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 pressu
re. 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.
