BioPHYSICAL chemistry

Let us determine what energies and wavelengths are relevant for transi-

tions between adjacent energy states. Consider a proton held to a spring

with k=500Nm−^1 :

(11.33)

E=Zω=(1.05 × 10 −^34 Js)(5.5 × 1014 s−^1 ) =5.8 × 10 −^20 J (11.34)

(11.35)

In general, transitions of vibrational states are always associated with

infrared light and involve energies that are much smaller than the energies

associated with the electronic levels.

Vibrational spectra

In small molecules, the vibrational modes of the

molecules are well defined and predictable. For

a molecule of Natoms, there are 3N–6 inde-

pendent modes if the molecule is nonlinear and

3 N–5 if it is linear. For example, CO 2 has four

vibrational modes, as shown in Figure 11.5.

Once the molecules become larger predicting

the vibrational modes becomes more complex

because of anharmonicities, the effects of mole-

cular rotation, and collisions. However, because

different groups of the molecule have certain

spectral features at characteristic frequencies,

the infrared spectrum can still often be used for

identification. Since proteins and DNA are built

from repeating units, their vibrational spectra will

reflect these repeating units, allowing assign-

ments of the individual vibrational bands. Typ-

ical values for these modes are: N–H stretch at

3200 –3500 cm−^1 ; N 6 H deformation at 1500 –

1600 cm−^1 ; and C 7 O stretch at 1600 –1800 cm−^1.

The C 7 O stretch is usually called the amide

I band and the N 6 H deformation is called the

amide II band.

The presence of a hydrogen bond will cause

shift of a vibrational mode to a lower frequency

λ

ν

==. μ

hc

34 m

ω

.

=.

×

=×

−

−

(^500) −
167 10

55 10

1

27

Nm 14 1

kg

s

230 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

(c)

Figure 11.5Some
vibrational modes
for carbon dioxide:
(a) two stretching
modes, (b) a
symmetrical and
antisymmetrical
mode, and (c) two
perpendicular modes.

(a)

(b)