  Spin-1 antiferromagnets are abundant in nature, but few theories or results
exist to understand their general properties and behavior, particularly in
situations when geometric frustration is present. Here we study the $S=1$
Kagome compound Na$_2$Ti$_3$Cl$_8$ using a combination of Density Functional
Theory, Exact Diagonalization, and Density Matrix Renormalization Group methods
to achieve a first principles supported explanation of exotic magnetic phases
in this compound. We find that the effective magnetic Hamiltonian includes
essential non-Heisenberg terms that do not stem from spin-orbit coupling, and
both trimerized and spin-nematic magnetic phases are relevant. The
experimentally observed structural transition to a breathing Kagome phase is
driven by spin--lattice coupling, which favors the trimerized magnetic phase
against the quadrupolar one. We thus show that lattice effects can be necessary
to understand the magnetism in frustrated magnetic compounds, and surmise that
Na$_2$Ti$_3$Cl$_8$ is a compound which cannot be understood from only
electronic or only lattice Hamiltonians, very much like VO$_2$.

Strongly Correlated Electrons;Materials Science