1. dia

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Chapter 3 STRUCTURE AND PROPERTIES OF MOLECULES
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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

• 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.
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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.
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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)

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Table of symmetry elements and operations
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Symmetry elements of water
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Demonstration of a tetragiroide
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Tetragiroide of methane
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Inversion center of trans-hydrogenperoxide
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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), XEEXX
• each X element has its inverse Y, XYYXE,
  YX-1, XY-1
  
• associativity (AB)CA(BC) A,B,C
• conjugate of an element YZXZ-1 and XZ-1YZ

.
X,Y,Z
A set of the group elements that are conjugated
each other builds a class of the group.
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Representations of point groups
The planar water molecule has two s symmetry
(mirror) planes, perpendicular each other. Their
crossing axis is a digir, C2. With a unit element
E the build the C2v point group.
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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.
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Representations are generally labelled with G.
The used notations
Indexing of these symbols
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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 sxz character is 1, the subscript is 1,
otherwise, if it is -1, the subscript is 2.
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Symmetry operations transform atoms in new
positions Proper operations (Cn, E) can be
regarded as rotations
Improper operations (Sn, s, i) are rotations
perpendicular reflections
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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.
The symmetry of the molecules plays important
role in the interpretation of molecular spectra.
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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
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
nucleus-nucleus repulsion (costant!). The 2nd-4th
terms give the potential energy operator. D is
the nabla operator.
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Solution of the Scrödinger equation exactly only
for

• Additional approximations (restrictions)
• Molecular wavefunction production of molecular
  orbital functions, depending on the cartesian and
  spin coordinates
• Paulis 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).

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• 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

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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, cij1 (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.
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Orbitals of water molecule. 1 eV 96.475 kJ
mol-1 (LCAO-MO calculations) Filled ring
region Empty ring - region Possible bond
participations 1b1 and 2a1
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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.
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Localized orbitals of water molecule (oxygen
orbitals), filled out-of-plane
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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.
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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).
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• 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)
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• 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.

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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.
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The intramolecular electron affinity of the
atoms is characterized by the electronegativity.
Under several definitions the widely used if that
of Mullikan
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.
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The structure of two-atomic molecules
The simplest molecules, suitable for studying
the chemical bond. Two equivalent atoms point
group
, cylindric form
Infinite gir, 2 operations, Infinite vertical
planes, Inversion center, Infinite giroid, 2
operations, Infinite vertical digirs. Special
labels for species of diatomic molecules.
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Hetero diatomic molecules have lower symmetry,
the symmetry elements of this group are only the
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 (s)
bond is cylindric, between the two atoms, maybe
s-s. s-p or p-p bond. This is the strongest
bond. The pi (p) bond is situated out of the
interatomic area, maybe p-p, p-d or d-d bond,
weaker than the sigma ones.
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Look again on the forms of the atomic
orbitals! Where are possible s or p bonds?
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Molecular orbitals for H2 from 1sA and 1sB atomic
orbitals. The antibonding orbitals are starred
().




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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 s
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( ). If one 2s
electron transits to a 2p orbital (according to
Hund's rule) then the electron configuration
changes to 1s22sp3 ( ). In space symmetric,
equienergetic orbitals, sp3 hybrids.
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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)
• Excitation of C (promotional energy needed),
  combined with four hydrogens (energy recovered)
• C is combined with four Hs, CH4 is in excited
  state, energy loss to lower 5S2 state.

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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.
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Delocalized systems
Organic compounds with conjugated double bonds
are special case of the double bonded molecules.
Beside the first s bonds each second chemical
bond is strengthened though a p bond. However
the electrons of the p bonds spread along the the
whole so-called conjugated system
The energy levels are
far over the s levels. The sp separation is a
good approach for describing the system.
Restrictions for the simple Hückel method (p
levels) 1. for overlap integrals
or
2. Hamilton matrix element
is denoted as
a, i and j on same atom (Coulomb integral) b, i
and j on vicinal atoms (resonance integral) is
zero, otherwise.
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Both constants have negative sign, a is of
higher absolute value. The eigenvalue equation
has the form
For the ethylene (ethene) molecule (only carbon
atoms are considered)
for bonding p orbital
The results
for antibonding p orbital
Extended Hückel theory (EHT) for heterocyclic
systems axahxb, bxykxyb, e.g. hN0.5,
kCN1.
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Application of the Hückel theory to benzene the
results of the eigenvalue equation are (b is
assumed as -75 kJ/mol)
Two levels are degenerated. The six b electrons
occupy the lowest E4, E5 and E6 levels. For one
carbon atom Eab. Without conjugation is the
total energy 6(ab)6a6b. With conjugation
E2(a2b)4(ab) i.e. E6a8b. The energy
decreased since 2b -150 kJ/mol, this is the
delocalization energy.
The Hückel theory is an acceptable approach for
such cases.
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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.
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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.
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The discussion of the nd (ngt3) and nf orbitals
is complicate. Our model is the 3d orbital.
Since n3, the maximal angular quantum number
l2, magnetic quantum number changes from m-2 to
m2. 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 and the
orbitals (transforming like 3 ) are
symmetric to the xy, xz and yz mirror planes
(3sh) 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 (6sd)
and to the inversion (i). Therefore they belong
to T2g. (as labels T and F are equivalent).
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Oh karaktertáblázat
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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.
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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.
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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.
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d electron configurations in strong octahedral
crystal field
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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.
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Comparison of the occupations of energy levels in
weak and strong crystal fields
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The ligand field theory is the application of
the molecular orbital theory to transition metal
complexes. It is very useful if the ligand-metal
bond is covalent (e.g. ,
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 (ngt3) and nf
orbitals it is necessary to modify this simple
model. The spin-orbital interactions play
important role in these cases.
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For comparison splitting of p2 electron energy
levels under effect of external fields
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Splitting of d2 electron levels in strong crystal
field
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Splitting of d2 electron levels in weak crystal
field
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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.
) 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
Jahn-Teller effect is observable also in the
electronic spectra of the transition metal
complexes. The spectral bands split or broaden.
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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.
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Rotational motion of diatomic molecules
The kinetic energy of the rotating bodies is
described by
I is the moment of inertia, L is the angular
moment, w 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
J is the rotational quantum number. The length of
the Lz component is determined by the MJ magnetic
quantum number
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Using the rigid rotator approach ( the atomic
distances do not change with the change of the
rotational energy),
with
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 BB'/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.
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Energy levels and spectral lines of a rigid
rotator
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According to the definition of the moment of
inertia for the rotational axis of a system of N
points
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
ro is the distance between the two atoms and
is the reduced mass of the molecule.
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Rotational spectra of the diatomic molecules
Substituting the eigenfunctions of the
rotational states into the expression of the
transition moment the following selection rules
may be derived
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)
where
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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
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NJ is the number of molecules on the J-th
level, 2J1 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.
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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
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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.
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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 ( ) are the
Stokes lines, the lines having higher frequencies
than 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,
give the frequencies of the rotational
lines.
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The Raman scattering
Flow chart of a Raman spectrometer
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The Raman lines appear if the polarizability
of the molecule changes during the transition.
The selection rules are
for equivalent atoms, e.g. H2. This is a
difference in comparison with the MW and IR
spectra (the selection rule is there
). For different atoms
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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
Boltzmanns distribution low.
O2
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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 thementioned equation,
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.
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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.
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Fo the simplest, spherical rotators the simplest
equation is valid
For symmetric top prolate molecules
For symmetric top oblate molecules
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).
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Selection rules for non-linear symmetric top
molecules
(IR)
(RA)
for linear symmetric top molecules K 0
(IR)
(RA)
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.
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N2O IR spectrum
rNO118 pm,
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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.
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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
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.
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The solution of the Schrödinger equation with
this Hamilton operator leads to
v is the vibrational quantum number, n 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.
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80
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.
81
The probability density distribution of the v1
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 v0 v1, the mechanical
model is a good approach. The predominant parts
of the molecules are at room temperature in
ground state (v0).
82
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
( 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.
83
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
can appear with law intensity. The density of the
overtone bands increases with increasing vi. The
energy of the anharmonic oscillator is (approach)
(
)
84
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 v0 states is
the dissociation energy (D) of the molecule, r
is the bond length.
85
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.
86

• 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.

87

• dihedral angle coordinate, i.e. change in the
  dihedral angle (out-of-plane deformation)
• - torsional coordinate, i.e. change in the
  torsion.

88
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
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.
89
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)
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
the l 's are the eigenvalues containing the
vibrational frequencies, E is a unit matrix.
90
The solutions are
The eigenvectors are columns vectors. Fitting
these column vectors each beside the other we
have the eigenvector matrix L
With the help of this matrix we can calculate
the normal coordinates
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.
91
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
or
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.
92
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.
93
Force constants of some stretching coordinates
(Fii /100 N m2)

94
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.
95
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 DJ0 transitions. The DJ1 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.
96
Acetonitrile IR spectrum
97
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.
98
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 a.
99
The IR spectra are measured in gas, liquid
(also in solution) and solid state. The classical
wayThe 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.
100
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 (a). 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
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102
The maximal value of r 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 34-66 vibrational modes. A1, A2 and
B1 modes are IR active, all modes are RA active
(see table).
103
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
1 for proper, and -1 for improper operations.
The number of the vibrational modes in the i-th
symmetry species is
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.
104
The formaldehyde molecule is planar, its plane
is the zy one. Applying the equation for
calculation of the mi values of the species B1
The full representation of the formaldehyde
molecule is
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.
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106
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
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-of-plane
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.
107
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
108
The full representation of the pyrazine
molecule is 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)
109
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).
110
The RA spectrum of the solid pyrazine . The bands
below 250 cm-1 are vibrations of the crystal
lattice.
111
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.
112
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 n , the in-plane deformations by b, the
out-of-plane deformations by g and the torsions
by t. 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-317 in-plane modes ( ) and
N-37 out-of-plane modes ( ).
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114
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.
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116
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
become greater and these terms
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.
117
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 multi-photon
effect. Two lasers with adequate intensity and
frequencies n1 (fixed) and n2 (tunable) irradiate
the sample. at the same time. If the frequency
difference is equal to the frequency of a
vibrational transition nin2-n1, then an
intense coherent radiation is observable at the
frequency 2n1-n2. 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.
118
The Raman amplification spectroscopy is an
absorption method. The sample is irradiated with
two lasers. Their frequencies are n1(fixed) and
n2 (tuneable). If n in2-n1, the molecule absorbs
light at n2 frequency and emits at n1 one. If
light is detected on n2 this is the inverse Raman
effect, Raman loss spectroscopy. If the light of
n1 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.
119
Neutron molecular spectroscopy (IINS, incoherent
inelastic neutron scattering) The wavenumber
region of thermal neutrons is comparable to that
of the molecular vibrations
k is the Boltzmann constant, T is the absolute
temperature, mn16749310-27 kg, the neutron mass,
l 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.
120
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.
121
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 mm2 surface.
122
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).
123
Potential energy function of the ethane rotation
124
Character table of the C3 point group
125
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 v0 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.
126
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 is the molar absorption coefficient, l is the
layer width, c is the chemical concentration (in
concentrated solutions the activity replaces the
concentration).
127
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 DS0 selection rule is here valid, the group
spin quantum number must not change during the
transition. This is strictly valid only for ls
coupling.
128
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 d