Cauchy-Schwarz integral screening different inequalites
As you have already stated in the question, (Ineq. 1) is the standard inequality used in Cauchy-Schwarz integral screening. (Ineq. 2) and (Ineq. 3) are not valid inequalities, which is clear if you select atomic orbitals such that a &amp; b are spatially disjoint from c &amp; d and the proposed upper bounds vanish. (Ineq. 4), (Ineq. 5), and (Ineq. 6) follow from a different use of the Cauchy-Schwarz inequality where the inner product is interpreted as being over the six spatial integrals and requires the Coulomb kernel to be pointwise non-negative so that you can take its element-wise square root,

$$ \begin{align} |g_{abcd}| &amp;= \left| \int d\vec{r} d\vec{r}' \phi_a(\vec{r}) \phi_b(\vec{r}) V(\vec{r},\vec{r}') \phi_c(\vec{r}') \phi_d(\vec{r}') \right| \\
&amp;= \left| \sqrt{V}\phi_a \phi_c \cdot \sqrt{V}\phi_b \phi_d \right| \\ &amp; \le \| \sqrt{V}\phi_a \phi_c \| \| \sqrt{V}\phi_b \phi_d \| = \sqrt{g_{aacc}} \sqrt{g_{bbdd}} \end{align}. $$  (Ineq. 4) is not useful for integral screening because it is strictly weaker than (Ineq. 1) since $g_{abab} \le g_{aabb}$ (for real orbitals). Whereas $g_{abab}$ vanishes as orbitals a &amp; b become spatially disjoint, $g_{aabb}$ only decays algebraically in the average distance between a &amp; b. (Ineq. 5) and (Ineq. 6) are not strictly weaker than (Ineq. 1), but they have the same problem of being weakly decaying bounds that don't screen very much. For example, they are tight and thus tighter than (Ineq. 1) when a=b and c=d, but this is a class of integrals that will only be screened out for extremely large spatial separations between a &amp; c.