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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. Asthe de Haas-van Alphen experiments reveal that the Pr-4f electrons in thesystems are sufficiently localized and irrelevant for the Fermi surface, wederive the low-energy effective tight-binding models consists of 196 orbitalsof conduction electrons so as to reproduce the first-principles electronicstructures 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 bythe conduction electrons. The obtained results indicate that the wave vector ofthe expected quadrupole order is Q=(0,0,0) for PrTi2Al20 while it isQ=(pi/a,pi/a,0) for PrV2Al20 as consistent with experimental observations inPrTi2Al20 and PrV2Al20 which exhibit ferro- and antiferro-quadrupole orders,respectively.

RKKY Interaction and Quadrupole Order in PrT2Al20 (T=Ti, V) Based on Effective 196 Orbital Model Extracted from First-Principles Calculation