Communication: Fragment-based Hamiltonian model of electronic charge-excitation gaps and gap closure

Despite many decades of effort, the ability to reliably model and predict the properties of technologically important materials remains a considerable theoretical and computational challenge. A principal difficulty is that existing models do not generalize well to novel materials—notable examples include nanoscale materials and chemically complex oxides—where underlying, competing quantum-mechanical effects play essential roles.

Model Hamiltonians such as the well-known Hubbard model are designed to capture global properties of materials such as phase diagrams or band gaps, by relying on the careful design and choice of model parameters. Among these properties, charge-excitation gaps have proven particularly difficult to model and are the focus of this communication. Fundamental questions about gaps that such a model could address include predicting conditions of strain, structural defects, ordering, composition, temperature, and spatial dimensionality governing gap opening and closure.

Historically, questions about charge-excitation gaps have been addressed solely with the states of electrons in a material, using model Hamiltonian approaches. An accepted view is that the charge-excitation gap Δμ closes when one-electron bands are wide enough to overcome the on-site Coulomb interaction and is often stated as

where U denotes the “Hubbard-U,”¹ and W is the average width of the one-electron and one-hole excitation bands.^2,3 An analogous statement exists for ceramics and covalent crystals.^4,U was originally defined by Hubbard as the on-site Coulomb repulsion between pairs of electrons, but is widely recognized as the difference in local values of chemical potentials.² For molecular systems, U corresponds to twice the absolute hardness η of Parr and Pearson,⁵ and in condensed phase systems to the Mott exciton pair-formation energy ϵ_M,³ i.e., U = 2η = 2ϵ_M.

The computation of Δμ in Eq. (1) requires the evaluation of global chemical potentials for adding (μ⁺) and removing (μ⁻) one electron from the system. These quantities can be expressed in terms of system energies for different numbers of electrons N: E_ζ = E(N = Z + ζ) for ζ = − 1, 0, or 1, where Z is the total nuclear charge. The expression for the gap energy Δμ becomes^3–7 

For a given model Hamiltonian, one can evaluate this expression in terms of model parameters in order to obtain a statement analogous to U − W in Eq. (1), making it possible to investigate specific properties of the gap, such as the possibility of gap closure for various materials and in different environments. This is the approach taken here, where we introduce and utilize a novel Fragment Hamiltonian (FH) model⁸ built upon charge states of fragments rather than electronic states of atoms as in traditional model Hamiltonians. Importantly, the term “fragment” is quite general here: it refers to an arbitrary collection of nuclei and electrons, bound together in any meaningful assembly—a molecule, protein motif, or metal surface, for example.

In this communication, we show that the FH model preserves the concept of electronic charge-transfer gaps while allowing for gap closure in systems where inter-fragment hopping is sufficiently strong. The gaps are expressed in terms of inter-fragment charge-transfer hopping integrals T and on-fragment parameters U^(FH) related to fragment-scale chemical potentials. These parameters are analogous to the electronic intra-band hopping parameters t and on-site Coulomb repulsion parameters U of the standard Hubbard (sH) model, but with the Hubbard quantum electronic states generalized to the quantum fragment states of the FH model. For relatively simple two-fragment and multi-fragment cases, we demonstrate that gap closure is enabled once T exceeds the threshold set by U^(FH). This is a remarkable result emerging from a relatively simple instantiation of the FH model, since it stands in contrast to the standard Hubbard model for 1d rings, for which Lieb and Wu (LW) proved that gap closure was impossible, regardless of the choices for t and U.^6,9

The paper is organized as follows. In Section II, we review examples from electronic model Hamiltonians and traditional atomistic models to set the stage for presenting an alternative, fragment-based view in Section III, based on the FH model. In Section IV, we derive the key results of the paper—expressions for charge-excitation gaps in the two-fragment and multi-fragment cases—followed by a summary and conclusions in Section V.

On the electronic structure side, how well a particular model Hamiltonian performs in predicting gap behavior varies considerably between high degrees of success and failure. In particular, some model Hamiltonians cannot capture gap closure at all. For instance, sH models² sometimes yield insulators, when it was expected that the underlying physical system should be metallic, as found in the seminal work of LW.^6,9 Specifically, LW modeled finite, one-dimensional (1d), periodic chains or rings within a single-band, nearest-neighbor-hopping Hubbard Hamiltonian. The charge-excitation gap never closes at any positive value of U. The sH model is the sum of single-band, tight-binding, and on-site U terms,^1,2,9

where $cmσ†$ and c_mσ are creation and annihilation operators, respectively, at site m with spin σ. $nmσ=cmσ†cmσ$ is the occupation number operator for an electron of spin σ at site m. t is an intra-band hopping integral and W is proportional to t. U is an input parameter for the model Hamiltonian that appears regardless of the charge-state of the system. Other varieties of models that utilize long-range hopping, higher spatial dimensions, and better treatment of electron correlation are required in order to capture expected gap closures.^2,10

Another analysis of charge excitations gaps is from Zaanen et al.,⁴ who analyze the conductivity gap for rocksalt NiO with an Anderson model Hamiltonian.¹¹ It is mentioned because it is also using a fragment or cluster approach qualitatively similar to our approach, but these systems require inequivalent fragments for their analyses.

On the atomistic side, we recognize that many variable-charge (vc) atomistic models utilize the concept of a Hubbard U, or equivalently, a local version of the absolute hardness.^5,12–17 The atom-energies are expressed as polynomials in numbers of electrons on each. First, let atoms with an integer number of electrons have energies ϵ_m(N_m). For nuclear charge Z_m = N_m, the relevant local chemical potentials are $μm+≡ϵm(Zm+1)−ϵm(Zm)$ for adding an electron and $μm−=ϵm(Zm)−ϵm(Zm−1)$ for removing an electron from m. Our generalization of U_m adopts the preferred definition as the difference in local chemical potentials for m,^2,18

At the quadratic level of approximation, the atom-energy E_m is given by^5,14,19–23

where $ϵm0$ is the isolated, neutral atom energy for m, and $μm≡ϵm+−ϵm−/2$ is the average chemical potentials for adding and removing an electron from m. Also μ_m = − χ_m, the negative of the Mulliken electronegativity.²⁴ We then find that Δμ_vc = U_m, meaning that the charge-excitation gap remains open at any bond length. The result should be expected based on the isolated-atom origins of the model which neglects any concept of hopping that might create a nonzero value of W.

Intermediate between purely classical and purely quantum mechanical models is quantum-mechanical/molecular mechanical²⁵ models. In these models, one region or “fragment” is treated quantum mechanically, while the other is treated classically. However, these models do not support charge transfer between the two regions and thus lose the gap property.

To achieve proper charge-excitation gap behavior, it is necessary to retain the rudiments of electronic states. To this end, we utilize a model Hamiltonian constructed at intrinsically atomic and larger (fragment) length scales, without explicitly constructing the states of the electrons. This model Hamiltonian is the recently developed FH,^23,26 which is designed to utilize the states and energies of material fragments directly. Fragments can be arbitrary collections of electrons and nuclei, such as atoms, pseudoatoms, functional groups, crystal lattices or sublattices, quantum dots, wires, sources, and drains. The central point of the FH model is that the Hamiltonian operators for fragments can be defined within a many-electron point of view.

The same questions concerning the opening and closing of charge-excitation gaps are addressed here within the FH model, for the two-fragment case. Our principal aim is to recover within the FH model the accepted view of gap closure in Eq. (1). The results found in this work suggest the possible value of this class of models in capturing a greater level of electronic structure information than traditional atomic Hamiltonian models. For this reason, it may provide some bridge between atomistic and electronic scales previously unavailable. We will focus on just the LW approach as a guide for our analyses. In this communication, we analyze charge-excitation gaps for finite systems based on a new class of model Hamiltonian that pertains to fragments of material, the recently developed FH.^23,26

As a motivating example for the FH model Hamiltonian, consider metal clusters. The charge-excitation gap changes with cluster size, decreasing approximately as the inverse of the cluster radius or one-third-power of the particle number.²⁷ These clusters can transition to a metallic state when they become sufficiently large. To see this, consider that macroscopically large clusters, i.e., bulk materials, are metallic while clusters of tens or even hundreds of atoms are not. Smaller clusters measured through photoionization experiments^28,29 generally show decreasing ionization potentials (−μ⁻) and increasing electrons affinities (−μ⁺) toward bulk values, as the cluster size increases. There is a distinct gap between the two which defines Δμ. Specifically, the small clusters may be considered as Mott-Hubbard insulators.^2,30 Bulk metals are observed to have a finite density of states at the Fermi level, indicating gap closure given by the condition Δμ ≤ 0.

Estimating Δμ in the FH model relies on states and energies of a system similar to those used by LW. However, these energies are determined from states and energies of the constituent fragments themselves, rather than from states and energies of single electrons. For this purpose, a new model Hamiltonian needs to be constructed which pertains directly to fragments. The guiding principle for the new Hamiltonian comes from a generalization of the atoms-in-molecules Hamiltonian of Moffitt.³¹ The FH model identifies a Hamiltonian operator for each fragment, but, in the FH version of this concept, the integer number of electrons associated with a particular fragment is allowed to fluctuate, as will be described later.

In quantitative terms, let each fragment m have N_m electrons on $Hˆm$⁠, the many-electron Hamiltonian that the fragment would be assigned if it were isolated from the other fragments. For a two-fragment system, the spin-free FH Hamiltonian operator reduces to⁸ 

where m and l label the “metal” and “ligand” fragments, respectively, N = N_m + N_l is the total number of electrons, and $Vˆml$ is the Coulomb interaction operator between the two fragments. $Hˆ(FH)$ contains all of the same operators that appear in the usual many-electron $Hˆ$ but are arranged differently than the customary “kinetic-external potential-electron-interaction” representation.

Concomitantly, the different basis wave functions applied to $Hˆ(FH)(N)$ must distribute different numbers of electrons on each fragment. Mixtures of these states yield eigenstates of the system each N of interest. Corresponding to the Hamiltonian decomposition in Eq. (6), special states are assumed to be available that computation of fragment energies for integer numbers of electrons.

Many-electron, antisymmetrized, normalized basis functions with prescribed electron distributions are expressed as |ζ_mζ_l〉, where the ζ_m = N_m − Z_m are deviations from the neutral fragments. These basis functions are motivated by those devised by Mulliken and others.^24,33,34 When the system is neutral (N = Z or ζ = 0), |00〉, | + − 〉, and | − + 〉 are used. N = Z corresponds to half-filling in LW.⁶ Note that, if orbital representations of the wave functions are made, particular choices of the expansion coefficients will reproduce either the Heitler-London or molecular orbital wave function. In this sense, our basis states interpolate between strongly and weakly correlated states. When the system is ionic, these traditional Mulliken states need to be augmented with the basis functions |0ζ〉 and |ζ0〉, for ζ = ± 1. The ground states are needed for each of these numbers of electrons, distributed between the fragments as indicated.

Applying these basis states to Eq. (6) yields a set of 2 × 2 Hamiltonian matrices whose matrix elements are the expectation values $〈ζζ′|Hˆ(FH)|ζ″ζ‴〉$⁠. In Table I, these are decomposed into expectation values for fragments and fragment-fragment interactions, $〈ζζ′|Hˆm|ζ″ζ‴〉≡Hζ″ζ‴ζζ′$ and $〈ζζ′|Vˆml|ζ″ζ‴〉=Vζ″ζ‴ζζ′$ for different values of the ζ’s. The fragment energies will be denoted by $ϵmζ≡〈ζζ′|Hˆm(ζ)|ζζ′〉=ϵm(Zm+ζ)$⁠, independent of ζ′; fragment-fragment interactions by V_ζζ′; and hopping integrals $Tζ″ζ‴ζζ′$ that are not decomposed.

Chemical potentials for adding or subtracting one electron from a chosen fragment^{1,2,4,9,18,32} thereby appear as a consequence. Let fragments have energies ϵ_m(N_m). For N_m equal to the nuclear charge Z_m, the relevant local chemical potentials are $μm+=ϵm(Zm+1)−ϵm(Zm)$ for adding an electron and $μm−=ϵm(Zm)−ϵm(Zm−1)$ for removing an electron from m. Favoring the preferred definition as the difference in local chemical potentials, our generalization of U to U^(FH) for site m is $Um(FH)=μm+−μm−$⁠.

Solutions to the associated eigenvalue problems have the structure

where $H̄$ is the average of the energies on the diagonal, ΔH is their difference, and T is the transfer integral between states, but is not the intra-band hopping integral t of Eq. (3). In each case, the negative-signed root $Eζ(−)=Eζ$ is the ground-state energy.

Two-fragment systems—For equivalent fragments, states with N = Z ± 1 retain a “particle-hole” symmetry in their solutions. The symmetry simplifies the ground-state energies to $Eζ=2ϵ0+sgn(ζ)μmζ+V0ζ−Tζ,$ where $Tζ≡T0ζζ0$⁠. Note that ΔH = 0 for this case. A critical difference between this energy and the analogous one in LW is that U^(FH) does not appear explicitly in the FH model because it is not in the fragment Hamiltonian itself. Summing E₋ and E₊ yields $E−+E+=4ϵ0+U(FH)+V0−+V0+−T−−T+$⁠, wherein U does appear. The ground state energy for N = Z is

where $Ṽ+−≡V+−+2T+−−+$⁠.

Combining these energies into Eq. (2) determines the gap to be

The combination $H̄++H̄−−2H̄0$ cancels that source of U_FH dependence. The surviving U^(FH) dependence appears solely in the transition term from the neutral state. The competition impacting gap closure includes both charge hopping integrals and fragment-fragment interactions for different charge states. How this competition plays out is difficult to assess in general. For this reason, we appeal to consideration of special cases.

Three special cases lend particular insight into the behavior of Δμ. First, at dissociation, all hopping and fragment interactions cease. The dissociation limit of Eq. (9) is $Δμ→U(FH)=2η$⁠, as it must. Second, assuming that the fragment interactions cancel and electron-hole production terms are negligible yields

Thus, for sufficiently large charge hopping integrals, it is possible to achieve the condition Δμ ≤ 0. Charge hopping in the FH model has replaced intra-band hopping in the sH model and is the central reason why the FH model is able to recover an analog to “U − W ≤ 0.” Third, note that the only difference between the periodic and non-periodic systems is a factor of two scaling of the hopping integrals. All other results and analyses remain the same for both types of systems.

A related corollary takes advantage of the consonance of definitions between U^(FH) in Eq. (4) and Δμ in Eq. (2). That is, U^(FH) is the local or fragment-level version of Δμ. This congruence permits the development of recursion relations as the number of fragments in the system is increased by powers of 2. For systems of 2^M fragments composed of two fragments composed of 2^M−1 fragments, the recursion relation becomes

where $U0(FH)$ is U^(FH) for a single, isolated fragment and the sum is over all subfragments contributing to the total system. If the isolated fragment is an atom, then U^(FH) is 2η.⁵ Implicit in this elementary treatment is that there are reorganization energies that need to be accounted for in a quantitative analysis. For instance, when two nonperiodic fragments are joined to form a ring, there is a reorganization energy from the ring periodicity.

Multi-fragment systems—Now we extend the results from the two-fragment-system to multi-fragment systems. The simplest conceivable estimate of the charge transfer gap ignores any charge fluctuations in any of the charge states. In one sense, only the ionic states exhibit charge-transfer hopping. No hopping is permitted in the neutral state as the analysis falls beyond the scope of this paper, as explained below. We also ignore the fragment-fragment interactions in this analysis.

The ionic ring energies for any number of equivalent fragments can be calculated in a manner very similar to the two-fragment case. One admits orthonormal states of the form |ζ00…0〉, |0ζ0…0〉, |0…0ζ0…0〉, etc., meaning that a single electron (ζ = + 1) or hole (ζ = − 1) resides on one fragment or another. If there are M fragments, the wave function for the ionic system is $|ψ〉=∑m=1M|ψm〉/M$⁠, where |ψ_m〉 = |0…ζ…0〉 with ζ on fragment m.

Solutions for the multi-fragment ring are those of a single-band tight-binding model with periodic boundary conditions. Only the ground-state solution is needed, which for the 1d periodic ring with nearest-neighbor hopping is E_ζ(M) = Mϵ₀ + sgn(ζ) μ^ζ − |T^ζ|. Any number of fragments, lattice spacing and spatial dimensionality dependencies in the hopping integrals are left implicit.

For the neutral state, the simplest trial state is the covalent one, |ψ〉 = |0…0〉, wherein each fragment is neutral. The energy of covalent state is just E₀(M) = Mϵ⁰. If even a single exciton was allowed, as was done for the two-fragment state, the electron and hole could appear on any pair of fragments, label them as m and m′. The state may be represented as |0…0ζ_m0…0 -ζ_m′0…0〉, where ζ_m means that there is an electron (+) or a hole (−) on fragment m. For a system of M fragments, there are M(M − 1) excitonic states, precluding analytical results. Furthermore, we neglected these fluctuations in the ionic states to begin with.

Calculating the charge-excitation gap for the 1d ring using the covalent state for the neutral system yields

The essential result of the two-fragment case from Eq. (10) is preserved. There is no cancellation of U^(FH) between the ionic terms and neutral terms as in the two-fragment case with excitonic states.

In summary, we have shown that the FH model Hamiltonian provides a representation of charge-excitation gaps wherein generalized Hubbard-U and t energy parameters can emerge naturally from the underlying electronic charge-transfer physics. Our analysis of two- and multi-fragment systems demonstrates the possibility of gap closure for positive-definite U^(FH) for arbitrary-size fragments. One-electron charge hopping—rather than the intra-band hopping of the standard Hubbard model—plays the key role in closing these gaps. We recover the oft-quoted assertion that charge excitation gaps can close in systems where one-electron hopping is sufficiently strong. These results suggest the value of the FH model in capturing essential electronic properties within atomic and larger-scale fragment-based models of real materials.

Work performed at Los Alamos National Laboratory was under the auspices of the U.S. Department of Energy, under Contract No. DE-AC52-06NA25396. Partial funding was provided by the Center for Materials at Irradiation and Mechanical Extremes, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award No. 2008LANL1026. Partial funding was also provided by Laboratory Directed Research and Development Program. Work performed at the University of New Mexico was under the auspices of the Center for Materials at Irradiation and Mechanical Extremes and the DoD/DTRA CB Basic Research Program under Grant No. HDTRA1-09-1-008. Thanks are extended to Anders Niklasson, Richard L. Martin, Ramamurthy Ramprasad, and Jian-Xin Zhu for commenting on the manuscript.