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The point is not really whether chloride or ammonia is a strong or weak field ligand, the point is $\ce{Co^3+}$ is $\mathrm{d^6}$, and virtually all "octahedral" $\mathrm{d^6}$ complexes are low spin - essentially some complexes of $\ce{Fe^2+}$ and a very small number of fluoro complexes of $\ce{Co^3+}$ are the only exceptions to the rule that all $\mathrm{... 2 This is a popular piece of confusion, because of a terseness in notation. Basically, we are secretly talking about two quantum numbers here: The total magnitude of the electron spin,$S$, and the relative orientation of the electron spins,$M_S$. As you correctly identify, for the$\alpha\beta + \beta\alpha$configuration you get some quantum number that is ... 2 I think the reason you're getting confused is because when we talk about "spin", we actually are referring to two separate observable values, the z-component of the spin (properly designated$\hat{S}_z$) and the square of the magnitude of the three-component spin vector (properly designated as$\hat{S}^2$). The commonly used values of$+\frac12$and$-\...
The spin operators for the $p^\textrm{th}$ MO are defined as follows: \begin{align} S_{x,p} &= \frac{\hbar}{2} \sum_{\tau,\tau'} c_{p,\tau}^\dagger \hat{\sigma}^x_{\tau,\tau'} c_{p,\tau'}, \\ S_{y,p} &= \frac{\hbar}{2} \sum_{\tau,\tau'} c_{p,\tau}^\dagger \hat{\sigma}^y_{\tau,\tau'} c_{p,\tau'}, \\ S_{z,p} &= \frac{\hbar}{2} \sum_{\tau,\tau'} c_{...