Effective Coulomb interactions within BEDT-TTF dimers

We calculate the parameters for Hubbard models of $\kappa \text{−}{\left(\text{BEDT-TTF}\right)}_{2}X$ and $\beta \text{−}{\left(\text{BEDT-TTF}\right)}_{2}X$. We use density-functional theory (DFT) to calculate the interactions between holes in dimers of the organic molecule bis(ethylenedithio)tetrathiafulvalene (BEDT-TTF) for 23 experimental geometries taken from a range of materials in both the $\beta$ and $\kappa$ polymorphs. We find that the effective Coulomb interactions are essentially the same for all of the compounds studied. We highlight the disagreement between our parametrization of the Hubbard model and previous results from both DFT and Hückel methods. We show that this is caused by the failure of an assumption made in previous calculations (which estimate the effective Coulomb interaction from the intradimer hopping integral). We discuss the implications of our calculations for theories of the BEDT-TTF salts based on the Hubbard model and use our calculated parameters to explain a number of phenomena caused by conformational disorder in these materials.

Figure 1

Schematic of the Hubbard models studied in this paper. The left panel shows the two-site extended Hubbard model. Here each site represents a monomer (single ET molecule). The Coulomb interactions ($U_m$ between electrons on the same monomer and $V_m$ between electrons on different monomers) and the hopping integral, $t$, are marked. The right panel shows the dimerized model. As the bonding orbital is filled only a single state is relevant, thus the only parameter is the effective Coulomb repulsion between electrons/holes in that state, $U_d$.

Figure 2

(Color online) The HOMO of ${\left(\text{ET}\right)}_2^{2+}$ (top) and charge neutral ${\left(\text{ET}\right)}_2$ (bottom), with nuclear positions from the crystal $\beta \text{−}{\left(\text{ET}\right)}_2{\text{I}}_3$. The HOMO of ${\left(\text{ET}\right)}_2^{2+}$ is the dimer bonding orbital and the HOMO of ${\left(\text{ET}\right)}_2$ is the antibonding orbital of the two ET HOMOs (cf. Fig. 4). The essential difference between the two lies in the relative phase of the orbital function on each molecule. The bonding orbital connects the ET molecules at the $\text{S}\cdots \text{S}$ contacts (cf. Fig. 8). In the antibonding orbital, there are nodes between the $\text{S}\cdots \text{S}$ contacts.

Figure 3

(Color online) The HOMO of ${\left(\text{ET}\right)}_2^{2+}$ (top) and charge neutral ${\left(\text{ET}\right)}_2$ (bottom), with nuclear positions from the crystal $\kappa \text{−}{\left(\text{ET}\right)}_2{\text{Cu}}_2{\left(\text{CN}\right)}_3$. The similarity of the nuclear structures and orbitals between this conformation and the $\beta$ conformation in Fig. 2 highlight the dimer as a common structural unit within two different packing motifs.

Figure 4

(Color online) The HOMO of a charge neutral ET monomer, with nuclear positions from the crystal $\beta \text{−}{\left(\text{ET}\right)}_2{\text{I}}_3$. This is the orbital from each molecule that contributes to the HOMO of the ${\left(\text{ET}\right)}_2$ and ${\left(\text{ET}\right)}_2^{2+}$ dimers.

Figure 5

Variation in Hubbard parameters found from DFT calculations with basis set, which improve from left to right. The test conformation is taken from the crystal $\kappa \text{−}{\left(\text{ET}\right)}_2\text{Cu}{\left(\text{NCS}\right)}_2$ (Ref. 25). All of the quantities except $t$ are well converged at TZP, the basis set chosen for all subsequent calculations. Indeed, ${\xi }_m$ is the only other quantity that changes significantly across the range of basis sets. However, $t$ is a relatively small quantity, on the order of its own variations with respect to basis-set size. Hence we conclude that solving the Hamiltonian (4) is not an accurate method for finding $t$.

Figure 6

(Color online) Intermonomer hopping term, $t$, for various ET dimers. The mean value for $\beta \text{−}{\left(\text{ET}\right)}_{2}X$ crystals is $t=0.54\pm 0.15\text{eV}$, although there is an apparent grouping of $\beta \text{−}{\left(\text{ET}\right)}_2{\text{I}}_3$ about $t\sim 0.64\text{eV}$ while the other $\beta$ crystals have $t\sim 0.35\text{eV}$. For $\kappa \text{−}{\left(\text{ET}\right)}_{2}X$ crystals, the value of $t=0.59\pm 0.10\text{eV}$. These results suggest that in terms of the delocalization effect of dimerization, either $\beta \text{−}{\left(\text{ET}\right)}_2{\text{I}}_3$ more closely resembles the $\kappa \text{-ET}$ crystals than its fellow $\beta$ crystals or $t$ is essentially the same for both crystal polymorphs and the low value of $t$ for $\beta \text{−}{\left(\text{ET}\right)}_2{\text{AuI}}_2$ and $\beta \text{−}{\left(\text{ET}\right)}_2{\text{IBr}}_2$ is unusual.

Figure 7

(Color online) The effective intradimer Coulomb repulsion, $U_d^{\left(v\right)}$, for various ET dimers. The $x$ axis separates the data by source crystal polymorph ($\beta$ or $\kappa$) and by the terminal ethylene-group conformation of each ET molecule in the dimer. $U_d^{\left(v\right)}$ does not change significantly across the different ET crystals examined. For $\beta \text{−}{\left(\text{ET}\right)}_{2}X$ crystals, $U_d^{\left(v\right)}=3.19\pm 0.07\text{eV}$. For $\kappa \text{−}{\left(\text{ET}\right)}_{2}X$ crystals, $U_d^{\left(v\right)}=3.23\pm 0.09\text{eV}$. The difference in $U_d^{\left(v\right)}$ between the two crystal polymorphs is $\sim 1\%$. Therefore, there is no significant dependence of $U_d^{\left(v\right)}$ on the dimer geometry associated with different crystal polymorphs.

Figure 8

ET molecules within the crystals studied occur in two conformations, denoted eclipsed and staggered. The difference between them lies in the relative orientation of the terminal ethylene groups.

Figure 9

(Color online) Dimer hole site energy, ${\xi }_d$, for various ET dimers. For $\beta \text{−}{\left(\text{ET}\right)}_{2}X$ crystals, the mean value is ${\xi }_d=4.45\pm 0.10\text{eV}$. For $\kappa \text{−}{\left(\text{ET}\right)}_{2}X$ crystals, the mean value is ${\xi }_d=4.46\pm 0.14\text{eV}$. The mean value for the whole data set is ${\xi }_d=4.45\pm 0.13\text{eV}$. The effect of conformation on ${\xi }_d$ is significantly larger for $\beta \text{−}{\left(\text{ET}\right)}_2{\text{I}}_3$ $\left(\sim 10\%\right)$ than it is for the other parameters. The variations in ${\xi }_d$ with dimer geometry associated with crystal polymorph and anion are $\sim 3\%$, similar to the relative variations in $U_d^{\left(v\right)}$ and $V_m^{\left(v\right)}$ across the whole data set.

Figure 10

(Color online) Intermonomer $V_m^{\left(v\right)}$ for various ET dimers. For $\beta \text{−}{\left(\text{ET}\right)}_{2}X$ crystals, $V_m^{\left(v\right)}=2.69\pm 0.13\text{eV}$ and for $\kappa \text{−}{\left(\text{ET}\right)}_{2}X$ crystals, $V_m^{\left(v\right)}=2.72\pm 0.09\text{eV}$. The mean value is $V_m^{\left(v\right)}=2.71\pm 0.10\text{eV}$. The difference in $V_m^{\left(v\right)}$ between the crystal polymorphs is $\sim 2\%$. Therefore, $V_m^{\left(v\right)}$, such as $U_d^{\left(v\right)}$, does not significantly depend on the geometry associated with crystal polymorph. The effect of ET conformation on the value of $V_m^{\left(v\right)}$ in the crystals $\beta \text{−}{\left(\text{ET}\right)}_2{\text{I}}_3$ and $\kappa \text{−}{\left(\text{ET}\right)}_2\text{Cu}\left[\text{N}{\left(\text{CN}\right)}_2\right]\text{I}$ is also similar to the effect on $U_d^{\left(v\right)}$. $V_m^{\left(v\right)}$ is lowest when the ET dimer has the staggered-staggered conformation and rises when either or both ET molecules are eclipsed.