^{1,a)}and B. J. Powell

^{1}

### Abstract

We calculate the effective Coulomb repulsion between electrons/holes and site energy for an isolated bis(ethylenedithio)tetrathiafulvalene (BEDT-TTF) molecule *in vacuo*. for 44 experimental geometries taken from a broad range of conformations,polymorphs, anions, temperatures, and pressures (the quoted “error” is one standard deviation). Hence we conclude that is essentially the same for all of the compounds studied. This shows that the strong (hydrostatic and chemical) pressure dependence observed in the phase diagrams of the BEDT-TTF salts is not due to . Therefore, if the Hubbard model is sufficient to describe the phase diagram of the BEDT-TTF salts, there must be significant pressure dependence on the intramolecular terms in the Hamiltonian and/or the reduction in the Hubbard due to the interaction of the molecule with the polarizable crystal environment. The renormalized value of is significantly smaller than the bare value of the Coulomb integral, , across the same set of geometries, emphasizing the importance of using the renormalized value of . The site energy (for holes), , varies only a little more than across the same set of geometries. However, we argue that this variation in the site energy plays a key role in understanding the role of disorder in bis(ethylenedithio)tetrathiafulvalene salts. We explain the differences between the and phases of on the basis of calculations of the effects of disorder.

We thank Julian Gale his for assistance with the SIESTA suite of programs and Noel Hush, Anthony Jacko, Ross McKenzie, Seth Olsen, Mark Pederson, Jenny Riesz, Jeff Reimers, Elvis Shoko, Weitao Yang, and, particularly, Laura Cano-Cortés and Jaime Merino for enlightening conversations. This work was supported by the Australian Research Council (ARC) under the Discovery scheme (Project No. DP0878523) and by a University of Queensland Early Career Research grant. B.J.P. was the recipient of an ARC Queen Elizabeth II Fellowship (Project No. DP0878523). Numerical calculations were performed on the APAC national facility under a grant from the merit allocation scheme.

I. INTRODUCTION

II. THE HUBBARD MODEL AND DENSITY FUNCTIONAL THEORY

III. COMPUTATIONAL METHODS

IV. GEOMETRY OPTIMIZATION, PSEUDOPOTENTIALS, BASIS SETS, AND CALCULATION METHODS

V. IMPURITY SCATTERING,POLYMORPHISM, AND CHEMICAL PRESSURE

VI. CONCLUSIONS

### Key Topics

- Electron transfer
- 50.0
- Hubbard model
- 22.0
- Density functional theory
- 18.0
- Basis sets
- 15.0
- Molecular conformation
- 13.0

## Figures

The eclipsed and staggered conformations of the ET molecule.

The eclipsed and staggered conformations of the ET molecule.

Dependence of the energy of an ET molecule on the charge of the molecule. Two different geometries are studied. The “frozen” geometry is relaxed from that found experimentally (Ref. 35) in in the neutral charge state and then held fixed during the SCF calculations at different charge states. For the “relaxed” data the nuclear geometry is relaxed separately for each charge state. The energies are exactly equal by definition in the charge neutral state; for other charge states the nuclear relaxation lowers energies. This reduces the curvature of , i.e., lowers and hence . It can be seen that for both the frozen and relaxed geometries the energy of the fractional charge states is extremely well described by the classical quadratic functions, Eq. (13). In contrast it is known that for the exact functional the energy of molecules with fractional charges is a linear interpolation between the integer charge states (Refs. 1, 23, and 25). We find that to an excellent approximation , whereas and are not good approximations to and . In contrast for the exact functional , , and . This incorrect result is a manifestation of the delocalization error of DFT and closely related to the band gap problem (Refs. 1, 25, and 27). These calculations use TZP basis sets and TM2 pseudopotentials.

Dependence of the energy of an ET molecule on the charge of the molecule. Two different geometries are studied. The “frozen” geometry is relaxed from that found experimentally (Ref. 35) in in the neutral charge state and then held fixed during the SCF calculations at different charge states. For the “relaxed” data the nuclear geometry is relaxed separately for each charge state. The energies are exactly equal by definition in the charge neutral state; for other charge states the nuclear relaxation lowers energies. This reduces the curvature of , i.e., lowers and hence . It can be seen that for both the frozen and relaxed geometries the energy of the fractional charge states is extremely well described by the classical quadratic functions, Eq. (13). In contrast it is known that for the exact functional the energy of molecules with fractional charges is a linear interpolation between the integer charge states (Refs. 1, 23, and 25). We find that to an excellent approximation , whereas and are not good approximations to and . In contrast for the exact functional , , and . This incorrect result is a manifestation of the delocalization error of DFT and closely related to the band gap problem (Refs. 1, 25, and 27). These calculations use TZP basis sets and TM2 pseudopotentials.

Orthographic projections of the isosurface of the HOMO of the ET molecule in experimental geometries taken from (a) eclipsed , (b) staggered , and (c) . Note the great similarity of the HOMOs corresponding to the experimental geometries from eclipsed and staggered and the small electronic density on the terminal ethylene group, which is involved in the change between the eclipsed and staggered conformations. Further, the HOMOs of the ET molecule in the experimental geometries taken from and are remarkably similar despite the fact that this geometry is taken from a different crystal polymorph with a different anion. This is consistent with our finding that the changes in the conformation of the ET molecule, in different polymorphs and in crystals with different anions, do not significantly affect . Color indicates the sign of the Kohn–Sham orbital. All isosurfaces are and calculated in the charge-neutral state. These calculations use TZP basis sets and TM2 pseudopotentials. Animations showing different isosurfaces are available online (Ref. 33).

Orthographic projections of the isosurface of the HOMO of the ET molecule in experimental geometries taken from (a) eclipsed , (b) staggered , and (c) . Note the great similarity of the HOMOs corresponding to the experimental geometries from eclipsed and staggered and the small electronic density on the terminal ethylene group, which is involved in the change between the eclipsed and staggered conformations. Further, the HOMOs of the ET molecule in the experimental geometries taken from and are remarkably similar despite the fact that this geometry is taken from a different crystal polymorph with a different anion. This is consistent with our finding that the changes in the conformation of the ET molecule, in different polymorphs and in crystals with different anions, do not significantly affect . Color indicates the sign of the Kohn–Sham orbital. All isosurfaces are and calculated in the charge-neutral state. These calculations use TZP basis sets and TM2 pseudopotentials. Animations showing different isosurfaces are available online (Ref. 33).

The effective Coulomb repulsion between electrons/holes on an ET monomer in the experimental geometries corresponding to different anions, conformations (eclipsed offset to the left, staggered to the right), temperatures, pressures, and crystal polymorphs. Note the limited range (3.9–4.4 eV) of the ordinate. We see that does not change significantly across the different crystals and has a mean value of . The calculations use TZP basis sets and TM2 pseudopotentials. Full details of the parametrization are given in the supplementary information (Ref. 33).

The effective Coulomb repulsion between electrons/holes on an ET monomer in the experimental geometries corresponding to different anions, conformations (eclipsed offset to the left, staggered to the right), temperatures, pressures, and crystal polymorphs. Note the limited range (3.9–4.4 eV) of the ordinate. We see that does not change significantly across the different crystals and has a mean value of . The calculations use TZP basis sets and TM2 pseudopotentials. Full details of the parametrization are given in the supplementary information (Ref. 33).

The site energy for electrons on an ET monomer in the experimental geometries corresponding to different anions, conformations (eclipsed offset to the left, staggered to the right), temperatures, pressures, and crystal polymorphs. The mean value is . The calculations use TZP basis sets and TM2 pseudopotentials. Full details of the parametrization are given in the supplementary information (Ref. 33).

The site energy for electrons on an ET monomer in the experimental geometries corresponding to different anions, conformations (eclipsed offset to the left, staggered to the right), temperatures, pressures, and crystal polymorphs. The mean value is . The calculations use TZP basis sets and TM2 pseudopotentials. Full details of the parametrization are given in the supplementary information (Ref. 33).

The site energy for holes on an ET monomer in the experimental geometries corresponding to different anions, conformations (eclipsed offset to the left, staggered to the right), temperatures, pressures, and crystal polymorphs. shows only slightly more variation than . However, small changes in are known to have significant effects on the superconducting state observed in ET salts (Refs. 66 and 67). (Because these ET molecules are quarter filled with holes in the salt, , rather than , is the relevant site energy to consider when discussing the role of disorder.) The mean value is . The calculations use TZP basis sets and TM2 pseudopotentials. Full details of the parametrization are given in the supplementary information (Ref. 33).

The site energy for holes on an ET monomer in the experimental geometries corresponding to different anions, conformations (eclipsed offset to the left, staggered to the right), temperatures, pressures, and crystal polymorphs. shows only slightly more variation than . However, small changes in are known to have significant effects on the superconducting state observed in ET salts (Refs. 66 and 67). (Because these ET molecules are quarter filled with holes in the salt, , rather than , is the relevant site energy to consider when discussing the role of disorder.) The mean value is . The calculations use TZP basis sets and TM2 pseudopotentials. Full details of the parametrization are given in the supplementary information (Ref. 33).

## Tables

Comparison of different methods of calculating the effective monomer on-site Coulomb repulsion for ET. No significant differences are found between and from the DFT calculations. This result has been confirmed for all of the geometries studied below. Further, no significant differences are found between the two conformations. is significantly larger than , in qualitative agreement with previous results from wave function based methods (c.f. Table II). The nuclear geometry is that seen “experimentally” in for both the eclipsed and staggered conformations. The calculations use TZP basis sets and TM2 pseudopotentials. All values are in eV.

Comparison of different methods of calculating the effective monomer on-site Coulomb repulsion for ET. No significant differences are found between and from the DFT calculations. This result has been confirmed for all of the geometries studied below. Further, no significant differences are found between the two conformations. is significantly larger than , in qualitative agreement with previous results from wave function based methods (c.f. Table II). The nuclear geometry is that seen “experimentally” in for both the eclipsed and staggered conformations. The calculations use TZP basis sets and TM2 pseudopotentials. All values are in eV.

Previously reported calculations of the Coulomb energy ( and ) of ET. These calculations were performed at various levels of theory, using various basis sets and with geometries taken from x-ray crystallography experiments on various different materials. is significantly larger than , consistent with our results. We make use of the following abbreviations in the above table: HF (Hartree–Fock), RHF (restricted Hartree-Fock), and VB (valence bond). Fortunelli and Painelli (Ref. 8) calculated for ET dimers in different charge states, . All energies are in eV.

Previously reported calculations of the Coulomb energy ( and ) of ET. These calculations were performed at various levels of theory, using various basis sets and with geometries taken from x-ray crystallography experiments on various different materials. is significantly larger than , consistent with our results. We make use of the following abbreviations in the above table: HF (Hartree–Fock), RHF (restricted Hartree-Fock), and VB (valence bond). Fortunelli and Painelli (Ref. 8) calculated for ET dimers in different charge states, . All energies are in eV.

Calculated bare and renormalized parameters for the Hubbard model for ET monomers under various geometry relaxation schemes and with different pseudopotentials, basis sets, and codes (see Sec. III). The “experimental” geometry is that reported for an ET molecule in (Ref. 35), measured at 298 K, with the H atom (not observed in x-ray crystallography) positions relaxed. The “frozen” coordinate system was relaxed in the charge neutral state and held fixed for other charge states. The “relaxed” geometry was optimized at every charge state. All geometry relaxations were carried out in calculations using TZP basis functions and TM2 pseudopotentials; we also carried out the relaxations using the other methods in the table and found no significant differences. The abbreviation pseudo. (for pseudopotential) is used in this table and others below. The root mean square error (RMSE) is taken from the fit to the classical Eq. (13). All values are in eV.

Calculated bare and renormalized parameters for the Hubbard model for ET monomers under various geometry relaxation schemes and with different pseudopotentials, basis sets, and codes (see Sec. III). The “experimental” geometry is that reported for an ET molecule in (Ref. 35), measured at 298 K, with the H atom (not observed in x-ray crystallography) positions relaxed. The “frozen” coordinate system was relaxed in the charge neutral state and held fixed for other charge states. The “relaxed” geometry was optimized at every charge state. All geometry relaxations were carried out in calculations using TZP basis functions and TM2 pseudopotentials; we also carried out the relaxations using the other methods in the table and found no significant differences. The abbreviation pseudo. (for pseudopotential) is used in this table and others below. The root mean square error (RMSE) is taken from the fit to the classical Eq. (13). All values are in eV.

Calculated bare and renormalized parameters for the Hubbard model for an ET molecule at the experimental geometry observed at ambient temperature and pressure in various polymorphs of . The changes in conformation due to the crystal packing structure do not have a large effect on the value of . The calculated is larger for the polymorph than the others explored, indeed varies by less than 1% among the other polymorphs. Note that there is a significant difference between the values of in the eclipsed and staggered conformations in the phase. This is consistent with the effects of conformational disorder on (Ref. 62). The calculations use TZP basis sets and TM2 pseudopotentials. All values are in eV.

Calculated bare and renormalized parameters for the Hubbard model for an ET molecule at the experimental geometry observed at ambient temperature and pressure in various polymorphs of . The changes in conformation due to the crystal packing structure do not have a large effect on the value of . The calculated is larger for the polymorph than the others explored, indeed varies by less than 1% among the other polymorphs. Note that there is a significant difference between the values of in the eclipsed and staggered conformations in the phase. This is consistent with the effects of conformational disorder on (Ref. 62). The calculations use TZP basis sets and TM2 pseudopotentials. All values are in eV.

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