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Interplay of dipoles and spins in $κ$-(BEDT-TTF)$_2X$, where $X=$ Hg(SCN)$_2$Cl, Hg(SCN)$_2$Br, Cu[N(CN)$_2$]Cl, Cu[N(CN)$_2$]Br, and Ag$_2$(CN)$_3$

We combine first principles density functional calculations with empirical relationships for the Coulomb interactions in the `monomer' model of $κ$-(BEDT-TTF)$_2X$. This enables us to calculate the parameters for the model of coupled dipolar and spin degrees of freedom proposed by Hotta [Phys. Rev. B $\textbf{82}$, 241104 (2010)], and Naka and Ishihara [J. Phys. Soc. Japan $\textbf{79}$, 063707 (2010)]. In all materials studied, retaining only the largest interactions leads to a transverse field Ising model of the dipoles. This quantifies, justifies and confirms recent claims that the dipoles are of crucial importance for understanding these materials. We show that two effects are responsible for a range of behaviors found in the dipoles in different $κ$-(BEDT-TTF)$_2X$ salts. (i) The inter-dimer hopping, $t_{b1}$, which gives rise to the "transverse field" in the Ising model for the dipoles ($H^T=2t_{b1}$), is between a third and a tenth smaller in the mercuric materials than for the mercury-free salts. (ii) The Ising model of dipoles is in the quasi-one-dimensional limit for the mercuric salts, but quasi-two-dimensional (between the square and isotropic triangular limits) for the mercury-free materials. Thus, the dimensionless critical field is much smaller in the mercuric compounds. Effect (ii) is much larger than effect (i). Simple explanations of both effects based on the band structures of the different salts are given. We show that dipolar order and even short-range dipolar correlations have a profound impact on the nature of the interdimer magnetic (superexchange) interactions. For example, dipole crystallization drives the materials towards quasi-one-dimensional magnetic interactions, which could be important for understanding the spin liquids found in some of these materials.