Molecular Spectroscopy

Chapter 14

Molecular Spectroscopy

ur most detailed knowledge of atomic and molecular structure has been obtained from spectroscopy—study of the emission, absorption and scattering O of electromagnetic radiation accompanying transitions among atomic or molecular energy levels. Whereas atomic spectra involve only electronic transitions, the spectroscopy of molecules is more intricate because vibrational and rotational degrees of freedom come into play as well. Early observations of absorption or emission by molecules were characterized as band spectra—in contrast to the line spectra exhibited by atoms. It is now understood that these bands reflect closely spaced vibrational and rotational energies augmenting the electronic states of a molecule. With improvements in spectroscopic techniques over the years, it has become possible to resolve individual vibrational and rotational transitions. This has provided a rich source of information on molecular geometry, energetics and dynamics. Molecular spectroscopy has also contributed significantly to analytical chemistry, environmental science, astrophysics, biophysics and biochemistry. In this chapter, we will focus on quantum-mechanical principles useful in spectroscopy. We will not take up any of the experimental techniques.

14.1

Vibration of Diatomic Molecules

A diatomic molecule with nuclear masses mA , mB has a reduced mass µ=

mA mB mA + m B

(14.1)

recalling the definition of reduced mass in Section 7.8. Solution of the electronic Schrödinger equation gives the energy as a function of internuclear distance Eelec (R). This plays the role of a potential energy function for motion of the nuclei 217

218

Chapter 14 Molecular Spectroscopy

Figure 14.1
Vibrational energies of a diatomic molecule, as approximated by a Morse oscillator.

V/hcDe

1

D0 De

0 –1

0

1 2 3 a(R – Re)

4

5

V (R), as sketched in Fig. 14.1. We can thus write the Schrödinger equation for vibration, in analogy with Eq (7.63), as
 h¯ 2 d 2 − + V (R) χ (R) = Eχ (R) (14.2) 2µ dR 2 If the potential energy is expanded in a Taylor series about R = Re , V (R) = V (Re ) + (R − Re )V
(Re ) + 21 (R − Re )2 V

(Re ) + · · ·

(14.3)

An approximation for this expansion has the form of a harmonic oscillator with V (R) ≈ 21 k(R − Re )2

(14.4)

The energy origin can be chosen so V (Re ) = 0. Also, at the minimum of the potential, V
(Re ) = 0. The best fit to the parabola (14.4) is obtained with a force constant set equal to  d 2 V (R)  k≈ (14.5) dR 2  R=Re

The force constant is a measure of the stiffness of a chemical bond. Larger values of k imply sharply curved potential energy functions and are often associated with deeper potential wells and stronger bonds. Molecular force constants are typically in the range of 200–2000 N/m, remarkably, not very different from those for bedsprings. From the solution for the harmonic oscillator, we identify the groundstate vibrational energy, with quantum number v = 0, as  k E0 = hω (14.6) ¯ = h¯ µ

Section 14.1

Vibration of Diatomic Molecules

219

The actual dissociation energy from ground vibrational state is then approximated by D0 ≈ De − 21 hω (14.7) ¯ or, expressed in wavenumber units, which we will use from now on, D0 ≈ De − 21 ν˜

cm−1

(14.8)

An improved treatment of molecular vibration must account for anharmonicity, deviation from a harmonic oscillator. Anharmonicity results in a finite number of vibrational energy levels and the possibility of dissociation of the molecule at sufficiently high energy. A very successful approximation for the energy of a diatomic molecule is the Morse potential:  2 V (R) = hcDe 1 − e−a(R−Re )

 a=

µω2 2hcDe

1/2 (14.9)

Note that V (Re ) = 0 at the minimum of the potential well. The Schrödinger equation for a Morse oscillator can be solved to give the energy levels 1 2 Ev = (v + 21 )hω ¯ e ¯ − (v + 2 ) hωx

(14.10)

or, expressed in wavenumber units, Ev = (v + 21 )ωe − (v + 21 )2 ωe xe hc

(14.11)

Spectroscopists generally write ωe in place of ν˜ for diatomic molecules. The anharmonicity ωe xe is usually tabulated as a single parameter. Higher vibrational energy levels are spaced closer together, just as in real molecules. The anharmonicity for a Morse oscillator is determined by xe =

a 2 h¯ ωe = 2µ ω 4De

(14.12)

More accurate characterization of the energy levels—beyond the Morse potential— might include additional anharmonic contribution of the form (v+ 21 )3 ωe ye . Vibrational transitions of diatomic molecules occur in the infrared, in the broad range of 50−15, 000 cm−1 . A molecule will absorb or emit radiation only if it has a nonzero dipole moment. Thus heteronuclear diatomic molecules such as HCl are infrared (IR) active, while homonuclear diatomics such as H2 and Cl2 are not. The quantum number of the highest bound vibrational level in a diatomic molecule is estimated by   ωe 1 vmax ≈ int (14.13) − 2 ωe xe 2 where “int” means the integer part of the expression. This can be found by equating the energy (14.11) to De or by setting dEv /dv ≈ 0.

220

Chapter 14 Molecular Spectroscopy

The selection rule for vibrational transitions for a harmonic oscillator is v = ∞ ±1, since the integral −∞ ψv x ψv
dx  = 0 only when v
= v ± 1 (see Exercise 5.4). The transition v = 1 ← v = 0 determines the fundamental vibrational frequency. For an anharmonic oscillator, overtone transitions such as 2 ← 0, 3 ← 0, etc. are also possible, usually with much weaker intensities than the fundamental.

14.2

Vibration of Polyatomic Molecules

A molecule with N atoms has a total of 3N degrees of freedom for its nuclear motions, since each nucleus can be independently displaced in three perpendicular directions. Three of these degrees of freedom correspond to translational motion of the center of mass. For a nonlinear molecule, three more degrees of freedom determine the orientation of the molecule in space and thus its rotational motion. This leaves 3N − 6 vibrational modes. For a linear molecule, there are just two rotational degrees of freedom, which leaves 3N − 5 vibrational modes. For example, the nonlinear molecule H2 O has three vibrational modes, while the linear molecule CO2 has four vibrational modes. The vibrations consist of coordinated motions of several atoms in such a way as to keep the center of mass stationary and nonrotating. These are called the normal modes. Each normal mode has a characteristic resonance frequency νi (expressed in cm−1 ), which is usually determined experimentally. To a reasonable approximation, each normal mode behaves as an independent harmonic oscillator of frequency νi . The normal modes of H2 O and CO2 are shown in Figs. 14.2 and 14.3. A normal mode will be infrared active only if it involves an oscillation of the dipole moment. All three modes of H2 O are

v1= 3 6 5 2 cm–1 v3= 3 7 5 6 cm–1 v2= 1595 cm–1 A1

B1

A1

Figure 14.2
Normal modes of H2 O: symmetric stretch, asymmetric stretch and bend, respectively. All are IR active.

v1= 1 3 8 8 cm–1 v3= 2 3 4 9 cm–1 Σg+ Figure 14.3


Σu+

v2= 667 cm–1 Πg

Normal modes of CO2 : all except ν1 are IR active.

Section 14.2

Vibration of Polyatomic Molecules

v1= 3 33 6.2 cm–1 A1

221

v2= 932. 5 cm–1A1

v3= 3443. 6 cm–1 E

Figure 14.4
Normal modes of NH3 : ν1 is a symmetric stretch, ν2 , a symmetric bend; ν3 and ν4 are doubly degenerate asymmetric stretches and bends. The symmetry species A1 and E are indicated.

v4= 1626. 1 cm–1 E

active. The symmetric stretch of CO2 is inactive because the two C–O bonds, each of which is polar, exactly compensate. Note that the bending mode of CO2 is doubly degenerate. Bending of adjacent bonds in a molecule generally involves less energy than bond stretching, thus bending modes generally have lower wavenumbers than stretching modes. Each normal mode transforms as one of the irreducible representations of the symmetry group of the molecule. Thus, the water molecule, with symmetry C 2v , has vibrations belonging to A1 and B1 . Carbon dioxide, with symmetry D∞h , has vibrations classsified as g+ , u+ , and g . The normal modes of ammonia, a C3v molecule, are shown in Fig. 14.4. All modes for NH3 are IR active. For molecules of any symmetry, the IR active modes of vibration can be determined from the character table of its symmetry group. Modes which are infrared active belong to the same irreducible representation as one of the cartesian coordinates x, y or z, shown in one of the right-hand columns of the character table. Check this for the C2v , C3v and D∞h molecules we have considered. Homonuclear diatomic molecules, such as N2 and O2 , have but a single vibrational mode of D∞h symmetry and are thus transparent to infrared radiation. By contrast, other molecules in the Earth’s atmosphere, such as CO2 and CH4 , can absorb in the infrared. This is the cause of the greenhouse effect. While almost all of the components of the atmosphere are transparent to the ultraviolet radiation from the Sun, much of the infrared which would be radiated back into space is trapped by the IR-absorbing greenhouse gases. In 1862, John Tyndall referred to infrared radiation “As a dam built across a river causes a local deepening of the stream, so our atmosphere, thrown as a barrier across the terrestrial rays, produces

222

Chapter 14 Molecular Spectroscopy

a local heightening of the temperature at the Earth’s surface.” This makes the Earth comfortably warm for most living things. However, it is generally agreed that, since the beginning of the Industrial Revolution, emissions of heat-trapping gases from burning coal and other fossil fuels might be intensifying the natural greenhouse effect to produce environmentally undesirable global warming. Infrared spectra of solutions are extensively used for identification and characterization of organic compounds. The features between 200 and 1400 cm−1 , known as the fingerprint region, are especially useful. These are absorptions due mainly to bending modes. Although it is difficult to assign the individual bonds producing this spectrum, each compound produces a characteristic pattern which can serve to identify the molecule. Fig. 14.5 shows the IR spectra of 1-propanol

TRANSMITTANCE %

100

50

1-propanol CH3CH2CH2OH 0 4000

3000

2000

1500

1000

500

WAVENUMBER cm–1

TRANSMITTANCE %

100

50

2-propanol CH3CHCH3 OH 0 4000

3000

2000

1500

1000

500

WAVENUMBER cm–1

Figure 14.5
IR spectra of 1-propanol and 2-propanol. (From the spectroscopic database for organic compounds on the website of the National Institute of Advanced Industrial Science and Technology (Tsukuba, Ibaraki, Japan): http://www.aist.go.jp/RIODB/SDBS/menu-e.html. Courtesy, SDBS, National Institute of Advanced Industrial Science and Technology.)

Section 14.3

Rotation of Diatomic Molecules

223

and 2-propanol. The broad absorption around 3200–3600 cm−1 is characteristic of the O–H bond stretch in alcohols, while the peaks around 2900 cm−1 are from C–H bond stretches. A very distinctive feature of carbonyl compounds is the C=O stretch around 1700 cm−1 . In condensed phases, rotational transitions cannot be individually resolved. Instead, they contribute to the broading of vibrational transitions into “bands”. This is actually a simplifying feature for purposes of using IR spectra as “fingerprints” to identify molecules.

14.3

Rotation of Diatomic Molecules

The rigid rotor model assumes that the internuclear distance R is a constant. This is not a bad approximation, since the amplitude of vibration is generally of the order of 1% of R. The Schrödinger equation for nuclear motion then involves the three-dimensional angular-momentum operator, written Jˆ rather than Lˆ when it refers to molecular rotation. The solutions to this equation are already known, and we can write Jˆ2 YJ M (θ, φ) = EJ YJ M (θ, φ) 2µR 2 J = 0, 1, 2 . . .

M = 0, ±1 · · · ± J

(14.14)

where YJ M (θ, φ) are spherical harmonics in terms of the quantum numbers J and M, rather than and m. Since the eigenvalues of Jˆ2 are J (J + 1)h¯ 2 , the rotational energy levels are h¯ 2 J (J + 1) (14.15) EJ = 2I The moment of inertia is given by 2 I = µRe2 = mA RA + mB RB2

(14.16)

where Re is the equilibrium internuclear separation, while RA and RB are the distances from nuclei A and B, respectively, to the center of mass. In wavenumber units, the rotational energy is expressed EJ = Be J (J + 1) cm−1 hc

(14.17)

where Be is the rotational constant. A rotational energy-level diagram is shown in Fig. 14.6. Each level is (2J +1)-fold degenerate. For small molecules, the rotational constant Be is generally in the range 0.1 to 10 cm−1 , thus pure rotational transitions are observed in the microwave or far-infrared region. Again, only polar molecules can absorb or emit radiation in the course of rotational transitions. The selection rules for rotational transitions are J = ±1, M = 0, ±1. If the molecule has nonzero electronic orbital angular momentum (term symbol other than ) such

Chapter 14 Molecular Spectroscopy ENERGY/cm–1

224

20B

12B 6B 2B 0 Figure 14.6
Rotational energies on wavenumber scale: EJ / hc = BJ (J + 1). Note (2J + 1)-fold degeneracy.

as NO, J = 0 transitions are also allowed. These selection rules are consistent with photons carrying one unit of angular momentum, so that the vector sum Jinitial + Jphoton can equal Jfinal . It must be possible for the quantum numbers Jinitial , Jphoton = 1 and Jfinal to form a triangle. At temperatures around 300 K, the great majority of diatomic molecules will occupy their v = 0 vibrational ground state. The thermal energy at 300 K corresponds to about kT / hc ≈ 200 cm−1 . Thus, for a molecule with vibrational constant ωe ≈ 2000 cm−1 , the Boltzmann factor is of the order of e−(E1 −E0 )/kT ≈ 5 × 10−5 By contrast, a significant number of rotational levels are occupied at room temperature. Taking account of the degeneracy gJ = 2J + 1, the Boltzmann distribution for rotational levels has the form NJ = const (2J + 1) e−Be J (J +1)hc/kT The maximum population occurs for the level  1/2 kT 1 Jmax = int − 2hcBe 2

(14.18)

(14.19)

Just as deviations from the harmonic model for molecular vibration become significant for larger values of v, the rigid rotor model for rotation requires correction for larger values of J . The physical effect is centrifugal distortion, which causes a rapidly rotating molecule to stretch and thereby increase its moment of inertia. Taking account of centrifugal distortion, the rotational energy can be approximated by EJ = Be J (J + 1) − DJ J 2 (J + 1)2 (14.20) hc

Section 14.4

Rotation-Vibration Spectra

225

Note that the increase in moment of inertia causes the rotational levels to be closer together, hence the minus sign. Centrifugal distortion constants DJ are typically of the order of 10−4 Be and are approximated by DJ ≈ 4Be3 /ωe2

(14.21)

(Spectroscopists often write DJ as De , risking confusion with the potential-well depth De in Fig. 14.1.) Centrifugal distortion is usually not very significant until rotational quantum numbers around J = 30 are reached. A more important effect is the dependence of the rotational constant on the vibrational quantum number. This can be parametrized as Bv = Be − αe (v + 21 )

(14.22)

where the vibration-rotation interaction constant αe is typically in the range of 0.1 to 1 cm−1 .

14.4

Rotation-Vibration Spectra

Taking into account corrections to the harmonic oscillator-rigid rotor model, vibrational-rotational energies for a diatomic molecule can be represented by EvJ = ωe (v + 21 ) − ωe xe (v + 21 )2 + Be J (J + 1) hc −αe (v + 21 )J (J + 1) − DJ J 2 (J + 1)2

(14.23)

ABSORPTION

Fig. 14.7 shows the vapor-phase infrared absorption spectrum of HCl, which involves rotational transitions occurring simultaneously with the fundamental vibrational transition v = 1 ← v = 0. Since the absorbed photon carries one unit of angular momentum, a pure vibrational transition is not possible. This accounts for the missing line labelled

v Figure 14.7
Infrared rotation-vibration spectrum of gaseous HCl in the region 2500−3100 cm−1 with ν00 at 2886 cm−1 .

226

Chapter 14 Molecular Spectroscopy

ν00 , which would be part of the Q-branch of the spectrum. A Q-branch can occur for non- diatomics and for nonlinear polyatomic molecules. The R-branch in Fig. 14.7 consists of transitions with J = +1 simultaneous with v = +1 vibrational transition. Specifically, the R(1) R(2) . . . lines represent the transitions J = 1 ← J = 0, J = 2 ← J = 1 . . . , respectively. Analogously, the P-branch represents transitions with J = −1: P(1) is J = 0 ← J = 1, etc. The spacing between successive lines in the P-branch are slightly greater than those in the R-branch, as determined by differences between B0 J (J + 1) and B1 J (J + 1) according to Eq (14.21). The variation in intensity of the absorption lines is caused by the relative populations in rotational levels of the ground vibrational state, consistent with Eq (14.18). The J = 2 level evidently has the maximum population. Finally, note that each line has an overlapping “satellite” about one-third as intense. This is caused by the 1 H37 Cl isotopomer present with about one-third the abundance of 1 H35 Cl (remember the atomic weight of Cl is approximately 35.5). Since the reduced mass of the 37 Cl isotopomer is larger, its vibrational constant ωe is smaller, so that the satellites appear to the left of the main peaks. In a deuterium-enriched sample of HCl we could also observe the 2 H35 Cl and 2 H37 Cl spectra. From accurate measurement of the HCl spectrum, the following spectroscopic constants can be assigned. For 1 H35 Cl, ωe = 2989.5, ωe xe = 51.88, Be = 10.5910, αe = 0.3027, all in cm−1 . For 1 H37 Cl, ωe = 2987.3, ωe xe = 51.81, Be = 10.5739, αe = 0.3021. For both, DJ ≈ 5 × 10−4 cm−1 .

14.5

Molecular Parameters from Spectroscopy

Table 14.1 lists spectroscopic constants for some common diatomic molecules. These are taken from several sources and are of varying accuracy. The National Institute of Standards and Technology (NIST) maintains an extensive online compendium of spectroscopic and other chemical data at: http: // webbook.nist.gov/chemistry/. Spectroscopic constants are expressed −1 ) = ν(Hz)/c in MHz and GHz rather than cm−1 . To convert units, note that ν(cm ˜ where c is the speed of light. A useful relation is ν˜ (cm−1 ) = 3.33564 × 10−5 ν(MHz)

(14.24)

The force constant for a molecule can be found from the vibrational constant by equating the energy quantities hω ¯ = hcωe . We find  k (14.25) ω = 2πc ωe = µ Thus, k = (2πc ωe )2 µ

(14.26)

Section 14.5 Molecular Parameters from Spectroscopy TABLE 14.1


227

Spectroscopic Constants for Diatomic Molecules

Molecule

Term

D0 /eV

ωe /cm−1

ωe xe /cm−1

Be /cm−1

αe /cm−1

1H 2

1+ g

4.4773

4395.2

1 H2 H

1+

117.9

60.86

2.99

4.5128

3812.3

2H 2

4.5553

35 Cl 2

1+ g 1+ g 3− g 1+ g 1+ g

90.91

45.66

2.00

3118.5

64.10

30.44

1.05

9.7598

2359.6

14.46

2.010

0.019

5.1156

1580.4

11.98

1.446

0.016

1.604

891.8

13.6

0.883

0.015

79 Br81 Br

1+

2.475

564.9

4.0

0.244

0.0015

127 I 2

1+ g

1.9708

323.3

1.08

0.0809

0.00032

1.5437

214.5

0.607

0.0374

0.00012

7 Li1 H

1+

2.429

1405.6

23.2

7.513

0.21

1 H19 F

1+

5.86

4138.5

90.07

20.94

0.77

1 H35 Cl

1+

4.436

2989.7

52.05

10.59

0.30

1 H79 Br

1+

3.775

2649.7

45.21

8.473

0.23

1 H127 I

1+

3.053

2309.5

39.73

6.511

0.18

12 C16 O

1+

11.108

2170.2

13.46

1.931

0.017

14 N16 O

2 1/2

6.50

1904.0

13.97

1.704

0.018

14 N 2 16 O 2 19 F 2

with µ=

mA mB M A MB = u mA + m B MA + M B

(14.27)

where u = 1.66054 × 10−27 kg, the atomic mass unit. MA and MB are the conventional atomic weights of atoms A and B (on the scale 12 C = 12). Putting in numerical factors k = 58.9 × 10−6 (ωe /cm−1 )2

MA MB N/m MA + M B

(14.28)

This gives 958.6, 512.4, 408.4 and 311.4 N/m for HF, HCl, HBr and HI, respectively. These values do not take account of anharmonicity. The equilibrium internuclear distance Re is determined by the rotational constant. By definition, h¯ 2 (14.29) hcBe = 2I Thus, h¯ Be = (14.30) 4πcI

228

Chapter 14 Molecular Spectroscopy

with I = µRe2 =

mA m B M A MB R2 = u Re2 kg m2 mA + m B e MA + M B

Thus,

 Re = 410.6

MA MB (Be /cm−1 ) pm MA + M B

(14.31)

(14.32)

For the hydrogen halides HF, HCl, HBr, HI, we calculate Re = 92.0, 127.9, 142.0, 161.5 pm, respectively.

14.6

Rotation of Polyatomic Molecules

A linear polyatomic molecule such as HCN, O=C=O or HC≡CH has a rotational spectrum closely analogous to that of a diatomic molecule, if one takes into account the more complicated form of the moment of inertia. Consider the most general case of a linear triatomic molecule:

RAB

RBC

mA

mB

mC

The moment of inertia is given by I=

2 + m m R2 + m m R2 mA mB RAB B C BC A C AC mA + m B + m C

(14.33)

where RAC = RAB + RBC . For a symmetric molecule, such as CO2 , with mA = mC = m and RAB = RBC = R, this reduces to I = 2mR 2 . For an unsymmetrical linear triatomic molecule such as OCS, the moment of inertia can be calculated from the rotational constant Be . But this does not provide enough information to determine the individual bond distances RCO and RCS , parameters of greater chemical interest. However, data from two different isotopomers of the molecule— 16 O12 C32 S and either 16 O12 C34 S, 16 O13 C32 S or 18 O12 C32 S—suffice to determine the two bond lengths RCO = 1.16 Å and RCS = 1.56 Å. A nonlinear molecule has three moments of inertia about three principal axes, designated IA , IB and IC . The classical rotational energy can be written E=

J2 J2 JA2 + B + C 2IA 2IB 2IC

(14.34)

where JA , JB , JC are the components of angular momentum about the corresponding principal axes. For example, the moment of inertia about the z-axis of a molecule can be found from  mi (xi2 + yi2 ) (14.35) Iz = i

where i runs over all the atoms in the molecule. An atom lying on the z-axis makes no contribution to Iz since xi = yi = 0.

Section 14.7

Electronic Excitations

229

For a spherical rotor, the three moments of inertia are equal: IA = IB = IC = I . Examples are CH4 and SF6 . The energy simplifies to J 2 /2I and the quantummechanical Hamiltonian is given by Jˆ2 Hˆ = 2I

(14.36)

The eigenvalues are EJ =

h¯ 2 J (J + 1) 2I

J = 0, 1, 2 . . .

(14.37)

just as for a linear molecule. For a three-dimensional rotor, space quantization of an angular momentum J gives not only 2J + 1 possible values of M, the component along the space-fixed z-axis, but also 2J + 1 possible values of K, which is the component along a molecule-fixed axis. Thus, the levels of a spherical rotor have degeneracies of (2J + 1)2 rather than the (2J + 1) for a linear rotor. A symmetric rotor has two equal moments of inertia, say, IC = IB  = IA . The molecules NH3 , CH3 Cl and C6 H6 are examples. The Hamiltonian takes the form Hˆ =

  JˆB2 + JˆC2 JˆA2 Jˆ2 1 1 JˆA2 + = + − 2IA 2IB 2IB 2IA 2IB

(14.38)

Since it its possible to have simultaneous eigenstates of Jˆ2 and its component JˆA along the molecule’s symmetry axis, the energies of a symmetric rotor have the form   J (J + 1) 1 1 K2 + − EJ K = 2IB 2IA 2IB J = 0, 1, 2 . . .

K = 0, ±1, ±2 · · · ± J

(14.39)

There is, in addition, the (2J + 1)-fold M degeneracy. A symmetric rotor is classified as prolate if IA IB , IC . A linear molecule is a degenerate case of a prolate-symmetric rotor, with IA = 0 and JA = 0. Dipole-allowed transitions of a symmetric rotor have the additional selection rule K = 0, since the electric dipole moment lies in the principal axis and cannot be accelerated to a different orientation within the molecule. When IA  = IB  = IC , the molecule is an asymmetric rotor. There is then no closed form for the rotational energy eigenvalues but they can be found by interpolation between the energy levels of the prolate and oblate symmetric rotors with IA < IB = IC and IA = IB < IC .

14.7

Electronic Excitations

The quantum states of molecules are composites of rotational, vibrational and electronic contributions. The energy spacings characteristic of these different degrees of freedom vary over many orders of magnitude, giving rise to very different

230

Figure 14.8
Schematic representation of the electronic ground state and an excited state of a diatomic molecule. Vibrational levels of the ground state are shown in red, those of the excited state, in blue. The rotational levels for v = 0 are also shown.

Chapter 14 Molecular Spectroscopy

J

spectroscopic techniques for studying rotational, vibrational and electronic transitions. Electronic excitations are typically of the order of several electron-volts, with 1 eV being equivalent to approximately 8000 cm−1 or 100 kJ mol−1 . As we have seen, typical energy differences are of the order of 1000 cm−1 for vibration and 10 cm−1 for rotation. Fig. 14.8 gives a general picture of the relative magnitudes of these energy contributions. Each electronic state has a vibrational structure, characterized by vibrational quantum numbers v, and each vibrational state has a rotational structure, characterized by rotational quantum numbers J and M. Every electronic transition in a molecule is accompanied by changes in vibrational and rotational states. Generally, in the liquid state, individual rotational transitions are not resolved, so that electronic spectra consist of broad bands from overlapping rotational transitions. Spectroscopy on the gas phase can often resolve individual rotational as well as vibrational transitions. When a molecule undergoes a transition to a different electronic state, the electrons can rearrange themselves much more rapidly than the nuclei. To a very good approximation, the electronic state can be considered to occur instantaneously, while the nuclear configuration remains fixed. This is known as the Franck-Condon principle. It has the same physical origin as the Born-Oppenheimer approximation, namely, the great disparity between the electron and nuclear masses. In classical terms, the nuclei begin in an equilibrium configuration in the lower electronic state. But after the transition, the nuclei generally find themselves in a nonequilibrium configuration of the new electronic state. Thus, they undergo vibrational motion with the original configuration serving as a turning point. Fig. 14.9 shows the energies of the ground and excited states of a diatomic molecule as functions of internuclear distance. Franck-Condon behavior is characterized by vertical transitions, in which R remains approximately constant as the molecule jumps from one potential curve to the other.

Section 14.7

Electronic Excitations

231

Figure 14.9
Most probable transition in absorption from the ground vibrational level according to FranckCondon principle. The vibrational wavefunctions are shown in blue.

In a vibrational state v = 0, the maximum of probability for the internuclear distance R is near the center of the potential well. For all higher values vibrational states, maxima of probability occur near the two turning points of the potential— where the total energy equals the potential energy. These correspond on the diagram to the endpoints of the horizontal dashes inside the potential curves. Transitions can actually occur to several excited vibrational levels in the vicinity of v
. The intensity of a transition between the levels v and v
of the ground (gnd) and excited (ex) electronic states, respectivley, depends on the Franck-Condon factor, the overlap of the two vibrational wavefunctions: Sv
v =

gnd

χvex
(R) χv (R) dR

(14.40)

which is a maximum for the vertical transition shown in Fig. 14.9. A molecule excited to a higher electronic state will, in most instances, spontaneously reemit a photon to return to its ground electronic state. This is called fluorescence if it occurs within a few nanoseconds after excitation. Often the molecule will first undergo a sequence of nonradiative decays to the lowest vibrational level of the excited state. The energy involved in this vibrational relaxation is usually converted to thermal energy in surrounding molecules. Transitions from the v
= 0 level of the excited state to the several vibrational levels of the ground state are also determined by Franck-Condon factors. The band structure of the fluorescence spectrum is approximately a mirror image of the absorption spectrum, as shown in Fig. 14.10. While the absorption spectrum is characteristic of the vibrational levels of the excited state, the fluorescence spectrum is characteristic of the vibrational levels of the ground state. Fluorescence is familiar in certain dyes which emit brilliant colors between red and green when irradiated with blue or ultraviolet light. Fluorescence spectroscopy is extensively applied in biochemistry and molecular biology. Fluorescent species bound to specific sites in biomolecules can be detected spectroscopically to study complexation and conformational phenomena.

Chapter 14 Molecular Spectroscopy

INTENSITY

232

WAVELENGTH Figure 14.10
Absorption and fluorescence spectra showing approximate mirrorimage patterns. Black arrows represent relaxation to the v
= 0 vibrational level of the excited state.

INT CR ERS OS YST SIN EM G

A Jablonski diagram, shown in Fig. 14.11, is a simplified representation of some possible absorption and emission processes in molecules. Assuming that the ground electronic state is a singlet, designated S0 , absorption of radiation can occur to several vibrational levels of the lowest excited singlet state S1 . Several things can then happen to the excited molecule. One possibility, which we described above for a diatomic molecule, is radiationless relaxation to the lowest vibrational level of S1 followed by emission of a photon, usually within several nanoseconds of the absorption. This is fluorescence, which returns the molecule to one of the

Energy

S1

T1

S0 Figure 14.11
Jablonski diagram showing radiative processes in color and radiationless processes in black. Short downward arrows represent vibrational relaxation.

Section 14.8

Lasers

233

vibrational levels of the ground electronic state. Another possible “fate” of the excited state is intersystem crossing, a radiationless transition of the molecule to one of the vibrational levels of its lowest electronic triplet state T1 . After relaxation to the lowest vibrational level of T1 , the molecule can again emit a photon to return to the ground state. Since triplet to singlet transitions are forbidden by electric dipole selection rules, the triplet state is metastable—it has a relatively long lifetime, typically of the order of 10−4 to 100 sec, or even longer. This delayed emission is known as phosphorescence. In contrast to fluorescence, phosphorescence persists even after the exciting radiation is discontinued. Transitions between singlet and triplet states, in both intersystem crossing and phosphorescence, are mediated by spin-orbit coupling. By this mechanism, states of differerent spin and orbital angular momentum can mix, so that the singlet acquires some triplet character and vice versa. Spin-orbit coupling becomes significant for organic compounds containing heavier atoms, such as P, S or Cl. Another possible process is internal conversion, radiationless transition to an electronic state of the same spin multiplicity, for example, S1 ← S2 , involving excited singlet states. This can be followed by delayed fluorescence. Chemical bonds in electronically excited molecules can also dissociate or rearrange themselves, thereby taking part in photochemical reactions.

14.8

Lasers

The development of lasers has revolutionized several branches of chemistry and spectroscopy, along with its well-known contributions to commerce and industry, not to mention CD players. LASER is an acronym for “Light Amplification by Stimulated Emission of Radiation.” The three fundamental processes involving the interaction of molecules with photons can be represented as generalized chemical reactions. Thus, absorption of a photon can be written X + hν → X∗ where X∗ indicates a molecule excited to a higher quantum state. The inverse process is spontaneous emission: X∗ → X + hν Also possible is stimulated emission: X∗ + hν → X + 2hν as first recognized by Einstein in 1917. What is remarkable is that stimulated emission produces a clone of the incoming photon. The two emerging photons are identical in frequency, direction and polarization. They are said to be coherent. By contrast, photons emitted by different molecules, though they might have the same frequency, will generally have relative phases which are random—they are

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said to be incoherent. The output of a laser is intense coherent radiation produced by stimulated emission. As we saw in Section 5.5, the rate of absorption between two molecular energy levels E1 and E2 is exactly equal to the rate of stimulated emission, for a given density of resonant photons, with ω = (E2 − E1 )/h. ¯ Whether there will be net absorption or emission depends on the relative populations, N1 and N2 , of the two levels. At thermal equilibrium, when the populations follow a Boltzmann distribution, with the lower level E1 more populated than the upper level E2 , a net absorption of radiation of frequency ω can occur. If the two populations are equal, there will be neither net absortion or emission since the rates of upward and downward transitions will exactly balance. Only if we can somehow contrive to achieve a population inversion, with N2 > N1 , can we achieve net emission, which amounts to amplification of the radiation (the “A” in LASER). Thus, to construct a laser, the first requirement is to produce a population inversion. Laser action can then be triggered by a few molecules undergoing spontaneous emission. The principle of laser action was suggested around 1953 by Townes and independently by Basov and Prokhorov. The earliest realizations were masers, producing microwave amplication by stimulated emission. The first successful optical laser was built by Theodore Maiman in 1960 using synthetic ruby. Crystalline ruby is mostly corundum, Al2 O3 , with approximately one out of every thousand Al3+ ions replaced by a Cr3+ ion. The relevant energy levels are those of chromium in a trigonal crystal field. The essentials of the ruby laser’s operation are represented in Fig. 14.12. A cylindrical rod of ruby is illuminated by a helical xenon flashlamp. Light in the blue and green regions of the spectrum is absorbed by Cr3+ ions, pumping a significant fraction of them from their 4 A2 ground state to several 4 F1 and 4 F1 excited states. The excited Cr 3+ ions subsequently undergo rapid nonradiative transitions to the metastable 2 E level (lifetime ≈ 4 × 10−3 sec) to set up a population inversion with respect to the ground state. A photon produced by spontaneous emission will then initiate stimulated emission of highly monochro-

Cr3+ in Al2O3 2E

4A 2

Figure 14.12
Essentials of a ruby laser. A cylindrical rod of ruby is illuminated by a helical xenon flashlamp which pumps a significant fraction of the Cr 3+ ions into the 4 F excited states. Nonradiative relaxation to the 2 E metastable state creates a population inversion which produces the red laser emission. The right-hand mirror on the ruby rod is partly transmitting to allow output of the laser beam.

Section 14.8

Lasers

235

matic red light of wavelength 694.3 nm. The ruby rod is made to act as a resonant optical cavity by placing mirrors at both ends so that the radiation bounces back and forth to produce a cascade of stimulated emissions. Since each photon leads to the emission of two photons, intensity of the laser transition will build up rapidly. The mirror on the right in Fig. 14.12 is partially transmitting (about 5%) to allow the laser beam to emerge. The ruby laser must actually operate in a pulsed mode since the population inversion is very difficult to sustain continuously. Note that the laser transition involves a change in spin multiplicity 4 A2 ←2 E. This becomes an allowed transition in atoms as heavy as Cr because of the significant spin-orbit coupling. The difference in spin multiplicity does, however, favor the metastabilty of the 2 E state. The ruby laser is the prototype of a three-level laser, the most rudimentary arrangement for laser action. A significant improvement is achieved in a four-level laser, for example, neodymium (Nd3+ ions) in YAG (yttrium aluminum garnet), with an energy-level arrangement similar to Fig. 14.13. This has the advantage that the lower metastable level E2 is initially unpopulated, so that population inversion with the higher metastable level E3 is much easier to maintain. The Nd-YAG laser can produce several wavelengths in the infrared, the 1064 nm band being the strongest. The intense laser output can be exploited in various nonlinear optical phenomena, including frequency doubling, which converts the radiation to green light of wavelength 532 mn. The radiation output of a laser is intense, highly monochromatic, sharply collimated and coherent. Laser action has been achieved with thousands of different active media, including solids, liquids and gases, with emitted frequencies spanning the infrared, visible and ultraviolet regions. Most elements in the gas state can be made to lase. Laser action is possible only at wavelengths for which the system has fluorescent emission. The upper levels in laser transitions must be metastable states, with lifetimes typically of the order of 10−3 sec. The laser output can be continuous wave (CW) or pulsed, with bursts as short as 1 picosecond (10−12 sec) and peak power in the gigawatt range. The use of pulsed lasers to study reactions and other fast molecular processes is currently a very active field of chemical research. Of greater interest in chemistry are lasers in which the active medium is a gas. The most common and least expensive gas laser is the helium-neon laser, which uses a mixture of 85% He and 15% Ne at pressures of the order of 10 torr. An energy-level diagram is shown in Fig. 14.14.An electric discharge excites He atoms to their 1s 2s 3 S and 1 S states, 19.8 and 20.6 eV above the ground state. This energy

E4 E3

E2 E1

Figure 14.13
Schematic energy-level diagram for fourlevel laser such as Nd-YAG. Nonradiative processes are shown as black arrows.

Figure 14.14
Heliumneon laser showing some relevant energy levels. Helium atoms are excited by electric discharge (blue arrows) and transfer energy to neon atoms via collisions (yellow arrows). The most important laser transitions are shown as rainbow arrows.

Chapter 14 Molecular Spectroscopy

ENERGY/eV

236

21

21S

3s2

20

23S

2s2

19

3p4

2p4

18 17

0

can be transferred by collision to Ne atoms, which very conveniently have excited states almost exactly matching the He levels. Transitions via electric discharge or molecular collisions are not restricted by electric dipole selection rules. The Ne energy levels labelled 3s2 and 2s2 thus become significantly populated, while the lower 3p4 and 2p4 levels are initially unpopulated. This inversion allows laser action at several frequencies, the most important being a red line at 632.8 nm and infrared transitions at 1152.3 and 3391.3 nm. The helium-neon laser can also be configured to produce green light at 543.5 nm, yellow at 594.1 nm, orange at 611.9 nm, and a multitude of other frequencies. The lower energy levels are depopulated principally by collisions of Ne atoms with the walls of the glass tube, thus maintaining a population inversion which allows the laser to operate in continuous mode. The carbon dioxide laser (see Fig. 14.15) also operates by collisional energy exchange, in this case with nitrogen gas. The N2 molecules are vibrationally excited by an electric discharge and transfer their energy to excite the ν3 antisymmetric stretching mode of CO2 . This is enabled by the near coincidence of the N2 v = 1 level at 2331 cm−1 with the CO2 (001) level (v3 = 1) at 2349 cm−1 . The strongest laser emission is the transition from (001) to (100), the first excited level (v1 = 1) of the symmetric stretch ν1 , producing intense infrared radiation at 10.59 µm. A second laser emission is (001) to (020) at 9.4 µm. Transitions between states of different symmetry are made possible by small anharmonicities in the vibrational Hamiltonian. The high-intensity infrared radiation of the CO2 laser makes it ideal for cutting metals and other materials in manufacturing operations—they essentially melt through a target. The CO2 laser is also used in skin surgery and other medical applications. In the argon-ion laser, argon gas at about 1 torr is ionized by an electric discharge to produce Ar + and Ar 2+ . These ions can then undergo laser transitions to lower

Section 14.8

Lasers

237

2000

9

5 0.

µm

1

(100)

(001) 9.4 µ m

ENERGY/cm–1

3000

(020)

1000

(010)

(000)

Figure 14.15
states (v1 v2 v3 ).

Carbon dioxide laser, based on population inversion of CO2 vibrational

electronic states. The two strongest emissions from Ar + produce a blue line at 488 nm and a green line at 514 nm. Exciplex lasers (also called excimer lasers) use reactive halogen atoms to form excited pseudo-molecules with noble-gas atoms. Molecules such as XeF∗ are stable only in excited electronic states and quickly dissociate after transition to the ground state. This makes possible a large population inversion and produces laser action in the ultraviolet region. A simple prototype for such behavior is the He∗2 excimer, which is an entry in Table 11.1. Dye lasers are based on organic dye molecules, such as Rhodamine 6G, in liquid solution or suspension. These have the advantage of being tunable over a broad range of wavelengths covering a large number of overlapping π ∗ ← π transitions. Dye lasers are often pumped by other lasers. Chemical lasers exploit chemical reactions to produce population inversion, thereby converting chemical energy directly into electromagnetic radiation. The first chemical laser was demonstrated by Kasper and Pimentel in 1965, based on the reaction Cl + H2 → HCl∗ + H. The reaction favors formation of HCl molecules in excited vibrational states. The resulting population inversion can produce laser action in the infrared region. Two devices of current interest are the hydrogen fluoride (HF) laser and the chemical oxygen-iodine laser (COIL), both being considered in connection with the Strategic Defense Initiative (“Star Wars”). The HF laser exploits the most exothermic reaction known, F + H2 → HF∗ + H, to produce excited HF molecules. Laser emission occurs at 2.7 µm in the infrared. The analogous DF laser (using 2 H rather than 1 H atoms) shifts the laser wavelength to 3.5 µm, where the atmosphere is more transparent. (Note that the wavelength is proportional to the square root of the reduced mass.) The COIL laser achieves a population inversion in iodine atoms by energy exchange with excited 1  oxygen

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Chapter 14 Molecular Spectroscopy

molecules: I + O∗2 → I∗ + O2 . The laser transition in iodine is 2 P3/2 ←2 P1/2 at 1.315 µm. Semiconductor lasers (laser diodes) now dominate in everyday applications, such as laser printers, CD players, bar-code readers, laser pointers, optical-fibre transmission and numerous other modern devices. These are compact, mobile, low cost and use very little power. A semiconductor laser is built around a junction of p-type and n-type layers. When electric current is passed through, electrons from the n-region fall into holes in the p-region, producing laser light from the junction. The most commmonly used semiconductors are gallium arsenide and other IIIV compounds. (Silicon-based semiconductors do not lase since their energy is dissipated as heat.) Aluminum gallium arsenide lasers, used in CD players, produce infrared radiation around 720 nm.

14.9

Raman Spectroscopy

The spectroscopic processes we have considered thus far involve absorption, stimulated emission and spontaneous emission of photons, accompanied by transitions among molecular energy states. It is also possible for photons to be scattered by molecules. Most scattering events are elastic, meaning that the energy of the photon is unchanged, although it can be deflected into a different direction. This process, in which the emitted photon has the same wavelength as the incident photon, is called Rayleigh scattering. The intensity of Rayleigh scattering is proportional to λ−4 and is thus more effective for shorter wavelengths—the blue end of the visible spectrum. The color of the sky is, in fact, caused by preferential scattering of the blue wavelengths in sunlight by molecules in the atmosphere. About 1 in every 107 scattering events involves inelastic scattering, in which the incident and emitted photons have different wavelengths. This is known as the Raman effect, discovered by C. V. Raman in 1928. The increase or decrease of photon energy is associated with a simultaneous change in the rotational-vibrational state of the molecule. Inelastic scattering of photons is exploited in Raman spectroscopy, a complementary technique to absorption and emission spectroscopy for studying the energy levels of molecules. Raman spectra use visible light rather than infrared or microwaves to obtain vibrational and rotational parameters. This is expecially useful for studying molecules in aqueous solution since the incident frequency can be chosen where there is no absorption by water. The development of lasers has revolutionized Raman spectroscopy. The high intensity and directionality of lasers can compensate for the low probability of Raman scattering. Stokes lines are those for which the energy of the scattered radiation is lower than the incident radiation, while anti-Stokes lines are those for which the energy of the scattered radiation is higher. The energy shift from the Rayleigh line is equal to some rotationalvibrational energy difference in the molecule. The Raman spectrum of CCl4 is shown in Fig. 14.16. Note that the Stokes and anti-Stokes lines are equally displaced from the Rayleigh line. Stokes lines are more intense, since anti-Stokes lines can occur only in molecules which are initially in excited vibrational states.

Raman Spectroscopy

INTENSITY

Section 14.9

239

RAYLEIGH

STOKES

ANTI-STOKES

16600

16400

16200

16000

15800

15600

15400 15200 15000 WAVENUMBER (cm–1)

14800

Figure 14.16
Vibrational Raman spectrum of CCl4 using 632.8-nm radiation. Note that the wavenumber scale increases right to left.

Because of the high monochromaticity of the incident laser beam, transitions differing by even a fraction of a wavenumber from the Rayleigh frequency can be detected, which makes pure rotational transitions accessible to laser-Raman spectroscopy. Infrared and Raman spectroscopy both measure vibrational energies of molecules but according to different selection rules. For a vibration to be IR active, the dipole moment of the molecule must change. For example, the asymmetric stretch ν3 and the bending mode ν2 in carbon dioxide are IR active, while the symmetric stretch ν1 is not (see Fig. 14.3). For a transition to be Raman active, the polarizability of the molecule must change in the vibration. This occurs in the symmetric stretch, as the molecule is expanded and compressed, but not in the other two modes. Thus, only ν1 in CO2 is Raman active. For molecules like CO2 possessing a center of inversion, it is observed that no normal mode can be both infrared and Raman active (it might be inactive to both). This is known as the mutual exclusion rule. In molecules lacking a center of symmetry, modes are likely to be active in both infrared and Raman spectroscopy. For example, all the vibrations in H2 O and NH3 are both IR and Raman active. For a molecule interacting with a radiation field, the perturbation of the Hamiltonian can be represented by   µ · E + 21 E · α · E = − V = −µ (14.41) µi Ei + 21 αij Ei Ej i

ij

where i, j run over cartesian coordinates x, y, z. This is a generalization of Eq (4.69), now including contributions quadratic in the electric field of the radiation interacting with the molecule’s polarizability tensor α . The two parts of the perturbation Hamiltonian (14.41) can be interpreted in terms of the quantum theory or radiation (Section 5.5). Terms linear in the electric field E produce singlephoton processes such as absorption and emission, while terms quadratic in E are the source of two-photon processes, including Rayleigh and Raman scattering. Scattering can be visualized as absorption of a photon as the the molecule makes a

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Chapter 14 Molecular Spectroscopy

transition to a “virtual” excited state, followed by reemission of the photon. If the incident radiation corresponds to an electronic transition frequency, the intermediate state is a real excited state of the molecule. This is known as resonance Raman spectroscopy and produces Raman scattering of much enhanced intensity. Resonance Raman spectroscopy is particularly useful in selectively studying molecules which are part of complex biological systems. Since the Raman effect involves two spin-one photons, the angular-momentum selection rule becomes J = 0, ±2. This gives rise to three distinct branches in the rotation-vibration spectra of diatomic and linear molecules: the O-branch (J = −2), the Q-branch (J = 0) and the S-branch (J = +2). All diatomic and linear molecules are Raman active. Raman spectroscopy can determine rotational and vibrational energy levels for homonuclear diatomic molecules, which have no infrared or microwave spectra. The dipole-moment components µx , µy , µz transform like the corresponding coordinates x, y, z. This determines the selection rules for IR transitions. Analogously, the components of the polarizability tensor αxx , αxy . . . transform like the cartesian quadratic forms x 2 , xy . . . . Accordingly, a normal mode will be Raman active if its irreducible representation is the same as one of the cartesian products x 2 , xy, etc. This is usually shown in the right-hand column of the group character table. For example, CCl4 and other tetrahedral molecules have normal modes of A1 , E and T2 symmetry. The Td character table (see Exercise 13.8) shows that these are all Raman active, while only T2 is IR active.

Problems 14.1. The Lennard-Jones potential for the interaction of two atoms has the form V (R) = 4 



  σ 12  σ 6 − R R

Determine the value of Re and De . What is the significance of the internuclear distance R = σ ? 14.2. From the spectroscopic constants in Table 14.1, calculate the force constant k and the internuclear distance Re for each of the hydrogen halides. The answers are given in the text. 14.3. Refer to the data on NO in Table 14.1. (i) Calculate the wavenumber of the J = 0 to J = 1 transition. (ii) Calculate the equilibrium internuclear distance in pm. (iii) Calculate the force constant in N/m. (iv) Taking account of the anharmonic correction, calculate the wavenumber of the v = 0 to v = 1 transition. 14.4. Estimate the number of bound vibrational states in each of the following molecules: H2 , HD and D2 .

Problems

241

14.5. From the spectroscopic results for 1 H35 Cl, predict ωe and Be for 1 H37 Cl,2 H35 Cl and 2 H37 Cl. 14.6. For N2 molecules in their ground electronic and vibrational states at 300 K, determine the most highly populated rotational level Jmax . 14.7. Derive a formula for the most highly populated level Jmax of a spherical rotor. Determine its value for CH4 at 300 K given that B = 5.24 cm−1 . 14.8. Derive Eqs (14.20) and (14.21) for centrifugal distortion in a diatomic molecule. Assume a molecule with reduced mass µ and equilibrium extension R obeying Hooke’s law with force constant k. Let the molecule be stretched from R to R + R by a centrigugal force m1 ω2 r1 + m2 ω2 r2 = 2µ ω2 R where ω is the angular velocity. √ Introduce the quantum-mechanical angular momentum using |J | = I ω = J (J + 1) h. ¯ Determine the rotational energy of the centrifugally-distorted molecule and identify the constant DJ . You may assume that R R so that 1 1 R ≈ − 2 2 2µ(R + R) 2µR µR 3 14.9. Referring to the character tables of C2v , C3v and D∞h , identify the symmetry species of infrared and Raman active normal modes of H2 O, NH3 and CO2 . 14.10. Referring to the D6h character table in Exercise 13.7, determine the symmetry species of normal modes in benzene C6 H6 which can be infrared and Raman active.