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Strictly the spherical transition
moments T(k,q) should have selection rules ΔS =
ΔΣ = 0, but as an extension PGOPHER relaxes this requirement and only
enforces the |ΔΩ| = q rule for ΔS ≠ 0
transitions. This will give some intensity to an otherwise
forbidden transition but does not give enough flexibility in all
cases, as several transition moments may be needed. (For exact
simulations give intensity to the spin forbidden transition by
using perturbations to mix in transitions to spin allowed states.)

For one photon ^{1}Σ
- ^{3}Σ transitions PGOPHER has enough flexibility as only one
parallel (q=0) and one
perpendicular (q=1)
component is required. The relative signs of these two are
significant and direct comparison with the results in the
literature requires care; there are two definitions for the
perpendicular component in use:

J.K.G.Watson, Can J Phys 46, 1637 (1968), equation
[23]:

μ(1) = 1/√2 < S=0 +- | μ(x) + i μ(y)
| S=1, -+, Σ=-1
>

J.T.Hougen, NBS
monograph 115, equation [3-27]:

μ(perp) = 1/√2 < S=1, -+, Σ=-1 | μ(x)
+ i μ(y) | S=0 +-
>

The only difference here is the order, but this is essential to the
definition. The two are related as follows: μ(1) = 1/√2 < S=0 +- | μ(x) + i μ(y)
| S=1, -+, Σ=-1
>

The complex conjugate can be ignored as μ(1) can be chosen to be
real. (in fact only the ratio μ(1)/μ(0) matters, and as shown
by Watson eq [25], this ratio is real.) PGOPHER uses the following definition for
T(1,1): = 1/√2 { < S=0 +- | μ(x) | S=1, -+, Σ=-1 > +
i < S=0 +- |
μ(y) | S=1, -+,
Σ=-1 > }

= 1/√2 { < S=1, -+, Σ=-1 | μ(x) | S=0 +- >*+ i < S=1, -+, Σ=-1 | μ(y) | S=0 +- >* }

= 1/√2 < S=1, -+, Σ=-1 | μ(x) - i μ(y) | S=0 +- >*

= -1/√2 < S=1, -+, Σ=+1 | μ(x) + i μ(y) | S=0 +- >*

= -μ(perp)*

= 1/√2 { < S=1, -+, Σ=-1 | μ(x) | S=0 +- >*+ i < S=1, -+, Σ=-1 | μ(y) | S=0 +- >* }

= 1/√2 < S=1, -+, Σ=-1 | μ(x) - i μ(y) | S=0 +- >*

= -1/√2 < S=1, -+, Σ=+1 | μ(x) + i μ(y) | S=0 +- >*

= -μ(perp)*

T(1,1) = -1/√2 < stateA,
Ω+1 | μ(x) + i μ(y) | stateB, Ω >

with stateA being the state in the bra as shown in the constants
window. If this is T(1,1) = -μ(perp) = μ(1)

but if the bra is T(1,1) = μ(perp) = -μ(1)

The difference is because with PGOPHER you can select
which state is on the left but e.g. Watson's definition always has
the singlet state on the
left, and swapping the states over introduces a minus sign. There
is no such problem with T(1,0) = μ(0) = μ(//).

Note on the definition of the strength parameter. For normal transitions, the strength is defined as the value of the matrix element:

<stateA, Λ+q| μ | stateB, Λ>

with the value of the <stateA,
-Λ-q| μ |
stateB, -Λ> matrix element derived by symmetry. For
forbidden transtions there are two symmetry related *q* = 0
matrix elements; the strength parameter is then the value of the
matrix element:

<stateA, Λ' Σ'| μ | stateB, Λ
Σ>