  Using ab initio calculations based on the correlated band theory, we have
investigated the quasi-one-dimensional chain system $\beta$-V$_2$O(PO$_4$),
showing both charge and spin orderings. Even in the uncorrelated region, the
pure transition from the tetragonal to the monoclinic structure leads to a
sizable charge difference between the two types of V ions, regardless of
magnetic orders. In the ferrimagnetic phase, inclusion of the on-site Coulomb
repulsion $U$ leads to a full orbital-polarization of V1 ($t_{2g}^{3\uparrow}$,
$S=\frac{3}{2}$) and V2 ($a_{1g}^{1\downarrow}e_g^{\prime{1\downarrow}}$,
$S=1$) above $U^c_{eff}\approx3.5$ eV, leading to local spin moments of 2.30
and --1.54 $\mu_B$, respectively, with small orbital moments of several
hundredth $\mu_B$. So, the net moment is nearly 1 $\mu_B$ per formula unit,
which is about 2--3 times larger than the experimental value. Our results show
significant variations, strongly depending on the strength of $U_{eff}$, in
energy differences between various magnetic states as well as a small magnetic
anisotropy. These results suggest that the substantial difference between the
calculated and experimental moments is attributed to quantum fluctuation of the
pyrochlore-like weakly linked V$_4$ tetrahedral structure. Our findings are
expected to provide a good platform to investigate the interplay among the
charge-, spin-, and lattice-degrees of freedom, and geometrical frustration.
