Revisiting Molecular Dissociation in Density Functional Theory: A Simple Model

Calculating Potential Energy Surfaces with Quantum Computers by Measuring Only the Density Along Adiabatic Transitions

We show that chemically accurate potential energy surfaces (PESs) can be generated from quantum computers by measuring only the density along an adiabatic transition between different molecular geometries. In lieu of using phase estimation, the energy is evaluated by performing line-integration using the inverted real-space time-dependent density functional theory Kohn-Sham (KS) potential obtained from the geometry-varying densities of the full wave function. The accuracy of this method depends on the validity of the adiabatic evolution itself and the potential inversion process (which is theoretically exact but can be numerically unstable), whereas the total evolution time is the defining factor for the precision of phase estimation. We examine the method with a one-dimensional system of two electrons for both the ground and first triplet states in first quantization, as well as the ground state of three- and four-electron systems in second quantization. It is shown that few accurate measurements can be utilized to obtain chemical accuracy across the full potential energy curve, with a shorter propagation time than may be required using phase estimation for a similar accuracy. We also show that an accurate potential energy curve can be calculated by making many imprecise density measurements (using a few shots) along the time evolution and smoothing the resulting density evolution. Finally, it is important to note that the method is able to classically provide a check of its own accuracy by comparing the density resulting from a time-independent KS calculation using the inverted potential with the measured density. This can be used to determine whether longer adiabatic evolution times are required to satisfy the adiabatic theorem.

When Is a Bond Broken? The Polarizability Perspective

The question of when a chemical bond can be said to be broken is of fundamental chemical interest but has not been widely studied. Herein we propose that the maxima of static polarizability along bond dissociation coordinates naturally define cutoff points for bond rupture, as they represent the onset of localization of shared electron density into constituent fragments. Examples of computed polarizability maxima over the course of bond cleavage in main-group and transition metal compounds are provided, across covalent, dative and charge-shift bonds. The behavior along reaction paths is also considered. Overall, the static polarizability is found to be a sensitive reporter of electronic structure reorganization associated with bond stretching, and thus can serve as a metric for describing bond cleavage (or diagnose the absence of a chemical bond).

Strengths and limitations of the adiabatic exact-exchange kernel for total energy calculations

We investigate the adiabatic approximation to the exact-exchange kernel for calculating correlation energies within the adiabatic-connection fluctuation-dissipation framework of time-dependent density functional theory. A numerical study is performed on a set of systems having bonds of different character (H2 and N2 molecules, H-chain, H2-dimer, solid-Ar, and the H2O-dimer). We find that the adiabatic kernel can be sufficient in strongly bound covalent systems, yielding similar bond lengths and binding energies. However, for non-covalent systems, the adiabatic kernel introduces significant errors around equilibrium geometry, systematically overestimating the interaction energy. The origin of this behavior is investigated by studying a model dimer composed of one-dimensional, closed-shell atoms, interacting via soft-Coulomb potentials. The kernel is shown to exhibit a strong frequency dependence at small to intermediate atomic separation that affects both the low-energy spectrum and the exchange-correlation hole obtained from the corresponding diagonal of the two-particle density matrix.

Secondary Kinetic Peak in the Kohn-Sham Potential and Its Connection to the Response Step

We consider a prototypical 1D model Hamiltonian for a stretched heteronuclear molecule and construct individual components of the corresponding KS potential, namely, the kinetic, the N - 1, and the conditional potentials. These components show very special features, such as peaks and steps, in regions where the density is drastically low. Some of these features are quite well-known, whereas others, such as a secondary peak in the kinetic potential or a second bump in the conditional potential, are less or not known at all. We discuss these features building on the analytical model treated in Giarrusso et al. J. Chem. Theory Comput. 2018, 14, 4151. In particular, we provide an explanation for the underlying mechanism which determines the appearance of both peaks in the kinetic potential and elucidate why these peaks delineate the region over which the plateau structure, due to the N - 1 potential, stretches. We assess the validity of the Heitler-London Ansatz at large but finite internuclear distance, showing that, if optimal orbitals are used, this model is an excellent approximation to the exact wave function. Notably, we find that the second natural orbital presents an extra node very far out on the side of the more electronegative atom.

Numerically stable inversion approach to construct Kohn-Sham potentials for given electron densities within a Gaussian basis set framework

We present a Kohn-Sham (KS) inversion approach to construct KS exchange-correlation potentials corresponding to given electron densities. This method is based on an iterative procedure using linear response to update potentials. All involved quantities, i.e., orbitals, potentials, and response functions, are represented by Gaussian basis functions. In contrast to previous KS inversion methods relying on Gaussian basis sets, the method presented here is numerically stable even for standard basis sets from basis set libraries due to a preprocessing of the auxiliary basis used to represent an exchange-correlation charge density that generates the exchange-correlation potential. The new KS inversion method is applied to reference densities of various atoms and molecules obtained by full configuration interaction or CCSD(T) (coupled cluster singles doubles perturbative triples). The considered examples encompass cases known to be difficult, such as stretched hydrogen or lithium hydride molecules or the beryllium isoelectronic series. For the stretched hydrogen molecule, potentials of benchmark quality are obtained by employing large basis sets. For the carbon monoxide molecule, we show that the correlation potential from the random phase approximation (RPA) is in excellent qualitative and quantitative agreement with the correlation potential from the KS inversion of a CCSD(T) reference density. This indicates that RPA correlation potentials, in contrast to those from semi-local density-functionals, resemble the exact correlation potential. Besides providing exchange-correlation potentials for benchmark purposes, the proposed KS inversion method may be used in density-partition-based quantum embedding and in subsystem density-functional methods because it combines numerical stability with computational efficiency.

Chemical potential, derivative discontinuity, fractional electrons, jump of the Kohn-Sham potential, atoms as thermodynamic open systems, and other (mis)conceptions of the density functional theory of electrons in molecules

Many references exist in the density functional theory (DFT) literature to the chemical potential of the electrons in an atom or a molecule. The origin of this notion has been the identification of the Lagrange multiplier μ = ∂E/∂N in the Euler-Lagrange variational equation for the ground state density as the chemical potential of the electrons. We first discuss why the Lagrange multiplier in this case is an arbitrary constant and therefore cannot be a physical characteristic of an atom or molecule. The switching of the energy derivative ("chemical potential") from -I to -A when the electron number crosses the integer, called integer discontinuity or derivative discontinuity, is not physical but only occurs when the nonphysical noninteger electron systems and the corresponding energy and derivative ∂E/∂N are chosen in a specific discontinuous way. The question is discussed whether in fact the thermodynamical concept of a chemical potential can be defined for the electrons in such few-electron systems as atoms and molecules. The conclusion is that such systems lack important characteristics of thermodynamic systems and do not afford the definition of a chemical potential. They also cannot be considered as analogues of the open systems of thermodynamics that can exchange particles with an environment (a particle bath or other members of a Gibbsian ensemble). Thermodynamical (statistical mechanical) concepts like chemical potential, open systems, grand canonical ensemble etc. are not applicable to a few electron system like an atom or molecule. A number of topics in DFT are critically reviewed in light of these findings: jumps in the Kohn-Sham potential when crossing an integer number of electrons, the band gap problem, the deviation-from-straight-lines error, and the role of ensembles in DFT.

Double and Charge-Transfer Excitations in Time-Dependent Density Functional Theory

Time-dependent density functional theory has emerged as a method of choice for calculations of spectra and response properties in physics, chemistry, and biology, with its system-size scaling enabling computations on systems much larger than otherwise possible. While increasingly complex and interesting systems have been successfully tackled with relatively simple functional approximations, there has also been increasing awareness that these functionals tend to fail for certain classes of approximations. Here I review the fundamental challenges the approximate functionals have in describing double excitations and charge-transfer excitations, which are two of the most common impediments for the theory to be applied in a black-box way. At the same time, I describe the progress made in recent decades in developing functional approximations that give useful predictions for these excitations. Expected final online publication date for the Annual Review of Physical Chemistry, Volume 73 is April 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

From Kohn-Sham to Many-Electron Energies via Step Structures in the Exchange-Correlation Potential

Accurately describing excited states within Kohn–Sham (KS) density functional theory (DFT), particularly those which induce ionization and charge transfer, remains a great challenge. Common exchange-correlation (xc) approximations are unreliable for excited states owing, in part, to the absence of a derivative discontinuity in the xc energy (Δ), which relates a many-electron energy difference to the corresponding KS energy difference. We demonstrate, analytically and numerically, how the relationship between KS and many-electron energies leads to the step structures observed in the exact xc potential in four scenarios: electron addition, molecular dissociation, excitation of a finite system, and charge transfer. We further show that steps in the potential can be obtained also with common xc approximations, as simple as the LDA, when addressed from the ensemble perspective. The article therefore highlights how capturing the relationship between KS and many-electron energies with advanced xc approximations is crucial for accurately calculating excitations, as well as the ground-state density and energy of systems which consist of distinct subsystems.

Exchange-Correlation Energy Densities and Response Potentials: Connection between Two Definitions and Analytical Model for the Strong-Coupling Limit of a Stretched Bond

We analyze in depth two widely used definitions (from the theory of conditional probability amplitudes and from the adiabatic connection formalism) of the exchange-correlation energy density and of the response potential of Kohn–Sham density functional theory. We introduce a local form of the coupling-constant-dependent Hohenberg–Kohn functional, showing that the difference between the two definitions is due to a corresponding local first-order term in the coupling constant, which disappears globally (when integrated over all space), but not locally. We also design an analytic representation for the response potential in the strong-coupling limit of density functional theory for a model single stretched bond.

Kinetic Correlation Functionals from the Entropic Regularization of the Strictly Correlated Electrons Problem

In this work, we study the entropic regularization of the strictly correlated electrons formalism, discussing the implications for density functional theory and establishing a link with earlier works on quantum kinetic energy and classical entropy. We carry out a very preliminary investigation (using simplified models) on the use of the solution of the entropic regularized problem to build approximations for the kinetic correlation functional at large coupling strengths. We also analyze lower and upper bounds to the Hohenberg-Kohn functional using the entropic regularized strictly correlated electrons problem.

Investigation of the Exchange-Correlation Potentials of Functionals Based on the Adiabatic Connection Interpolation

We have studied the correlation potentials produced by various adiabatic connection models (ACMs) for several atoms and molecules. The results have been compared to accurate reference potentials (coupled cluster and quantum Monte Carlo results) as well as to state-of-the-art ab initio DFT approaches. We have found that all the ACMs yield correlation potentials that exhibit a correct behavior, quite resembling scaled second-order Görling-Levy (GL2) potentials and including most of the physically meaningful features of the accurate reference data. The behavior and contribution of the strong-interaction limit potentials have also been investigated and discussed.

Asymptotic behavior of the Hartree-exchange and correlation potentials in ensemble density functional theory

We show that the Hartree-exchange and correlation potentials of ensemble systems can have unexpected features, including non-vanishing asymptotic constants and non-trivial screening of electrons. These features are demonstrated here on Li, C, and F.

Local and global interpolations along the adiabatic connection of DFT: a study at different correlation regimes

Interpolating the exchange-correlation energy along the density-fixed adiabatic connection of density functional theory is a promising way to build approximations that are not biased toward the weakly correlated regime. These interpolations can be performed at the global (integrated over all spaces) or at the local level, using energy densities. Many features of the relevant energy densities as well as several different ways to construct these interpolations, including comparisons between global and local variants, are investigated here for the analytically solvable Hooke's atom series, which allows for an exploration of different correlation regimes. We also analyze different ways to define the correlation kinetic energy density, focusing on the peak in the kinetic correlation potential.

One-particle Green's function of interacting two electrons using analytic solutions for a three-body problem: comparison with exact Kohn-Sham system

For a three-electron system with finite-strength interactions confined to a one-dimensional harmonic trap, we solve the Schrödinger equation analytically to obtain the exact solutions, from which we construct explicitly the simultaneous eigenstates of the energy and total spin for the first time. The solutions for the three-electron system allow us to derive analytic expressions for the exact one-particle Green's function (GF) for the corresponding two-electron system. We calculate the GF in frequency domain to examine systematically its behavior depending on the electronic interactions. We also compare the pole structure of non-interacting GF using the exact Kohn-Sham (KS) potential with that of the exact GF to find that the discrepancy of the energy gap between the KS system and the original system is larger for a stronger interaction. We perform numerical examination on the behavior of GFs in real space to demonstrate that the exact and KS GFs can have shapes quite different from each other. Our simple model will help to understand generic characteristics of interacting GFs.

Response Potential in the Strong-Interaction Limit of Density Functional Theory: Analysis and Comparison with the Coupling-Constant Average

Using the formalism of the conditional amplitude, we study the response part of the exchange-correlation potential in the strong-coupling limit of density functional theory, analyzing its peculiar features and comparing it with the response potential averaged over the coupling constant for small atoms and for the hydrogen molecule. We also use a simple one-dimensional model of a stretched heteronuclear molecule to derive exact properties of the response potential in the strong-coupling limit. The simplicity of the model allows us to unveil relevant features also of the exact Kohn-Sham potential and its different components, namely the appearance of a second peak in the correlation kinetic potential on the side of the most electronegative atom.

Exact Time-Dependent Exchange-Correlation Potential in Electron Scattering Processes

We identify peak and valley structures in the exact exchange-correlation potential of time-dependent density functional theory that are crucial for time-resolved electron scattering in a model one-dimensional system. These structures are completely missed by adiabatic approximations that, consequently, significantly underestimate the scattering probability. A recently proposed nonadiabatic approximation is shown to correctly capture the approach of the electron to the target when the initial Kohn-Sham state is chosen judiciously, and it is more accurate than standard adiabatic functionals but ultimately fails to accurately capture reflection. These results may explain the underestimation of scattering probabilities in some recent studies on molecules and surfaces.

How Interatomic Steps in the Exact Kohn-Sham Potential Relate to Derivative Discontinuities of the Energy

Accurate density functional calculations hinge on reliable approximations to the unknown exchange-correlation (xc) potential. The most popular approximations usually lack features of the exact xc potential that are important for an accurate prediction of the fundamental gap and the distribution of charge in complex systems. Two principal features in this regard are the spatially uniform shift in the potential, as the number of electrons infinitesimally surpasses an integer, and the spatial steps that form, for example, between the atoms of stretched molecules. Although both aforementioned concepts are well known, the exact relationship between them remained unclear. Here we establish this relationship via an analytical derivation. We support our result by numerically solving the many-electron Schrödinger equation to extract the exact Kohn-Sham potential and directly observe its features. Spatial steps in the exact xc potential of a full configuration-interaction (FCI) calculation of a molecule are presented in three dimensions.

Charge transfer in time-dependent density functional theory

Charge transfer plays a crucial role in many processes of interest in physics, chemistry, and bio-chemistry. In many applications the size of the systems involved calls for time-dependent density functional theory (TDDFT) to be used in their computational modeling, due to its unprecedented balance between accuracy and efficiency. However, although exact in principle, in practise approximations must be made for the exchange-correlation functional in this theory, and the standard functional approximations perform poorly for excitations which have a long-range charge-transfer component. Intense progress has been made in developing more sophisticated functionals for this problem, which we review. We point out an essential difference between the properties of the exchange-correlation kernel needed for an accurate description of charge-transfer between open-shell fragments and between closed-shell fragments. We then turn to charge-transfer dynamics, which, in contrast to the excitation problem, is a highly non-equilibrium, non-perturbative, process involving a transfer of one full electron in space. This turns out to be a much more challenging problem for TDDFT functionals. We describe dynamical step and peak features in the exact functional evolving over time, that are missing in the functionals currently used. The latter underestimate the amount of charge transferred and manifest a spurious shift in the charge transfer resonance position. We discuss some explicit examples.

Quantum Singwi-Tosi-Land-Sjölander approach for interacting inhomogeneous systems under electromagnetic fields: Comparison with exact results

For inhomogeneous interacting electronic systems under a time-dependent electromagnetic perturbation, we derive the linear equation for response functions in a quantum mechanical manner. It is a natural extension of the original semi-classical Singwi-Tosi-Land-Sjölander (STLS) approach for an electron gas. The factorization ansatz for the two-particle distribution is an indispensable ingredient in the STLS approaches for the determination of the response function and the pair correlation function. In this study, we choose an analytically solvable interacting two-electron system as the target for which we examine the validity of the approximation. It is demonstrated that the STLS response function reproduces well the exact one for low-energy excitations. The interaction energy contributed from the STLS response function is also discussed.

From the Kohn–Sham band gap to the fundamental gap in solids. An integer electron approach

The upshift Δ of the level at the bottom of the conduction band (the LUMO) from the neutral N-electron crystal to the negative N + 1 system, and therefore the fundamental gap ε_LUMO(N + 1) − ε_HOMO(N) = I − A, can be calculated simply and cheaply from the response part of v_xc.

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.

Origin of the step structure of molecular exchange–correlation potentials

The step structure of exact exchange–correlation potentials is linked to the properties of the average local electron energy (ALEE).

Reduction of Electronic Wave Functions to Kohn-Sham Effective Potentials

A method for calculating the Kohn-Sham exchange-correlation potential v(XC)(r) from a given electronic wave function is devised and implemented. It requires on input one- and two-electron reduced density matrices and involves construction of the generalized Fock matrix. The method is free from numerical limitations and basis-set artifacts of conventional schemes for constructing v(XC)(r) in which the potential is recovered from a given electron density, and is simpler than various many-body techniques. The chief significance of this development is that it allows one to directly probe the functional derivative of the true exchange-correlation energy functional and to rigorously test and improve various density-functional approximations.

Kinetic and interaction components of the exact time-dependent correlation potential

The exact exchange-correlation (xc) potential of time-dependent density functional theory has been shown to have striking features. For example, step and peak features are generically found when the system is far from its ground-state, and these depend nonlocally on the density in space and time. We analyze the xc potential by decomposing it into kinetic and interaction components and comparing each with their exact-adiabatic counterparts, for a range of dynamical situations in model one-dimensional two-electron systems. We find that often, but not always, the kinetic contribution is largely responsible for these features that are missed by the adiabatic approximation. The adiabatic approximation often makes a smaller error for the interaction component, which we write in two parts, one being the Coulomb potential due to the time-dependent xc hole. Non-adiabatic features of the kinetic component were also larger than those of the interaction component in cases that we studied when there is negligible step structure. In ground-state situations, step and peak structures arise in cases of static correlation, when more than one determinant is essential to describe the interacting state. We investigate the time-dependent natural orbital occupation numbers and find the corresponding relation between these and the dynamical step is more complex than for the ground-state case.

Strong Correlations in Density-Functional Theory: A Model of Spin-Charge and Spin-Orbital Separations

It is known that the separation of electrons into spinons and chargons, the spin-charge separation, plays a decisive role when describing one-dimensional (1D) strongly correlated systems [ Phys. Rev. B 2012 , 86 , 075132 ]. In this paper, within the density-functional theory (DFT) formalism, we extend the investigation by considering a model for the third electron fractionalization: the separation into spinons, chargons and orbitons, the last associated with the electronic orbital degree of freedom. Specifically, we deal with two exact constraints of exchange-correlation (XC) density-functionals: (i) the constancy of the highest occupied (HO) Kohn-Sham (KS) eigenvalues upon fractional electron numbers and (ii) their discontinuities at integers. By means of 1D discrete Hubbard chains and 1D H2 molecules in the continuum, we find that spin-charge separation yields almost constant HO KS eigenvalues, whereas the spin-orbital counterpart can be decisive when describing derivative discontinuities of XC potentials at strong correlations.

The derivative discontinuity of the exchange–correlation functional

Manifestations of the derivative discontinuity of the energy in density functional theory are demonstrated in simple systems in chemistry and physics.

Challenging adiabatic time-dependent density functional theory with a Hubbard dimer: the case of time-resolved long-range charge transfer

Adiabatic TDDFT fails for time-resolved charge-transfer in a simple model, even when linear response spectra is accurate.

Orbital localization, charge transfer, and band gaps in semilocal density-functional theory

We derive an exchange energy functional of generalized gradient form with a corresponding potential that changes discontinuously at integer particle numbers. The functional is semilocal, yet incorporates key features that are connected to the derivative discontinuity of Kohn-Sham density-functional theory. We validate our construction for several paradigm systems and explain how it addresses central well-known deficiencies of antecedent semilocal methods, i.e., the description of charge transfer, properly localized orbitals, and band gaps. We find, e.g., an improved shell structure for atoms, eigenvalues that more closely correspond to ionization energies, and an improved description of band structure where localized states are lowered in energy.

Dynamics of Charge-Transfer Processes with Time-Dependent Density Functional Theory

We show that whenever an electron transfers between closed-shell molecular fragments, the exact correlation potential of time-dependent density functional theory develops a step and peak structure in the bonding region. This structure has a density dependence that is nonlocal both in space and in time that even the exact adiabatic ground-state exchange-correlation functional fails to capture it. For charge-transfer between open-shell fragments, an initial step and peak vanish as the charge-transfer state is reached. The inability of usual approximations to develop these structures leads to inaccurate charge-transfer dynamics. This is illustrated by the complete lack of Rabi oscillations in the dipole moment under conditions of resonant charge transfer for an exactly solvable model system. The results transcend the model and are applicable to more realistic molecular complexes.

Strong correlation in Kohn-Sham density functional theory

We use the exact strong-interaction limit of the Hohenberg-Kohn energy density functional to approximate the exchange-correlation energy of the restricted Kohn-Sham scheme. Our approximation corresponds to a highly nonlocal density functional whose functional derivative can be easily constructed, thus transforming exactly, in a physically transparent way, an important part of the electron-electron interaction into an effective local one-body potential. We test our approach on quasi-one-dimensional systems, showing that it captures essential features of strong correlation that restricted Kohn-Sham calculations using the currently available approximations cannot describe.

Correct description of the bond dissociation limit without breaking spin symmetry by a random-phase-approximation correlation functional

A correlation functional that is termed exact-exchange random phase approximation (EXX-RPA) functional and is obtained with the exact frequency-dependent exchange kernel via the fluctuation-dissipation theorem is shown to correctly describe electron pair bonds in the dissociation limit without the need to resort to symmetry breaking in spin space. Because the functional also yields more accurate electronic energies for molecules in their equilibrium geometry than standard correlation functionals, it combines accuracy at equilibrium bond distances and in dissociation processes with a correct description of spin, something all commonly employed correlation functionals fail to do. The reason why the EXX-RPA correlation functional yields distinctively and qualitatively better results than RPA approaches based on Hartree-Fock and time-dependent Hartree-Fock is explained.