Static Correlation Errors Research Articles

Accurate Diels-Alder reaction energies from efficient density functional calculations.

We assess the performance of the semilocal PBE functional; its global hybrid variants; the highly parametrized empirical M06-2X and M08-SO; the range separated rCAM-B3LYP and MCY3; the atom-pairwise or nonlocal dispersion corrected semilocal PBE and TPSS; the dispersion corrected range-separated ωB97X-D; the dispersion corrected double hybrids such as PWPB95-D3; the direct random phase approximation, dRPA, with Hartree-Fock, Perdew-Burke-Ernzerhof, and Perdew-Burke-Ernzerhof hybrid reference orbitals and the RPAX2 method based on a Perdew-Burke-Ernzerhof exchange reference orbitals for the Diels-Alder, DARC; and self-interaction error sensitive, SIE11, reaction energy test sets with large, augmented correlation consistent valence basis sets. The dRPA energies for the DARC test set are extrapolated to the complete basis set limit. CCSD(T)/CBS energies were used as a reference. The standard global hybrid functionals show general improvements over the typical endothermic energy error of semilocal functionals, but despite the increased accuracy the precision of the methods increases only slightly, and thus all reaction energies are simply shifted into the exothermic direction. Dispersion corrections give mixed results for the DARC test set. Vydrov-Van Voorhis 10 correction to the reaction energies gives superior quality results compared to the too-small D3 correction. Functionals parametrized for energies of noncovalent interactions like M08-SO give reasonable results without any dispersion correction. The dRPA method that seamlessly and theoretically correctly includes noncovalent interaction energies gives excellent results with properly chosen reference orbitals. As the results for the SIE11 test set and H2(+) dissociation show that the dRPA methods suffer from delocalization error, good reaction energies for the DARC test set from a given method do not prove that the method is free from delocalization error. The RPAX2 method shows good performance for the DARC, the SIE11 test sets, and for the H2(+) and H2 potential energy curves showing no one-electron self-interaction error and reduced static correlation errors at the same time. We also suggest simplified DARC6 and SIE9 test sets for future benchmarking.

Singlet-triplet energy gaps for diradicals from particle-particle random phase approximation.

The particle-particle random phase approximation (pp-RPA) for calculating excitation energies has been applied to diradical systems. With pp-RPA, the two nonbonding electrons are treated in a subspace configuration interaction fashion while the remaining part is described by density functional theory (DFT). The vertical or adiabatic singlet-triplet energy gaps for a variety of categories of diradicals, including diatomic diradicals, carbene-like diradicals, disjoint diradicals, four-π-electron diradicals, and benzynes are calculated. Except for some excitations in four-π-electron diradicals, where four-electron correlation may play an important role, the singlet-triplet gaps are generally well predicted by pp-RPA. With a relatively low O(r(4)) scaling, the pp-RPA with DFT references outperforms spin-flip configuration interaction singles. It is similar to or better than the (variational) fractional-spin method. For small diradicals such as diatomic and carbene-like ones, the error of pp-RPA is slightly larger than noncollinear spin-flip time-dependent density functional theory (NC-SF-TDDFT) with LDA or PBE functional. However, for disjoint diradicals and benzynes, the pp-RPA performs much better and is comparable to NC-SF-TDDFT with long-range corrected ωPBEh functional and spin-flip configuration interaction singles with perturbative doubles (SF-CIS(D)). In particular, with a correct asymptotic behavior and being almost free from static correlation error, the pp-RPA with DFT references can well describe the challenging ground state and charge transfer excitations of disjoint diradicals in which almost all other DFT-based methods fail. Therefore, the pp-RPA could be a promising theoretical method for general diradical problems.

Static correlation and electron localization in molecular dimers from the self-consistent RPA andGWapproximation

We investigate static correlation and delocalization errors in the self-consistent $GW$ and random-phase approximation (RPA) by studying molecular dissociation of the ${\mathrm{H}}_{2}$ and LiH molecules. Although both approximations contain topologically identical diagrams, the nonlocality and frequency dependence of the $GW$ self-energy crucially influence the different energy contributions to the total energy as compared to the use of a static local potential in the RPA. The latter leads to significantly larger correlation energies, which allow for a better description of static correlation at intermediate bond distances. The substantial error found in $GW$ is further analyzed by comparing spin-restricted and spin-unrestricted calculations. At large but finite nuclear separation, their difference gives an estimate of the so-called fractional spin error normally determined only in the dissociation limit. Furthermore, a calculation of the dipole moment of the LiH molecule at dissociation reveals a large delocalization error in $GW$ making the fractional charge error comparable to the RPA. The analyses are supplemented by explicit formulas for the $GW$ Green's function and total energy of a simplified two-level model providing additional insights into the dissociation limit.

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.

A new exchange–correlation functional free of delocalization and static correlation errors

Predicting the correct binding curves of H2(+) and H2 systems presents a great challenge in current applications of electronic density functional theory. Here we report a new functional for the exchange-correlation energy based on the weighted density approximation and the classical mapping method. With the exact sum rule for the exchange-correlation hole and accurate correlation functions of uniform electrons as the input, the new functional is free of delocalization and static correlation errors. It yields the exact results for any one-electron systems and the correct asymptotic limit of the binding energy between hydrogen atoms.

Benchmark tests and spin adaptation for the particle-particle random phase approximation

The particle-particle random phase approximation (pp-RPA) provides an approximation to the correlation energy in density functional theory via the adiabatic connection [H. van Aggelen, Y. Yang, and W. Yang, Phys. Rev. A 88, 030501 (2013)]. It has virtually no delocalization error nor static correlation error for single-bond systems. However, with its formal O(N(6)) scaling, the pp-RPA is computationally expensive. In this paper, we implement a spin-separated and spin-adapted pp-RPA algorithm, which reduces the computational cost by a substantial factor. We then perform benchmark tests on the G2/97 enthalpies of formation database, DBH24 reaction barrier database, and four test sets for non-bonded interactions (HB6/04, CT7/04, DI6/04, and WI9/04). For the G2/97 database, the pp-RPA gives a significantly smaller mean absolute error (8.3 kcal/mol) than the direct particle-hole RPA (ph-RPA) (22.7 kcal/mol). Furthermore, the error in the pp-RPA is nearly constant with the number of atoms in a molecule, while the error in the ph-RPA increases. For chemical reactions involving typical organic closed-shell molecules, pp- and ph-RPA both give accurate reaction energies. Similarly, both RPAs perform well for reaction barriers and nonbonded interactions. These results suggest that the pp-RPA gives reliable energies in chemical applications. The adiabatic connection formalism based on pairing matrix fluctuation is therefore expected to lead to widely applicable and accurate density functionals.

Exchange-correlation energy from pairing matrix fluctuation and the particle-particle random-phase approximation

We formulate an adiabatic connection for the exchange-correlation energy in terms of pairing matrix fluctuation. This connection opens new channels for density functional approximations based on pairing interactions. Even the simplest approximation to the pairing matrix fluctuation, the particle-particle Random Phase Approximation (pp-RPA), has some highly desirable properties. It has no delocalization error with a nearly linear energy behavior for systems with fractional charges, describes van der Waals interactions similarly and thermodynamic properties significantly better than particle-hole RPA, and eliminates static correlation error for single-bond systems. Most significantly, the pp-RPA is the first known functional that has an explicit and closed-form dependence on the occupied and unoccupied orbitals and captures the energy derivative discontinuity in strongly correlated systems. These findings illlustrate the potential of including pairing interactions within a density functional framework.

Variational fractional-spin density-functional theory for diradicals

Accurate computation of singlet-triplet energy gaps of diradicals remains a challenging problem in density-functional theory (DFT). In this work, we propose a variational extension of our previous work [D. H. Ess, E. R. Johnson, X. Q. Hu, and W. T. Yang, J. Phys. Chem. A 115, 76 (2011)], which applied fractional-spin density-functional theory (FS-DFT) to diradicals. The original FS-DFT approach assumed equal spin-orbital occupancies of 0.5 α-spin and 0.5 β-spin for the two degenerate, or nearly degenerate, frontier orbitals. In contrast, the variational approach (VFS-DFT) optimizes the total energy of a singlet diradical with respect to the frontier-orbital occupation numbers, based on a full configuration-interaction picture. It is found that the optimal occupation numbers are exactly 0.5 α-spin and 0.5 β-spin for diradicals such as O(2), where the frontier orbitals belong to the same multidimensional irreducible representation, and VFS-DFT reduces to FS-DFT for these cases. However, for diradicals where the frontier orbitals do not belong to the same irreducible representation, the optimal occupation numbers can vary between 0 and 1. Furthermore, analysis of CH(2) by VFS-DFT and FS-DFT captures the (1)A(1) and (1)B(1) states, respectively. Finally, because of the static correlation error in commonly used density functional approximations, both VFS-DFT and FS-DFT calculations significantly overestimate the singlet-triplet energy gaps for disjoint diradicals, such as cyclobutadiene, in which the frontier orbitals are confined to separate atomic centers.

Failure of the random-phase-approximation correlation energy

The random phase approximation (RPA) is thought to be a successful method; however, basic errors have been found that have massive implications in the simplest molecular systems. The observed successes and failures are rationalized by examining its performance against exact conditions on the energy for fractional charges and fractional spins. Extremely simple tests reveal that the RPA method satisfies the constancy condition for fractional spins that leads to correct dissociation of closed-shell molecules and no static correlation error (such as in H${}_{2}$ dissociation) but massively fails for dissociation of odd electron systems, with an enormous delocalization error (such as H${{}_{2}}^{+}$ dissociation). Other methods related to the RPA, including the Hartree-Fock response (RPAE) or range-separated RPA, can reduce this delocalization error but only at the cost of increasing the static correlation error. None of the RPA methods have the discontinuous nature required to satisfy both exact conditions and the full unified condition (e.g., dissociation of H${{}_{2}}^{+}$ and H${}_{2}$ at the same time), emphasizing the need to go beyond differentiable energy functionals of the orbitals and eigenvalues.

Dispersion, static correlation, and delocalisation errors in density functional theory: An electrostatic theorem perspective

Dispersion, static correlation, and delocalisation errors in density functional theory are considered from the unconventional perspective of the force on a nucleus in a stretched diatomic molecule. The electrostatic theorem of Feynman is used to relate errors in the forces to errors in the electron density distortions, which in turn are related to erroneous terms in the Kohn-Sham equations. For H(2), the exact dispersion force arises from a subtle density distortion; the static correlation error leads to an overestimated force due to an exaggerated distortion. For H(2)(+), the exact force arises from a delicate balance between attractive and repulsive components; the delocalisation error leads to an underestimated force due to an underestimated distortion. The net force in H(2)(+) can become repulsive, giving the characteristic barrier in the potential energy curve. Increasing the fraction of long-range exact orbital exchange increases the distortion, reducing delocalisation error but increasing static correlation error.

The benchmark of Gutzwiller density functional theory in hydrogen systems

AbstractWe propose an approximate form of the exchange‐correlation energy functional for the Gutzwiller density functional theory. It satisfies certain physical constraints in both weak and strong electron correlation limits. We benchmark the Gutzwiller density functional approximation in the hydrogen systems, where the static correlation error is shown to be negligible. The good transferability is demonstrated by applications to the hydrogen molecule and some crystal structures. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012

Open-shell reduced density matrix functional theory

Open-shell reduced density matrix functional theory is established by investigating the domain of the exact functional. For spin states that are the ground state, a particularly simple set is found to be the domain. It cannot be generalized to other spin states. A number of conditions satisfied by the exact density matrix functional is formulated and tested for approximate functionals. The exact functional does not suffer from fractional spin error, which is the source of the static correlation error in dissociated molecules. We prove that a simple approximation (called the Buijse-Baerends functional, Müller or square root functional) has a non-positive fractional spin error. In the case of the H atom the error is zero. Numerical results for a few atoms are given for approximate density and density matrix functionals as well as a recently developed range-separated combination of both.

Discontinuous Nature of the Exchange-Correlation Functional in Strongly Correlated Systems

Standard approximations for the exchange-correlation functional have been found to give big errors for the linearity condition of fractional charges, leading to delocalization error, and the constancy condition of fractional spins, leading to static correlation error. These two conditions are now unified and extended to states with both fractional charge and fractional spin to give a much more stringent condition: the exact energy functional is a plane, linear along the fractional charge coordinate and constant along the fractional spin coordinate with a line of discontinuity at the integer. Violation of this condition underlies the failure of all known approximate functionals to describe the gaps in strongly correlated systems. It is shown that explicitly discontinuous functionals of the density or orbitals that go beyond these currently used smooth approximations is the key for the application of density functional theory to strongly correlated systems.

Fractional spins and static correlation error in density functional theory

Electronic states with fractional spins arise in systems with large static correlation (strongly correlated systems). Such fractional-spin states are shown to be ensembles of degenerate ground states with normal spins. It is proven here that the energy of the exact functional for fractional-spin states is a constant, equal to the energy of the comprising degenerate pure-spin states. Dramatic deviations from this exact constancy condition exist with all approximate functionals, leading to large static correlation errors for strongly correlated systems, such as chemical bond dissociation and band structure of Mott insulators. This is demonstrated with numerical calculations for several molecular systems. Approximating the constancy behavior for fractional spins should be a major aim in functional constructions and should open the frontier for density functional theory to describe strongly correlated systems. The key results are also shown to apply in reduced density-matrix functional theory.

Insights into Current Limitations of Density Functional Theory

Density functional theory of electronic structure is widely and successfully applied in simulations throughout engineering and sciences. However, for many predicted properties, there are spectacular failures that can be traced to the delocalization error and static correlation error of commonly used approximations. These errors can be characterized and understood through the perspective of fractional charges and fractional spins introduced recently. Reducing these errors will open new frontiers for applications of density functional theory.