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A reasonably general way to write the term in the Hamiltonian arising from the interaction of a molecule with an external electric field is of the form:

This is written in spherical tensor
form, with* k *being the* *rank of the tensors
involved. *p* is a projection quantum number (|*p*|
≤ *k*), reflecting the projection of a tensor of rank k
onto the space fixed *z* axis*. *Limiting the sum
to *k* = 1 corresponds to the common electric dipole case,
in which case **T**^{1}(*μ*) is the electric
dipole moment operator (in a space fixed frame) and **T**^{1}(*E*)
is the electric field. The Zeeman Hamiltonian is similar, with
electric field (*E*) replaced by magnetic field (*B*)
and the electric dipole being replaced by the magnetic dipole.
More exotic transitions can have contributions from other ranks,
including Raman (typically *k* = 0 and 2), quadrupole (*k* = 2)
and multiphoton (*k* ≤ number of photons)
transitions. For all these cases the matrix element of the above
operator is required to calculate intensities of transitions and
shifts in external fields, which can be written as:

…*F* implies various the
spin angular momenta adding up to *F*, though these will be
absent in the simplest cases with no spins where *J* = *F*.
We use *K* to illustrate the handling of molecule fixed
quantum numbers; for linear molecules read this as Ω instead
of *K*. η is used to represent all the vibrational and
electronic quantum numbers, apart from electronic spin; in PGOPHER this essentially
corresponds to the identity of the state object.

The field, **T*** ^{k}*(

Assuming **T*** ^{k}*(

with a similar multiplying factor
for the reduced matrix element introduced for each additional
spin. To evaluate the last term on the right hand side we must
transform the operator from space to body fixed components. This
can be done using Wigner *D* matrices:

where *q *is used to label
the projections of the tensor onto the body fixed axis system. The
matrix element with all the spins removed is:

_{} corresponds
to
the
(transition)
dipole
moment
in the body fixed axis system, and its numerical value corresponds
directly to the value input when using a spherical transition
moment. For calculating Stark and Zeeman shifts the above
equations (adapted as required for additional quantum numbers) can
be used directly to evaluate the required matrix elements for the
overall Hamiltonian matrix.

The intensity of a transition is
proportional to the square of the transition moment worked out
above. Two steps are required for this. Firstly, the above
expresses the matrix element in terms of the basis states, but a
transformation to the true states using the eigenvectors from the
matrix diagonalization to find the energy levels is required.
(This is molecule specific, so is not discussed further here, but
complicates the situation in cases like asymmetric tops and linear
molecules that are not a pure Hund's case (a), where the line
strength will involve a linear combination of the transition
moments above.) Secondly, in the absence of external fields, the
different *M* levels are degenerate, so the sum over all
possible transitions between the degenerate upper and lower state
levels is required. This leads naturally to the definition of the
line strength,* S*, in the literature*:*

The sum over *p* corresponds
to the sum over possible polarisations of the light; under
standard conditions, such as spontaneous fluorescence, the
unweighted sum as above is appropriate. However, to allow PGOPHER to handle
relative intensities between different ranks of transitions in
multiphoton spectroscopy, a different line strength is introduced,
*S*_{pol}, defined as the sum over upper and lower
state *M* for a single polarization, *p* = 0:

With this definition of *S*_{pol},
the contributions from different ranks of transition moment tensor
in a multiphoton spectrum can simply be added together, assuming
linearly polarised light is used. (Different weighting factors
must be used for circularly polarized light.)

The standard line strength for one
photon transitions comes from summing over the 2*k*+1
components. These must be independent of polarization (as
spontaneous fluorescence is unpolarized) which gives:

This also implies *S* = 3 *S*_{pol}
for single photon transitions.

For quantitative calculations of
absorption cross-section this has units, specifically the square
of the units used for the value in the transition moment object.
The built in conversions (see below) assume Debye for electric
dipole (transition) moments and Bohr or nuclear magneton for
magnetic dipole moments.

The definition of *S*_{pol} above
needs modification in the presence of a field, where the sum over
*M* levels is not performed, and each *M'* - *M*
transition is calculated separately.* S*_{pol} values
are still used internally and displayed, but the interpretation
needs care as the values displayed will depend on the polarization
of light selected. The default is random polarization, but this
can be changed by adding a Polarization
object under the Simulation object.
A scale factor is added so that the *S*_{pol} values
for the components in the presence of a field add up to the *S*_{pol}
value in the absence of a field. For one photon transitions this
is 1/3 for random polarization, 1 for parallel polarization and
½ for perpendicular polarization, so the nearest equivalent
of *S* in the presence of a field is obtained by dividing
the displayed *S*_{pol} by these values.

The line strengths, *S,* are
also Hönl-London factors, with some caveats. Consider a
simple parallel transition in a singlet state of a linear
molecule. This has *J* = *F*, *K* = Λ
and the above equations give:

For *R* branch transitions
within a Σ state, *J'* = *J* + 1
and Λ = 0 so the above becomes:

For an R branch of a perpendicular Π – Σ band:

The Hönl-London factors are *J* + 1
and *J* + 2 respectively, so Hönl-London
factors can be obtained by setting the transition moment value(s)
to 1 provided there is only a single transition moment
contributing to any given transition.To find the Hönl-London
factor for a particular transition given this setup, set the `IntensityUnits`
of the simulation to `HonlLondon` before simulating; in
the linelist window obtained by right
clicking on a transition the "Strength" column will then be the
Hönl-London factor. Note that there are several different
definitions of Hönl-London factors in the literature; see A
Hansson and J.K.G. Watson, J. Molec. Spectrosc **233** 169
(2005) for a discussion of this point for linear molecules. The
values calculated as above are consistent with the recommendations
of those workers.

Caveats: The number displayed is
simply 3 *S*_{pol} as defined above, so is correct
for single photon transitions, but care must be taken for
multiphoton transitions. For ranks (*k*) other than one a
scaling by (2*k*+1)/3 is in principle appropriate, but it
depends on the polarization of the light. As discussed above, for
linear polarization the *S*_{pol} values must be
added, rather than *S*, and a different rule is required for
circularly polarized light. In addition, setting the transition
moments to 1 may not be possible if more than one molecular
transition moment contributes to a given transition, as the
relative signs and magnitudes will affect the computed value for *S*_{pol}.
In particular quantum mechanical interference can take place so,
for example, if two transition moments contribute, setting both to
+1 will typically give a completely different spectrum to values
of +1 and –1.

The transitions
window displays the make up of *S*_{pol} but
can, for simpler cases, be switched to show *S.* The default
is to display:

i.e. *S*_{pol}^{1/2
} - note the inclusion of the 2*k*+1 factor. If a
only a single transition moment is involved in the selected
transitions, or all the transition moments have the same rank,
then a Hönl-London check box will be enabled and checking it
will remove the 2*k*+1 factor.

The “original” transition matrix pane shows the above matrix element between the basis states contributing to the selected total angular momenta and symmetries. The “transformed” transition matrix comes by transforming the original matrix with the eigenvectors from the matrix diagonalization for the two manifolds involved. The transformation must be done before the matrix elements are squared to allow for interference between transition moments.

Note that this reduced matrix element is not necessarily symmetric, even if the transition operator is Hermitian. This can be seen by considering:

Considering *p *= 0
specifically we expect:

and comparing equations implies

Reduced matrix elements off diagonal
in total angular momentum by an odd integer will therefore change
sign, depending on the order chosen. The vibronic part of a
transition moments is defined with respect to a specific order of
bra and ket and, depending on the ordering of the calculation, the
matrix element with the bra and ket swapped might be required,
which is calculated by swapping the bra and ket and taking the
complex conjugate of the result. For Hemitian operators in the
presence of a field, this is fine as the full matrix element,
including the *M* dependence, is calculated. In the absence
of a field the matrix element is also multiplied by (-1)^{J}^{-}^{J'}.
(In versions before 10 this factor was omitted, which could lead
to a spurious *J* dependence to the relative sign of a
transition moment which could lead to incorrect simulations in the
presence of interfering transition moments of different types,
though many such set-ups did not trigger the problem.)

Non-Hermitian operators, such as the
T^{1}_{1}(μ) dipole used for perpendicular
bands, require additional consideration, as the required matrix
element is then of the Hermitian conjugate operator, which
typically has matrix elements differing by (only) a phase factor.
If `FixTransitionPhase`
(introduced in version 10) is set then this factor is included in
the calculation as required. The default is to omit this to ensure
compatibility with older data files. This omission is harmless in
almost all circumstances as any single intensity calculation will
only need one specific order of bra and ket so the omitted phase
factor can be included by changing the sign of the affected
transition moments.