Density-functional theory cannot be constrained to completely overcome self-interaction error Glenn Moynihan,1, ∗ Gilberto Teobaldi,2, 3 and David D. O’Regan1
arXiv:1608.07320v1 [physics.chem-ph] 25 Aug 2016
1
School of Physics, CRANN and AMBER, Trinity College Dublin, Dublin 2, Ireland 2 Stephenson Institute for Renewable Energy and Department of Chemistry, The University of Liverpool, L69 3BX Liverpool, United Kingdom 3 Beijing Computational Science Research Center, Beijing 100094, China (Dated: August 29, 2016)
In approximate density functional theory (DFT), the self-interaction error is a pervasive electron delocalization associated with underestimated insulating gaps. It exhibits a predominantly quadratic energy-density curve that is amenable to correction using computationally efficient, constraintresembling methods such as DFT + Hubbard U (DFT+U ). Constrained DFT (cDFT) exactly enforces conditions on DFT by means of self-consistently optimized Lagrange multipliers, and its use to automate DFT+U type corrections is a compelling possibility. We show that constraints beyond linear order are incompatible with cDFT. For DFT+U , we overcome this by separating its Hubbard U parameters into linear and quadratic terms. For a one-electron system, the resulting generalized DFT+U method can recover the exact subspace occupancy and free-energy, but neither the exact total-energy nor the exact ionization potential, for reasonable parameters. Approximate functionals thus cannot be systematically corrected by constraining their ground-states. PACS numbers: 71.15.-m, 31.15.E-, 71.15.Qe, 71.15.Dx
Approximate density-functional theory (DFT) [1, 2] underlies much of contemporary quantum-mechanical atomistic simulation, providing a widespread and valuable complement to experiment [3, 4]. The predictive capacity of DFT is severely limited, however, by systematic errors [5, 6] exhibited by computationally tractable exchange-correlation functionals such as the local-density approximation (LDA) [7] or generalized-gradient approximations [8]. Perhaps the most prominent of these pathologies is the delocalization or many-electron selfinteraction error (SIE) [7], which is due to spuriously curved rather than correctly piecewise-linear total-energy profiles with respect to the total electron number [5, 9]. This gives rise to the underestimated fundamental gaps characteristic of practical DFT [10], underestimated polarizabilities [11], charge-transfer energies [12, 13], and activation barriers [14]. While the construction of viable, closed-form approximate functionals free of pathologies such as SIE is extremely challenging [15], the simple, quadratic energy-density profile of the latter suggests the potential for its correction. A number of very elegant methods have been developed to target manyelectron [16–18] or one-electron [7, 19, 20] self-interaction, but none yet have shown the combination of computational feasibility, ease of use, and general applicability necessary for very widespread adoption. The removal of orbital-dependencies in the SIE-correcting potential [7, 15], numerical instabilities [21] and explicit two-centre exchange integrals [16] would be desirable. An established, computationally very efficient correction scheme for SIE, appropriate to systems in which it may be primarily attributed to identifiable subspaces, is DFT+U [6, 22–26]. Originally developed to restore the
Mott-Hubbard physics absent in the LDA description of correlated-electron oxides [25], a simplified formulation of DFT+U [6, 22, 26] is now routinely applied to a very diverse range of systems [27–31]. DFT+U replaces any quadratic interacting contributions to the energy dependence on the density-matrices of selected subspaces, contributions which represent repulsive SIEs, by linear contributions. The corrective functional is given by X UI I,σ
2
Iσ Tr n ˆ −n ˆ Iσ n ˆ Iσ , where n ˆ Iσ = Pˆ I ρˆσ Pˆ I , (1)
where ρˆσ is the Kohn-Sham density-matrix for spin σ, and Pˆ I is a projection operator for the subspace I. DFT+U attains the status of an automatible, firstprinciples method when it is provided with calculated Hubbard parameters U I [22, 26, 27, 32, 33] (particularly at their self-consistency [29, 34, 35]) which may be thought of as subspace-averaged SIEs quantified in situ. The subspaces are pre-defined for corrective treatment, having been deemed responsible for the dominant SIEs. A further level of self-consistency over subspaces is possible using Wannier functions [36]. DFT+U effectively adds a set of penalty functionals promoting integer eigenvalues of n ˆ Iσ , and replicates the effect of a derivative discontinuity in the energy due to each subspace I, by adding an occupancy-dependent potential vˆIσ = U I (Pˆ I /2 − n ˆ Iσ ). The potential acts repulsively on eigenvectors of n ˆ Iσ with occupancy less than one half, and attractively for those with greater than one-half. Constrained density-functional theory (cDFT) [37, 38] makes exact the use of penalty functionals in DFT, optimizing their pre-factors as Lagrange multipliers in order to drive their values to zero. In cDFT, specified ex-
2
● ● ● ●10-1 ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ●● ●● -15.0 10-3 ●●● ●● ● ● ● -15.2 ●●● 10-5 ● ● Total Energy 10-7 -15.4 ●● C2(Vc) ● dC2/dVc -15.6 ● 10-9 2 C /dV2 ● ● d 2 c ● ● -15.8●●● 10-11
0
100
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300
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| d nC2/dVnc (eV-n) |
Total Energy (eV)
-14.8
500
Lagrange Multiplier Vc (eV)
FIG. 1. (Color online) The constrained total-energy of the stretched H2+ system, with a target occupancy of Nc = 0.5 e per fixed atom-centred 1s-orbital subspace, against the cDFT Lagrange multiplier Vc . Also shown is the constraint functional C2 = (N − Nc )2 , averaged over two atoms, and its first and second derivatives, which rapidly fall off with Vc .
| Response Function | (eV-n)
pectation values are targeted [37, 39, 40], and the lowest energy state consistent with those values, an excited state of the unconstrained system, is located. cDFT has long provided an accessory to DFT+U by enabling the first-principles calculation of Hubbard U parameters [25, 32, 41, 42]. It is also effective for treating SIE indirectly, in systems comprising well-separated fragments, for which it may be used to impose the integer occupancies at which SIE is typically minimized [38, 43]. Moving beyond the Kohn-Sham equations, cDFT has also been used to construct efficient, minimal bases for configuration interaction calculations, with considerable accuracy gains [44]. These demonstrations of the utility of cDFT for ameliorating SIE, together with the constraint-like profile of the successful DFT+U functional, where the U I resemble Lagrange multipliers, suggest the potential use of cDFT to directly enforce conditions absent in the underlying approximate functional, such as subspace SIE freedom, in a fully automated fashion. However, as we shall demonstrate, such a possible use of cDFT may be rigorously excluded. One of the simplest conceivable SIE correcting P constraint functional is the quadratic form C2 = I (N I − Nc )2 , where N I = Tr[ˆ ρPˆ I ] is the total occupancy of a particularly error-prone subspace I, and NcI is its targeted value, neglecting the spin index σ for concision. For simplicity, this SIE constraint is a functional of the total subspace occupancy, rather than the occupancy eigenvalues, but the the distinction becomes irrelevant for single-orbital sites. For symmetry-related subspaces with NcI = Nc for all I, the total-energy of the system simply becomes W = EDFT +Vc C2 , where Vc is a common cDFT Lagrange multiplier. This gives P rise to a constraining potential of the form vˆc = 2Vc I (N I − Nc )Pˆ I , making explicit its dependence on the constraint non-satisfaction. This, in turn,P implies externally applied interaction given by fˆc = 2Vc I Pˆ I Pˆ I , which is identical to that generated by DFT+U , or Eq. 1, when Vc = −U I /2. Fig. 1 demonstrates that this simple non-linear constraint is not viable, since the total-energy W does not attain an extremum characteristic of constraint satisfaction even for very large values of Vc . The system in question is H2+ , the canonical system for studying one-electron SIE [45, 46], here stretched to an internuclear distance of 4 a0 and simulated using the PBE functional [8] with a hard norm-conserving pseudopotential. The PBE total-energy error exceeds 1 eV at this bondlength, and the two atom-centred PBE 1s orbital subspaces together over-count the occupancy by 24% overlap, accounting for spillage. In Fig. 1, a target occupancy of Nc = 0.5 e is applied to these subspace, which necessitates a repulsive constraint and a positive Vc . The same qualitative outcome is provided by Nc > NPBE and Vc < 0 eV, however. As is clear in Fig. 1, an automated optimization of Vc would drive it to infinite magnitude, with W and N I tending only asymptotically to their lim-
10-2
χ0(Vc) dN/dVc d2N/dV2c
10-4
d3N/dV3c
10-6 10-8 10-10 0
100
200
300
400
500
Lagrange Multiplier Vc (eV)
FIG. 2. (Color online) The magnitude of the cDFT subspace non-interacting response χ0 , calculated by means of Eq. 3, and the interacting density response functions, which are calculated as the first to third derivatives of a polynomial fit to the average 1s orbital subspace occupancy. The interacting responses vanish with Vc , which drives the latter to infinite magnitude as W and N I approach their asymptotic limits.
iting values. The key to the failure of this constraint is the vanishing self-consistent density response functions dn N/dVcn = dn+1 W/dVcn+1 [47], as Vc is increased and the constraint approaches satisfaction, as seen in Fig. 2. Motivated by non-viability of the constraint C2 , we may ask whether non-linear constraints of the form Cn = (N − Nc )n are generally unsatisfiable for any order n 6= 1 and for any target choice Nc 6= NDFT . The n = 0 case is trivial, and the constraint is ill-defined for n < 0 since there the total-energy diverges upon constraint satisfaction. Cn becomes imaginary for non-integer n with negative (N − Nc ), so that we may limit our discussion to integer n ≥ 2. We begin by analyzing the derivatives of the total-energy W with respect to Vc . Fol-
3 lowing Ref. [47] for the self-consistent cDFT problem, the Hellmann-Feynman theorem provides that the first derivative is simply the constraint functional itself, i.e., dW/dVc = Cn , so that the total-energy always attains a stationary point upon constraint satisfaction, that is if N = Nc . The second derivative directly follows, as dCn d2 W n−1 dN = = n (N − Nc ) . dVc2 dVc dVc
(2)
The derivative of order m generally involves response functions up to order m−1, and positive integer powers of (N − Nc ) which may vanish, depending on m and n, but not diverge. The cDFT response functions are gainfully expanded in terms of the intrinsic response, specifically the subspace non-interacting response defined by χ0 = Tr[(dN/dˆ vKS )Pˆ ], where vˆKS is the Kohn-Sham potential. The first-order response function is thus expressed, by means of the chain rule via vˆKS = (δW/δ ρˆ) − Tˆkinetic , while noting derivatives commutation, as δN dˆ vKS δN d δW dN = Tr = Tr (3) dVc δˆ vKS dVc δˆ vKS dVc δ ρˆ δN δ dW δN δCn = Tr = Tr δˆ vKS δ ρˆ dVc δˆ vKS δ ρˆ δN ˆ n−1 n−1 P = n (N − Nc ) χ0 . = n (N − Nc ) Tr δˆ vKS
functions of the same type but of lower order, which vanish at stationary points, plus a single term which is proportional to a generally non-vanishing mixed response function dm−1 χ0 /dVcm−1 but multiplied by the vanishing (N − Nc )n−1 . Thus, at stationary points, all response functions vanish due to their dependence on vanishing lower order response functions, beginning with dN/dVc . This serves as an inductive proof that response functions at all orders vanish with vanishing Cn , as illustrated in th Fig. 2. Then, since each term in the (m+1) energy derivative is always proportional to response functions of at most order m (and non-divergent powers of (N −Nc )), so too do the energy derivatives at all orders go to zero, as demonstrated in Fig. 1, proving the conjecture. In order to circumvent this, we begin by expanding the C2 functional as C2 = −2Nc Vc (N − Nc ) − Vc (Nc2 − N 2 ), ignoring summation for concision. This now resembles a generalized DFT+U functional, comprising separate linear and quadratic Hubbard U parameters, as first proposed in Ref. [48] but now augmented with a cDFT target. This suggests an alternative approach for achieving constraint satisfaction and, by extension, correcting for SIE. Recasting this in the notation of DFT+U , by simple change of variables, we arrive at a marriage of cDFT and DFT+U in which the Hubbard U parameters are interpreted as Lagrange multipliers. This takes the form C=
At any stationary point for n ≥ 2, this response function and therefore d2 W/dVc2 = n2 (N − Nc )2n−2 χ0 both vanish. The second energy derivative is thus not useful as a stationary point discriminant, and we must move to higher derivatives. The second-order response is given by d2 N n−2 dN = n (n − 1) (N − Nc ) χ0 dVc2 dVc n−1 dχ0 + n (N − Nc ) , dVc
(4)
which vanishes trivially at stationary points for n > 2. When n = 2, it vanishes with the aid of the vanishing first-order response dN/dVc . The third-order energy derivative is expressed in terms of response functions as 2 dN d3 W n−2 = n (n − 1) (N − Nc ) dVc3 dVc 2 d N n−1 + n (N − Nc ) dVc2 3n−4 2 = 2n2 n2 − n (N − Nc ) χ0 dχ 0 2n−2 , + n2 (N − Nc ) dVc
(5)
and also clearly vanishes at stationary points. In general, the cDFT response function dm N/dVcm at each order comprises terms proportional to response
X UI 1
I
2
X U2I N I − Nc + Nc2 − N I , 2
(6)
I
where the DFT+U functional, Eq. 1, is recovered by setting U1 = U2 and Nc = 0. The constraining potential is modified to vˆI = U1I Pˆ /2 − U2I n ˆ I , so that the characteristic occupancy eigenvalue dividing attractive from repulsive corrective potentials is changed from 1/2 to U I /2U2I . Fig. 3 shows the total-energy W of the aforementioned H2+ system, thereby constrained, against U1 and U2 , as defined in Eq. 6 for a subspace target occupancy Nc = Nexact = 0.602 e, which is the population of the exact system calculated in the basis of a PBE 1s orbital. The zero of the energy is the exact energy and, while the totalenergy is maximized along the heavy white line where the constraint is satisfied, it does not provide a unique global maximum, and remains approximately 0.92 eV below the exact energy. To illustrate why the energy is degenerate on a line, it suffices to show that the Hessian of the constraint functional Hij = ∂ 2 W/∂Ui ∂Uj , is singular, implying a vanishing curvature along an eigenvector in the space of U1 and U2 . The Hessian Hij is very conveniently calculated in terms of the response functions, as 1 dN/dU1 −dN 2 /dU1 |H| = 2 dN/dU2 −dN 2 /dU2 dN dN dN dN =N − = 0. (7) dU1 dU2 dU1 dU2
4
6
6 -0.92
5
5 Quadratic U2 (eV)
Quadratic U2 (eV)
-0.94
4 -0.96
3
-0.98
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4 2
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1 0
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-1
1
1
-2 -3
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2 3 4 Linear U1 (eV)
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FIG. 3. (Color online) The constrained total-energy of stretched H2+ , against the Lagrange multipliers U1 and U2 defined in Eq. 6. The per-subspace target occupancy is set to Nc = Nexact , and the energy of the exact system is set to zero. The constraint is satisfied, and the total-energy is maximized, along the heavy white line. The eigenvalue of the occupied Kohn-Sham state matches the total-energy along the thin solid line, and it matches the exact energy along the dotdashed line. The linear-response Hubbard U2 , together with the U1 value needed to correspondingly recover the exact subspace density, are shown with dashed lines.
The dashed lines in Fig. 3 indicate the linear-response Hubbard U2 , using a method adapted from Ref. [26], and the corresponding value of U1 needed to recover the exact subspace density. These values are 4.84 eV and 3.16 eV , respectively. The thin solid line shows where the eigenvalue of the occupied Kohn-Sham state matches the total-energy, while the dot-dashed line shows where the eigenvalue is exactly correct, at Eexact = −14.859960 eV . The heavy white line of Fig. 4 shows the constrained free-energy obtained by setting Nc = 0 e in Eq. 6, which corresponds to a generalized DFT+U total-energy. The energy scale is, again, set relative to the exact energy. The dashed lines indicate the linear-response Hubbard U2 and the corresponding value of U1 that recover the exact energy which, in this case, occur at 4.84 eV and 4.44 eV , respectively. The dot-dashed line, where U1 = U2 , corresponds to a traditional DFT+U calculation, and it intersects with the exact energy at U = 3.85 eV. The intersection of the heavy white lines from Figs. 3 and 4 thus yield the pair of Hubbard parameters at which the freeenergy equals the exact energy, for the exact subspace occupancy. This point lies at approximately, U1 = 5.73 eV
0 0
1
2 3 4 Linear U1 (eV)
5
6
FIG. 4. (Color online) As in Fig. 2 but for the free-energy obtained by setting Nc = 0 e in Eq.6. This corresponds to a generalized DFT+U total-energy. The heavy white line shows the intercept with the exact energy, and the dashed line shows the linear-response U2 together with the corresponding U1 needed to recover that energy. The thin white line (repeating the heavy line in Fig. 2) indicates where the exact 1s orbital density is recovered. The dot-dashed line follows U1 = U2 .
and U2 = 6.98 eV. There, the occupied Kohn-Sham eigenvalue lies at approximately 4 eV lower than the exact energy, which is a remarkably large deviation for a one-electron system given that the exact subspace occupancy and free-energy are achieved. The eigenvalue can be made to match the free-energy, but it does so at the rather unphysical U1 = U2 = −9 eV , which is not shown. Thus, one may attain, using the free-energy, the exact energy and exact occupancy, or the Kohn-Sham one-electron eigenvalue, but not all three simultaneously. To conclude, quadratic or higher-order constraints are required to change the inter-electron interaction in DFT, and thus to systematically correct SIE. We have proven, however, both analytically and by means of stringent numerical tests, that non-linear density constraints are impossible to satisfy. Progress may be made by decoupling the constraining terms at different orders in the density, as we have shown, but the resultant non-uniqueness of stationary points requires the introduction of additional physics, such as linear-response theory for U2 . Interestingly, the free-energy of this functional, equivalent to a generalized DFT+U energy, allows both the exact occupancy and energy to be recovered for a reasonable pair of parameters, improving upon the performance of con-
5 ventional DFT+U . Our conclusion that an SIE-inflicted ground-state cannot be systematically excited to state that is less so, by means of constraints, is reinforced by the fact that it is the target-independent free-energy, and not total-energy, that agrees with the exact one. The additional freedom provided by a third parameter, perhaps responsible for a constant energy shift as suggested in Ref. [48], could make up the difference. The gross disagreement of the occupied Kohn-Sham eigenvalue with the exact energy for all relevant calculations, in any case, reveals a glaring failure of DFT+U type functionals to simultaneously reconcile the energy and the ionization potential with that of the exact system. It appears that we must decide whether to use DFT+U , even generalized, to either correct the total-energy or the eigenspectrum, since it cannot achieve both for a one-electron system. This work was enabled by the Royal Irish Academy – Royal Society International Exchange Cost Share Programme (IE131505). GT acknowledges support from EPSRC UK (EP/I004483/1 and EP/K013610/1). GM and DDO’R wish to acknowledge support from the Science Foundation Ireland (SFI) funded centre AMBER (SFI/12/RC/2278). All calculations were performed on the Lonsdale cluster maintained by the Trinity Centre for High Performance Computing. This cluster was funded through grants from Science Foundation Ireland.
∗
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