Error Decomposition in Dissociation Processes within Density-Corrected DFT: An Ensemble Perspective

Abstract

Density functional theory (DFT) is widely used for electronic structure calculations but can fail to describe bond dissociation correctly, even for simple systems such as H₂ ⁺ and H₂. These failures are commonly attributed to delocalization error in H₂ ⁺ and static correlation in H₂. More generally, dissociation errors are closely linked to deviations from ideal fractional charge and fractional spin ensemble behavior. Here, we investigate bond dissociation within density-corrected DFT (DC-DFT) for NaCl, H₂ ⁺, He₂ ⁺, and H₂, using Hartree–Fock (HF) and alternative localized densities. We seek to understand why DC-DFT improves the dissociation behavior of some systems, such as NaCl, while providing little benefit in others, such as H₂ ⁺. We show that for H₂ ⁺, a localized density can yield substantially more accurate dissociation energetics than the exact HF density when it is used within DC-DFT. Although formally less accurate, the localized density compensates for the functional’s delocalization error and recovers the correct asymptotic behavior. This result demonstrates that an approximate density can sometimes yield a more accurate energy than an exact density when used with an approximate functional. In both NaCl and H₂ ⁺, localization compensates for the energetic consequences of the convex fractional-charge behavior of the functional by effectively driving the system toward an ensemble end point corresponding to an integer charge. We further show that the connection between dissociation errors and ensemble behavior is preserved within DC-DFT and that HF densities worsen the dissociation of strongly correlated systems such as H₂.

I. Introduction

Density Functional Theory (DFT) within the Kohn–Sham (KS) or the generalized KS (GKS) formalism is an exact method for describing the ground state of an electronic system. , In practice, the performance of DFT depends on the approximate exchange–correlation (XC) functionals. Despite the great success of DFT, standard local and semilocal functionals suffer from some severe systematic drawbacks. −

While the exact XC functional is unknown, a number of exact conditions that it must satisfy have been identified. Among those conditions is the behavior of the exact functional at fractional electron numbers. Fractional charges can be described using an ensemble perspective: an open system interacting with an electron bath forms a statistical mixture of pure (integer electron number) states, resulting in fractional electron occupation. Alternatively, fractional charges can also arise by considering a supermolecular system composed of infinitely separated subsystems. In this picture, the additional electron is distributed among degenerate integer-electron ground states. Averaging over all possible placements yields subsystem densities corresponding to arbitrary fractional charges, while the energy per subsystem remains the weighted average of the integer electron energies.

Both perspectives result in the piecewise linearity condition, stating that the curve of the energy E(N) versus particle number N should be a series of straight-line segments with discontinuities in the derivative (chemical potential) at integer N.

In practice, most approximate functionals deviate from linearity and provide convex energy curves, which have been attributed to the infamous delocalization error. ,−

Many important properties, such as charge-transfer excitations, − transition states in chemical reactions, , the dissociation of molecular ions, , and the binding energies of charge-transfer complexes, are incorrectly described because many commonly used density functionals deviate from piecewise linearity, suffer from self-interaction errors (one- or many-electron), and fail to exhibit the correct –1/r asymptotic behavior of the XC potential. While these deficiencies are related, they are not equivalent.

Another important exact condition is the spin constancy condition. For fractional-spin ensembles corresponding to the low-energy manifold around the equilibrium spin, the exact functional yields energies degenerate with the underlying pure spin states, so the total energy remains constant within this range. Standard approximate functionals violate this condition, causing significant static correlation errors, as fractional-spin states appear in systems with static correlation. Together with the fractional-charge condition, this requirement defines the flat-plane condition.

While most DFT calculations are not done with fractional electrons or spins, it is established that the behavior of functionals at fractional charges or spins affects the prediction of different properties. It has been demonstrated that the error in dissociation of H₂ ⁺ and H₂ directly matches the error (namely, the deviation from linearity) of hydrogen atoms with fractional charges or spin, respectively. Thus, correcting the behavior of the ensemble in approximate density functionals is a key step in improving the description of dissociation processes.

Range-separated hybrid (RSH) functionals, which incorporate the correct 1/r asymptotic behavior, significantly reduce delocalization errors. Optimally tuned RSH functionals, where the range-separation parameter is adjusted to uphold the ionization potential (IP) theorem, have been shown to yield superior results for various properties. − Furthermore, it has been demonstrated that the improvements in RSH functionals, particularly when optimally tuned, restore piecewise linearity for ensembles with fractional charges.

While RSH functionals correct many of the ailments described above, they do not correct static correlation errors and fractional spin errors in DFT. Tuning of range-separated double hybrids has been recently applied to enforce both the piecewise linearity and spin constancy conditions for the description of molecular dissociation, demonstrating a reduction of both errors despite the presence of a nonphysical maximum in the dissociation curve.

An additional practical approach to correcting errors in density functionals is through density-corrected DFT (DC-DFT). In DC-DFT, the total error is decomposed into contributions from using an approximate functional, Ẽ, and from using an approximate density, ñ, as follows: ΔE = Ẽ[ñ] – E[n] = Ẽ[ñ] – Ẽ[n] + Ẽ[n] – E[n] = ΔE _D + ΔE _F where E and n are the exact energy and density, respectively. Here, ΔE _F represents the functional-driven error and ΔE _D denotes the density-driven error.

It has been demonstrated that the dominant errors in most calculations are functional-driven. Cases where the density-driven errors are dominant are referred to as abnormal. In many abnormal cases, where density-driven errors are dominant, performing calculations using the Hartree–Fock (HF) density instead of the density obtained from self-consistent field (SCF) KS calculations proves to be a practical way of improving DFT calculations. This is true, for example, for the dissociation of heteronuclear species at large distances, reaction barriers, ions in solution, and more. −

In contrast, for some systems, such as the dissociation of H₂ ⁺ and H₂, the use of the HF density leads to only a marginal improvement and can even worsen the results. It therefore remains unclear why DC-DFT yields improvement for certain delocalization-error cases, such as NaCl, while providing little or no improvement in cases such as H₂ ⁺. Moreover, it is not well understood how these improvements, when present, are reflected in the overall ensemble picture. Addressing these issues, and in particular, the role of density localization in the successful description of dissociation within DC-DFT, constitutes the central motivation for the present study.

In this communication, we study the performance of DC-DFT across several representative dissociating systems at the dissociation limit, including NaCl, H₂ ⁺, He₂ ⁺, and H₂, and examine the role of density localization in determining its performance. To do so, we follow the density-evaluation procedure implied by DC-DFT, namely, evaluating approximate functionals on HF-derived densities, and analyze how the results depend on whether the supplied density is localized or delocalized at dissociation. Our goal is to assess the extent to which the success of DC-DFT in dissociation problems originates from the tendency of HF to produce localized densities. When needed, localized densities are generated using constrained DFT (cDFT).

II. Results and Discussion

We demonstrate that in systems where standard DFT exhibits spurious charge transfer, employing an HF density within DC-DFT can lead to improved predictions, provided the HF density is localized. However, when the HF density itself remains delocalized, the DC-DFT offers only marginal improvements. As we show below, imposing a localized density in such cases can nevertheless lead to a significant improvement in the large-distance energetics.

Moreover, we examine how the behavior of approximate XC functionals at fractional charge or spin relates to their ability to describe dissociation processes. Errors in binding energy (BE) curves are closely linked to deviations from linearity in fractional charge or spin ensembles. In systems where static correlation errors dominate, such as the dissociation of H₂, employing a non-SCF density as done in the DC-DFT procedure increases deviations from linearity in the fractional spin ensemble. Consequently, the BE curves exhibit greater deviations from the exact behavior.

It is well established that standard functionals, when used self-consistently, yield an incorrect NaCl dissociation curve in the gas phase. Instead of dissociating into neutral Na and Cl fragments, the system spuriously favors fractional charges due to delocalization errors. For instance, PBE predicts Na and Cl atoms with +0.4 and –0.4 charges, respectively, according to Mulliken population analysis. ,

The DC-DFT procedure has been shown to improve the dissociation curve for NaCl. We revisit this well-studied example because it provides a clear illustration of the relationship between density localization and DC-DFT performance, which will prove to be useful for interpreting the behavior of H₂ ⁺ discussed below.

To understand this behavior, we begin with the underlying HF solutions. At dissociation, restricted HF (RHF) converges to a heterolytic solution, yielding charged fragments. In contrast, unrestricted HF (UHF) produces the expected homolytic solution at large separations, yielding two neutral atoms. Symmetry breaking between the two solutions occurs at the Coulson-Fischer (CF) point, located at 5.8 Å as shown in Figure S1 of the Supporting Information (SI).

In order to maintain the homolytic solution along the dissociation coordinate, HF must be constrained by means of cDFT. This results in a curve that is repulsive at short distances and decays to zero at long distances, converging to the proper dissociation limit.

The two HF solutions are distinguishable using Mulliken population analysis, as shown in Figure S2. This analysis clearly demonstrates the difference between heterolytic and homolytic dissociations along the dissociation coordinate.

Following the notation introduced by Kim et al., we use the expression HF-DFT to denote the evaluation of a functional using a density obtained from HF. For example, HF-PBE corresponds to evaluating the PBE functional on an HF density rather than on a self-consistent KS density.

When HF is employed in the DC-DFT procedure, the lowest-energy HF solution is typically used. Here, we additionally examine alternative HF solutions obtained through SCF convergence or cDFT constraints to gain insights into the role of density localization.

Before the CF point, the UHF and RHF solutions are effectively identical. Beyond the CF point, UHF breaks spin symmetry and produces the correct homolytic solution at the dissociation. Below, we refer to HF­(heterolytic) or HF­(homolytic) to indicate which solution is used in DC-DFT, while HF­(SCF) refers to an unconstrained SCF calculation that may converge to either solution, depending on the fragment separation.

Figure shows PBE, HF­(SCF)-PBE, and HF­(heterolytic)-PBE dissociation curves, along with the corresponding fractional-charge behavior. As shown in the figure, PBE substantially underestimates the BE; this error directly parallels its deviation from linearity along the fractional-charge axis, predicting Na and Cl charges of approximately ± 0.4 at the dissociation limit.

1.

Dissociation profile of NaCl. Left panel: Dissociation energy curves obtained with self-consistent PBE (blue), HF­(SCF)-PBE (yellow), and HF­(heterolytic)-PBE (green). Right panel: Energy versus fractional charges for the NaCl system, with Cl occupation on the top axis indicating the transition from Cl to Cl^– and Na occupation on the bottom axis indicating the transition from Na to Na⁺. HF-PBE denotes PBE calculations using HF density.

For HF-PBE, improvement is observed relative to PBE at intermediate distances, followed by a jump, as previously shown. This jump to zero occurs at the CF point, where the HF solution spontaneously switches to the homolytic (unrestricted) solution, similar to the behavior reported previously for He₂ ⁺. Alternatively, we can converge the HF solution on the heterolytic (restricted) solution at larger distances and use this density in the HF­(heterolytic)-PBE calculation, as shown in the figure. This curve gradually increases until it crosses the zero to positive BE. It has been suggested to use this curve until it reaches zero and then switch to the HF triplet density to improve the DC-DFT procedure. As we show below, these previously proposed procedures can be naturally understood in terms of the localization properties of the underlying densities, providing additional insight into the origin of the successes and limitations of DC-DFT in dissociation problems.

As the HF solutions can converge to either charged or neutral fragments, they correspond at large separations to the ensemble description at the end points. The DC-DFT approach is effective for NaCl at large distances (beyond the CF point) as the underlying HF density becomes localized and correctly dissociates into neutral fragments. Consequently, the density approaches one of the ensemble end points shown in the figure, allowing the approximate functional to recover the correct dissociation behavior.

While DC-DFT has been demonstrated to be useful for NaCl, this is not the situation in the case of homonuclear dissociation processes like H₂ ⁺. Despite the fact that at moderately stretched geometries (e.g., 5 Å), H₂ ⁺ has been classified as an “abnormal” system, the DC-DFT procedure has almost no effect on its dissociation. ,, It has been argued that the dissociation of H₂ ⁺ is not density sensitive. , This conclusion is based on the observation that evaluating the functional on the HF density, which is exact for the one-electron H₂ ⁺ system, produces little change relative to the self-consistent result. However, additional insight can be gained by considering densities beyond the ground-state HF density. As we show below, approximate functionals can exhibit a pronounced dependence on the localization character of the density. In particular, at large separations, evaluating an approximate functional on a localized density can yield substantially different results from those obtained with the exact delocalized density. This observation suggests that, for functionals suffering from delocalization error, the success of density correction may not be determined solely by the proximity of the density to the exact one. In the present case, a density that differs from the exact density yields a more accurate energy because it compensates for the error of the approximate functional. More generally, our results suggest that an approximate functional can sometimes be improved through the use of an alternative approximate density whose error partially compensates for the functional error.

In the dissociation limit, the correct behavior is that the two systems, namely, H^+0.5···H^+0.5 and H⁺¹···H⁰ (or any other charge distribution) should be degenerate, as previously explained by Yang et al. by considering a supermolecular system of infinitely separated replicas: averaging over all possible placements of the “extra” electron among degenerate integer-electron ground states yields subsystem densities corresponding to arbitrary fractional charges, yet the energy per subsystem remains the weighted average of the integer-electron energies, resulting in piecewise-linear energy behavior versus fractional electron number.

In HF, H₂ ⁺ can dissociate homolytically into H^+0.5···H^+0.5, which we refer to as the symmetric dissociation, or it can dissociate heterolytically into H⁺¹···H⁰. These two solutions differ at short distances and approach degeneracy at large distances, as demonstrated in Figure S3.

The performance of DC-DFT, using the SCF homolytic solution, is demonstrated in Figure . The left panel of the figure shows the dissociation curves of H₂ ⁺ obtained with various methods. As demonstrated in the figure, the HF curve correctly converges to the correct dissociation limit at large distances as the HF approximation is exact for one-electron systems. The right panel of the figure shows that for fractional charges, the HF energy curve is piecewise linear.

2.

Dissociation profile of H₂ ⁺. Left panel: Dissociation energy curves obtained by using different functionals. Right panel: Energy versus fractional charges for the H₂ ⁺ system, with one hydrogen occupation on the top axis indicating the transition from H⁺ to H and another hydrogen occupation on the bottom axis indicating the transition from H to H⁺.

In contrast, local and semilocal density functionals such as BLYP and LDA also dissociate to a symmetric solution, with a charge of +0.5 on each fragment. ,, However, as the right panel of Figure shows, all of these functionals deviate significantly from the exact, linear behavior. Consequently, there is no degeneracy between the two solutions at large distances as there should be, resulting in the erroneous BE curves seen in the figure. The deviation from linearity is directly related to the deviation in the BE.

Remarkably, this is not only true for the SCF density; we next study the behavior of the different functionals using the HF density, as done in the DC-DFT procedure. Here as well, we observe a correspondence in the deviation at fractional charges and the deviation of the BE. It has been previously reported that the DC-DFT results only in a small improvement for the case of H₂ ⁺. Indeed, this can be seen from the behavior at the fractional charge ensemble; HF-LDA results in a slight improvement at the ensemble (0.78 kcal/mol), which in turn reflects the same slight improvement at large distances of the dissociation. HF-BLYP improves more upon the BLYP results (2.91 kcal/mol) in the ensemble, which again directly relates to the behavior of the functional at large distances at the dissociation limit. Indeed, the DC-DFT treatment for this example exhibits only a slight improvement. The reason is that the underlying HF density in this case is symmetric; when DFT inherits a symmetric HF density, it also predicts a symmetric solution, which, because of the deviation from piecewise linearity, deviates significantly.

A different behavior is obtained when an asymmetric HF solution is used within the DC-DFT procedure. Using this density results in a curve that is repulsive at short distances, as shown in Figure , but largely corrects the energies near the dissociation limit. Formally, the asymmetric (localized) density is not the finite-R HF ground-state density of H₂ ⁺. However, the splitting between the asymmetric (localized) and symmetric solutions vanishes exponentially with R, such that the two become practically degenerate within HF at large separations. In contrast, PBE does not treat these densities equivalently because of its convex fractional-charge behavior. As a result, evaluating PBE on a localized density substantially improves the large-distance energetics, even though this density does not correspond to the symmetric ground-state solution.

3.

Dissociation profile of H₂ ⁺ computed with HF-PBE; using the symmetric solution, HF­(sym)-PBE (green), the asymmetric solution, cHF­(asym)-PBE (orange), and a HF­(SCF)-PBE curve that converges to the symmetric solution at short distances, and transitions to the asymmetric solution (yellow). In addition, HF energies are shown as a pink curve.

This observation explains why the standard DC-DFT procedure provides little improvement for H₂ ⁺. By construction, it employs the symmetric HF density, which equally distributes the charge between the fragments and, therefore, inherits the delocalization error of the approximate functional. In contrast, localized densities recover the correct asymptotic behavior, leading to a substantial reduction of the dissociation error, similar to what is observed for heteronuclear systems such as NaCl. The key difference is that for NaCl, localization corresponds to the exact neutral dissociation products, whereas for H₂ ⁺, the localized density is not the exact finite-separation density.

The H₂ ⁺ example provides a broader lesson regarding the density correction. Although the localized density is formally less accurate than the exact HF density, it yields a more accurate energy when used with an approximate functional because it compensates for the functional’s delocalization error. This suggests that the success of DC-DFT in charge-transfer problems is closely related to the localization properties of the supplied density and that assessing density sensitivity solely through a comparison with the HF density may overlook important effects. More generally, errors in an approximate functional can sometimes be partially compensated for by an alternative approximate density.

Other homonuclear dissociation curves, such as Li₂ ⁺ and Na₂ ⁺, behave similarly to H₂ ⁺ as a result of similar behavior at the HF level, as demonstrated in Figures S5–S10 of the SI. In contrast, the He₂ ⁺ example resembles the NaCl example, as discussed below.

The performance of the different functionals and the behavior at fractional charges for the He₂ ⁺ system are shown in Figure S11. Different functionals exhibit jumps between different solutions, as previously reported. , Here, as well, we observe that the deviation from piecewise linearity in the fractional-charge ensemble is maintained for the symmetric solution; when the solution contains partial charges on the fragments, the deviation is dictated by the error in the ensemble.

The two HF solutions, namely, the symmetric and the asymmetric solutions, are shown in Figure S12 of the SI. Here, unlike the H₂ ⁺, the two solutions cross already at moderate separations, and thus, we expect HF SCF to converge to the symmetric solution at short distances and to the asymmetric solution at larger distances, just as in the NaCl example. Indeed, SCF-HF results start in the symmetric solution close to equilibrium and converge to the localized asymmetric solution at dissociation, as shown in the figure. As in the H₂ ⁺ case, the energy improvement should therefore be associated with the use of a localized density.

Figure shows the dissociation curves of He₂ ⁺ calculated using self-consistent PBE, HF, HF-PBE, and the localized PBE solution obtained by means of cDFT. As shown in the figure, the PBE dissociation curve results in an artificial maximum and continues to decrease until a jump at 3.8 Å, where the PBE SCF solution spontaneously switches to the asymmetric solution, as indicated by the Mulliken analysis given in Figure S13 of the SI. Even though we are dealing with a homonuclear dissociation, we demonstrate that HF-PBE results are significantly improved over the PBE results, and gradually increase to the correct zero BE as it uses the HF localized density. Notably, a similar improvement in the dissociation limit can be achieved using the localized PBE solution, further demonstrating the crucial role of density localization.

4.

Dissociation profile of He₂ ⁺. HF (pink), PBE­(SCF) (blue), cPBE­(asym) (teal), and HF-PBE (yellow) with CCSD­(T) (black) as reference.

This example further emphasizes that for these delocalization-dominated dissociation errors, the energetic improvement is primarily associated with the localization of the density. In this sense, the improvement is not unique to HF: a localized density obtained from PBE, for example, can reduce the large-distance energy error, as it avoids the spuriously delocalized fractional-charge solution. Thus, the role of HF in HF-DFT for dissociation problems depends on whether the HF density provides the localization required to overcome the delocalization error of the approximate functional.

We note that, whereas in DC-DFT a more favorable large-distance energy can be obtained by supplying a localized density, a correct dissociation curve can also be obtained with an RSH functional. In RSH, improvement is achieved by reducing the deviation from piecewise linearity. The dissociation energy curves obtained using the LRC-ωPBE functional with different values of ω, the range-separation parameter, are shown in Figures S14 and S15 of the SI for the case of H₂ ⁺ and NaCl, respectively. Here as well, the errors in the dissociation curves directly match the deviation from linearity in the ensemble picture. Moreover, it has been demonstrated that ΔE _D decreases as the range-separation parameter increases. Examining these examples, we find that the relation between ΔE _D and the deviation from linearity depends on the nature of the dissociation. Restoring piecewise linearity with a properly tuned RSH functional removes the spurious energetic preference for fractional-charge distributions. For heteronuclear systems, because the two end points are not degenerate, the self-consistent solution then converges to the lower-energy localized end point, and ΔE _D decreases. For the homonuclear systems considered here, however, the self-consistent density is already the symmetry-preserving delocalized density. The remaining dissociation error originates from the failure of the approximate functional to make the localized and delocalized charge distributions degenerate and is therefore not captured by ΔE _D. With a properly tuned RSH functional, localized and delocalized density solutions converge to nearly the same energy, while the self-consistent solution remains delocalized. This behavior is illustrated in Figure , where cDFT was used to generate symmetric and asymmetric LRC-ωPBE solutions. Using a functional with the correct piecewise-linear behavior, such as a properly tuned RSH, resolves the dissociation problem in both H₂ ⁺ and NaCl. However, when using a functional that suffers from delocalization error, the dissociation error can nevertheless be substantially reduced by employing a localized density. In both cases, the localized density mitigates the energetic consequences of the functional’s convex fractional-charge behavior.

5.

Symmetric and asymmetric LRC-ωPBE dissociation curves for He₂ ⁺ (top) and Li₂ ⁺ (bottom). The tuned ω values were 1.000 bohr^–1 and 0.403 bohr^–1, respectively.

Next, we studied the dissociation of the neutral H₂ molecule, with the results shown in Figure . Here, the two end points of the atomic spins are degenerate, and any fractional spin point should be degenerate as well, resulting in a straight line connecting the two edges for the exact functional. Approximate functionals deviate significantly from this behavior, as previously reported. The behavior of functionals at large distances in dissociation has been connected to the behavior of the functionals at fractional spins, as demonstrated for the cases of the LDA, PBE, and HF functionals.

6.

Dissociation profile of H₂. Left panel: Dissociation energy curves obtained using different functionals. Right panel: Energy versus fractional spin for H atom, showing the transition in fractional spin from -0.5 to 0.5 for one hydrogen atom and the opposite transition to the other H atom, as indicated by the bottom and top axes, respectively.

In this example, within the restricted HF formalism, using a density other than the SCF density is demonstrated to result in higher energies and a larger deviation from the ideal straight-line behavior. Importantly, the deviation is evaluated relative to the end points of the curve, which themselves shift when the HF density is used. Thus, applying the DC-DFT procedure to H₂ typically deteriorates the underlying density-functional results. Indeed, as shown in the figure, the deviation of HF-PBE from a straight line is larger than the deviation of PBE by 6 kcal/mol, and the deviation of HF-LDA from a straight line is larger than the LDA deviation by 4 kcal/mol. This error is also evident in the dissociation curve, further demonstrating the connection between dissociation and ensemble behavior, even when using a density that is not the SCF density.

III. Conclusions

To summarize, we evaluated the performance of DC-DFT for different dissociation processes in systems such as NaCl, H₂ ⁺, He₂ ⁺, and H₂, focusing on the role of the charge and spin ensembles. For systems suffering from spurious charge transfer, DC-DFT improves the dissociation behavior when the supplied density is localized, whereas little improvement is obtained when the density remains delocalized. The analysis of H₂ ⁺ provides a broader lesson: when approximate density functionals are employed, assessing density sensitivity solely through comparison to the HF density may overlook important effects. Although the HF density is exact for H₂ ⁺, alternative densities can still provide valuable insights and substantially improve the energetics when they compensate for deficiencies of the approximate functional. This suggests that when approximate functionals are employed, the most useful density is not always the one closest to the exact density, but one whose errors partially compensate for deficiencies of the functional.

In contrast, for systems dominated by strong static correlation, such as the dissociation of H₂, modifying the density does not provide a remedy, and DC-DFT worsens the results. This indicates that, unlike the charge-transfer cases discussed above, the dominant error does not originate from delocalization and thus cannot be compensated for through the choice of the localized density. More generally, across all systems studied, errors in the BE curves are closely linked to deviations from the ideal fractional-charge and fractional-spin conditions. These ensemble constraints, therefore, provide a unifying framework for understanding both the successes and limitations of DC-DFT.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpca.6c01739.

• Computational details; constrained-DFT procedure; additional Hartree–Fock dissociation curves and Mulliken population analyses; symmetric and asymmetric density solutions for NaCl, H₂ ⁺, He₂ ⁺, Li₂ ⁺, and Na₂ ⁺; additional fractional-charge ensemble curves; range-separated hybrid dissociation curves; numerical data supporting the main-text figures (PDF)

The authors declare no competing financial interest.