  Many-body localization (MBL) is well characterized in Fock space. To quantify
the degree of this Fock space localization, the multifractal dimension $D_q$ is
employed; it has been claimed that $D_q$ shows a jump from the delocalized
value $D_q=1$ in the ETH phase (ETH: eigenstate thermalization hypothesis) to a
smaller value $0<D_q<1$ at the ETH-MBL transition, yet exhibiting a conspicuous
discrepancy from the fully localized value $D_q=0$, which indicate that
multifractality remains inside the MBL phase. Here, to better quantify the
situation we employ, instead of the commonly used computational basis, the
one-particle density matrix (OPDM) and use its eigenstates (natural orbitals)
as a Fock state basis for representing many-body eigenstates $|\psi\rangle$ of
the system. Using this basis, we compute $D_q$ and other indices quantifying
the Fock space localization, such as the local purity $S$, which is derived
from the occupation spectrum $\{n_\alpha\}$ (eigenvalues of the OPDM). We
highlight the statistical distribution of Hamming distance $x_{\mu\nu}$
occurring in the pair-wise coefficients $|a_\mu|^2|a_\nu|^2$ in $S$, and
compare this with a related quantity reported in the literature.
