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Spin-1 antiferromagnets are abundant in nature, but few theories or resultsexist to understand their general properties and behavior, particularly insituations when geometric frustration is present. Here we study the $S=1$Kagome compound Na$_2$Ti$_3$Cl$_8$ using a combination of Density FunctionalTheory, Exact Diagonalization, and Density Matrix Renormalization Group methodsto achieve a first principles supported explanation of exotic magnetic phasesin this compound. We find that the effective magnetic Hamiltonian includesessential non-Heisenberg terms that do not stem from spin-orbit coupling, andboth trimerized and spin-nematic magnetic phases are relevant. Theexperimentally observed structural transition to a breathing Kagome phase isdriven by spin--lattice coupling, which favors the trimerized magnetic phaseagainst the quadrupolar one. We thus show that lattice effects can be necessaryto understand the magnetism in frustrated magnetic compounds, and surmise thatNa$_2$Ti$_3$Cl$_8$ is a compound which cannot be understood from onlyelectronic or only lattice Hamiltonians, very much like VO$_2$.

Spin--lattice coupling and the emergence of the trimerized phase in the $S=1$ Kagome antiferromagnet Na$_2$Ti$_3$Cl$_8$