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The color function for tetraquark configuration $(qq)(\bar{q}\bar{q})$ is shown following: $$ \zeta_1=\frac{1}{2\sqrt6}[(rb+br)(\bar{b}\bar{r}+\bar{r}\bar{b})+(gr+rg)(\bar{g}\bar{r}+\bar{r}\bar{g})+(gb+bg)(\bar{b}\bar{g}+\bar{g}\bar{b})+2(rr)(\bar{r}\bar{r})+2(bb)(\bar{b}\bar{b})+2(gg)(\bar{g}\bar{g})] $$

for $|6\bar{6}\rangle$ color $\text{SU}(3)$ representation.

However, for example, why is the term $(rb+br)\bar{r}\bar{r}$ impossible in $\zeta_{1}$?

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    $\begingroup$ ? The $\zeta_1$ color function is color neutral: all colors are neutralized by the respective anticolor. But your "for example" instance has unsaturated b and $\bar r$ indices, so it is not colorless. $\endgroup$ – Cosmas Zachos Feb 19 at 15:36

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