA)
for linear symmetric top molecules:
DJ  1
(IR)
(RA)
DJ  2
J  K  J
K=0
The description of the energy levels of the
asymmetric top molecules is very complicate.
There do not exist solutions for these rotators in
closed mathematical form.
NON  140o
N2O IR spectrum
rNO=118 pm, NON  140o
N2O Raman spectrum
Pay attention on the double density of the RA spectral
lines comparing to the IR ones (in the RA spectrum only
the DJ=-2 transitions appear) and the maxima of the line
intensities.
The vibration of molecules
Vibrational motion of diatomic molecules
The vibration of the molecule is in first approach independent
of its rotation.
Further approach is the harmonic oscillator model, i.e.
harmonic vibrations are assumed.
Hamilton operator of a diatomic molecule with a reduced
mass
2 d2 1
Ĥ  
 kq 2
2μ dq 2 2
q is the displacement coordinate (in vibrational equilibrium its
value is zero), k is the force constant of the harmonic
vibration; the first term is the operator of the kinetic, the
second term is the operator of the potential energy of the
oscillator.
The solution of the Schrödinger equation with this
Hamilton operator leads to
1

E v  h v  
2

v ,1,2,...
v is the vibrational quantum number,  is the oscillator
frequency.
The next figure contains the forms of the harmonic
oscillator wavefunctions (dashed lines) and the
probability distribution functions (full lines).
The wavefunctions of odd vibrational quantum
numbers are antisymmetric, while those of even
quantum numbers are symmetric. All probability
distribution functions are symmetric.
Equidistant energy
levels of the harmonic
oscillator and the curve
of the potential energy
function V (dashed line)
of a diatomic molcule,
as function of the
distance of atoms r.
The probability density distribution of the v=1 state
is very similar to the classical mechanical model of
the vibrations. According to the classical model the
system has also two points with maximal staying
time. Since the most important transition is v=0
v=1, the mechanical model is a good approach.
The predominant parts of the molecules are at room
temperature in ground state (v=0).
Vibrational spectra of diatomic molecules
The spectra are recorded applying both infrared and
Raman spectroscopy.
Infrared spectra are measured in practice only with
Fourier transform spectrometers.
From the definition of the transition moment the
selection rule is for IR spectra
Dv  1 (+: absorption, -: emission)
Predominantly absorption spectra are recorded, the
measurement of the emission spectra is difficult. The
vibrational transition is infrared active if the molecule
has permanent dipole moment (necessary condition, as
for the rotational spectra). Therefore the X2 type
molecules have not IR spectra.
Raman spectra are measured classically with
perpendicularly incident laser light applying a
monochromator, or with the introduction of the laser light in
a FT spectrometer (in this case the light source is replaced
with the exciting monochromatic laser beam). The
selection rules are like in case of IR spectra. Since the
Raman activity depends on the change in the components
(
of the probability tensor
the X2 molecules are Raman active.
)
The real vibrations are
anharmonic. Therefore the
selection rule is not strickt. Overtones: Dv  vi  v j j  0 , i  2,3,...
can appear with law intensity. The density of the overtone
bands increases with increasing vi. The energy of the
anharmonic oscillator is (approach):
2

1
1 

E v  h  v    x v   
2
2  


Increasing the ambient temperature the population of the
higher levels increase and the bands belonging to the
excitations from these levels also appear in the spectrum
(overtones, "hot bands"). A considerably excitation leads to
the dissociation of the molecule. The energy difference of
the v= and the v=0 states is the dissociation energy (D)
of the molecule, r is the bond length.
Vibrations of polyatomic molecules
An N-atomic molecule has 3N degrees of freedom.
Three of them are translations, three of them are rotations
(for linear molecules only two), the other 3N-6 (for linear
molecules 3N-5) are vibrational degrees of freedom.
For the description of the vibrational motions of
polyatomic molecules three coordinate types are used.
Each is fixed to the molecule, i.e. they are internal
coordinates.
1. Cartesian displacement coordinates (r). They have
zero values in their equilibrium positions. An N-atomic
molecule has 3N Cartesian displacement coordinates.
Instead of these coordinates sometimes the so-called
mass weighted coordinates (q) are applied. The Cartesian
displacement coordinates are multiplied with the square
root of the mass of the corresponding atoms.
2. Chemical internal coordinates (S). These are the
changes in the geometric parameters of the molecule.
Four types of chemical internal coordinates exist:
-stretching coordinate, i.e. change in bond length;
-bending coordinate , i.e. change in the valence angle
(in-plane deformation);
- dihedral angle coordinate, i.e. change in the dihedral
angle (out-of-plane deformation);
- torsional coordinate, i.e. change in the torsion.
-dihedral angle coordinate, i.e. change in the dihedral angle
(out-of-plane deformation);
- torsional coordinate, i.e. change in the torsion.
3. Normal coordinates (Q). Applying these are
coordinates the Schrödinger equation of the vibrational
motion of molecules separates into 3N-6 (3N-5)
independent equations. Each depends only on one
normal coordinate and is therefore relatively easily
solvable.
It seems, the application of the normal coordinates is
the most reasonable for the solution of the vibrational
problems. Using normal coordinates the equations of
the kinetic and potential energies have the form in the
framework of the classical mechanical harmonic model:
2V  4 c
2
3 N 6
2
~i Q
2
i 1
2
i
2T 
3 N 6
2

Q
 i
i 1
Since the spectra contain information only about the
vibrational frequencies we have not information about
the normal coordinates. This coordinates can be
determinated only by further calculations.
The S and the q (or r) coordinates are applied in the real
calculations. The potential and kinetic energies have the
forms (in vector-matrix formulations):
2V  q,fq
2T  q ,g 1q
2V  S, FS
2T  S , G 1S
q and S are column vectors of dimension 3N-6, f and F
are the force constant matrices (they are the unknown
quantities), g and G are the inverse kinetic energy
matrices, they depend only on the atomic masses and
geometric parameters of the molecule. The solution of
the equation of motion lead to the eigenvalue equation
GF  E  0
the  's are the eigenvalues containing the vibrational
frequencies, E is a unit matrix.
The solutions are
 i  42 c 2 ~
2
i  1,2,...,3N  6
The eigenvectors are columns vectors. Fitting these
column vectors each beside the other we have the
eigenvector matrix L:
GFL  LΛ
With the help of this matrix we can calculate the
normal coordinates:
Q  L1S
Since the S coordinates are known it is possible to
calculate their value and the direction of the atomic
displacements in the normal coordinates. The movements
belonging to the individual normal coordinates are the
vibrational modes (or normal modes) of the molecule, the
corresponding frequencies are the fundamental or normal
frequencies.
If the F matrix is known, the frequencies are calculable.
The F matrix was calculated formerly with the help of the
frequencies and isotopomer frequencies of the molecule.
Today, with the development of the quantum chemistry and
the computer technology the calculation of F matrices is
already possible. The basis of these calculations are the
equations
  2E 

Fij  
 S S 
 i j 0
or
  2E 

f ij  
 q q 
 i j 0
the 0 subscript refers to the equilibrium position. The
differentiation is either once analytical and one numerical
or twice analytical. The result is the f matrix that is
transformed into the F matrix.
The values of the calculated force constants depend on
the chemical quality of the atoms belonging to the S
coordinate, the type of the chemical bonds and the
applied quantum chemical method. Since the greatest
part of the errors is systematic the calculated force
constants are fitted to the measured frequencies by
multiplication with scale factors. Chemically similar
compounds have transferable scale factors. The
calculation of force constants is a very good tool for the
interpretation of vibrational spectra.
The change of the diagonal elements of the force
constant matrix with the quality of the atoms and the
strength of the bonds is well observable on their values.
Force constants of some stretching coordinates
(Fii /100 N m2)
Vibrational spectra of polyatomic molecules
The vibrational spectra of polyatomic molecules are
recorded as IR or RA ones. The spectra consist of
bands. This has several reasons:
1. the interaction of the vibration with the rotation;
2. intra- and intermolecular interactions;
3. the translational energy of the molecules;
4. the Fermi resonance.
The vibrational spectra contain three types of
information: frequencies, intensities and band shapes.
The vibration-rotation interaction. The change in the vibrational
state of the molecule may go together with the change in the
rotational state. Therefore rovibrational lines appear shifted from
the vibrational frequency both left and right with the frequencies of
the rotational term differences. This is in the gas (vapour) phase
observable.
Example: a part of the IR vapour spectrum of acetonitrile. The
vibrational frequency is 920 cm-1. The line belonging to DJ=-1
build the P branch. The Q branch belongs to the DJ=0
transitions. The DJ=+1 lines build the R branch. If J increases the
moment of inertia also increases, therefore the rotational constant
decreases: the lines of the R branch are more dense than that of
the P branch. Since the population of the higher rotational levels is
smaller the intensities in the R branch are smaller than in the P
branch. Band contours (shapes) appear in the vapour spectra of
large molecules at medium resolution instead of the individual
lines (the spectrometer builds averages). Sometimes the Q band
does not appear for symmetry reasons.
Acetonitrile IR spectrum
~
 /cm 1
The rotational structure is complicated through the
Coriolis vibrational - rotational interaction (an inertial force
between translation and rotation).
Inter- and intramolecular interactions. The interactions
change the energy levels and since the environments of
the individual molecules are not the same, their
frequencies shift individually from the frequency of the
separated molecule (in condensed phases).
Doppler effect appear as a result of the velocity
distribution of the molecules in gas phase.
Fermi resonance bands appear in the case of the
accidental coincidence of two bands with the same
symmetry. Their intensities try to equilize and the bands
move away from one another.
Infrared spectra
The selection rules are the same as for the diatomic
molecules. If the molecule has symmetry elements, this
selection rules become sharper. Infrared active vibrational
modes have the same symmetry like the translations of the
molecule. On the character table T labels the translations, R
stands for the rotations and the elements of the polarizability
tensor are denoted by .
The IR spectra are measured in gas, liquid (also in
solution) and solid state. The classical way:The spectra are
measured generally in solid state, using 0.1-0.2 % of the
substance in KBr. This mixture is pressed to transparent KBr
discs. The substances have strong absorption in liquid
phase, therefore very thin layers are necessary. The same
problem arises in solution: the solvents have also strong
absorption in some regions of the IR.
New methods of total reflection combined also with
microscope make easy the measurements in solid state,
direct measurement of the compound.
Raman spectra
The selection rule is similar like for diatomic moecules. If
the molecule has symmetry, the selection rule becomes
sharper. Only those vibrational modes are Raman active
that belongs to symmetry species common with at least one
of the elements of the polarizability tensor (). If a molecule
has a symmetry center, the IR and Raman activities
mutually exclude each other.
There is a special possibility of the Raman spectroscopy
for more information. Supplementing a Raman
spectrometer with a polarizer, the detected intensities of the
spectral bands depend on the direction of the polarizer. The
incident light is polarized in the xz plane. The scattered light
is analyzed both in parallel and in perpendicular polarizer
directions. The depolarization ratio of a spectral band is



z
Y
sample
direction of polarization
polarizer
I
I
I
incident light
scattered light
X
The maximal value of  is 0.75. The bands belonging to
the vibrational modes of the most symmetric species
are polarized, i.e. their depolarization ratio is smaller
than 0.75. This is a good information for the
assignment of these bands (assignment, i.e. the
interpretation of the band).
Example 1: The formaldehyde molecule (4 atoms) has
34-6=6 vibrational modes. A1, A2 and B1 modes are IR
active, all modes are RA active (see table).
The table contains the character table of the
formaldehyde molecule, the rotation (R) and the translation
(T) are also given. This table will be applied also for the
calculation of the number of vibrational mode belonging to
the individual symmetry species. This is possible using the
characters cj of the R symmetry operations:
 p
c j  1  2 cos 2 
 n
p  1,2,..., n  1
+1 for proper, and -1 for improper operations.
The number of the vibrational modes in the i-th
symmetry species is
1
mi   n jc j (R )cij  ri
h j
h is the total number of the symmetry operations, nj is the
number of atoms that are not moved under the effect of the
Rj operation, ri is the number of non-vibrational degrees of
freedom belonging to the i-th species (rotations and
translations), the cij values are elements of the character
table.
The formaldehyde molecule is planar, its plane is the
zy one. Applying the equation for calculation of the mi
values of the species B1
1
m B  4 * 3 * 1  2 * ( 1) * ( 1)  2 * 1 * 1  4 * 1 * ( 1)  2  1
4
The full representation of the formaldehyde molecule is
1
G  3A1  B1  2B2
The vibrational modes belonging to A1 preserve the
symmetry of the molecule (first three formations).
A2 modes are antisymmetric to the molecular plane (zy)
similarly antisymmetric to the perpendicular plane (zx).
Rotation only,no active modes.
B1 modes are perpendicular to molecular (yz) plane, since
N atomic planar molecule has N-3 o.o.p. modes only the
forth from belongs here.
B2 modes are planar, antisymmetric motions, fifth and sixth
forms, the last 2 from the 2N-3 planar modes.
H
C
O
y
H
z
+
The vibrational modes belonging to A1 preserve the symmetry
of the molecule. The first three formations are of this kind.
The modes belonging to A2 must be antisymmetric to the
molecular plane (zy) since , and must be similarly also
antisymmetric to the perpendicular plane since . This is possible
if only the molecule rotates around the z axis. Therefore mA2  0
In species B1 yz has also a character -1, the other mirror
plane, however, has a +1 character. Only one mode, the fourth
belongs to here. N atomic planar molecule with has N-3 out-ofplane modes, this is the only o.o.p. mode of formaldehyde.
The vibrational modes of the B2 species move again in the
molecular plane. They are, antisymmetric to the perpendicular
mirror plane. The last two modes belong to this species.
A planar molecule has 2N-3 in-plane modes and under the 6
formations 5 are in-plane modes (A1 + B2).
Under the mentioned conditions the modes of the species A1,
B1 and B2 are IR active and all vibrational modes are RA active.
Example 2.The pyrazine molecule (1,4-diazine)
belongs to the D2h point group. The molecule is
planar in the xy plane. Its character table:
N
N
The full representation of the pyrazine molecule is
G  5Ag  4 B1g  2 B2g  B3g  2Au  2 B1u  4 B2u  4 B3u
10 vibrational modes are IR active, 12 modes are RA
active, 2 do not appear in the spectra. Since the
molecule has a symmetry center, the IR active modes
do not appear in the RA and vice versa.
Solid state infrared spectrum of pyrazine (KBr tablet)
Infrared vapour spectrum of pyrazine. The molecule is an
asymmetric top. The band shape icharacterizes the direction of
the transition.
Z ~ B1u ~ maximal moment of inertia ~ C band (very strong Q
branch).
Y ~ B2u ~ medium moment of inertia ~ B band (no Q branch).
X ~ B3u ~ minimal moment of inertia ~ A band ( weak Q
branch).
The RA spectrum of the solid pyrazine . The bands
below 250 cm-1 are vibrations of the crystal lattice.
The RA spectrum of the pyrazine melt. It is a polarized
RA spectrum. Curve 1 is recorded with parallel, curve 2
with perpendicular polarizer. Find the polarized bands
belonging to the A1g species.
The next table gives the quantum chemically calculated and
measured normal frequencies and the types of the normal
modes. The individual vibrational modes have order numbers,
the fundamental modes of the parent compounds and of the
substituted molecules can be compared in this way. Beside
frequencies also the characters of the vibrational modes are
calculated ab initio. Their characters show the weight of the
participation of the individual chemical internal coordinates.
The Ring (rg) and CH motions are distingushed. The
stretching modes are labelled by  , the in-plane deformations
by , the out-of-plane deformations by g and the torsions by .
The labels p and dp denotes the polarized and depolarized
bands, respectively. A, B, and C denote the observed IR
vapour band types. Values in parentheses are results of other
measurements. The molecule has 2N-3=17 in-plane modes ( )
and N-3=7 out-of-plane modes ( ).
The motions in several vibrational modes are determined
practically only by one chemical internal coordinate of a
chemical group. These modes are called group modes, the
corresponding bands and frequencies are called group bands
and group frequencies, respectively. The pyrazine molecules
have 4 frequencies above 3000 cm-1, these are CH valence
(or stretching) frequencies (only CH stretching coordinates
move in them). Several other groups have also characteristic
frequencies.
If a group has the form XY2, the two XY stretchings are
coupled. If the stretchings are in phase, this is a symmetric
(s) vibration, if they are in opposite phase, this is an
antisymmetric (as) vibration. The frequency of antisymmetric
modes is always higher than that to the symmetric ones.
Under the vibrational modes belonging to the same group the
valence frequencies are highest, lower are the in-plane
deformation ones, the out-of-plane modes have the lowest
frequencies.
Non-linear spectroscopy
The Raman spectrometers use low energy laser light as light
sources having frequencies far from the frequencies of
electron transitions. Only one laser is applied. Applying other
conditions special phenomena are observable.
Applying high energy laser as light source the non-linear terms
in p  po  E  1 E2 ... become greater and these terms
2
determine the induced
dipole moment. The lines appear in the
scattered light. This is the hyper Raman effect.
If the frequency of the high energy laser source fall into an
electron transition bond the spectrum changes absolutely. As
result of the interaction some Raman lines disappear, other
more intense and/or shift. This is the resonance Raman effect.
The strong lines are intense also in diluted solutions and are
therefore suitable for quantitative analysis.
If the laser energy is extremely high, as a result of the
excitation the population of the excited state is greater than
that of the ground state ("inversion" of the population). Some
lines become extremely strong, their intensity is comparable
to the intensity of the scattered light at the laser frequency.
This is the stimulated laser effect.
The coherent anti-Stokes Raman effect (CARS) is a multiphoton effect. Two lasers with adequate intensity and
frequencies 1 (fixed) and 2 (tunable) irradiate the sample.
at the same time. If the frequency difference is equal to the
frequency of a vibrational transition: i=2-1, then an
intense coherent radiation is observable at the frequency
21-2. In contrary to the Raman effect here the
fluorescence does not disturb this effect. This effect can
follow fast processes (ns, ps). It is also applicable in
quantitative analysis.
The Raman amplification spectroscopy is an absorption
method. The sample is irradiated with two lasers. Their
frequencies are 1(fixed) and 2 (tuneable). If  i=2-1,
the molecule absorbs light at 2 frequency and emits at
1 one. If light is detected on 2 this is the inverse
Raman effect, Raman loss spectroscopy. If the light of 1
frequency is measured, this is the Raman gain
spectroscopy.
The light sources of non-linear methods are pulse
lasers (with ns, ps pulses). These lasers are suitable for
the measurement of short lifetimes of states.
Neutron molecular spectroscopy
(IINS, incoherent inelastic neutron scattering)
The wavenumber region of thermal neutrons is comparable to
that of the molecular vibrations:
E kT
h
~
 


hc hc 2m n c2
k is the Boltzmann constant, T is the absolute temperature,
mn=16749310-27 kg, the neutron mass,  is the wavelength, c is
the light velocity in vacuum. The frequency region of the
thermal neutrons is 5 - 4500 cm-1. The incoherent neutrons
interact with the molecules through inelastic scattering
(absorption), if their frequency are equal to one of vibrational
fundamentals of the molecule. The cross section of the
interaction is very high in the case of hydrogen (79.7 barn, 1
barn = 10-28 m2). A lot of elements have cross section between
1 and 10 barn. The cross section of 12C and 16O is zero.
This method is very effective in measurement of
vibrational spectra of molecules with high hydrogen
content. The selection rules differ from that of the optical
spectroscopic methods, therefore the transitions that are
forbidden in IR and RA may appear.
Tunneling electron spectroscopy
(IETS, inelastic electron tunnelling spectroscopy).
It is based on the quantum mechanical tunnel effect:
particles can cross an energy barrier without an
excitation, depending on their mass and the height of
the barrier. The molecules are absorbed on an
insulator. The insulator layer is placed between two
metal plates. Under electric tension the molecules can
receive energy from the tunnel electrons with about 1
% probability. This absorption is measurable with a
very complicate instrument. The measurement is very
sensitive: 2 pg substance can be detected on 20 m2
surface.
Large amplitude motions
If a molecule has more than one energy minimums the
motions between these minimums are called large
amplitude motions. The minimums are not always
energetically equivalent.
The internal rotation is a large amplitude motion of a
torsional coordinate. The rotation of the ethane molecule
around its C-C axis is a good example. During a rotation of
360o it has three maximums (eclipsed) and three
minimums (straggled), see the next figure. This motion has
a periodic potential. The symmetry in the maximums and in
the minimums are high, in all other positions it is C3 (see
next table).
Potential energy function of the ethane rotation
Character table of the C3 point group
According to the symmetry two energy level series exist:
the A and the E. The energy barriers are the differences
between the maximal energy and the v=0 levels. For the
ethane molecule they are 12.25 kJ mol-1. Here is also
possible the quantum mechanical tunnel effect.
The inversion is a transition between two energetically
equivalent states in the case of non-planar configurations
through a planar intermediate state. The ammonia
inversion is one of the most known cases.
The large amplitude motions of the non-planar 4-, 5- and
6-membered rings known from the organic chemistry.
The existence of more than one energy minima causes
splittings in the vibrational spectrum.
Electronic transitions in molecules
The electronic transitions in molecules are not in
connection with any molecular motions. The energy
differences are higher, the times of transitions are shorter
than in the case of vibrational motions. The electronic
transitions may be coupled with vibrational and rotational
transitions (vibronic and rovibronic transitions, respectively).
Therefore the electronic spectra have vibrational
(rovibrational) structure.
The electronic spectra are measured in solutions. Their
intensity is recorded in absorbance. According to the
Lambert-Beer law A  l c
 is the molar absorption coefficient, l is the layer width, c is
the chemical concentration (in concentrated solutions the
activity replaces the concentration).
The excitation of the electrons
The molecular energy depends on the molecular geometry.
Diatomic molecules have only one parameter, the atomic
distance, the potential energy curve is two-dimensional. For
polyatomic molecules the energy function builds a
hypersurface. So our model remain the diatomic molecule.
Exciting an electron of the molecule it comes to a new
state. This is either an antibonding or a dissociative level. If
the electron comes to a lower state it may emit photon(s).
The probability transitions is determined above all by the
transition moment, the change in the dipole moment and the
symmetry of the ground and excited states are important.
The DS=0 selection rule is here valid, the group spin
quantum number must not change during the transition. This
is strictly valid only for ls coupling.
The following viewpoints are also acceptable for both
absorption and emission:
1. If the molecule is excited the molecule remains for the
longest time in position of the maximal displacement of
vibration.
2. The electronic transition is faster than the motion of the
atomic core. The atomic cores do not change their positions
during the excitation: Franck-Condon principle. If the
equilibrium nuclear distance do not change during excitation
(rg=ro), according to the Franck-Condon principle the 0 0
transition is the most probable (see figure). If the nuclear
distance increases during excitation (rg>ro), the 1 0 transitions
are favoured. In first case the positions of the maximal
displacements belong to the same nuclear distance. In second
case these positions are shifted and same position belongs to
other vibrational levels. Other electronic transitions have lower
transitional probability.
Effects of Franck-Condon principle
Types of electronic transitions
The principal types of electronic transitions are
- transitions between bonding and antibonding levels,
- d d transitions,
-charge transfer (CT) transitions.
Bonding to antibonding transitions are observable in the
UV (or in the VIS) region and they are possible between
levels (orbitals) of  electrons and non-bonded (n)
electron levels: *
 and *
n transitions. The next
figure is a simplified diagram.
Possible relative , , n, * and * levels, and the most
frequent transitions
E
*
*
n


The two transition types are distinguishable in solution
with the aid of the solvent effect.
The polar is the solvent the stronger decreases the
energy levels. The energy of the more polar level
decreases more. In the *  transitions the excited
state, in the * n transitions the ground state is more
polar. The *
 bands shift to the visible
(bathochromic shift), the * n ones to the ultraviolet
(hypsochromic shift). With increasing acidity of the
solvent the intensities of the * n bands decrease
(hydrogen bond formation) and at a given pH the bands
disappear.
Solvent effect of p*
p and p*
n bands
*
*
DE

*
DE
*
DE
DE
n

solvent polarity
n
solvent polarity
The d d transitions are in case of the transition
metal complexes important. The transitions are
possible between the splitted d levels The
corresponding bands appear in the VIS or in the
NIR regions. This kind complexes give the colours
of the solutions of the transition metal complexes.
The next figure is the electronic excitation
spectrum of the Ti H2O 3
ion in the aqueous solution
6
of TiCl3. The Ti3+ ion has one 3d electron and its
hexaquo ion is an octahedral complex. The
spectrum has an intense band at 20.300 cm-1 and
a weak shoulder at 17.400 cm-1. Since =20.300 cm1, Dq=2.030 cm-1.
1S
1G
d
2
3P
1D
3F
electron-electron
interaction
1A
1g
1A
1g
1T
1g
1E
g
1T
2g
3T
1g
1
Eg
1T
2g
3A
2g
3T
2g
3T
1g
crystal field
effect
3
Visible spectrum of the Ti H2O 6 transition metal complex
(octahedral symmetry)
The charge transfer (CT) transitions are possible if the
polarity of the molecule increases extremely during the
transition. The bond is very intense.
This is a result of an inter- or intramolecular electron
jump. The CT is frequent in excitation of transition metal
complexes (ion-ligand transfer) and also in simple
molecules, e.g. nitrobenzene. The next figure presents
the electron excitation spectrum of nitrobenzene. The
intense band at 39.800 cm-1 (251 nm) is a CT band.
CT band in the
electron
excitation
spectrum of
benzene
The vibronic transitions are mostly in the vapour spectra
observable. The figure shows electron excitation spectrum
of the benzene vapour. Beside the vibronic bands also
combinations and overtones appear. The 0 0 transition is
forbidden, but with low intensity appears.
The solvent spectrum
of benzene (solution
in n-hexane) contains
less vibronic bands.
Hot bands appear with increasing temperature: the
populations of the higher energy levels increase and they
may be also ground states for further excitations.
The electron excitation spectrum of the iodine vapour
contains also several overtones and combinations. In high
resolution also the rovibronic lines are observable.
Iodine
vapour
spectrum
in low and
high
resolution
The excited state and its decay
The molecule cannot remain in excited state for a longer
time (the excited state has a lifetime). It loses the received
energy either with the emission of a photon (spontaneous
emission, radioactive decay) or with nonradiative decay.
The way of the nonradiative decay may be different:
- energy transfer during collisions increasing the internal
energy (vibration, rotation) of another molecule,
- solutes may be interact with the solvent molecule
increasing its energy,
- a photodissociation process,
- the excited molecule acts as reactant in a chemical
reaction.
If the molecule emits after the photon absorption a photon
immediately, this is the phenomenon of the fluorescence.
The molecule absorbs a photon according the FranckCondon principle, then comes to the vi=0 level with
nonradiative decay and from this level it may arrive to the
vibrational levels of the ground state with fluorescence. The
absorption and the fluorescence spectra are mirror images of
each other.
The fluorescence is a spontaneous emission. Its frequency
is always lower than that of the absorption. The analytical
fluorescence indicators absorb in the UV or in the VIS region
near to the UV. The emitted light appears in the VIS. The
intensity of the florescence is often very high (e.g.
condensed aromatics). The fluorescence is very useful in the
detection and quantitative analysis of small quantities of
fluorescent substances beside non-fluorescent ones.
Fluorescence:
Since rg>ro, the 1 0
transitions is here
favoured.
The phosphorescence is a spontaneous light emission
with a delay of some seconds or minutes. It occurs if the
spin-orbital (jj) coupling is strong.
The multiplicity of the ground state is singlet (S), the
excited state is triplet (T). The potential energy curves may
cross each other in excited states. The molecule arrives
the excited state during absorption. From this state it loses
its energy in non-radiative decay until the cross-point of
the two curves. Here is the geometry of the two states the
same. From this point the molecule is in triplet state since
this has the lower energy. Continuing the non-radiative
decay the molecule arrives the vi=0 level on this curve.
Since the ground state is singlet, the excited state is triplet
and the S=0 selection rule is valid, the molecule is in an
energy hole.
Since this rule is valid strictly for ls coupling and here is
also the jj coupling important, the validity of the rule is not
so strong, the S T transition has a finite probability.
Electron spin resonance (see later) measurements found
such kind molecules paramagnetic. The phosphorescence
of solid substances may be very strong, see e.g. the
computer and television screens.
The process of
absorption and
phosphorescence
The excitation of the molecule may lead to its
dissociation. The next figure shows the process of
dissociation. The vibrational levels pile up in the region of
the long bondlength. There are two possibilities: either
the molecule arrives the horizontal part of the potential
curve or its excited state is dissociative (e.g. in the case
of diatomic molecules). A continuum is observed in the
spectrum in both cases since the molecular energy over
the level of dissociation is kinetic energy. The kinetic
energy, however, is not quantized. The compounds of the
noble gas elements have a dissociative state as ground
state and a bonding state as excited one (noble gas
excitations, excimer lasers, ).
Excitation with
dissociation
The excited state is
dissciative, like in case
of diatomic molecules
The electron excitation spectra and the substituent effect
An electron transition may be forbidden, however, the
corresponding band may appear in the spectrum if the
joining vibrational transitions are allowed. Their superposition can make the transitions allowed and one can find
the band in the spectrum.
The band intensities in the electron excitation spectra are
given with the molar absorption coefficient.
Dissociative ground level, excimer lasers
Predissociation
The electron excitation spectra and the substituent effect
An electron transition may be forbidden, however, the
corresponding band may appear in the spectrum if the
joining vibrational transitions are allowed. Their superposition can make the transitions allowed and one can find
the band in the spectrum.
The band intensities in the electron excitation spectra are
given with the molar absorption coefficient.
The integrated intensity of a band is
A   ( )d
band
The oscillator strength is a quantum chemical quantity
that also the characterizes the band intensity:
4m c
f  A e o ln (10 )  1.44  10  19  A
N e2
A
dm3 cm 1s  1mol 1
me is the electron mass, e is the absolute value of its
charge. The oscillator strength is proportional to the
square of the transition moment:
4m e  2
f
P
2 2
3 e
The average lifetime of the excited state is proportional to
the reciprocal of f. The energy levels of electrons and the
transition moments are calculable quantum chemically.
There are atomic groups in the investigated compound that
are responsible for the absorption in the UV or VIS regions.
These groups are the chromofors. The transitions may have
the forms *  and * n and
* n. If the electrons are
localized the corresponding bands fall into the high frequency
part of the UV region or in the far UV. Localized systems are
results of the hybridization.
If the highest occupied orbital (HOMO) and the lowest
unoccupied orbital (LUMO) come closer the band shows a
bathochromic effect (“red shift”). If these levels diverge, the
band shows hypsochromic effect (“blue shift”).
Bathochromic shift is a change of spectral band position of
a molecule to a longer wavelength (lower frequency).
Hypsochromic shift is a change of spectral band position of a
molecule to a shorter wavelength (higher frequency).
Characteristic bands of some chromofors in solution
The band positions are determined essentially by the
chromofors but the substituent effects and the solvents
have also influences on these.
The inductive effect is a direct electrostatic effect that
influenced the electron distribution in the molecule. The
reason is the dipole character of the substituent. The
character and strength of the effect depend on the
direction and value of the dipole moment. The electron
attracting groups show -I effect, the electron repulsing
ones +I effect.
The order of the groups with +I effect is
 CH 3  C 2 H 5  CH(CH 3 )2  C(CH 3 )3  S   O 
The order of the groups with -I effects is
 F   NO2  OH  Cl   NH2  Br  I  C  O  COOH  CN  SH   R3N
The groups containing free electron pairs but no double
bonds are called auxochromic groups if they join groups
with conjugated double bonds. Follow the displacements of
the electrons with -I and +I groups subtituted benzene ring.
X
X
X
-
-
-
+
+
-I
+
X
X
X
+
+
+
-
-
+I
-
The delocalized electron system treats into conjugation
with the groups containing double bond(s) or free electron
pair(s). This mesomeric effect results also electron shifts.
The order of the groups with +M effect is:
 F  Cl   Br  OH  OCH3   NH2  O
The order of the groups with -M effect is:
 NO2  CHO  COCH3  COOH  COO  CN  SO2NH2
The mesomeric effects on substituted benzenes can
follow on the next figure, it introduces the possible
mesomeric forms.
Here one can here folllow the formation of the
chinoidal structures and the shifts of the double
bonds in directions of the substituents.
X
X+
X
X+
X+
-
-
+M
O-
O
+
N
O-
O
+
N
-O
O+
-O
O-
-O
+
N
+
N
+
N
+
-M
+
O-
Investigating e.g. the substituent effect of aniline we
find +M > -I, the positive effect dominates. The NH2
group directs the next substituent in o- and ppositions, under +M effect the corresponding C atoms
have negative signs.
For nitrobenzene, however, the substituent effects
are -I, -M, i.e. the nitro group has a negative electron
effect. This means, the next substituent is directed into
the less negative meta positions.
A quantitative measure of the substituent effect is its
Hammett constant. This is the shift of the pKa value of
a substituted benzoic acid to the pKa of the benzoic
acid. The pKa is the negative decimal logarithm of the
ionization constant.
Measurement and application of the electron excitation
spectra
The UV and VIS spectrometers
instruments.
are dispersive
In the UV region the light source is in general a
deuterium discharging lamp. The optics is made of quartz,
the detector is a PMT with quartz window.
The light source in the VIS is a tungsten or a halogen
lamp, the optics is made of glass, the detector is a PMT.
Since several spectrometers work in both UV and VIS,
their optics is quartz and only the light sources and the
detectors are changed according the spectral region. They
are double beam instruments, the sample and the
reference are measured parallelly.
The recorded spectrum is the wavenumber or wavelength
function of the absorbance. Since the accuracy of the
absorbance is in these regions very good the UV and VIS
spectrometry is suitable and applicable for quantitative
analysis. Since there exist characteristic bands, also
multicomponent analyses may carry out.
An interesting application of these measurements is the
determination of the ionization constants. Recording the
spectra as function of the solvent acidity we get a series of
spectra. If in a given pH region the spectrum is constant this
is a spectrum of an ion. With the knowledge of the spectra of
the ions and their mixtures the ionization constant is
calculable:
An interesting application of these measurements is
the determination of the ionization constants.
Recording the spectra as function of the solvent
acidity we get a series of spectra. If in a given pH
region the spectrum is constant this is a spectrum of
an ion. With the knowledge of the spectra of the ions
and their mixtures the ionization constant is calculable:

XH  X  H

Ka 
A X  A H
A XH
pK a  lg A XH  lg A X   pH
Look at the acidity dependence of the pyrazine
spectrum. The spectra cross each other in the same
point. This is an isobestic point: at this wavelength is
the absorbance independent of the solvent acidity.
Acidity
dependence of
the pyrazine
spectrum
Ultraviolet photoelectron spectroscopy (UPS)
The photoelectron spectroscopy is based on the inelastic
scattering of a particle with particle change. The high
energy photon collides with the molecule in high vacuum
and ionizes it. If the energy of the photon is greater than the
ionization energy (I) of the molecule the difference appears
as kinetic energy:
1
2
h 
2
mev  I
The UV photons may have energy for the external
ionization, for internal ionization X-ray photons (XPS) are
necessary. Since the electrons are situated on different
orbitals there are several ionization energies. Besides,
also the
vibrational states may parallelly change.
Therefore
h 
1
mev2  I  DEv  DEr
2
v labels the vibrational, r the rotational energy.
The changes in the rotational energy are very small in
comparison with I, the spectrometers cannot resolve these
changes. If the vibrational state does not change during the
ionization we speak about adiabatic ionization, all other are
vertical ionizations.
According Kopman's theorem the absolute value of the
ionization energy is equal to the orbital energy of the emitted
electron. This is a good approach and this is the basis of the
interpretation of the photoelectron spectra.
The light source is a He discharging lamp, its 20.21 eV line
is used. The analyzer separates the electrons according to
their velocity, the detector is a special PMT.
The first derivative of the intensity - electron energy function
has "spectrum like" shape. The figure shows the UPS
spectrum of nitrogen completed with its XPS spectrum. The
interpretation was based on Kopman's theorem.
The dispersion of light
The dispersion of the refractive index
The dispersion of the light is the frequency dependence
of the refractive index (n), that is the ratio of the light
velocity in vacuum (c) and in a medium (v):
c and
n
v  
v
where  is the frequency and  is the wavelength of
light. The refractive index is a function of the relative
permittivity (εr) and the relative permeability (μr) of the
medium:
n  rr  r
Let z the direction of the light and x the direction of
the transversal elongation of the electric field vector E,
the electric wave has the form (i is the imaginary unit,
Eo is the amplitude)

 nz 
E  E o exp i 2 t  
c 


If the light penetrates into the medium, the wave
amplitude decreases exponentially, therefore
n z


 nz 
E  E o exp   2 k  exp i 2 t  
c 
c 



nk is a damping factor.
The complex refractive index is n̂  n  in k
Comparing this result with the Lambert-Beer law
nk 
c ln 10

~
4
c is the concentration of the solute in the solution. The
imaginary refractive index is proportional to the molar
absorption coefficient ()and depends on the light frequency.
The real part of the complex refractive index depends
also on the frequency. Let the frequency of the absorbed
light o, and let Dpo the amplitude of the light induced dipole
moment, then a good approach is (a is a constant ):
Dpo 
a
2o  2
According considering the definition of molar polarization,
the molar refraction is (r was substituted by n2)
RM
n2  1 M
 2
n 2 
the dispersional formula for n is
Ci A i
n 2 1 N


3 i  o2 ,i   2
n2  2
(sum over all components)
N is the number of particles in unit volume, o labels the
absorption maxima, Ci's are constant, Ai is Einstein's
absorption probability that is proportional to the square of
the transition moment:
Pi2
Ai 
6 0 2
The next figure shows the dispersion curve of the real
refractive index. If the refractive index increases with the
frequency the dispersion is normal, if it decreases (at the
absorption maxima) it is anomalous.
The shaded areas indicate the corresponding absorption
bands.
Electron excitation with polarized light
A molecule (a substance) is optically active if it rotates the
plane of the linear polarized light. We regard this light as the
resultant of two counter-rotating circularly polarized lights (L
and R) having different velocity in the medium (see figure).
Therefore the refractive indices are different, the plane of
polarization rotates, the substance is birefringent. The angle
of rotation characterizes the effect:
D  2 l
nL  nR

Here if nL>nR, the substance rotates to right, l is the
length of the sample.
The specific rotation is
   D
lc
c is the concentration.
The definition of the molar rotation is
M  103 M 
M is the molar mass.
Angle of rotation
The optical rotatory dispersion (ORD) is the frequency
dependence of the optical activity. If the rotational angle
decreases with the frequency in the region of the
anomalous dispersion of the optical activity, this is the
positive Cotton effect, and the opposite case is called
negative Cotton effect.
The circular dichroism (CD) is observable if the
molecule absorbs the L and R circular polarized light
with different intensity. These are the cases of the
asymmetric carbon (very often) and nitrogen (rarely)
atoms and some asymmetric transition metal
complexes. The measure of CD is the difference of
the two molar absorption coefficients:
D   R   L
The CD spectrum is the frequency dependence of .
The form of the spectrum is affected by the chromofors,
and also by the substituent effects of other groups.
Both ORD and CD give information about the
symmetry centers of the molecule. The ORD gives
information from all these centers, the CD only from the
environment of the chromofors if there are optical active
centers in it. Both methods help in the determination of
the absolute spatial configuration of molecules.
ORD and CD spectra are recorded on spectropolarimeters. See nex figure! It presents the ORD
spectra of molecules with negative (dashed) and
positive (full) Cotton effect.
Positive Cotton effect
Structure
dependence
reflects in
ORD
CD spectra of
molecules with
negative (dashed)
and positive (full)
Cotton effect. The
different configuration is clear
reflected in the CD
spectra.
Mass spectroscopy (MS)
The principle and instrumentation of mass spectroscopy
The mass spectroscopy is based on the ionization of the
molecules. At first it was applied only for the separation of the
atomic isotopes already in the second decade of our century.
It has two important applications:
- the structure elucidation of the molecules,
- in the chemical analysis, coupled to chromatograph (GCMS).
The mass spectrometers have three principal components:
the ionizator, the analyzer and the detector.
The most popular method of the ionization is the electron
impact (EI). The electrons are produced through thermal
electron emission (W, Ta or Mo cathodes) and accelerated by
an electric field. The kinetic energy of the electrons depends
on the strength of this field.
The measurement is realizable only in high vacuum (10-410-6 Pa). This vacuum is necessary for the collision of the
electrons with the molecules.
Low energy electrons only ionize the molecule, electrons
having higher energy cause the dissociation of chemical
bonds and ionize the new particles, the fragments. This is
the process of fragmentation. The fragments characterize
the molecule.
There are multi-ionizations possible, fragments may have
negative charges or they may be also neutral.
In the process of chemical ionization (CI) the first step is
the yielding of a high energy gas plasma (CI plasma) with
electron impact. As plasma gas mostly ammonia, isobutane
or methane are used. During the CI both positive and
negative ions are produced. The pressure in the ionization
chamber is about 10 Pa.
The fast atomic bombardment (FAB). The molecules are
bombarded with fast Ar or Xe atoms or Cs+ ions. This is a
soft ionization: only positive ions are yielded, fragments are
not produced.
Several other ionization methods are elaborated, some of
them escpecially for the GC-MS coupling.
The most important analyzer types apply either electric
and/or magnetic fields for separation of the ions.
The magnetic mass analyzer is based on the effect of the
magnetic field (B) on moving ions: the field B forces the ions
on a ring path.
Magnetic mass analyzer
Let the ion move with velocity v, mass m and charge z
on a ring with radius r. The ions are accelerated with an
electric field U. The force forced the ion on the ring path.
This force is equal to the centripetal force:
mv 2
zvB 
r
In this way the potential energy of the ion is transformed to
kinetic energy:
2
mv
zU 
2
The mass belonging to unit charge (i.e. the absolute
value of the electron charge) is therefore
B2 r 2
m/ z 
2U
If both U and B are constant the ions with charge +1 (z=e)
come to different orbitals according their masses. The
detector is either photo plate, film or PMT. If the detector is a
PMT the field strength B is changed to forced each ion one
after the other to the same orbital. Of course, in this case the
signals of the ions appear separately, each after the other.
The resolution of the instrument (m/Dm) is about 5.000, the
upper limit of the measurement is m/e=1.500 Dalton/charge
unit.
The double focused spectrometer corrects the uncertainty in
the ionic velocity (Boltzmann distribution). Energy and pulse
filtrations are applied here. Therefore the resolution increases
to over 100.000, the limit to m/e= 50.000.
The quadrupole mass
spectrometer consists of four
parallel metal bars.
Opposite bars are equally,
neighbouring are differently
charged (+ or -) under dc.
voltage U. An ac. voltage
(V)is superimposed on U
having frequency .
The particle moves between the bars parallel to them. If
its velocity is enough to across the system during a half
frequency period, it leaves the system. The mass limit
depends on the frequency:
m/z
5 .7 V
4 2  2
This is a typical mass filter. The resolution is low, only 1,
the mass limit is m/e=4.000. It is very often applied in GCMS measurements.
The time-of-flight analyzer (TOF) accelerates all ions with
the same U dc. voltage. The time of flying of the distance s
depends on the mass m / z  2 U t 2
s2
The time-of-flight is measured. The resolution is about
10.000, the mass limit is m/e>200.000.
The ion-cyclotron mass spectrometer (ICR-MS) or Fourier
transform MS (FT-MS) is based on the effect of field that B
forced the ions on ring orbitals. Irradiating these ions with a
wide range radio frequency perpendicular to B, they receive
selectively energy from the radiation. Therefore the orbital
radius increases and the rotation become in phase with the
radiation. The result is an induced voltage that can be detected
with the electrodes standing perpendicular to the direction of
both B and the irradiation. The Fourier transform of the
detector signal (free induction decay, FID) gives the mass
spectrum. For small fragments the resolution is very high (over
106) but increases with increasing mass.
Ion-cyclotron mass
spectrometer
The tandem mass spectrometers consist at least of two
coupled mass spectrometers (MS/MS). An ion is selected
from the fragment ions of the first MS and is lead to the
second one. Here a second fragmentation can follow for the
better identification of the primary fragment. This can be
important in the environmental analysis and in metabolism
studies.
Applications of the mass spectroscopy
The following peak types may appear in a mass spectrum:
- molecule peak M/e, if the field energy is at least equal to
ionization energy of the ion;
- fragment peaks: they indicate the fragments of the
molecule ion but as results of possible rearrangments in the
molecule peaks of groups may also appear some those are
originally not present in the molecule;
- multi-peaks: they belong to m/z (z>e) and give the
possibility of measuring molecules having very high molecular
masses;
- metastable peaks: the peaks of ions having shorter lifetime
than the time-of-flight from the ion source to the detector. If
an ion with mass m2 is formed from the ion with mass m1 a
diffuse peak appears at
m2
m* 
2
m1
An important application of the mass spectroscopy is the
chemical structure elucidation. The formation of the
predominant part of the fragments is interpretable with
chemical reactions since during the reaction functional
groups split from the molecule. In this way we can draw
conclusions on the structure of the molecule. The first of the
next figures presents the mass spectrum of 2methylpentane, while the next one is the mass spectrum of
its isomer the n-hexane. Both molecules have the same
molecule masses, 86 Dalton.
The peaks m/e=71 and m/e=43 dominate in the mass
spectrum of 2-methylpentane. The first peak shows the
splitting of the molecule into a methyl group (m/e=15) and an
n-amyl group after realignment (m/e=71). The second peak
refers to the splitting of the molecule into two equal parts: two
n-propyl ions (2x43). Both spectra contain also the peak of
the ethyl group (m/e=29).
The peaks m/e=43 and m/e=57 dominate in the mass
spectrum of n-hexane. The first peak shows the splitting of
the molecule into two n-propyl (CH3-CH2-CH2) ions:
43+43=86, the C3-C4 bond splits in this case. Besides, the
stronger m/e=57 peak shows the splitting of the molecule
into an ethyl group and an n-butyl group (86-29=57).
Mass spectrum and
decompositon
rections of thiophene
The ionization energy is determinable using mass
spectroscopy. In this case electrons with very low energy
dispersion are necessary for the ionization (soft ionization).
So the molecule peak becomes more intense. Also
dissociation energies, heats of formations of ions and
radicals are determinable. The mass spectrometers give
always vertical energies (the vibrational state of the
molecules changes during the ionization). Since also
negative ions are produced in mass spectrometers the
electron affinities are also determinable.
The MS has the advantage of high sensitivity and the
fastness. Its drawback is that it is only in gas phase
applicable. The newer ionization methods: electrospray,
thermospray,
MALDI
(matrix-assisted
laser
desorption/ionization, soft ionization) and sample preparation
methods remove this drawback. These methods expanded
the applicability of the MS to substances with high molar
masses: MS is very often used in biochemistry (biopolymers)
and in the plastic research (synthetical polymers).
High resolution (m/m>1,000) mass spectra are used in the
environmental analysis for the determination of the isotopic
composition of the compounds. This analysis give possibilities
to determine the origin of the substance, e. g. gold. The
technology
control
of
medicament
production
by
contamination analysis can unfold falsificatons or frauds.
Paramagnetic properties of molecules
Paramagnetic molecules
If the spin and magnetic moments are not balanced, the
molecule shows paramagnetism. From the viewpoint of
chemistry the unbalanced spin magnetic moment is important.
Free radicals, triplet state molecules, transiton metal
complexes with unpaired d electrons are paramagnetic.
A good example of the paramagnetic compounds is the
oxygen molecule having two unpaired electrons, i.e. it is a
biradical. The high spin complexes are examples of
paramagnetic transition metal complexes.
Radicals are formed during several chemical reactions (e.g.
polymerization). The measurement of the paramagnetism
helps in the investigation of reaction mechanisms (the process
is of radical or ionic mechanism).
The paramagnetic balance is used for the paramagnetic
measurements. This is a very sensitive balance. An external
magnetic field (electromagnet) acts on the paramagnetic
substance. This force is compensated through an other
electromagnet that acts on an iron bar. The necessary current
is measured.
The result is suitable to decide about the paramagnetism of
the substance but it does not give information about the
reason of the paramagnetism.
This is only possible with application of the method of
electron paramagnetic (electron spin) resonance.
Electron paramagnetic resonance (EPR)
Electron spin resonance (ESR)
Both names are used but considering the physical reasons of
the effect, EPR is better.
The essence of the magnetic resonance is the following. The
molecule, having a magnetic moment m and precessing with
angular velocity  around the constant magnetic induction
vector B, is irradiated in the direction perpendicular to B with
an electromagnetic wave having frequency .
The two, originally degenerated electron spin levels split
under effect of the field B. The spin magnetic moment in the
favoured direction z may be either -B or +B (B is the Bohr
e
magneton,
 
B
2me
Since the energy of the dipole in magnetic field is -mB, the
splitting of the two levels is
DE   B B  (  B B)  2 B B
E
E
m s = +1/2
m = +1/2
s
+ B
B
h
0
m = -1/2
s
- B B
B
m = -1/2
s
If this energy difference is equal to the energy of the radio
frequency photon then the molecule absorbs it (resonance):
DE  h  2 B B
The selection rule is Dm s  1
The molecule is excited from the lower (ms=-1/2)
to the higher (ms=+1/2) state. The induction B and
the frequency are proportional
B
h

2 B
We have absorption if only the magnetic transition
moment is not zero and its direction is the same then
that of the external B.
If any atom has paramagnetic nucleus the magnetic
field of the nuclear magnetic moment is added to or
subtracted from the field B. Therefore the original spin
levels split if the unpaired electron has finite density on
the place of the nucleus. If the magnetic moment of the
nucleus has two possible values (e.g. hydrogen) the
number of the levels will be doubled.
Splitting of the electron level under the effect of nucleus and
possible absorptions
E
E
m = +1/2
s
M =+1/2
I
m = +1/2
s
M = -1/2
I
h
M = -1/2
I
0
m = -1/2
s
M =+1/2
I
m = -1/2
s
M +1/2
I
-1/2
B
The nuclear magnetic moment decreases the external field:
B  aMI 
h

2 B
a is the hyperfine coupling constant, the MI is the
nuclear magnetic quantum number. The possible values
of MI are for the hydrogen atom  1/ 2 . The real situation
is more complicate. The external field indicates a counter
field. The local field
B'  (1  )B
interacts, <1 is the shielding factor. Therefore in reality
DE  h  2 B B'
and
B' 
h

2 B
Flow chart of an ESR spectrometer
K
D
M
S
M
C
The sample (S) is positioned between the poles of a
constant magnet (M). The value of the field B is changable
through changing the electric current in the coil C. The
radiofrequency source (about 10 GHz) is a klystron (K)
producing fixed frequency and the magnetic field is changed.
D is the radiofrequency detector
Molecular structure
and ESR spectrum
of the free radical
ion of 1,4benzoquinone in
solution
An unpaired electron is on both oxygens. The structure is
symmetrical (point group D2h). 16 possible configurations exist
for the four hydrogen protons: 1 configuration with MI=2, 4 with
MI=1, 6 with MI=0, 4 with MI=-1 and 1 with MI=-2, i.e. like nk ,
k=0,1,...,n. The number of possible configurations gives the
statistical weight (relative intensity) of the levels. The relative
intensities are well seen on the figure.
()
The nuclear spin effect and the hyperfine structure must
be considered at the interpretation of the ESR spectra.
Since not all nuclear spin quantum numbers have the
value 1/2, one has to act also on the number of splitted
levels. E.g. 14N has the nuclear spin quantum number +1,
therefore three projections on the z axis are possible, MI =
1, 0 or -1.
The most important application of the ESR spectroscopy
is the discovery of free radicals. Since in reactions with
radical mechanism the radicals are very active, their
concentrations may be very low. Nevertheless their
discovery is important since it gives information about the
reaction mechanism. Therefore the ESR spectroscopy is
important in the investigation of polymerization and the
oxidative reactions (through peroxides) of conjugated
compounds.
A great deal of free radicals is formed under high energy
irradiation (X ray or g ray) of solid substances. The
lifetimes of the free radicals increase at low temperatures
embedded in solid matrices since their mobilities
decrease. The matrix, however, influences the spectrum.
Free radicals are formed in catalysis on the surface of
the catalyst. Similarly, free radicals are formed also in the
fermentation processes.
Free radicals are coupled to the molecules in biological
investigations. From differences between ESR spectrum of
the "spin-marked" and the free molecule conclusions can
be drawn about the structure of the molecule.
At fluorescence processes triplet states exist and in
homogenous magnetic field the energy levels split
according the spin quantum numbers. Three levels
appear: , and , . The formation of these levels was
proved by ESR.
Nuclear magnetic resonance spectroscopy (NMR)
The nuclear magnetic resonance
An important part of the atomic nuclei have magnetic
moment. In case of odd atomic mass (A) this magnetic
quantum number (I) is odd mutiple of half. If A is even
and the atomic number (Z) is is odd, the magnetic
quantum is integer. If both A and Z are even, the
magnetic moment and also the I nuclear spin quantum
number are zero.
Properties of some nuclei applied in NMR spectroscopy
The z component of the nuclear magnetic moment is
M I, z  g a M N
M  -I, - I  1,..., I  1, I
M is the magnetic nuclear quantum number; its values depend
on the possible values of I, altogether 2I+1 different values are
possible; ga is the Landé factor of the nucleus; µN is the
nuclear magneton.
The E energy of the nuclear magnetic moment is in a
magnetic field B
E  M I B  g a  N MB
The selection rule is deducable from the expression of the
magnetic transition moment
DM  1
and therefore
DE  g a  N B
The energy transition is possible with the absorption of
an electromagnetic wave having equal frequency to the
Larmor one. This is a resonance. The resonance
frequency is
ga
0 
B
2
ga is the gyromagnetic ratio of the nucleus
The splitting of the nuclear levels in a magnetic field B.
In case if a molecule is placed in a magnetic field B then
the chemical environment (the electron cloud) shields the
magnetic field, and a so-called local field acts on the
nucleus. This change is
B  B
 is the shielding factor. The shielding factor is positive
in diamagnetic environment (this is the general case).
The diamagnetic field is generated by an external field.
DB  DE  Dp  DT  Dl  Dm
The local magnetic induction will be
Bl  B(1  )
Therefore the Larmor frequency changes to
ga
ga

Bl  (1  ) B  (1  ) r
2
2
The nucleus, however, absorbes at r frequency. There are
two possibilities for solving this problem. Either, the
frequency will changed to a value where Bl corresponds to
r, or the external B induction will increased up to a  that will
be equal to r.
The 0 frequency is not measurable since a nucleus without
environment does not exist. Therefore it was looked for a
molecule that produce extremely large electron cloud, i.e.
extremely large diamagnetic field around the nuclei during
the measurement. The molecule tetramethylsilane (TMS)
has this property. The diamagnetism of the electron cloud
around the protons exceeds the diamagnetism of almost all
hydrogens connected to carbon atoms. The signal itself is
very intense since it is produced from 12 protons in similar
position. The TMS is applied as reference not only in 1H
measurements but also in 13C ones. For 31P measurements
the 85% aqueous solution of phosphoric acid is used.
The chemical environment of the nucleus can be
theoretically characterized by the frequency shift (D) that is
necessary for shifting the constant frequency (s, the
standard reference frequency, e.g. that of TMS) from the
signal of the investigated proton to s. A relative frequency
scale is applied in the practice since (this is usually some
tesla):
D
6   s
  10
 10
0
0
6
Here 0 is the high frequency of the NMR spectrometer,
some hundred MHz. The chemical shift () increases with
increasing paramagnetism (practically with decreasing
diamagnetism) relative to the reference (TMS), i.e. the
local field increases with . Increasing 0 decreases .
There exists also an other scale, with opposite direction
  10  
The 1H chemical shifts depend on the solvents. Very strong
electronic effects can generate chemical shifts over 10 ppm.
Spin-spin interaction
The magnetic fields of the spins interact with one another in
external magnetic field (similar to EPR)). The energy levels of
the nucleus split on the effect of the other nuclei of the
molecule. Diatomic (AB) system (B nucleus act on A):
If several equvalent nuclei act on a nucleus A (ABn
system) the splitting depends on the relative positions of
the spins of the nuclei B.
The weight of the states (degree of degeneration) is
n
determined for the k-th level by  k  . 1H-NMR spectrum of
 
ethylbenzene (very often also the integrated intensities
are presented.)
. One can observe on this figure three line groups. The ratio of
the total intensities of the line groups is 5:2:3. Since the signal
intensities of all the protons is equal, the first very intense line
belongs to the five protons of the phenyl group.
The quartet is the signal of the methylene group, while the triplet
belongs to the methyl group. According to the spectrum there is
not coupling between the protons of the phenyl group and the
other protons of the molecule under the condition of the
measurement. The chemical shifts of the phenyl protons are
approximately the same.
Looking at quartet of the methylene group it reflects the effect of
the methyl group. The effect on all protons of CH3 are equivalent
(AB3 system). Therefore n=3, and so the relative intensities of the
lines are 1:3:3:1.
The coupling of the methylene and methyl group is mutual.
Therefore we have an AB2 system, the CH2 group action on the
CH3 one. Therefore the line intensity ratio for the methyl signal
group is 1:2:1.
The freqeuncy shift affected by the spin-spin coupling is
independent of the applied frequency and magnetic
induction. However, applying an NMR spectrometer of
higher 0 frequency, the frequency shifts seem smaller on
the  scale.
The interactions between the different moments and
magnetic moment are characterized with coupling constants.
The interactions between the nuclear spins are represented
with spin-spin coupling constants. For an AB system they are
symmetric, JAB=JBA. Their unit is Hz. The energy of such a
coupled system is in case if the chemical shifts are higher
than the splitting caused by the spin-spin coupling is
h
E  E 0  g a  N B (1   i )M i   J ij M i M j
2 i j
i
E0 stands for the energy of the magnetic field
without (chemical) environment. If this energy
equation is valid, the selection rules are for an ABn
system
DM A  1
DM B  0
i
i  1,2,..., n
If the chemical shift and the spin-spin coupling are
commensurable, the selection rule is
D M i  1
i
Considering the selection rules, the frequency of the
magnetic transition will be
g a  B B(1   i )
i 
  J ij M j   i0   J ij M j
h
j i
j i
the zero superscript refers to the state without spinspin coupling.
The closer the coupled nuclei the greater the spin-spin
coupling constant.
The calculation of the splitting is relatively easy for an AB
system if the mentioned energy equation is valid (otherwise
the calculation is very complicate). Let
D   0A   0B
   0A   0B
J  J AB
q  J 2  D2
The possible energy levels are under these conditions:
The observable frequencies are ( DM A  1
DM B  1 ):
1
1
 A1  (q    J )
 A 2  (q    J )
2
2
1
1
( q    J )


 B1  ( q    J )
B2
2
2
If D  J , then q  D , if D  0, then q  J . The first
case results two doublets, the second one results
triplet with the intensity ratio 1:2:1 The line intensities
are in general case
2qJ
I 1
q2  J2
the upper sign refer to the two inner lines, while the
lower one to the two outer lines. The next figure
intoduces the positions of the lines and their intensities
as function of J/D, in case of two non-equivalent nuclei.
The value of the J coupling constants does not exceed 25
Hz in 1H-NMR spectroscopy and can have both positive and
negative sign. However, in case of other nuclei it can be
even 1000 Hz. The values of the coupling constants (like the
chemical shifts) are found in the special NMR literature or
on the internet.
The nuclei those are symmetrically equivalent are called
chemically equivalent.
Magnetically equivalent are the nuclei those have only one
type of spin-spin coupling constants with their vicinal
groups.
Nuclei are isochronic, if their chemical shits are identical.
The chemically equivalent nuclei are at the same time also
isochronic, however, the magnetic equivalent atoms are not
by all means.
The difluoromethane (CH2F2) has two 1H like the two
19F nuclei are isochronic and chemically equivalent. All
JHF coupling constants are identical.
In 1,1-difluoroethylene the 1H atoms are chemically
equivalent, similarly to its 19F atoms. However, they are
magnetically not equivalent, since the 1H atoms have not
the same coupling constants with the 19F atoms in Z and
E positions.
13C-NMR
spectroscopy
The 13C resonance has some specialities in comparison
with the 1H-NMR spectroscopy.
1. The natural abundance of the 13C nucleus is low,
therefore the signal is small and 13C -13C coupling does
not exist.
2. The magnetic moment of the 13C nucleus is small,
therefore its relative NMR sensitivity is also small.
The 13C-NMR measurements became popular only with
the appearance of the Fourier transform NMR
spectrometers (see later). The 13C-NMR measurement
has some advances in comparison to the 1H-NMR ones.
1. It gives direct information about the carbon skeleton of
the molecule.
2. The spin-spin coupling does not disturb the spectrum,
the inductive and mesomeric effects are easier
observable.
3. The 1H-NMR spectrum is confused in the case of large
molecules with several hydrogen atoms, the 13C-NMR
spectrum gives separated signals of every carbon atoms.
4. If the molecule does not contain hydrogen atoms this
spectrum gives even in this case structural information.
5. 13C-13C coupling does not exist, the 13C-1H couplings
are easily eliminable using the method of the wide range
double resonance (spin decoupling). The 13C neighbouring
protons are irradiated with a second wide range
radiofrequency radiation in the region of the proton
absorptions (saturation). As result the multiplet bands
become singlets.
13C-NMR
spectrum of imidazole. The signal of the carbon
atom 2 is the line 1, the atoms 4 and 5 are equivalent
(there is a fast proton exchange between the two
tautomers), their signal is denoted by 2. The spectrum was
reorded with spin decoupling.
The 13C-NMR spectrum of ethylbenzene. In contrary to
its 1H-NMR spectrum it consists of singlets only. The
assignment of the lines is the following. 1: C1, 2: C2 and
C6, 3: C3 and C5, 4: C4, 5: CH2, 6: CH3. The +I effect of
the ethyl group is well observable on the shifts of the
benzene carbon lines (for benzene C=128.9 ppm).
The lines in ethylbenzene spectrum have different
intensities. The intensities depend on the difference in the
populations of the ground and excited states and their
relaxation time (T1) is decisive.
The next table contains some 13C-NMR shifts. The shift
increases with increasing polarity of the functional group
(similarly to the 1H-NMR shifts). Extraordinary high is the
shift of the 13C in -COOH and =C= groups.
Some
13C-NMR
shifts in functional groups
Recording NMR spectra
The early instruments applied the continuous wave
(CW) technique. They used fixed radio frequency (RF)
and the magnetic induction (B) was changed. In this way
the individual transitons were measured each after the
other.
The new instruments apply the Fourier transform
method (FT-NMR), i.e. excitation with pulses + Fourier
transformation (PFT). The nuclear spins precessing in a
field B in direction z with frequency o are excited with a
wide range (D) RF radiation for a short (some seconds)
time tr. The pulses are repeated in a period time tp.
The high frequency pulse creates a rotating field Bi, the
effective field at the nucleus is Beff. This induction turns
the magnetic moment from z to y direction (see figure).
The detected signal is the y component of the magnetic
moment M, My. This signal contains all the possible
frequency components - depending on the type and
position of the nucleus in the molecule. During the decay of
M the y component decreases.
The signal detected in time is called free induction decay
(FID). Ceasing the pulse the magnetic moments relax and
My trend to zero. The meaurement is repeated to get a
better signal-to-noise ratio (accumulation of the signals).
The NMR samples are overwhelming liquids, solutions.
The solvents are in 1H- and 13C-NMR spectroscopy
deuterated liquids. Solid state NMR measurements are
also possible.
a: Induction B turns the magnetic moment from z to y
b: During the decay of M the y component decreases
c: Signal detected in time: free induction decay (FID)
FT-NMR spectrometer: radio frequency source (RFS), pulse
amplifier (PA), pulse programmer (PP), superconducting magnet
(SM), radio frequency receiver (RFR), memory for accumulation
(MA), computer for control, data acquisition and data elaboration
(PC) and plotter (P). The applied RF is some hundred MHz.
RFS
PA
PP
RFR
SM
sample
MA
PC
P
The Overhauser effect
The NMR spectroscopy is a very effective, flexible and
versatile method. One of its special methods is the mentioned
wide range double resonance for spin decoupling. Some other
methods will be introduced in the next sections.
The essence of the nuclear Overhauser effect (NOE) is the
following. If we saturate a nucleus (turn its magnetic moment)
this effect accelerates the relaxation processes of nuclei
coupled to it. Therefore the resonance line of the second
nucleus becomes sharper, it may absorb more energy from
the Bi field without getting saturated. Such kind nuclei are e.g.
1H and 13C.
It is important in case of NOE to eliminate all external
paramagnetic contaminants from the sample (e.g. dissolved
oxygen).
The 13C-1H coupling offers four possibilities for the spins (
labels the ground state,  the excited one): 1: , 2:, 3: 
and 4:. The spins of the protons are saturated through
double resonance and therefore the originally forbidden 4 1
and 3 2 transitions become allowed with P41 and P32 finite
probabilities. Let be the probabilities of the originally allowed
transitions Po. The intensities of the 13C resonance lines
increase according to the equation
I NOE

P32  P41
gH 
  I(1  )
 I1 
 P32  P41  2Po g C 
I is the original intensity, g is the magnetogyric ratio, 
is the NOE factor. If the two magnetogyric ratios are of
different signs the intensity decreases.
Relaxation processes
The FT technique gives the possibility of the determination
of the relaxation times of both the individual nuclei (T1) and
the chemically equivalent nuclei (T2). These relaxation times
give information about the chemical structure of the molecule
beside the chemical shift, the coupling constants and the line
intensities. Since the carbon nuclei are in closer connection
with the skeleton of the organic compounds, their relaxation
is of higher interest.
The relaxation is the result of the local magnetic fields
induced by the disordered molecular motions. From the
viewpoint of NMR only the frequency components in
magnitudes of some MHz to some hundred MHz are
important. The disordered motions are characterizable by
their correlation times (c).
The correlation time (c) is the average lifetime of a type
of motion. For translations this is the average time
between two collisions, for rotations the time of a turning.
The correlation times of small molecules are in the
magnitude of 10-12 - 10-13 s, for medium size molecules
(100 - 300 Daltons) 10-10 s. The corresponding
frequencies are the reciprocals of these values.
We shall deal with two types of relaxation: the spinlattice relaxation and the spin-spin relaxation
Spin-lattice relaxation (longitudinal relaxation).
The motions in the environment (in the "lattice") of the nuclear
spins influence in the field B the saturation of the levels
splitted according to the magnetic quantum number M. If M
has only two possible values (+1/2 and -1/2), i.e. I=1/2, the
differential equation of the relaxation process is
dn
 2P(n e  n )
dt
n  N  N
N+ is the population of M=+1/2 (the lower) level, N- is that of
M=-1/2, the subscript e refers on the equilibrium, P is the
average probability of the processes there and back, the
coefficient 2 refers to the fact that whenever a spin turns,
the difference in the populations changes with 2. The
solution
n e  n  (n e  n )o
 t 
exp   
 T1 
T1 
1
2P
T1 is the time constant of the spin-lattice relaxation.
The nuclear spins turn in the inducing field Bi during the
pulse time tr with the angle
g a Bi t r

2
The amplitude of the pulse is gaBi/2, the angle of the
rotation depends on the pulse length tr , ga is the
gyromagnetic ratio of the nucleus.
Spin-spin relaxation ( transverse relaxation).
The precessions of the chemically equivalent nuclei may
induce vibrating magnetic fields of one other at their places.
The frequencies of the fields are equal to the Larmor
frequencies of the inducing nuclei. The result is the mutual
change in the direction of the nuclear magnetic moments
relatively to the polarizing field B. Their energies, however,
remain unaltered.
Exciting the nuclear spins with short Bl magnetic pulses
their phases become coherent (all spins are of equal both in
values and directions). This state remains only for a short
time, the spins relax with a time constant T2. The local fields
affect on this process therefore the measured effective time
constant is
1
1 ga
(B  B  )


*
T2 T2 2
The relaxation causes line broadening, the average
FWHH is
1
*
D 1 / 2 
T
2  T2  T1
*
T2
Measurements of the relaxation effects
1.The measurement of the spin-lattice relaxation
The relaxation is influenced by several facts:
- the viscosity (temperature and concentration effect),
- the paramagnetic compounds (e.g. oxygen) must be
eliminated,
- molecular diffusion from the "effective area" into the
ineffective one and vice versa,
-the evaporation (T1 of vapours is smaller, low temperature
is necessary).
The measurements are carried out periodically. The
periods are called sequences. The description of the
sequence means the detailing of the period.
The inversion recovery technique
The sequence is: 180o –  - 90o. This means: the first pulse
turns the nuclear spins 180o from the direction of the vector B
(a). After the pulse, time  is the waiting period. During this
time the spins relax partly into the starting direction. After this
time a pulse in direction x turns the remained z component of
the magnetic moment in direction y (b) and so it becomes
measurable (My). Changing  we get the decay curve (c).
If <T1, the FID is negative, if the FID=0, that is the
halftime =T1*ln2, at 5T1 the starting state is practically
restored (Me=Mz).
A modification of this technique is the following. It uses
sequence 90o – t - 180o -  - 90o - t. The spins are turned
first 90o in direction y(Me) around x, the spins are turned
180o to –y around z, then changing the relaxation time 
(M), the Me signal is turned 90o from z to y. The difference
of this signal M and the equilibrium signal (Me) is
measured, both are in y direction. The difference signal
starts from 2Me and finishes at zero.
Look at the repetition of the sequence for measuring the full
relaxation curve! The relaxation of the different 13C-NMR
peaks are different, see the different time-intensity
dependences of the peak lengths.
2. The measurement of the spin-spin relaxation
We deal only with the spin-echo technique, in detail. Its
sequence is 90o -  - 180o - (echo) - td. The spins are
turned 90o in direction y (a). During the time  the signals
spread in the plane xy since their precession frequencies
are greater or smaller than the nominal one (b). A 180o
pulse turns the spins in the xy plane around the x axis.
During this second  time the spins come closer each to
the other, this is the echo (c). After td>5T2 the starting
state will be restored. Changing  the FID will be
measured. The signals change with exp ( 2) / T2* . The
relaxation time is 2 because of the echo.
a: turn to y,
b: speading
c: echo
Two-dimensional NMR spectroscopy
The principle of the two dimensional (2D) NMR technique
is the change of two different properties (two different time
scales exist) during the measurement. Therefore two
different Fourier transformations are possible and two
frequency domains are yielded.
1. J, spectroscopy
The measurement consists of three sections. During the
preparation the nuclear spins are turned around x into y with
a 90o pulse. During the evolution the spin system changes
under the effect of different factors: spin-spin relaxation,
inhomogenity of the external magnetic field, Larmor
precession, spin-spin coupling. The first two factors
influence the line width.
Line width: D 1 / 2
1

T2*
T2*  T2  T1
This method is a 2D spin-echo technique. Its sequence is
similar to the original spin-echo one: 90xo - t1 / 2 -180xo - t1 / 2. - t2
Consequently, the spins are rotated always around x (90o
then 180o, figs. a and b).
A simple AB spin system is characterized by the resonance
frequencies A and B and their coupling constant J. The
lines of the doublet of nuclear spin A are denoted in the figure
by A1 and A2, respectively. In figure b the relative positions of
the lines are shown at times a, b, c and d (a). The vectors A1
and A2 move clockwise around the z axis according to the
length of time t1 (their angle is j). Their angle of precession
is labeled by F on b. It was supposed (A1)> (A2) (this fact
is labeled by + and -).
At time d the two vectors have symmetric positions to y.
Their phase differences are their angle to y. Figure c.
shows the change in phase of one of the vectors as
function of the time t1. It is clear from the figure: the
frequency of the phase modulation is the coupling constant
J. After the Fourier transformation of this t1 function one
gets the frequency function (F1) [here the frequencies are
denoted usually by F].
Besides the 1D function (F2) we become also the J(F1)
function. The result is a 2D data matrix.
The J, spectroscopy is a very good tool for the separation
of the lines originated from the chemical shifts and the spinspin couplings.
a: sequence
b: evaluation of the signals,
a-b-c-d
c: Positions of A1 and A2 vectors
and signal intensity as function of
the t1 time, period: 2/J
time domain t1
frequency domain F1
Besides the 1D function (F2) we become also the J(F1)
function. The result is a 2D data matrix. The next figure (a)
presents 5 NMR lines and the data matrix. The data points
belong to the same F2 frequency and are positioned along
a 45o straight line. After a transformation these lines are
positioned vertically (b). So we have the chemical shifts
without the couplings. The (c) on the next figure shows the
possible 2D presentations: the panorama diagram (this is
used more frequently) and the contour diagram. The (d) is
an example: The 1D spectrum and the two 2D
representations are shown.
a: 2D spectrum and its
b: F1 – F2 projection
c: J(F1),(F2)
diagrams: panorama
and contour
d: J, spectra of 1-nbutylbromide
2. Correlation spectroscopy (COSY).
This is a very important method for the discovering the
nuclei that are coupled to one another. The COSY
sequence is presented on the next figure (a). The t1 time
is changed and the FID signal is detected during t2 for all
t1. The result of the two Fourier transformations is a ,
spectrum. The contour diagram contains spots those
point to the correlation of the nuclei. The diagonal signals
are "autocorrelation" signals, the off-diagonal signals are
important (b).
The (c) is an example, 1H-NMR 2D-COSY spectrum of
o-nitroaniline. Ha and Hb, Hb and Hc, Hc and Hd are
correlated, they are in vicinic position each to the other.
There is not correlation between the amino protons and
the ring protons.
Diffraction methods in the molecular structure
elucidation
Introduction to the diffraction methods
We dealt with the collisions of particles with molecules
(atoms). Not only the inelastic collisions yield information
about the molecular structure but also the elastic ones.
The incident particle beam scatters on the atoms of the
molecule. The overwhelming part of this beam is coherent. In
the practice photons, electrons and neutrons are applied.
The common is in the scattered radiations of all these
particles, they contain some information about the structure
of the investigated substance.
The applied photons are high energy X-ray beams. Their
calculated moving mass is smaller than the mass of the
electron. Electric and magnetic fields propagate with the Xray radiation. Photons have neither electric charge, nor spin.
This electric field forces the electron on vibration. The
vibrating electric charges induce electromagnetic fields
around them, that's frequency and phase are equal to those
of the inducing field. Theoretically also the nuclei may be
excited but this effect is negligible since their mass is too
large.
Therefore X-ray diffraction is suitable to determine the
electron density distribution. The resulting atomic distances
are distances between the charge centers of electron
density distributions of the atoms. The method is suitable for
the determination of the atomic distances in crystalline
phase. Its drawback is the uncertainty in the determination
of the hydrogen atom positions.
The mass of the electrons is essentially smaller than that of
the nuclei. They have electric charge and spin, their velocity is
smaller than the velocity of light. Therefore they can interact
with the electron clouds of the atoms and molecules and
scatter from them.
The intensity of the electron scattering is some million times
more intense than that of the X-ray. The scattered intensities,
however, decrease essentially faster with the scattering angle
than that of the X-ray scattering. Owing to the high intensity of
the scattered electron beam it is suitable for the determination
of the molecular structure in gas phase.
The yielded atomic distances are the distances of the
average positions of the atoms (the molecule vibrates).
The neutrons are electrically neutral, but they have
nuclear spin (their spin quantum number is 1/2), so they
have magnetic moments. They interact with magnetic
dipoles, scatter both on nuclei and electrons. Above all
their interaction with the nuclei is important for the structure
determination of magnetic substances having ordered
structure.
The scattered intensity of neutrons is smaller than even
that of the photon scattering. Therefore the neutron
scattering is suitable only for the investigations in
condensed matter. The determined atomic distances are
essentially the averaged distances between the charge
centers of the nucleus density distributions.
The electron diffraction method is applied predominantly
for determination of the geometry of near isolated
molecules (gas phase), while X-ray and neutron diffractions
are applied predominantly in solid state (Chapter 4).
Scatterings on isolated molecules
Our scattering model is restricted to the following
conditions:
1. the wavelength of the particle does not change
during the scattering,
2. each particle scatters only ones,
3. the intensity of the wave does not decrease during
the scattering,
4. only coherent elastic scattering is possible.
If a particle with mass m is scattered on a particle with mass
M, its direction changes. The situation can follow on the next
figure. The impulse of the particle is before the scattering
I o  k o
after the scattering
I  k
The vectors k are wavevectors, their absolute values are
2
~
k  2 

The particles scatter under an angle  along the
surface of a cone. The particles detected on a surface
perpendicular to the axis of the cone form a ring on the
surface (next figure). In the coordinate system fixed to
the mass center of the molecule the atom j has the
position vector rj, the point of detection has the position
vector R. Since R  r j, we assume R  rj  rj  R  R .
Considering the definition of s (see figure), and
assuming k o  k , we have

s  2 k o sin
2
The amplitude of the scattered radiation depends on the
type and the scattering vector of the atom. It is, however,
practically independent of the chemical environment of the
atom. The wavefunction of the j-th atom depends on the
atomic scattering function f(s), depending only on the type
of atom:
exp( ik o R )
yj  A
R
f j (s ) exp(isrj )
A is a constant. The further approach is the independent
atom model: the scatterings of the individual atoms do
no affect the scatterings of other atoms. According to
this approach the wavefunction of the molecule is
N
  yj
j1
N is the number of atoms in the molecule.
The intensity is proportional to * . The atoms of the
molecule are regarded as fixed to one another but it must
be considered the different orientations of the molecules in
gas phase. These calculations resulted the intensity of the
scattered radiation
N N
sin srjk
I(s )  K   f j (s )f k (s )
j1 k 1
srjk
( )
rjk is the difference of the two position vectors, K is a
constant. The intensity distribution by directions is
characterized by the scalar products srjk. The form of
the distribution figure is characteristic and depends on
the atomic distances. Omitting from the last equation,
the terms j=k characterizing the atoms, all other terms
depend on the atomic distances and
form the
molecular scattering:
I m (s )  K   f j (s )f k (s )
N N
j1 k 1
sin (srjk )
srjk
j k
The atomic scattering function for X-ray photons is
approached by f jj (s)    j (r' ) exp( isr' )dr'
j is the electron density around the atom j, r' is the position
vector centered on the nucleus. These functions decrease
with s, that of the hydrogen atom decreases drastically:
The atomic scattering functions are for the electrons more
complicate. Considering the scattering of both the nucleus
and the electrons we have


C
f (s )  2 Z j  f jj (s )
s
e
j
C is a constant, Z is the atomic number.
The atomic scattering functions of the neutrons consist
of two parts. The first term describes the scattering on the
nucleus, the second one characterizes the magnetic
interaction with the electron cloud. The first term is
important for magnetic systems.
Electron diffraction in gas phase (ED)
As it was mentioned a regularity is observable in the
scattering of disordered molecules in gas phase that
approach the isolated molecule model. The electron
diffraction is applied for the determination of the geometric
parameters of molecules since the scattered electron beam
has high intensity. The ED is suitable also for the
determination of intramolecular motions and charge
densities.
The measurement needs high vacuum. The electrons are
accelerated with some ten thousand volts. Their narrow,
some tenth mm of diameter electron beam crosses the
similarly narrow beam of the investigated molecules. The
scattered electrons are detected on a plane photo film or
observed on a screen.
Some problems arise in course of the evaluation of the
measurements. The real scattered intensity consists of three
parts. The first two, the incoherent scattering and the atomic
scattering form the background intensity, Ig, that is not
periodic. The molecular scattering Im, however, is periodic
(next figure (a).
The Ig is very intense at small scattering angles (small
values of s) and therefore a rotating sector is applied at this
values for decreasing the intensity. Look at its form!
For the evaluation of the results the periodic part of the
function must be separated from the background. This is a bit
subjective, difficult iteration process. We can follow it on the
example of sulfuryl chloride. Next figure (a) shows the results
of two measurements in different s regions. Passing all the
measured values a smooth curve is drawn as background.
After them the reduced molecular intensity is calculated:
I m (s )
M(s ) 
I g (s )
In practice the function sM(s) is used (next figure b). The
experimental values (dots) are compared with the
calculated theoretical model (connected curve). The
theoretical model is changed up to the difference between
the experimental and calculated functions becomes minimal
(iteration).
The lower curve in b of the figure shows this difference.
For very low s values the experimental function is not
determinable. This part is substituted by the theoretical
values. The Fourier transform of
M(s) is the radial
distribution function, f(r). Part c of the figure presents this
result. The dashed curve is the experimental radial
distribution function, the connected one is the theoretical
curve. The maxima of the f(r) function give the atomic
distances (c).
The character of the measured and calculated geometric
parameters
The structure, i.e. the geometric parameters of the
isolated molecules are determinable with spectroscopic
methods (rotational spectroscopy) or with electron
diffraction (see before). These data are also calculable
using quantum chemical methods. The resulted structures
differ fundamentally from one another. The structures of
molecules in crystalline phase are determinable with X-ray
diffraction. Let us evaluate the different types of
geometries of isolated molecules.
The quantum chemical methods (QCH) are suitable for
calculation of molecular energies. Changing the atomic
distances in the directions of decreasing interatomic forces
and decreasing molecular energy the calculations lead to
an energy minimum. This is the process of geometry
optimization. The energy corresponds the minimum of the
potential energy curve, and is below the energy of the v=0
vibrational state. The programs apply Born-Oppenheimer
approach, and calculate with harmonic forces. The results
depend on the applied quantum chemical approach and
basic functions. The model is the rigid molecule. This
method yields the re geometry (e: equilibrium).
The molecule rotates and vibrates during the
experiments. Therefore the measured parameters differ
from the equilibrium values. Above all the effects of
vibrations are important. These differences can
considered with the probability density functions of the
vibrations and the kinetic energy distribution of the
molecules in the vibrational states (Boltzmann’s
distribution). In this way the p(r) probability density
functions of the r atomic distances are the results. The
average (or estimated) value of r in thermal equilibrium is
the position of the center mass of p(r):
p(r )
rg   r
dr
r
0

The center mass of area below p(r)/r curve is the
effective atomic distance:
p(r )
0 r r dr

r 
p(r )
0 r dr

The radial distribution function f(r) maxima define the
ED measurements resulted ra geometry. These atomic
distances must be corrected for the stretchings of the
non-rigid rotator (centrifugal distorsion).
The rotational constants Ao, Bo and Co are directly
calculable from the rotational spectra of the molecule.
The geometric parameters calculated from these
constants form the ro geometry.
Considering also the effect of vibrations, a harmonic (H)
correction is necessary. The corrected rotational constants
are Az, Bz and Cz. Applying these constants, we have the rz
geometry.
If the geometry of the vibrational ground state is
o
r
calculated from ED data this is the a geometry.
From the rotational constants one can have only three
independent geometric parameters. If more parameters
exist the rotational constants of isotopomers are necessary
as additive parameters (isotopic correction, I), and so we
get the rs geometry.
Correcting the rotational constants for anharmonicity
(AH) the Ae, Be and Ce constants are resulted and the re
geometry may be calculated. The quantum chemical
calculations result also re geometry.
The name of the geometry refers always to the model and
not on the values of the parameters.
The connections between the different geometry models
are as follows:
SP
ED
ro
rs
I
H
A oBoC o
rz
A zBzCz
AH
A eB C
e e
H
rg
ra
AH
re
QCH
The state with vibrational zero energy (v=0) is theoretically
equal to the state of absolute zero (T=0 K). This is
expressed by the zero superscripts of ro and rgo . If the
molecular vibrations are excited the molecular geometry is
denoted with rv. In thermal equilibrium the different
vibrational states have wi weights (the i subscript denotes a
possible state in thermal equilibrium).
thermal
equilibrium
v> 0
v = 0 T= 0 K
 wr
i i vi
r

r
g
rv
rz
~~
ro
ro
g
r
a