MOLECULAR VIBRATIONS

contribute an electron that they can formnegativeions, stably but weakly binding an extra electron. This even includes H², the negative hydrogen ion. Hydrogen is in the odd position of having some properties in common with the alkali metals and some in common with the halogens.

The characterization of atomic electron states in terms of the four quantum numbers n,‘,m, andms, together with the Pauli exclusion principle, thus allows us to understand why Na is chemically similar to K, Mg is chemically similar to Ca, and so forth. These chemical periodicities, according to which the periodic table is arranged, are con- sequences of the way electrons fill in the allowed “slots” when they combine with nuclei to form atoms.

Of course, there is a great deal more that can be said about the periodic table. For a rigorous treatment of atomic structure, we must refer the reader to textbooks on atomic physics. As mentioned earlier, however, we can understand lasers without a more detailed understanding of atomic and molecular physics.

force is quantum mechanical; we will not attempt to explain it but will simply accept the result (2.3.1) and consider its consequences.

For simplicity, let us assume that the nuclei can move only in one dimension.

The total energy of a diatomic system (i.e., the sum of kinetic and potential energies) is then

E¼¹₂m₁x_₁²þ¹₂m₂x_₂²þ¹₂k(x₂x₁x₀)², (2:3:3) where the dots denote differentiation with respect to time, that is,x_ ¼dx=dt. In terms of the reduced mass

m¼m₁m₂

M , (2:3:4)

whereM¼m1þm2is the total mass, and the center-of-mass coordinate X¼m₁x₁þm₂x₂

M , (2:3:5)

we may write (2.3.3) as (Problem 2.2)

E¼¹₂MX_²þ¹₂mx_²þ¹₂k(xx0)²: (2:3:6) The first term is just the kinetic energy associated with the center-of-mass motion. We ignore it and focus our attention on the internal vibrational energy

E¼¹₂mx_²þ¹₂k(xx₀)²: (2:3:7) The vibrational motion of a diatomic molecule must clearly be one dimensional, and so we lose nothing in the way of generality by restricting ourselves to one-dimensional vibrations from the start [Eq. (2.3.3)].

The quantum mechanics of the motion associated with the energy formula (2.3.7) has much in common with that for the hydrogen atom electron. The most important result is that the allowed energiesEof the oscillator are also quantized. The quantized

m₁ m₂ F F F F

x₀ x x

(a) (b) (c)

Figure 2.4 (a) When the two nuclei of a diatomic molecule are separated by the equilibrium distance x0, there is no force between them. If their separationxis larger thanx0, there is an attractive force (b), whereas whenxis less thanx₀the force is repulsive (c). The internuclear force is approximately harmonic, that is, springlike.

2.3 MOLECULAR VIBRATIONS 27

energies are given by

En¼hv(nþ¹₂), n¼0, 1, 2, 3,. . ., (2:3:8) with

v¼ ffiffiffiffiffiffiffiffiffi k=m

p : (2:3:9) This formula is clearly quite different from Bohr’s formula for hydrogen. The quantum mechanical energy spectrum for a harmonic oscillator is simply a ladder of evenly spaced levels separated byhv(Fig. 2.5). The ground level of the oscillator corresponds ton¼0. However, an oscillator in its ground level is not at rest at its stable equilibrium point x¼x₀. Even the lowest possible energy of a quantum mechanical oscillator has finite kinetic and potential energy contributions. At zero absolute temperature, where classically all motion ceases, the quantum mechanical oscillator still has a finite energy¹₂hv. For this reason the energy¹₂hv is called the zero-point energy of the harmonic oscillator.

Of course, real diatomic molecules are not perfect harmonic oscillators, and their vibrational energies do not satisfy (2.3.8) precisely. Figure 2.6 shows the sort of potential

E₄ = — E

9 2 E₃ = —7

2 E₂ = — 5

2 E₁ = — 3

2 E₀ = —1

2 hw hw hw hw hw

Figure 2.5 The energy levels of a harmonic oscillator form a ladder with rung spacinghv.

V(x)

Harmonic oscillator

Real diatomic molecule

x₀

x (Internuclear separation)

Figure 2.6 The potential energy function of a real diatomic molecule is approximately like that of a harmonic oscillator for values ofxnearx₀.

energy functionV(x) that describes the bonding of a real diatomic molecule. The Taylor series expansion of the functionV(x) about the equilibrium pointx0is

V(x)¼V(x0)þ(xx0) dV dx _x¼x

0

þ1

2(xx0)² d²V dx²

x¼x0

þ1

6(xx₀)³ d³V dx³

x¼x0

þ : (2:3:10)

HereV(x0) is a constant, which we put equal to zero by shifting the origin of the energy scale. Also (dV/dx)x¼x0¼0 because, by definition,x¼x0is the equilibrium separation, at which the potential energy is a minimum. Furthermore (d²V/dx²) atx¼x0is positive ifx0is a point of stable equilibrium (Fig. 2.6). Thus, we can replace (2.3.10) by

V(x)¼¹₂k(xx0)²þA(xx0)³þB(xx0)⁴þ , (2:3:11) whereA,B,. . .are constants andk¼(d²V=dx²)_x¼x0.

From (2.3.10) we can conclude thatanypotential energy function describing a stable equilibrium [i.e., (dV=dx)_x¼x0 ¼0, (d²V=dx²)_x¼x0 .0] can be approximated by the harmonic oscillator potential (2.3.1) for small enough displacements from equilibrium.

Of course, what is “small” is determined by the constantsA,B,. . .in (2.3.11), that is, by the shape of the potential function V(x). If the terms involving third and/or higher powers of x 2x0in (2.3.11) are not negligible, however, we have what is called an anharmonic potential. The energy levels of an anharmonic oscillator do not satisfy the simple formula (2.3.8).

Real diatomic molecules have vibrational spectra that are usually only slightly anhar- monic. In conventional notation the vibrational energy levels of diatomic molecules are written in the form

E_v¼hcv_{e v}þ¹₂

x_evþ¹₂₂

þy_evþ¹₂₃

h þ i

, (2:3:12) where

v¼0, 1, 2, 3,. . ., (2:3:13)

andv_eis in units of “wave numbers,” i.e., cm²¹;cv_eis the same asv_/2p_¼nin this notation. If the anharmonicity coefficientsxe,ye,. . .are all zero, we recover the harmonic oscillator spectrum (2.3.8). Numerical values ofv_e,x_e,y_e,. . .are tabulated in the litera- ture.⁴Values ofv_e,xe,ye,. . .are given for several diatomic molecules in Table 2.1. The deviations from perfect harmonicity are small untilvbecomes large, that is, until we climb fairly high up the vibrational ladder. The level spacing decreases asvincreases, in contrast to the even spacing of the ideal harmonic oscillator (Fig. 2.7).

4A standard source is G. Herzberg, Molecular Spectra and Molecular Structure. Volume I, Spectra of Diatomic Molecules, Robert E. Krieger, Malabar, FL, 1989.

2.3 MOLECULAR VIBRATIONS 29

† For a simple check on our theory, consider the two molecules hydrogen fluoride (HF) and deuterium fluoride (DF). These molecules differ only to the extent that D has a neutron and a proton in its nucleus and H has only a proton. Since neutrons have no effect on molecular bond- ing, we expect HF and DF to have the same potential functionV(x) and therefore the same “spring constant”k. According to (2.3.9), therefore, we should have

v^HF_e

v^DF_e ¼ m^DF m^HF 1=2

, (2:3:14)

wherem^DFandm^HFare thereducedmasses of DF and HF, respectively, so that m^DF

m^HF¼ m^Dm^F m^Dþm^F

m^Hm^F

m^Hþm^F(2)(19)

2þ191þ19

(1)(19)1:90: (2:3:15) From the value ofvefor HF given in Table 2.1, therefore, we calculate

v^DF_e (4138:52)(1:90)¹⁼²¼2998:64 cm¹, (2:3:16) and indeed this is very close to the tabulated valueve¼2998.25 cm²¹for DF.⁴

Regarding the zero-point energy of molecular vibrations, consider a transition in which there is a change in both the electronic (e) and the vibrational (v) states of a diatomic molecule.

E

E_v+1 E_v E_v–1

Figure 2.7 The vibrational energy level spacing of a real (anharmonic) diatomic molecule decreases with increasing vibrational energy.

TABLE 2.1 Vibrational Constants of the Ground Electronic State for a Few Diatomic Molecules

Molecule ve(cm²¹) xe ye

H2 4395.24 0.0268 6.6710²⁵

O₂ 1580.36 0.00764 3.4610²⁵

CO 2170.21 0.00620 1.4210²⁵

HF 4138.52 0.0218 2.3710²⁴

HCl 2989.74 0.0174 1.8710²⁵

The transition energy is approximately

DEe⁰v⁰,ev¼Ee⁰Eeþhc v_e0v⁰þ¹₂

v_evþ¹₂

, (2:3:17)

where the unprimed and primed labels refer to the initial and final states, respectively, andveand ve⁰are the vibrational constants associated with the two electronic states. Now suppose that one of the nuclei of the diatomic molecule is replaced by a different isotope, for example, HF is replaced by DF. The electronic energy levels are approximately unchanged by this replacement, but the vibrational constantsveandve⁰are changed torveandrve⁰, whereris the square root of the ratio of reduced masses of the two molecules, as in our example above comparing HF and DF.

For the second, isotopically different molecule, then, the transition energy (2.3.17) is replaced by DE_eⁱ0v⁰,ev¼Ee⁰Eeþhc rv_e0v⁰þ¹₂

rv_evþ¹₂

: (2:3:18)

The vibrational spectra of the two isotopic molecules for the same electronic transitionEe!Ee⁰

therefore differ by

DEⁱ_e0v⁰,evDEe⁰v⁰,ev¼hc(r1) v_e0v⁰þ¹₂

v_evþ¹₂

, (2:3:19) and in particular, forv¼0!v⁰¼0,

DEⁱ_e00,e0DEe⁰0,e0¼¹₂hc(r1)(ve⁰ve): (2:3:20) This is nonzero because of zero-point energy, that is, it would vanish if the energy levels of a harmonic oscillator were given byEn¼nhv instead of En¼(nþ¹₂)hv. The zero-point energy of molecular vibrations was confirmed in this way by R. S. Mulliken (1924), who compared the observed vibrational spectra of B¹⁰O¹⁶and B¹¹O¹⁶. This was before the quantum mechanical derivation by Heisenberg (1925) of the formula (2.3.8) for the energy

levels of a harmonic oscillator. †

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