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 2pz orbital, the calculation gives the number 0.74 atomic
units along the z direction and 0 along the x
and y axes.
For molecules, there are also selection
rules reflecting the symmetry of molecules
(Figure 14.10) that arise from the conservation of momentum. Homonuclear molecules
may possess parity. The parity of a molecular orbital is even, denoted by g from the
German word gerade for even, if the sign of
the orbital is unchanged for a vector passing through the center. Likewise, the parity
is odd, denoted by u for 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 always zero
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 represented 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.
Y Y Y
l
lfm f
li m i
l
l
*
sin
,
0
2
0
10
0
π
π
θ θ φ
∫
∫
=
d d
unless f f li
= ± 1
302
PART 2
QUANTUM MECHANICS AND SPECTROSCOPY
πg
σu
σg
πu
Figure 14.10 The parity of an orbital may be even
(g) or odd (u) for homonuclear molecules.
*
Figure 14.11 The Franck–Condon
principle states that optical transitions
correspond to vertical transitions
between electronic states with no
change of nuclear coordinates.
9781405124362_4_014.qxd 4/29/08 13:09 Page 302
(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 2pz orbital, the calculation gives the number 0.74 atomic
units along the z direction and 0 along the x
and y axes.
For molecules, there are also selection
rules reflecting the symmetry of molecules
(Figure 14.10) that arise from the conservation of momentum. Homonuclear molecules
may possess parity. The parity of a molecular orbital is even, denoted by g from the
German word gerade for even, if the sign of
the orbital is unchanged for a vector passing through the center. Likewise, the parity
is odd, denoted by u for 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 always zero
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 represented 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.
Y Y Y
l
lfm f
li m i
l
l
*
sin
,
0
2
0
10
0
π
π
θ θ φ
∫
∫
=
d d
unless f f li
= ± 1
302
PART 2
QUANTUM MECHANICS AND SPECTROSCOPY
πg
σu
σg
πu
Figure 14.10 The parity of an orbital may be even
(g) or odd (u) for homonuclear molecules.
*
Figure 14.11 The Franck–Condon
principle states that optical transitions
correspond to vertical transitions
between electronic states with no
change of nuclear coordinates.
9781405124362_4_014.qxd 4/29/08 13:09 Page 302