Many-electron Self-interaction Error Research Articles

Self-interaction corrected SCAN functional for molecules and solids in the numeric atom-center orbital framework.

Semilocal density-functional approximations (DFAs), including the state-of-the-art SCAN functional, are plagued by the self-interaction error (SIE). While this error is explicitly defined only for one-electron systems, it has inspired the self-interaction correction method proposed by Perdew and Zunger (PZ-SIC), which has shown promise in mitigating the many-electron SIE. However, the PZ-SIC method is known for its significant numerical instability. In this study, we introduce a novel constraint that facilitates self-consistent localization of the SIC orbitals in the spirit of Edmiston-Ruedenberg orbitals [Rev. Mod. Phys. 35, 457 (1963)]. Our practical implementation within the all-electron numeric atom-centered orbitals code FHI-aims guarantees efficient and stable convergence of the self-consistent PZ-SIC equations for both molecules and solids. We further demonstrate that our PZ-SIC approach effectively mitigates the SIE in the meta-generalized gradient approximation SCAN functional, significantly improving the accuracy for ionization potentials, charge-transfer energies, and bandgaps for a diverse selection of molecules and solids. However, our PZ-SIC method does have its limitations. It cannot improve the already accurate SCAN results for properties such as cohesive energies, lattice constants, and bulk modulus in our test sets. This highlights the need for new-generation DFAs with more comprehensive applicability.

Reducing the Many-Electron Self-Interaction Error in the Second-Order Screened Exchange Method.

Preserving the beneficial properties of the second-order screened exchange (SOSEX) method, such as its freedom from one-body self-correlation error, and its seamless treatment of long-range dispersion interaction, we construct new nonlocal functionals by down-scaling the higher-order terms in the SOSEX series to reduce the many-body self-correlation error in molecular systems. Our down-scaled SOSEX (dsSOSEX) and scaled equation SOSEX (seSOSEX) approaches deliver considerably more accurate noncovalent interaction energies, reaction energies, and barrier heights than the original SOSEX method. The dsSOSEX approach improves on the description of medium- and long-range correlation, for example, in isogyric processes, while the seSOSEX approach improves on the description of processes with short- and medium-range rearrangement in the electronic structure, such as atomization. The He2+ potential energy curve shows that the many-body self-correlation error is slightly smaller in the dsSOSEX method than in the seSOSEX approach. Furthermore, the H2 potential energy curve reveals that the static correlation error can also be reduced at medium range by the dsSOSEX and seSOSEX schemes. As our dsSOSEX method provides a better description of medium- and long-range correlation, it generally leads to more accurate energetics than the seSOSEX method; hence it should be preferred over the latter. We compare our results to other empirical and nonempirical direct random phase approximation (dRPA) and beyond-dRPA methods and also discuss the possibilities for further development.

On the many-electron self-interaction error of the semilocal exchange hole based meta-GGA level range-separated hybrid with the B88 hybrids

Delocalization error or many electron self-interaction error related to the fractional occupation number is studied within the framework of the density matrix expansion based meta-GGA level range-separated hybrid functional. The present study mainly focuses on the performance of that functional with the BECKE88 exchange based local, global and range-separated hybrids. It is observed that using the Kohn–Sham kinetic energy dependent exchange hole, several properties relating to the fractional occupation number are significantly improved without hindering the atomization energy accuracy. The present systematic assessment and functional performance for G2/148 test set and (non-) hydrogen barrier heights indicate that beyond the BECKE88 based range-separated functional, present meta-GGA level range-separated functional can be used for further molecular simulations.

Accurate computation of X-ray absorption spectra with ionization potential optimized global hybrid functional.

This manuscript presents the first consistent ionization potential (IP) optimized global hybrid functional to accurately estimate the vertical ionization and excitation energies of the inner-shell electrons in molecules. The new method fulfills the IP theorem that the Kohn-Sham eigenvalues of all the occupied orbitals (including the core orbitals) are good approximations to the exact vertical ionization energies. The accuracy of the one-particle spectrum is essential to enabling the one-particle density functional theory (DFT) to provide accurate results. Compared to its precursor, the range-separated hybrid functional CAM-QTP00, the new method is more computationally efficient. The IP theorem enables the new method to provide inner-shell ionization energies measured by X-ray photoelectron spectroscopy, and it can further accurately simulate the X-ray absorption spectrum (XAS, or NEXAFS). The simulated spectra can be compared to the experiment directly without shifting. In addition, the new method reduces the delocalization error (many-electron self-interaction error) which is a severe problem in DFT.

Combining Density Functional Theory and Green's Function Theory: Range-Separated, Nonlocal, Dynamic, and Orbital-Dependent Hybrid Functional.

We present a rigorous framework which combines single-particle Green's function theory with density functional theory based on a separation of electron-electron interactions into short- and long-range components. Short-range contribution to the total energy and exchange-correlation potential is provided by a density functional approximation, while the long-range contribution is calculated using an explicit many-body Green's function method. Such a hybrid results in a nonlocal, dynamic, and orbital-dependent exchange-correlation functional of a single-particle Green's function. In particular, we present a range-separated hybrid functional called srSVWN5-lrGF2 which combines the local-density approximation and the second-order Green's function theory. We illustrate that similarly to density functional approximations, the new functional is weakly basis-set dependent. Furthermore, it offers an improved description of the short-range dynamic correlation. The many-body contribution to the functional mitigates the many-electron self-interaction error present in many density functional approximations and provides a better description of molecular properties. Additionally, we illustrate that the new functional can be used to scale down the self-energy and, therefore, introduce an additional sparsity to the self-energy matrix that in the future can be exploited in calculations for large molecules or periodic systems.

Where Does the Density Localize? Convergent Behavior for Global Hybrids, Range Separation, and DFT+U.

Approximate density functional theory (DFT) suffers from many-electron self-interaction error, otherwise known as delocalization error, that may be diagnosed and then corrected through elimination of the deviation from exact piecewise linear behavior between integer electron numbers. Although paths to correction of energetic delocalization error are well-established, the impact of these corrections on the electron density is less well-studied. Here, we compare the effect on density delocalization of DFT+U (i.e., semilocal DFT augmented with a Hubbard U correction), global hybrid tuning, and range-separated hybrid tuning on a diverse test set of 32 transition metal complexes and observe the three methods to have qualitatively equivalent effects on the ground state density. Regardless of valence orbital diffuseness (i.e., from 2p to 5p), ligand electronegativity (i.e., from Al to O), basis set (i.e., plane wave versus localized basis set), metal (i.e., Ti, Fe, Ni), and spin state, or tuning method, we consistently observe substantial charge loss at the metal and gain at ligand atoms (∼0.3-0.5 e or more). This charge loss at the metal is preferentially from the minority spin, leading to increasing magnetic moment as well. Using accurate wave function theory references, we observe that a minimum error in partial charges and magnetic moments occurs at higher tuning parameters than typically employed to eliminate energetic delocalization error. These observations motivate the need to develop multifaceted approximate-DFT error correction approaches that separately treat density delocalization and energetic errors to recover both correct density and orbital energy-derived properties.

One- and many-electron self-interaction error in local and global hybrid functionals

Electronic self-interaction poses a fundamental challenge in density-functional theory. It greatly limits, e.g., the physical interpretation of eigenvalues as electron removal energies. We here investigate whether local hybrid functionals that are designed to be free from one-electron self-interaction lead to occupied Kohn-Sham eigenvalues and orbitals that approximate photoemission observables well. We compare the local hybrid results to the ones from global hybrid functionals that only partially counteract the self-interaction, and to the results that are obtained with a Perdew-Zunger-type self-interaction correction. Furthermore, we check whether being nominally free from one-electron self-interaction translates into a reduced many-electron self-interaction error. Our findings show that this is not the case for the local hybrid functionals that we studied: In practice they are similar to global hybrids in many respects, despite being formally superior. This finding indicates that there is a conceptual difference between the Perdew-Zunger way and the local hybrid way of translating the one-electron condition to a many-electron system. We also point out and solve some difficulties that occur when using local hybrid functionals in combination with pseudopotentials.

Can short- and middle-range hybrids describe the hyperpolarizabilities of long-range charge-transfer compounds?

The hyperpolarizabilities of five prototypical and four recently synthesized long-range charge-transfer (CT) organic compounds are calculated using short- and middle-range (SR and MR) hybrid functionals. These results are compared with data from MP2 and other DFT methods including GGAs, global hybrids, long-range corrected functionals (LC-DFT), and optimally tuned LC-DFT. Although it is commonly believed that the overestimation of hyperpolarizabilities associated with CT excitations by GGA and global hybrid functionals is the result of their wrong asymptotic exchange potential, and that LC-DFT heals this issue, we show here that SR and MR functionals yield results similar to those from LC-DFT. Hence, the long-range correction per se does not appear to be the key element in the well-known improved description of hyperpolarizabilities by LC-DFT. Rather, we argue that the inclusion of substantial amounts of Hartree-Fock exchange, which reduces the many-electron self-interaction error, is responsible for the relatively good results afforded by range separated hybrids. Additionally, we evaluate the effects of solvent and frequency on hyperpolarizabilities computed by SR and MR hybrids and compare these predictions with other DFT methods and available experimental data.

Tuned range-separated hybrid functionals in the symmetry-adapted perturbation theory.

The aim of this study is to present a performance test of optimally tuned long-range corrected (LRC) functionals applied to the symmetry-adapted perturbation theory (SAPT). In the present variant, the second-order energy components are evaluated at the coupled level of theory. We demonstrate that the generalized Kohn-Sham (GKS) description of monomers with optimally tuned LRC functionals may be essential for the quality of SAPT interaction energy components. This is connected to the minimization of a many-electron self-interaction error and exemplified by two model systems: polyacetylenes of increasing length and stretching of He 3 (+). Next we provide a comparison of SAPT approaches based on Kohn-Sham and GKS description of the monomers. We show that LRC leads to results better or comparable with the hitherto prevailing asymptotically corrected functionals. Finally, we discuss the advantages and possible limitations of SAPT based on LRC functionals.

Accurate description of torsion potentials in conjugated polymers using density functionals with reduced self-interaction error

We investigate the torsion potentials in two prototypical π-conjugated polymers, polyacetylene and polydiacetylene, as a function of chain length using different flavors of density functional theory. Our study provides a quantitative analysis of the delocalization error in standard semilocal and hybrid density functionals and demonstrates how it can influence structural and thermodynamic properties. The delocalization error is quantified by evaluating the many-electron self-interaction error (MESIE) for fractional electron numbers, which allows us to establish a direct connection between the MESIE and the error in the torsion barriers. The use of non-empirically tuned long-range corrected hybrid functionals results in a very significant reduction of the MESIE and leads to an improved description of torsion barrier heights. In addition, we demonstrate how our analysis allows the determination of the effective conjugation length in polyacetylene and polydiacetylene chains.

On the relationship between bond-length alternation and many-electron self-interaction error

Predicting accurate bond-length alternations (BLAs) in long conjugated molecular chains has been a major challenge for electronic-structure theory for many decades. While Hartree-Fock (HF) overestimates BLA significantly, second-order perturbation theory and commonly used density functional theory (DFT) approaches typically underestimate it. Here, we discuss how this failure is related to the many-electron self-interaction error (MSIE), which is inherent to both HF and DFT approaches. We use tuned long-range corrected hybrids to minimize the MSIE for a series of polyenes. The key result is that the minimization of the MSIE alone does not yield accurate BLAs. On the other hand, if the range-separation parameter is tuned to yield accurate BLAs, we obtain a significant MSIE that grows with chain length. Our findings demonstrate that reducing the MSIE is one but not the only important aspect necessary to obtain accurate BLAs from density functional theory.

Hole Localization in Molecular Crystals from Hybrid Density Functional Theory

We use first-principles computational methods to examine hole trapping in organic molecular crystals. We present a computational scheme based on the tuning of the fraction of exact exchange in hybrid density functional theory to eliminate the many-electron self-interaction error. With small organic molecules, we show that this scheme gives accurate descriptions of ionization and dimer dissociation. We demonstrate that the excess hole in perfect molecular crystals forms self-trapped molecular polarons. The predicted absolute ionization potentials of both localized and delocalized holes are consistent with experimental values.

Many-electron self-interaction and spin polarization errors in local hybrid density functionals

Errors for systems with noninteger occupation have been connected to common failures of density functionals. Previously, global hybrids and pure density functionals have been investigated for systems with noninteger charge and noninteger spin state. Local hybrids have not been investigated for either of those systems to the best of our knowledge. This study intends to close this gap. We investigate systems with noninteger charge to assess the many-electron self-interaction error and systems with noninteger spin state to assess the spin polarization error of recently proposed local hybrids and their range-separated variants. We find that long-range correction is very important to correct for many-electron self-interaction error in cations, whereas most full-range local hybrids seem to be sufficient for anions, where long-range-corrected density functionals tend to overcorrect. On the other hand, while all hitherto proposed long-range-corrected density functionals show large spin polarization errors, the Perdew-Staroverov-Tao-Scuseria (PSTS) functional performs best of all local hybrids in this case and shows an outstanding behavior for the dependence of the energy on the spin polarization.

Can short-range hybrids describe long-range-dependent properties?

Long-range-corrected hybrids, which incorporate all of the long-range exact exchange interaction, improve performance for a host of molecular properties. The long-range portion of exact exchange is both computationally and formally problematic in solids, and screened hybrids therefore eliminate it. While screened hybrids give similar results to their parent global hybrids for many molecular properties, one may worry that they perform poorly for those properties that are improved by the long-range-correction procedure. In this paper, we show that at least for the Heyd-Scuseria-Ernzerhof (HSE) screened hybrid, this is not the case; for many properties improved by long-range-correction, screened hybrids and global hybrids deliver essentially the same results. We suggest that this is because screened hybrids and global hybrids have fundamentally the same many-electron self-interaction error. We also introduce some small revisions to our computational implementation of the HSE screened hybrid, and we recommend these revisions for future applications of HSE.

Development of exchange-correlation functionals with minimal many-electron self-interaction error

New exchange-correlation functionals that address the important issue of many-electron self-interaction are developed. This is carried out by considering the performance of the functional on systems with fractional numbers of electrons at the same time as more standard thermochemical tests. The inclusion of Coulomb-attenuated exchange in the functional is facilitated by use of the adiabatic connection coupled with a short-range and long-range splittings. The new functionals have a good performance on thermochemistry and a much improved description of the total energy versus number of electrons and henceforth a much smaller many-electron self-interaction error.

Tests of functionals for systems with fractional electron number

In the exact theory, the ground state energy of an open system varies linearly when the electron number is changed between two adjacent integers. This linear dependence is not reproduced by common approximate density functionals. Deviation from linearity in this dependence has been suggested as a basis for the concept of many-electron self-interaction error (SIE). In this paper, we quantify many-electron SIE of a number of approximations by performing calculations on fractionally charged atoms. We demonstrate the direct relevance of these studies to such problems of common approximate functionals as instabilities of anions, spurious fractional charges on dissociated atoms, and poor description of charge transfer. Semilocal approximations have the largest many-electron SIE, which is only slightly reduced in typical global hybrids. In these approximations the energy versus fractional electron number curves upward, while in Hartree-Fock theory the energy curves downward. Perdew-Zunger self-interaction correction [Phys. Rev. B 23, 5048 (1981)] significantly reduces the many-electron SIE of semilocal functionals but impairs their accuracy for equilibrium properties. In contrast, a long-range corrected hybrid functional can be nearly many-electron SIE-free in many cases (for reasons we discuss) and at the same time performs remarkably well for many molecular properties.

Many-electron self-interaction error in approximate density functionals

One of the most important challenges in density functional theory (DFT) is the proper description of fractional charge systems relating to the self-interaction error (SIE). Traditionally, the SIE has been formulated as a one-electron problem, which has been addressed in several recent functionals. However, these recent one-electron SIE-free functionals, while greatly improving the description of thermochemistry and reaction barriers in general, still exhibit many of the difficulties associated with SIE. Thus we emphasize the need to surpass this limit and shed light on the many-electron SIE. After identifying the sufficient condition for functionals to be free from SIE, we focus on the symptoms and investigate the performance of most popular functionals. We show that these functionals suffer from many-electron SIE. Finally, we give a SIE classification of density functionals.