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I just recently began studying the Molecular Orbital Theory in Chemistry and I just learnt about the paramagnetic and diamagnetic nature of molecules based on their unfilled and filled orbitals. Now if I talk about the $H_{2}^{+}$ ion and the $H_{2}^{-}$ ion, both of them have an unfilled orbital.

In case of $H_{2}^{+}$, the electronic configuration is: $\sigma 1s^1$ and case of $H_{2}^{-}$, the electronic configuration is: $\sigma 1s^2 \sigma^{*}1s^{1}$. This shows that both the ions are paramagnetic. Now my question is, how to tell which is more paramagnetic than the other, is there a way to quantify the paramagnetic properties, and if yes, how?

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The paramagnetic properties of atoms and molecules are mostly determined by the properties of the electron (in principle, a rotating molecule also is susceptible to a magnetic field, but the interaction associated with this effect is negligible in most cases). In addition to electron spin angular momentum, an electron can also have orbital angular momentum. To understand which energy level will have the strongest interaction with a magnetic field, you need to know a little bit about angular momentum coupling. The example you give is actually rather simple, because s states have no orbital angular momentum and - ignoring interactions between different states - the strength of the interaction is only determined by the electron spin and roughly the same for both ions. Note also that the Lorentz force on these ions is about four orders of magnitude larger than the magnetic interaction of the electron spin with a magnetic field.

If you want to know more about this, you could search for "Zeeman effect".

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