MODULE 15 (701)
Selection Rules for Electronic Transitions
We start with transitions between atomic states, then generalize
to molecular states.
Selection rules for electric dipole transitions may be defined 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.
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We first revisit parity.
We met parity earlier when we looked at the symmetry properties
of the two orbitals that we obtained for the hydrogen moleculeion, 1sg and 2su , g = “gerade” and u = “ungerade”.
These characteristics were obtained by performing a parity
operation on the orbitals, which means looking at the sign of the
orbital when the coordinates of a point are inverted through the
center of symmetry.
In a Cartesian system, an eigenfunction having even parity would
satisfy the equality
 ( x,  y,  z )   ( x, y, z )
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An orbital with odd parity satisfies the equality
 ( x,  y,  z )   ( x, y, z )
EVEN
ODD
1.0
1.0
0.8
0.5
sin x
sin x
0.6
0.0
0.4
-0.5
0.2
0.0
-1.0
-2
-2
-2
-1
x
-1
-1
-2
-2
-1
-1
0
x
All eigenfunctions that are bound-state solutions to timeindependent Schrödinger equations for a potential that can be
written as V(r) have a definite parity, either g or u.
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Such eigenfunctions are for systems that are constrained by a
centro-symmetric Coulomb potential.
Examples are hydrogenic atoms, homonuclear diatomic molecules,
and some polyatomics. These all 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, q, f).
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When we do this
r  r, q   q , f    f
z
y
q
f
r
r
q
x
x
f
If we apply these parity transformations to some of the
hydrogenic wavefunctions we find that
 nlm (r, q ,  f )  (1)l nlm (r,q , f )
l
l
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 nlm (r, q ,  f )  (1)l nlm (r,q , f )
l
l
We see that the OAM quantum number, l determines the parity of
the wavefunction.
If l is even, the parity is even. If l odd, the parity is odd (the sign
of the wavefunction changes).
What about the parity of the dipole moment operator ˆ if in the
matrix element?
The position vector (r) changes into its negative when the signs of
the Cartesian coordinates are changed.
Therefore the parity of r, and hence ˆ if is odd.
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Recall the transition moment dipole:
 fi  f ˆ fi i
Since the operator is odd, the whole integrand will be odd if the
two wavefunctions have the same parity (both even; both odd).
Then the integral will vanish and the transition rate will be zero.
Considering the schematic electric dipole transition:
1  h  2
For it to be effective, the two states involved must be of different
parity.
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According to our analysis of the behavior of the hydrogenic
wavefunctions the parities of wavefunctions are determined by the
factor (-1)l.
Thus l must change by 1 for the transition to have a non-zero rate
(to be allowed).
This generates one part of the Laporte selection rule
“For a transition to be allowed, Dl = ±1.”
Dl = 0, or Dl = ±2 changes do not change the parity and are
therefore not allowed.
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Theory states that photons emitted in electric dipole transitions
have angular momentum of 1 in atomic units (ħ).
Thus to conserve angular momentum during an electric dipole
transition, the total angular momentum of the atom must change
according to the rule Dj = 0, ±1.
The Dj = 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 Dj selection rule forbids transitions Dl = ±3, ±5, (acceptable
by parity) which would lead to too large changes in total AM.
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These rules forbid some ED transitions that have favorable energy
correspondence.
Formally forbidden transitions can have non-zero rates (104 to 106
reduction) because of the second order influences of temporal
fluctuations in magnetic dipoles/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.
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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.
Where parity is definable (molecules with inversion symmetry) gto-g and u-to-u transitions are forbidden, as are singlet-to-triplet
and similar intersystem crossing processes.
Polyatomic organic and organometallic molecules are discussed
in terms of their light-absorbing residues (chromophores).
In such molecules the transitions (absorption and emission) can
be identified as arising from a particular grouping of atoms.
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Examples of chromophores are carbonyl, nitro, phenyl, naphthyl,
The wavelength regions in which transitions occur are typical for
the chromophores, irrespective of the molecular environment.
The MOs of such systems have designations such as HOMO,
HOMO-1, LUMO, LUMO+1, ...
Labels such as s, , d, n, *, s* d*, … occur which inform about
the bonding/anti-bonding nature of the orbital and about its
electron density distribution.
The lowest energy (longest wavelength) transition is HOMO-toLUMO, but which orbital type is HOMO and which is LUMO can
vary from compound to compound, and solvent to solvent.
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The carbonyl chromophore is a good example.
The plots show the results of a HF 6-31G** calculation for H2CO.
The HOMO-to-LUMO
LUMO
transition is *<-n, and the
HOMO-1 to LUMO is *<-.
*
HOMO
The *<-n (or n,*)
transition that occurs in
simple carbonyls near 290
nm thus involves the
transfer of electron density
n
from the O atom to the C
atom (the * orbital

spreads over both atoms
whereas the n-orbital is
HOMO-1
localized on O).
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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 the
state generated by this transition will be less polar than that
generated by the n,* transition.
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This electron density redistribution during *<-n transitions
causes interesting solvent effects.
In H-bonding and polar solvents the lone pair on O will either be
involved in H-bonding, or will induce specific solvent orientations.
These interactions stabilize the ground state (lower energy).
During a transition the electron redistribution occurs much faster
than nuclei can respond, thus the instantaneous excited state is in
an unfavorable solvent environment.
This raises the transition energy.
Such influences are not effective in the *< transition, and the
energy is hardly influenced by the nature of the solvent.
The solvent-induced shift in the transition energy can be large
enough to cause a change in the state ordering, i.e. the *<
can become the lowest energy transition.
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Another major difference between these two types of transitions
is that *< is allowed while the *<n is forbidden for symmetry
reasons.
To a good approximation the n-orbital confined on the O atom is
O2py (the z-direction 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 orthogonally disposed and the
integral in the matrix element will vanish.
The transition dipole moment is thus zero and the transition is
forbidden.
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Forbidden transitions are rarely completely forbidden.
Magnetic dipole and electric quadrupole oscillations can assist, as
can coupling between electronic and vibrational motions.
The *< transition is fully allowed since it involves orbitals of
the same symmetry.
Its transition dipole is directed along the inter-nuclear axis.
As a result of this transition the * orbital becomes occupied, the
bond order is lowered and the bond weakens.
The resulting s framework acquires torsional freedom.
Thus the excited state of such species may be perpendicular,
whereas the ground state was planar--photoisomerization.