Metal-organic frameworks are nano-porous adsorbents of relevance to gas separation and catalysis, and separation of oxygen from air is essential to diverse industrial applications. The ferrous salt of 2,5-dihydroxy-terephthalic acid, a metal-organic framework of the MOF74 family, can selectively adsorb oxygen in a manner that defies the classical picture: adsorption sites either do or do not share electrons over a long range. Here we propose, and then justify phenomenologically and computationally, a mechanism. Charge-transfer-mediated adsorption of electron acceptor oxygen molecules in the metal-organic framework, which is a quasi-one-dimensional electron-donor semiconductor, drives and is driven by quasi-one-dimensional metal–insulator–metal transitions that localize or delocalize the quasi-one-dimensional electrons. This mechanism agrees with the empirical evidence, and predicts a class of nano-porous semiconductors or metals and potential adsorbents and catalysts in which chemistry and metal–insulator–metal transitions intertwine.
Most adsorbents—materials essential to chemical industry—are porous solids of high surface area1. Metal-organic frameworks (MOFs)2,3,4 is a vast and rapidly expanding class of synthetic adsorbents. MOFs find applications in gas separation5 and storage6,7,8, just as do their prototypes, zeolites. The geometry of a MOF interconnects the metal cations and ligands (linkers) in a three-dimensional nano-porous lattice. The vertices of the lattice are the metal cations and its bridges are the linkers. If the cations are redox-labile transition metals, the MOF can participate in electron transfer-driven (that is redox) chemistry9,10,11, unlike the redox-rigid zeolites. Separation of O2 from air12,13 is an indispensable step in industrial applications14. Since O2 is the only major prominent redox-labile component in dry and clean air15, a separation strategy can rely on differences between adsorption of the redox-rigid gases and O2 over a redox-labile adsorbent9,11,16. Fe2(DOBDC), the ferrous salt of 2,5-dihydroxy-terephthalic acid H4DOBDC (Fig. 1a), is a MOF of the MOF74 family (Fig. 1e) and is also a redox-active adsorbent whose reactive exposed Fe sites17 selectively bind O29 (Fig. 1d, inset). In particular, Fe2(DOBDC) is an adsorbent suitable for O2 separation at temperatures . However, the material adsorbs O2 irreversibly within the pressure range of ref. 9 and above the temperature TA, which is below the temperatures that are desirable in the industrial separation applications9.
Reference 99 establishes experimentally an atypical pattern of O2 adsorption in Fe2(DOBDC) that has neither been previously seen nor understood in the MOF chemistry. O2 chemisorbs in Fe2(DOBDC) in the form of a superoxide adduct, (DOBDC) for 0≤x≤1 (to be denoted π-O2(1−x)), at the temperatures , and the type of structure of the MOF remains intact under an O2 load. The π-O2(1−x) fully desorbs its O2 and regenerates back into Fe2(DOBDC) on lowering the O2 partial pressure at . Remarkably, π-O2(1−x) converts exothermically at T≈TA to the peroxide adduct, (DOBDC) for (to be denoted σ-O2(1−x)), while losing about half of its O2. As such, σ-O2(1−x) neither desorbs nor chemisorbs O2. The O2 coordinates the exposed Fe site symmetrically in π-O2(1−x) (Fig. 1b, inset) but asymmetrically in σ-O2(1−x) (Fig. 1f, inset). The experimental structural data at low temperatures9 indicate that no more than one O2 molecule per a metal site chemisorbs immediately at an Fe site at the distance about 2 Å away and the O–O bonds elongate, and, if any additional O2 molecules physisorb, they remain at least twice as far away from an Fe site and their O–O bonds remain as long as that in a free O2. The green-coloured semiconductor Fe2(DOBDC) has the apparent band gap of about 0.5 eV but Mössbauer spectroscopy9 suggests that the black π-O2(1−x) is a mixed-valence compound whose FeII–III sites are identical, that is, it appears to be a metal. In contrast, Fe2(DOBDC) remains a semiconductor on adsorption of the redox-rigid N2, and physisorption of O2 or N2 in the homologous Zn2(DOBDC) does not affect its optical response at visible wavelengths. These experiments suggest that the distributed and intermittent flow of the electrons along the MOF must play a crucial role during the adsorption of O2 in the redox-labile Fe2(DOBDC) but not in the redox-rigid Zn2(DOBDC). Unlike the redox properties, the geometries of these materials9,18 are nearly identical. Hence, the differences in their redox properties must originate in the electronic structure—the other side of the material’s chemical identity.
This report shows that O2 adsorption in Fe2(DOBDC) is quite an unusual (but likely not uncommon) redox chemical reaction for nano-porous materials. It is often taken for granted that charge transfer confined to the individual adsorption sites controls redox chemistry of nano-porous materials. It turns out that Fe2(DOBDC) is a strongly correlated quasi-one-dimensional (1D) material that undergoes a sequence of metal–insulator–metal transitions intertwined with O2 adsorption-desorption, and the charge transfer during O2 adsorption-desorption in Fe2(DOBDC) involves cooperation of many interconnected adsorption sites rather than a single site. The parameter TA controls the activation barrier for this process. To the best of our knowledge, this report is the first description of this type of redox chemistry in nano-porous materials.
Metal–insulator transitions and adsorption-desorption
The MOF Fe2(DOBDC), as a nano-porous material, is a sponge of a definite geometry and maximal porosity. The MOF’s geometry defines the boundary that delimits two dual regions: the network of the pores where the adsorbate can flow and the space where the electrons and atoms of the MOF concentrate. As the electron acceptors O219 adsorb from or desorb into the pores and the electrons flow out or into the MOF, the charge can flow along the boundary of the MOF (Fig. 1e) according to the likelihoods of the formal redox events that connect distinct sites and involve the metal or the linker as Fig. 1c explains. Because of possible metal–insulator transitions, the charge flow can fracture into or reconnect from disconnected null dimensional atom-like, one-dimensional chain-like and two dimensional surface-like conducting components within the geometry of the MOF during adsorption of O2. Since the properties of metals, insulators and metal–insulator transitions differ qualitatively in null, one or higher dimensions20, so must redox chemistry, as the chemistry of charge flow. It remains to be shown that the processes in Fig. 1b,d,f (insets) constrain the changes in the charge distribution during O2 adsorption to these in Fig. 1b,d,f, and that the consequences of these constraints are consistent with the empirical evidence of ref. 9.
A model and its validation
To arrive at and validate a model of O2 adsorption, this report relies on quantum chemistry calculations as detailed in Methods section. The total energy, the geometries, the harmonic vibrational frequencies and the electronic band structure are modelled within the numerical spin-unrestricted density functional theory (DFT) using the PW91+U approximation to the exchange-correlation energy as implemented in refs 21, 22 and as further detailed in Methods section (Density functional theory calculations). The optical absorption spectrum of ref. 9 is interpreted in terms of a model crystal field Hamiltonian and the PW91+U method as detailed in Methods sections (Density functional theory calculations and Crystal field Hamiltonian: top-down approach). The low-lying excited states of Fe2+ and [Fe−O2]2+ in the midst of the crystal field are computed in the CIS(D), state-averaged CASSCF MP2, and TD-DFT approximations as implemented in ref. 22 and as detailed in Methods section (Crystal field Hamiltonian: bottom-up approach). The electron correlation effects in the extended system beyond the PW91+U level of theory are treated within the density matrix renormalization group (DMRG)23 method as implemented in ref. 24 and as detailed in Methods sections (Model of interacting electrons and DMRG calculations).
The model geometries of Fe2(DOBDC) and π-O2, which are computed with PW91+U at U=0, are enclosed in Supplementary Data 1 and 2. As Table 1 shows, the model geometries agree with the experimental data at a level no worse than 0.3 and 0.1 Å, respectively. The computed and experimental band gaps in Fe2(DOBDC) are matched, and the computed optical excitation energies also agree with the experiment9. Since the PW91+U band structure theory at U=0 underestimates the electron–electron interaction in the strongly correlated semiconductor Fe2(DOBDC), it also underestimates the band gap. As U increases, the band gap increases achieving the empirical value of about 0.5 eV at the repulsion U of about 0.5 eV (Supplementary Fig. 1). Since the band structure of Fe2(DOBDC) at U=0 is gapless in contradiction with the experiment, but the gap opens on introduction of the electron–electron repulsion U=0.5 eV, it exemplifies a correlated insulator. The computed binding energy of −0.56 eV of O2 in π-O2 also agrees satisfactorily with the experimental saturation heat of adsorption at −0.68 eV. The binding of O2 in π-O2 in the O–O bond-down configuration is more favourable than in the end-down configuration (by 0.3 eV), which is consistent with the experiment. The self-consistent magnetic structure features the absolute magnetization of 4.1μB per Fe and the total magnetization of zero, which is consistent with the experiment assuming that the single-ion anisotropy effects at the Fe sites are neglected7,9.
The electron flow is low dimensional
To establish that the electrons flow from Fe2(DOBDC) to O2 during adsorption, it is sufficient to recognize that the elongation of the O–O bonds signals the electron transfer to O225, and the O–O bonds, computed and experimental in π-O2 and experimental in σ-O2,9 are longer than the bond in the free O2 by 0.09(0.05) and 0.4 Å, respectively. The DFT calculations further corroborate that every O2 in π-O2 hosts 0.44 e excess electrons (Table 2).
If the electrons flow to an O2, then this flow originates from the Fe site just underneath it. To elucidate the mechanism underlying this flow, the following model is considered. If the intersite electron flow is discarded, then the effect of the Fe2(DOBDC) crystal on the Fe site is that of a polarizing crystal field. A numerical density functional theory estimates this crystal field. For the strongly correlated redox-labile Fe site in the presence of the polarizing crystal field, the bottom-up wavefunction-based quantum chemistry (Methods Section (Crystal field Hamiltonian: bottom-up approach)) predicts the energy levels as well as the shapes of the electronic distributions in these levels. In contrast, the experimental optical absorption spectrum and geometry9 constrain in a top-down approach the form of and the values of the adjustable parameters within the crystal-field Hamiltonian (Methods Section (Crystal field Hamiltonian: top-down approach)). The electronic states that are obtained by the bottom-up and top-down approaches agree at the Fe site in Fe2(DOBDC) as shown in Fig. 2f; Supplementary Fig. 2. The lowest excitations that are centered at an Fe site occur between the ground 5A0 and the excited 5A1, 5A2, 5A3 and 5A4 crystal field multiplets (Fig. 2a–e), which descend from the ground state multiplet 3d6 5D of the free Fe2+.
In [Fe−O2]2+ in the crystal field, the electron-acceptor antibonding molecular orbital πg of O2 splits into the components π′ and π′′ (out and in the plane of the adsorption site), and π′ and the electron-donor molecular orbital a0 match in shape (Fig. 2a,h). The ground and first excited states of an [Fe−O2]2+ site at the geometry of π-O2 are and , respectively. The first excitation energy is about 1 eV (see Supplementary Table 1). For the PW91+U model for the extended system π-O2 (Fig. 2f,h), the projected densities of electronic states on the valence 3d orbitals of Fe and 2p orbitals of O in O2 further elucidate the nature of the Fe–O2 chemisorption bond. The molecular orbitals π′ and π′′ at O2 form bonding and antibonding combinations with the a0,1,2,3,4 orbitals at the Fe site so that all the bonding orbitals and some antibonding orbitals are occupied. As O2 adsorbs, the initial high density of the occupied 3d-like states of Fe (Fig. 2f) decreases, while the density of the 2p-derived states at O2 increases (Fig. 2h) just below the Fermi level. The donation-backdonation three-centre bond between O2 and Fe forms in the direction of the overlap between a0 and π′, and during adsorption the electrons flow from a0 at Fe to π′ at O2. The binding of O2 in π-O2 is reminiscent to that of ethylene in donation-backdonation π-complexes26. In fact, the ground states of the dioxygen and ethylene, and 1Ag H2C=CH2, are equivalent through a deformation that fuses the protons of the ethylene into their neighboring carbon nuclei. This analogy reflects the observation of the π-complexes7,9 in Fe2(DOBDC).
If electrons flow from the Fe underneath the O2, then this flow can originate from an adjacent Fe site within the 1D chain of Fe sites or from an adjacent 1D chain of Fe sites through the linker (Fig. 1c). To examine these possibilities, the electronic band structure is calculated using a PW91+U method (Methods section (Density functional theory calculations)). Every crystal field multiplet 5A0,1,2,3,4 corresponds to a triple of a0,1,2,3,4-like electronic bands (Fig. 2g) in the semiconductor FeII(DOBDC), while the antibonding combinations of a0 and π′ define the valence band in π-O2 (Fig. 2f,h). The dispersion within and the splitting of these bands quantify the extent and direction of the electron delocalization upon excitations. The bands are dispersionless away from the trigonal direction, and their hybridization with the DOBDC-centred electronic states is negligible, while the couplings of the bands along the adjacent metal ion chains are two orders of magnitude smaller than the widths of the bands themselves; the interchain coupling is particularly small for the states surrounding the Fermi level. Therefore, the electrons in these bands are 1D along the parallel 1D metal ion chains in the trigonal direction (Fig. 1d). More pronounced deviations from the 1D character arise at the energy scale around 1.1 eV of the bands, which hybridize through the linker; the deviation is also present to a much lesser degree in the other bands away from the Fermi level. The valence bands in π-O2 are 1D along the trigonal direction around the Fermi level (Fig. 1b). The lowest electronic excitations in π-O2(1−x) in the band structure picture are intersite excitations that involve the partially filled bands rather than the intrasite excitations. The projected density of states on valence orbitals of the DOBDC further suggests that the coupling through the linker is very small in Fe2(DOBDC) and in π-O2 at the energies just below the Fermi level where the band is concentrated (Fig. 2h), but the states of the DOBDC become involved if the band is sufficiently depleted of its electrons.
The trends in the geometry of the linker further suggest that the electrons do not flow along the linker in Fe2(DOBDC) and π-O2 (Fig. 1b,d), but the electrons must flow along the linker during conversion of π-O2 to σ-O2 (Fig. 1f). The formal anion DOBDC4−, as a 2,5-quinolate anion, is a reducer. Its prototype, the 2,5-benzene quinolate, is the active agent in the reductive developers in the silver halide photography due to the capacity to transfer electrons from Ag0 to nearby Ag+ (ref. 27). Similarly, DOBDC4− may oxidize (Fig. 1c) while transferring the electrons between the Fe sites in the adjacent chains. An indicator of the involvement of the DOBDC is the shortening of the C1–O′ and C2–C3 bonds (Fig. 1a). This distortion is ruled out (Table 1) at a level no worse than 0.03 Å in the redox rigid Mg2(DOBDC) and Zn2(DOBDC) as well as in Fe2(DOBDC) and π-O2. However, the distortion is manifest in σ-O2(1−x). Indeed, Table 1 shows that those bond lengths of the free reduced form of the ligand DOBDC4−, which are meant to transfer (C1–C2, C2–C3, C3–C1, C1–O′), do transfer to the redox unremarkable Mg2(DOBDC) and Zn2(DOBDC) no worse than 0.02 Å for the calculated equilibrium geometries. The transferability is at no worse than 0.01 and 0.03 Å for the calculated geometry of the redox-suspect Fe2(DOBDC) and its superoxide adduct π-O2. The geometry of the DOBDC in σ-(O2)1−x agrees better (<0.05 Å) with that of the oxidized form of the DOBDC.
The changes in the oxidation state of DOBDC are also characterized by shifts in the frequencies of the vibrational modes. The C1–C2 and C1–C3 stretches must shift redwards as the bond orders decrease while the C1–O′ and C2–C3 stretches must shift bluewards as the bond orders increase from the quinolate to the semiquinone forms of DOBDC. Reference 2828 establishes a degree of transferability of the deformation modes in the IR spectrum of the benzene ring of the free DOBDC to the IR spectrum of Ni2(DOBDC), a homologue of Fe2(DOBDC). As Fig. 3 shows, characteristic shifts of the C1–C2 and C1–C3 stretches redwards as well as the C2–C3 and C1–O′ stretches bluewards in the IR spectra of the quinolate and semiquinone forms of DOBDC also do not support any significant changes in its oxidation state in Fe2(DOBDC) and π-O2. The experimental IR spectra of Fe2(DOBDC) and π-O2 (Fig. 3) in the frequency region occupied by the benzene ring modes correlate better with the computed IR spectrum of the quinolate form than with the computed spectrum of the semiquionone form of DOBDC. Although the experimental spectra show blue and red shifts from Fe2(DOBDC) to π-O2 (1,130 to 1,124 cm−1, 1,156 to 1,151 cm−1, 1,188 to 1,197 cm−1, 1,244 to 1,267 cm−1), which could by themselves signify oxidation of DOBDC, other and more drastic changes in the spectrum, which oxidation of the linker induces in the computed spectra (Fig. 3) are mute in the experimental spectrum. Hence, this analysis by itself does not favour any significant change in the oxidation state of the ligand from Fe2(DOBDC) to π-O2. Thus, the trends in the band structure, geometry and vibrational spectroscopy suggest that the electronic structure is 1D during formation of π-O2.
Consequences of the electron flow being low dimensional
To show that the consequences of the electronic structure being 1D are the anomalies observed in the O2 adsorption experiments, a model is considered. Adsorbing O2 applies the electron chemical potential at the exposed Fe sites to the electrons in the 1D bands at the chemical potential μe, so that O2 acquires and the band loses an average of electrons per site as the equilibrium condition requires. The lattice Hamiltonian of a three-band generalized Hubbard model20,
describes the electrons within the triple of the bands at a given or μe in a finite chain of the Fe sites subject to the open boundary conditions at the left and right ends of the chain. Here, r=(λξ) indexes the distinct Fe sites, λ=0, 1, 2 labels the translationally distinct sites within a unit cell (that are related by the third-order twist axis), while 0<ξ≤ L/3 labels the unit cells in a chain of L Fe sites; are the intersite hopping parameters; and U and are the intra- and intersite electron-electron repulsion parameters, respectively. arσ and are the Fermion creation and annihilation operators for the spin channels σ=↑,↓, and and nr=nr↑+nr↓. Then ηr=2−‹nr› is the quantum average and . The parameters in the Hamiltonian equation (1) are obtained from the PW91 band structure and the experimental UV/vis spectrum as detailed in Methods sections (Density functional theory calculations and Model of interacting electrons).
As changes during O2 adsorption, the charge distribution at a site r deviates from its average in a pattern telling of the mechanics of O2 adsorption but also defined by the interaction between the electrons in a 1D partially filled band20. Since the latter cannot be disregarded, the problem of the electronic motion in the bands belongs to a methodologically challenging domain, which, however, can be accessed numerically within the density matrix renormalization group23. The total energy, the electron chemical potential and the charge distribution ηr for the ground and first excited states of the Hamiltonian equation (1) are computed for the chains of Fe sites at a conserved and even total number of the electrons using DMRG23 as detailed in Methods section (DMRG calculations).
To show that metal–insulator–metal transitions occur at about , , the charge excitation gap Δ≥0 is calculated using DMRG for a range of and U. Δ measures the minimal energy to displace the electrons. A partially filled ideal 1D conduction band can open a gap Δ>0 for the charge excitations so that the state of the electrons is insulating only at particular and rational fillings of the band; otherwise Δ=0 so that the state of the electrons is metallic (that is a Luttinger liquid)20. As shown in Fig. 4a, Δ peaks at about , , which corresponds to the insulating states. Lower values of Δ correspond to ‘metallic’ states. The charge compressibility (that is, the derivative of the volume of the electrons versus the chemical potential, Supplementary Fig. 3) diverges before the excitation gap opens, which further confirms that the gap openings are metal–insulator transitions.
To show that the process for a small enough δη >0 is activationless for or but the significant activation barrier of E≠=0.5Δ emerges near , it is sufficient to note that there have to be thermally accessible charge transfer excited states for the charge to flow to O2. The concentration of the thermally activated charge carriers scales as since the thermal excitations take place between the chemical potential level in the middle of the gap (up to the accuracy logarithmic in T) and the states at the energies above Δ. Defects in the distribution of O2 in π-O2(1−x) set a finite length scale above which ideal chains of Fe sites terminate. The gap Δ decreases towards its asymptotic value as the length increases. At the U =0.5 eV, E≠ are 367, 214, 101 K for the chains of 24, 48, 96 Fe sites at , respectively. This temperature range is consistent with the empirical TA of 220 K. To explore the sensitivity of the model equation (1) on the parameter U, Fig. 5 compares the computed charge gaps. Although the magnitude of the gap (and TA) depends on U, the fractional filling of the conduction band at which the metal–insulator–metal transition occurs is independent of . Therefore, the qualitative conclusions about O2 adsorption driving the metal–insulator–metal transition still stand even if U, which enters equation (1), were in error or assumed a range.
To show that a pronounced pattern of FeII-like and FeIII-like sites emerges past the metal–insulator–metal transition at around , the distribution is computed (Fig. 4b). In the ‘metals’, ηr exhibits a charge density wave modulation whose period is incommensurate with the lattice, the cost for displacing a charge Δ ≈0, and all Fe sites are nearly equivalent in a finite chain. However, the ground-state ηr (Fig. 4b) exhibits charge density wave modulation, whose period is commensurate with the lattice of Fe sites in the insulating state near where the cost for displacing the charge is finite. Therefore, the maxima and minima of ηr follow the Fe sites for , and the amplitude of the charge oscillations reaches about 0.6 e. For the insulator metallizes once again, but the modulation at around defines the new periodic background for the remaining electrons.
A mechanism of adsorption of O2 in Fe2(DOBDC) is now manifest. The initial O2 adsorption triggers the insulator–metal transition above the O2 coverage at which the applied electron chemical potential due to O2 exceeds the gap in the correlated semiconductor Fe2(DOBDC)20. The minimum work to desorb η electrons from to is −μeη. If the work, kT , due to pressure change from P0 to P1 can overcome μeη, then O2 adsorption is reversible. μe changes slowly with in the metallic state for where the charge distribution remains homogeneous in average. However, μe jumps to a lower value past the metal–insulator–metal transition because of the stabilization of the pattern of the distinct sites, and with ηr′<ηr′′, which minimizes the electron repulsion. If , the activation barrier hinders the metal–insulator–metal transition at , and the O2 adsorption halts at . In contrast, if , the charge transfer to O2 proceeds to where the pattern of weaker and stronger bound emerges, O2 desorbs from but remains at . The further pumps the electrons and converts to σ-O2 while the DOBDC partially oxidizes.
This mechanism implies further observable consequences. The optical conductivity σ(ω) of Fe2(DOBDC)+O2 must be suppressed at the frequencies , but at it should show a scaling law σ(ω) ∝ (Δ−ω)3K−4 where the exponent is determined by the Luttinger liquid parameter characteristic of a 1D system20. Conversion of the weakly bound π-O2 to strongly bound σ-O2 is undesirable in the context of the separation. If the mechanism is correct, the conversion is an activated process whose rate should decrease if the length of Fe chains unterminated by defects is decreased or if the electron repulsion U at Fe sites is changed (Fig. 4a; Supplementary Fig. 4), given that all other factors are fixed.
Unlike non-porous low-dimensional materials, whose electronic structure is accessed by penetrating radiation at long range, the nano-porous materials offer the unique advantage that their geometry and long-range electronic structure can also be accessible at short range through by a more diverse probe: a chemical reaction. This report shows how O2 chemisorption, a probe reaction, reflects the profound changes in the electronic structure throughout the material. Chemistry of electron acceptors other than O2 in Fe2(DOBDC) can be similarly anomalous. Many heterogeneous catalytic reactions are redox processes1. If an electronically 1D MOF is a catalyst, then its chemical profile is likely anomalous. The material Fe2(DOBDC) belongs to a broader class of nano-porous materials whose geometry is 1D29. Those materials in this class, which are also electronically 1D, should exhibit anomalous chemical profiles, which may follow the pattern that is described in this report. The electronic structure of a nano-porous material may further delocalize and become two dimensional if the linkers become conductors. By tailoring the geometry of the electronic distribution through varying the chemistry of the charge transfer network in a MOF and adjusting external conditions, the chemistry of the material could be modulated.
Density functional theory calculations
The initial magnetic structure of Fe2(DOBDC) in the self-consistent DFT calculations is taken to be consistent with the magnetic susceptibility measurements7,9, which point to the magnetic order such that the spin 2 magnetic moments at Fe sites are parallel within a helical Fe chain and are antiparallel in the helical chains of the opposite chirality. The spin-orbit coupling is neglected. The geometries of Fe2(DOBDC) and π-O2 are obtained by minimizing the total ground state energy with respect to the atomic displacements. The residual total force and the stress at the optimized geometries are below 0.1 eV·Å−1 and 1·10−4 eV·Å−3, respectively. The term ‘band structure’ denotes here the spin-unrestricted mean field PW91+U band structure. The PW91+U mean field model corrects the Perdew-Wang 91 (PW91) generalized gradient approximation (that is, PW91+U at U=0) for the underestimation of the Hubbard repulsion U>0 between the strongly correlated d-electrons at Fe sites. The value of U is set by the condition that the experimental and computed band gaps match (Supplementary Fig. 1). The mean field of the ultrasoft pseudopotentials21 replaces the electrons in the ionic cores Fe[Ne], C[He], C[He] and H. The Kohn-Sham Bloch wavefunctions and the electron densities of the remaining electrons are then expanded into the Fourier series up to the plane wave kinetic energy cutoffs of 476.2 and 5,714.4 eV, respectively. The numerical integration over the reciprocal space is carried over 1a × 1b × 4c Monkhorst-Pack quadratures in the total energy and geometry optimization calculations. In the band structure calculations, the reciprocal space integration is carried over the 4a × 4b × 8c, and then over the 1a × 1b × 71c Monkhorst-Pack grids, after it is determined that the dispersion in the ab directions is much smaller than that along the c direction. The Gaussian occupation number smearing of 0.136 eV is applied when the band structure is metallic. The projected densities of states are computed with respect to the projections on the valence states of the atoms.
Equilibrium geometry and vibrational frequencies of the ligand
The equilibrium geometries and harmonic vibrational frequencies for the quinolate and semiquinone forms of the DOBDC are computed using the PW91 approximation to the exchange-correlation energy as implemented in ref. 22. The Kohn-Sham orbitals are expanded over the Pople’s gaussian basis set 6−311++G(3df, 3pd). The numerical integration in the exchange-correlation contributions to the energy, forces and frequencies is carried over atomic-centred Euler-Lebedev quadratures with 99 × 590 points along the radial and the angular directions per atom, respectively. The maximum forces at the equilibrium structures do not exceed 10−3 eV·Å−1.
Crystal field Hamiltonian: top-down approach
The ground state configurations of the free isotropic ion Fe2+ is 3d6. The electron–electron interaction arranges 210 degenerate states of the 3d6 configuration into 16 multiplets19. The restriction of the Hamiltonian onto the Hilbert space spanned by the multiplets of the configuration 3d6 (ref. 30) is
The two-particle Hamiltonian acts on the states of the configuration 3d6 as
where the Slater-Condon parameters31, , and set the strength of the electron–electron interaction. The Hee is block diagonal in the basis |LMLSMSι› of the eigenstates of the total orbital and electronic spin angular momenta operators L2, L3, S2 and S3; the quantum number ι discriminates the terms of identical L and S in the configuration. The crystal field operator contains only the s-, d-, h-moments, , in its multipole expansion at an Fe site,
The matrix elements of the unit spherical tensors are computed standardly30. For the purpose of the phenomenological analysis, the crystal field spherical moments are written in the form
where RA and nA are the radius of the sphere centered at Fe0, and the position at this sphere of an ion A in the crystal structure, Yl,q are the surface spherical harmonics. Effective charges ZA are computed using the Bader population analysis32. The phenomenological parameters ρ2, ρ4, are constrained by the experimental UV/vis absorption spectrum of ref. 9. First, the parameters ρ2, ρ4 are found by the minimization of the least-square deviation between the spectrum of the crystal-field Hamiltonian and the transitions in the UV/vis absorption spectrum. Second, starting from the Hamiltonian at the optimal ρ2 and ρ4, the eight independent parameters and are determined. Thus, determined the real and the imaginary parts of the parameters are
The real and the imaginary parts of thus determined Hamiltonian in the basis |2ML200› for ML = −2, −1, 0, 1, 2 are
The angular distributions,
of the electrons at an Fe site in the crystal field states are then defined by the eigenvectors cm(5A0,1,2,3,4) of the crystal field Hamiltonian in the basis of the d-orbitals.
Crystal field Hamiltonian: bottom-up approach
To further corroborate the phenomenological picture and establish the electronic distribution in the low-lying electronic states, the low-lying excited states of Fe2+ and [Fe−O2]2+ are computed in the CIS(D), and state-averaged CASSCF MP2 approximations, as well as using the linear response time-dependent density functional theory in the adiabatic PW91 approximation. The molecular orbitals in these methods are expanded over the Dunning basis set aug—cc—pvtz of Gaussian functions as implemented in ref. 22. The single-particle external potential,
represents the crystal field that the Fe site experiences; here the summation runs over the scaled Bader charges λCFZA of the ions within the ball centred at Fe site of the 60 Bohr radius; the screening exponent α is 0.035 Bohr−1. The external potential is taken the same in the Fe2(DOBDC) and in π-O2 at a vanishingly small O2 load. The reference in the CIS(D) approximations is a stable unrestricted Hartree-Fock determinant. The initial orbitals that generate the active space in the CASSCF and CASSCF MP2 methods are the natural orbitals of the unrestricted Hartree-Fock density matrix. The six-electron active space of the Fe2+ ion includes the five a0, a1, a2, a3, a4 3d-like orbitals. The eight-electron active space of the [Fe−O2]2+ includes seven orbitals that arise from the a0, a1, a2, a3, a4 orbitals of the Fe2+ and the two 1πg-like orbitals of O2.
Model of interacting electrons
The kinetic energy matrix of the three-band model is
and its Fourier transform is the kinetic energy in the reciprocal space,
The parameters ?, t1, t2, t3 are obtained by matching the PW91 GGA band structure along the direction ZΛΓΛZ (Supplementary Fig. 5) to the three independent unitary invariants of the 3 × 3 matrix, which are taken as the trace,
the trace of the square,
and the determinant,
Thus determined hopping matrix elements are listed in Table 3. The interaction is taken to be given by the Ohno–Klopman extrapolation formula,
where the on-site electron repulsion is U, and a0 is the Bohr radius. On the lattice, the distances between the 1 · Fe and 3+ · Fe, , sites are R12 =3.2 , R13 =5.2, R14 =7.1 Å, respectively. The exponential screening at the length scale of l, no matter how long, in the bulk material, ensures that the Coulomb interaction is short range in the macroscopic limit; l is set to R14 here. The exchange integrals, which are much smaller than the Coulomb integrals in the atomic case31, are neglected.
The total energies and the charge distribution in the ground and low-lying excited states of the model equation (1) are computed using DMRG approach. The maximum number of the states spanning the DMRG Hilbert space is increased from the initial 20 to 200 in six sweeps. The truncation errors do not exceed 10−7 eV. The ground state at conserved N and L includes the total spin S=0 state. The gaps between excited states at given L and N are then determined by computing the ground and the first excited states at fixed S=0, 1 and N.
The derivatives of the ground state energy in the expression of the chemical potential,
and the charge compressibility,
are computed numerically,
at a given L.
How to cite this article: Maximoff, S. N. and Smit, B. Redox chemistry and metal–insulator transitions intertwined in a nano-porous material. Nat. Commun. 5:4032 doi: 10.1038/ncomms5032 (2014).
We thank E. Bloch, C.M. Brown, J.R. Long, G.J. Long, W. Queen, B. Wiers and J. Zadrozny for stimulating discussions on the subject of this report and for being open about their experimental results, as well as M.P. Head-Gordon, J.R. Long and J. Neaton for sharing their comments on the manuscript. The work was funded by the Center for Gas Separations Relevant to Clean Energy Technologies, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under award number DE-SC0001015. This work used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.