By ab initio methods, we have characterized the potential curves of the unusual, doubly excited valence van der Waals states, Be(2p pi(+1)2p pi(-1) P-3(J)). RG[(3)Sigma(-)], where RG=He, Ne. Similar to the Mg(3p pi(+1)3p pi(-1) P-3(J)). RG[(3)Sigma(-)] states (RG=He, Ne, Ar, Kr) which were characterized experimentally and theoretically earlier, these Be(2p pi 2p pi P-3(J)). RG[(3)Sigma(-)] states are much more strongly bound than their singly excited Be(2s2p pi P-3(J)). RG[(3)Pi] analogs, and even much more strongly bound than the analogous Be+(2s S-2). RG[(2)Sigma(+)] ground-state ions. This is attributed to the lack of a large Be(2s) electron cloud with density along the internuclear axis, so that quadrupole/induced-dipole and dispersion attraction forces can proceed to much smaller internuclear distances before repulsion sets in. The BeHe[(3)Sigma(-)] state is also almost five times more bound than the BeNe[(3)Sigma(-)] state, despite the fact that the polarizability of the He atom is only one-half that of the Ne atom. This is again attributed to minimization of repulsive forces, since strong Be(2p pi)/RG(np pi) exchange repulsion is completely absent when RG=He. The fact that the bond strengths of the Be(2p pi 2p pi). He[(3)Sigma(-)] and Be+(2p pi). He[(2)Pi] states are quite similar, and only about 20% less than the bond strength of the free Be++. He[(1)Sigma(+)] ion, is consistent with this interpretation. All of the strongly bound Be . RG neutral and ionic states calculated here have bond strengths greater than, and bond lengths smaller than, their Mg . RG analogs. This is because the Be electron clouds are all smaller than their Mg analogs, so that, again, all attractive forces can proceed to smaller distances before repulsion sets in.