BioPHYSICAL chemistry

According to the properties of spherical harmonics:

(14.26)

Complete integration of these terms will of course give the exact dipole

moment for any transitions. For example, for a transition from the 1s

orbital to the 2pzorbital, the calculation gives the number 0.74 atomic

units along the zdirection and 0 along the x

and yaxes.

For molecules, there are also selection

rules reflecting the symmetry ofmolecules

(Figure 14.10) that arise from the conserva-

tion of momentum. Homonuclear molecules

may possess parity. The parity of a mole-

cular orbital is even, denoted by gfrom the

German word geradefor even, if the sign of

the orbital is unchanged for a vector pass-

ing through the center. Likewise, the parity

is odd, denoted by ufor ungerade, if the sign

changes. Since the dipole operator has odd

parity, the dipole moment will be exactly

equal to zero if the wavefunctions have the

same parity (because the integrand will be

odd and the integral of an odd function is alwayszero

if the limits are equal but of opposite sign).

The Franck–Condon principle

Biological systems have a large number of vibrational

states and the transitions can be to any of these states.

The Franck–Condon principle states that only certain

transitions are observed. During a transition the nuclei

can be considered to be not moving; thus the electron can

be pictured as making a transition between two electronic

states without any involvement of contributions from

nuclei. The electrons are pictured as existing in states

that are formed by vibronic oscillators that represent the

bonds between atoms in the molecule. The transition

occurs for a precise configuration of the nuclei without

any alteration of the nuclear geometry. This is repres-

ented as a strictly vertical transition from the lower state

to the higher state (Figure 14.11). The level marked with

the asterisk is the most probable transition state because

of the matching of the wavefunctions.

YYYlfm f*llli m i, sin l

0

2

0

10 0

ππ

∫∫ θθφdd= unlessffli=±^1

302 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

πg σu

πu σg

Figure 14.10The parity of an orbital may be even
(g) or odd (u) for homonuclear molecules.

*

Figure 14.11The Franck–Condon
principle states that optical transitions
correspond to vertical transitions
between electronic states with no
change of nuclear coordinates.