Cauchy-Schwarz integral screening different inequalites

Question

Eq. (9.12.23) in [1] states that:

$$\sum_{abcd}c_{ab}c_{cd}g_{abcd}>0\tag{MEST 9.12.23}\label{91223}$$

with $c_{ab}=c_ac_b$ for $c_a$ being some distribution coefficients and $g_{abcd}$ being the two-electron integrals in AO basis. Now for $c=a$ and $d=b$ giving:

$$c_{ab}c_{ab}>0$$

Which seem obvious as long as $c$ are real numbers. It can thus be seen that to fulfill Eq. (\ref{91223}) that:

$$g_{abab}>0 \tag{MEST 9.12.24}$$

Which fulfill the requirements for the Cauchy-Schwarz inequality, thus giving:

$$\left| g_{abcd} \right| \leq \sqrt{g_{abab}}\sqrt{g_{cdcd}}\tag{MEST 9.12.25}$$

Now since $c_{ab}=c_{a}c_{b}$ this implies that:

$$c_{aa}c_{bb}>0\tag{1}$$

which in turns further implies that:

$$g_{aabb}>0\tag{2}$$ Which surely also fulfill the requirements for the Cauchy-Schwarz inequality, giving:

$$\left| g_{abcd} \right| \leq \sqrt{g_{aabb}}\sqrt{g_{ccdd}}\tag{3}$$

Now further doing some index juggleling will lead to the following six different inequilities:

$$\left| g_{abcd} \right| \leq \sqrt{g_{abab}}\sqrt{g_{cdcd}}\tag{Ineq. 1}\label{ineq1}$$

$$\left| g_{abcd} \right| \leq \sqrt{g_{acac}}\sqrt{g_{bdbd}}\tag{Ineq. 2}$$

$$\left| g_{abcd} \right| \leq \sqrt{g_{adad}}\sqrt{g_{cbcb}}\tag{Ineq. 3}$$

$$\left| g_{abcd} \right| \leq \sqrt{g_{aabb}}\sqrt{g_{ccdd}}\tag{Ineq. 4}$$

$$\left| g_{abcd} \right| \leq \sqrt{g_{aacc}}\sqrt{g_{bbdd}}\tag{Ineq. 5}$$

$$\left| g_{abcd} \right| \leq \sqrt{g_{aadd}}\sqrt{g_{ccbb}}\tag{Ineq. 6}$$

I think I have ever only seen Eq. (\ref{ineq1}) described before, surely this could be because the rest are obvious extensions, that I just haven't thought of as being obvious before. Or that I have made some flaws in my reasoning. In practise are all of the inequilites used? If not anyone got clues about why not? Especially Ineq. 4-6 is somewhat different from Ineq. 1-3.

[1] : Molecular Electronic-Structure Theory, Trygve Helgaker, Poul and Jørgensen Jeppe Olsen

Answer 1

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 & b are spatially disjoint from c & 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,

$$ \begin{align} |g_{abcd}| &= \left| \int d\vec{r} d\vec{r}' \phi_a(\vec{r}) \phi_b(\vec{r}) V(\vec{r},\vec{r}') \phi_c(\vec{r}') \phi_d(\vec{r}') \right| \\ &= \left| \sqrt{V}\phi_a \phi_c \cdot \sqrt{V}\phi_b \phi_d \right| \\ & \le \| \sqrt{V}\phi_a \phi_c \| \| \sqrt{V}\phi_b \phi_d \| = \sqrt{g_{aacc}} \sqrt{g_{bbdd}} \end{align}. $$ (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 & b become spatially disjoint, $g_{aabb}$ only decays algebraically in the average distance between a & 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 & c.