Documentation: clarify that `dipole` and `dipole-x` expect the same numerical TDM magnitude for perpendicular transitions

[absurd]

Summary

The documentation for the dipole keyword is ambiguous about whether it expects Cartesian body-fixed components or reduced spherical tensor components. For perpendicular transitions (|ΔΛ|=1), these differ by a factor of √2, leading to a factor of 2 error in Einstein A coefficients and lifetimes.

Numerical testing confirms that dipole and dipole-x with the same Cartesian input values produce identical results. However, multiple documentation sources suggest they should differ:

• Eq. 42 in the paper defines spherical tensor components with the standard √2 factor (μ₊₁ = ∓(1/√2)(μ_x ± iμ_y))
• The readthedocs fields page describes dipole-x as "related to the Cartesian-representation," implying dipole uses a different (spherical/Lambda) representation
• The quadrupole documentation explicitly states "Duo requires quadrupole moment curves to be provided in the spherical irreducible representation," which creates a natural inference that dipole works the same way

The molpro_duo() subroutine in diatom.f90 does apply a -sqrt(0.5) factor when converting dipole-x input, but this is exactly cancelled by the |x⟩,|y⟩ → |+Λ⟩,|-Λ⟩ unitary transformation, making the net conversion just a sign flip.

Test case

Two-state CH model (X²Π ↔ B²Σ⁻, ΔΛ=1) with the same TDM curve:

Input method	Einstein A (s⁻¹)
dipole + raw Cartesian μ_x	4.592×10⁶
dipole-x + raw Cartesian μ_x	4.592×10⁶
dipole + μ_x/√2	2.296×10⁶

Impact

This ambiguity is causing users to unnecessarily divide their ab initio Cartesian TDMs by √2 before providing them to Duo, resulting in lifetimes that are 2× too large. This came up in a Reddit thread where at least one user was affected, and they mention a published calculation that may also have this error.

Suggestion

A note in the dipole documentation clarifying that the Cartesian body-fixed component should be used directly would prevent this confusion. Something like:

For perpendicular transitions (|ΔΛ|=1), dipole expects the same numerical value as the Cartesian body-fixed component (e.g. μ_x). Do not apply the √2 conversion to spherical tensor components; this is handled internally. dipole and dipole-x accept the same numerical values.

[Trovemaster]

It is a good point, thank you very much. I will think about making this less confusing. Adding a note is a good idea.