Supporting information for: Electronic chemical. potentials of porous metal-organic frameworks

Transcription

1 Supporting information for: Electronic chemical potentials of porous metal-organic frameworks Keith T. Butler, Christopher H. Hendon, and Aron Walsh Department of Chemistry, University of Bath, Claverton Down, Bath, BA2 7AY, UK Supporting Information Density functional theory 1,2 (DFT) is commonly used to calculate the electron (band) energies of solid-state systems; however, in most routinely applied DFT techniques these energies are calculated relative to an internal (system dependent) reference level, thus alignment of the energies between materials is problematic. The vacuum level of the electrostatic potential just beyond the surface of a material is commonly used as a reference level through which the energy levels of different materials can be aligned. The application of DFT to the surfaces of solid-state materials is well developed and can be used to calculate a local vacuum level; however, in the case of MOFs the direct modelling of surface structures using first-principles electronic structure techniques is rendered impossible by the large crystal sizes of the systems. Other techniques exist for electron energy level alignment, but all rely on the formation of a clean surface or interface. 3,4 In our technique we exploit the porous nature of the MOF structures to access a reference electrostatic potential within the pore of the material, analogous to the vacuum level at the surface of a solid. To whom correspondence should be addressed S1

2 The electrostatic potential which we use to align the energy levels is the spherical average of the total electrostatic potential (ionic and Hartree terms) at the centre of the pore (Φ av ), sampled with a radius of 2 Å. The Kohn-Sham energy levels (eigenvalues) are then aligned relative to this potential: ε absolute KS = ε KS Φ av. (1) Following Janak s theorem, 5 the highest occupied eigenvalue is a good representation of the vertical ionisation potential (IP). While this relationship is only satisfied in finite systems with the exact exchange-correlation functional, in the solid-state, the Kohn-Sham eigenvalue is equilivalent to the electron removal energy for all functionals. Indeed the equivalence of the Kohn-Sham and quasi-particle ([N+1] + [N-1] - 2E[N]) band gaps is well documented. 6 The python script to calculate the spherical average of the potential (also capable of calculating planar and macroscopic averages) from the standard VASP output is available from an online repository. 7 A 2-D plot of the electrostatic potential in MIL-125 is presented in Figure Figure 1. In addition to the calculated IP, the corresponding electron affinity (EA) can be computed using the band gap (E g ) from the standard relation EA = IP - E g. All electronic and electrostatic properties for each framework were calculated using DFT with the HSE06 8 exchange-correlation functional, which has been shown to perform consistently with inorganic semiconductors and hybrid metal-organic materials The IP, EA and E g, as well as statistical variance of the electrostatic potential within the spherical averaging section and the principle components of the electric field tensor across the sampling region, are presented in Table Table 1. Visualisations of the structures, densities and potentials in this work were made using the code VESTA. 14 S2

3 Table S1: Summary of calculated metal-organic framework properties, including the calculated Hartree potential at the pore centre (Φcent), ionisation potential (IP), electron affinity (EA), band gap (Eg), as well as the variance (Var) and electric field tensor (E) associated with the electrostatic potential sampling probe within the the structural pore. Framework Φcen.(eV) IP (ev) EA (ev) Eg (ev) Var (V) Exx, Eyy, Ezz (V/Å) COF , ,0 CPO-27-Mg , , HKUST-1 [a] , , HKUST-1 [b] , , HKUST-1 [c] , , MIL , , MOF , , ZIF , , [a] - closed shell singlet configuration (Cu-Cu δ bond) [b] - ferromagnetic configuration (triplet) [c] - antiferromagnetic configuration (singlet) S3

4 Figure S1: Linear scan of the spherical average electrostatic potential across the pore of MIL-125. The scan is along the < 020 > direction. The error bars show the variance within each spherical average section. References (1) Hohenberg, P.; Kohn, W. Phys. Rev. 1964, 155, (2) Kohn, W.; Sham, L. J. Phys. Rev. 1965, 140, (3) Alkauskas, A.; Broqvist, P.; Pasquarello, A. Phys. Rev. Lett. 2008, 101, (4) Van de Walle, C. G.; Neugebauer, J. Nature 2003, 423, (5) Janak, J. F. Phys. Rev. B 1978, 18, (6) Sham, L. J.; Schlüter, M. Phys. Rev. Lett. 1983, 51, (7) (Accessed 05/08/2013). (8) Heyd, J.; Scuseria, G. E.; Ernzerhof, M. J. Chem. Phys. 2003, 118, (9) Walsh, A. J. Phys. Chem. Lett. 2010, 1, (10) Hummer, K.; Harl, J.; Kresse, G. Phys. Rev. B 2009, 80, (11) Kim, Y.-S.; Hummer, K.; Kresse, G. Phys. Rev. B 2009, 80, S4

5 (12) Wróbel, J.; Kurzydłowski, K.; Hummer, K.; Kresse, G.; Piechota, J. Phys. Rev. B 2009, 80, (13) Hendon, C. H.; Tiana, D.; Fontecave, M.; Sanchez, C.; D arras, L.; Sassoye, C.; Rozes, L.; Mellot-Draznieks, C.; Walsh, A. J. Am. Chem. Soc. 2013, 135, (14) Momma, K.; Izumi, F. J. Appl. Cryst. 2011, 44, S5