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We develop the cluster self-consistent field method incorporating bothelectronic and lattice degrees of freedom to study the origin of ferromagnetismin Cs$_{2}$AgF$_{4}$. After self-consistently determining the harmonic andanharmonic Jahn-Teller distortions, we show that the anharmonic distortionstabilizes the staggered x$^{2}$-z$^{2}$/y$^{2}$-z$^{2}$ orbital andferromagnetic ground state, rather than the antiferromagnetic one. Theamplitudes of lattice distortions, Q$_{2}$ and Q$_{3}$, the magnetic couplingstrengthes, J$_{x,y}$, and the magnetic moment, are in good agreement with theexperimental observation.

Anharmonic effect on lattice distortion, orbital ordering and magnetic properties in Cs2AgF4