The Chemistry Maths Book, Second Edition

12.5 The harmonic oscillator 349

EXAMPLE 12.10The vibrations of diatomic molecules

The vibrations of a diatomic molecule are often modelled in terms of the Morse

potential

(12.30)

where (Figure 12.2) Ris the distance between the

nuclei,R

e

is the distance at equilibrium (the equi-

librium bond length), D

e

is the dissociation energy

of the molecule and ais a constant (the vibrations of

the molecule can be visualized in terms of a ball

rolling forwards and backwards in the ‘potential

well’ in Figure 12.2).

A stable molecule in its ground or low-lying excited

vibrational states undergoes only small displace-

ments,R 1 − 1 R

e

, from equilibrium. Then, expanding the

potential-energy functionV(R) as a power series in

(R 1 − 1 R

e

),

(12.31)

≈ 1 a

2

D

e

(R 1 − 1 R

e

)

2

The force acting between the nuclei of the molecule is (see Section 5.7, equation

(5.57)),

(12.32)

Therefore, for small displacements, differentiation of (12.31) gives

F 1 ≈ 1 − 2 a

2

D

e

(R 1 − 1 R

e

) (12.33)

Ifk 1 = 12 a

2

D

e

andx 1 = 1 (R 1 − 1 R

e

), the force isF 1 ≈ 1 −kx, and the vibrations of the molecule

are (approximately) simple harmonic.

Equation (12.29) can be written in the standard form (12.3) of a homogeneous linear

equation with constant coefficients,

(12.34)

dx

dt

k

m

x

2

2

+= 0

F

dV

dR

=−

VDaRR aRR=−−−+













ee e

2233

()()

VR D e

aR R

()

()

=−













−−

e

e

1

2

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Figure 12.2