462 12 MOLECULAR SPECTROSCOPY 1: ROTATIONAL AND VIBRATIONAL SPECTRA

Different relative nuclear spin orientations change into one another only veryslowly, so an H2 molecule with parallel nuclear spins remains distinct from one withpaired nuclear spins for long periods. The two forms of hydrogen can be separated byphysical techniques, and stored. The form with parallel nuclear spins is called ortho-hydrogen and the form with paired nuclear spins is called para-hydrogen. Becauseortho-hydrogen cannot exist in a state with J = 0, it continues to rotate at very lowtemperatures and has an effective rotational zero-point energy (Fig. 12.22). This energy is of some concern to manufacturers of liquid hydrogen, for the slow conver-sion of ortho-hydrogen into para-hydrogen (which can exist with J = 0) as nuclearspins slowly realign releases rotational energy, which vaporizes the liquid. Techniquesare used to accelerate the conversion of ortho-hydrogen to para-hydrogen to avoidthis problem. One such technique is to pass hydrogen over a metal surface: themolecules adsorb on the surface as atoms, which then recombine in the lower energypara-hydrogen form.

The vibrations of diatomic molecules

In this section, we adopt the same strategy of finding expressions for the energy levels,establishing the selection rules, and then discussing the form of the spectrum. Weshall also see how the simultaneous excitation of rotation modifies the appearance ofa vibrational spectrum.

12.8 Molecular vibrations

Key point The vibrational energy levels of a diatomic molecule modelled as a harmonic oscillator

depend on a force constant kf (a measure of the bond’s stiffness) and the molecule’s effective mass.

We base our discussion on Fig. 12.23, which shows a typical potential energy curve (asin Fig. 10.1) of a diatomic molecule. In regions close to Re (at the minimum of thecurve) the potential energy can be approximated by a parabola, so we can write

V = kf x2 x = R − Re (12.27)

where kf is the force constant of the bond. The steeper the walls of the potential (thestiffer the bond), the greater the force constant.

To see the connection between the shape of the molecular potential energy curveand the value of kf , note that we can expand the potential energy around its minimumby using a Taylor series, which is a common way of expressing how a function variesnear a selected point (in this case, the minimum of the curve at x = 0):

V(x) =V(0) +0

x + 0

x2 + · · · (12.28)

The notation (. . .)0 means that the derivatives are first evaluated and then x is setequal to 0. The term V(0) can be set arbitrarily to zero. The first derivative of V iszero at the minimum. Therefore, the first surviving term is proportional to the squareof the displacement. For small displacements we can ignore all the higher terms, andso write

V(x) ≈0

x2 (12.29)DEF

d2V

dx2

ABC12

DEFd2V

dx2

ABC12

DEFdV

dx

ABC

Parabolicpotential energy

12

A

A

B

B

AB

(–1)J

Changesign ifantiparallel

Ch

ang

e si

gn

Rotateby 180°

Fig. 12.21 The interchange of two identicalfermion nuclei results in the change in signof the overall wavefunction. The relabellingcan be thought of as occurring in two steps:the first is a rotation of the molecule; thesecond is the interchange of unlike spins(represented by the different colours of thenuclei). The wavefunction changes sign inthe second step if the nuclei haveantiparallel spins.

J = 1

J = 0

Lowest rotational stateof ortho-hydrogen

Lowest rotational stateof para-hydrogen

Thermalrelaxation

Fig. 12.22 When hydrogen is cooled, themolecules with parallel nuclear spinsaccumulate in their lowest availablerotational state, the one with J = 1.They can enter the lowest rotational state (J = 0) only if the spins change their relativeorientation and become antiparallel. This is a slow process under normalcircumstances, so energy is slowly released.

12.8 MOLECULAR VIBRATIONS 463

Therefore, the first approximation to a molecular potential energy curve is a parabolicpotential, and we can identify the force constant as

kf =0

[12.30]

We see that, if the potential energy curve is sharply curved close to its minimum, thenkf will be large. Conversely, if the potential energy curve is wide and shallow, then kf

will be small (Fig. 12.24).The Schrödinger equation for the relative motion of two atoms of masses m1 and

m2 with a parabolic potential energy is

− + kf x2ψ = Eψ (12.31)

where meff is the effective mass:

meff = (12.32)

These equations are derived in the same way as in Further information 9.1, but here theseparation of variables procedure is used to separate the relative motion of the atomsfrom the motion of the molecule as a whole.

The Schrödinger equation in eqn 12.31 is the same as eqn 8.23 for a particle of massm undergoing harmonic motion. Therefore, we can use the results of Section 8.4 towrite down the permitted vibrational energy levels:

Ev = (v + )$ω ω =1/2

v = 0, 1, 2, . . . (12.33)Vibrational energylevels of a diatomicmolecule

DEFkf

meff

ABC12

Effective massm1m2

m1 + m2

12

d2ψdx 2

$2

2meff

Formal definition ofthe force constant

DEFd2V

dx2

ABC

Mo

lecu

lar

po

ten

tial

en

erg

yParabola

Internuclear separation, R

Re

Fig. 12.23 A molecular potential energycurve can be approximated by a parabolanear the bottom of the well. The parabolicpotential leads to harmonic oscillations. At high excitation energies the parabolicapproximation is poor (the true potential isless confining), and it is totally wrong nearthe dissociation limit.

Po

ten

tial

en

erg

y, V

Displacement, x

Increasingkf

Fig. 12.24 The force constant is a measure ofthe curvature of the potential energy closeto the equilibrium extension of the bond. A strongly confining well (one with steepsides, a stiff bond) corresponds to highvalues of kf.

A note on good practice Distinguisheffective mass from reduced mass. Theformer is a measure of the mass that ismoved during a vibration. The latteris the quantity that emerges from theseparation of relative internal andoverall translational motion. For adiatomic molecule the two are thesame, but that is not true in generalfor vibrations of polyatomicmolecules. Many, however, do notmake this distinction and refer toboth quantities as the ‘reduced mass’.

464 12 MOLECULAR SPECTROSCOPY 1: ROTATIONAL AND VIBRATIONAL SPECTRA

The vibrational terms of a molecule, the energies of its vibrational states expressed aswavenumbers, are denoted ô(v), with Ev = hcô(v), so

ô(v) = (v + )# # =1/2

(12.34)

The vibrational wavefunctions are the same as those discussed in Section 8.5.It is important to note that the vibrational terms depend on the effective mass of the

molecule, not directly on its total mass. This dependence is physically reasonable for,if atom 1 were as heavy as a brick wall, then we would find meff ≈ m2, the mass of the lighter atom. The vibration would then be that of a light atom relative to that of a stationary wall (this is approximately the case in HI, for example, where the I atombarely moves and meff ≈ mH). For a homonuclear diatomic molecule m1 = m2, and theeffective mass is half the total mass: meff = m.

• A brief illustration

An HCl molecule has a force constant of 516 N m−1, a reasonably typical value for a

single bond. The effective mass of 1H35Cl is 1.63 × 10−27 kg (note that this mass is very

close to the mass of the hydrogen atom, 1.67 × 10−27 kg, so the Cl atom is acting like

a brick wall). These values imply ω = 5.63 × 1014 s−1, ν = 89.5 THz (1 THz = 1012 Hz),

# = 2987 cm−1, λ = 3.35 μm. These characteristics correspond to electromagnetic radia-

tion in the infrared region. •

12.9 Selection rules

Key points The gross selection rule for infrared spectra is that the electric dipole moment of the

molecule must change when the atoms are displaced relative to one another. The specific selection

rule is ΔV = ±1.

The gross selection rule for a change in vibrational state brought about by absorptionor emission of radiation is that the electric dipole moment of the molecule must changewhen the atoms are displaced relative to one another. Such vibrations are said to be infrared active. The classical basis of this rule is that the molecule can shake the electromagnetic field into oscillation if its dipole changes as it vibrates, and vice versa(Fig. 12.25); its formal basis is given in Further information 12.2. Note that themolecule need not have a permanent dipole: the rule requires only a change in dipolemoment, possibly from zero. Some vibrations do not affect the molecule’s dipole moment (for instance, the stretching motion of a homonuclear diatomic molecule),so they neither absorb nor generate radiation: such vibrations are said to be infraredinactive. Homonuclear diatomic molecules are infrared inactive because their dipolemoments remain zero however long the bond; heteronuclear diatomic molecules areinfrared active.

• A brief illustration

Of the molecules N2, CO2, OCS, H2O, CH2=CH2, and C6H6, all except N2 possess at least

one vibrational mode that results in a change of dipole moment, so all except N2 can

show a vibrational absorption spectrum. Not all the modes of complex molecules are

vibrationally active. For example, the symmetric stretch of CO2, in which the O–C–O

bonds stretch and contract symmetrically, is inactive because it leaves the dipole

moment unchanged (at zero). •

12

Vibrational terms ofa diatomic molecule

DEFkf

meff

ABC1

2πc12

Fig. 12.25 The oscillation of a molecule,even if it is nonpolar, may result in anoscillating dipole that can interact with the electromagnetic field.

12.10 ANHARMONICITY 465

Weak infrared transitions can be observed from homonuclear diatomic moleculestrapped within various nanomaterials. For instance, when incorporated into solidC60, H2 molecules interact through van der Waals forces with the surrounding C60

molecules and acquire dipole moments, with the result that they have observable infrared spectra.

Self-test 12.6 Which of the molecules H2, NO, N2O, and CH4 have infrared activevibrations? [NO, N2O, CH4]

The specific selection rule, which is obtained from an analysis of the expression forthe transition moment and the properties of integrals over harmonic oscillator wave-functions (as shown in Further information 12.2), is

Δv = ±1 (12.35)

Transitions for which Δv = +1 correspond to absorption and those with Δv = −1correspond to emission. It follows that the wavenumbers of allowed vibrational transitions, which are denoted Δôv+ 1––

2for the transition v + 1 ← v, are

Δôv+ 1––2= ô(v + 1) − ô(v) = # (12.36)

As we have seen, # lies in the infrared region of the electromagnetic spectrum, so vibrational transitions absorb and generate infrared radiation.

At room temperature kT/hc ≈ 200 cm−1, and most vibrational wavenumbers aresignificantly greater than 200 cm−1. It follows from the Boltzmann distribution thatalmost all the molecules will be in their vibrational ground states initially. Hence, thedominant spectral transition will be the fundamental transition, 1 ← 0. As a result,the spectrum is expected to consist of a single absorption line. If the molecules areformed in a vibrationally excited state, such as when vibrationally excited HF mole-cules are formed in the reaction H2 + F2 → 2 HF*, the transitions 5 → 4, 4 → 3, . . .may also appear (in emission). In the harmonic approximation, all these lines lie atthe same frequency, and the spectrum is also a single line. However, as we shall nowshow, the breakdown of the harmonic approximation causes the transitions to lie atslightly different frequencies, so several lines are observed.

12.10 Anharmonicity

Key points (a) The Morse potential energy function can be used to describe anharmonic motion.

(b) A Birge–Sponer plot may be used to determine the dissociation energy of the bond in a

diatomic molecule.

The vibrational terms in eqn 12.34 are only approximate because they are based on a parabolic approximation to the actual potential energy curve. A parabola cannot be correct at all extensions because it does not allow a bond to dissociate. At high vibrational excitations the swing of the atoms (more precisely, the spread of the vibrational wavefunction) allows the molecule to explore regions of the potential energy curve where the parabolic approximation is poor and additional terms in theTaylor expansion of V (eqn 12.28) must be retained. The motion then becomes anharmonic, in the sense that the restoring force is no longer proportional to the displacement. Because the actual curve is less confining than a parabola, we can anticipate that the energy levels become less widely spaced at high excitations.

Specific vibrationalselection rule