Chapter 3 STRUCTURE AND PROPERTIES OF MOLECULES

Chapter 3

STRUCTURE AND PROPERTIES

OF MOLECULES

The molecule is a set of atoms that are in strong chemical connection to one another building a new substance.

Three types of the strong interactions exist between atoms^: 1. Several individual atoms build the system. Each of them

add electrons to the full system. These electrons are

delocalized and move practically without resistance in the system: the metal bond was built.

2. One of the interacting atoms has low first ionization energy (e. g. an alkali metal), the second one has high electron affinity (e.g. a halogen element): easy electron transfer from the first to the second atom. Ion pair, they attract each other: we have an ionic bond.

3. Atoms with open valence shells share a part of their valence electrons, a new electron pair is formed. They

build a chemical bond, the covalent bond. Another possibilty:

an atom has a non-bonded electron pair, the other an electron pair gap. The electron pair will be common and build a

chemical bond, the dative bond.

The shared electron pair of the molecule moves on molecular orbitals (MO).

Polar bond: the participition of the electron pair between the two atoms is unequal.

Delocalized orbital: the bonding electrons are shared under more than two atoms.

Molecule: finite number of atoms with the exclusion of polymers.

Model: isolated molecule.

Symmetry elements and symmetry conditions

Object is symmetric if there exist an operation bringing it in equivalent position. Equivalent: covers the original one.

The operations fulfilling this conditions are symmetry operations. Symmetry operations belong to symmetry elements of the object.

Symmery elements are mirror planes,

symmetry centers (inversion), symmetry axes (girs),

reflection-rotation axes (giroids)

Table of symmetry elements and operations

H H O

1 2

C₂

X

yz

xz z

y

Symmetry elements of water

S

Demonstration of a tetragiroide

C

H H

H H

S

Tetragiroide of methane

Inversion center of trans-hydrogenperoxide

Point groups

Symmetry operations build algebraic groups (G). An algebraic group is a heap of objects, properties or ideas, characterized as follows.

- A group operation exists. The group is closed for it, the result is member of the group.

- a unit element exists (E), X*E=E*X=X

- each X element has its inverse Y, X*Y=Y*X=E, Y=X^-1, X=Y^-1

- associativity: (A*B)*C=A*(B*C) A,B,C

- conjugate of an element: Y=Z*X*Z^-1 and X=Z^-1*Y*Z

X,Y,Z

 G

.

 G

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  symmetry (mirror)

planes, perpendicular each other. Their crossing axis is a digir, C₂. With a unit element E the build the C_2v 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 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 _ij coefficients.

The gir character is +1, the species is A. If it is -1, the species is B. If _xz character is +1, the subscript is 1, otherwise, if it is -1, the subscript is 2.

Symmetry operations transform atoms in new positions:

Proper operations (C_n, E) can be regarded as rotations:

Improper operations (S_n, , 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: _j for the j^th operation.

1 ,...,

2 , 1 2

cos 2

1   



 



 



 p n

n p

j





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:

 



















i

N

i n

i

n

i

j ij

n

e i

r

e Z Z r

e Z r

e H m

1

2

1 1

2 1

2 1

2 2

ˆ 2

   





 



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 2^nd-4^th terms give the potential energy operator.  is the nabla operator.

Solution of the Scrödinger equation: exactly only for

H

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 (_i) functions,depending only on their own Cartesian coordinates (x_i).

-Hartree-Fock (HF) method in Roothaan (HFR) representation : orbital functions expanded into

series using basic functions. Usually atomic orbitals are applyed in praxis. Linear combinations of atomic orbitals as molcular orbitals: LCAO-MO.

Solution of the eigenvalue equations for using the listed approaches:

Self consistent field (SCF) method, it is iterative.

Estimating or assuming values for linear coefficents for linear combinations, energy is calculated. With

this energy new coeffcients can calculated, with them one have new energy value, etc., until the deviation between the energies of two successive steps arrives the wanted limit.

Shorthand: LCAO-SCF

The symmetry of molecular orbitals

Molecular orbitals have symmetry. The orbital functions maybe symmetric: their sign does not change under the under effect of the symmetry operation, _ij=+1 (character table);

antisymetric, their sign changes under the effect of the symmetry operation, _j=-1 (character table);

The symmetry of the molecular orbitals are denoted according to their symmetry species, but lower case letters are used.

If some belongs to the same species, they are numbered beginning with them of lowest energy and are used as

coefficients.

z

x

-557.3 eV

z

1a1 2a

1 -36.3 eV

z

x

1b1 -19.3 eV

z

3a1 -15.2 eV

z y

x

1b2 -11.9 eV

x

Orbitals of water x

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, 2p_x and 2p_y orbitals of the oxygen atom build the chemical bonds (2a₁ and 1b₁). There exist also real delocalized molecular orbitals, e.g. those of the aromatic rings. Here is difficult to form localized orbitals.

The covalent bond The characteristics of the chemical bond

Influences on the formation of the molecular orbitals: (look also at the Hamilton operator)

- kinetic energy: smaller free space for electrons higher;

- electron-electron repulsion: increases their distance;

- electron-nucleus attraction: acts on the electron;

- nulceus-nucleus repulsion: important role in the formation of molecular geometry;

- spin-spin electron interaction: with parallel spin repulsive, with opposite spin attractive (Pauli principle, Hund rule).

Results for localized elecron pairs:

-Try to avoid one another;

-Try to expanding their possible area,

-Try to come as close to the nucleus as possible The molecular geometry is the result of the listed effects.

Formation of the molecular orbital: the electron clouds of the atoms approach one another.

Hybridization: mixing of the atomic orbitals  overlap integral, measure (grade) of mixing (atoms A and B):





 d

S

^ 

Mixing of atomic orbitals: chemical bond.

Extreme cases:

- there is not mixing of atomic orbitals, e.g. water 1a₁ orbital;

-the participations are equivalent, like H-H bond in hydrogen molecule, with 1s orbitals.

During the approaching of the atomic orbitals two levels build, these molecular orbitals:

-the energy of one is lower than those of the atomic orbials, it is localized between the two atoms, this is the bonding molecular orbital;

- the energy of the other is increased, it has a nodal surface, is wide spreaded, this is the antibonding molecular orbital.

The intoduced model is valid only in case of bonds with s-s atomic orbitals.

The more atomic orbitals with nearly the same energy levels are combined in the bond, the greater the

deformation of the original atomic orbitals.

The description of the molecular orbitals is possible only as the linear combination of several atomic orbitals.

If only elements with atomic number lower than 10 take part in the molecule, the deformation of the atomic

orbitals is small.

The attractive force between the interatomic electron

clouds and the atomic cores is greater than the repulsive force between the atomic cores (nucleus and inner

electrons). This is the fundamental reason of the formation of chemical bonds.

The intramolecular electron affinity of the atoms is characterized by the electronegativity. Under several definitions the widely used if that of Mullikan:





X  

2 1

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.

Special labels are applied for symmetry species of diatomic molecules: the Greek letters instead of the corresponding Latin ones.

The sigma ( bond is cylindric, between the two

atoms, maybe s-s. s-p or p-p bond. This is the strongest bond.

The pi () bond is situated out of the interatomic area, maybe p-p, p-d or d-d bond, weaker than the sigma

ones.

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.

C

Look again on the forms of the atomic orbitals!

Where are

possible or  bonds?

Molecular orbitals for H₂ from 1s_A

and 1s_B atomic orbitals. The antibonding orbitals are starred (*).

   _{A B}

g 1s 1s

S 1 2

1 

 



   _{A B}

u s s

S 1 1

1 2

* 1 

 



_g^

_u^

Hybridization

A molecular orbital is called hybrid orbital if an atom takes part in it with more then one orbitals. Measure: participation of the atomic orbitals in the molecular wavefunction.

Hybridization is possible only in case of bonds.

The central atom contacts n equivalent atoms or atom groups. Results: n equivalent orbitals arranging

symmetrically in space and determine the structure.

Example. The ground state of the C atom is 1s²2s²p²( ).

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.



3P_o

5S2

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 ):

Therefore methan has tetrahedral structure. From one carbon to methan with hybridization (2 ways):

1. Excitation of C (promotional energy needed),

combined with four hydrogens (energy recovered);

2. C is combined with four H’s, CH₄ is in excited state, energy loss to lower ⁵S₂ state.

In the MO theory the hybridization means the forming of equivalent orbitals. Beside this sp³ hybrid orbitals

they are formed with ethene (C₂H₄): sp² hybrid and also with ethine (C₂H₂): sp hybrid.

The substitution demages these hybrid orbitals since their equivalence disappears.

The hybridization is important in case of complex

compounds of transition elements. Their d orbitals can form hybrid molecular orbitals with the ligands.

E.g.: spd² determines a square structure [(PtCl₄)²-] , sp³d a trigonal bipyramide (PCl₅), sp³d² an octaheder (SF₆) , etc.

Delocalized systems

Organic compounds with conjugated double bonds are special case of the double bonded molecules.

Beside the first  bonds each second chemical bond is strengthened though a ^bond. However the

electrons of the  bonds spread along the the whole so- called conjugated system The energy levels are

far over the  levels. The separation is agood approach for describing the system.

Restrictions for the simple Hückel method ( ^levels):

1. for overlap integrals ^S_ij ^{ 0 i  j} or ^S_ij ^{1 i  j} 2. Hamilton matrix element ^H_ij ^



, i and j on same atom (Coulomb integral);

, i and j on vicinal atoms (resonance integral);

is zero, otherwise.

is denoted as

Both constants have negative sign,  is of higher absolute value. The eigenvalue equation has the form

 0

 ES H

For the ethylene (ethene) molecule (only carbon atoms are considered):

 E 

  E 

The results: ^E1    for bonding  orbital







2 

E for antibonding  orbital

Extended Hückel theory (EHT) for heterocyclic systems:

_x=+h_x, _xy=k_xy*e.g.h_N=0.5, k_CN=1.

Application of the Hückel theory to benzene: the results of the eigenvalue equation are ( is assumed as -75 kJ/mol):

EE E E E E

1

2 3

4 5

6

2 2

  

  

 

 

  

 

Two levels are degenerated. The six  electrons occupy the lowest E₄, E₅ and E₆ levels. For one carbon atom

E=Without conjugation is the total energy

6*()=. With conjugation  i.e.

. The energy decreased since = -150 kJ/mol, this is the delocalization energy.

The Hückel theory is an acceptable approach for such cases.

For the point of view of reactivity of molecules two energy levels are important:

The electron density on the highest occupied molecular orbital (HOMO) is nearly proportional to the reactivity in electrophylic reactions.

The electron density on the lowest unoccupied molecular orbital (LUMO) is nearly proportional to the reactivity in nucleophylic reactions.

These limit levels play also important role in the

development of the chemical and spectroscopic properties of the molecule.

The advanced quantum chemical methods result better approximations, like post-Hartree-Fock and density

functional methods (DFT: density functional theory).

The d orbitals complicate these calculations.

Complex compounds of the transition metals

The d orbitals are important since several transition metals play role in catalysts and enzymes. Their description is more complicate than that of the molecules with atoms below atomic number 10.

Even a simple theory is a good tool in this field.

Bethe's crystal field theory is simple, old, but suitable also in our days.

The ligands with their negative charges (ion or dipole) connect the central ion (having positive charge). The bond is relatively weak. Practically the central ion determines the molecular structure. The electric field acts on the crystal field, the spin-orbital interaction and the internal magnetic field take also part in the Hamilton operator of the molecule.

The discussion of the nd (n>3) and nf orbitals is complicate. Our model is the 3d orbital.

Since n=3, the maximal angular quantum number l=2, magnetic quantum number changes from m=-2 to m=2.

Octahedral complexes with six equvalent ligands

(sp³d² hybrids) are discussed here. They belong to the O_h 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 (3_h) and are also symmetrical to the x, y and z digirs (3C₂), therefore they belong to the symmetry species E_g. The three other d orbitals, d_xy, d_xz and d_yz are symmetric to the six axis-axis bisectors

(6C₂) and to the mirror planes determined by a bisector and an axis (6_d) and to the inversion (i). Therefore

they belong to T_2g. (as labels T and F are equivalent).

2 2 y

dx

3 _ 3d_z2

z²  r²

O_h karaktertáblázat

The originally five times degenerated energy level splits into two groups. The ligands connecting to the central ion are positioned on the coordinate axes. The t_2g orbitals (orbitals are labelled similarly to their symmetry species only small letters are used) are situated between the coordinate axes, while the e_g orbitals are centred on them. Therefore the ligands repulse the e_g orbitals, so their energy is higher than that of the t_2g 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 

The crystal field theory gives the order of the orbital energies but only as expressions, their values are not calculable.

The measure of the splitting in octahedral crystal field is labelled by 10Dq. Using the experimental data the Dq

becomes calculable (q is the ratio of two matrix elements, D is a coefficient in the description of the crystal field).

The shift of the band system by the ligands is not taken into account. Therefore the average energy of the d orbitals is always 0 D. The splitting is influenced by two effects:

1. The crystal field (metal ion - ligand, d orbital symmetry) effect.

2. The mutual repulsion of the d electrons.

First effect stronger: strong crystal field, second effect stronger: weak crystal field.

In a strong crystal field the electrons occupy the energy levels according to the increasing energy.

Therefore the t_2g orbitals are occupied at first, and the e_g ones only later.

The energy of the t_2g orbitals is 4Dq lower than average, while that of the e_g orbitals is 6Dq higher than average.

The t_2g orbitals are the bonding ones, the e_g's are the antibonding ones.

d electron configurations in strong octahedral crystal field

The situation is more complicate in weak crystal fields. The repulsion of the electrons split the orbitals into several levels.

Sometimes these levels are very close. The electrons occupy the orbitals according Hund's rule. At first all orbitals are occupied by one electron. After the occupation of all orbitals in this way the second electrons join stepwise the first ones with opposite spins.

Octahedral complexes with weak and strong crystal fields differ in the case of d⁴, d⁵, d⁶ and d⁷ configurations.

The group spin quantum numbers of these configurations are for weak crystal fields high, they are high spin states. For strong

crystal fields the group spin quantum number is in these cases low, they are low spin states. These two types of states are

distinguishable by magnetic measurements.

E

eg

t 2g

d1 d2 d3 d4 d5 d6 d7 d4 d5 d6 d7 d8 d9 d10

strong crystal field weak crystal field

Comparison of the occupations of energy levels in weak and strong crystal fields

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.

MnO₄^{ Fe}

 

p2

1S

1D

3P

1S o

1D 2

3P 2

3P 1 3P

o

MJ

0

2

-2 2

-2

1 -1 0

H atom level electrostatic

interaction magnetic

interaction external magnetic field

For comparison: splitting of p² electron energy levels under effect of external fields

d2

eg

t 2g

(eg)2

excitation of 2 electrons

excitation of 1 electron

(eg)(t 2g)

(t 2g)2 10 Dq

1A1g 1Eg 3A2g

1T1g 1T2g 3T1g 3T2g

1A1g 1Eg 1T2g 3T1g

10 Dq 10 Dq

crystal field

effect configurations

strong field electron-electron interactions

Splitting of d² electron levels in strong crystal field

d 2

1S 1G

3P 1D

3F

electron-electron interaction

crystal field effect

1A1g 1A1g 1T1g 1Eg 1T2g 3T1g 1Eg 1T2g 3A2g 3T2g 3T1g

Splitting of d² electron levels in weak crystal field

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.

 

Fe

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 L 2 I 1

2 E 1

T

2 2

r   



I is the moment of inertia, L is the angular moment,  is the angular velocity. The quantum chemical problem is calculation of the operator eigenvalues (similar to the problem of the H atom). Here the eigenvalues of the angular moment are quantified:

J is the rotational quantum number. The length of the L_z component is determined by the M_J magnetic

quantum number

J M

J M

L_z  _J   _J 





J

L     ^* 

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

^{J 1} ^B'J^{J 1}

I J E 2

2

r     

I ' 2

B

2

with 

B’ has energy dimension. Since the experimental data appear in MHz or cm^-1 units, the rotational constants are used in forms B'/h (MHz) or 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.

Er

6B

4B

2B

J 3

2

1 0

~ 2B

Energy levels and spectral lines of a rigid rotator

According to the definition of the moment of inertia for the rotational axis of a system of N points

2 i N

1 i

ir m

I







m is the mass of the atom, r is its perpendicular distance from the axis.

The moment of inertia for a diatomic molecule and its axis crossing the mass center and perpendicular to the valence line has the form

₂

r

I  

r_o is the distance between the two atoms and

2 1

2 1

m m

m m

 

 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:

1 M

and 1

J   _J  









 _i p _jd



 ^*

P

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)

    









hc B E -

~  E^{ri, r,j}       

where

cI 8

h hc

B B ₂

 

 

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:

 



 







 kT

exp E 1

J 2 N

N_{J o} ^r,J

N_J is the number of molecules on the J-th level, 2J+1 is the degree of degeneration according to the magnetic

quantum number. N_J has a maximum (see the spectra down).

The rotational spectra can be measured recording

microwave (MW), far infrared (FIR) or Raman (RA) spectra.

SS S CD E

FS FM RR P

Flow chart of a microwave spectrometer 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.

Far infrared spectroscopy. FT spectrometers are applied.

The optical material is polyethylene, the beam splitter is polyethylene-terephtalate foil.

The molecule must have a permanent dipole moment, since otherwise the transition moment is zero. Therefore the

diatomic molecules with two equivalent atoms have not pure rotational MW or IR spectra. This is the pure rotational IR

spectrum of H³⁵Cl. The H-³⁵Cl distance is calculable form the line distances.

Raman spectroscopy. Raman spectroscopy is a special method of the rotational and the vibrational spectroscopy.

This is a scattering spectrum. Spectrum lines are observed in the direction perpendicular to exciting light (a VIS or NIR laser beam) beside the original signal

The effect is called Raman scattering, the spectral lines are lines of the Raman spectrum. The series that appear at lower frequencies than the that of the exciting beam ( ) 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.

~o



~o



i o

i ~

~   





 

o

o i

laser beam sample

Raman scattering

+

-

~

~ ~

~



_o

~

anti-Stokes lines

Stokes lines

Flow chart of a Raman

spectrometer

The Raman scattering

The Raman lines appear if the polarizability of the molecule changes during the transition.

E α

p   



The selection rules are

2 J  



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 ^{J  1}

2 , 1 J   



O₂

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-³⁵Cl spectrum are 20.7 cm^-1. Using

thementioned equation, kg m².

Therefore the bond length is 129 pm. Similarly, taking into account the line distance in the Raman spectrum of oxygen (11.5 cm^-1) and it equivalence with 8B, the bond length in the oxygen molecule is 121 pm.

10 27

703 .

2

I   ^

cI 8

h hc

B B ₂

 

  ⁱ²

N 1 i

ir m

I







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:

    









hc B E -

~  E^{ri, r,j}       

For symmetric top prolate molecules

   





E_r  hc B   A  B

For symmetric top oblate molecules

   





E_r  hc B   C  B

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: ^{ J K  J}

J  1 K  0 (IR)

J   1 2, K  0 (RA)

for linear symmetric top molecules: K = 0

J  1

J  2

(IR) (RA)

The description of the energy levels of the asymmetric top molecules is very complicate.

There do not exist solutions for these rotators in closed mathematical form.

N₂O IR spectrum rNO=118 pm,

NON140^o

NON  140^o

N₂O Raman spectrum

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

kq2

2 1 dq2

d2 2μ

Hˆ  2 

q is the displacement coordinate (in vibrational equilibrium its value is zero), k is the force constant of the harmonic

vibration; the first term is the operator of the kinetic, the second term is the operator of the potential energy of the oscillator.

The solution of the Schrödinger equation with this Hamilton operator leads to

,...

2 , 1 , 2 v

v 1 h

E_v  



 

 





v is the vibrational quantum number,  is the oscillator frequency.

The next figure contains the forms of the harmonic oscillator wavefunctions (dashed lines) and the

probability distribution functions (full lines).

The wavefunctions of odd vibrational quantum numbers are antisymmetric, while those of even quantum numbers are symmetric. All probability distribution functions are symmetric.

Equidistant energy

levels of the harmonic oscillator and the curve of the potential energy function V (dashed line) of a diatomic molcule, as function of the

distance of atoms r.

The probability density distribution of the v=1 state is very similar to the classical mechanical model of the vibrations. According to the classical model the system has also two points with maximal staying time. Since the most important transition is v=0 v=1, the mechanical model is a good approach.

The predominant parts of the molecules are at room temperature in ground state (v=0).

Vibrational spectra of diatomic molecules

The spectra are recorded applying both infrared and Raman spectroscopy.

Infrared spectra are measured in practice only with Fourier transform spectrometers.

From the definition of the transition moment the selection rule is for IR spectra

v  1

Predominantly absorption spectra are recorded, the measurement of the emission spectra is difficult. The vibrational transition is infrared active if the molecule

has permanent dipole moment (necessary condition, as for the rotational spectra). Therefore the X₂ type

molecules have not IR spectra.

Raman spectra are measured classically with perpendicularly incident laser light applying a

monochromator, or with the introduction of the laser light in a FT spectrometer (in this case the light source is replaced with the exciting monochromatic laser beam). The selection rules are like in case of IR spectra. Since the Raman

activity depends on the change in the components of the probability tensor the X₂ molecules are Raman active.

The real vibrations are anharmonic. Therefore the

selection rule is not strickt. Overtones:

can appear with law intensity. The density of the overtone bands increases with increasing vi. The energy of the

anharmonic oscillator is (approach):

(

,...

3 , 2 i , 0 j v

v

v _i  _j  



)











 



 

 

 



 

 



2

2 1 2

E_v h v 1 x v

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:













 ^{3 6}

1 6 2

3 1

2 2 2

2 ~ 2

4

2 ^N

i

i N

i

i

i Q T Q

c

V   

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.