BioPHYSICAL chemistry

compared to the same group that does not have the hydrogen bond.
Because the secondary structure has well-defined hydrogen-bonding
patterns, this results in highly characteristic vibrational spectra. For
example, the amide I mode has typical frequencies at: 1650 – 1660 cm−^1
for an αhelix, 1630 –1640 cm−^1 for a βsheet, and 1680 –1700 cm−^1 for
no hydrogen bond.
The effect of hydrogen bonding can be understood by reviewing the
properties of vibrational states. For a mass, m, attached to a wall by a spring
with a characteristic constant, k, the mass will vibrate with a vibrational
frequency ω(eqn 11.7):

(11.36)

We can represent a simple bond between two atoms as a spring, with
a spring constant kattached to two masses, m 1 and m 2. In this case, we
need to include the contribution of the mass of the second atom to the
vibrational frequency ω. This is done through the reduced mass μ:

(11.37)

then

(11.38)

The effect of hydrogen reducing the vibrational frequency can be modeled
as decreasing the bond strength or increasing the effective mass. The sen-
sitivity of the vibrational frequency to such effects allows identification of
the contributions that individual atoms make to the vibrational spectra
through isotopic substitution.
Consider a CO bond in a protein. Normally it is composed of^12 C^16 O and
a typical bond strength would be 1900 N m−^1. This leads to a vibrational
frequency of:

(11.39)

(11.40)

(11.41)

A

().

=

××−

−

−

1

23 10

1900

(^101) 114 10
1
π cm s^26
Nm
kgg
= 2167 cm−^1

A== =

νω

ccππμc

1 k

2

1

2

μ=.

• 
  

=

×

+

=×

mm

mm

CO m

CO

p

12 16

12 16

606 ( .1 67 10×=×−−^27 kg) 1 14. 1026 kg

ω
μ

=

k

μ
μ

=

+

=+

mm

mm m m

12

12 1 2

11 1

or

ω=
k
m

CHAPTER 11 VIBRATIONAL MOTION 231