  The electronic structures of four Laves phase iron compounds (e.g. YFe$_2$,
ZrFe$_2$, LuFe$_2$ and HfFe$_2$) have been calculated by the state-of-the-art
full potential electronic structure code. The magnetic moments collapse under
hydrostatic pressure. This feature is found to be universal in these materials.
Its electronic origin is provided by the sharp peaks in the density of states
near the Fermi level. It is shown that a first order quantum phase transition
can be expected under pressure in Y(Zr, or Lu)Fe$_2$, while a second order one
in HfFe$_2$. The bonding characteristics are discussed to elucidate the
equilibrium lattice constant variation. The large spontaneous volume
magnetostriction gives one of the most important character of these compounds.
Invar anomalies in these compounds can be partly explained by the current work
when the fast continuous magnetic moment decrease at the decrease of the
lattice constant was properly considered. This work may remind the
experimentalists of these "old" compounds and exploration of the quantum
properties under high pressures are greatly encouraged.
