Forbidden Transitions in Linear Molecules

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 ¹Σ - ³Σ 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 >

= 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)*

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):

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 ³Σ we have:

T(1,1) = -μ(perp) = μ(1)

but if the bra is ¹Σ (reversing the states) the definition of μ(1) and μ(⊥) does not change but the definition of T(1,1) does, giving:

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>

with Λ > 0, or Σ > 0 if Λ = 0. This definition was introduced in version 7.1.450, and versions before this gave incorrect values for some q = 0 matrix elements.