Strong electron correlation from partition density functional theory.

Standard approximations for the exchange-correlation functional in Kohn-Sham density functional theory (KS-DFT) typically lead to unacceptably large errors when applied to strongly correlated electronic systems. Partition-DFT (PDFT) is a formally exact reformulation of KS-DFT in which the ground-state density and energy of a system are obtained through self-consistent calculations on isolated fragments, with a partition energy representing inter-fragment interactions. Here, we show how typical errors of the local density approximation (LDA) in KS-DFT can be largely suppressed through a simple approximation, the multi-fragment overlap approximation (MFOA), for the partition energy in PDFT. Our method is illustrated on simple models of one-dimensional strongly correlated linear hydrogen chains. The MFOA, when used in combination with the LDA for the fragments, improves LDA dissociation curves of hydrogen chains and produces results that are comparable to those of spin-unrestricted LDA, but without breaking the spin symmetry. MFOA also induces a correction to the LDA electron density that partially captures the correct density dimerization in strongly correlated hydrogen chains. Moreover, with an additional correction to the partition energy that is specific to the one-dimensional LDA, the approximation is shown to produce dissociation energies in quantitative agreement with calculations based on the density matrix renormalization group method.

Split electrons in partition density functional theory.

Partition density functional theory is a density embedding method that partitions a molecule into fragments by minimizing the sum of fragment energies subject to a local density constraint and a global electron-number constraint. To perform this minimization, we study a two-stage procedure in which the sum of fragment energies is lowered when electrons flow from fragments of lower electronegativity to fragments of higher electronegativity. The global minimum is reached when all electronegativities are equal. The non-integer fragment populations are dealt with in two different ways: (1) An ensemble approach (ENS) that involves averaging over calculations with different numbers of electrons (always integers) and (2) a simpler approach that involves fractionally occupying orbitals (FOO). We compare and contrast these two approaches and examine their performance in some of the simplest systems where one can transparently apply both, including simple models of heteronuclear diatomic molecules and actual diatomic molecules with two and four electrons. We find that, although both ENS and FOO methods lead to the same total energy and density, the ENS fragment densities are less distorted than those of FOO when compared to their isolated counterparts, and they tend to retain integer numbers of electrons. We establish the conditions under which the ENS populations can become fractional and observe that, even in those cases, the total charge transferred is always lower in ENS than in FOO. Similarly, the FOO fragment dipole moments provide an upper bound to the ENS dipoles. We explain why and discuss the implications.

Using quantum annealers to calculate ground state properties of molecules.

Quantum annealers are an alternative approach to quantum computing, which make use of the adiabatic theorem to efficiently find the ground state of a physically realizable Hamiltonian. Such devices are currently commercially available and have been successfully applied to several combinatorial and discrete optimization problems. However, the application of quantum annealers to problems in chemistry remains a relatively sparse area of research due to the difficulty in mapping molecular systems to the Ising model Hamiltonian. In this paper, we review two different methods for finding the ground state of molecular Hamiltonians using Ising model-based quantum annealers. In addition, we compare the relative effectiveness of each method by calculating the binding energies, bond lengths, and bond angles of the H3 + and H2O molecules and mapping their potential energy curves. We also assess the resource requirements of each method by determining the number of qubits and computation time required to simulate each molecule using various parameter values. While each of these methods is capable of accurately predicting the ground state properties of small molecules, we find that they are still outperformed by modern classical algorithms and that the scaling of the resource requirements remains a challenge.

The Importance of Being Inconsistent.

We review the role of self-consistency in density functional theory (DFT). We apply a recent analysis to both Kohn-Sham and orbital-free DFT, as well as to partition DFT, which generalizes all aspects of standard DFT. In each case, the analysis distinguishes between errors in approximate functionals versus errors in the self-consistent density. This yields insights into the origins of many errors in DFT calculations, especially those often attributed to self-interaction or delocalization error. In many classes of problems, errors can be substantially reduced by using better densities. We review the history of these approaches, discuss many of their applications, and give simple pedagogical examples.

Non-additive non-interacting kinetic energy of rare gas dimers.

Approximations of the non-additive non-interacting kinetic energy (NAKE) as an explicit functional of the density are the basis of several electronic structure methods that provide improved computational efficiency over standard Kohn-Sham calculations. However, within most fragment-based formalisms, there is no unique exact NAKE, making it difficult to develop general, robust approximations for it. When adjustments are made to the embedding formalisms to guarantee uniqueness, approximate functionals may be more meaningfully compared to the exact unique NAKE. We use numerically accurate inversions to study the exact NAKE of several rare-gas dimers within partition density functional theory, a method that provides the uniqueness for the exact NAKE. We find that the NAKE decreases nearly exponentially with atomic separation for the rare-gas dimers. We compute the logarithmic derivative of the NAKE with respect to the bond length for our numerically accurate inversions as well as for several approximate NAKE functionals. We show that standard approximate NAKE functionals do not reproduce the correct behavior for this logarithmic derivative and propose two new NAKE functionals that do. The first of these is based on a re-parametrization of a conjoint Perdew-Burke-Ernzerhof (PBE) functional. The second is a simple, physically motivated non-decomposable NAKE functional that matches the asymptotic decay constant without fitting.

Constructing a non-additive non-interacting kinetic energy functional approximation for covalent bonds from exact conditions.

We present a non-decomposable approximation for the non-additive non-interacting kinetic energy (NAKE) for covalent bonds based on the exact behavior of the von Weizsäcker (vW) functional in regions dominated by one orbital. This covalent approximation (CA) seamlessly combines the vW and the Thomas-Fermi functional with a switching function of the fragment densities constructed to satisfy exact constraints. It also makes use of ensembles and fractionally occupied spin-orbitals to yield highly accurate NAKE for stretched bonds while outperforming other standard NAKE approximations near equilibrium bond lengths. We tested the CA within Partition-Density Functional Theory (P-DFT) and demonstrated its potential to enable fast and accurate P-DFT calculations.

Partition-DFT on the water dimer.

As is well known, the ground-state symmetry group of the water dimer switches from its equilibrium Cs-character to C2h-character as the distance between the two oxygen atoms of the dimer decreases below RO-O∼2.5 Å. For a range of RO-O between 1 and 5 Å, and for both symmetries, we apply Partition Density Functional Theory (PDFT) to find the unique monomer densities that sum to the correct dimer densities while minimizing the sum of the monomer energies. We calculate the work involved in deforming the isolated monomer densities and find that it is slightly larger for the Cs geometry for all RO-O. We discuss how the PDFT densities and the corresponding partition potentials support the orbital-interaction picture of hydrogen-bond formation.

Numerical density-to-potential inversions in time-dependent density functional theory.

We treat the density-to-potential inverse problem of time-dependent density functional theory as an optimization problem with a partial differential equation constraint. The unknown potential is recovered from a target density by applying a multilevel optimization method controlled by error estimates. We employ a classical optimization routine using gradients efficiently computed by the discrete adjoint method. The inverted potential has both a real and imaginary part to reduce reflections at the boundaries and other numerical artifacts. We demonstrate this method on model one-dimensional systems. The method can be straightforwardly extended to a variety of numerical solvers of the time-dependent Kohn-Sham equations and to systems in higher dimensions.

Ground-State Charge Transfer: Lithium-Benzene and the Role of Hartree-Fock Exchange.

Most approximations to the exchange-correlation functional of Kohn-Sham density functional theory lead to delocalization errors that undermine the description of charge-transfer phenomena. We explore how various approximate functionals and charge-distribution schemes describe ground-state atomic-charge distributions in the lithium-benzene complex, a model system of relevance to carbon-based supercapacitors. To understand the trends, we compare Hartree-Fock (HF) and correlated post-HF calculations, confirming that the HOMO-LUMO gap is narrower in semilocal functionals but widened by hybrid functionals with large fractions of HF exchange. For semilocal functionals, natural bond orbital (NBO) and Mulliken schemes yield opposite pictures of how charge transfer occurs. In PBE, for example, when lithium and benzene are <1.5 Å apart, NBO yields a positive charge on the lithium atom, but the Mulliken scheme yields a negative charge. Furthermore, the partial charges in conjugated materials depend on the interplay between the charge-distribution scheme employed and the underlying exchange-correlation functional, being critically sensitive to the admixture of HF exchange. We analyze and explain why this happens, discuss implications, and conclude that hybrid functionals with an admixture of about one-fourth of HF exchange are particularly useful in describing charge transfer in the lithium-benzene model.

Fragment-based treatment of delocalization and static correlation errors in density-functional theory.

One of the most important open challenges in modern Kohn-Sham (KS) density-functional theory (DFT) is the correct treatment of systems involving fractional electron charges and spins. Approximate exchange-correlation functionals struggle with such systems, leading to pervasive delocalization and static correlation errors. We demonstrate how these errors, which plague density-functional calculations of bond-stretching processes, can be avoided by employing the alternative framework of partition density-functional theory (PDFT) even using the local density approximation for the fragments. Our method is illustrated with explicit calculations on simple systems exhibiting delocalization and static-correlation errors, stretched H2 (+), H2, He2 (+), Li2 (+), and Li2. In all these cases, our method leads to greatly improved dissociation-energy curves. The effective KS potential corresponding to our self-consistent solutions displays key features around the bond midpoint; these are known to be present in the exact KS potential, but are absent from most approximate KS potentials and are essential for the correct description of electron dynamics.

Density-based partitioning methods for ground-state molecular calculations.

With the growing complexity of systems that can be treated with modern electronic-structure methods, it is critical to develop accurate and efficient strategies to partition the systems into smaller, more tractable fragments. We review some of the various recent formalisms that have been proposed to achieve this goal using fragment (ground-state) electron densities as the main variables, with an emphasis on partition density-functional theory (PDFT), which the authors have been developing. To expose the subtle but important differences between alternative approaches and to highlight the challenges involved with density partitioning, we focus on the simplest possible systems where the various methods can be transparently compared. We provide benchmark PDFT calculations on homonuclear diatomic molecules and analyze the associated partition potentials. We derive a new exact condition determining the strength of the singularities of the partition potentials at the nuclei, establish the connection between charge-transfer and electronegativity equalization between fragments, test different ways of dealing with fractional fragment charges and spins, and finally outline a general strategy for overcoming delocalization and static-correlation errors in density-functional calculations.

Current density partitioning in time-dependent current density functional theory.

We adapt time-dependent current density functional theory to allow for a fragment-based solution of the many-electron problem of molecules in the presence of time-dependent electric and magnetic fields. Regarding a molecule as a set of non-interacting subsystems that individually evolve under the influence of an auxiliary external electromagnetic vector-scalar potential pair, the partition 4-potential, we show that there are one-to-one mappings between this auxiliary potential, a sharply-defined set of fragment current densities, and the total current density of the system. The partition electromagnetic (EM) 4-potential is expressed in terms of the real EM 4-potential of the system and a gluing EM 4-potential that accounts for exchange-correlation effects and mutual interaction forces between fragments that are required to yield the correct electron dynamics. We prove the zero-force theorem for the fragmented system, establish a variational formulation in terms of action functionals, and provide a simple illustration for a charged particle in a ring.

More methemoglobin is produced by benzocaine treatment than lidocaine treatment in human in vitro systems.

The clinical use of local anesthetic products to anesthetize mucous membranes has been associated with methemoglobinemia (MetHba), a serious condition in which the blood has reduced capacity to carry oxygen. An evaluation of spontaneous adverse event reporting of MetHba submitted to FDA through 2013 identified 375 reports associated with benzocaine and 16 reports associated with lidocaine. The current study was performed to determine the relative ability of benzocaine and lidocaine to produce methemoglobin (MetHb) in vitro. Incubation of 500µM benzocaine with whole human blood and pooled human liver S9 over 5h resulted in MetHb levels equaling 39.8±1.2% of the total hemoglobin. No MetHb formation was detected for 500µM lidocaine under the same conditions. Because liver S9 does not readily form lidocaine hydrolytic metabolites based on xylidine, a primary metabolic pathway, 500µM xylidine was directly incubated with whole blood and S9. Under these conditions MetHb levels of 4.4±0.4% were reached by 5h. Studies with recombinant cytochrome P450 revealed benzocaine to be extensively metabolized by CYP 1A2, with 2B6, 2C19, 2D6, and 2E1 also having activity. We conclude that benzocaine produces much more MetHb in in vitro systems than lidocaine or xylidine and that benzocaine should be more likely to cause MetHba in vivo as well.

Stretching Bonds without Breaking Symmetries in Density Functional Theory.

Kohn-Sham density functional theory (KS-DFT) stands out among electronic structure methods due to its balance of accuracy and computational efficiency. However, to achieve chemically accurate energies, standard density functional approximations in KS-DFT often need to break underlying symmetries, a long-standing "symmetry dilemma". By employing fragment spin densities as the main variables in calculations (rather than total molecular densities, as in KS-DFT), we present an embedding framework in which this symmetry dilemma is understood and partially resolved. The spatial overlap between fragment densities is used as the main ingredient to construct a simple, physically motivated approximation to a universal functional of the fragment densities. This "overlap approximation" is shown to significantly improve semilocal KS-DFT binding energies of molecules without artificially breaking either charge or spin symmetries. The approach is shown to be applicable to covalently bonded molecules and to systems of the "strongly correlated" type.

Fragment-based time-dependent density functional theory.

Using the Runge-Gross theorem that establishes the foundation of time-dependent density functional theory, we prove that for a given electronic Hamiltonian, choice of initial state, and choice of fragmentation, there is a unique single-particle potential (dubbed time-dependent partition potential) which, when added to each of the preselected fragment potentials, forces the fragment densities to evolve in such a way that their sum equals the exact molecular density at all times. This uniqueness theorem suggests new ways of computing the time-dependent properties of electronic systems via fragment-time-dependent density functional theory calculations. We derive a formally exact relationship between the partition potential and the total density, and illustrate our approach on a simple model system for binary fragmentation in a laser field.

π donation and its effects on the excited-state lifetimes of luminescent platinum(II) terpyridine complexes in solution.

Introducing electron-donating groups extends the excited-state lifetimes of platinum(II)-terpyridine complexes in fluid solution. Such systems are of interest for a variety of applications, viz., as DNA-binding agents or as components in luminescence-based devices, especially sensors. The complexes investigated here are of the form [Pt(4'-X-T)Y](+), where 4'-X-T denotes a 4'-substituted 2,2':6',2″-terpyridine ligand and Y denotes the coligand. The π-donating abilities of -X and -Y increase systematically in the orders -NHMe < -NMe2 < -(pyrrolidin-1-yl) and -CN < -Cl < -CCPh, respectively. The results presented include crystal structures of two new 4'-NHMe-T complexes of platinum, as well as absorption, emission, and excited-state lifetime data for nine complexes. Excited-state lifetimes obtained in deoxygenated dichloromethane vary by a factor of 100, ranging from 24 µs for [Pt(4'-pyrr-T)CN](+) to 0.24 µs for [Pt(4'-ma-T)Cl](+), where ma-T denotes 4'-(methylamino)-2,2':6',2″-terpyridine and pyrr-T denotes 4'-(pyrrolidin-1-yl)-2,2':6',2″-terpyridine. Analysis of experimental and computational results shows that introducing a simple amine group on the terpyridine and/or a π-donating coligand engenders the emitting state with intraligand charge-transfer (ILCT) and/or ligand-ligand charge-transfer (LLCT) character. The excited-state lifetime increases when the change in orbital parentage lowers the emission energy, suppresses quenching via d-d states, and encourages delocalization of the excitation onto the ligand(s). At some point, however, the energy is low enough that direct vibronic coupling to the ground-state surface becomes important, and the lifetime begins to decrease again.

Fragment occupations in partition Density Functional Theory.

The exact ground-state energy and density of a molecule can in principle be obtained via Partition Density Functional Theory (PDFT), a method for calculating molecular properties from Kohn-Sham calculations on isolated fragments. For a given choice of fragmentation, unique fragment densities are found by requiring that the sum of fragment energies be minimized subject to the constraint that the fragment densities sum to the correct molecular ground-state density. We investigate two interrelated aspects of PDFT: the connections between fragment densities obtained via different choices of fragmentation, for which we find "near-additivity", and the nature of their corresponding fragment occupations. Whereas near-integer occupations arise for very large inter-fragment separations, strictly integer occupations appear for small inter-fragment separations. Cases where the fragment chemical potentials cannot be equalized lead to fragment occupations that lock into integers.

Density functional resonance theory: complex density functions, convergence, orbital energies, and functionals.

Aspects of density functional resonance theory (DFRT) [D. L. Whitenack and A. Wasserman, Phys. Rev. Lett. 107, 163002 (2011)], a recently developed complex-scaled version of ground-state density functional theory (DFT), are studied in detail. The asymptotic behavior of the complex density function is related to the complex resonance energy and system's threshold energy, and the function's local oscillatory behavior is connected with preferential directions of electron decay. Practical considerations for implementation of the theory are addressed including sensitivity to the complex-scaling parameter, θ. In Kohn-Sham DFRT, it is shown that almost all θ-dependence in the calculated energies and lifetimes can be extinguished via use of a proper basis set or fine grid. The highest occupied Kohn-Sham orbital energy and lifetime are related to physical affinity and width, and the threshold energy of the Kohn-Sham system is shown to be equal to the threshold energy of the interacting system shifted by a well-defined functional. Finally, various complex-scaling conditions are derived which relate the functionals of ground-state DFT to those of DFRT via proper scaling factors and a non-Hermitian coupling-constant system.

Density functional resonance theory of unbound electronic systems.

Density functional resonance theory (DFRT) is a complex-scaled version of ground-state density functional theory (DFT) that allows one to calculate the in-principle exact resonance energies and lifetimes of metastable anions. In this formalism, the energy and lifetime of the lowest-energy resonance of unbound systems is encoded into a complex "density" that can be obtained via complex-coordinate scaling. This complex density is used as the primary variable in a DFRT calculation, just as the ground-state density would be used as the primary variable in DFT. As in DFT, there exists a mapping of the N-electron interacting system to a Kohn-Sham system of N noninteracting particles. This mapping facilitates self-consistent calculations with an initial guess for the complex density, as illustrated with an exactly solvable model system.

Molecular binding energies from partition density functional theory.

Approximate molecular calculations via standard Kohn-Sham density functional theory are exactly reproduced by performing self-consistent calculations on isolated fragments via partition density functional theory [P. Elliott, K. Burke, M. H. Cohen, and A. Wasserman, Phys. Rev. A 82, 024501 (2010)]. We illustrate this with the binding curves of small diatomic molecules. We find that partition energies are in all cases qualitatively similar and numerically close to actual binding energies. We discuss qualitative features of the associated partition potentials.