I am studying the example 9.2 from Zetilli's Quantum Mechanics book which is about the Stark Effect. There is a week electric field $\mathcal{E}$ directed along the positive z-axis and we want to study the effect of such a field on the ground state of an Hydrogen atom (ignoring the spin effect) and also find and approximation for the polarizability of it.
First we find an expression for the energy shift, which is given by
$$ \Delta E = e^{2} \mathcal{E}^{2} \sum_{nlm \neq 100} \frac{\vert{\langle nlm \vert \hat{Z}\vert 100\rangle} \vert^{2}}{E_{100}^{(0)}-E_{nlm}^{(0)}} $$
Then in order to find the polarizability $\alpha$ the book writes this equation below
$$\alpha = \frac{-2\Delta E}{\mathcal{E}^{2}}$$
Which I don't understand where it comes from. I know that $\hat{P}= \alpha E $ but I don't know how to proceed from there.
After this we use an inequality based on the energies of the states $E_{100}^{(0)}$ and $E_{200}^{(0)}$ and to find an upper limit for $\alpha$ and get to this result (where $a_{0}$ is the Bohr radius)
$$ \alpha \leq \frac{16}{3} a_{0} \sum_{nlm \neq 100} \vert{\langle nlm \vert \hat{Z}\vert 100\rangle} \vert^{2}$$
This sum is equal to $\langle 100 \vert \hat{Z}^{2}\vert 100 \rangle$, but when I calculate the value from this expression using the radial wave function $R_{10}$ and I substitute this result in the inequality I get
$$\alpha \leq \frac{a_{0}^{1/2}}{3}\sqrt{\frac{\pi}{2}}$$
While the author get to this result
$$\alpha \leq a_{0}^{3} \frac{16}{3}$$
I don't know where is my mistake.