In which case, effective nuclear charge is minimum?

The correct option is A. Be

Effective nuclear charge $Z_{eff}=Z-S$

where Z $=$ atomic number and S $=$ screening constant

$=$ 0.35 per electron for electron in nth orbit.
$=$ 0.85 per electron for electrons in (n - 1)th orbit
$=$ 1.00 per electron for electrons in (n - 2)th, (n - 3)th, (n - 4)th orbit
$=$ 0.30 per electron in 1s-orbital (when alone)

$Be$: $1s^{2}2s^2$; one valence-electron in 2s is screened by one electron in 2s-orbital (nth orbit) and two electrons in 1s orbital ((n-1)th orbit)

$\therefore S=0.35+2\times 0.85=2.05$

$\therefore Z_{eff}=4-2.05=1.95$ (given)

$B{e}^{+}:1s^{2}2s^1;2s$ - electron is screened by two electrons in 1s-orbital ((n -1)th orbital)

$S=2\times 0.85=1.70$

$\therefore Z_{eff}=4-1.70=2.30$

$B{e}^{2+}:1s^2$; one electron in 1s-orbital (alone exists) is screened by another electron in same orbit. Thus,

$S=0.30$

$\therefore Z_{eff}=4-0.30=3.70$

$B{e}^{3+}:1s^1$ no-screening (single electron)

$S=0.00$

Thus, $Z_{eff}=4$

Thus, $Z_{effective}:Be<B{e}^{+}<B{e}^{2+}<B{e}^{3+}$

Hence, the correct option is $\text{A}$