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Timestamp: 2019-04-19 18:46:19+00:00

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Anions and radicals are important for many applications including environmental chemistry, semiconductors, and charge transfer, but are poorly described by the available approximate energy density functionals. Here we test an approximate exchange-correlation functional based on the exact strong-coupling limit of the Hohenberg-Kohn functional on the prototypical case of the He isoelectronic series with varying nuclear charge Z<2, which includes weakly bound negative ions and a quantum phase transition at a critical value of Z, representing a big challenge for density functional theory. We use accurate wavefunction calculations to validate our results, comparing energies and Kohn-Sham potentials, thus also providing useful reference data close to and at the quantum phase transition. We show that our functional is able to bind H− and to capture in general the physics of loosely bound anions, with a tendency to strongly overbind that can be proven mathematically. We also include corrections based on the uniform electron gas which improve the results.
Density functional theory (DFT), Hohenberg and Kohn (1964) in its Kohn-Sham (KS) formulation, Kohn and Sham (1965) has been a real breakthrough for electronic structure calculations. The key idea of KS DFT is an exact mapping Kohn and Sham (1965) between the physical, interacting, many-electron system and a model system of non-interacting fermions with the same density, allowing for a realistic treatment of the electronic kinetic energy. All the complicated many-body effects are embedded in the so-called exchange-correlation (xc) energy functional. Although, in principle, the exact xc functional is unique (or “universal”), in practice a large number of approximations has been developed in the last thirty years, often targeting different systems, different properties, and different phenomena. Common practice for DFT users is nowadays to consult the (rather extensive) benchmark literature to choose the approximate xc functional most suitable for the problem at hand. This reflects the intrinsic difficulty of building a general approximation able to recognize and capture, for each class of systems or process, the many-body effects relevant for its description.
The mainstream strategies to construct approximate functionals consist of making an ansatz for the dependence of the xc functional on the relevant “ingredients” such as the local density, the local density gradients, the KS kinetic orbital energy, the KS orbitals, etc. Perdew et al. (2005) The ansatz can be constructed in order to fulfill as many exact constraints as possible given the ingredients used. Perdew et al. (2005) Some authors also introduce a (sometimes very large) number of parameters to be fitted to a specific data set (for recent reviews, see, e.g., Refs. Cramer and Truhlar, 2009; Cohen, Mori-Sánchez, and Yang, 2012; Peverati and Truhlar, 2014 ).
In recent years, an exact piece of information on the exact exchange-correlation functional, namely the limit of infinite correlation, Seidl, Gori-Giorgi, and Savin (2007); Gori-Giorgi, Vignale, and Seidl (2009); Gori-Giorgi, Seidl, and Vignale (2009) has become available. The “strictly-correlated-electrons” (SCE) functional, that utilizes this information, has a highly non-local dependence on the density, but its functional derivative (yielding the KS potential) can be easily constructed via a rigorous and physically transparent shortcut. Malet and Gori-Giorgi (2012); Malet et al. (2013) The SCE functional becomes exact in the limit in which the electron-electron interaction dominates over the electronic kinetic energy, and it has been successfully applied to model low-density quantum wires Malet and Gori-Giorgi (2012); Malet et al. (2013) and quantum dots. Mendl, Malet, and Gori-Giorgi (2014) In those systems, the SCE functional has been shown capable of capturing the physics of charge localization without introducing magnetic order or any other symmetry breaking. In other words, the SCE functional achieved what was often regarded as practically impossible: making non-interacting electrons behave as strongly-correlated ones, showing that restricted KS DFT with the appropriate functionals can yield results beyond mean-field theory.
with Z<2. Accurate wavefunction calculations Baker et al. (1990) have shown that when the nuclear charge Z is lowered and crosses a critical value, Zcrit≈0.91103, a quantum phase transition occurs from a bound to an unbound two-electron system. Thus, with this simple hamiltonian we can explore a whole class of very loosely bound anions, including the quantum phase transition at Zcrit.
It is worth mentioning that the failure of standard DFT approximations to bind anions properly is often attributed to the self-interaction error (SIE). However, despite being self-interaction free, the Hartree-Fock (HF) method fails for H−, yielding a negative binding energy for the second electron in contradiction to experiment. Hotop and Lineberger (1975) Thus in this case, it is correlation that stabilizes the system, so SIE is not the only problem.
In this work we test the KS SCE functional for the hamiltonian of Eq. (1), focusing on the anions close to the quantum phase transition. We compare our results (including energies, densities and KS potentials) with those from a very accurate wavefunction treatment and from standard approximate xc functionals. We also consider local corrections to KS SCE, and we analyze some exact properties of the density and of the KS potential at Zcrit.
The ± sign depends on whether l is even or odd to assure the proper symmetry of the basis functions under exchange of the two electrons (t→−t). The powers n, l, m, and j are chosen to duplicate the first several leading terms in the behavior of the exact wavefunction of a helium-like ion near the 3-particle coalescence, which is given by the Fock expansion. Fock (1954, 1958); Morgan III (1986) This composite basis was used in Ref. Baker et al., 1990 to obtain compact and highly accurate representations of the wavefunction of the sole bound state of the helium isoelectronic sequence for values of Z between Zcrit≃0.9110289 and 1. Table 1 shows the approximately optimal values of k and c used for several values of Z in the present paper.
Table 1: Approximately optimal values of k and c used for several values of Z in the accurate wavefunctions. The first row is from Ref. Baker et al., 1990 .
We quickly review some basic aspects of Kohn-Sham density functional theory, as this helps in clarifying the concepts behind the less familiar SCE functional, introduced in the next subsection.
where uH is the Hartree potential and vxc is the exchange-correlation potential. The density is obtained as ρ(r)=∑i|ψi(r)|2, with the sum running over the occupied orbitals, and the KS equations are solved self-consistently.
The simplest approximation for Exc[ρ] is the local density approximation (LDA), defining the exchange-correlation energy as a functional of the local density alone. The next levels of refinement are the generalized-gradient approximations (GGA), obtained by including the gradient of the local density ∇ρ, and the meta-GGA functionals which use also the local Laplacian of the density ∇2ρ and/or the local kinetic energy density τ(r)=∑i|∇ψi(r)|2. For the special case of the two-electron systems considered here, the Hartree-Fock method becomes equivalent to KS DFT with the exact exchange functional, as the non-local HF exchange potential reduces to a local one-body potential. For systems with higher electron number the non-local Hartree-Fock exchange can be transformed into a local potential via the optimized effective potential method, yielding a well defined orbital-dependent functional (called exact exchange). Hybrid functionals are obtained by mixing a fraction of single determinant exchange with GGA or metaGGA functionals, and are normally computed using a non-local potential, a treatment outside the KS framework.
which has a simple physical meaning: as the position r of one electron fixes all the relative distances, the net electron-electron repulsion acting on an electron at r becomes a function of r alone, and can be represented as the gradient of a one-body potential. Equation (18) is a very powerful shortcut to compute the functional derivative of the highly non-local functional VSCEee[ρ] of Eqs. (13)-(16).
Figure 1: The KS SCE approximation from the point of view of the adiabatic connection of Eq. (25). We have reported Wλ[ρ] as a function of λ for a typical weakly-correlated and a typical strongly-correlated system. The area between Wλ[ρ], the λ-axis and the vertical lines corresponding to λ=0 and λ=1 gives the exchange-correlation energy Exc[ρ]. The KS SCE approximates Wλ[ρ] with its value at λ→∞ for all λ, and thus the xc energy with the area of the rectangle limited by W∞[ρ], the λ-axis, and the vertical lines corresponding to λ=0 and λ=1.
This inequality becomes even stronger when we minimize the right-hand side by solving self-consistently the KS SCE equations. The SCE functional is also self-interaction free, as VSCEee[ρ]=0 for any one-electron density. Thus, as EKSSCEN=2≤EexactN=2 and EKSSCEN=1=EexactN=1, the self-consistent KS SCE method will certainly bind all the anions of the He isoelectronic series that are physically bound, and its error will always be towards overbinding, providing a lower bound for Zcrit, which, however, turns out to be not very tight (see Sec. III).
In Fig. 1 we show, schematically, Wλ[ρ] as a function of λ for a weakly- and a strongly-correlated system. The area between Wλ[ρ], the λ-axis and the vertical lines corresponding to λ=0 and λ=1 gives the exchange-correlation energy Exc[ρ]. The KS SCE approximates Wλ[ρ] with its value at λ→∞ for all λ, and thus the exchange-correlation energy with the area of the rectangle limited by W∞[ρ], the λ-axis, and the vertical lines corresponding to λ=0 and λ=1. This is evidently a good approximation only when the system is very correlated.
with boundary condition vSCE(r→∞)=0. Notice that vSCE(r) has the correct asymptotic behavior of the Hartree plus xc potential, vSCE(r→∞)∼1/r, since f(r→∞)→0. This is true for the general N-electron case also, since the correct (N−1)/r asymptotic leading term can be similarly derived Seidl, Gori-Giorgi, and Savin (2007) from Eq. (18).
At high densities, the SCE energy is very far from the exact one, and at low densities it becomes asymptotically exact. This is also true, more generally, for the functional VSCEee[ρ], which approaches the exact Hartree plus exchange correlation functional when the density is scaled as ργ(r)=γ3ρ(γr) and γ→0. Here we have set d0≈0.891687, which is the value from the Perdew-Wang-92 LDA parametrization. Perdew and Wang (1992) We denote this method KS SCE+LDA.
where vdee(rs) is obtained by subtracting from Eq. (33) the kinetic correlation contribution tc=−ddrs(rsϵxc). We call this approximation KS SCE+LVee,d.
Before presenting and discussing the KS SCE results, we extend the work of Umrigar and Gonze Umrigar and Gonze (1994) by studying the accurate densities and KS potentials obtained from the wavefunctions of Sec. II.1 close to the quantum phase transition. The densities and KS potentials (obtained by inversion of the KS equations) Umrigar and Gonze (1994) for selected values of Z≤1 are shown in Fig. 2.
Figure 2: Accurate r2ρ(r) and vKS(r) for various anions with nuclear charge Z of the He isoelectronic series.
where δ is an arbitrary small positive number and C±(δ) are constants depending on δ.
with a=√2(−Z+N−1). This decay agrees to leading order with Eq. (36). The accurate density at the quantum phase transition together with the decays from Eqs. (36) and (38) are displayed in Fig. 3, where in both cases the proportionality constant has been adjusted to match the accurate density at the end of the radial grid (r≈100). Notice that Eq. (37) implies that for the exact KS system (which yields the exact ground-state density) the equality ϵHOMO=−Ip also holds at Z=Zcrit, when Ip=0.
Figure 3: Comparison of the long-range behaviour of r2ρ(r) at Zcrit obtained from the asymptotic decay expressions in Eqs. (36) and (38) with the almost exact result obtained from the wavefunction in Section II.1.
From Fig. 2 we see that the KS potentials have a bump at intermediate length scale. This bump increases for smaller Z as can be expected from the asymptotic first order contribution at large r, vKS(r→∞)=(1−Z)/r that will be positive for Z<1. The bump is present also for the Hydrogen anion, where this first order contribution vanishes.
In Fig. 4 we show the correlation potentials for selected values of Z. We see that, as was found in Ref. Umrigar and Gonze, 1994 , the accurate correlation potential close to the nucleus has a nearly quadratic behavior. In Refs. Qian, 2007; Qian and Sahni, 2007 it has been shown that the linear term in the correlation potential is due to the kinetic contribution, which, thus, turns out to be very small.
Figure 4: Accurate correlation potential vc(r) for various Z.
For the He isoelectronic series with Z≤2 we solved self-consistently the restricted KS equations with various approximate functionals. Calculations for the HF (or exact exchange) method, KS LDA, KS SCE, and KS SCE with the two local corrections of Sec. II.4 were performed with a numerical code developed in our group. We chose the Perdew-Wang-92 functional (PW92) Perdew and Wang (1992) LDA parametrization. To compare our calculations with the available standard approximations we have further performed restricted KS-DFT calculations with the Amsterdam Density Functional package (ADF). te Velde et al. (2001); ?; ? From the GGA class of functionals we chose PBE, Perdew, Burke, and Ernzerhof (1996) from the metaGGA class the revTPSS Tao et al. (2003) functional and for the hybrid functional we chose B3LYP. Becke (1993); Stephens et al. (1994); Lee, Yang, and Parr (1988); Vosko, Wilk, and Nusair (1980) If not mentioned otherwise, all ADF calculations were carried out in the even-tempered (ET) QZ3P basis supported by 3 diffuse s-functions with the parametrization of Hydrogen. Chong et al. (2004) To assess the quality of the basis set we also performed KS-LDA (PW92 functional) calculations with the ADF package and compare them to our numerical solution of the KS equations. To assess the quality of the basis set we also performed KS-LDA calculations with the ADF package (PW92 functional) and compare them to our numerical solution of the KS equations.
We define the critical nuclear charge Zcrit for the various DFT approximations to be the value of Z at which either the ionization energy Ip=EN−1−EN becomes smaller than 0 or the HOMO eigenvalue ϵHOMO becomes positive, whichever is larger. Although the equality ϵHOMO=−Ip does not hold in general for approximate functionals, we invoke the HOMO eigenvalue criterion to avoid the conceptual and numerical issue of occupying orbitals with a positive eigenvalue already discussed.
Table 2 shows the predicted Zcrit for the quantum phase transition together with the corresponding ionization energy Ip=EN=1−EN=2 and the HOMO energies for the various approximations. Of the DFT approximations considered only the SCE functionals (SCE and SCE with local corrections) and the hybrid functional are able to bind the Hydrogen anion. The hybrid functional however, yields an unphysical description of the bound anion as we will further discuss below. Remarkably, all the standard functionals at different levels of approximation yield a similar value of Zcrit≈1.2. This shows that the nonlocality encoded in the SCE functional is able to capture different many-body effects than the standard approximations.
Table 2: Zcrit with corresponding negative ionization energy −Ip=EN=2−EN=1 and HOMO energies for various approaches.
As already discussed in Sec. II.3, the KS SCE self-consistent results yield a lower bound to the total energy, which for these systems is not very tight as can be seen from the underestimation of Zcrit≈0.7307 versus the actual value Zcrit≈0.9110289. This is due to the inherent strong correlation nature of the electrons in the SCE formulation that results in underestimating the electron-electron repulsion energy. Self-consistently thus, the KS-SCE densities become quite compact until the kinetic energy starts to dominate in Eq. (19). This is manifested in Fig. 5 where the density of H− is displayed for several methods, the KS-SCE yielding the most compact density. Physically, this is due to the fact that the two electrons, being perfectly correlated, can avoid each other as much as possible and can get much closer to the nucleus to lower the total energy.
Figure 5: r2ρ(r) for H− and various approaches.
Figure 6: r2ρ(r) and vKS for various Z for the KS-SCE (above) and KS-SCE+LDA (below) method.
The local corrections to the SCE functional improve considerably the predicted Zcrit and give a more realistic description of the electronic interactions. We observe that the KS-SCE+LDA density is too spread out compared to the accurate data (e.g. Fig. 5). This can be attributed to the self-interaction error that is introduced by the LDA correction which is obvious from Eq. (33) – the energy densities do not vanish for a density integrating to 1.
In Fig. 5 we display also the Hartree-Fock density. It is possible to do this because the HOMO eigenvalue is negative even though EN=1<EN=2. (If the electron number were treated as a variational parameter, the minimum energy would be attained for N<2.) We see that the HF density resembles the accurate density more closely than the density from other functionals considered. This supports the point of view of Ref. Kim, Sim, and Burke, 2013 , and the general idea of using HF densities as input for DFT energies in the case of negative ions, Lee, Furche, and Burke (2010); Kim, Sim, and Burke (2011) even when HF does not bind the last electron.
For the hybrid functional in the ET-QZ3P+3diffuse basis we obtain a negative ϵHOMO and EN=1>EN=2 for H−. Formally the hybrid thus binds the Hydrogen anion. When inspecting the density however, one observes that it escapes partially from the nucleus, as shown in Fig. 7. When removing the 3 diffuse basis functions from the basis set to prevent the density accumulation in the outside regions, we obtain a value of Zcrit in between that from HF and conventional DFT, as expected.
Figure 7: r2ρ(r) for H− with the B3LYP functional and a ET-QZ3P plus 3 diffuse function basis. It displays a second unphysical maximum in the density.
We now discuss the Kohn-Sham and exchange-correlation potentials for the self-consistent densities, displayed in Figs. 6 and 8.
Figure 8: vKS(r) and vxc(r) for H− and various approaches.
We see that the SCE total Kohn-Sham potential does not develop the bump for Z=1, but only for smaller nuclear charges when the interelectronic repulsion dominates over the weaker nuclear attraction (Fig. 6). As already observed in Fig. 5, this corresponds to a very compact density. For larger distances, the SCE potential is in good agreement with the accurate one, as expected from the absence of the self-interaction error in the SCE. From Fig. 8 we also see that the SCE potential is quadratic close to the nucleus, as can be easily proven analytically from Eq. (28), since when r→0 we have f(r→0)→∞, so that v′SCE(r→0)=0. This is in agreement with the findings of Refs. Qian, 2007; Qian and Sahni, 2007 , as there is no kinetic contribution in the SCE potential.
Although the SCE functional approximates exchange and correlation together, in Fig. 9 we show the SCE correlation potential alone, obtained by subtracting from the xc SCE potential the exchange potential constructed from the self-consistent KS SCE densities. We see that the SCE correlation potential is always negative, in contrast to the exact one. The positive part of the exact correlation potential is mainly due to kinetic correlation effects Buijse, Baerends, and Snijders (1989); Gritsenko, van Leeuwen, and Baerends (1997) that are missed in the bare SCE.
Figure 9: The self-consistent correlation potentials vc(r) from the bare KS SCE method.
At least qualitatively, the bump for H− in the total KS potential is captured by the KS-SCE with the two local corrections, though the bump is too pronounced, particularly in KS SCE+LDA. This is also responsible for the overestimation of Zcrit and can be partially attributed to the self-interaction error. However, the self-interaction error present in the KS-SCE+LDA approach is substantially different from the self-interaction error in standard KS-LDA or KS-GGA. In KS-LDA and GGA the self-interaction error manifests in the wrong asymptotic decay of the KS potential (−Z−Nr instead of −Z−N+1r). KS-SCE has the correct −Z−N+1r decay and this is not altered by the exponentially vanishing LDA contribution upon going from KS-SCE to KS-SCE-LDA. The KS SCE+LVee,d is more attractive at short distance than the exact KS potential, achieving error compensation with the overestimation of the bump (less severe than in the KS-SCE+LDA method), which results in a good estimate for Zcrit≈0.9012. Of the methods studied, the KS-SCE approach with the local corrections is the one in which the HOMO energy deviates the least from the corresponding EN−EN−1 (see Tab. 2).
The HF (or exact exchange) potential is also shown in Fig. 8, although, once more, we have to keep in mind that in this case EN=2>EN=1, so that the system is not really physically bound.
Figure 10: ϵHOMO vs. N in the Hydrogen nuclear potential for various approaches.
The KS DFT results with the standard functionals at fractional electron numbers can be easily obtained by giving fractional occupation to the HOMO. Vydrov, Scuseria, and Perdew (2007); Gaiduk, Firaha, and Staroverov (2012) As discussed in the Introduction, here we consider the challenging case of the restricted KS method, where, for singlet N=2 systems, as we increase the occupancy Q of the HOMO orbital we should observe a jump in its energy at Q=1. Notice that in the restricted KS method the conditions regarding the spin degree of freedom Mori-Sanchez, Cohen, and Yang (2009) are automatically fulfilled, so that the gap at Q=1 is the same as the “Mott gap” for 1/2 spin-up and 1/2 spin-down electrons.
Table 3: Maximum number of electrons Qmax bound in the Hydrogen nuclear potential for methods unable to bind H−.
Finally, Fig. 10 allows for a determination of the maximum number of electrons Qmax bound by the conventional DFT approaches. The results are compiled in Tab. 3. We observe, similarly to Zcrit, that the predicted value of Qmax is insensitive to the level of approximation of the standard functionals, further supporting the idea behind the model potential of Ref. Gaiduk, Firaha, and Staroverov, 2012 .
Finally, our study also provides reference data for the anions of the He isoelectronic series close to and at the quantum phase transition that can be valuable to test the accuracy of new DFT approximations (see, e.g., Ref. Bleiziffer et al., 2013 which presents correlation potentials from RPA approaches that are good approximations to the true correlation potential).
We are very grateful to Erik van Lenthe for his assistance with the ADF package, and to A. J. Cohen and P. Mori-Sanchez for pointing out an error in Fig. 10 concerning the Hartree-Fock curve. This work was supported by the Netherlands Organization for Scientific Research (NWO) through a Vidi grant and by the NSF through grant DMR-0908653.
Not supported by ADF te Velde et al. (2001); ?; ?
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