Ebook Physical chemistry (4th edition) Part 2

(BQ) Part 2 book Physical chemistry has contents: Rotational and vibrational spectroscopy, electronic spectroscopy of molecules, magnetic resonance spectroscopy, statistical mechanics, experimental kinetics and gas reactions, chemical dynamics and photochemistry,...and other contents.

The Basic Ideas of Spectroscopy

Einstein Coefficients and Selection Rules

Schro¨dinger Equation for Nuclear Motion

Rotational Spectra of Diatomic Molecules

Rotational Spectra of Polyatomic Molecules

Vibrational Spectra of Diatomic Molecules

Vibration–Rotation Spectra of Diatomic Molecules

Vibrational Spectra of Polyatomic Molecules

The types of spectroscopic transitions that can occur are limited by selectionrules As in the case of atoms, the principal interactions of molecules with electro-magnetic radiation are of the electric dipole type, and so we will concentrate onthem Magnetic dipole transitions are about 10 times weaker than electric dipoletransitions, and electric quadrupole transitions are about 10 times weaker Al-though the selection rules limit the radiative transitions that can occur, molecularcollisions can cause many additional kinds of transitions Because of molecularcollisions the populations of the various molecular energy levels are in thermalequilibrium

Rotational and Vibrational Spectroscopy

Calculation of the energy of light

IR, infrared; UV, ultraviolet The abbreviations for powers of 10 are given inside the back cover of the book IUPAC Report,

“Names, Symbols, Definitions, and Units for Quantities in Optical Spectroscopy,” 1984.

Calculate the energy in joules per quantum, electron volts, and joules per mole of photons

13.1 The Basic Ideas of Spectroscopy

13.1 THE BASIC IDEAS OF SPECTROSCOPY

When an isolated molecule undergoes a transition from one quantum eigenstate

with energy to another with energy , energy is conserved by the emission or

absorption of a photon The frequency of the photon is related to the difference

in energies of the two states by Bohr’s relation,

where we have used the symbol ˜ ( 1/ ) introduced in Chapter 9 for the

transi-tion energy in (SI unit m , but usually cm is used) The wave

number ˜ is the number of waves per unit length If , the process is

pho-ton emission; if , the process is photon absorption The frequency range

of photons, or the electromagnetic spectrum, is classified into different regions

ac-cording to custom and experimental methods as outlined in Table 13.1 By

mea-suring the frequency of the photon, we can learn about the eigenstates of the

molecule being studied This is called molecular spectroscopy

The frequency of the photon in the absorption or emission process often tells

us the kinds of molecular transitions that are involved In the radio-frequency

region (very low energy), transitions between nuclear spin states can occur

(see Chapter 15) In the microwave region, transitions between electron spin

states in molecules with unpaired electrons (Chapter 15) and, in addition,

transi-tions between rotational states can take place In the infrared region, transitransi-tions

between vibrational states take place (with and without transitions between

rota-tional states) In the visible and ultraviolet regions, the transitions occur between

electronic states (accompanied by vibrational and rotational changes) Finally, in

the far ultraviolet and X-ray regions, transitions occur that can ionize or dissociate

molecules

radiant energy density

spectral radiant energy density as a function of frequency

rate of absorption

Einstein coefficient for stimulated absorption.

13.2 EINSTEIN COEFFICIENTS AND SELECTION RULES

in the far infrared and microwave regions

The spectrum of a molecule consists of a series of lines at the frequencies sponding to all the possible transitions Let us consider the transition from state

corre-1 to state 2 The strength or intensity of a spectral line depends on the number ofmolecules per unit volume that were in the initial state (the population density

of that state) and the probability that the transition will take place Einstein tulated that the rate of absorption of photons is proportional to the density of theelectromagnetic radiation with the right frequency The isthe radiant energy per unit volume, so it is expressed in J m (See Section 9.16.)

of the radiant energy of a particular frequency; it is given by

Thus, is expressed in J s m The energy density at the frequency required to cite atoms or molecules from to is represented by ( ) Thus Einstein’s pos-tulate about the of photons is summarized by the rate equation

ex-d

d

is m kg (Note that can be taken as dimensionless or expressed in m )There is a minus sign because decreases when electromagnetic radiation isabsorbed Note that d /d d /d

1 2

1 2

ν

B21N2 ( ρν∼ ν∼12)

N N

21 2 spont

Definition of Einsteincoefficients

rate of spontaneous emission

Einstein coefficient for spontaneous emission.

rate of stimulated emission

Einstein coefficient for stimulated emission.

Excited atoms or molecules do not remain in excited states indefinitely, and

Einstein postulated two processes for their return to the initial state, namely,

spon-taneous emission and stimulated emission, as illustrated in Fig 13.1 The

is given by (here is the population density of state 2)

d

(13 7)d

is s The rate of spontaneous emission is independent of the radiation density,

and the radiation is emitted in random directions with random phases

Stimulated emission is quite different in that its rate is proportional to ( ),

and the electromagnetic wave that is produced adds in phase and direction (i.e.,

coherently) to the stimulating wave The is indicated

by the rate equation

d

d

feature in stimulated emission is that it amplifies the radiation density

Accord-ing to equation 13.8, incident light with frequency causes more radiation to

be produced with exactly the same frequency and direction as long as there are

molecules in state 2 As we will discuss later in more detail, this is the basis for

a which is the acronym for “light amplification by stimulated emission of

radiation.”

Equations 13.6–13.8 have been written for the three separate processes, but

of course all three can occur in a system at the same time so that the whole rate

equation is

This rate equation leads to several interesting conclusions The first is that the

three Einstein coefficients are related to each other This can be seen by

consider-ing the equilibrium situation in which d /d d /d 0 When the system

is in equilibrium, equation 13.9 can be solved for the equilibrium spectral radiant

energy density ( ) to obtain

( / )When the system is in equilibrium, the ratio / is given by the Boltzmann

distribution (Section 16.1) When is the energy of the higher level and is the

energy of the lower level, the shows that

Since is positive, most of the atoms or molecules will be in the lower

energy level at thermal equilibrium If the system is exposed to electromagnetic

radiation with frequency , where , the equilibrium distribution

can be written as

ˆ ˆ

3 12

Planck’s blackbody distribution law

This means that irradiation of a two-level system can never put more atoms or molecules in the higher level than in the lower level.

population inversion.

transition dipole moment

quantum mechanical dipole moment operator

冱 冮

8 ( / )

1because they both apply to a system at equilibrium Comparison of equation 13.13with equation 13.14 indicates that

(13 15)and

8

(13 16)Thus a measurement of any one of the three Einstein coefficients yields all three.The second conclusion from equation 13.9 is that the time course of the irradi-ation can be calculated Since , these symbols can be replaced by , andsince there is no , can be replaced by can be replaced by ,where , and equation 13.9 can be integrated (see Problem 13.4)

This may

be a surprise, but the significance of the conclusion is that laser action cannot beachieved with a two-level system In order to obtain laser action, stimulated emis-sion must be greater than the rate of absorption so that amplification of radiation

of a particular frequency is obtained This requires that

Since , laser action can be obtained only when This tion is referred to as a The way population inversion can beachieved is discussed in the next chapter

situa-Quantum mechanics provides the means to calculate (and ) betweenstates and in terms of the transition dipole moment (and ) is pro-

(13 20)where the sum is over all the electrons and nuclei of the molecule, is the charge,and is the position of the th charged particle To understand how the transition

Molecules and Radiation

23 24

2

selection rules,

lifetime

total radiative lifetime

13.2 Einstein Coefficients and Selection Rules

moment enters, we can think of the molecule interacting with the electric field of

the radiation because of a transient or fluctuating dipole moment given by

equa-tion 13.19

From equation 13.19, we see that if the transition dipole moment vanishes

(usually because of symmetry), the spectral line has no intensity The rules

gov-erning the nonvanishing of are called and these allow us to

make sense out of observed molecular spectra

If the transition moment from state to state is nonzero and there is enough

population in the initial state, then the spectral line will be seen in the spectrum

The quantum mechanical derivation of the relationship between the Einstein

co-efficients and the transition probability is too advanced for this book;* however,

the final results are given here When the ground state and excited states have

degeneracies of and , the Einstein coefficient is given by

16

(13 21)3

This equation indicates that the rate of spontaneous emission, , increases

rapidly with frequency; as a matter of fact, this rate is negligible in the microwave

and infrared regions, and so only absorption spectra are measured In the visible

and ultraviolet regions spontaneous emission is significant, and both emission and

absorption spectra are measured The Einstein coefficient is given by

2

(13 22)3

If the rate of spontaneous emission is negligible, the net rate of absorption is

given by

This shows that if the populations of the two states are equal, there will be no net

absorption of radiation

We can also think of as a measure of the lifetime of state 2 Consider

molecules in (excited) state 2 with no radiation field present (and so no stimulated

emission) The molecules will make a transition to state 1, emitting a photon

fre-quency ˜ , with a probability Every time this occurs, decreases After

a time , the number of molecules per unit volume in state 2 is given by

where we have defined the Actually, if a molecule in state 2 can

also make transitions to states 3, 4, (with photons of frequency ˜ ˜ ),

1

(13 25)

If other decay processes besides radiative transitions are possible (such as

non-radiative transitions) we must add those rates to equation 13.25 to get the total

decay rate (inverse lifetime)

Radiati e lifetimes and transition moments

2 2

tr rot vib

The radiative lifetime of a hydrogen atom in its first excited level (2p) is 1 6 10 s What

is the magnitude of the electronic transition moment for this transition? The

tional

vibra-R

¨ 13.3 SCHRODINGER EQUATION FOR NUCLEAR MOTION

We saw in Chapter 11 that the Schro¨dinger equation for a molecule can be treated

in the Born–Oppenheimer approximation so that the isthat for fixed nuclei, while the contains the ki-netic energy operator of the nuclei and the electronic energy (as a function of thenuclear coordinates) as the potential energy operator:

(13 27)where the translational and rotational Hamiltonians contain only kinetic energyterms, while the vibrational Hamiltonian contains ( ), the potential energy de-pending on the internuclear distances These internuclear distances are thecoordinates of the molecule

If the Hamiltonian is the sum of three terms, one for each kind of motion,then the wavefunction can be written as a product of wavefunctions:

(13 28)

rot vib

vib

2 r

1 r

r

2

vibrational degrees of freedom.

termvalues.

13.4 Rotational Spectra of Diatomic Molecules

13.4 ROTATIONAL SPECTRA OF DIATOMIC MOLECULES

The translational wavefunction is that for a free particle (or particle in a very large

box) with a mass equal to the mass of the molecule The translational eigenvalues

are very closely spaced and cannot be probed in molecular spectroscopy, so we

will neglect them in our discussions

To understand the number of coordinates required to describe a polyatomic

molecule, consider the following The total number of coordinates needed to

de-scribe the locations of the atoms in a molecule is 3 However, to describe

the internal motions in a molecule, we are not interested in its location in space,

and so the three coordinates required to specify the position of the center of mass

of the molecule can be subtracted, leaving 3 3 coordinates To describe the

rotational motions of a molecule, we are interested in its orientation in a

coordi-nate system The orientation of a diatomic or linear molecule with respect to a

coordinate system requires two angles, so this leaves 3 5 coordinates to

de-scribe the internal motions The orientation of a nonlinear polyatomic molecule

with respect to a coordinate system requires three angles, so this leaves 3 6

coordinates to describe the internal motions These 3 5 or 3 6 internal

motions are referred to as

ˆ

To sum up, for a diatomic molecule, depends only on two angles, and

ˆ(see equation 9.153); depends only on , the internuclear separation For

ˆpolyatomic molecules, is more complex, depending on 3 6 coordinates

for nonlinear molecules and 3 5 coordinates for linear molecules We will now

turn to a description of the rotational and vibrational eigenstates of both diatomic

and polyatomic molecules

To a first approximation the rotational spectrum of a diatomic molecule may be

understood in terms of the Schro¨dinger equation for rotational motion of the rigid

rotor (equation 9.142) The wavefunctions are the spherical harmonics ( ),

and there are two quantum numbers and for molecular rotation The energy

eigenvalues are given by

¯

2

where is the moment of inertia (Section 9.11) Since the energy does not depend

on , the rotational levels are (2 1)-fold degenerate

In spectroscopy it is standard to express the energies of various levels in wave

numbersbydividing by andreferringtothesevaluesas Termvalues

are usually given in cm , but the SI unit for a term value is m A tilde will be used to

˜indicate the wave numbers in cm Rotational term values are represented by ( )

/ , so that the rotational term values for a diatomic molecule are given by

( 1)

8

Relative population Energy

(e)

e e e

a rigid diatomic molecule are given in Fig 13.2 in terms of the rotational constant.According to the Born–Oppenheimer approximation (Section 11.1), thewavefunction for a molecule in the electronic state , the vibrational state ,and having a particular set of rotational quantum numbers can be written as

a product The transition moment for an electric dipole transition from

a rotational state to a rotational state of the same electronic state istherefore given by

dipole moment to emit or absorb radiation in making a transition between different

E

hc D

3

ⴱ

centrifugal distortion constant

13.4 Rotational Spectra of Diatomic Molecules

The integral over the vibrational coordinate yields the permanent dipole moment

in that particular vibrational state For simplicity, we will write it as , so that the

final result for the integral is

A molecule has a rotational spectrum only if this integral is nonzero

This is expected from the fact that a rotating dipole produces

an oscillating electric field that can interact with the oscillating field of a light

wave A homonuclear diatomic molecule such as H or O does not have a dipole

moment, so it does not show a pure rotational spectrum Heteronuclear diatomic

molecules do have dipole moments, so they do have rotational spectra Polyatomic

molecules are discussed in the next section To find the specific selection rules we

need to find the conditions on the quantum numbers that make the integral in

equation 13.38 nonzero For a linear molecule it can be shown that the transition

moment is nonzero for

This selection rule may be understood in the same way as that for atoms (Section

10.14) Since a photon has one unit of angular momentum, and angular

momen-tum must be conserved in emission or absorption, the angular momenmomen-tum of a

molecule must change by a compensating amount

The frequencies ˜ of the absorption lines due to 1 are given by the

difference between rotational term values (equation 13.33):

ferent species of a given molecule, because the moments of inertia of isotopically

substituted molecules are different

We have been talking about diatomic molecules as if they are rigid rotors,

but of course they are not As the rotational motion increases, the chemical bond

stretches due to centrifugal forces, the moment of inertia increases, and,

conse-quently, the rotational energy levels come closer together This may be taken into

account by adding a term to equation 13.33:

˜

cen-trifugal distortion is taken into account, the frequencies ˜ of the absorption lines

due to 1 are given by

Internuclear distance from rotational spectra

/

/ /

In early measurements of the pure rotational spectrum of H Cl, Czerny found that thewave numbers of absorption lines are given by

where is the quantum number of the lower state What is the internuclear distance in

H Cl? What is the value of the centrifugal distortion constant?

˜From equation 13.41, 10 397 cm Since

˜

we have

˜8

129 pm(The reduced mass of H Cl is given in Example 9.21.) The centrifugal distortion constant

vibra-We have discussed the selection rules that determine the transitions that cangive rise to absorption or emission, but we already noted that there is another fac-tor that determines the observed intensities, namely, the population of the initialstate given by the Boltzmann distribution (equations 13.11 and 16.2) The fraction

of the molecules in the th energy state is given by

(13 42)e

where is the denominator If the energy of a state is large compared with , theprobability of finding a molecule in that state at equilibrium will be small Because

of degeneracy (Section 9.7), many states of a molecule may have the same energy,and these degenerate states make up the energy When energy levels areused, the Boltzmann distribution can be written

e

(13 43)e

J J

i

hcJ J B kT J

kT

kT

J

i i i

2

moment of inertia

13.5 Rotational Spectra of Polyatomic Molecules

13.5 ROTATIONAL SPECTRA OF POLYATOMIC MOLECULES

where is the degeneracy (Section 9.7) of the th level As discussed earlier, the

component of the angular momentum in a particular direction is equal to ¯ ,

where may have values of ( 1) 0 , where is the rotational

quantum number Thus, there are in all 2 1 different possible states with

quan-tum number In the absence of an external electric or magnetic field the energies

are identical for these various sublevels, and so the th energy level is said to have

a degeneracy of 2 1 The rotational energy in the absence of an external

elec-tric or magnetic field, ignoring in equation 13.41, is given by ( 1)

so that, using equation 13.42, the fraction of molecules in the th rotational level

is given by

(2 1) e

(13 44)According to this equation, the number of molecules in level increases with

at low values, goes through a maximum, and then, because of the exponential

term, decreases as is further increased The lines in the spectrum at the bottom of

Fig 13.2 have been labeled with the rotational quantum number of the upper of

the two states involved The intensities of the lines are proportional to the

popu-lations in the lower state involved in the transition

For molecules with larger moments of inertia , the rotational energies

are smaller, in fact, small compared with The quantum numbers may

be-come quite large before e becomes appreciably different from unity For

small quantum numbers populations are proportional to the degeneracies, since

There is a complication in rotational spectroscopy that we will not be able

to discuss The statistics of nuclear spin affect the number of degenerate states at

each level, and therefore the intensities of the rotational lines The use of the

Boltzmann distribution alone is an oversimplification.*

Although homonuclear diatomic molecules do not have permanent electric

dipole moments and do not exhibit pure rotational spectra, they do show

rota-tional Raman spectra (Section 13.9), and their electronic and vibrarota-tional spectra

show rotational fine structure

For the treatment of its pure rotational spectrum we may consider a polyatomic

molecule to be a rigid framework with fixed bond lengths and angles equal to their

mean values For a polyatomic molecule the about a particular

axis that passes through the center of mass of the molecule is simply the sum of

the moments due to the various nuclei about that axis:

(13 45)where is the perpendicular distance of the nucleus mass from the axis

The rotation of a polyatomic molecule can be described in terms of moments

of inertia taken relative to three mutually perpendicular axes The moment about

the axis is

Z(c)

X(a) Y(b)

b a

prolate oblate asymmetric

and the moments of inertia about these axes are called the

, , and The axes are designated by a, b, and c and are fixed withrespect to the molecule and rotate with it The principal moments of inertia aboutthese axes are always labeled so that The principal axes can often

be assigned by inspecting the symmetry of the molecule The momental ellipsoid

is constructed as follows Lines are drawn from the center of mass of the molecule

in various directions with length proportional to ( ) , where is the moment

of inertia about that line as an axis Any symmetry operation of a molecule mustapply to its momental ellipsoid

The principal moments of inertia are used to classify molecules, as shown inTable 13.2 If all three principal moments of inertia are equal, the molecule is a

top If two principal moments are equal, the molecule is atop A molecule is a top (cigar shaped) if the two larger moments areequal The molecule is an top (discus shaped) if the two smaller momentsare equal The molecule is an top if all three principal moments areunequal

The quantum mechanical Hamiltonian operator for the rotational motion ofpolyatomic molecules is found by first writing the classical mechanical energy interms of angular momentum operators Since we know how to convert classicalangular momentum to its quantum mechanical form, we can then find the quan-tum Hamiltonian and solve the Schro¨dinger equation The last part turns out to

be straightforward for all the cases except the asymmetric top We will not discussthe latter

In classical mechanics the rotational energy of a rotor with one degree offreedom is

where is the angular velocity in radians per second, is the moment of inertia,

and is the angular momentum For an object that can rotate in three dimensions

the classical expression for the rotational kinetic energy is

(13 49)Since we will want to convert this to a quantum mechanical expression, it is more

convenient to express it in terms of the angular momentum , where

represents a direction,

(13 50)

in which the components of the total angular momentum about the three principal

axes are given by

(13 51)(13 52)(13 53)The total angular momentum is given by

(13 54)The expressions for the energies of spherical tops, linear molecules, and sym-

metric tops are as follows

For a spherical top, , the momental ellipsoid is a sphere, and

2 8 3

CH , is

(13 57)where is the bond length and is the mass of each of the four atoms arranged

R R

r R R

Moments of inertia of an octahedral symmetric top molecule

Meaning of the tum number

quan-Derive the expressions for the moments of inertia and of the octahedral symmetric

top molecule AB C shown in the diagram

The quantum number determines the component of the angular momentum

along the axis of the symmetric top; this is the angular momentum of rotation

about the symmetry axis When 0 there is no rotation about the symmetry

axis, and the rotation is about the axis perpendicular to the symmetry axis, that is,

end-over-end rotation When has its maximum value ( or ), most of the

molecular rotation is about the symmetry axis (see Fig 13.4)

The specific selection rules for rotational spectra of symmetric top molecules

are 1 and 0 The reason there cannot be a change in quantum

number is that the dipole vector of the molecule is oriented along the principal

axis The electromagnetic field of radiation can affect the rotation of the dipole,

but it cannot affect the rotation of the molecule about its principal axis because

there is no dipole moment perpendicular to the principal axis

The pure rotational spectroscopy of molecules has enabled the most

pre-cise evaluations of bond lengths and bond angles The spectrum of a polyatomic

molecule gives at most three principal moments of inertia; since usually more than

three bond lengths and angles are involved, isotopically different molecules must

be studied, and it must be assumed that isotopically different molecules have the

same set of bond lengths and bond angles In effect, a number of simultaneous

equations are solved for the internuclear distances and angles

To vacuum line

Pre-amp

Lock-in amplifier

Square-wave modulator

Recorder

scope

Oscillo-Klystron

Klystron power supply

Stabilizer

Frequency standard

Frequency counter

Comment:

Microwa e spectroscopy of gases at low pressures can be used to determine rotational frequencies to one part per million since the lines are ery sharp Separate lines are obtained for molecules with different isotopic compositions Since moments of inertia can be determined so accurately, bond lengths and bond angles can be determined with unprecedented precision.

Block diagram of a Stark-modulated microwave spectrometer

v

v

Figure 13.5

These spectra are in the microwave region Microwave radiation is produced

by special electronic oscillators called klystrons Monochromatic radiation is duced, and the frequency may be varied continuously over wide ranges The usualexperimental arrangement is shown in Fig 13.5 Microwave radiation is transmit-ted down in a waveguide that contains the gas being studied The intensity of theradiation at the other end of the waveguide is measured by use of a crystal diodedetector and amplifier The oscillator frequency is swept over a range, and thetransmitted intensity is presented on an oscilloscope or a recorder as a function

pro-of frequency

According to the Heisenberg uncertainty principle, the accuracy with which

an energy level may be determined is inversely proportional to the time themolecule is in this level Hence, to obtain sharp rotational lines of a gas, thepressure must be maintained sufficiently low so that the average time betweencollisions is long compared with the period of a rotation Usually it is neces-sary to determine microwave spectra at pressures below 10 Pa to reduce theline-broadening effects of collisions

The lines in the microwave spectrum are split if the molecules being studiedare in an electric field This so-called Stark effect is due to the interaction of thedipole moment of the gaseous molecule and the electric field Since the splitting

is proportional to the permanent dipole moment, the magnitude of the dipolemoment may be derived from the spectrum

V/eV

3.0 2.5

0.2

2.0 1.5 1.0 0.5 0

e 2

e 2

2

3

3 e 3

1 2 1/2

1 2

e

v

v

v

Potential energy curve for a diatomic molecule At internuclear distances in

the neighborhood of the equilibrium distance , the curve is nearly parabolic, as indicated

by the dashed line The parabolic approximation fails at higher excitation energies (See

Computer Problem 13.G.)

force constant.

Taylor series

13.6 Vibrational Spectra of Diatomic Molecules

13.6 VIBRATIONAL SPECTRA OF DIATOMIC MOLECULES

The harmonic oscillator was discussed in Sections 9.9 and 9.10, but in Chapter

12 we saw that the potential energy curves of diatomic molecules are not exactly

parabolic However, as shown in Fig 13.6, the potential energy curve for a

di-atomic molecule is approximately parabolic in the vicinity of the equilibrium

in-ternuclear distance The potential energies indicated by the dashed line are

given by the parabola

where is the We have seen this earlier as equation 9.107

It is difficult to solve the Schro¨dinger equation for the exact form of ( ),

but we can expand ( ) in a about the equilibrium separation :

ometry, and the second term is zero since d /d is zero at the minimum of the

potential energy curve The third term is given by equation 13.64 If all higher

terms are neglected as giving small corrections, then we have approximated the

exact ( ) by a harmonic potential, and we can solve the resulting Schro¨dinger

equation In Section 9.10, we discussed the solutions of the Schro¨dinger

equa-tion for the simple harmonic oscillator There we saw that the energy levels are

given by

where (1/2 )( / ) and is the red mass of the diatomic molecule (see

Section 9.11) It is standard in spectroscopy to give the energy in terms of wave

numbers, so we divide by :

R Thus, the selection rule for a diatomic molecule is that

a molecule will show a ibrational spectrum only if the dipole moment changes with internuclear distance.

R , ,

The vibrational frequencies for many diatomics are of the order of 1000 cm ,with higher values for molecules with hydrogen atoms or strong bonds, and lowervalues for molecules with heavy atoms or weak bonds

Not all diatomic molecules have an infrared (vibrational) absorption trum To determine which transitions are possible in a vibrational spectrum, wemust use equation 13.35 for the electric dipole transition moment Since the dipolemoment for a diatomic molecule, which is given by equation 13.37, depends onthe internuclear distance, we expand this dipole moment in a Taylor series about

spec-:

1

2For a molecule in a given electronic state, the transition dipole moment for a vi-brational transition is given by

1

2The first term is equal to zero because the vibrational wavefunctions for differentare orthogonal The second term is nonzero if the dipole moment depends onthe internuclear distance

Homonuclear diatomic molecules, such as H and N , have zero dipole ment for all bond lengths and therefore do not show vibrational spectra In gen-eral, heteronuclear diatomic molecules do have dipole moments that depend oninternuclear distance, so they exhibit vibrational spectra

mo-The integral in the second term of equation 13.69 vanishes unless

1 for harmonic oscillator wavefunctions According to this specific tion rule, a harmonic oscillator would have a single vibrational absorption oremission frequency In general, we would expect the second and higher deriva-tives of the dipole moment with respect to internuclear distance to be small;after all, if the dipole moment were due to fixed charges a variable distanceapart, then ( / ) and higher derivatives would be equal to zero Althoughthese higher derivatives are small, they do give rise to overtone transitions with

selec-2 3 , with rapidly diminishing intensities

These can be seen in the vibrational absorption spectrum of HCl representedschematically in Fig 13.7 The strongest absorption band is at 3 46 m; there is amuch weaker band at 1 76 m and a very much weaker one at 1 198 m Theseare the overtone transitions 0 to 2, and 0 to 3 The vibrationalenergy levels of Cl are shown in Fig 13.8

Second overtone

0 →3 1.20 m

8333 cm–1

m /

4 µ

µ λ

First overtone

0 → 2 1.76 m

5682 cm–1µ

Fundamental

0 →1 3.46 m

2890 cm–1µ

/ / 0

0

/

/ 0

35 2

v v

v v v v v

v v

v v

v v

“Stick” representation of the vibrational absorption spectrum of H Cl The

relative intensities of the lines fall off five times as fast as indicated

The potential energy curve for Cl calculated with the Morse potential

(equation 13.82) with every fifth vibrational level from 0 to 40 (See Computer

Problem 13.B.)

13.6 Vibrational Spectra of Diatomic Molecules

冱 冱 冱 冱

For a harmonic oscillator, equation 13.42 indicates that the fraction of the

molecules in the th energy level is given by (note that the levels are

nondegenerate)

ee

e

(13 70)e

The denominator is a geometric series with 1 for which the sum is given by

1

(13 71)1

so that

1

1 eThus, the fraction of the molecules in the th vibrational state is given by

Populations of ibrational states for different temperatures

Calculation of ibrational absorption frequencies

.

4302/ (4302/ )

4 302 0

4 302 1

4 302 2 2

4 302 3 3

2 151 0

2 151 1

2 151 2 2

2 151 3 3

What fractions of H Cl molecules are in the 0 1 2 and 3 states at ( ) 1000 K and ( )

At room temperature this relation predicts that the ratio of the population

of H Cl in 1 to that in 0 is 8 9 10 Therefore, the molecules with

1 and higher do not contribute to the spectrum

Figure 13.6 shows that equation 13.67 is not sufficient to represent the energylevels of a diatomic molecule; if equation 13.67 did apply, the overtones would be

at integral multiples of the fundamental When the Schro¨dinger equation is solvedfor equation 13.65 truncated after the cubic term, it is found that the energy levelsare given by an equation of the form

where ˜ is the vibrational wave number, and are anharmonicity constants,*and 0 1 2 When the third term in equation 13.74 can be ignored, thefrequencies ˜ of absorption lines due to 1 are given by

y

( ) For the harmonic oscillator approximation, the frequencies in wave numbers are

given by ˜ , where is the vibrational quantum number in the higher level in 0

( ) For the anharmonic oscillator approximation, the frequencies in wave numbers

˜ 52 819 cm , the frequencies are given by the following table:

See Fig 13.7 and Computer Problem 13.I

The potential energy

of a diatomic molecule as a tion of the internuclear distance.Only the 0 vibrational level isshown The dissociation energy that

func-we are primarily concerned with inthis chapter is the spectroscopicdissociation energy

Potential energy curves for the ground electronic states of H and H with

the zero-point vibrational levels shown

equilibrium dissociation energy

spectroscopic dissociation energy

13.6 Vibrational Spectra of Diatomic Molecules

⫺

⫺

Figure 13.9

Figure 13.10

from the minimum in the potential energy curve But now we will be dealing with the

measured from the zeroth vibrational level

The relationship between these two dissociation energies is shown in Fig 13.9

The potential energy curves for H and H are shown in Fig 13.10 along

with their respective spectroscopic dissociation energies, (H ) and (H )

2 0

e

2 0

e

i 2 i

2

Dissociation Energies for H (g) and

H (g) and Ionization Potentials for

Thus, the zero-point levels of H and H are separated by 15.4259 eV, as shown

in Fig 13.10 The potential energy curves for H and H at infinite internucleardistance are separated by the ionization potential of a hydrogen atom in its groundstate The ionization potential calculated in Example 10.4 can be corrected for thefinite mass of the nucleus:

di-kJ mol The vibrational parameters for a number of diatomic molecules are given inTable 13.4 According to equation 13.74 the energy of the ground state of a di-atomic molecule is given by

The Morse potential for H Cl

c

a a

2

e e

e e e

a potential energy function for the whole range of values The

is a simple function that provides an approximate potential energy as a function

of internuclear distance in terms of the equilibrium dissociation energy andother spectroscopic properties:

˜

(13 86)4

Since actual potential energy curves differ from the Morse equation, this is not anexact relation, but it is useful when the dissociation energy of an excited molecule,for example, is not known

y

V/eV

6 5

0.1

4 3 2 1 0

J

J

J R

c a

Plot of the potential energy of H Cl versus internuclear distance according

to the Morse equation The actual potential energy curve has a slightly different shape

(See Computer Problem 13.H.)

The equilibrium dissociation energy in m is given by

At high resolution, each of the absorptions in the vibrational spectrum in Fig 13.7

is found to have a complicated structure that results from simultaneous changes

in rotational energy Because of this structure, molecular spectra are often

re-ferred to as The fundamental vibration band for HCl ( 0 1)

is shown in Fig 13.12

When a molecule in a state with vibrational quantum number and rotational

quantum number makes a spectral transition to another state, the vibrational

quantum number changes to 1 (according to the harmonic oscillator selection

rules), and the rotational quantum number can change to 1 or remain the

same The possible transitions are shown in Fig 13.13 The transitions with

1 give rise to lines in the branch of the spectrum, and the transitions with

1 give rise to lines in the branch of the spectrum The intensities of

the lines in these branches reflect the thermal populations of the initial rotational

states The branch, when it occurs, consists of lines corresponding to 0

y

P J'' = 0

J' = 0

ν

v'' v'

1 2 3 4 5 6

1 2 3 4 5 6

P R

–6 –5 –4 –3 –2 –1

2700 2750 2800 2850 2900 2950 3000 3050

1 2 3 4 5 6 7 8 9 10

2nd ed Copyright 1977, Pergamon Press on behalf of IUPAC.)

Vibrational and rotational energy levels for a diatomic molecule and thetransitions observed in the vibration–rotation spectrum when the transition betweenand is allowed In the spectrum shown at the bottom, the relative heights of the spectrallines indicate relative intensities of absorption

v v

E , J hc G F J

B B

vibration–rotation coupling constant.

13.7 Vibration–Rotation Spectra of Diatomic Molecules

⫺

v v

v v

Generally, these transitions are forbidden, except for molecules such as NO, which

have orbital angular momentum about their axes

The energies of the levels in Fig 13.13 are given to the accuracy we need here

which expresses the energy of a level as the sum of the first two terms of the

vibra-tional term value (equation 13.74) and the first term of the rotavibra-tional term value

˜(equation 13.40) Now it is necessary to put a subscript on since the rotational

˜constant depends on the vibrational quantum number Since is inversely pro-

portional to the moment of inertia , it varies as , where is the equilibrium

internuclear distance varies with the vibrational state in 1 is slightly

larger than in 0; therefore,

The dependence of the rotational constant on the vibrational quantum

num-ber is generally represented by

where is the

Now let us consider a vibrational transition from 0 to 1 A molecule

with 0 can have various values, and in going to 1, the value of can go

to 1 or 1 because the selection rule is 1 In the vibrational ground

state, equation 13.87 indicates that the energy is given by

˜

where is the rotational constant when 0 When the molecule absorbs a

photon and 1 and 1, the energy of the upper state is given by

˜

where is the rotational constant when 1 These transitions lead to the

branch of the vibration–rotation spectrum, and the absorption frequencies are

is the center of the vibration–rotation band where there is no absorption because

0 is forbidden If , then these frequencies are equally spaced

When the molecule absorbs a photon and 1 and 1, the energy

of the upper state is given by

Population of rotational states of H Cl at 300 K

J

N N J

0

1 23 2

1 2(5 007 10 ) 0

˜ergy relative to the ground state is represented by

In Section 13.3, we saw that 3 5 coordinates are required to describe the nal motions of a diatomic or linear molecule and 3 6 coordinates are requiredfor a nonlinear polyatomic molecule The different types of vibrational motionthat are possible can be described in terms of whichare described below For a diatomic molecule, 3 5 1, and so there is a singledegree of vibrational freedom and a single normal mode For a linear triatomicmolecule, such as CO , 3 5 4, and so there are four normal modes This

( ) Normal modes of vibration of the symmetrical linear triatomic molecule

CO ( ) Normal modes of vibration of the nonlinear triatomic molecule H O The vectors

representing the magnitudes of the oxygen vibrations have been increased relative to those

means that there are four types of vibrational motion Figure 13.14 provides a

schematic representation of four types of vibrational motion for a symmetrical

linear triatomic molecule and gives the vibrational frequencies in wave numbers

for CO For a nonlinear triatomic molecule, such as H O, 3 6 3, and so

there are three normal modes (Fig 13.14 ) NH , CH , and N O have 6, 9, and

12 normal modes of vibration

To see what normal modes of vibration are, we first consider the vibration of

polyatomic molecules from a classical mechanical viewpoint The kinetic energy

of a polyatomic molecule is given by

where and so on are the values of the coordinates at the equilibrium geometry

of the molecule Since these are independent of time, the kinetic energy becomes

of several variables:

1

(13 98)2

Since is the potential energy at the equilibrium configuration, it is a constantwhich we can set equal to zero, and the terms in ( / ) are all equal to zerobecause the potential energy is a minimum at the equilibrium configuration bydefinition If we neglect terms higher than quadratic, equation 13.98 can be written

1

(13 99)2

so that the total energy is given by

to as and the corresponding 3 6 or 3 5 vibrations arereferred to as

In a normal mode of vibration, the nuclei move in phase (i.e., the nucleipass through the extremes of their motion simultaneously) The motions of thenuclei in a normal mode are such that the center of mass does not move, andthe molecule as a whole does not rotate This means that different atoms movedifferent distances Each normal mode has a characteristic vibration frequency.Sometimes several modes have identical vibration frequencies and are referred

to as degenerate modes It can be shown that any vibrational motion of a atomic molecule can be expressed as a linear combination of normal modes ofvibrations

b

The gross selection rule is still that the displacements

of a normal mode must cause a change in dipole moment in order to be

spectro-scopically acti e in the infrared.

i i

1 2 1

1/2 2

Turning now to the quantum mechanical treatment of a molecule, the

vibra-tional Hamiltonian obtained from equation 13.101 is simply a sum of terms, one

for each coordinate:

This indicates that the vibrational wavefunctions for the molecule can be written

as the product of harmonic oscillator wavefunctions, one for each coordinate

We have seen in equation 9.116 that the eigenvalues for the harmonic oscillator

are given by ( ) ˜ so that the total vibrational energy of a polyatomic

molecule is

The frequency of a normal mode depends both on the force constant for the

mode and on the reduced mass for the mode: 2 ( / )

The four normal modes of CO are shown in Fig 13.14 The first normal

mode is a symmetrical stretching vibration in which the carbon atom remains

fixed The third normal mode is an asymmetrical stretching vibration The other

two normal modes are orthogonal bending vibrations The lower vibration

fre-quency for the bending vibrations indicates that it is generally easier to bend a

molecule than to stretch it Figure 13.14 shows the three normal modes of

vibra-tion of H O As indicated in the diagrams, the displacements of various atoms in

a normal mode are not equal, but depend on the masses and force constants

For a polyatomic molecule, some normal modes of vibration are

spectroscop-ically active and some are not

Of the four normal-mode vibrations for CO the symmetric stretch is not

ac-tive in the infrared, but the other vibrations are Since CO is linear and

symmetri-cal in its equilibrium state, it does not have a dipole moment, and the symmetrisymmetri-cal

stretching vibration does not create one The asymmetric stretch and bending

vi-brations produce a changing dipole moment The three normal modes of H O are

all active in the infrared because the magnitude of the dipole moment changes in

each type of vibration

The specific selection rule for vibrational spectroscopy is that 1 in the

harmonic oscillator approximation In addition, combination bands are formed in

which two or more vibrational modes change simultaneously

The frequencies, in cm , of the strongest bands for H O vapor are

sum-marized in Table 13.5 The weaker bands in the spectrum are the overtones and

combinations shown in the table As shown in Table 13.5, the vibrations are not

harmonic, and so the overtones are not exact multiples, and the combinations are

not exact sums

One of the vibrational motions of a polyatomic molecule may be an internal

rotation If there is an appreciable potential energy barrier for an internal rotation

about some bond, there will be an oscillation about the mean position For

exam-ple, in ethylene, CH CH , there is a large potential energy barrier for internal

rotation, so that there are only small oscillations about the C C bond In some

Intensity Interpretation

1

2 2 1 3

1 1

1 1 1

1

1

1 1

1 1

identifica-The infrared spectrum of a molecule may be considered to be made up ofseveral regions

Hydrogen stretching vibrations, 3700–2500 cm These vibrations occur athigh frequencies because of the low mass of the hydrogen atom If an OHgroup is not involved in hydrogen bonding (Section 11.10), it usually has afrequency in the vicinity of 3600–3700 cm Hydrogen bonding causes thisfrequency to drop by 300–1000 cm or more The NH absorption falls inthe 3300- to 3400-cm range, and the CH absorption falls in the 2850- to3000-cm range For SiH, PH, and SH, it is approximately 2200, 2400, and

2500 cm Triple-bond region, 2500–2000 cm Triple bonds have high frequencies be-cause of the large force constants The C C group usually causes absorptionbetween 2050 and 2300 cm , but this absorption may be weak or absent be-cause of the symmetry of the molecule The C N group absorbs near 2200–

2300 cm Double-bond region, 2000–1600 cm Absorption bands of substituted aro-matic compounds fall in the range 2000–1600 cm and are a good indicator

of the position of the substitution Carbonyl groups, C O, of ketones, hydes, acids, amides, and carbonates usually show strong absorption in thevicinity of 1700 cm Olefins, C C, may show absorption in the vicin-ity of 1650 cm The bending of the C N H bond also occurs in thisregion

alde-Single-bond stretch and bend region, 500–1700 cm The region 500–

1700 cm is not diagnostic for particular functional groups, but it is a useful

“fingerprint” region, since it shows differences between similar molecules.Organic compounds usually show peaks in the region between 1300 and

1475 cm because of the bending motions of bonds to hydrogen plane bending motions of olefinic and aromatic CH groups usually occurbetween 700 and 1000 cm

Applying the selection rule that a normal mode will ha e a ibrational spectrum

only if the dipole moment changes in the ibration may be difficult for the normal

modes of a polyatomic molecule Fortunately, that information is in the character

table for the symmetry group An example of a character table was gi en at the

end of the preceding chapter (see Table 12.4) That character table for C shows

that for any molecule in this symmetry group some of the ibrational modes will

be infrared acti e Water is an example, as shown by Fig 13.14b.

When a sample is irradiated with monochromatic light, the incident radiation may

be absorbed, may stimulate emission, or may be scattered A part of the scattered

radiation is referred to as the It is found that some photons

lose energy in scattering from a molecule in the sample and emerge with a lower

frequency; these photons produce what are referred to as in the

spec-trum of the scattered radiation A smaller fraction of the scattered photons gains

energy in striking a molecule in the sample and emerges with a higher frequency;

these photons produce what are referred to as in the spectrum of

the scattered radiation Only a very small fraction (usually less than 1 part in 10 )

of the incident radiation is scattered, and the frequency shifts may be quite small;

since lasers can produce very intense radiation that is highly monochromatic, they

are used as the radiation source

The interpretation of Raman spectra is based on the conservation of energy,

which requires that when a photon of frequency is scattered by a molecule in

a quantum state with energy and the outgoing photon has a frequency , the

molecule ends up in quantum state f with energy :

(13 104)or

where the shift in frequency is labeled and the shift in wave number is labeled

˜ Notice that Raman spectroscopy is different from absorption or emission

spectroscopy in that the light need not coincide with a quantized energy

difference in the molecule Therefore, any frequency of light can be used Since

many final states are possible, of both higher and lower energy than the initial

state, many Raman spectral lines can be observed A typical experimental

appa-ratus is shown in Fig 13.15

The frequency shifts seen in Raman experiments correspond to vibrational

or rotational energy differences, so this kind of spectroscopy gives us information

on the vibrational and rotational states of molecules

The Raman effect arises from the induced polarization of scattering molecules

that is caused by the electric vector of the electromagnetic radiation Some

as-pects of the Raman effect can be understood classically First we will consider

an isotropic molecule, that is, one that has the same optical properties in all

M2 M1

Laser

Photo tube Amplifier

In order for a ibrational mode

to be Raman acti e, the polarizability must change during the ibration, and for

0

Apparatus for obtaining Raman spectra Mirrors M and M are required

to obtain laser action

where is the The polarizability has units of dipole moment divided

by electric field strength, that is, C m/V m C m /J For an isotropic moleculethe vectors and point in the same direction, and the polarizability is a scalar.The polarizability of a molecule that is rotating or vibrating is not constant, butvaries with some frequency (for example, a vibration or rotation frequency)according to

where is the equilibrium polarizability and is its maximum variation Sincethe electric field of the impinging electromagnetic radiation varies with time ac-cording to

In order for a molecular motion to be Raman active, the polarizability mustchange when that motion occurs (that is, 0)

a rotation to be Raman acti e, the polarizability must change as the molecule

ro-tates in an electric field.

.

E

E

alters the electronic structure The polarizability of an atom or a spherical

ro-tor (Section 13.5) does not change in a rotation; indeed, we cannot even talk

about the rotation of an atom Thus, spherical rotors do not have a rotational

Raman effect All other molecules are anisotropically polarizable; that means

that the polarization is dependent on the orientation of the molecule in the

electric field

When a molecule is anisotropic, the application of an electric field in a

particular direction induces a moment in a different direction In this case is

a tensor, and the induced dipole moment is given by

(13 110)which is expressed by the following matrix equation (see Appendix D.8):

(13 111)

This is equivalent to the following set of algebraic equations:

(13 112)(13 113)(13 114)Thus, each component ( ) of the induced dipole moment can depend

on each component ( ) of the electric field Only six of the nine

coeffi-cients of the polarizability are independent, since it can be shown that ,

The quantum mechanical theory for the selection rules for the Raman effect is

more complicated than for pure rotational and vibrational spectra because Raman

scattering is a kind of two-photon process: The incident photon is absorbed and

the leaving photon is emitted by the molecule in a single quantum process

The specific selection rules for rotational Raman transitions are as follows for

linear and symmetric top molecules:

Linear molecules 0 2

where is the component of the angular momentum along the principal

sym-metry axis The 0 applies in vibration–rotation transitions The fact that

2 for linear molecules is a result of the fact that the polarizability of

a molecule returns to its initial value twice in a 360 revolution, as shown in

Fig 13.16 The 0 is a result of the fact that the dipole of a symmetric top

molecule is along the principal axis, so there cannot be a component of the dipole

moment perpendicular to this axis

The frequencies of the Stokes lines ( 2) in the rotational Raman

spec-trum of a linear molecule are given by

These lines appear at lower frequencies than the exciting line and are referred

to as the branch The relative intensities of these lines are determined by thepopulations of the initial states, as we have discussed for the vibration–rotationspectra in the infrared

The frequencies of the anti-Stokes lines ( 2) in the rotational Ramanspectrum are given by

˜

The lines appear at higher frequencies and are referred to as the branch Inaddition, there is a branch for 0 The , , and branches correspond

to the , , and branches of infrared spectroscopy

The pure rotational Raman spectrum of CO is shown in Fig 13.17 Notice thelarge number of initially populated rotational states, since the rotational splitting

is small compared with

As noted above, for a molecule to have a vibrational Raman spectrum it isnecessary for the polarizability to change as the molecule vibrates The polariz-abilities of both homonuclear and heteronuclear diatomic molecules change asthe molecule vibrates, so both types of molecules have vibrational Raman spectra

in contrast to infrared vibrational spectra The specific selection rule for the brational Raman effect is 1 The vibrational transitions are accompanied

vi-by rotational Raman transitions with the specific selection rules 0 2, as

Theoretical rotation–vibration Raman spectrum of a linear molecule The

effects of nuclear spin statistics have been omitted from this illustration (From W A

before The vibrational Raman spectra of homonuclear diatomic molecules are

of special interest because they yield force constants and rotational constants that

are not available from infrared absorption spectroscopy

Figure 13.18 shows the theoretical rotation–vibration Raman spectrum for

1 0 1 and 2 0 1 for a linear molecule It is assumed that there

is a small population in the first excited vibrational state The center series of lines

is the rotational Raman spectrum of the molecule Since homonuclear diatomic

molecules give spectra of this type, Raman spectroscopy provides the possibility,

not available in microwave or infrared spectroscopy, of determining their

inter-nuclear distances and force constants

For a polyatomic molecule, some normal modes will be Raman active and

some will not, depending on what happens to the polarizability ellipsoid when the

generalized coordinate for the normal mode has changed This is most easily seen

for a linear symmetric molecule, such as CO , in which the principal axes of

the polarizability ellipsoid are coincident with the symmetry axes of the molecule

In these molecules, only the symmetrical stretching normal mode is Raman active

This particular mode is not active in the infrared (Section 13.8) According to the

for molecules with a center of symmetry, fundamental sitions that are active in the infrared are forbidden in the Raman scattering, and

tran-vice versa However, there are some vibrations that are forbidden in both spectra

The torsional vibration of ethylene is neither infrared nor Raman active; this is

the vibration in which ethylene is twisted out of its planar equilibrium structure

Benzene has thirty normal modes of vibration, and eight of them are totally

inac-tive in both infrared and Raman; these are referred to as spectroscopically dark

vibrations

I( )δ

I( )~

~ νδ ν

2 2

1 2

interfer-ometer with source S, beam splitter

B, movable mirror M , fixed mirror

M , and detector D Rays 1 and 2

are recombined at the detector, but

they have a path difference , which

causes interference

Intensity ( )

mea-sured at the detector as a function of

path difference for monochromatic

radiation from the source

␦

␦

resonance Raman spectroscopy.

13.10 SPECIAL TOPIC: FOURIER TRANSFORM INFRARED

par-The sensitivity of infrared absorption measurements can be greatly increased

by using a method involving Fourier transforms A Michelson interferometer isbuilt into the spectrometer, along with a dedicated computer The construction ofthe Michelson interferometer is shown in Fig 13.19 The infrared radiation thathas been transmitted by the sample, designated as the source S, is split into tworays by B, which is usually a very thin film of germanium supported on a potas-sium bromide substrate The beam splitter transmits half of the infrared radiationfrom the sample and reflects half toward a movable mirror M The transmittedray is reflected from a stationary mirror M When the two rays reach the de-tector D, there is interference because of the path difference If the radiationfrom the sample is monochromatic, the intensity measured at the detector ( )

con-of path difference are obtained from actual infrared spectra These measurementscan be used to calculate ( ˜ ) by use of a Fourier transformation, which gives

1

2

where (0) is the intensity for zero path difference This integration is carried out

by the dedicated computer in the spectrometer, and the spectrum ( ˜ ) is ted out To a first approximation, the resolution is inversely proportional to thedistance moved by mirror M , but there are problems with making this distancegreater than about 5 cm, which gives a resolution of about 0 1 cm

The advantage of a Fourier transform spectrometer is that it makes use of

the radiation at all wave numbers from the source for all of the time of

record-ing Fourier transforms are also used in nuclear magnetic resonance spectroscopy

(Section 15.10) and in determining the structures of crystals by X-ray diffraction

(Section 23.6)

The electromagnetic spectrum is divided into regions by the modes of

de-tection, but these regions are also characterized by the ranges of photon

energy they involve The absorption of a photon leads to different types

of energetic changes in molecules in different regions of the spectrum

Consideration of the equations for Einstein’s stimulated absorption,

spontaneous emission, and stimulated emission shows that irradiation

of a two-level system can never put more atoms or molecules in the

higher level than in the lower level

The Hamiltonian of a molecule can, to a good approximation, be

sep-arated into translational, vibrational, and rotational contributions When

this is satisfactory, the wavefunction can be written as the product of

trans-lational, vibrational, and rotational wavefunctions

The rotational lines for a rigid diatomic molecule are equally spaced, and

the spacing yields the moment of inertia To have a rotational spectrum,

a molecule must have a permanent dipole moment

Diatomic molecules are not harmonic oscillators, and the deviation from

equal spacing of vibrational lines yields anharmonicity constants To

have a vibrational spectrum, a molecule must have a dipole moment that

changes with internuclear distance

Vibration–rotation spectra yield vibration–rotation coupling constants

Vibrational spectra of polyatomic molecules with atoms can be

de-scribed in terms of 3 5 coordinates for linear molecules and 3 6

coordinates for nonlinear molecules There is a Hamiltonian for each of

these normal modes of vibration

When light is scattered by molecules, part of the scattered light emerges

with lower frequency (Stokes lines) and a smaller part emerges with a

higher frequency (anti-Stokes lines) The incident light need not coincide

with a quantized energy difference in the molecule Laser light sources

are used because of their brightness

In order for a vibrational mode to be Raman active, the polarizability

must change as the molecule vibrates, and for a rotation to be Raman

active, the polarizability must change as the molecule rotates

Spectroscopic measurements yield properties of gas molecules that have

many scientific and practical applications