  We report an extraordinary pressure dependence of the magnetic interactions
in the metal-organic system [(CuF$_2$(H$_2$O)$_2$)$_2$pyrazine]. At zero
pressure, this material realizes a quasi-two-dimensional (Q2D) spin-1/2
square-lattice Heisenberg antiferromagnet. By high-pressure, high-field
susceptibility measurements we show that the dominant exchange parameter is
reduced continuously by a factor of 2 upon compression. Above 18 kbar, a phase
transition occurs, inducing an orbital re-ordering that switches the
dimensionality, transforming the Q2D lattice into weakly coupled chains (Q1D).
We explain the microscopic mechanisms for both phenomena by combining detailed
x-ray and neutron diffraction results with quantitative modeling using
spin-polarized density functional theory.
