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Advanced Chemical Physics

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Advanced Physical ChemistrySpectroscopy

– Electronic spectroscopy (basics in quantum mechanics)– Vibrational spectroscopy (IR+Raman)– Time resolved spectroscopy– Surface spectroscopy– Single molecule spectroscopy– Photoelectrons spectroscopy

Advanced Topics in Thermodynamics and Kinetics– Liquid-solid interfaces (wetting, contact angle)– Molecules at Interfaces (Langmuir films, self-assembled layers)– Catalysis (Chemisorption, kinetics, mechanisms)– Structure and dynamics in liquids

Books: 1. Modern Spectroscopy, J. M. Hollas, John Wiley&Sons 2. Molecular Vibrations, Wilson, Decious and Cross, Dover

Publications Inc. 3. Physical Chemistry of Surfaces, A.W. Adamson and Cast, Wiley-

Interscience Publication. 4. “An Introduction to the Liquid State" by P.A. Egelstaff, Oxford

University Press.

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Electronic Spectroscopy

• Quantum mechanics- Born-Oppenheimer approximation• Molecular symmetry• Electromagnetic radiation and its interaction with atoms

and molecules• Coupling of angular momenta• Classification of electronic states and selection rules.• Vibronic spectra, Franck-Condon principle and selection

rules• Non Born-Oppenheimer effects, radiationless transitions.

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Quantum mechanics- Born-Oppenheimer approximation

In 1924 Louis de Broglie recognized the similarity that exists between Fermat’s principle of least time, which

governed the propagation of light, and Maupertuis’s principle of least action, which governed the

propagation

of particles. He proposed that with any moving body there is associated a wave and that the momentum

of the particle and the wavelength are related by: p=h/.

It can be shown that as a result of this relation one obtains also the Heisenberg uncertainty principle:

p x ≥h.

Hence in order that an electron will reside in a radius around a nuclei a standing wave must exist in which

rn

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In quantum mechanics we deal with the solution of the Schrödinger Equation,

which is an equation for the spatial and temporal behavior of the de Broglie

waves:

The Schrödinger Equation is given by:

Vzyxm2H

tiH

2222222

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The equation can be separated into two parts (and making it one dimensional):

)t(E)t(dt

di

)x(E)x()x(V)x(dx

d

m2 2

2

Hence:

tiE

e)x(Ct,x

The time independent equation has the form of a standing wave. The time dependent

part results in a phase, which does not effect the probability which is given by

(x,t) *(x,t), when *(x,t) is the complex conjugate of (x,t).

Most of the spectroscopy requires only to understand the standing wave

approximation.

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In the case of molecules, the Hamiltonian is very complex because it contains the kinetic energies and

interactions between many particles- all the nuclei and all the electrons.

For the hydrogen atom, and for hydrogen-like ions with a single electron in the field of a nucleus with

charge +Ze, the Hamiltonian is given by

r4

Ze

2H

0

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For polyelectronic atom the Hamiltonian becomes

i ji ij0

2

i0

2

i

2i r4

e

r4

Ze

2H

where the summation is over all electrons i. Because of the last term, the Hamiltonian cannot be

broken down into a sum of contributions from each electron and the Schrödinger equation can no

longer be solved exactly. To overcome this one uses the Hartree-Fock approximation, not to be

discussed here.

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In the case of polyatomic molecules, in addition to the electron-nuclei and electron-

electron terms in the potential energy we have also the nucleus-nucleus repulsion in

the kinetic energy we have in addition to the electronic kinetic energy also the nuclei

kinetic energy. Hence,

H=Te+Tn+Ven+Vee+Vnn

In the Born Oppenheimer approximation one assumes that since the nuclei are much

heavier than the electrons, the electrons adjust instantly to any change in the nuclei

configuration (a “classical” way of thinking) and therefore one can solve the equation

for the electrons at each fixed configuration of the nuclei (can you see why this is

“wrong” in quantum mechanics?). Hence the nuclei configuration is serving as

parameters in the electronic equation.

Now the hamiltonian has the form of:

H= He+Hn

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Therefore it is possible now to solve the equation separately for the electrons and the

nuclei so that:

)Q,q()Q(E)Q,q(H 0ee0ee

where He=Te+Ven+Vee and Hn=Tn+Vnn+Ee when Ee contribute to the potential

energy of the nuclei due to the solution of the equation of the electrons.

So now the equation for the nuclei is given by:

)Q(E)Q(H nnnn

This means that: )Q()Q,q()Q,q( ne

and E=Ee+En

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In order to solve the Schr ödinger equation the following operation has to be performed:

In order that the integral will be no zero it is required that the expression iiH will be

symmetric in , hence I must be either symmetric or antisymmetric with regards to the

symmetry of H.

Hence if we know the symmetry of H we know something about the property of the

wavefunction.

iiiii

iiiiiiii EEH

EdEdH

:obtain and sidesboth from ontegrate wenow

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Molecular Symmetry

Elements of Symmetry-

n-fold axis of symmetry Cn

Plane of symmetry, v; h; d (d=dihedral)

Center of inversion, i

n-fold rotation-reflection axis of symmetry, Sn h x Cn=Sn

The identical elements, I

Conditions for chirality- A molecule is chiral if it does not have

any Sn symmetry element with any value of n.

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Point groups- All elements of symmetry which any molecule may have constitute a point

group.

Cn point group contains Cn axis of symmetry and Cn2, Cn

3,.. Cnn-1

Cnv point group contains a Cn axis and n planes of symmetry all of which contain

the Cn axis.

Dn point group contains a Cn axis and n C2 axes.

Cnh contains Cn axis and a h plane perpendicular to Cn

Dnd point group contains a Cn axis, an S2n axis, n C2 axes perpendicular to

Cn and nd planes (ethan in staggered configuration belongs to D3d).

Dnh point group contains a Cn axis, nC2 axes, a h plane and n other

planes. (benzen belong to D6h point group).

Td point group contains four C3 axes, three C2 axes, and six d planes. (tetrahedral).

Character table

Character table of a point group summarized the way a state is transformed under a certain

symmetry operator belonging to this group.

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Electromagnetic radiation and its interaction with atoms and molecules

Electromagnetic radiation, means that energy is moving in the form of both electric and

magnetic energies. The perturbed the irradiated region by oscillating fileds, where the

electric (E) and magnetic (H) fields are perpendicular to each other. For a radiation

travelling in the direction x the E and H vectors are in in the direction of y and z

respectively.

)2sin(

)2sin(

kxtAH

kxtAE

z

y

where A is the amplitude. Because k is the same for each component, they are in-phase.

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The plane of polarization is conventionally taken to be the

plane containing the direction of E and that of the propagation,

this is the plane xy.

This is because the interaction of radiation with matter is much

stronger through E than through H.

En

Em

E

n

m

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The rate of change of population Nn of state n due to induced absorption is given by

)(

mnmn BN

dt

dN

Where Bnm is a the Einstein coefficient and the spectral radiation density is given by:

Similarly, induced emission changes the population of Nn by

)(

nmmn BN

dt

dN

where Bnm=Bmn. For spontaneous emission nmnn AN

dt

dN

1)exp(

8)(

3

kThc

hc

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In the presence of radiation that has an energy of ~ wavenumbers, all processes occurring at

once. If the radiation is on, the population reaches equilibrium and

0AN)~(BNNdt

dNnmnnmnm

n

At equilibrium the populations in the two states are related through the Boltzmann distribution

kT

Eexp

g

g

N

N

m

n

m

n

From the above equations one obtains the ration between Anm and Bnm

nm3

nm B~hc 8A

This equation illustrates that the spontaneous emission increases rapidly, relative to the induced

emission, as the excitation energy increases.

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Electric field can interact with atoms or molecules through the dipole operation (first

order).

The dipole is defined as i

iik kq when qi is the charge of the i-th particle, ki is

the coordinate of the i-th particle and k=x,y,z.

The dipole transition moment has the form of

dkR mk*n

nmk

and the transition probability is given by:

k

2nmk

2nm RR It is related to Bnm by:

2nm

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3

nm Rh34

8B

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In absorption experiment NlII )(exp0 when is the cross section for the process, N

are the density of molecules and l is the length of the measuring system. The absorbance is

defined as:

cI

IA )(log 0

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Often one uses the term max however the physical meaningful value is the integrated area under

the peak assuming that Nn<<Nm (no stimulated emission).

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The relation between Bnm and the molar absorptivity is:

10ln

B~hN~d)~( nmnmA

~

~

2

1

.

NA is the Avogadro number.

If the absorption is due to electronic transition, sometimes one uses oscillator strength as a

measure for the strength of the transition

~d)~(eN

10lncm4f

2

1

~

~2

A

2e0

nm

The quantity fnm is dimensionless and is the ratio of the strength of the transition to that of an

electric dipole transition between two states of an electron oscillating in three dimensions in a

simple harmonic way. The maximum value of fnm is usually one.

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Short Summary

The “dream” of any quantum chemist is to present the hamiltonian in the form of:

H=h1+h2+ h3+….

When hi is hydrogen atom like hamiltonian.

The solution is given than as:

E=E1+E2+E3+…

Alternatively one tries to write the hamiltonian as sum of hamiltonians with the terms that couple them as “off-diagonal” terms.

....321

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Short Summary-continue

The transition dipole moment

ezey,ex, dR mnm*nmn

Because of symmetry considerations the functionin the integral must be symmetric.Since the dipole moment is always anti-symmetric ,in order that a dipole transition will be “allowed”, one of the wave functions must be symmetric and the other anti-symmetric.

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Line width

Natural line broadening

If the excited state is populated in excess of its Boltzmann population, the excited state will decay

to a lower state until the Boltzmann population is regained. The process is described by:

when k is the first-order rate constant and 1/k=.

is the time taken for Nn to fall to 1/e of its initial value and

is referred to as the lifetime of the state n. If spontaneous emission is the

only decay process than k=Anm.

The Heisenberg uncertainty principle relates the life time to the energy width E .

nn kN

dt

dN

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From all the above one can show that:

2nm3

0

34

nm Rhc34

64A

And from the uncertainty relation we obtain:

2nm

30

33R

hc34

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Hence, the linwidth dependence on indicate that it increases

very fast with the energy. This natural linewidth in typically very

small compared to other sources for line broadening.

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Doppler broadening

The relation of the transition frequency to the velocity of the absorbance relative

to the radiation source is given by:

1a

a c

v1

where c is the speed of light.

Because there is a Maxwell distribution values of velocities for a given temperature,

the characteristic broadening is given by:

2

1

m

2lnkT2

c

where m is the mass of the absorbance.

This broadening is inhomogeneous since not all of the absorbance

are absorbing at the same frequency.

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Pressure broadening:

Collisions shorten the lifetime of excited states.

If is the mean time between collision and each collision results

in a transition between two states,

the result line broadening is 12 .

This broadening produces usually Lorenzian line shape,

however for low frequency transitions a non symmetrical line shape may appear.

Laurenzian line shape

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0 2

2)(

g

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Coupling of angular momenta

In the case of hydrogen atom the electronic wavefunction has the form:

),(Y)r(R),,r( mn . The

mY functions are known as the angular

wave functions or, because they describe the distribution of over the

surface of a sphere of radius r, spherical harmonics.

For , the quantum number n=1,2,3…is the main quantum number and ℓ

is the azimuthal quantum number associated with the discrete orbital

angular momentum values, and mℓ is known as the magnetic quantum

number which results from the space quantization or the orbital angular

momentum. These quantum numbers can take the values:

ℓ= 0,1,2 …,(n-1); mℓ=0,±1,±2,….., ±ℓ

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The function mY can be factorized further to give

)imexp()(2),(Y mm 21

The functions )(m are the associated Legendre polynomials

that are independent on Z, the nuclear charge number, and

therefore the same for all one-electron atoms.

Each electron in an atom has two possible kinds of angular

momenta, one due to the orbital motion and the other to its spin

motion. The magnitudes of the angular momenta are:

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1ss and 1

The total angular momentum of an electron is given by

s-1.......-s s,j when sj .

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There are two limits for coupling of angular momenta-

1. Coupling of the orbital angular momentum and spin of the electron to produce j.

This limits assumes that the coupling between electrons (spin and orbital angular

momenta) is small. It is called the jj coupling approximation and it is useful only

in the case of heavy atoms.

2. Coupling between the all orbital angular momenta of the electrons is strong and

between the spins is also appreciable. This is the Russel-Suanders coupling

approximation and is the most useful one (also called sometimes the LS coupling

scheme).

In this latter case we define for two electrons 212121 ,.....,1,L and

For L=0,1,2,3 the terms o the atoms are labeled S,P,D,F….

For example the 2p13d1 configuration gives rise to P,D, and F terms.

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In a similar way S=s1+s2, s1+s2-1, …. , │s1-s2 │.

The value 2S+1 is called the multiplicity and so for S=0, 2S+1=1

it is called singlet and for S=1, 2S+1=3 and the multiplicity is a triplet.

We can now define the “total angular momentum ” which is given by J=L+S.

J is restricted to the values: J=L+S, L+S-1, ….., │L-S│ .

Hence if we have two atoms C 1s22s22p13d1 and Si 1s22s22p63s23p13d1

electronic configurations the following terms can be obtained:

(the assignment is (2S+1)LJ)

1P1, 3P0,

3P1, 3P2,

1D1,3D1,

3D2, 3D3,

1F3,3F2,

3F3, 3F4

The terms relate to electronic states.

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Coupling between two equivalent electrons –

Here one has to take into account the Pauli exclusion principle

that is in danger unless the two electrons have different values

of either mℓ or ms.

For example we consider carbon in the ground configuration

1s22s22p2. We have to consider only the 2p electrons (n=2 ℓ=1).

For one electron we have ℓ1=1 and (mℓ)1=+1,0,-1 and s1=1/2,

(ms)1=+1/2 or -1/2 and similarly for the second electron. The

Pauli exclusion principle requires that the pair of quantum

number mℓ ms cannot simultaneously have the same values for

the two electrons. Hence it can be shown that only three terms

arise- 1D,3P,1S.

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Hund formulated two rules:

1. O the term arising from equivalent electrons, those with the

highest multiplicity lie lowest on energy.

2. Of these, the lowest is that with the highest value of L.

The splitting of a term by spin-orbit interaction is proportional to

J: EJ-EJ-1=AJ where EJ is the energy corresponding to J.

If A is positive, the component of the smallest value of J lies

lowest in energy and the multiplet is said to be normal.

If A is negative the multiplet is inverted.

1. Normal multiplets arise from equivalent electrons when

a partially filled orbital is less than half full.

2. Inverted multiplets arise from equivalent electrons when

a partially filled orbital is more than half full.

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Classification of electronic states and selection rules.

Electronic Spectroscopy of Diatomic Molecules

Molecular orbitals

In the molecular orbitals (MO) approach is to consider the nuclei, without their electrons,

at a distance apart which equal to the internuclear equilibrium distance, and to construct

MOs around them from linear combination of the atomic orbitals (AO).

Electrons are then fed into the MOs in pairs.

Hence the molecular orbital = ∑i cii when i are the atomic orbitals.

For the MOs to be different from the atomic orbitals, three conditions must exist:

1. The energies of the AOs must be comparable

2. The AOs should overlap as much as possible

3. The AOs must have the same symmetry properties with respect to certain symmetry

element of the molecule.

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For homonuclear diatomic (lets take N2 as an example) = c11+ c22.

The 1s AOs satisfy condition (1), since their energies are identical, but not condition

(2).

On the other hand, the 2s AOs satisfy all conditions.

However if we take 2s AO with 2px (z is the axis connecting the two atoms)

the overlap between the two AOs cancel.

From two s AOs it is possible to form two MO with cylindrical symmetry

in respect to the internuclei axis. This will be a MO.

Two pz AOs also form a MO. Two px or py AO form MOs.

When two identical atoms interact it can be shown that two identical AOs

will form two MOs of the form: 212

12

and their energies will be E±= E± when

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The function with the + sign is symmetric and its energy is lower than

the energy of the antisymmetric wavefunction by 2

(since the two atom attract each other and therefore is negative).

Symmetric and antisymmetric functions, in respect to the inversion

through the center of the molecule, are assigned as g and u symmetries respectively.

It is common to note * and * for bonding and antibonding characters

of the MOs.

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For example, the ground configuration of the fourteen-electron

nitrogen molecule is:

2g4

u2*

u2

g2*

u2

g p2p2s2s2s1s1

Note that bonding orbitals have u symmetry, while bonding

orbitals have g.

There is a general rule that the bonding character of an electron in a

bonding orbital is approximately cancelled by the antibonding

character of an electron in an abtibonding orbital.

Therefore we define:

Bond order=1/2 net number of bonding electrons.

Hence, for nitrogen the bond order is three.

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For heteronuclear diatomic, when the two atoms are similar (NO,

CO, CN etc.) the treatment is the same. When the atoms are very

different (HCl for example) the MO method can be applied but

than because of the difference in energies between the coupled

AOs, the parameter will usually be very small and therefore the

mixing between the AOs will be small. The measure for the

energy is the ionization potential from each AO.

For example in HCl the chlorine atom has configuration that is

KL3s23p5, but only the 3p electrons have comparable energy with

the hydrogen 1s electon (ionization energies of 12.967 and 13.598

eV respectively). Of the 3p orbitals it is only 3pz which has the

correct symmetry for linear combination with the hydrogen 1s

orbital.

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Classification of electronic states

For all diatomic the coupling approximation that best describes electronic states is

thatwhich is analogous to the Russel-Saunders approximation in atoms (LS). The orbital

angular momenta of all the electrons are coupled to give the resultant L and all the

electron spin momenta produce S.

However, unless one of the nuclei is highly charged, the coupling between L and S is

sufficiently weak, that instead of being coupled to each other, they couple instead to the

electrostatic field produced by the two nuclei. This situation is called Hund’s case a.

The vector L is so strongly coupled to the electrostatic field and the consequence

frequency of precession about the internuclear axis is so high that the magnitude of L is

not defined, L is not a “good ” quantum number. Only the projection of L on the

internuclear axis, , is defined, where =0,1,2,3. All electronic states with >0 are

doubly degenrate (clock wise and anti clockwise rotation). The value of are sympolized

by ,,,, … corrsponfing to =0,1,2,3,4..

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The coupling of S to the internuclear axis is due to the magnetic field along the axis due to

the orbital motion of the electrons. The components of S along the internuclear axis is

. The quantum number is analogous to Ms in an atom and can take the values:

=S,S-1,…,-S.

S remains a good quantum number and for states with >0, there are 2S+1 components

corresponding to the number if values can take. So for example a state can be 3.

The components of the total angular momentum along the internuclear axis is .

For example for =1 and =1,0,-1 the three components of 3 have =2,1,0 and are

symbolized by: 32, 31,

30. Spin-orbit interaction splits the components so that the

energy level after interaction is shifted by AE where A is the spin-orbit

couplog constant.

If one of the nuclei is heavy (high Z) LS coupling may be strong enough to couple L and S

to J (Hund’s case c). Now the states will be labeled according to the value of .

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Selection rules

For atoms the selection rules were specified entirely by L,S and J. In diatomic the

quantum numbers , S and are not sufficient. We must use one (for

heteronuclear) or two (for homonuclear) symmetry properties of the electronic wave

function. [There is a relation between the number degree of freedom and the number

of quantum numbers required to describe the system].

The first symmetry is g or u, which indicates that e is symmetric or antisymmetric

respectively to inversion through the center of the molecule (this is for homonuclear

molecules only). The second element of symmetry applies to all diatomic and

concerns the symmetry of e with respect to reflection across any (v) plane

containing the internuclear axis, If e is symmetric, the state is labeled + and if it is

antisymmetric it is labeled -. This is usually used for states. In states there is

double degeneracy due to the + and – states but the label is not often used.

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Since we consider electric dipole type electronic transitions

the selection rules are:

1. .-or -not but

allowed are ons transiti-,-,- examplefor 1,0

2. S=0. This restriction does not hold for very heavy atoms,

for example in I2.

3. =0; =0,±1 for transitions between multiplet components.

4. ; This relevant only for - transitions so that only

+-+ or --- are allowed.

5. gu; No g to g or u to u transitions.