Search

Standard approximations for the exchange-correlation (XC) 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 the \textit{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 generalized overlap approximation (GOA), for the partition energy in PDFT. Our method is illustrated on simple models of one-dimensional strongly-correlated linear hydrogen chains. The GOA, when used in combination with the LDA for the fragments, improves the LDA dissociation curves of hydrogen chains and produces results that are comparable to those of spin-unrestricted LDA, but without breaking the spin symmetry. GOA 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, the approximation is shown to produce dissociation energies in quantitative agreement to calculations based on the Density Matrix Renormalization Group method.

Accurate first-principles calculations for the energies, charge distributions, and spin symmetries of many-electron systems are essential to understand and predict the electronic and structural properties of molecules and materials. Kohn-Sham density functional theory (KS-DFT) stands out among electronic-structure methods due to its balance of accuracy and computational efficiency. It is now extensively used in fields ranging from materials engineering to rational drug design. 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 resolved for the case of stretched molecules. 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 semi-local KS-DFT binding energies of molecules without artificial symmetry breaking.

We explore a variety of unsolved problems in density functional theory, where mathematicians might prove useful. We give the background and context of the different problems, and why progress toward resolving them would help those doing computations using density functional theory. Subjects covered include the magnitude of the kinetic energy in Hartree-Fock calculations, the shape of adiabatic connection curves, using the constrained search with input densities, densities of states, the semiclassical expansion of energies, the tightness of Lieb-Oxford bounds, and how we decide the accuracy of an approximate density., Comment: 21 pages, 6 figures

Partition Density Functional Theory (P-DFT) 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-integral 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 2 and 4 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 implications., Comment: 11 pages, 12 figures

Inverse Kohn-Sham (iKS) problems are needed to fully understand the one-to-one mapping between densities and potentials on which Density Functional Theory is based. They are also important to advance computational schemes that rely on density-to-potential inversions such as the Optimized Effective Potential method and various techniques for density-based embedding. Unlike the forward Kohn-Sham problems, numerical iKS problems are ill-posed and can be unstable. We discuss some of the fundamental and practical difficulties of iKS problems with constrained-optimization methods on finite basis sets. Various factors that affect the performance are systematically compared and discussed, both analytically and numerically, with a focus on two of the most practical methods: the Wu-Yang method (WY) and partial-differential-equation constrained-optimization (PDE-CO). Our analysis of the WY and PDE-CO highlights the limitation of finite basis sets and the importance of regularization. We introduce two new ideas that will hopefully contribute to making iKS problems more tractable: (1) A correction to the WY method that utilizes the null space of the relevant Hessian matrices; and (2) A finite potential basis-set implementation of the PDE-CO method. We provide an overall strategy for performing numerical density-to-potential inversions that can be directly adopted in practice. We also provide an Appendix with several examples that can be used for benchmarking.

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 H+3and 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 accuracy of charge-transfer excitation energies, solvatochromic shifts and other environmental effects calculated via various density embedding techniques depend critically on the approximations employed for the non-additive non-interacting kinetic energy functional, $T_{\scriptscriptstyle\rm s}^{\scriptscriptstyle\rm nad}[n]$. Approximating this functional remains an important challenge in electronic structure theory. To assist in the development and testing of approximations for $T_{\scriptscriptstyle\rm s}^{\scriptscriptstyle\rm nad}[n]$, we derive two virial relations for fragments in molecules. These establish separate connections between the non-additive kinetic energies of the non-interacting and interacting systems of electrons, and quantities such as the electron-nuclear attraction forces, the partition (or embedding) energy and potential, and the Kohn-Sham potentials of the system and its parts. We numerically verify both relations on diatomic molecules.

A truly isolated atom always has an integer number of electrons. If placed in contact with a far-away metallic reservoir, a {\em range} of metallic chemical potentials $\mu$ will lead to an identical number of electrons, $N$, on the atom. We formulate a density embedding method in which the range of $\mu$ leading to integer $N$ decreases due to finite-distance interactions between the metal and the atom. The typical $N(\mu)$ staircase function is smoothed out due to these finite-distance interactions, resembling finite-temperature effects. Fractional occupations on the atom occur only for sharply-defined $\mu$'s. We illustrate the new method with the simplest model system designed to mimic an atom near a metal surface. Because calculating fractional charges is important in various fields, from electrolysis to catalysis, solar cells and organic electronics, we anticipate several potential uses of the proposed approach.

We find the numerically exact partition potential for 1-D systems of interacting electrons designed to model diatomic molecules. At integer fragment occupations, the kinetic contribution to the partition potential develops sharp features in the internuclear region that nearly cancel corresponding features of exchange-correlation. They occur at locations that coincide with those of well-known features of the underlying molecular Kohn-Sham potential. For non-integer fragment occupations, we demonstrate that the fragment Kohn-Sham gaps determine the kinetic part of the partition potential. Our results highlight the importance of non-additive noninteracting kinetic and exchange-correlation energy approximations in density-embedding methods at large internuclear separations and the importance of non-additive noninteracting kinetic energy approximations at all separations.

The ground-state energy and density of four low-energy conformations of the formic acid dimer were calculated via Partition Density Functional Theory (PDFT). The differences between isolated and PDFT monomer densities display similar deformation patterns for primary and secondary hydrogen bonds among all four dimers. In contrast, the partition potential shows no transferable features in the bonding regions. These observations highlight the global character of the partition potential and the cooperative effect that occurs when a dimer is bound via more than one hydrogen bond. We also provide numerical confirmation of the intuitive (but unproven) observation that fragment deformation energies are larger for systems with larger binding energies.

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 Weizs\"{a}cker (vW) functional in regions dominated by one orbital. This covalent approximation (CA) seamlessly combines the vW and the Thomas-Fermi (TF) 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.

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 PBE functional. The second is a simple, physically-motivated non-decomposable NAKE functional that matches the asymptotic decay constant without fitting.

The inverse problem of Kohn-Sham density functional theory (DFT) is often solved in an effort to benchmark and design approximate exchange-correlation potentials. The forward and inverse problems of DFT rely on the same equations but the numerical methods for solving each problem are substantially different. We examine both problems in this tutorial with a special emphasis on the algorithms and error analysis needed for solving the inverse problem. Two inversion methods based on partial differential equation constrained optimization and constrained variational ideas are introduced. We compare and contrast several different inversion methods applied to one-dimensional finite and periodic model systems., Comment: 62 pages, 22 figures

We review the role of self-consistency in density functional theory. 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, many of their applications, and give simple pedagogical examples., Comment: submitted for publication to the Annual Review of Physical Chemistry

As is well known, the ground-state symmetry group of the water dimer switches from its equilibrium $C_{s}$-character to $C_{2h}$-character as the distance between the two oxygen atoms of the dimer decreases below $R_{\rm O-O}\sim 2.5$ \AA{}. For a range of $R_{\rm O-O}$ between 1 and 5 \AA{}, 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 $C_s$ geometry for all $R_{\rm O-O}$. We discuss how the PDFT densities and the corresponding partition potentials support the orbital-interaction picture of hydrogen-bond formation.

The non-additive non-interacting kinetic energy is calculated exactly for fragments of H$_2$, Li$_2$, Be$_2$, C$_2$, N$_2$, F$_2$, and Na$_2$ within partition density-functional theory. The resulting fragments are uniquely determined and their sum reproduces the Kohn-Sham molecular density of the corresponding XC functional. We compare the use of fractional orbital occupation to the usual PDFT ensemble method for treating the fragment energies and densities. We also compare Thomas-Fermi and von Weiz{\"a}cker approximate kinetic energy functionals to the numerically exact solution and find significant regions where the von Weiz{\"a}cker solution is nearly exact.

Conceiving a molecule as composed of smaller molecular fragments, or subunits, is one of the pillars of the chemical and physical sciences, and leads to productive methods in quantum chemistry. Using a fragmentation scheme, efficient algorithms can be proposed to address problems in the description of chemical bond formation and breaking. We present a formally exact time-dependent density-functional theory for the electronic dynamics of molecular fragments with variable number of electrons. This new formalism is an extension of previous work [Phys. Rev. Lett. {\bf 111}, 023001 (2013)]. We also introduce a stable density-inversion method that is applicable to time-dependent and ground-state density-functional theory, and their extensions, including those discussed in this work., Comment: 9 pages, 3 figures

Density functional approximations to the exchange-correlation energy of Kohn-Sham theory, such as the local density approximation and generalized gradient approximations, lack the well-known integer discontinuity, a feature that is critical to describe molecular dissociation correctly. Moreover, standard approximations to the exchange-correlation energy also fail to yield the correct linear dependence of the ground-state energy on the number of electrons when this is a non-integer number obtained from the grand canonical ensemble statistics. We present a formal framework to restore the integer discontinuity of any density functional approximation. Our formalism derives from a formula for the exact energy functional and a new constrained search functional that recovers the linear dependence of the energy on the number of electrons., Comment: 5 pages, 2 figures

We show that the energetics and lifetimes of resonances of finite systems under an external electric field can be captured by Kohn--Sham density functional theory (DFT) within the formalism of uniform complex scaling. Properties of resonances are calculated self-consistently in terms of complex densities, potentials and wavefunctions using adapted versions of the known algorithms from DFT. We illustrate this new formalism by calculating ionization rates using the complex-scaled local density approximation and exact exchange. We consider a variety of atoms (H, He, Li and Be) as well as the hydrogen molecule. Extensions are briefly discussed., Comment: 5 pages, 5 figures. This document is the unedited Author's version of a Submitted Work that was subsequently accepted for publication in J.Phys.Chem.Lett., copyright (c) American Chemical Society after peer review. To access the final edited and published work see http://pubs.acs.org/doi/abs/10.1021/jz401110h

One of the most important open challenges in modern Kohn-Sham (KS) density-functional theory (DFT) is the correct treatment of fractional electron charges and spins. Approximate exchange-correlation (XC) functionals struggle to do this in a systematic way, 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 with simple local and semi-local functionals for the fragments. Our method is illustrated with explicit calculations on the two paradigm systems exhibiting delocalization and static-correlation, stretched H$_2^+$ and H$_2$. We find in both cases our scheme leads to dissociation-energy errors of less than 3%. The effective KS potential corresponding to our self-consistent solutions display 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.

Using the Runge-Gross theorem that establishes the foundation of Time-dependent Density Functional Theory (TDDFT) 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 pre-selected 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 time-dependent properties of electronic systems via fragment-TDDFT 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., Comment: 5 pages, 2 figures

This paper has been withdrawn by the authors; the main conclusion is incorrect, as some of the crucial calculations were not properly converged., Comment: This paper has been withdrawn

Aspects of Density Functional Resonance Theory (DFRT) [Phys. Rev. Lett. \textbf{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, $\theta$. In Kohn-Sham DFRT, it is shown that almost all $\theta$-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 a 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., Comment: 20 pages, 3 figures

The derivative discontinuity in the exact exchange-correlation potential of ensemble Density Functional Theory (DFT) is investigated at the specific integer number that corresponds to the maximum number of bound electrons, $J_{max}$. A recently developed complex-scaled analog of DFT is extended to fractional particle numbers and used to study ensembles of both bound and metastable states. It is found that the exact exchange-correlation potential experiences discontinuous jumps at integer particle numbers including $J_{max}$. For integers below $J_{max}$ the jump is purely real because of the real shift in the chemical potential. At $J_{max}$, the jump has a non-zero imaginary component reflecting the finite lifetime of the $(J_{max}+1)$ state., Comment: 8 pages, 6 figures

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 [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.

Density Functional Resonance Theory (DFRT) is a complex-scaled version of ground-state Density Functional Theory (DFT) that allows one to calculate the resonance energies and lifetimes of metastable anions. In this formalism, the exact 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 non-interacting particles in DFRT. This mapping facilitates self consistent calculations with an initial guess for the complex density, as illustrated with an exactly-solvable model system. Whereas DFRT yields in principle the exact resonance energy and lifetime of the interacting system, we find that neglecting the complex-correlation contribution leads to errors of similar magnitude to those of standard scattering close-coupling calculations under the bound-state approximation., Comment: 5 pages, 4 figures

The ab-initio calculation of resonance lifetimes of metastable anions challenges modern quantum-chemical methods. The exact lifetime of the lowest-energy resonance is encoded into a complex "density" that can be obtained via complex-coordinate scaling. We illustrate this with one-electron examples and show how the lifetime can be extracted from the complex density in much the same way as the ground-state energy of bound systems is extracted from its ground-state density.

Partition density functional theory is a formally exact procedure for calculating molecular properties from Kohn-Sham calculations on isolated fragments, interacting via a global partition potential that is a functional of the fragment densities. An example is given and consequences discussed., Comment: Revised version. 5 pages, 3 figures

The recently proposed Partition Theory (PT) [J.Phys.Chem.A 111, 2229 (2007)] is illustrated on a simple one-dimensional model of a heteronuclear diatomic molecule. It is shown that a sharp definition for the charge of molecular fragments emerges from PT, and that the ensuing population analysis can be used to study how charge redistributes during dissociation and the implications of that redistribution for the dipole moment. Interpreting small differences between the isolated parts' ionization potentials as due to environmental inhomogeneities, we gain insight into how electron localization takes place in H2+ as the molecule dissociates. Furthermore, by studying the preservation of the shapes of the parts as different parameters of the model are varied, we address the issue of transferability of the parts. We find good transferability within the chemically meaningful parameter regime, raising hopes that PT will prove useful in chemical applications., Comment: 12 pages, 16 figures

Systems whose underlying classical dynamics are chaotic exhibit signatures of the chaos in their quantum mechanics. We investigate the possibility of using time-dependent density functional theory (TDDFT) to study the case when chaos is induced by electron-interaction alone. Nearest-neighbour level-spacing statistics are in principle exactly and directly accessible from TDDFT. We discuss how the TDDFT linear response procedure can reveal the mechanism of chaos induced by electron-interaction alone. A simple model of a two-electron quantum dot highlights the necessity to go beyond the adiabatic approximation in TDDFT., Comment: 8 pages, 4 figures

We illustrate the main features of a recently proposed method based on ensemble density functional theory to divide rigorously a complex molecular system into its parts [M.H. Cohen and A. Wasserman, J. Phys. Chem. A 111, 2229 (2007)]. The illustrative system is an analog of the hydrogen molecule for which analytic expressions for the densities of the parts (hydrogen "atoms") are found along with the "reactivity potential" that enters the theory. While previous formulations of Chemical Reactivity Theory lead to zero, or undefined, values for the chemical hardness of the isolated parts, we demonstrate they can acquire a finite and positive hardness within the present formulation., Comment: 8 pages, 8 figures

Phase shifts for single-channel elastic electron-atom scattering are derived from time-dependent density functional theory. The H$^-$ ion is placed in a spherical box, its discrete spectrum found, and phase shifts deduced. Exact-exchange yields an excellent approximation to the ground-state Kohn-Sham potential, while the adiabatic local density approximation yields good singlet and triplet phase shifts., Comment: 5 pages, 4 figures, 1 table

We show that under well-defined conditions the Hohenberg-Kohn theorem (HKT) can be extended to the lowest-energy resonance of unbound systems. Using the Gel'fand Levitan theorem, the extended version of the HKT can also be applied to systems that support a finite number of bound states. The extended version of the HKT provides an adequate framework to carry out DFT calculations of negative electron affinities., Comment: 4 pages, 3 figures

A method is given that extracts accurate Rydberg excitations from LDA density functional calculations, despite the short-ranged potential. For the case of He and Ne, the asymptotic quantum defects predicted by LDA are in less than 5% error, yielding transition frequency errors of less than 0.1eV., Comment: 4 pages, 6 figures, submitted to Phys. Rev. Lett

Linear response time-dependent density functional theory is used to study low-lying electronic continuum states of targets that can bind an extra electron. Exact formulas to extract scattering amplitudes from the susceptibility are derived in one dimension. A single-pole approximation for scattering phase shifts in three dimensions is shown to be more accurate than static exchange for singlet electron-He$^+$ scattering., Comment: 5 pages, 2 figures, J. Chem. Phys. accepted

To construct an N-representable time-dependent density-functional theory, a generalization to the time domain of the Levy-Lieb (LL) constrained search algorithm is required. That the action is only stationary in the Dirac-Frenkel variational principle eliminates the possibility of basing the search on the action itself. Instead, we use the norm of the partial functional derivative of the action in the Hilbert space of the wave functions in place of the energy of the LL search. The electron densities entering the formalism are $N$-representable, and the resulting universal action functional has a unique stationary point in the density at that corresponding to the solution of the Schr\"{o}dinger equation. The original Runge-Gross (RG) formulation is subsumed within the new formalism. Concerns in the literature about the meaning of the functional derivatives and the internal consistency of the RG formulation are allayed by clarifying the nature of the functional derivatives entering the formalism., Comment: 9 pages, 0 figures, Phys. Rev. A accepted. Introduction was expanded, subsections reorganized, appendix and new references added