MODULE 15_03
Selection Rules for Electronic Transitions
We shall develop the argument for transitions between atomic states, and then generalize to the
molecular case. In a formal sense we can attribute the selection rules as
a set of conditions that apply to the quantum numbers of the eigenfunctions of the initial and
final states. If a pair of eigenfunctions possesses quantum numbers that do not conform to the
conditions, then the matrix element of the electric dipole moment becomes zero.
We can use a symmetry argument to elaborate this.
First we need to understand about parity. We met parity when we considered the symmetry
properties of the two orbitals that we obtained for the hydrogen molecule-ion, 1g and 2u,
where the g and u designations refer to “gerade” (even) and “ungerade” (odd).
These
characteristics were obtained by performing a parity operation on the orbitals, which means
looking at the sign of the orbital when the electron coordinates are inverted. In a Cartesian
system, an eigenfunction having even parity would satisfy the equality
 ( x,  y,  z )   ( x, y, z )
(15.1)
One with odd parity satisfies the equality
 ( x,  y,  z )   ( x, y, z )
Figures 15.1 and 15.2 show even and odd functions, respectively.
FIG 15.1
f ( x)
x
1
(15.2)
1
f ( x)
1
 3.142
x
3.142
FIG 15.2
All eigenfunctions that are bound-state solutions to time-independent Schrödinger equations for
a potential that can be written as V(r) have a definite parity, either even or odd.
The
eigenfunctions of those systems that are constrained by a centro-symmetric Coulomb potential,
such as hydrogenic atoms, homonuclear diatomic molecules, and some polyatomics exhibit
parity. Let us examine this for the hydrogenic (one-electron) atom wavefunctions we have
worked with earlier. To see this we need to transform our coordinates into spherical polar
(r, when
r  r,     ,     
The two 2-dimensional representations in Figure 15.3 show this.
Now carry out these parity transformations on several of the wavefunctions we have met in
earlier modules, when we find that
 nlm (r,  ,   )  (1)l nlm (r, ,  )
l
l
(15.3)
Thus the parity is determined by the orbital angular momentum quantum number, l. If l is even,
the parity is even; if it is odd, the parity is odd (the sign in front of the wavefunction changes).
2
y
z


r
r

x
x

FIGURE 15.3
Now we have to consider the parity of the dipole moment operator ˆ if (= -er) in the matrix
element.
The position vector changes into its negative when the signs of the Cartesian
coordinates are changed. Therefore the parity of r, and hence ˆ if is odd.
Looking again at the
matrix element  fi  f ˆ fi i we see that if the initial and final state eigenfunctions are of the
same parity (both even or both odd) the integrand will be odd since the operator is of odd parity.
The integral of an odd function vanishes because the contribution from one volume element will
be exactly cancelled by the contribution from the diametrically opposite element (see Figure
15.2). If this is the case the transition rate will be zero. For the integrand not to vanish in the
matrix element the parities of the two state eigenfunctions must be different.
Thus for an electric dipole transition from an initial state to a final state to be successful, it must
involve a change of parity (even to odd, or vice versa), and since the parities of wavefunctions
are determined by the factor (-1)l, clearly l must change by 1 if the transition is to have a nonzero rate. This generates the selection rule that l  1 for a transition to be allowed. This is
one statement of the Laporte Selection Rule. The changes l  0, or  2 do not change the parity
and are therefore not allowed.
Quantum electrodynamics theory [e.g. Richard Feynman] tells us that photons carries angular
momentum, in addition to linear momentum. It states that a photon emitted in an electric dipole
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transition has an angular momentum of 1 in atomic units ( ). Thus in order to conserve angular
momentum as a result of an electric dipole transition, the total angular momentum of the atom
must change according to the rule j  0,  1 . The j = 0 case is understood by allowing for a
change in the orientation in space of the total angular momentum vector when the transition
occurs. The j selection rule serves also to rule out l  3, 5 transitions, which are acceptable
according to parity but not allowed because they would lead to too large changes in total angular
momentum.
The selection rules discussed above prohibit potential electric dipole transitions, but the
transition may still occur at a much-reduced rate (by ca 104) through emission of radiation from
an oscillating magnetic dipole moment, brought about by temporal fluctuations in the orientation
of magnetic dipoles. Even lower rate (by ca 106) transitions can be induced by oscillating
electric quadrupoles.
Electric dipole transitions between states of different multiplicity are forbidden because spin is
not conserved during such events. This is because the perturbing influence is an electric field,
which has no influence on magnetic moments due to electron spin. Where the initial and final
states are orthogonal due to their spin functions, the perturbing hamiltonian is unable to mix in
states of differing multiplicity and the matrix element vanishes. We shall see later that the spin
conservation rule is also a requisite for collisional interactions between molecules. However, the
details differ from the photon case.
The selection rules for diatomic and larger molecules bear some similarities to those for atomic
species, but complications arise due to nuclear rotation and nuclear spin. The techniques of
Group Theory are usefully employed in determined the selection rules for diatomics and higher.
In the cases where parity is definable (molecules having inversion symmetry) g-to-g and u-to-u
transitions are forbidden, as are singlet-to-triplet and similar intersystem crossing processes.
Large molecules such as organics and organometallics are usually discussed in terms of the lightabsorbing elements (chromophores) they contain. In such molecules the radiative transitions
(absorption and emission) can be identified as arising from a particular grouping of atoms.
Examples are carbonyl, nitro, phenyl residues, etc.
The wavelength regions in which these
transitions occur are typical for the chromophores, irrespective of the molecular environment.
Molecular orbitals of such systems have designations such as HOMO, HOMO-1, LUMO,
LUMO+1, and so forth and we find other kinds of labels such as , , , n, *, * *, which
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inform about the bonding/anti-bonding nature of the orbital and about its electron density
distribution. The lowest energy (longest wavelength) transition is HOMO-to-LUMO, but which
orbital type is HOMO and which is LUMO can vary from compound to compound, and solvent
to solvent. The carbonyl chromophore is a good example. Figure 15.4 shows the results of a
Hartree-Fock 6-31G** calculation of the formaldehyde molecule. The top panel is designated as
LUMO, the center is HOMO and the bottom is HOMO-1. Scrutiny of the surfaces gives the
clear indication that the HOMO has a large amount of electron density in a pair of lobes on the
O-atom, and the remainder on the C-H -bonds. Thus the HOMO seems to be an n-orbital with
the electron lone pair located therein. On the other hand HOMO-1 has all its electron density in
a bonding -orbital composed of the same symmetry p-orbitals from the C and O atoms. In the
LUMO the O-associated density has its node in the xz plane whereas yz was the nodal plane for
the n-orbital. The LUMO is therefore most likely antibonding , or *. Thus we see that the
HOMO-to-LUMO transition is of  *  n character, and the HOMO-1 to LUMO is of
 *   character. The  *  n or often, n,*)transition that occurs in simple carbonyls near
290 nm thus involves the transfer of electron density from the O atom to the C atom, because the
antibonding orbital spreads over both atoms whereas the n-orbital is localized on O. The
HOMO-1 to LUMO absorption transition will occur at shorter wavelengths (higher energies)
than the above, and its character will be  *   (or This affects the electron density at C
and O much less, and thus the state produced by this transition will be less polar than that
generated by the n,* transition.
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FIGURE 15.4 Top panel is LUMO, middle is HOMO, and bottom is HOMO-1.
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This electron density redistribution during  *  n transitions creates an interesting solvent
effect. In hydrogen-bonding solvents, or polar solvents in general, the lone pair on O will either
be involved in H-bonding, or will induce particular solvent molecule orientations due to its
dipolar nature. These interactions will stabilize the ground state. However, during a transition
the electronic distribution shifts much more rapidly than the nuclei can respond to (see later), and
so the immediately formed excited state is in an unfavorable solvent environment. This means
that the transition energy will be raised.
Such influences are not effective in
the  *   transition, and the energy is hardly influenced by the nature of the solvent. In some
cases the shift in the  *  n transition energy is so large that it can cause a change in the normal
ordering of the states, i.e. the  *   can be the lowest energy transition.
Another major difference between these two types of transitions is that the  *   one is
allowed while the  *  n is forbidden. This arises from the symmetry properties of the orbitals.
To a good approximation the non-bonding orbital confined on the O atom is O2py (the zdirection by convention is along the bond). The * orbital has a node in the yz plane and it can
be approximately described as an LCAO-MO of the type: c1O 2 px  c2 2 px . The n (2p) and *
orbitals are thus orthogonally disposed and the integral in the matrix element will vanish. The
transition dipole moment is thus zero and the transition is forbidden. Of course, even forbidden
transitions are allowed to some extent. This is because the forbiddenness is an aspect of the use
of a simplified hamiltonian; the use of higher order approximations may help to relax the
constraints on the transition. Other sources of transition intensity may arise from the coupling of
electronic and vibrational motions in the molecule. We shall encounter this in more detail later.
The  *   transition on the other handinvolves orbitals of the same symmetryand the
transition is allowed with its transition dipole directed along the internuclear axis. As a result of
this transition in the carbonyl chromophore under consideration, the anti-bonding orbital
becomes occupied, thus reducing the C=O bond order. In molecules such as alkenes, this causes
the -bond to weaken and the resulting C-C framework to acquire torsional motion. Thus the
excited state of such species may be perpendicular, whereas the ground state was planarphotoisomerization.
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