Molecular Spectroscopy

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I. OPTICAL TRANSITIONS IN MOLECULES: INTRODUCTION

The origin of spectral lines in molecular spectroscopy is the absorption, emission, and

scattering of a proton when the energy of a molecule changes. In atomic spectroscopy,

all transitions reflect changes in the configuration of electrons. In molecules, in addition,

there are energy states corresponding to nuclei vibrations and rotations. In consequence,

the molecular spectra are more complex than atomic spectra and contain information on

the molecular structure and the bond strength. They also provide a way of determining

a variety of molecular properties, like dipole and quadrupole moments and the quantum

numbers characterizing all molecular degrees of freedom. The molecular spectroscopy is

also important for astrophysical and environmental science, for investigation of chemical

reactions, for biology, and in many other areas of science and technology which needs detailed

investigation of properties of microscopic atomic and molecular objects.

In the Born-Oppenheimer approximation the energy of a molecule can be presented as

sum of electronic energy Eel, vibrational energy Evib and rotational energy Erot energy.

E = Eel + Evib + Erot (1)

The energy difference corresponding to the excitation of electrons Eel = hel in this

approximation is much larger that the energy difference corresponding to the molecular

vibration Evib = hvib which is large that the energy difference corresponding to the

molecular rotations Erot = hrot:

hel hvib hrot (2)

Usually, transitions within the rotation energy levels manifold belong to the far infrared

and microwave spectral region, transitions within the vibrational energy levels manifold

belong to the infrared spectral region, and the transitions between the electronic energy levels

belong to the visible, or ultraviolet spectral region. In general, the vibrational transitions

result in changes in the rotational mode and the electronic transitions result in changes in

the rotational and vibrational modes as well. Selection rules determine whether transitions

are allowed or not.

As discussed above the Time-Dependent Perturbation Theory shows that the probability

of absorbing a light photon by a molecule is proportional to

Wk0 |k|Z |0|2 E2z , (3)

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where k|Z |0 is the matrix element of the transition dipole moment:

k|Z |0 =

kz0dq (4)

and 0 and k denote the wave functions of the molecular initial and excited states, respec-

tively.

Absorbing the photon a molecule passes from its initial (ground) state to the excited state.

Here we assume that the electric vector of light E (the light polarization vector) is parallel

to the laboratory Z axis. As shown in eq. (3) the rate of the transition is proportional to

E2Z , and therefore, to the intensity I of the incident radiation and is also proportional to the

squire of the Z component of the dipole moment matrix element k|z|0. In general, thetotal molecular dipole moment is a vector which can be presented as

~ =

i

qi~ri, (5)

where qi is the charge of the i-th particle (electron, or nucleus), ~ri is the radius-vector of the

i-the particle, and summation is proceeded over all electrons and nuclei of the molecule.

This matrix element in eq. (4) depends on the symmetry of both molecular states which

are indicated by the corresponding quantum numbers. For some particular quantum num-

bers of the initial and excited molecular states the matrix element k|Z |0 can be equalto zero, the corresponding optical transition is called forbidden transition. The relation-

ship between the quantum numbers of the initial and excited molecular states for which the

transition matrix element is not zero are known as transition selection rules.

II. OPTICAL TRANSITIONS AND SPECTRAL LINES

A. Einstein Coefficients

In emission spectroscopy a molecule undergoes a transition from a high energy state

E1 to a state of lower energy E2 and emits the excess energy as a photon. In absorption

spectroscopy the absorption of nearly monochromatic incident radiation is monitored as

function of the radiation frequency. Anyway, the energy of the emitted, or absorbed photon

h is given by the Bohr formula

h = E1 E2 (6)

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These are two convenient units which are often used by spectroscopists for characterizing

a transition energy:

=c

and =

c, (7)

where the wavelength is measured in nanometers, microns, or Angstroms (1A = 0, 1nm)

and the units of wavenumber are almost always chosen as reciprocal centimeters (cm1).

It is easy to see that = 1/.

Emission and absorption spectroscopy give the same information about energy level sep-

aration and only practical considerations generally determine which technique is employed.

As known from the Einstein theory of radiation spontaneous decay of the excited state

E1 can be described by the equation

dN1dt

= A12 N1 (8)

where N1 is population of the level 1 and A12 is the Einstein coefficient of spontaneous

emission.

This equation has the evident solution

N1(t) = N1(t = 0) eA12t (9)

The life time of the state is defined as the time needed for a reduction of the level

population by a factor of 1/e. Thus,

=1

A12(10)

If the molecule is initially in its lower state 2 interacting with incident radiation the

transition rate of the stimulated absorption from the state 2 to the state 1 is

w = B21 , (11)

where d is the radiation energy density and B21 is the Einstein coefficient of stimulated

absorption.

If the molecule is initially in its upper state 1 interacting with incident radiation popu-

lation of the state 2 through stimulated emission transition obeys the equation

w = B12 , (12)

where B12 is the Einstein coefficient of stimulated emission.

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Relationship between all three Einstein coefficients is as follows

B21B12

=g1g2

and A12 =

(8h3

c3

)B12 (13)

where g1 and g2 are the multiplicities of degeneracy of the corresponding states. If the

degeneracy of both states is the same, we have

B21 = B12 (14)

B. The Intensity of Spectral Lines: Absorption

In absorption spectroscopy the ratio of the transmitted light intensity to the incident

light intensity at a given frequency is called transmittance, T of the sample:

T =I

I0(15)

According to the Lambert-Beer law, the transmitted light intensity varies with the

sample length l as

I = I0e N l, (16)

where is an absorption cross section and N is the number of molecules per volume (con-

centration). Adequate units are: cm2 for , cm3 for N , and and cm for l.

Another form of eq. (16) which is widely used in laboratory practice is

I = I010 c l, (17)

where is the extinction coefficient and c is a molar concentration :

c =n

V=

number of molecules

NAV= N/NA, (18)

where NA is the Avogadro number, NA 6.022 1023mol1. The appropriate units aremole/liter ([molL1]) for c and [Lmol1cm1] for .

Each of the coefficient and can be determined from experimental data

ln(I

I0) = c l ln(10) = N l, (19)

where the I, and are function of the light frequency .

In case if the exponent factor = N l in eq.(16) is small compare to unity 1 theexponential function can be expanded over . Keeping in this expansion only first two terms

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one comes to the important for practice particular case called low optical density of the

sample:

I = I0(1 N l). (20)

Integrating the expression in eq.(20) over the light frequency within the absorption peak,

one obtains the integrated cross section < >

< >=

d = Bmnh0c

, (21)

where Bmn is the Einstein absorption coefficient and 0 is the center of the molecular ab-

sorption line.

Thus the Einstein coefficient Bmn can be directly determined from experiment.

C. Spectral Line Shape

1. Lifetime Broadening

It is found that spectroscopic lines from gas-phase samples are not initially sharp.The

same is true for the spectra from solid state and solutions. There are several reasons for the

spectral line broadening. One them is due to quantum mechanical effects. Particularly, if

the quantum mechanical system is changing with time it is impossible to specify the energy

levels exactly.If the system survives in a quantum state for a time , the energy of the level

in principle cannot be known with accuracy better than spontaneous decay spontaneous

decay

E h

(22)

This is fundamental uncertainty relation for energy. In principle, no excited state

has infinite lifetime, thus all excited states are subject of the lifetime broadening and the

shorter the lifetimes of the states involved in a transition, the broader the corresponding

spectral lines. These are two main processes which are responsible to the finite lifetime of

excited states.

Firstly, there is the spontaneous decay which is proportional to the corresponding Einstein

coefficient A. The intensity the spontaneous decay is proportional to the square of the matrix

element of interaction with electromagnetic modes of vacuum and in principle cannot

be changed. However, for some particular pairs of the energy states these matrix elements

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can be very small,