We emphasize that Method #2 can evaluate the exact determinant of the orbital overlap matrix (Step (i ′ )-(iii ′ )) at the same order of computational cost as in Method #1, but does not rely on approximating the orbital determinants. More importantly, by design, the key advantage of Method #2 is the decomposition of the summation in Eq. ( ) into two cheaper factorized summations (A JK and B J C K ) as in Step (iv ′ ) and (v ′ ). Therefore, the