  Electronic states and quadrupole orders in the Pr 1-2-20 systems PrT2Al20
(T=Ti, V) are investigated on the basis of the first-principles calculations.
The effective 196 orbital model is derived to reproduce the first-principles
electronic structures of LaT2Al20 (T=Ti, V) without contribution from the Pr 4f
electrons which are considered to be well localized and is employed to
calculate the Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions between
quadrupole and octupole moments of the Pr ions. Within the random phase
approximation for the RKKY Hamiltonian, the most divergent susceptibility is
found to be the quadrupole one for the wave vector Q = (0,0,0) in the case of
PrTi2Al20 while that for Q = (pi/a,0,pi/a) in the case of PrV2Al20 as
consistent with experimental observations in the both cases which exhibit the
ferro-quadrupole (FQ) and the antiferro-quadrupole (AFQ) orders, respectively.
We also discuss the ordered states using the mean-field approximation and find
that, in the case of PrTi2Al20, the 1st-order phase transition to the O20 FQ
order with a tiny discontinuity takes place as predicted by the Landau theory.
In the case of PrV2Al20, the system exhibits two distinct O22 AFQ orders, AFQ-I
and AFQ-II, and shows subsequent two phase transitions, the 2nd-order one from
normal to AFQ-I and the 1st-order one from AFQ-I to AFQ-II, that may be
responsible for the double transitions observed by specific heat measurements.
