  Electronic states and quadrupole orders in the 1-2-20 systems PrT2Al20 (T=Ti,
V) are investigated on the basis of the first-principles band calculation. As
the de Haas-van Alphen experiments reveal that the Pr-4f electrons in the
systems are sufficiently localized and irrelevant for the Fermi surface, we
derive the low-energy effective tight-binding models consists of 196 orbitals
of conduction electrons so as to reproduce the first-principles electronic
structures of LaT2Al20 (T=Ti, V) without contribution from the 4f electrons.
Based on the effective models, we calculate the Ruderman-Kittel-Kasuya-Yosida
(RKKY) interaction between the quadrupole moments of the Pr ions mediated by
the conduction electrons. The obtained results indicate that the wave vector of
the expected quadrupole order is Q=(0,0,0) for PrTi2Al20 while it is
Q=(pi/a,pi/a,0) for PrV2Al20 as consistent with experimental observations in
PrTi2Al20 and PrV2Al20 which exhibit ferro- and antiferro-quadrupole orders,
respectively.
