Effective Coulomb interactions within BEDT-TTF dimers
aa r X i v : . [ c ond - m a t . s t r- e l ] J u l Effective Coulomb interactions within BEDT-TTF dimers
Edan Scriven ∗ and B. J. Powell Centre for Organic Photonics and Electronics, School of Mathematics and Physics, University of Queensland 4072, Australia
We calculate the effective Coulomb interactions between holes in dimers of the organic moleculeBEDT-TTF in vacuo . We use density functional theory (DFT) to parameterise Hubbard models for β and κ phase organic charge transfer salts. We focus on the intra-dimer Coulomb repulsion, U ( v ) d ,and the inter-monomer Coulomb repulsion, V ( v ) m . We find that U ( v ) d = 3 . ± .
09 eV and V ( v ) m =2 . ± .
10 eV for 23 experimental geometries taken from a range of materials in both the β and κ polymorphs. The quoted error is one standard deviation over the set of conformations studied. Weconclude that U ( v ) d and V ( v ) m are essentially the same for an isolated dimer with the geometries presentin all of the compounds studied. We highlight the disagreement between our parameterisation ofthe Hubbard model and previous results from both DFT and H¨uckel methods and show that this iscaused by the failure of an assumption made in previous calculations (that U ( v ) m ≫ V ( v ) m , where U ( v ) m is the effective intra-monomer Coulomb repulsion). We discuss the implications of our calculationsfor theories of the BEDT-TTF salts based on the Hubbard model on the 2D anisotropic triangularlattice and explore the role of conformational disorder in these materials. PACS numbers:
I. INTRODUCTION
Layered organic charge transfer saltsof the form (ET) X , where ET isbis(ethylenedithio)tetrathiafulvalene or BEDT-TTFand X is a monovalent anion, exhibit a variety ofunusual phenomena due to the strong electronic corre-lations present in these materials. These phenomenainclude unconventional superconductivity with a smallsuperfluid stiffness, a Mott insulator, a spin liquid, strongly correlated and unconventional metallicstates, and a pseudogap. Experimentally, one can tunebetween these phases by varying the temperature andpressure (including both hydrostatic and ‘chemical’pressure, i.e., varying the anion, X ). DFT, as implemented with current approximateexchange-correlation functionals, does not capture sev-eral important aspects of the physics of strongly-correlated electronic systems. For example, DFT bandstructure calculations of ET crystals, produce ahalf-filled valence band and hence a metallic state. How-ever, these calculations do not recover the Mott insulat-ing state or the other strongly correlated effects that areobserved experimentally. Therefore, efforts have focusedon the application of effective low-energy Hamiltionians,such as Hubbard models. However, in molecular crys-tals, the effective parameters for such low energy Hamil-tonians may be calculated from studying the properties ofsingle molecules or small molecular clusters, which maybe accurately described by DFT.
ET salts occur in a variety of crystal packing struc-tures. In the β and κ polymorphs the ET molecules formdimers. Intradimer dynamics are often integrated out ofeffective low energy models of β -(ET) X and κ -(ET) X .In these charge transfer salts, each dimer donates oneelectron to the anion layers to form a half-filled system.Both H¨uckel and DFT calculations have found that the dimers form an anisotropic triangular lattice inwhich each lattice site is a dimer. However, there is astrong effective Coulomb repulsion, U d , between two elec-trons on the same dimer, which must be included in theeffective low energy description. The electronic inter-actions within an ET dimer are stronger than those be-tween an ET molecule and its next-nearest neighbourson the crystal lattice. Therefore, these materials havebeen widely studied on the basis of Hubbard models. In order to explain the observed physics these theoriesassume that both chemical and hydrostatic pressure re-duce U d /W , where W is the bandwidth. Therefore, animportant task for the field is to understand how thisratio varies with chemical and hydrostatic pressure.Previously, the on-site Coulomb repulsion term inthe Hubbard model, U d , has been estimated from bothH¨uckel and density functional calculations under the assumptions that the intra-monomer Coulomb repulsion U m → ∞ , and the inter-monomer Coulomb repulsion V m →
0. We will showbelow that this assumption is incorrect and leads to asystematic underestimate of U d .Disorder plays a number of important roles in or-ganic superconductors. Increasing the degree of disor-der leads to a suppression of the superconducting criticaltemperature, T c , which is correlated with a rise in theresidual resistivity. Further disorder can cause a viola-tion of Matthiessen’s rule via impurity assisted tunnellingin the interlayer direction. In κ -(ET) Cu[N(CN) ]Br the degree of disorder canbe increased by increasing the rate at which the sam-ple is cooled, which leads to a suppressionof T c by ∼ Two hypothesis have beenproposed for the source of this disorder: terminal ethy-lene group disorder and disorder in the anionlayer. Therefore it is important to estimate the scat-ering rate caused by terminal ethylene disorder in thismaterial.Even more dramatic effects are observed in β -(ET) I .Variations of the pressure as the sample is cooled canchange the ambient pressure T c from 1 K (for samplescooled at ambient pressure; known as the β L phase) to 7K (for samples cooled at P & β H phase). In this materialclear differences in the terminal ethylene groups in the β H and β L phases are observed via x-ray scattering. Thusit has been argued that the terminal ethylene disorder isresponsible for the differences in the critical temperaturesbetween the β H and β L phases. Therefore, we also present calculations of the effectivesite energy for holes, ξ d , for the β - and κ - phase salts.This allows us to study the effects of impurity scatteringcaused by conformational disorder in the terminal ethy-lene groups of the ET molecule.In this paper we present DFT calculations for ETdimers in vacuum. In Sec. II we describe the compu-tational method by which we calculate these energies.In Sec. III we discuss the problem for the isolated dimerand review the parameterisation of the two-site extendedHubbard model from the total energies of the relevant(ET) charge states. In Sec. IV we report and discussthe resulting values of U ( v ) d , V ( v ) m and ξ d . In Sec. V wedraw our conclusions. II. COMPUTATIONAL METHODS
We used DFT to calculate the total energies of ETdimers in various conformations and charge states. Weused the SIESTA implementation of DFT, with thePBE exchange-correlation functional, a triple- ζ plussingle polarisation (TZP) basis set (except where we ex-plicitly specify otherwise) and basis functions consistingof Sankey type numerical atomic orbitals. The orbitalfunctions were confined to a radius r c from their cen-tres, which slightly increases the energy of the orbital.The specified maximum allowed increase in energy dueto this cutoff was 2 mRy. The convergence of the in-tegration mesh was determined by specifying an effec-tive plane-wave cutoff energy of 250 Ry. The initialspin moments on each atom were arranged antiferromag-netically wherever possible. We used pseudopotentialsconstructed according to the improved Troullier-Martins(TM2) method. Nuclear positions for C and S atoms were obtainedfrom x-ray crystallography.
H atoms,which are not observed in x-ray scattering experiments,were relaxed by the conjugate-gradient method. To-tal DFT energy differences between the relevant chargestates [ E (1) − E (0)] and [ E (2) − E (0)] were equatedwith the corresponding analytical expressions in Eq. (5)to determine the Hubbard parameters. We focus on these‘experimental’ geometries rather than performing a fullrelaxation for a number of reasons. Firstly, there are small differences in the reported geometries for differentET salts, and one would like to understand the effectof these. Secondly, the experiments effectively ‘integrateover’ all of the relevant charge states and therefore pro-vide an ‘average’ conformation. Thirdly, the experimentsnaturally include the effects on the molecular conforma-tion due to the crystalline environment, which are absentfrom in vacuo calculations. III. THE TWO SITE EXTENDED HUBBARDMODEL
In calculations of the effective Coulomb in-teraction (the Hubbard U ) in molecular solidsit is important to recognise that their are twocontributions. That is, the effectiveCoulomb interaction on a ET dimer may be written as U d = U ( v ) d − U ( p ) d (1)where U ( v ) d is the value of U d for the dimer cluster in vac-uum, and U ( p ) d is the reduction in U d from the polarisablecrystalline environment. Calculating U ( p ) d for ET salts isa highly non-trivial problem due to the large size of theET molecule relative to the intermolecular spacing. Be-low we present results of DFT calculations for U ( v ) d ofdimers in the conformations found in a wide range of κ and β phase ET salts. Similar results hold for U m , the ef-fective Coulomb repulsion between two holes on the samemonomer and V m the effective Coulomb interaction be-tween two holes on neighbouring monomers. Below wewill primarily discuss the vacuum contributions to theseterms, U ( v ) m and V ( v ) m .The effective vacuum intradimer Coulomb energy, U ( v ) d , is given by (see, e.g., Ref. 13) U ( v ) d = E (0) + E (2) − E (1) , (2)where E ( q ) is the ground state energy of the dimer in vac-uum containing q holes, i.e., with charge + q . Similarly,the effective site energy for holes is given by ξ d = E (0) − E (1) . (3)Below we calculate E ( q ) via density functional methods.It is also interesting to consider intradimer dynamics,which can be described via a two site extended Hubbardmodel, ˆ H = X iσ ξ mi ˆ n iσ − t X σ (cid:16) ˆ h † σ ˆ h σ + h.c. (cid:17) + X i U mi ˆ n i ↑ ˆ n i ↓ + V m ˆ n ˆ n (4)where ˆ h ( † ) iσ annihilates (creates) a hole on site (monomer) i in spin state σ , ξ mi is the site energy for holes on site2 IG. 1: The HOMO of (ET) (top) and charge neutral(ET) (bottom), with nuclear positions from the crystal β -(ET) I . The HOMO of (ET) is the dimer bonding orbitaland the HOMO of (ET) is the antibonding orbital of the twoET HOMOs (cf. Fig. 3). The essential difference between thetwo lies in the relative phase of the orbital function on eachmolecule. The bonding orbital connects the ET molecules atthe S · · · S contacts (cf. Fig. 6). In the antibonding orbital,there are nodes between the S · · ·
S contacts. i , ˆ n iσ is the number operator for spin σ holes on site i ,ˆ n i = P σ ˆ n iσ , t is the intradimer hopping integral, U mi isthe effective on-site (monomer) Coulomb repulsion, and V m is the intersite Coulomb repulsion.The lowest energy eigenvalues of Hamiltonian (4) foreach charge state are E (0) = 0 , (5a) E (1) = ξ m − p t + (∆ ξ m ) , (5b)and E (2) = 2 ξ m + 13 (cid:0) U m + V m − A cos θ (cid:1) (5c)where ξ m = ( ξ m + ξ m ), A = 12 t + (∆ U m ) + ( U m − V m )( U m − V m ) + 3(∆ ξ m ) , cos 3 θ = ( U m − V m )[18 t − (2 U m − U m − V m )( U m − U m + V m ) − ξ m ) ] / A .∆ ξ m = ξ m − ξ m , U m = ( U m + U m ), and ∆ U m = U m − U m . FIG. 2: The HOMO of (ET) (top) and charge neutral(ET) (bottom), with nuclear positions from the crystal κ -(ET) Cu (CN) . The similarity of the nuclear structures andorbitals between this conformation and the β conformationin Fig. 1 highlight the dimer as a common structural unitwithin two different packing motifs.FIG. 3: The HOMO of a charge neutral ET monomer, withnuclear positions from the crystal β -(ET) I . This is the or-bital from each molecule that contributes to the HOMO ofthe (ET) and (ET) dimers. We have previously calculated ξ m and U ( v ) m from theone site Hubbard model for an ET monomer for the ex-perimental observed conformations in all the materialsdiscussed below, therefore one may solve Eqs. (5) for t and V m taking ξ m and U ( v ) m from the monomer calcula-tions. The case of two holes with different site energiesand on-site Coulomb repulsion may be solved by a generalmethod for diagonalising cubic matrix eigensystems. Incases where the two molecules in a dimer have the same3eometry (e.g., by symmetry), ξ m = ξ m = ξ m and U m = U m = U m and the eigenvalues simplify to E (1) = ξ m − t (6a) E (2) = 2 ξ m + 12 (cid:16) U m + V m − p t + ( U m − V m ) (cid:17) (6b)in which case the solution is straightforward.In the limit U m = V m = 0 the two site Hubbard modelhas two solutions: the bonding state | φ bσ i = | φ σ i + | φ σ i and the antibonding state | φ bσ i = | φ σ i − | φ σ i , where | φ iσ i = ˆ h iσ | i is a single electron state centred on the i th monomer and | i is the (particle) vacuum state.In Figs. 1 and 2 we plot the HOMOs of ETdimer for the conformations found in β -(ET) I and κ -(ET) Cu (CN) respectively, in both the charge neutraland the 2+ states. It can be seen that these dimer or-bitals are the antibonding and bonding hybrids of theET monomer HOMO (shown in Fig. 3), respectively.The most important difference between the orbital ge-ometries lies in the S · · · S intermolecular contacts, whichcontain nodes in the antibonding orbital, but are con-nected in the bonding orbital. Thus the DFT picture ofthe (ET) system is remarkably similar to the molecularorbital description of a diatomic molecule, but with the‘covalent bond’ between the two monomers rather thanbetween two atoms. IV. CALCULATION OF THE HUBBARDMODEL PARAMETERSA. Basis set convergence
We tested the basis set convergence of the DFTcalculations using the conformation observed in κ -(ET) Cu(NCS) as the test case, with single- ζ (SZ),single- ζ plus polarisation (SZP), double- ζ (DZ), double- ζ plus polarisation (DZP) and TZP basis sets. We alsocalculated the monomer parameters, U m and ξ m , in eachbasis, using the method we previously applied to the ETmonomers. The Hubbard model parameters in each ba-sis set are reported in Fig. 4. The values of all parametersare well-converged in the TZP basis, except t . t is an or-der of magnitude smaller than the other parameters, andon the order of both the variation of the other param-eters among the basis sets tested and the uncertaintyassociated with the calculation method. This suggeststhat extracting t from band structure calculations isa more accurate and reliable method of estimating thehopping integrals in these systems. B. Variation of the intra-dimer Coulomb repulsion
Now we consider variation of the Hubbard model pa-rameters across the conformations found in different ma-terials, beginning with U ( v ) d . In Fig. 5 we show the values FIG. 4: Variation of Hubbard parameters found from DFTcalculations with basis set, which improve from left toright. The test conformation is taken from the crystal κ -(ET) Cu(NCS) . All of the quantities except t are well-converged at TZP, the basis set chosen for all subsequent cal-culations. Indeed, ξ m is the only other quantity that changessignificantly across the range of basis sets. However, t is arelatively small quantity, on the order of its own variationswith respect to basis set size. Hence we conclude that solvingthe Hamiltonian (4) is not an accurate method for finding t .FIG. 5: The effective intra-dimer Coulomb repulsion, U ( v ) d , forvarious ET dimers. The x -axis separates the data by sourcecrystal polymorph ( β or κ ), and by the terminal ethylenegroup conformation of each ET molecule in the dimer. U ( v ) d does not change significantly across the different ET crystalsexamined. For β -(ET) X crystals, U ( v ) d = 3 . ± .
07 eV. For κ -(ET) X crystals, U ( v ) d = 3 . ± .
09 eV. The difference in U ( v ) d between the two crystal polymorphs is ∼ U ( v ) d on the dimergeometry associated with different crystal polymorphs. IG. 6: ET molecules within the crystals studied occur intwo conformations, denoted eclipsed and staggered. The dif-ference between them lies in the relative orientation of theterminal ethylene groups. of U ( v ) d for the conformations observed experimentally ina variety of ET crystals. Of particular note are the threedata points corresponding to different possible confor-mations of β -(ET) I . In the ET molecule the terminalethylene groups may take two relative orientations knownas the staggered and eclipsed conformations (cf. Fig. 6). U ( v ) d is smallest when both ET molecules are in the stag-gered conformation. Conversely, the largest U ( v ) d for thiscrystal occurs when both ET molecules are eclipsed, withintermediate U ( v ) d values for the case with one staggeredand one eclipsed ET molecule. This trend is repeated inthe κ -phase crystals, where two data sets (correspondingto different temperatures at which the nuclear positionswere determined) for κ -(ET) Cu[N(CN) ]I provide datafor both conformations.The mean value of U ( v ) d for the β phase crystals is 3.19 ± U ( v ) d for the κ phase crystals is3.23 ± U ( v ) d is ∼ U ( v ) d takesthe same value, 3.22 ± β and κ phase ETsalts. This result is significantly larger than the value of U ( v ) d obtained from H¨uckel calculations ( ∼ C. Variations in site energy and the role of disorder
As reviewed in the introduction, a number of exper-iments have shown that disorder has strong effects onboth the normal state and superconducting propertiesof organic charge transfer salts.
Therehas been relatively little work on the effect of the ran-dom U Hubbard model. Conclusions drawn from studiesin one dimension cannot be straightforwardly gen-eralised to higher dimensions. Mutou used dynami-cal mean field theory to study the metallic phase of the FIG. 7: Dimer hole site energy, ξ d , for various ET dimers.For β -(ET) X crystals, the mean value is ξ d = 4 . ± . κ -(ET) X crystals, the mean value is ξ d = 4 . ± . ξ d = 4 . ± . ξ d is significantly larger for β -(ET) I ( ∼ ξ d with dimer geometry associated with crys-tal polymorph and anion are ∼ U ( v ) d and V ( v ) m across the whole data set. random U Hubbard model. However, he did not con-sider the effect a random U on either superconductivityor the Mott transition, which are the primary concerns inthe organic charge transfer salts. However, Mutou con-cluded that for small impurity concentrations Kondo-likeeffects mean that the random U Hubbard model is signifi-cantly different from the virtual crystal approximation tothe random U Hubbard model, which describes the sys-tem in terms of an average U . The only study we areaware of that discusses superconductivity in the random U Hubbard model treats the negative U model, which isnot realistic for the organic charge transfer salts. Litakand Gy¨orffy studied a model where some sites have U = 0 and others have a negative U . They find thatsuperconductivity is suppressed above at certain criticalconcentration of U = 0 sites. Therefore, it is not clearwhat implications our finding of small changes in U ( v ) d and hence U d has for the physics of the organic chargetransfer salts. However, it is interesting to ask what rolethis plays in the observed role of disorder in suppressingsuperconductivity and driving the system towards theMott transition. To understand the role of conformational disorder interms of an effective Hamiltonian built up from ETdimers one must also understand the effect of confor-mational disorder of the effective dimer site energy (forholes), ξ d . This is straightforwardly found from the DFTcalculations described above via Eq. (3) and the results5 IG. 8: Intradimer V ( v ) m for various ET dimers. For β -(ET) Xcrystals, V ( v ) m = 2 . ± .
13 eV and for κ -(ET) X crystals, V ( v ) m = 2 . ± .
09 eV. The mean value is V ( v ) m = 2 . ± . V ( v ) m between the crystal polymorphsis ∼ V ( v ) m , like U ( v ) d , does not significantlydepend on the geometry associated with crystal polymorph.The effect of ET conformation on the value of V ( v ) m in thecrystals β -(ET) I and κ -(ET) Cu[N(CN) ]I is also similarto the effect on U ( v ) d . V ( v ) m is lowest when the ET dimer hasthe staggered-staggered conformation, and rises when eitheror both ET molecules are eclipsed. are reported in Fig. 7. The effective scattering rate dueto conformational disorder is given by ~ τ = X i N i πD ( E F ) | ∆ i ξ d | , (7)where i labels the type of impurity (both staggered ormixed; the ground state conformation is both eclipsed), N i is number of impurities of type i , D ( E F ) is the densityof states at the Fermi level, and ∆ i ξ d is the differencebetween ξ d for i type impurities and ξ d of eclipsed dimers.In quasi-2D systems, D ( E F ) is simply related to thecyclotron electron mass by the relation D ( E F ) = m c π ~ (8)and in the presence of interactions Luttinger’s theorem for a Fermi liquid produces D ( E F ) = m ∗ π ~ (9)where m ∗ is the effective mass. From quantum os-cillation measurements, Wosnitza et al . found that m ∗ /m e = 4.2 in β -(ET) I , where m e is the electronrest mass. From Shubnikov-de Haas measurements in κ -(ET) Cu[N(CN) ]Br Caulfield et al . found that m ∗ /m e = 6.4. The scattering rate τ can be found from mea-surement of the interlayer residual resistivity, ρ , by therelation ρ = π ~ e m ∗ ct ⊥ τ (10)where c is the interlayer lattice constant taken fromthe relevant x-ray scattering measurements and t ⊥ is the interlayer hopping integral, which has previouslybeen estimated from experimental data for both κ -(ET) Cu[N(CN) ]Br (Ref. 59) and β -(ET) I (Ref. 31).Using these parameters we calculated the scattering ratein both the β L and β H phases of β -(ET) I from thelow temperature values of ρ reported by Ginodman etal .. The scattering rate due to conformational impu-rities, τ − c is then τ − c = τ − H − τ − L , where τ L ( τ H ) isthe quasiparticle lifetime in the β L ( β H ) phase. Givenour calculated values of ∆ i ξ d an ∼
8% concentrationof staggered impurities would be required to cause thisscattering rate. From a similar calculation comparingthe residual resistivity measured in a single sample of κ -(ET) Cu[N(CN) ]Br cooled at different rates we findthat a ∼
2% concentration of staggered impurities wouldbe sufficient to explain the rise increase in residual re-sistivity observed in the experiment utilising the fastestcooling over that performed with the slowest cooling rate.X-ray scattering experiments find that 3 ±
3% of theET molecules are in the staggered conformation at 9 K,which is entirely consistent with our result. However,Wolter et al .’s argument that this impurity concentra-tion is too small to cause the observed effects of disorderin not sustained by the above calculations. Rather wefind that all of the suppression in T c is entirely consis-tent with this degree of disorder. D. Variations in inter-molecular Coulomb repulsion
In Fig. 8 we show the distribution of the calculatedvalues of V ( v ) m . The mean value of V ( v ) m for the β phasecrystals is 2.69 ± κ phase crystals is 2.72 ± ∼ V ( v ) m is essen-tially the same across all of the conformations studied,with a mean value of 2.71 ± U ( v ) d based on both the Huckelmethod and DFT have assumedthat U ( v ) m → ∞ and V ( v ) m = 0. Substituting these condi-tions into Eqs. (2) and (6) yields U ( v ) d = 2 t . Literaturevalues of U ( v ) d based on this approximation are presentedin Table I for comparison with our DFT results. It canbe seen that this assumption yields values of U ( v ) d thatare significantly smaller than those we have calculatedabove (cf. Fig. 5). However, we have previously found that U ( v ) m = 4 . ± . rystal Method U ( v ) d (eV) β -(ET) I H¨uckel β -(ET) IBr H¨uckel β -(ET) ICl H¨uckel β -(ET) I H¨uckel β -(ET) CH(SO CF ) H¨uckel β -(ET) [OsNOCl ] H¨uckel κ -(ET) Cu[N(CN) ]Cl DFT κ -(ET) Cu[N(CN) ]Br H¨uckel κ -(ET) Cu(NCS) H¨uckel κ -(ET) Cu(NCS) H¨uckel κ -(ET) Cu(NCS) H¨uckel κ -(ET) Cu(NCS) DFT κ -(ET) Cu(NCS) DFT κ -(ET) Cu[N(CN) ]Br H¨uckel κ -(ET) Cu (CN) H¨uckel κ -(ET) Cu (CN) DFT κ -(ET) Cu (CN) DFT κ -(ET) I H¨uckel κ -(ET) I H¨uckel U ( v ) d for various β - and κ -phase ET salts. These values were obtained from bothH¨uckel and density functional methods under the assump-tions U ( v ) m → ∞ and V ( v ) m = 0, which yields U ( v ) d = 2 t . Theseestimates substantially underestimate the actual value of U ( v ) d (see Fig. 5) as U ( v ) m ∼ V ( v ) m . The two site extended Hubbardmodel produces values of t on the same order of magnitudeas these H¨uckel calculations. One should also note the widescatter between the different H¨uckel calculations, even be-tween different studies of the same material. results we see that U ( v ) m /V ( v ) m ∼ .
5, in contradiction tothe assumption that U ( v ) m ≫ V ( v ) m . Hence U ( v ) m ≫ | t | .If we instead make the assumption U ( v ) m ≃ V ( v ) m ≫ | t | ,then Eqs. (2) and (6) give U ( v ) d ≈
12 ( U ( v ) m + V ( v ) m ) . (11)Substituting in the mean values of U ( v ) m and V ( v ) m gives U ( v ) d = 3.41 eV. This result is close (within 6%) to ourcalculated value of U ( v ) d . Therefore, this is a reasonableapproximation for the ET salts. Further, this shows thatthe result that U ( v ) d does not vary significantly because ofchanges in conformation between different salts or poly-morphs is a consequence of the fact that neither U ( v ) m or V ( v ) m vary significantly because of changes in conforma-tion between different salts or polymorphs. V. CONCLUSIONS
The effective Coulomb repulsion terms in the Hubbardmodel are essentially the same for all of the ET con-formations studied. We found that U ( v ) d = 3 . ± . V ( v ) m = 2 . ± .
10 eV. The value of U ( v ) d is sig-nificantly larger than previous estimates based on the extended H¨uckel formalism or DFT under the assump-tions U ( v ) m → ∞ and V ( v ) m = 0. This can be under-stood because we have shown that U ( v ) m ∼ V ( v ) m and hence U ( v ) d ≈ ( U ( v ) m + V ( v ) m ).The lack of variation of U ( v ) d between the two poly-morphs and when the anion is changed is interesting inthe context of theories of these organic charge transfersalts based on the Hubbard model. These theories require U d /W to vary significantly as the anion is changed (chem-ical pressure) and under hydrostatic pressure. Thereforeour results show that either U ( p ) d or W must vary signif-icantly under chemical and hydrostatic pressure, or elsethese theories do not provide a correct description of the β and κ phase organic charge transfer salts. This is par-ticularly interesting as fast cooling has been shown todrive κ -(ET) Cu[N(CN) ]Br to the insulating side of themetal-insulator transition. We have also studied the effects of conformational dis-order on these parameters, which is found to be quitesmall, consistent with the often subtle effects of con-formational disorder observed in these materials. Thelargest changes are found in the geometries taken from β -(ET) I , which shows the strongest effects of confor-mational disorder. It is also interesting that we found asystematic variation in U ( v ) d is caused by conformationaldisorder. As there has been relatively little work on therandom U Hubbard model it is difficult to speculate whateffects this has on the low temperature physics of the or-ganic charge transfer salts at present.Given that DFT band structure parameterisations ofthe interdimer hopping integrals have recently been re-ported for several organic charge transfer salts, theoutstanding challenge for the parameterisation of theHubbard model in these systems is the accurate calcula-tion of U ( p ) d . The bandwidth in both the β (Ref. 10) and κ (Ref. 11,12) phase salts is around 0.4-0.6 eV. Therefore,our finding that U ( v ) d is significantly larger than has beenrealised previously shows that U ( p ) d must be significantas if U d ≃ U ( v ) d then all of these materials would be wellinto the Mott insulating regime. Thus U ( p ) d must signifi-cantly reduce U d in order for the, observed, rich phase di-agram to be realised. This is consistent with comparisonsof DMFT calculations to optical conductivity measure-ments on κ -(ET) Cu[N(CN) ]Br x Cl − x , which suggestthat U d = 0 . Further, U ( p ) d may be quite sensitiveto the crystal lattice and therefore may be important forunderstanding the strong dependence of these materialson chemical and hydrostatic pressure. Acknowledgments
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