Tight constraints on the exchange-correlation potentials of degenerate states Paul W. Ayers and Mel Levy Citation: The Journal of Chemical Physics 140, 18A537 (2014); doi: 10.1063/1.4871732 View online: http://dx.doi.org/10.1063/1.4871732 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/18?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Exchange-correlations in a dilute quasi-two-dimensional electron gas at finite temperature AIP Conf. Proc. 1447, 731 (2012); 10.1063/1.4710211 Experiments on metastable states of three-dimensional trapped particle clusters Phys. Plasmas 15, 040701 (2008); 10.1063/1.2903549 The performance of time-dependent density functional theory based on a noncollinear exchange-correlation potential in the calculations of excitation energies J. Chem. Phys. 122, 074109 (2005); 10.1063/1.1844299 Time-dependent density functional theory based on a noncollinear formulation of the exchange-correlation potential J. Chem. Phys. 121, 12191 (2004); 10.1063/1.1821494 Emphasizing the exchange-correlation potential in functional development J. Chem. Phys. 114, 3958 (2001); 10.1063/1.1342776
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THE JOURNAL OF CHEMICAL PHYSICS 140, 18A537 (2014)
Tight constraints on the exchange-correlation potentials of degenerate states Paul W. Ayers1,a) and Mel Levy2,3,4,a) 1
Department of Chemistry and Chemical Biology, McMaster University, Hamilton, Ontario L8S4M1, Canada Department of Chemistry, Duke University, Durham, North Carlina 27708, USA 3 Department of Physics, North Carolina A&T State University, Greensboro, North Carolina 27411, USA 4 Department of Chemistry, Tulane University, New Orleans, Louisiana 70118, USA 2
(Received 31 December 2013; accepted 7 April 2014; published online 25 April 2014) Identities for the difference of exchange-correlation potentials and energies in degenerate and nondegenerate ground states are derived. The constraints are strong for degenerate ground states, and suggest that local and semilocal approximations to the exchange-correlation energy functional are incapable of correctly treating degenerate ground states. For degenerate states, it is possible to provide both local (pointwise) equality and global inequality constraints for the exchange-correlation potential in terms of the Coulomb potential. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4871732] I. INTRODUCTION
Degenerate ground states have been problematic in DFT from its inception. The original Hohenberg-Kohn proof applied only to nondegenerate ground states1 and while that proof is easily extended, the first fully rigorous treatment of degenerate ground states in DFT came not from a HohenbergKohn-style proof-by-contradiction, but from the constrained search formulation.2 Since then, much work has been done to understand the structural nuances associated with degenerate ground states, especially when those ground states are induced by the symmetry of the system.3–17 It is known, for example, that the Kohn-Sham potentials of some (perhaps most) of the ground-state densities of a symmetry-induced degeneracy do not have the full symmetry of the system. In addition, for degenerate ground states, many ground-state densities seem not to be noninteracting-v-representable (as is evident from holes below the Fermi level18 ). It seems that many approximate functionals artificially break the degeneracy of the ground-state energy level, so that some groundlevel ensemble-v-representable densities have lower energies than others. In this paper, we focus on the exchange-correlation energy and potential in degenerate states. Section II presents inequalities that can be deduced from the variational principle. These constraints on the exact exchange-correlation functional are especially interesting for degenerate states (Sec. III). The key result in this section is an inequality relating the exchange-correlation potentials of degenerate states to the Coulomb potentials, Eq. (13). Since the Coulomb energy is an inherently nonlocal functional of the electron density, this implies that writing the approximate exchangecorrelation energy in terms of just the electron density and its derivatives is inherently limited, and entirely unsuitable for systems with degenerate ground states. Section IV focuses a) Authors to whom correspondence should be addressed. Electronic ad-
on equalities, including the pointwise equality (28), that the exchange-correlation potentials of degenerate states must satisfy. II. IDENTITIES BETWEEN GROUND STATE DENSITIES
Let ρ (a) (r) and ρ (b) (r) be N-electron ensemble-vrepresentable19 ground-level densities for the systems with external potentials v (a) (r) and v (b) (r).19 The variational principle for the energy indicates that1, 19 Fλ [ρ (b) ]+ ρ (b) (r)vλ(a) (r)dr ≥ Fλ [ρ (a) ]+ ρ (a) (r)vλ(a) (r)dr . (b) (b) (a) (b) (b) (a) Fλ [ρ ]+ ρ (r)vλ (r)dr ≥ Fλ [ρ ]+ ρ (r)vλ (r)dr (1) Here, λ is the parameter that turns on the electron-electron repulsion potential in the adiabatic connection.20–23 The λ = 0 case corresponds to the noninteracting (Kohn-Sham24 ) reference system. The λ = 1 case corresponds to the fully interacting system of physical interest. So that all of the ensemble-v-representable ground-level densities have the same energy, we will choose to use the constrained-search functional2 extended to ensembles25 (or an equivalent functional26, 27 derived from the Legendre transform or continuity arguments). As stressed by Yang et al., functionals that violate these conditions are not size-consistent in the sense that the functional is not additive for separated systems.28 Adding Eq. (1) gives19 (2) (ρ (b) (r) − ρ (a) (r)) vλ(a) (r) − vλ(b) (r) dr ≥ 0, the fact that the inequality must be strict when the two external potentials differ by more than an additive constant directly implies the Hohenberg-Kohn theorem. For two or more systems, one has the result19 Nsystems (π{a}) (r) − vλ(a) (r) dr ≥ 0, (3) ρ (a) (r) vλ
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where π {a} permutes the labels for the states. In terms of the more explicit, two-line, notation for permutations, Nsystems 1 2 ··· Nsystems − 1 , π {1} π {2} π {Nsystems − 1} π {Nsystems } (4) there is a similar result for the classical Coulomb interaction between densities. For two densities, (ρ (a) (r) − ρ (b) (r))(ρ (a) (r ) − ρ (b) (r )) drdr |r − r | = (ρ (a) (r) − ρ (b) (r)) vJ(a) (r) − vJ(b) (r) dr ≥ 0.
(5)
For two or more densities, one has an expression similar to inequality (3), but the sign of the inequality is reversed: (π{a}) (r) − vJ(a) (r) dr ≤ 0. (6) ρ (a) (r) vJ a
From inequalities (3) and (6), we may write: (π{a}) (π{a}) (r)−vJ (r)− vλ(a) (r)−vJ(a) (r) dr ≥ 0. ρ (a) (r) vλ
III. IDENTITIES FOR DEGENERATE GROUND STATES
A useful, but simple, constraint is obtained when all the electron densities in (10) correspond to the same degenerate ground state energy level; the left-hand-side of the inequality then vanishes. A more stringent condition can be derived directly from inequalities (3) and (5). Using the fact that the external potentials are equal for a degenerate ground state, 0≤
a
=
a
(π{a}) (a) (r) dr ρ (a) (r) vλ=0 (r) − vλ=0 (π{a}) (π{a}) (a) (r)+vxc (r)− vJ(a) (r)−vxc (r) dr. ρ (a) (r) vJ (12)
Thus, for a degenerate ground state, (π{a}) 0≤− (r) − vJ(a) (r) dr ρ (a) (r) vJ a
≤
(π{a}) (a) (r) − vxc (r) dr. ρ (a) (r) vxc
(13)
a
a
(7) An especially compact result is obtained in the noninteracting (λ = 0) limit. Substituting vs (r) ≡ vλ=0 (r) = vλ=1 (r) + vJ (r) + vxc (r)
(8)
in inequality (7) gives a constraint on the exchangecorrelation potential, (π{a}) (a) (π{a}) (a) (r)− vλ=1 (r)+vxc (r) dr ≥ 0. ρ (a) (r) vλ=1 (r)+vxc a
(9) This may be rewritten as (π{a}) (a) (r) dr − ρ (a) (r) vλ=1 (r) − vλ=1 a . (π{a}) (a) (a) ρ (r) vxc (r) − vxc (r) dr ≤
(10)
a
When only two densities are being considered, this reduces to the simple formula (a) (b) − (ρ (b) (r) − ρ (a) (r)) vλ=1 (r) − vλ=1 (r) dr . (11) (a) (b) (a) (b) ≤ (ρ (r) − ρ (r)) vxc (r) − vxc (r) dr Relationships (10) and (11) are equivalent to the variational principle for the energy, so they will be satisfied by the approximate ground-level densities obtained by minimizing the energy of an approximate exchange-correlation energy functional. Violations of Eq. (10) indicate that densities from the approximate Kohn-Sham calculations do not correspond to energy minimizations.29 If accurate electron densities for a set of external potentials are known, however, then inequality (10) is a strong constraint. This equation, then, provides a new way to use accurate electron densities from exactly solvable systems (e.g., Hooke’s atom or spherium)30–33 or accurate ab initio data.
Equality rarely, if ever, holds in the first line of Eq. (13). For two densities, this reduces to the simple expression 0≤− ≤
(ρ (b) (r) − ρ (a) (r)) vJ(a) (r) − vJ(b) (r)
(a) (b) (ρ (b) (r) − ρ (a) (r)) vxc (r) − vxc (r) dr.
(14)
Because the Coulomb potential is a nonlocal, nonnearsighted, functional of the electron density, it seems doubtful that any local functional for the exchange-correlation energy/potential will satisfy Eq. (14). This coincides with the observation that, in spatially degenerate states, the exchange-correlation potential must cancel a term arising from the Coulomb energy.11, 13, 16 As in the nondegenerate case, relationships (13) and (14) rely on nothing more than the variational principle and the defining relationships between the fundamental density functionals. If an approximate functional predicts that two densities are degenerate, then these relationships will be automatically satisfied. The more interesting case arises when electron densities are known to be degenerate, but are predicted to be nondegenerate by an approximate calculation. Sometimes the degeneracy is clear from symmetry; for example, all of the electron densities from an atomic multiplet should have the same energy, but this is rarely achieved by approximate exchange-correlation functionals.6, 34–37 Other times the degeneracy arises because all ensemble averages of degenerate ground-state densities should have the same energy. Suppose that one has g degenerate ground state densities. These densities could be known to have the same energy from symmetry arguments, or they could be densities that were found to be degenerate from a computation with an approximate exchange-correlation functional. Any convex linear combination of these electron densities should
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also be degenerate. Define ρ (r) = (a)
1=
g
Using Eqs. (20) and (21), one has g
lim (vJ [ρ; αr] + α −1 vxc [ρα ; r]), lim α −1 vλ=1 [ρα ; r] = −
pi(a) ρi (r)
i=1
pi(a)
α→0
.
(15)
pi(a) ≥ 0
i=1
Inequality (13) must hold for every possible choice of the {pi(a) }. This provides a very strong constraint on the exchangecorrelation potential. All of these results can be extended to spin-resolved form as long as the exact functional (e.g., the spin-extended Legendre transform38, 39 ) gives equal energies for all possible spin states. In this case, the variational principle for the energy implies that (π{a}) (a) (r) dr, (16) ρσ(a) (r) vλ;σ (r) − vλ;σ 0≤ a
σ =α,β
where vλ;σ (r) are the spin-external potentials at a given point on the adiabatic connection. Using the convenient matrixvector notation for the spin,40, 41 p(r) · q(r) = pσ (r)qσ (r), (17) σ =α,β
all of the previous results can be extended to electron spin densities by simply “decorating” the density and potentials with vectors. Equations (13)–(15) then provide strong constraints on the form of the spin-resolved exchange-correlation potentials, vxc;σ (r) =
δExc [ρα , ρβ ] . δρσ (r)
(18)
These constraints are especially relevant since approximate exchange-correlation potentials usually give very poor results for ensemble densities with fractional spin.15, 42–46 IV. COORDINATE-SCALING CONDITIONS AND POINTWISE EQUALITIES
The preceding results provide integral inequalities for the exchange-correlation energy. There are also strong, rdependent, equality constraints that can be derived. Consider the coordinate-scaling behavior of the electron density,47 ρα (x, y, z) = α 3 ρ(αx, αy, αz),
(19)
and the Kohn-Sham and Coulomb potentials, vλ=0 [ρα ; r] = α 2 vλ=0 [ρ; αr],
(20)
vJ [ρα ; r] = αvJ [ρ; αr].
(21)
Consider the behavior of the external potential in the lowdensity limit (α → 0): lim α→0
vλ=1 [ρα ; r] vλ=0 [ρα ; r] − vJ [ρα ; r] − vxc [ρα ; r] = lim . α α α→0
(22)
α→0
(23) because the contribution from the Kohn-Sham potential vanishes in this limit. This is a pointwise equality constraint linking the external potential to the exchange-correlation potential. Using this constraint with inequality (2) gives
(b) ρα (r) − ρα(a) (r) vλ=1 ρα(a) ; r 0 ≤ lim α −1 α→0
−vλ=1 ρα(b) ; r dr,
(24)
which simplifies to (b) ρα (r) − ρα(a) (r) (vJ [ρ (a) ; αr] − vJ [ρ (b) ; αr])dr lim α→0
≤ − lim α
−1
(b) ρα (r)−ρα(a) (r) vxc ρα(a) ; r −vxc ρα(b) ; r dr.
α→0
(25) Note that this constraint has a similar form to inequality (14), which was only valid for degenerate ground states. However, inequality (25) is true for the approximate densities obtained from any approximate exchange-correlation potential with correct coordinate-scaling behavior. A similar, but more general, inequality can be derived from (3). Unsurprisingly, even stronger result can be derived for degenerate ground states. Consider the convex linear combination of any two degenerate ground state densities, ρ (a) (r) = (1 − a)ρ (0) (r) + aρ (1) (r)
0 ≤ a ≤ 1.
(26)
Then because the external potential is the same for all these densities, one has
(a) ;r ∂v ρλ=1 , (27) 0= ∂a and thus, ∂vxc [ρ (a) ; r] ∂vλ=0 [ρ (a) ; r] ∂vJ [ρ (a) ; r] = − (28) ∂a ∂a ∂a This pointwise equality is stronger than inequality (14). V. SUMMARY
Equality (28) and inequality (13) are the key results of this work. These constraints are very stringent, and seemingly impossible to satisfy when exchange-correlation functionals are approximated in terms of just the electron density and its derivatives. Inequality (13) links the exchange-correlation functional to the Coulomb potential, and is especially easy to use for that reason. All of our key inequalities are derived directly from the variational principle, Eq. (1); all of our equalities are derived using the equality of the energy of degenerate ground states. Failure to satisfy these constraints is associated with incorrect ground state densities (Sec. II) or artificial breaking of the degeneracy of the system so that some of the ensemblev-representable ground-level densities of the system will
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have lower energies than others (Secs. III and IV). Recent work suggests that density functional approximations that overstabilize or understabilize certain mixed states can give qualitatively wrong chemical predictions.11, 13, 15–17, 28, 42, 48–55 It is interesting that constraint (13) is automatically satisfied if the Kohn-Sham potentials for all of the degenerate ground-level densities are equal. This does not seem to be true, in general, but it does recommend that theories which produce fully symmetric Kohn-Sham potentials be investigated more fully.3–5, 8–10 Such approaches, however, might not suffice for “accidental” degeneracies. ACKNOWLEDGMENTS
P.W.A. acknowledges financial support from Sharcnet, NSERC, and the Canada Research Chairs. P.W.A. acknowledges interesting discussions with Andreas Savin and Weitao Yang on the “symmetry dilemma” in density-functional theory. M.L. thanks Professor Weitao Yang and Professor David Beratan for thieir warm hospitality at Duke University. 1 P.
Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). Levy, Proc. Natl. Acad. Sci. U.S.A. 76, 6062 (1979). 3 A. Gorling, Phys. Rev. A 47, 2783 (1993). 4 A. Gorling, Phys. Rev. Lett. 85, 4229 (2000). 5 F. Della Sala and A. Gorling, J. Chem. Phys. 118, 10439 (2003). 6 A. D. Becke, J. Chem. Phys. 117, 6935 (2002). 7 E. R. Johnson, R. M. Dickson, and A. D. Becke, J. Chem. Phys. 126, 184104 (2007). 8 A. K. Theophilou, Int. J. Quantum Chem. 61, 333 (1997). 9 A. K. Theophilou, Int. J. Quantum Chem. 69, 461 (1998). 10 A. K. Theophilou, and P. Papaconstantinou, J. Mol. Struct.: THEOCHEM 501–502, 85 (2000). 11 A. Savin and J. M. Seminario, in Recent Developments and Applications of Modern Density Functional Theory (Elsevier, New York, 1996), p. 327. 12 P. Gori-Giorgi and A. Savin, J. Phys. Conf. Series 117, 12017 (2008). 13 A. Savin, Chem. Phys. 356, 91 (2009). 14 A. Nagy, S. Liu, and L. Bartolloti, J. Chem. Phys. 122, 134107 (2005). 15 A. J. Cohen, P. Mori-Sanchez, and W. T. Yang, Science 321, 792 (2008). 16 R. Merkle, A. Savin, and H. Preuss, J. Chem. Phys. 97, 9216 (1992). 17 Y. K. Zhang and W. T. Yang, J. Chem. Phys. 109, 2604 (1998). 18 M. Levy and J. P. Perdew, NATO ASI Ser., Ser. B 123, 11 (1985). 19 M. Levy, Phys. Rev. A 26, 1200 (1982). 20 J. Harris and R. O. Jones, J. Phys. F 4, 1170 (1974). 2 M.
J. Chem. Phys. 140, 18A537 (2014) 21 D.
C. Langreth, and J. P. Perdew, Solid State Commun. 17, 1425 (1975). Gunnarsson, and B. I. Lundqvist, Phys. Rev. B 13, 4274 (1976). 23 D. C. Langreth, and J. P. Perdew, Phys. Rev. B 15, 2884 (1977). 24 W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). 25 S. M. Valone, J. Chem. Phys. 73, 4653 (1980). 26 E. H. Lieb, Int. J. Quantum Chem. 24, 243 (1983). 27 P. W. Ayers, Phys. Rev. A 73, 012513 (2006). 28 W. T. Yang, Y. K. Zhang, and P. W. Ayers, Phys. Rev. Lett. 84, 5172 (2000). 29 L. O. Wagner, E. M. Stoudemire, K. Burke, and S. R. White, Phys. Rev. Lett. 111, 093003 (2013). 30 C. Filippi, C. J. Umrigar, and M. Taut, J. Chem. Phys. 100, 1290 (1994). 31 J. Cioslowski, and K. Pernal, J. Chem. Phys. 113, 8434 (2000). 32 M. Seidl, Phys. Rev. A 75, 062506 (2007). 33 N. R. Kestner and O. Sinanoglu, Phys. Rev. 128, 2687 (1962). 34 D. J. Wales, Mol. Phys. 103, 269 (2005). 35 S. Pittalis, S. Kurth, and E. K. U. Gross, J. Chem. Phys. 125, 084105 (2006). 36 S. Pittalis, S. Kurth, S. Sharma, and E. K. U. Gross, J. Chem. Phys. 127, 124103 (2007). 37 A. D. Becke, A. Savin, and H. Stoll, Theor. Chim. Acta 91, 147 (1995). 38 P. W. Ayers and W. T. Yang, J. Chem. Phys. 124, 224108 (2006). 39 A. Holas and R. Balawender, J. Chem. Phys. 125, 247101 (2006). 40 J. Garza, R. Vargas, A. Cedillo, M. Galvan, and P. K. Chattaraj, Theor. Chem. Acc. 115, 257 (2006). 41 P. Perez, E. Chamorro, and P. W. Ayers, J. Chem. Phys. 128, 204108 (2008). 42 A. J. Cohen, P. Mori-Sanchez, and W. T. Yang, J. Chem. Phys. 129, 121104 (2008). 43 K. Capelle, G. Vignale, and C. A. Ullrich, Phys. Rev. B 81, 125114 (2010). 44 R. Haunschild, T. M. Henderson, C. A. Jimenez-Hoyos, and G. E. Scuseria, J. Chem. Phys. 133, 134116 (2010). 45 E. R. Johnson, W. T. Yang, and E. R. Davidson, J. Chem. Phys. 133, 164107 (2010). 46 D. H. Ess, E. R. Johnson, X. Q. Hu, and W. T. Yang, J. Phys. Chem. A 115, 76 (2011). 47 M. Levy and J. P. Perdew, Phys. Rev. A 32, 2010 (1985). 48 P. Mori-Sanchez, A. J. Cohen, and W. T. Yang, J. Chem. Phys. 125, 201102 (2006). 49 A. J. Cohen, P. Mori-Sanchez, and W. T. Yang, Phys. Rev. B 77, 115123 (2008). 50 P. Mori-Sanchez, A. J. Cohen, and W. T. Yang, Phys. Rev. Lett. 100, 146401 (2008). 51 P. Mori-Sanchez, A. J. Cohen, and W. T. Yang, Phys. Rev. Lett. 102, 066403 (2009). 52 A. Ruzsinszky, J. P. Perdew, G. I. Csonka, O. A. Vydrov, and G. E. Scuseria, J. Chem. Phys. 126, 104102 (2007). 53 J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, V. N. Staroverov, and J. M. Tao, Phys. Rev. A 76, 040501 (2007). 54 D. J. Tozer, N. C. Handy, and A. J. Cohen, Chem. Phys. Lett. 382, 203 (2003). 55 B. Braida, P. C. Hiberty, and A. Savin, J. Phys. Chem. A 102, 7872 (1998). 22 O.
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THE JOURNAL OF CHEMICAL PHYSICS 140, 18A537 (2014)
Tight constraints on the exchange-correlation potentials of degenerate states Paul W. Ayers1,a) and Mel Levy2,3,4,a) 1
Department of Chemistry and Chemical Biology, McMaster University, Hamilton, Ontario L8S4M1, Canada Department of Chemistry, Duke University, Durham, North Carlina 27708, USA 3 Department of Physics, North Carolina A&T State University, Greensboro, North Carolina 27411, USA 4 Department of Chemistry, Tulane University, New Orleans, Louisiana 70118, USA 2
(Received 31 December 2013; accepted 7 April 2014; published online 25 April 2014) Identities for the difference of exchange-correlation potentials and energies in degenerate and nondegenerate ground states are derived. The constraints are strong for degenerate ground states, and suggest that local and semilocal approximations to the exchange-correlation energy functional are incapable of correctly treating degenerate ground states. For degenerate states, it is possible to provide both local (pointwise) equality and global inequality constraints for the exchange-correlation potential in terms of the Coulomb potential. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4871732] I. INTRODUCTION
Degenerate ground states have been problematic in DFT from its inception. The original Hohenberg-Kohn proof applied only to nondegenerate ground states1 and while that proof is easily extended, the first fully rigorous treatment of degenerate ground states in DFT came not from a HohenbergKohn-style proof-by-contradiction, but from the constrained search formulation.2 Since then, much work has been done to understand the structural nuances associated with degenerate ground states, especially when those ground states are induced by the symmetry of the system.3–17 It is known, for example, that the Kohn-Sham potentials of some (perhaps most) of the ground-state densities of a symmetry-induced degeneracy do not have the full symmetry of the system. In addition, for degenerate ground states, many ground-state densities seem not to be noninteracting-v-representable (as is evident from holes below the Fermi level18 ). It seems that many approximate functionals artificially break the degeneracy of the ground-state energy level, so that some groundlevel ensemble-v-representable densities have lower energies than others. In this paper, we focus on the exchange-correlation energy and potential in degenerate states. Section II presents inequalities that can be deduced from the variational principle. These constraints on the exact exchange-correlation functional are especially interesting for degenerate states (Sec. III). The key result in this section is an inequality relating the exchange-correlation potentials of degenerate states to the Coulomb potentials, Eq. (13). Since the Coulomb energy is an inherently nonlocal functional of the electron density, this implies that writing the approximate exchangecorrelation energy in terms of just the electron density and its derivatives is inherently limited, and entirely unsuitable for systems with degenerate ground states. Section IV focuses a) Authors to whom correspondence should be addressed. Electronic ad-
on equalities, including the pointwise equality (28), that the exchange-correlation potentials of degenerate states must satisfy. II. IDENTITIES BETWEEN GROUND STATE DENSITIES
Let ρ (a) (r) and ρ (b) (r) be N-electron ensemble-vrepresentable19 ground-level densities for the systems with external potentials v (a) (r) and v (b) (r).19 The variational principle for the energy indicates that1, 19 Fλ [ρ (b) ]+ ρ (b) (r)vλ(a) (r)dr ≥ Fλ [ρ (a) ]+ ρ (a) (r)vλ(a) (r)dr . (b) (b) (a) (b) (b) (a) Fλ [ρ ]+ ρ (r)vλ (r)dr ≥ Fλ [ρ ]+ ρ (r)vλ (r)dr (1) Here, λ is the parameter that turns on the electron-electron repulsion potential in the adiabatic connection.20–23 The λ = 0 case corresponds to the noninteracting (Kohn-Sham24 ) reference system. The λ = 1 case corresponds to the fully interacting system of physical interest. So that all of the ensemble-v-representable ground-level densities have the same energy, we will choose to use the constrained-search functional2 extended to ensembles25 (or an equivalent functional26, 27 derived from the Legendre transform or continuity arguments). As stressed by Yang et al., functionals that violate these conditions are not size-consistent in the sense that the functional is not additive for separated systems.28 Adding Eq. (1) gives19 (2) (ρ (b) (r) − ρ (a) (r)) vλ(a) (r) − vλ(b) (r) dr ≥ 0, the fact that the inequality must be strict when the two external potentials differ by more than an additive constant directly implies the Hohenberg-Kohn theorem. For two or more systems, one has the result19 Nsystems (π{a}) (r) − vλ(a) (r) dr ≥ 0, (3) ρ (a) (r) vλ
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where π {a} permutes the labels for the states. In terms of the more explicit, two-line, notation for permutations, Nsystems 1 2 ··· Nsystems − 1 , π {1} π {2} π {Nsystems − 1} π {Nsystems } (4) there is a similar result for the classical Coulomb interaction between densities. For two densities, (ρ (a) (r) − ρ (b) (r))(ρ (a) (r ) − ρ (b) (r )) drdr |r − r | = (ρ (a) (r) − ρ (b) (r)) vJ(a) (r) − vJ(b) (r) dr ≥ 0.
(5)
For two or more densities, one has an expression similar to inequality (3), but the sign of the inequality is reversed: (π{a}) (r) − vJ(a) (r) dr ≤ 0. (6) ρ (a) (r) vJ a
From inequalities (3) and (6), we may write: (π{a}) (π{a}) (r)−vJ (r)− vλ(a) (r)−vJ(a) (r) dr ≥ 0. ρ (a) (r) vλ
III. IDENTITIES FOR DEGENERATE GROUND STATES
A useful, but simple, constraint is obtained when all the electron densities in (10) correspond to the same degenerate ground state energy level; the left-hand-side of the inequality then vanishes. A more stringent condition can be derived directly from inequalities (3) and (5). Using the fact that the external potentials are equal for a degenerate ground state, 0≤
a
=
a
(π{a}) (a) (r) dr ρ (a) (r) vλ=0 (r) − vλ=0 (π{a}) (π{a}) (a) (r)+vxc (r)− vJ(a) (r)−vxc (r) dr. ρ (a) (r) vJ (12)
Thus, for a degenerate ground state, (π{a}) 0≤− (r) − vJ(a) (r) dr ρ (a) (r) vJ a
≤
(π{a}) (a) (r) − vxc (r) dr. ρ (a) (r) vxc
(13)
a
a
(7) An especially compact result is obtained in the noninteracting (λ = 0) limit. Substituting vs (r) ≡ vλ=0 (r) = vλ=1 (r) + vJ (r) + vxc (r)
(8)
in inequality (7) gives a constraint on the exchangecorrelation potential, (π{a}) (a) (π{a}) (a) (r)− vλ=1 (r)+vxc (r) dr ≥ 0. ρ (a) (r) vλ=1 (r)+vxc a
(9) This may be rewritten as (π{a}) (a) (r) dr − ρ (a) (r) vλ=1 (r) − vλ=1 a . (π{a}) (a) (a) ρ (r) vxc (r) − vxc (r) dr ≤
(10)
a
When only two densities are being considered, this reduces to the simple formula (a) (b) − (ρ (b) (r) − ρ (a) (r)) vλ=1 (r) − vλ=1 (r) dr . (11) (a) (b) (a) (b) ≤ (ρ (r) − ρ (r)) vxc (r) − vxc (r) dr Relationships (10) and (11) are equivalent to the variational principle for the energy, so they will be satisfied by the approximate ground-level densities obtained by minimizing the energy of an approximate exchange-correlation energy functional. Violations of Eq. (10) indicate that densities from the approximate Kohn-Sham calculations do not correspond to energy minimizations.29 If accurate electron densities for a set of external potentials are known, however, then inequality (10) is a strong constraint. This equation, then, provides a new way to use accurate electron densities from exactly solvable systems (e.g., Hooke’s atom or spherium)30–33 or accurate ab initio data.
Equality rarely, if ever, holds in the first line of Eq. (13). For two densities, this reduces to the simple expression 0≤− ≤
(ρ (b) (r) − ρ (a) (r)) vJ(a) (r) − vJ(b) (r)
(a) (b) (ρ (b) (r) − ρ (a) (r)) vxc (r) − vxc (r) dr.
(14)
Because the Coulomb potential is a nonlocal, nonnearsighted, functional of the electron density, it seems doubtful that any local functional for the exchange-correlation energy/potential will satisfy Eq. (14). This coincides with the observation that, in spatially degenerate states, the exchange-correlation potential must cancel a term arising from the Coulomb energy.11, 13, 16 As in the nondegenerate case, relationships (13) and (14) rely on nothing more than the variational principle and the defining relationships between the fundamental density functionals. If an approximate functional predicts that two densities are degenerate, then these relationships will be automatically satisfied. The more interesting case arises when electron densities are known to be degenerate, but are predicted to be nondegenerate by an approximate calculation. Sometimes the degeneracy is clear from symmetry; for example, all of the electron densities from an atomic multiplet should have the same energy, but this is rarely achieved by approximate exchange-correlation functionals.6, 34–37 Other times the degeneracy arises because all ensemble averages of degenerate ground-state densities should have the same energy. Suppose that one has g degenerate ground state densities. These densities could be known to have the same energy from symmetry arguments, or they could be densities that were found to be degenerate from a computation with an approximate exchange-correlation functional. Any convex linear combination of these electron densities should
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J. Chem. Phys. 140, 18A537 (2014)
also be degenerate. Define ρ (r) = (a)
1=
g
Using Eqs. (20) and (21), one has g
lim (vJ [ρ; αr] + α −1 vxc [ρα ; r]), lim α −1 vλ=1 [ρα ; r] = −
pi(a) ρi (r)
i=1
pi(a)
α→0
.
(15)
pi(a) ≥ 0
i=1
Inequality (13) must hold for every possible choice of the {pi(a) }. This provides a very strong constraint on the exchangecorrelation potential. All of these results can be extended to spin-resolved form as long as the exact functional (e.g., the spin-extended Legendre transform38, 39 ) gives equal energies for all possible spin states. In this case, the variational principle for the energy implies that (π{a}) (a) (r) dr, (16) ρσ(a) (r) vλ;σ (r) − vλ;σ 0≤ a
σ =α,β
where vλ;σ (r) are the spin-external potentials at a given point on the adiabatic connection. Using the convenient matrixvector notation for the spin,40, 41 p(r) · q(r) = pσ (r)qσ (r), (17) σ =α,β
all of the previous results can be extended to electron spin densities by simply “decorating” the density and potentials with vectors. Equations (13)–(15) then provide strong constraints on the form of the spin-resolved exchange-correlation potentials, vxc;σ (r) =
δExc [ρα , ρβ ] . δρσ (r)
(18)
These constraints are especially relevant since approximate exchange-correlation potentials usually give very poor results for ensemble densities with fractional spin.15, 42–46 IV. COORDINATE-SCALING CONDITIONS AND POINTWISE EQUALITIES
The preceding results provide integral inequalities for the exchange-correlation energy. There are also strong, rdependent, equality constraints that can be derived. Consider the coordinate-scaling behavior of the electron density,47 ρα (x, y, z) = α 3 ρ(αx, αy, αz),
(19)
and the Kohn-Sham and Coulomb potentials, vλ=0 [ρα ; r] = α 2 vλ=0 [ρ; αr],
(20)
vJ [ρα ; r] = αvJ [ρ; αr].
(21)
Consider the behavior of the external potential in the lowdensity limit (α → 0): lim α→0
vλ=1 [ρα ; r] vλ=0 [ρα ; r] − vJ [ρα ; r] − vxc [ρα ; r] = lim . α α α→0
(22)
α→0
(23) because the contribution from the Kohn-Sham potential vanishes in this limit. This is a pointwise equality constraint linking the external potential to the exchange-correlation potential. Using this constraint with inequality (2) gives
(b) ρα (r) − ρα(a) (r) vλ=1 ρα(a) ; r 0 ≤ lim α −1 α→0
−vλ=1 ρα(b) ; r dr,
(24)
which simplifies to (b) ρα (r) − ρα(a) (r) (vJ [ρ (a) ; αr] − vJ [ρ (b) ; αr])dr lim α→0
≤ − lim α
−1
(b) ρα (r)−ρα(a) (r) vxc ρα(a) ; r −vxc ρα(b) ; r dr.
α→0
(25) Note that this constraint has a similar form to inequality (14), which was only valid for degenerate ground states. However, inequality (25) is true for the approximate densities obtained from any approximate exchange-correlation potential with correct coordinate-scaling behavior. A similar, but more general, inequality can be derived from (3). Unsurprisingly, even stronger result can be derived for degenerate ground states. Consider the convex linear combination of any two degenerate ground state densities, ρ (a) (r) = (1 − a)ρ (0) (r) + aρ (1) (r)
0 ≤ a ≤ 1.
(26)
Then because the external potential is the same for all these densities, one has
(a) ;r ∂v ρλ=1 , (27) 0= ∂a and thus, ∂vxc [ρ (a) ; r] ∂vλ=0 [ρ (a) ; r] ∂vJ [ρ (a) ; r] = − (28) ∂a ∂a ∂a This pointwise equality is stronger than inequality (14). V. SUMMARY
Equality (28) and inequality (13) are the key results of this work. These constraints are very stringent, and seemingly impossible to satisfy when exchange-correlation functionals are approximated in terms of just the electron density and its derivatives. Inequality (13) links the exchange-correlation functional to the Coulomb potential, and is especially easy to use for that reason. All of our key inequalities are derived directly from the variational principle, Eq. (1); all of our equalities are derived using the equality of the energy of degenerate ground states. Failure to satisfy these constraints is associated with incorrect ground state densities (Sec. II) or artificial breaking of the degeneracy of the system so that some of the ensemblev-representable ground-level densities of the system will
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have lower energies than others (Secs. III and IV). Recent work suggests that density functional approximations that overstabilize or understabilize certain mixed states can give qualitatively wrong chemical predictions.11, 13, 15–17, 28, 42, 48–55 It is interesting that constraint (13) is automatically satisfied if the Kohn-Sham potentials for all of the degenerate ground-level densities are equal. This does not seem to be true, in general, but it does recommend that theories which produce fully symmetric Kohn-Sham potentials be investigated more fully.3–5, 8–10 Such approaches, however, might not suffice for “accidental” degeneracies. ACKNOWLEDGMENTS
P.W.A. acknowledges financial support from Sharcnet, NSERC, and the Canada Research Chairs. P.W.A. acknowledges interesting discussions with Andreas Savin and Weitao Yang on the “symmetry dilemma” in density-functional theory. M.L. thanks Professor Weitao Yang and Professor David Beratan for thieir warm hospitality at Duke University. 1 P.
Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). Levy, Proc. Natl. Acad. Sci. U.S.A. 76, 6062 (1979). 3 A. Gorling, Phys. Rev. A 47, 2783 (1993). 4 A. Gorling, Phys. Rev. Lett. 85, 4229 (2000). 5 F. Della Sala and A. Gorling, J. Chem. Phys. 118, 10439 (2003). 6 A. D. Becke, J. Chem. Phys. 117, 6935 (2002). 7 E. R. Johnson, R. M. Dickson, and A. D. Becke, J. Chem. Phys. 126, 184104 (2007). 8 A. K. Theophilou, Int. J. Quantum Chem. 61, 333 (1997). 9 A. K. Theophilou, Int. J. Quantum Chem. 69, 461 (1998). 10 A. K. Theophilou, and P. Papaconstantinou, J. Mol. Struct.: THEOCHEM 501–502, 85 (2000). 11 A. Savin and J. M. Seminario, in Recent Developments and Applications of Modern Density Functional Theory (Elsevier, New York, 1996), p. 327. 12 P. Gori-Giorgi and A. Savin, J. Phys. Conf. Series 117, 12017 (2008). 13 A. Savin, Chem. Phys. 356, 91 (2009). 14 A. Nagy, S. Liu, and L. Bartolloti, J. Chem. Phys. 122, 134107 (2005). 15 A. J. Cohen, P. Mori-Sanchez, and W. T. Yang, Science 321, 792 (2008). 16 R. Merkle, A. Savin, and H. Preuss, J. Chem. Phys. 97, 9216 (1992). 17 Y. K. Zhang and W. T. Yang, J. Chem. Phys. 109, 2604 (1998). 18 M. Levy and J. P. Perdew, NATO ASI Ser., Ser. B 123, 11 (1985). 19 M. Levy, Phys. Rev. A 26, 1200 (1982). 20 J. Harris and R. O. Jones, J. Phys. F 4, 1170 (1974). 2 M.
J. Chem. Phys. 140, 18A537 (2014) 21 D.
C. Langreth, and J. P. Perdew, Solid State Commun. 17, 1425 (1975). Gunnarsson, and B. I. Lundqvist, Phys. Rev. B 13, 4274 (1976). 23 D. C. Langreth, and J. P. Perdew, Phys. Rev. B 15, 2884 (1977). 24 W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). 25 S. M. Valone, J. Chem. Phys. 73, 4653 (1980). 26 E. H. Lieb, Int. J. Quantum Chem. 24, 243 (1983). 27 P. W. Ayers, Phys. Rev. A 73, 012513 (2006). 28 W. T. Yang, Y. K. Zhang, and P. W. Ayers, Phys. Rev. Lett. 84, 5172 (2000). 29 L. O. Wagner, E. M. Stoudemire, K. Burke, and S. R. White, Phys. Rev. Lett. 111, 093003 (2013). 30 C. Filippi, C. J. Umrigar, and M. Taut, J. Chem. Phys. 100, 1290 (1994). 31 J. Cioslowski, and K. Pernal, J. Chem. Phys. 113, 8434 (2000). 32 M. Seidl, Phys. Rev. A 75, 062506 (2007). 33 N. R. Kestner and O. Sinanoglu, Phys. Rev. 128, 2687 (1962). 34 D. J. Wales, Mol. Phys. 103, 269 (2005). 35 S. Pittalis, S. Kurth, and E. K. U. Gross, J. Chem. Phys. 125, 084105 (2006). 36 S. Pittalis, S. Kurth, S. Sharma, and E. K. U. Gross, J. Chem. Phys. 127, 124103 (2007). 37 A. D. Becke, A. Savin, and H. Stoll, Theor. Chim. Acta 91, 147 (1995). 38 P. W. Ayers and W. T. Yang, J. Chem. Phys. 124, 224108 (2006). 39 A. Holas and R. Balawender, J. Chem. Phys. 125, 247101 (2006). 40 J. Garza, R. Vargas, A. Cedillo, M. Galvan, and P. K. Chattaraj, Theor. Chem. Acc. 115, 257 (2006). 41 P. Perez, E. Chamorro, and P. W. Ayers, J. Chem. Phys. 128, 204108 (2008). 42 A. J. Cohen, P. Mori-Sanchez, and W. T. Yang, J. Chem. Phys. 129, 121104 (2008). 43 K. Capelle, G. Vignale, and C. A. Ullrich, Phys. Rev. B 81, 125114 (2010). 44 R. Haunschild, T. M. Henderson, C. A. Jimenez-Hoyos, and G. E. Scuseria, J. Chem. Phys. 133, 134116 (2010). 45 E. R. Johnson, W. T. Yang, and E. R. Davidson, J. Chem. Phys. 133, 164107 (2010). 46 D. H. Ess, E. R. Johnson, X. Q. Hu, and W. T. Yang, J. Phys. Chem. A 115, 76 (2011). 47 M. Levy and J. P. Perdew, Phys. Rev. A 32, 2010 (1985). 48 P. Mori-Sanchez, A. J. Cohen, and W. T. Yang, J. Chem. Phys. 125, 201102 (2006). 49 A. J. Cohen, P. Mori-Sanchez, and W. T. Yang, Phys. Rev. B 77, 115123 (2008). 50 P. Mori-Sanchez, A. J. Cohen, and W. T. Yang, Phys. Rev. Lett. 100, 146401 (2008). 51 P. Mori-Sanchez, A. J. Cohen, and W. T. Yang, Phys. Rev. Lett. 102, 066403 (2009). 52 A. Ruzsinszky, J. P. Perdew, G. I. Csonka, O. A. Vydrov, and G. E. Scuseria, J. Chem. Phys. 126, 104102 (2007). 53 J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, V. N. Staroverov, and J. M. Tao, Phys. Rev. A 76, 040501 (2007). 54 D. J. Tozer, N. C. Handy, and A. J. Cohen, Chem. Phys. Lett. 382, 203 (2003). 55 B. Braida, P. C. Hiberty, and A. Savin, J. Phys. Chem. A 102, 7872 (1998). 22 O.
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