"Hamiltonian operator has no effect on the spin function" what does it mean?
Because the operator $\mathrm d/\mathrm dx$ only acts on $x$ and not on $y$, we can write

$$\frac{\mathrm d}{\mathrm dx}[f(x)g(y)] = g(y) \left[\frac{\mathrm d}{\mathrm dx}f(x)\right]$$

Likewise, in this context, the Hamiltonian operator $\hat{H}$ only acts on $(x,y,z)$ and not $m_s$. This is explained by Levine1 in the paragraph immediately preceding this


  To a very good approximation, the Hamiltonian operator for a system of electrons does not involve the spin variables but is a function only of spatial coordinates and derivatives with respect to spatial coordinates [...]


Is there something about this paragraph which you don't understand? This is pretty much all there is to be said on the topic.




Levine, I. N. Quantum Chemistry (7th ed.), p 268