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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. T hree types of the strong interactions exist between atoms :. Several individual atoms build the system. Each of them

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Chapter 3



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 delocalizedandmove 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)


Point groups

  • Symmetry operations build algebraic groups (G). An algebraic groupis 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 inverseY, X*Y=Y*X=E, Y=X-1, X=Y-1
  • associativity: (A*B)*C=A*(B*C)A,B,C
  • 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.


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 elementE 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 sxz 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, s, 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.

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:

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. Dis thenabla operator.


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;
  • 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 basicfunctions. 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.


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


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 delocalizedmolecular 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):


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:

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:

, 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.


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.


Look again on the forms of the atomic orbitals!

Where are possible s or p bonds?


Molecular orbitals for H2 from 1sA and 1sB atomic orbitals. The antibonding orbitals are starred (*).



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.


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 onecarbon to methan with hybridization (2 ways):
  • Excitation of C (promotional energy needed), combined with four hydrogens (energy recovered);
  • 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 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 agood approach for describing the system.

Restrictions for the simple Hückel method (p levels):

1. for overlap integrals


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.


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:

ax=a+hxb, bxy=kxy*b, e.g.hN=0.5, kCN=1.


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 E=a+b.Without conjugation is the total energy 6*(a+b)=6a+6b. With conjugationE=2*(a+2b)+4*(a+b) i.e.

E=6a+8b. The energy decreased since 2b= -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 methodsresult 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 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).


The originally five times degenerated energy level splitsinto two groups. The ligandsconnecting 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 ligandsrepulse the eg orbitals, so their energy is higher than thatof 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 takeninto 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.


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. Forstrong 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.


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 (n>3) and nf orbitals it is necessary to modify this simple model. The spin-orbital interactions play important role in these cases.


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.


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

I is the moment of inertia, L is the angular moment, wis 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 MJmagnetic quantum number


Using the rigid rotator approach ( the atomic distances do not change with the change of the rotational energy),


B’ has energy dimension. Since the experimental data appear in MHz or cm-1 units, the rotational constants are used informs 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.


According to the definition of the moment of inertia for the rotational axis ofa 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 moleculeandits 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.


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)



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:


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


Far infrared spectroscopy. FT spectrometersare 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( ) 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.


The Raman scattering

Flow chart of a

Raman spectrometer


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


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.



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-35Clspectrum 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.


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:

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).


Selection rules

for non-linear symmetric top molecules:



for linear symmetric top molecules: 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.


N2O IR spectrum

rNO=118 pm,


N2O Raman spectrum

Pay attention on the double density of the RA spectral lines comparing to the IR ones (in the RA spectrum only theDJ=-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

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

v is the vibrational quantum number, nis 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 theselection ruleis 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.


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 excitingmonochromatic 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):




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:

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):

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 eigenvaluescontaining the vibrational frequencies, E is a unit matrix.


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.


If the F matrix is known, the frequencies are calculable. The F matrix was calculatedformerly 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


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 theirvalues.


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. TheDJ=+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.


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 oneanother.


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.


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 ruleis 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


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 34-6=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 cjof 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, njis the number of atoms that are not moved under the effect of the Rjoperation, 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 valuesof the species B1

The full representation of the formaldehyde molecule is

The vibrational modes belonging to A1preserve 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.


The vibrational modes belonging to A1 preserve the symmetry of the molecule. The first three formations are of this kind.

The modes belonging to A2must 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.


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:


The full representation of the pyrazine molecule is

10 vibrational modes are IR active, 12 modes are RAactive, 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. Thestretching modes are labelled byn, the in-plane deformations by b, the out-of-plane deformations byg 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-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 coordinateof achemical 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 arecoupled. If the stretchings are in phase, this is a symmetric (s) vibration, if they are in opposite phase, this is anantisymmetric (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 termsin 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. Asresult 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 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: ni=n2-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.


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 i=n2-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.


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, mn=16749310-27 kg, the neutron mass, lis 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 hydrogencontent. 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 about1 % 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.


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).


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 configurationsthrough a planar intermediate state. Theammonia 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

ais 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 principlethe00 transition is the most probable (see figure). If the nuclear distance increases during excitation (rg>ro), the 10 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.


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) ofp electrons and non-bonded (n) electron levels: p* pand p*n transitions. The next figure is a simplified diagram.

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 p* p transitions the excited state, in the p* n transitions the ground state is more polar. The p* pbands shift to the visible (bathochromic shift), the p*n ones to the ultraviolet (hypsochromic shift). With increasing acidity of the solvent the intensities of the p*n bands decrease (hydrogen bond formation) and at a given pH the bands disappear.


The dd 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 ion in the aqueous solution 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 cm-1, Dq=2.030 cm-1.


Visible spectrum of the 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 nextfigure presents the electron excitation spectrum of nitrobenzene. The intense band at 39.800 cm-1 (251 nm) is a CT band.


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 00 transition is forbidden, but with low intensity appears.


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.


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 collisionsincreasing 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 asreactant 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 Franck-Condon 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 indicatorsabsorb 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.



Since rg>ro, the 10 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 ST 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 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 potentialcurveor 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, ).


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 super-position 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 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 super-position 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

The oscillator strength is a quantum chemical quantity that also the characterizes the band intensity:

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:

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 p* pand p*n and s*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 showshypsochromic 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 shorterwavelength (higher frequency).


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

The order of the groups with -I effects is


The groups containing free electron pairs but no double bonds are called auxochromic groups if they join groupswith conjugated double bonds. Follow thedisplacements of the electrons with -I and +I groups subtituted benzene ring.


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:

The order of the groups with -M effect is:

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.


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 p- positions,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 are dispersive instruments.

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:

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.


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:

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


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 theoremthe 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):


where nis the frequency and lis the wavelength of light. The refractive index is a function of the relative permittivity (εr) and the relative permeability (μr) of the medium:


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, Eois the amplitude)

If the light penetrates into the medium, the wave amplitude decreases exponentially, therefore

nk is a damping factor.

The complex refractive index is

Comparing this result with the Lambert-Beer law

c is the concentration of the solute in the solution.The imaginary refractive index is proportional to the molar absorption coefficient (a)and depends on the light frequency.


The real part of the complex refractive index depends also on the frequency. Let the frequency of theabsorbed lightno, and let Dpo the amplitude of the light induced dipole moment, then a good approach is (a is a constant ):

According considering the definition of molar polarization, the molar refraction is (er was substituted by n2)


the dispersional formula for n is

(sum over all components)

rN is the number of particles in unit volume, no labels the absorption maxima, Ci\'s are constant, Ai is Einstein\'s absorption probability that is proportional to the square of the transition moment:

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.


Electron excitation with polarized light

A molecule (a substance) is optically activeif 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:

Here if nL>nR, the substance rotates to right,lis the length of the sample.

The specific rotation is

c is the concentration.


The definition of the molar rotationis

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:


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 CDonly 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 spectro-polarimeters. See nex figure! Itpresents the ORD spectra of molecules with negative (dashed) and positive (full) Cotton effect.


CD spectra of molecules with negative (dashed) and positive (full) Cotton effect.

CD spectra of molecules with negative (dashed) and positive (full) Cotton effect.

Positive Cotton effect


CD spectra of molecules with negative (dashed) and positive (full) Cotton effect. The different con-figuration 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 (GC-MS).

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 (10-4-10-6 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.


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:

In this way the potential energy of the ion is transformed to kinetic energy:

The mass belonging to unit charge (i.e. the absolute value of the electron charge) is therefore


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 n.

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:

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 GC-MS 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

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.


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


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 2-methylpentane, while the next one is the mass spectrum of its isomer the n-hexane. Both molecules have the same moleculemasses, 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 ann-butyl group (86-29=57).


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 mand precessing with angular velocity waround the constant magnetic induction vector B, is irradiated in the direction perpendicular to Bwith an electromagnetic wave having frequency n.

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 -mB or +mB (mB is the Bohr magneton,

Since the energy of the dipole in magnetic field is -mB, the splitting of the two levels is


If this energy difference is equal to the energy of the radio frequency photon then the molecule absorbs it (resonance):


The selection rule is

The molecule is excited from the lower (ms=-1/2) to the higher (ms=+1/2) state. The induction B and the frequencynare proportional

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 fieldB. 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.


The nuclear magnetic moment decreases the external field:

a is the hyperfine coupling constant, the MIis thenuclear magnetic quantum number. The possible values of MI are for the hydrogen atom . The real situation is more complicate. The external field indicates a counter field. The local field

interacts, <1 is the shielding factor. Therefore in reality



Flowchart of an ESR spectrometer

The sample (S) is positioned between the poles of a constant magnet (M). The value of the fieldB 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 structureandESR spectrum of the free radical ion of 1,4-benzoquinone 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 , 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 orgray) 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 processestriplet 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 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; µNis the nuclear magneton.


The E energy of the nuclear magnetic moment is in a magnetic field B

The selection rule is deducable from the expression of the magnetic transition moment

and therefore

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

a is the gyromagnetic ratio of the nucleus


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

  • 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.

The local magnetic induction will be

Therefore the Larmor frequency changes to


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 thatwill 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 () 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):

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 n0 decreases d. There exists also another scale, with opposite direction


The 1Hchemical shiftsdepend 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 determined for the k-th level by . 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 CH3areequivalent (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


E0stands for the energy of the magnetic field without (chemical) environment. If thisenergy equation is valid, the selection rules are for an ABn system

If the chemical shift and the spin-spin coupling are commensurable, the selection rule is

Considering the selection rules, the frequency of the magnetic transition will be

the zero superscript refers to the state without spin-spin 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

The possible energy levels are under these conditions:


The observable frequencies are ( ):

If , then , if , then . 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

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/, 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.


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 no are excited with a wide range (Dn) 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 Mthe 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.


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 processesof 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 (alabels the ground state, bthe excited one): 1:aa , 2:ab, 3:ba and 4:bb. The spins of the protons are saturated through double resonance and therefore the originally forbidden 41 and 32 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 is the original intensity, gis the magnetogyric ratio,h 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 (tc).


The correlation time(tc) 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 spin-lattice 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 fieldB 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

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


T1 is the time constant of the spin-lattice relaxation.

The nuclear spins turn in the inducing field Bi during the pulse time trwith the angle

The amplitude of the pulse is gaBi/2p, 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 Blmagnetic 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

The relaxation causes line broadening, the average FWHH is


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 – t - 90o. This means: the first pulse turns the nuclear spins 180o from the direction of the vector B (a). After the pulse, timetis the waiting period. During this time the spins relax partly into the starting direction. After this time a pulse in direction xturns the remained z component of the magnetic moment in direction y (b) and so it becomes measurable (My). Changing twe get the decay curve (c).


If t<T1, the FID is negative, if the FID=0, that is the halftime t=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 - t - 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 t (Mt), the Me signal is turned 90o from z to y. The difference of this signal Mt 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 - t - 180o - t(echo) - td. The spins are turned 90o in direction y (a). During the time t the signalsspread 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 t time the spins come closer each to the other, this is the echo (c). After td>5T2 the starting state will be restored. Changing tthe FID will be measured. The signals change with . The relaxation time is 2t because of the 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,d 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:

This method is a 2D spin-echo technique. Its sequence is similar to the original spin-echo one: . 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 nA and nB 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 Fon b. It was supposed n(A1)> n(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,d spectroscopy is a very good tool for the separation of the lines originated from the chemical shifts and the spin-spin couplings.


a: sequence

b: evaluation of the signals,


c: Positions of A1 and A2 vectors and signal intensity as function of the t1 time, period: 2/J

time domain t1

frequencydomain 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),d(F2) diagrams: panorama and contour

d: J,d spectra of 1-n-butylbromide


2. Correlation spectroscopy (COSY).

This is a very important method for the discovering the nuclei that are coupled to oneanother. 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 d,dspectrum. 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 theelastic 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 X-ray 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 cloudsof 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


after the scattering

The vectors kare wavevectors, their absolute values are

The particles scatter under an angle Jalong 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, we assume. Considering the definition of s(see figure), and assuming, we have


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 functionf(s), depending only on the type of atom:

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 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

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:


The atomic scattering function for X-ray photons is approached by

rj 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 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 magneticinteraction 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 scatteringIm, however, is periodic(next figure (a).

The Igis very intenseat 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 differentsregions. Passing all the measured values a smooth curve is drawn as background. After them the reduced molecular intensity is calculated:


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 ofthe figure shows this difference. For very low s valuesthe 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 cof 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):


The center mass of area below p(r)/r curve is the effective atomic distance:

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 rogeometry.


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 calculated from ED data this is the 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:


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 and . 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).