Confused about unit conversion involving natural units

Hello! I have an expression whose natural units are Joules, but all the terms are expressed in terms of cm##^{-1}## (it is for an atomic transition). I have a term in the expression whose units are 1/A (angstrom) and I am not sure how to convert it to what I need. On one hand, if I were to go from 1/A to 1/cm, I would need to multiply that term by ##10^8##. On the other hand, if I want to convert to J, I need to multiply by ##\hbar## c, then convert from J to cm##^{-1}##, which gives ##15927759.569##. The difference between the 2 approaches is ##2\pi##, but I am not sure why. Should I actually use hc instead of ##\hbar##c? The issue is that the formula is defined using ##\hbar## and I am not sure what I am doing wrong. For reference I am talking about equation 1 in this paper (##\alpha_5##, ##A_1## and ##A_2## are unitless). Thank you!

You have implicit conversion factors somewhere. You need to make them explicit. (Well, you don't need to, but the chances of screwing up are smaller if you do)

kelly0303 said:

On one hand, if I were to go from 1/A to 1/cm, I would need to multiply that term by ##10^8##.

Correct.

kelly0303 said:

Should I actually use hc instead of ##\hbar##c?

Yes. What spectroscopists refer to as wave number is ## 1/\lambda ##, not ## k = 2 \pi / \lambda ##, as theorists often do. In terms of energy (## E=hc/\lambda ##), ##\rm 1 ~eV = 8065.5~cm^{-1} = 1.6022 \times 10^{-19}~J##.