  We present a systematically improvable tensor hypercontraction (THC)
factorization based on interpolative separable density fitting (ISDF). We
illustrate algorithmic details to achieve this within the framework of Becke's
atom-centered quadrature grid. A single ISDF parameter $c_\text{ISDF}$ controls
the tradeoff between accuracy and cost. In particular, $c_\text{ISDF}$ sets the
number of interpolation points used in THC, $N_\text{IP} = c_\text{ISDF}\times
N_\text{X}$ with $N_\text{X}$ being the number of auxiliary basis functions. In
conjunction with the resolution-of-the-identity (RI) technique, we develop and
investigate the THC-RI algorithms for cubic-scaling exact exchange for
Hartree-Fock and range-separated hybrids (e.g., $\omega$B97X-V) and
quartic-scaling second- and third-order M{\o}ller-Plesset theory (MP2 and MP3).
These algorithms were evaluated over the W4-11 thermochemistry (atomization
energy) set and A24 non-covalent interaction benchmark set with standard
Dunning basis sets (cc-pVDZ, cc-pVTZ, aug-cc-pVDZ, and aug-cc-pVTZ). We
demonstrate the convergence of THC-RI algorithms to numerically exact RI
results using ISDF points. Based on these, we make recommendations on
$c_\text{ISDF}$ for each basis set and method. We also demonstrate the utility
of THC-RI exact exchange and MP2 for larger systems such as water clusters and
$\text{C}_{20}$. We stress that more challenges await in obtaining accurate and
numerically stable THC factorization for wavefunction amplitudes as well as the
space spanned by virtual orbitals in large basis sets and implementing
sparsity-aware THC-RI algorithms.
