We now turn to our final quantum mechanical topic, molecular ...

209

Lecture 34 We now turn to our final quantum mechanical topic, molecular spectroscopy. Molecular

spectroscopy is intimately linked to quantum mechanics. Before the mid 1920's when quantum mechanics was developed, people had begun to develop the science of molecular spectroscopy. However, its early use was purely qualitative, based solely on the observation that molecules and atoms, when exposed to light, absorbed and emitted it in patterns that were unique to each species. The interpretation of these spectra in terms of the physical parameters that describe the geometries and bonding of the molecules that produce them had to wait until the development of quantum mechanics. The close interrelation between spectroscopy and quantum mechanics is shown by the fact that Robert Mulliken, who was the foremost practitioner and proponent of Molecular Orbital theory in its early years, was spurred on by an attempt to understand the details of molecular spectra. It is further shown by the fact that Gerhard Herzberg, the foremost practitioner of molecular spectroscopy from the middle of the '30s to the 70's, used to keep a bound copy of Mulliken's papers by his desk. In fact, I keep a bound copy of Mulliken's major papers, edited by Norman Ramsey, Herzberg's collaborator, by my desk.

Molecular spectroscopy, which is the study of the interaction of light with atoms and molecules, is one of the richest probes into molecular structure. Radiation from various regions of the electromagnetic spectrum yields different information about molecules. For example, microwave radiation is used to investigate the rotation of molecules and yields moments of inertia and therefore bond lengths. Infrared radiation is used to study the vibrations of molecules, which yields information about the stiffness or rigidity of chemical bonds. Visible and ultraviolet radiation is used to investigate electronic states of molecules, and yields

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information about ground and excited state vibrations, electronic energy levels, and bond strengths. We will do a fairly detailed treatment of rotational and vibrational spectroscopy and a somewhat briefer treatment of electronic spectroscopy. In our treatment we will begin with the basic interpretation of molecular spectra, and finish with the quantum mechanical basis of these interpretations.

The features of the electromagnetic spectrum that are of interest to us are summarized below. The absorption of microwave radiation is due to transition between rotational energy levels; the absorption of infrared radiation is due to transitions between vibrational levels,

Region Microwave

Far Infrared Infrared

Visible and Ultraviolet

Frequency/Hz 109-1011

1011-1013 1013-1014 1015-1016

Wavelength/m 3x10-1-3x10-3

Wave number/cm-1

0.033-3.3

3x10-3-3x10-5

3.3-330

3x10-5-3x10-6

330-4000

9 x 10-7-3x10-8 11,000-3.3x105

Energy/ J *molecule-1

6.6x10-256.6x10-23

6.6x10-236.6x10-21

6.6x10-218.0x10-20

2.2x10-196.6x10-18

Molecular Process

Rotation of polyatomic molecules

Rotation of small molecules

Vibration of flexible bonds

Electronic transitions

accompanied by transitions between rotational levels; and the absorption of visible and ultraviolet radiation is due to transitions between electronic energy levels, accompanied by simultaneous transitions between vibrational and rotational levels. The frequency of radiation absorbed is calculated from the energy difference between the upper and lower states in the transition, given by

E = Eu - El = h where Eu and El are the energies of the upper and lower states respectively.

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For example, if we have an absorption with wavenumber, = 1.00 cm-1, we calculate E

as follows. Remember that = 1/ (cm) . Therefore E is related to by

and so

E = h = hc = hc

E = hc = (6.626 x 10-34 Js)(3.00 x 1010 cm/s)(1.00cm-1) = 1.99 x 10-23 J.

TO WHAT TYPE OF MOLECULAR PROCESS WILL THIS RADIATION CORRESPOND?

We will begin with molecular rotation. A rigid rotor is the simplest model of molecular

rotation. We discussed the quantum-mechanical properties of a rigid rotor before, but will review

the pertinent results here. The energy of rotation of a rigid rotor is all kinetic energy. For a motion

with circular symmetry, it is convenient to express this energy in terms of the angular variables L,

angular momentum, and I, moment of inertia. The moment of inertia for a diatomic molecule is

given by

I = ? R02 where ? is the reduced mass of the molecule and R0 is the bond length of the molecule. The

angular momentum L is given by

L = I ,

where is the angular frequency. When the Schr?dinger equation for this problem is solved, it is

found that the angular momentum is quantized according to the equation

J2

=

J(J

+ 1) 2

where J is the spectroscopic notation for the total angular momentum of a molecule, and that the

angular momentum in the z direction is quantized according to the equation

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Jz

=

M,

where M = J, J-1, ..., -J. The energy of the rigid rotor, in the absence of an external field, depends

on J only, is also quantized and is given by

2

EJ

=

2I

J(J + 1),

J = 0,1,2,...

Since there are 2J+1 values of M for each energy EJ, the energy states are 2J+1 fold degenerate.

When microwave or far infrared radiation shines on a rotating molecule, the J value can

either increase or decrease, and the energy of the light absorbed is given by E = Eu - El . An

important characteristic of all molecular spectra is that the choice of upper and lower states

is not completely free but is in fact extremely limited. The rules that govern the choices of

upper and lower states are called selection rules. These selection rules are determined by time

dependent perturbation theory. For the pure rotational transitions of a diatomic rigid rotor

the selection rule is J = ?1. In other words, when a diatomic rigid rotor absorbs light, its

rotational quantum number can increase by 1 or decrease by 1. These are the only possible

transitions that occur. (The reason that I specify diatomic rigid rotors is that the selection rules

depend in part on symmetry, and molecules with other symmetries have different selection rules.)

In addition, for pure rotational transitions the molecule must have a permanent dipole

moment. Thus molecules like H2 and I2 will have no pure rotational spectra, while HI or CO,

which have permanent dipole moments, will exhibit rotational spectra. For molecules that do have

permanent dipole moments, the larger the dipole moment, the more strongly the molecule will

absorb light.

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The consequence of this selection rule is that the rotational absorption spectrum of a

rigid rotor is a series of evenly spaced lines in the microwave or far infrared region. To see

this we begin with the energy eigenvalues,

2

EJ

=

2I

J(J + 1),

J = 0,1,2,...

According to our selection rule, for absorption J = +1. The energy change for this transition is

2

2

E

=

E J +1 - E J

=

2I

(J + 1)(J + 2) - 2I

J(J + 1)

2

= (J + 1), J = 0,1,2,.... 2I

The wavenumber change for this absorption is given by

=

E hc

=

h 4 2cI

(J

+ 1),

J = 0,1,2

It is typical in spectroscopy to write the energy in the form

EJ = hcBJ(J + 1) where B , the rotational constant of the molecule, has units of wavenumbers and is given by

B

=

h 8 2cI

In this form the wavenumber of the rotational absorptions is given by

= 2B(J + 1), J = 0,1,2,...

Thus for the lowest energy rotational transition, J = 0 to J = 1, the absorption occurs at a

wavenumber of 2B . The second transition, J = 1 to J = 2, occurs at a wavenumber of 4 B . The

third transition occurs at a wavenumber of 6 B and so on. Thus the rotational spectrum consists

of a series of equally spaced lines separated by 2 B . [Illustrate] Lets check on our assertion that

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the rotational spectrum is in the microwave region, by calculating the value of B for the rotation

of HI.

h B = 8 2cI The moment of intertia I is given by ?R02, where ? is the reduced mass in kg and R0 is the bond length in meters. For HI, the reduced mass is given by

? HI

=

mH mI mH + mI

=

1AMUx127AMU 1AMU + 127AMU

x1.66x10-27

kg AMU

= 1.65x10-27 kg

and the bond length is 160.4 pm, so the moment of intertia is

I = ?R02 = 1.65 x 10-27 kg (160.4 x 10-12 m)2 = 4.25 x 10-47 kg m2,

and

B

=

(8

2

6.6262x10-34 Js )(3.00x1010 cm / s)(4.25x10-47

kg

m2

)

=

6.58cm-1

Our first rotational transition will occur at a wavenumber of 2 B = 13.16 cm-1, which is in the far

infrared region, as predicted.

A more typical application is to determine the value of B from the spacing of a

rotational spectrum, and use this to determine the moment of inertia and therefore the bond length. For example, the microwave spectrum of 39K127I consists of a series of lines whose spacing is almost constant at 3634 MHz. Calculate the bond length of 39K127I. The rotational spacing for

KI is given in Hz, a unit of frequency. Our equation for the rotational wavenumber is

= 2B(J + 1)

The relation between wavenumbers and frequency is = c

where c is the speed of light in cm/s. Thus our rotational absorption frequency becomes

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= 2cB ( J +1) ,

and our rotational spacing becomes =2cB . Thus

B

=

2c

=

3634x106 s-1 2x3.00x1010

=

6.05x10-2

cm-1

From this we can use our formula for B to calculate the moment of intertia for KI,

I

=

h 8 2cB

=

(8

2

6.6262x10-34 )(3.00x1010 )(.06057)

=

4.618x10-45

kgm2

.

The reduced mass of 39K127I is

? KI

=

39AMUx127AMU 39AMU + 127AMU

x1.67x10-27

kg AMU

= 4.983x10-26 kg.

Thus R0 = (I/?)1/2 = 3.04 x 10-10 M.

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Lecture 35-37

The simplest model of a vibrating molecule is the harmonic oscillator. The harmonic oscillator

assumes

that

the

potential

energy

function

for

the

vibration

is

given

by

V

=

1 2

kx2

,

where

k,

the

vibrational force constant, is a measure of the stiffness of the bond. The energy eigenvalues for

the harmonic oscillator are quantized and are given by

En

=

(n

+

1 2

)h

,

n = 0,1,2,...

1/ 2

F I where = 1 k . HG KJ 2 ?

Transitions between vibrational levels are subject to the selection rules n = ?1 and that the dipole moment must change during a vibration. Once again for diatomic molecules this means that homonuclear diatomics will have no vibrational absorption spectrum. The implication of this selection rule together with the formula for the energy eigenvalues of the harmonic oscillator is that the pure vibrational absorption spectrum of a diatomic molecule is a single absorption whose

1/ 2

F I frequency is given by = 1 k . To see this note that for absorption n = +1, so HG KJ 2 ?

E = En+1 - En = (n + 1 + 1/2) h - (n + 1/2) h = h Thus the spectrum consists of a single line with frequency . Determining the infrared frequency allows us to determine the force constant k of the bond. For example, the infrared spectrum of 39K35Cl has a single intense line at 378.0 cm-1. What is the force constant? Our first step is to convert the wavenumber to frequency, using = c . This yields

= 378 cm-1 x 3.00 x 1010 cm/s = 1.134 x 1013 s-1.

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