Physical Chemistry Third Edition

938 22 Translational, Rotational, and Vibrational States of Atoms and Molecules

depend on only 3n−5 of the new coordinates for a linear molecule, or on 3n−6of

them for a nonlinear molecule.

The new coordinates are linear combinations of theq’s:

wi

∑^3 n

j 1

cijqj (i1, 2,...,3n−5(6)) (22.4-14)

where thec’s are constants and where we introduce the notation 3n−5(6) to indicate

3 n−5 for a linear molecule and 3n−6 for a nonlinear molecule. We do not discuss

the process by which thecijcoefficients are chosen. The outcome is that

VVe+

1

2

3 n∑− 5 ( 6 )

i 1

κiw^2 i (22.4-15)

Thewcoordinates are callednormal coordinates. The constantκiis the effective force

constant for the coordinatewi.

The classical vibrational energy can be written

EvibKvib+Vvib

1

2

3 n∑− 5 ( 6 )

i 1

[

Mi

(

dwi

dt

) 2

+κiw^2 i

]

+Ve (22.4-16)

where theM’s are effective masses for the new coordinates. Equation (22.4-16) is a sum

of classical harmonic oscillator energy expressions. According to classical mechanics,

each normal coordinate oscillates independently with a characteristic classical fre-

quency given by

νi

1

2 π

√

κi

Mi

(22.4-17)

The motions of the normal coordinates are thenormal modesof vibration. Since each

normal coordinate is a linear combination of the Cartesian coordinates of the nuclei,

each normal mode corresponds to a concerted motion of some or all of the nuclei.

When the quantum mechanical Hamiltonian for vibration is constructed from

Eq. (22.4-16) there are 3n−5or3n−6 terms, each one of which is a harmonic

oscillator Hamiltonian operator. The variables can be separated, and the vibrational

Schr ̈odinger equation is solved by a vibrational wave function that is a product of

3 n−5or3n−6 factors:

ψvibψ 1 (w 1 )ψ 2 (w 2 )··· 

3 n−∏ 5 ( 6 )

i 1

ψi(wi) (22.4-18)

Eachψifactor is a harmonic oscillator wave function. The energy is a sum of harmonic

oscillator energy eigenvalues:

Evib

3 n∑− 5 ( 6 )

i 1

hνi

(

vi+

1

2

)



3 n∑− 5 ( 6 )

i 1

hc ̃νi

(

vi+

1

2

)

(22.4-19)

wherev 1 ,v 2 , ... are vibrational quantum numbers, one for each normal mode, andν 1 ,

ν 2 , and so on, are the frequencies of Eq. (22.4-17). The quantities ̃νi, ̃νi, and so on

are the frequencies divided by the speed of light, usually specified in cm−^1. Just as in

classical mechanics, each normal mode oscillates independently of the others.