Kinetic and electron-electron energies for convex sums of ground state densities with degeneracies and fractional electron number Mel Levy, James S. M. Anderson, Farnaz Heidar Zadeh, and Paul W. Ayers Citation: The Journal of Chemical Physics 140, 18A538 (2014); doi: 10.1063/1.4871734 View online: http://dx.doi.org/10.1063/1.4871734 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 Molecular aniline clusters. I. The electronic ground state J. Chem. Phys. 132, 174303 (2010); 10.1063/1.3419505 The electronic ground-state energy problem: A new reduced density matrix approach J. Chem. Phys. 125, 064101 (2006); 10.1063/1.2222358 Ground state, growth, and electronic properties of small lanthanum clusters J. Chem. Phys. 120, 5104 (2004); 10.1063/1.1647060 Electronic ground states of the V 2 O 4 +/0/− species from multireference correlation and density functional studies J. Chem. Phys. 120, 4207 (2004); 10.1063/1.1643891 Density functional theory with approximate kinetic energy functionals applied to hydrogen bonds J. Chem. Phys. 106, 8516 (1997); 10.1063/1.473907

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THE JOURNAL OF CHEMICAL PHYSICS 140, 18A538 (2014)

Kinetic and electron-electron energies for convex sums of ground state densities with degeneracies and fractional electron number Mel Levy,1,2,3,a) James S. M. Anderson,4 Farnaz Heidar Zadeh,4 and Paul W. Ayers4,a) 1

Department of Chemistry, Duke University, Durham, North Carolina 27708, USA Department of Physics, North Carolina A&T State University, Greensboro, North Carolina 27411, USA 3 Department of Chemistry, Tulane University, New Orleans, Louisiana 70118, USA 4 Department of Chemistry and Chemical Biology, McMaster University, Hamilton, Ontario, Canada 2

(Received 1 January 2014; accepted 7 April 2014; published online 25 April 2014) Properties of exact density functionals provide useful constraints for the development of new approximate functionals. This paper focuses on convex sums of ground-level densities. It is observed that the electronic kinetic energy of a convex sum of degenerate ground-level densities is equal to the convex sum of the kinetic energies of the individual degenerate densities. (The same type of relationship holds also for the electron-electron repulsion energy.) This extends a known property of the Levy-Valone Ensemble Constrained-Search and the Lieb Legendre-Transform refomulations of the Hohenberg-Kohn functional to the individual components of the functional. Moreover, we observe that the kinetic and electron-repulsion results also apply to densities with fractional electron number (even if there are no degeneracies), and we close with an analogous point-wise property involving the external potential. Examples where different degenerate states have different kinetic energy and electron-nuclear attraction energy are given; consequently, individual components of the ground state electronic energy can change abruptly when the molecular geometry changes. These discontinuities are predicted to be ubiquitous at conical intersections, complicating the development of universally applicable density-functional approximations. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4871734] I. INTRODUCTION

Existing approximations to the energy functional in density functional theory (DFT) are often inadequate for degenerate ground states.1–6 If one could design a functional that treated degenerate states correctly, then the ground-state energies of all the states in an atomic multiplet would be the same and problems with fractional electron number and fractional spin would be resolved. The latter problems, in particular, seem to be the source of many of the failures of modern-day density-functional approximations: they are associated with the many-electron self-interaction error, the tendency to excessively delocalize electrons, and the tendency to overestimate static correlation energy in highly correlated cases.6–11 Many properties of the exact density functionals are known, and these properties have been useful for developing new approximate exchange-correlation energy functionals. There has been less study, however, of the subuniversal properties of functionals and, in particular, how the fundamental density functionals behave for degenerate systems. With this in mind, we derive simple results of this type in Secs. II–IV; other results on degenerate ground states may be found in Refs. 1–6 and 12. Numerical tests in Sec. V show that although different degenerate states have the same energy (by definition), they often have different values for components of the energy. Consequently, the values of energy-components sometimes change discontinuously in response to changes in molecular geometry. a) [email protected] and [email protected]

0021-9606/2014/140(18)/18A538/7/$30.00

We restrict ourselves to electronic systems, so the N-electron Hamiltonian has the form, Hˆ =

N 

1 1 1  . − ∇i2 + v(ri ) + 2 2 |r − rj | i i=1 i=1 j =1 N

N

(1)

j =i

Atomic units are used throughout. The results we will derive are valid along the entire constant-density adiabatic connection pathway that connects the noninteracting Kohn-Sham reference system to the physical system of interest.13–15 For clarity of notation, however, we will usually restrict ourselves to the physical system and the noninteracting Kohn-Sham system (whose associated functionals are specified with the subscript s). The total energy is written as  (2) Ev [ρ] = F [ρ] + ρ(r)v(r)dr and F[ρ], the original Hohenberg-Kohn functional or one of its reformulations, is usually decomposed either as the sum of the total kinetic energy, T[ρ], and the electron-electron repulsion energy, Vee [ρ], F [ρ] = T [ρ] + Vee [ρ],

(3)

or in the manner of Kohn and Sham, F [ρ] = Ts [ρ] + J [ρ] + Exc [ρ],

(4)

where Ts [ρ] is the Kohn-Sham kinetic energy, J[ρ] is the classical Coulomb self-repulsion energy, and Exc [ρ] is the exchange-correlation energy.16

140, 18A538-1

© 2014 AIP Publishing LLC

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When one extends the original Hohenberg-Kohn functional to non-v-representable and degenerate densities, one may choose F[ρ] to be convex, so it is always true that     F wi ρi ≤ wi F [ρi ], (5) i

Combining this with the known linearity relation for the F[ρ], Eq. (7), gives the desired results,     (12) wi ρi = wi T [ρi ], T 

i

Vee

where (6)

i

There are several approaches in the literature that define F[ρ] as a convex functional, including (a) the ensembleconstrained search definition, (b) the Legendre-transform definition, and (c) continuity arguments.17–21 All of these approaches give the same functional. The pure-state constrained-search F[ρ],17 which extended the original functional of Hohenberg and Kohn to include degeneracies, is not convex in that exceptions to Eq. (5) have been found with its use.21–23 But this pure-state functional does nevertheless satisfy Eq. (5) in the vast majority of cases. II. KEY MATHEMATICAL IDENTITIES

If all the electron densities in Eq. (5) are ground-state electron densities of the same external potential, then the equality in Eq. (5) holds.4 That is, for an ensemble average of degenerate ground states,     F (7) wi ρi = wi F [ρi ] i

T

 

 wi ρi +Vee

i

 

i

 wi ρi =



i

wi T [ρi ]+



i

wi Vee [ρi ].

i

(8) Consider the virial relation for an ensemble-average density,24, 25    wi ρi (r) r · ∇v(r)dr i

= 2T

 

 wi ρi + Vee

 

i

 wi ρi .

(9)

i

The left-hand side of this equation is a linear functional of the electron density. Then, from the virial theorem for the individual components of the ensemble, we also have    wi ρi (r)r · ∇v(r)dr = wi (2T [ρi ] + Vee [ρi ]) i

i

(10) and therefore,        wi ρi +Vee wi ρi = wi (2T [ρi ] +Vee [ρi ]). 2T i

i



 wi ρi =

i

0 ≤ wi  1 = wi .

or

i

i

(11)

i



wi Vee [ρi ].

(13)

i

These identities hold for convex linear combinations of degenerate ground-state densities at any point along the adiabatic connection pathway. Equalities analogous to identities (12) and (13) apply to densities with fractional electron number, even in the absence of degeneracies. See identities (40) and (41). The fundamental equations (12) and (13) may also be derived from the Hohenberg-Kohn theorem for ensembles.26 That is, the ensemble of degenerate densities determines the ensemble of degenerate wave functions. Evaluating the kinetic and electron repulsion energies, separately, of this wave function ensemble gives Eqs. (12) and (13). In fact, the form of this result is very general, and applies to any observable corresponding to a mathematically well-behaved operator,26, 27      ˆ i = Q wi ρi = wi i |Q| wi Q[ρi ]. (14) i

i

i

This result also extends to functionals of the 1-electron reduced density matrix. When only two degenerate densities, corresponding to real-valued ground-state wavefunctions, contribute to the ensemble, then the identities (12)-(14) hold for all exact functionals, including the pure-state constrained-search functional.17 This is because any ensemble average of these degenerate densities, ρ (a) (r) = (1 − a) ρ (0) (r) + aρ (1) (r)

0 ≤ a ≤ 1,

(15)

is pure-state v-representable, with complex wavefunction19, 26 1

1

| (a)  = (1 − a) 2 | (0)  + ia 2 | (1) .

(16)

This complex linear combination is, of course, also a ground state of the Hamiltonian. The Coulomb repulsion energy functional,  ρ(r)ρ(r ) 1 (17) drdr J [ρ] = 2 |r − r | is strictly convex because its second functional derivative, |r − r |−1 , is always positive. This means that the exchangecorrelation potential energy, Vxc [ρ] = Vee [ρ] − J [ρ] = Ex [ρ] + Vc [ρ]

(18)

is strictly concave functional in the interior of the set28 of degenerate-state densities,     Vxc (19) wi ρi > wi Vxc [ρi ]. i

i

It is probably impossible to design a local or GGA type functional that fulfills Eq. (13); J[ρ] is inherently nonlocal, so

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the compensating term that one must add to J[ρ] to fulfill Eq. (13) must also be nonlocal. The need for nonlocal functionals to compensate J[ρ] seems to have been first noted in the 1990s.3, 29 If one uses the ensemble-constrained search or the Legendre transform to define the Kohn-Sham kinetic energy, then Ts [ρ] is a nonconcave functional;17–19, 21 it is strictly convex within the set of degenerate densities except when one is lucky enough to find that some of the degenerate electron densities of the interacting system are also degenerate in the noninteracting system. Equation (12) then implies that Tc [ρ] = T[ρ] − Ts [ρ] is a nonconvex functional within the set of degenerate densities,     Tc (20) wi ρi ≥ wi Tc [ρi ]. i

i

Adding inequality (19) and (20) reveals that the exchangecorrelation energy, Exc [ρ] = Vxc [ρ] + Tc [ρ], is strictly concave in the interior of the set28 of degenerate densities,     Exc (21) wi ρi > wi Exc [ρi ]. i

in Eq. (24) and integrating gives 

∂ρλ(a) (r)

(a) (a) (a) vs (r) − vJ (r) − vxc (r) ∂λ

λ=1

(0) (r) ∂ρ (0) (r) λ = (1 − a) vs(0) (r) − vJ(0) (r) − vxc dr ∂λ λ=1  (1) ∂ρ (r)

(1) (1) (r) λ (26) +a vs (r) − vJ(1) (r) − vxc dr ∂λ 



λ=1

This equation can be simplified using the following coordinate-scaling identities,24, 25

 ∂ρλ (r) −2Ts [ρ] = vs (r) dr, (27) ∂λ λ=1

 J [ρ] =

i

It seems plausible that Eq. (21) is generally true (i.e., without the restriction to degenerate states), but the authors know no proof that it is true, nor any counter example that it is not true.

dr

vJ (r)

∂ρλ (r) ∂λ

dr,

(28)

∂ρλ (r) = 3ρ(r) + r · ∇ρ(r), ∂λ λ=1

(29)

λ=1

and the integration-by-parts identity,   w(r) (3ρ(r) + r · ∇ρ(r)) dr = − ρ(r)r · ∇w(r)dr. (30)

III. COROLLARIES FROM COORDINATE-SCALING

Consider the coordinate scaling of an arbitrary convex linear combination of any two degenerate ground-state electron densities, ρ (0) (r) and ρ (1) (r), ρλ(a) (r) = (1 − a) ρλ(0) (r) + aρλ(1) (r).

(22)

(It is straightforward to generalize the following derivations for convex linear combinations of three or more degenerate ground-state densities, but for clarity we will discuss explicitly only the two-density case.) In Eq. (22), we are using the traditional notation for coordinate-scaling of the electron density: ρλ(a) (x, y, z) = λ3 ρ (a) (λx, λy, λz). Differentiating both sides with respect to the coordinate scaling parameter gives ∂ρλ(a) (r) ∂ρλ(0) (r) ∂ρλ(1) (r) = (1 − a) +a . (23) ∂λ ∂λ ∂λ λ=1

λ=1

One obtains,  −2Ts [ρ (a) ] − J [ρ (a) ] +

(a) ρ (a) (r)r · ∇vxc (r)dr





= (1 − a) −2Ts [ρ ] − J [ρ ] + (0)

(0)

ρ (r)r · (0)

(0) ∇vxc (r)dr



 (1) (1) (1) (1) + a −2Ts [ρ ] − J [ρ ] + ρ (r)r · ∇vxc (r)dr . (31) Using the definition of the F[ρ], F [ρ] = Ts [ρ] + J [ρ] + Exc [ρ]

(32)

and the ensemble condition,4, 27

λ=1

and, using the fact that all three systems have the same external potential gives (a) (0) (r) (r) ∂ρ ∂ρ v (a) (r) λ = (1 − a) v (0) (r) λ ∂λ ∂λ λ=1 λ=1 (1) ∂ρ (r) + av (1) (r) λ (24) . ∂λ λ=1

Using the definition of the Kohn-Sham potential in terms of the Coulomb and exchange-correlation potentials, vs (r) = v(r) + vJ (r) + vxc (r)

(25)

F [ρ (a) ] = (1 − a)F [ρ (0) ] + aF [ρ (1) ],

(33)

one can eliminate either Ts [ρ] or J[ρ] from Eq. (31). This gives two identities,  (a) −Ts [ρ (a) ] + Exc [ρ (a) ] + ρ (a) (r)r · ∇vxc (r)dr

 (0) (0) (0) (0) = (1−a) −Ts [ρ ] + Exc [ρ ] + ρ (r)r · ∇vxc (r)dr

 (1) + a −Ts [ρ (1) ] + Exc [ρ (1) ] + ρ (1) (r)r · ∇vxc (r)dr , (34)

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 J [ρ ] + 2Exc [ρ ] + (a)

(a)

(a) ρ (a) (r)r · ∇vxc (r)dr

 (0) (r)dr = (1−a) J [ρ (0) ] + 2Exc [ρ (0) ] + ρ (0) (r)r · ∇vxc



+ a J [ρ ] + 2Exc [ρ ] + (1)

ρ (r)r ·

(1)

(1)

(1) ∇vxc (r)dr

.

(35) Both of these equalities must hold for all possible choices of degenerate state densities, ρ (0) (r) and ρ (1) (r), and all 0 ≤ a ≤ 1. The second equality constraint, Eq. (35), provides an especially convenient test for exchange-correlation functionals because it relies only upon the readily computable Coulomb energy. (Using the first equality constraint, Eq. (34), requires performing an ensemble-constrained-search or a Legendre transform procedure to compute Ts [ρ (a) ].30–33 ) As before, both these equations can be extended to the spin-resolved case by “decorating” the densities and potentials with vectors. For example, Eq. (35), in explicit form, becomes       (a) (r)dr ρσ(a) (r)r ·∇vxc;σ J ρα(a) +ρβ(a) +2Exc ρα(a) , ρβ(a) + σ =α,β

    = (1 − a) J ρα(0) + ρβ(0) + 2Exc ρα(0) , ρβ(0) +

 

(0) (0) ρσ (r)r · ∇vxc;σ (r)dr

    + a J ρα(1) + ρβ(1) + 2Exc ρα(1) , ρβ(1) +

·

(1) ∇vxc;σ (r)dr

.

 ρ (r)r ·

+

(0)

∇vc(0) (r)dr

+ a − 2Ts [ρ (1) ] − J [ρ (1) ] − Ex [ρ (1) ]  +

ρ (1) (r)r · ∇vc(1) (r)dr ,

(38)

which can be rewritten in a form similar to Eq. (37), J [ρ (a) ] − (1 − a)J [ρ (0) ] − aJ [ρ (1) ] + Ex [ρ (a) ] − (1 − a)Ex [ρ (0) ] − aEx [ρ (1) ]

= −2 Ec [ρ (a) ] − (1 − a)Ec [ρ (0) ] − aEc [ρ (1) ]   (a) − ρ (r)r · ∇vc(a) (r) − (1 − a)ρ (0) (r)r (39)

The left-hand side of Eq. (39) is equal to the electron-electron repulsion energies of the Kohn-Sham Slater determinants; this is a nonlocal functional of the Kohn-Sham density matrix. The right-hand side must also be nonlocal. Nevertheless, the partitioning in Eq. (39) should help in the approximation of the correlation energy functional.



ρσ(1) (r)r

= (1 − a) − 2Ts [ρ (0) ] − J [ρ (0) ] − Ex [ρ (0) ]

 · ∇vc(0) (r) − aρ (1) (r)r · ∇vc(1) (r) dr.

σ =α,β

 

(cf. Eq. (28)), one can derive an equation analogous to Eq. (31),  −2Ts [ρ (a) ] − J [ρ (a) ] − Ex [ρ (a) ] + ρ (a) (r)r · ∇vc(γ ) (r)dr

(36)

σ =α,β

Equations (35) and (36) again show that the exchangecorrelation energy functional must delicately cancel a term from the Coulomb energy functional.1 Rearranging Eq. (35), J [ρ (a) ] − (1 − a)J [ρ (0) ] − aJ [ρ (1) ]

= −2 Exc [ρ (a) ] − (1 − a)Exc [ρ (0) ] − aExc [ρ (1) ]   (a) (a) − ρ (r)r · ∇vxc (r) − (1 − a)ρ (0) (r)r  (0) (1) · ∇vxc (r) − aρ (1) (r)r · ∇vxc (r) dr.

(37)

The left-hand side of this equation is inherently nonlocal, and cannot be accurately modeled with a local or semilocal “gradient-type” functional.29 Local and gradient-type exchange-correlation functionals are therefore incapable of satisfying Eq. (37). Equation (34) implies that the same is true of the kinetic energy: unless the approximate kinetic energy functional is fully nonlocal, it will be unable to resolve degenerate ground states. Given the recent interest in functionals with 100% exact exchange,34–42 one might hope that at least the correlation energy functional would be semi-local. This does not seem to be true. Using the fact that the exchange energy satisfies the same coordinate-scaling condition as the Coulomb energy24

IV. EXTENSIONS TO FRACTIONAL ELECTRON NUMBER AND SPIN

We now observe that the key results in Eqs. (12) and (13), and subsequent expressions, also apply to densities with fractional electron number, even if there are no degeneracies. Specifically, for a convex linear combination of the groundstate densities of the N and N − 1 electron systems of the same Hamiltonian, Vee [aρ (N) (r) + (1 − a)ρ (N−1) (r)] = aVee [ρ (N) (r)] + (1 − a)Vee [ρ (N−1) (r)],

(40)

T [aρ (N) (r) + (1 − a)ρ (N−1) (r)] = aT [ρ (N) (r)] + (1 − a)T [ρ (N−1) (r)].

(41)

Thus the result that Perdew, Parr, Levy, and Balduz, derived for F[ρ] when the density has a noninteger number of electrons,4, 43, 44 is here extended to its kinetic and electronrepulsion components. An interesting corollary to these identities is obtained by differentiating both sides with respect to a. Using the chain rule for functional derivatives, Eq. (40) then implies the

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identity,  δVee [ρ] δρ(r) = Vee [ρ

J. Chem. Phys. 140, 18A538 (2014)

ρ=aρ (N) +(1−a)ρ (N−1)

(N)

(N) ρ (r) − ρ (N−1) (r) dr

] − Vee [ρ (N−1) ].

(42)

Together with the fact that the second and higher derivatives of expression (40) with respect to a are zero, Eq. (42) is an alternative way of indicating that Vee [ρ] is linear on the domain of ground-state densities. Analogous results hold for the kinetic energy in the fractional electron number case (Eq. (41)) and for Eqs. (12)-(14). More generally, if ρi(N) (r) is the ground-state density of a (possibly, but not necessarily, degenerate) N-electron system and ρj(N+{−1,0,1}) (r) is a ground-state density for the same system with N − 1, N, or N + 1 electrons, then   Q aρi(N) + (1 − a)ρj(N+{−1,0,1})     (43) = aQ ρi(N) − (1 − a)Q ρj(N+{−1,0,1}) and    δQ[ρ] ρi(N) (r)−ρj(N+{−1,0,1}) (r) dr δρ(r) ρ=aρi(N) +(1−a)ρj(N+{−1,0,1})     = Q ρi(N) − Q ρj(N+{−1,0,1}) ,

(44)

where Q is any property which is defined by the expectation value of the ground-level density matrix for the interacting system. Special cases of this general form have been derived by other researchers.45 Equations like Eq. (44) can also be viewed as constraints on DFT-based chemical reactivity indicators46, 47 for degenerate states.48–50 Almost all of the results from the preceding sections can be extended to spin-density functional theory51, 52 by replacing ρ(r) by ρ α (r) and ρ β (r) in the arguments of the functionals, provided that the spin-resolved F[ρ α , ρ β ] is defined as convex functional (e.g., using the ensemble-constrained search53 or the Legendre transform54, 55 ). If the only degeneracy present in the system is a spin-degeneracy, however, then all the degenerate states have the same total density and the Coulomb energy is the same for all of the degenerate states. In this case, the inequality in Eq. (19) becomes an equality:     wi [ρα,i , ρβ,i ] = wi Vxc [ρα,i , ρβ,i ], (45) Vxc i

i

and equality is mathematically allowed, although unlikely to be achieved, in Eqs. (20) and (21).56 Equation (45) is a condition for the exchange-correlation functional that provides a stringent test for approximate functionals. The failure to satisfy this constraint may explain why present-day approximate exchange-correlation functionals fail badly for systems with fractional spin.6, 36, 57–63 The constraint (44) is applicable to all of the approaches to spin-DFT we know,53–55, 63–65 and seems difficult to satisfy using approximate functionals. Spinfree formulations based on the on-top pair density66–68 should fulfill many—perhaps all—of our constraints on degenerate spin-densities. This is not a popular formulation for approximate density functionals, but our results suggest it should be considered more seriously.

V. NONDEGENERACY OF ENERGY COMPONENTS IN DEGENERATE STATES: MATHEMATICAL ARGUMENTS AND NUMERICAL TESTS

If, for an approximate functional, all the T’s are the same on both sides of Eq. (12) and all the Vee ’s are the same on both sides of Eq. (13), then these identities are automatically satisfied. It is therefore of interest to learn when all the degenerate states have identical values for the energy components:   (46) ρi (r)v(r)dr = ρj (r)v(r)dr, F [ρi ] = F [ρj ],

(47)

Vee [ρi ] = Vee [ρj ],

(48)

T [ρi ] = T [ρj ].

(49)

Lieb states that Eqs. (46) and (47) hold for degenerate ground states in a spherical potential as long as there is no accidental degeneracy. (That is, the degeneracy is 2L + 1, where L is the total angular momentum quantum number of the ground state.)19 The equalities also seem to hold true in other cases where there are symmetry-related degeneracies (e.g., noninteracting electrons bound by Coulomb potentials). Do these equalities always hold for degenerate states? We now show that they do not. For a diatomic molecule, the virial theorem can be written as dE = 0, (50) 2T (R) + Vee (R) + Vne (R) + R dR or, equivalently, as dE T (R) = −E(R) − R , (51) dR dE Vee (R) + Vne (R) = 2E(R) + R . (52) dR Here R is the internuclear distance, Vnn is the nuclear-nuclear repulsion contribution, and Vne is the nuclear-electron attraction energy,  Vne (R) = ρ(R, r)v(R, r)dr. (53) Consider a bond distance at which the ground-state energy curve E0 (Rcross ), crosses an excited state curve E1 (Rcross ). At this accidental degeneracy, the energies of the two states are equal, E0 (Rcross ) = E1 (Rcross ),

(54)

but the slopes of the potential energy curves are not, ∂E0 (Rcross ) ∂E1 (Rcross ) = . (55) ∂R ∂R Using Eqs. (54) and (55) in Eq. (51) shows that the kinetic energy of these two states is also different, T0 (Rcross ) = T1 (Rcross ).

(56)

This means that the value of the kinetic-energy density functional changes discontinuously at a point of accidental degeneracy on a potential energy surface. This is not that surprising, as the electron density also changes discontinuously here.

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Linear response theory is invalid for degenerate states. Equation (49) is not generally true for degenerate ground states. In the noninteracting limit, we can say something similar for Eqs. (46)-(48). Consider the case where the Kohn-Sham potential is equal to the external potential for H+ 2, 1 1  −  .   vs (R, r) = − r − 0, 0, − 1 R r − 0, 0, 1 R 2 2

(57)

If we occupy the N-lowest states of this potential, the corresponding virial relation is,  N  dEn 2Ts (R) + ρ(R, r)vs (R, r) + R = 0. (58) dR n=1 Consider a case where the highest occupied Kohn-Sham state is accidentally degenerate, EN (Rcross ) = EN+1 (Rcross ),

(59)

∂EN+1 ∂EN = . (60) ∂R ∂R By the same argument that was used to derive Eq. (56), Eqs. (46) and (47) are not true when λ = 0. Even when the Kohn-Sham system is degenerate, different degenerate states can have different values for Ts . This gives rise to a discontinuity in Ts (R), as shown in Figure 1. While it seems very unlikely that Eqs. (46)-(49) would be invalid in the noninteracting limit, but valid for real chemical systems, we did confirm these inequalities for several diatomic molecules. Degenerate electronic states can clearly have different values for the kinetic energy, electron-electron repulsion energy, electron-nuclear attraction potential, and F[ρ]. The quantities are, in general, discontinuous functions of the molecular geometry. Figure 2 shows a representative example: for the lowest-lying doublet state of the B2 + molecule, F[ρ] is a discontinuous function of the internuclear

FIG. 1. The Kohn-Sham kinetic energy (Ts ; ●) and the noninteracting electronic energy (Es ; ) as a function of internuclear separation for ten electrons bound in the Kohn Sham potential of the hydrogen molecule ion, Eq. (57). This demonstrates that Ts is not always a continuous function of molecular geometry and demonstrates that different degenerate ground-state densities can have different values for Ts . This plot is constructed from the exact solution for the hydrogen molecule ion.70

FIG. 2. The Hohenberg-Kohn functional (F[ρ], ●) and the potential energy (, including the nuclear-nuclear repulsion) in the B+ 2 molecule, as a function of bond length. The region of the potential energy curve shown is near the level-crossing of the 2σ u and the 1π u orbitals. This demonstrates that F is not always a continuous function of molecular geometry and demonstrates that different degenerate ground-state densities can have different values for F. These curves are constructed from full configuration interaction calculations with the cc-pVDZ basis set,71 using the GAMESS-US program.72, 73

separation. While it is difficult to see in Figure 1, it is clear from Figure 2 that discontinuities in the energy components occur at curve-crossings between potential energy surfaces, as expected. VI. CLOSING REMARKS

We have shown that the total kinetic energy and the electron-electron repulsion energy of an ensemble of degenerate-state densities are equal to the ensemble average of the kinetic energies and electron-electron repulsion energies of the contributing densities, Eqs. (12) and (13). This extends a result that is known for the whole F[ρ] to its components. It also proves that the exchange-correlation potential energy and the total exchange-correlation energy (including the kinetic energy contribution) are concave functionals of degenerate-state densities, cf. Eqs. (19)–(21). In Sec. IV, we were able to extend these results, even in the absence of degeneracies, to fractional numbers of electrons and to fractional spins because the key identities hold for convex sums of ground-state densities of the same system with different numbers of electrons, provided the total number of electrons does not differ by more than one (see Sec. IV). No (semi)local functional seems capable of reproducing these traits of the exact functional, demonstrating certain inherent limitations of the generalized gradient approximation functionals’ form. While the kinetic energy and the electron-electron repulsion energies of all of the possible degenerate ground state densities are sometimes the same, different ground-state densities can have different values for the components of the energy, even though the sum of all these components is necessarily equal. Consequently, even though the electronic energy is a continuous (albeit not necessarily differentiable) function of molecular geometry, components of the energy are generally discontinuous functions of molecular geometry. For

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18A538-7

Levy et al.

J. Chem. Phys. 140, 18A538 (2014)

example, the kinetic energy, electron-electron repulsion energy, and their sum all change discontinuously when the B2 + molecule is stretched (see Figure 2). More exhaustive numerical investigations of diatomic molecules reveal that this is a general feature: B2 + is not a special case, though often the discontinuities occur at chemically irrelevant geometries.69 By theoretical arguments, such discontinuities are expected to be ubiquitous at conical intersections. We close by observing that a stringent pointwise identity exists that is analogous to expressions (40) and (41). The external potential, v[ρ (N+a) ; r], is independent of a. Defining, for 0 < a < 1, ρ (N+a) (r) = (1 − a)ρ (N) (r) + aρ (N+1) (r),

(61)

the constancy of the external potential, Eq. (25), and the fact both the interacting and noninteracting chemical potentials are independent of a for 0 < a < 1 imply that ∂vJ [ρ (N+a) ; r] ∂vxc [ρ (N+a) ; r] + ∂a ∂a

(N+a) ∂ δTs [ρ (N+a) ] ∂vs [ρ ; r] =− = ∂a ∂a δρ(r)

(62)

ACKNOWLEDGMENTS

P.W.A. acknowledges financial support from NSERC and computer time from Sharcnet. J.S.M.A. and F.H.Z. acknowledge graduate (F.H.Z. and J.S.M.A.) and postdoctoral (J.S.M.A.) fellowships from NSERC. The authors had several helpful discussions with Professor Andreas Savin, who made insightful comments about the entire manuscript and, in particular, suggested studying the hydrogen molecule ion. M.L. thanks Professors Weitao Yang and David Beratan for thieir warm hospitality at Duke University. 1 A.

Savin, Chem. Phys. 356, 91 (2009). Gori-Giorgi and A. Savin, J. Phys. Conf. Series 117, 12017 (2008). 3 A. Savin and J. M. Seminario, in Recent Developments and Applications of Modern Density Functional Theory (Elsevier, New York, 1996), p. 327. 4 W. T. Yang, Y. K. Zhang, and P. W. Ayers, Phys. Rev. Lett. 84, 5172 (2000). 5 Y. K. Zhang and W. T. Yang, J. Chem. Phys. 109, 2604 (1998). 6 A. J. Cohen, P. Mori-Sanchez, and W. T. Yang, Science 321, 792 (2008). 7 P. Mori-Sanchez, A. J. Cohen, and W. T. Yang, Phys. Rev. Lett. 102, 066403 (2009). 8 P. Mori-Sánchez, A. J. Cohen, and W. T. Yang, Phys. Rev. Lett. 100, 146401 (2008). 9 A. Ruzsinszky, J. P. Perdew, G. I. Csonka, O. A. Vydrov, and G. E. Scuseria, J. Chem. Phys. 126, 104102 (2007). 10 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). 11 J. P. Perdew, A. Ruzsinszky, J. M. Tao, V. N. Staroverov, G. E. Scuseria, and G. I. Csonka, J. Chem. Phys. 123, 062201 (2005). 12 J. P. Perdew and E. Sagvolden, Can. J. Chem. 87, 1268 (2009). 13 D. C. Langreth and J. P. Perdew, Solid State Commun. 17, 1425 (1975). 14 O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 13, 4274 (1976). 15 D. C. Langreth and J. P. Perdew, Phys. Rev. B 15, 2884 (1977). 16 W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). 17 M. Levy, Proc. Natl. Acad. Sci. U.S.A. 76, 6062 (1979). 18 S. M. Valone, J. Chem. Phys. 73, 4653 (1980). 19 E. H. Lieb, Int. J. Quantum Chem. 24, 243 (1983). 20 P. W. Ayers, Phys. Rev. A 73, 012513 (2006). 21 M. Levy and J. P. Perdew, NATO ASI Ser., Ser. B 123, 11 (1985). 22 J. P. Perdew and M. Levy, Phys. Rev. B 31, 6264 (1985). 23 E. H. Lieb, NATO ASI Ser., Ser. B 123, 31 (1985). 2 P.

24 M.

Levy and J. P. Perdew, Phys. Rev. A 32, 2010 (1985). K. Ghosh and R. G. Parr, J. Chem. Phys. 82, 3307 (1985). 26 M. Levy, Phys. Rev. A 26, 1200 (1982). 27 P. W. Ayers, J. Math. Chem. 43, 285 (2008). 28 That is, equality holds only in the trivial case where the ensemble has only one component (so that there is only one nonzero weight). 29 R. Merkle, A. Savin, and H. Preuss, J. Chem. Phys. 97, 9216 (1992). 30 Q. Zhao, R. C. Morrison, and R. G. Parr, Phys. Rev. A 50, 2138 (1994). 31 Q. Zhao and R. G. Parr, J. Chem. Phys. 98, 543 (1993). 32 Q. Wu and W. T. Yang, J. Chem. Phys. 118, 2498 (2003). 33 F. Colonna and A. Savin, J. Chem. Phys. 110, 2828 (1999). 34 P. Mori-Sanchez, A. J. Cohen, and W. Yang, J. Chem. Phys. 124, 091102 (2006). 35 A. J. Cohen, P. Mori-Sanchez, and W. T. Yang, J. Chem. Phys. 126, 191109 (2007). 36 A. D. Becke, J. Chem. Phys. 138, 074109 (2013). 37 A. D. Becke, Abstr. Pap. Am. Chem. Soc. 242 (2011). 38 E. R. Johnson, and J. Contreras-Garcia, J. Chem. Phys. 135, 081103 (2011). 39 A. D. Becke, and E. R. Johnson, J. Chem. Phys. 127, 124108 (2007). 40 A. D. Becke, J. Chem. Phys. 122, 064101 (2005). 41 A. D. Becke, J. Chem. Phys. 139, 021104 (2013). 42 A. D. Becke, J. Chem. Phys. 138, 161101 (2013). 43 J. P. Perdew, R. G. Parr, M. Levy, and J. L. Balduz, Jr., Phys. Rev. Lett. 49, 1691 (1982). 44 Y. K. Zhang and W. T. Yang, Theor. Chem. Acc. 103, 346 (2000). 45 J. K. Percus, Phys. Rev. A 60, 2601 (1999). 46 P. Geerlings, F. De Proft, and W. Langenaeker, Chem. Rev. 103, 1793 (2003). 47 P. A. Johnson, L. J. Bartolotti, P. W. Ayers, T. Fievez, and P. Geerlings, in Modern Charge Density Analysis, edited by C. Gatti and P. Macchi (Springer, New York, 2012), p. 715. 48 C. Cardenas, P. W. Ayers, and A. Cedillo, J. Chem. Phys. 134, 174103 (2011). 49 P. Bultinck, C. Cardenas, P. Fuentealba, P. A. Johnson, and P. W. Ayers, J. Chem. Theory Comput. 9, 4779 (2013). 50 P. Bultinck, C. Cardenas, P. Fuentealba, P. A. Johnson, and P. W. Ayers, J. Chem. Theory Comput. 10, 202 (2014). 51 U. Von Barth and L. Hedin, J. Phys. C 5, 1629 (1972). 52 A. K. Rajagopal and J. Callaway, Phys. Rev. B 7, 1912 (1973). 53 J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). 54 P. W. Ayers and W. T. Yang, J. Chem. Phys. 124, 224108 (2006). 55 A. Holas and R. Balawender, J. Chem. Phys. 125, 247101 (2006). 56 P. Gori-Giorgi, R. C. Morrison, P. W. Ayers, and A. Savin, “The variation of energy components with respect to spin-polarization” (unpublished). 57 A. J. Cohen, P. Mori-Sanchez and W. T. Yang, J. Chem. Phys. 129, 121104 (2008). 58 K. Capelle, G. Vignale, and C. A. Ullrich, Phys. Rev. B 81, 125114 (2010). 59 R. Haunschild, T. M. Henderson, C. A. Jimenez-Hoyos, and G. E. Scuseria, J. Chem. Phys. 133, 134116 (2010). 60 E. R. Johnson, W. T. Yang, and E. R. Davidson, J. Chem. Phys. 133, 164107 (2010). 61 D. H. Ess, E. R. Johnson, X. Q. Hu, and W. T. Yang, J. Phys. Chem. A 115, 76 (2011). 62 D. G. Peng, X. Q. Hu, D. Devarajan, D. H. Ess, E. R. Johnson, and W. T. Yang, J. Chem. Phys. 137, 114112 (2012). 63 C. R. Jacob and M. Reiher, Int. J. Quantum Chem. 112, 3661 (2012). 64 P. W. Ayers and P. Fuentealba, Phys. Rev. A 80, 032510 (2009). 65 M. Higuchi and K. Higuchi, Phys. Rev. B 69, 035113 (2004). 66 A. D. Becke, A. Savin, and H. Stoll, Theor. Chim. Acta 91, 147 (1995). 67 J. P. Perdew, A. Savin, and K. Burke, Phys. Rev. A 51, 4531 (1995). 68 J. P. Perdew, M. Ernzerhof, K. Burke, and A. Savin, Int. J. Quantum Chem. 61, 197 (1997). 69 J. S. M. Anderson and P. W. Ayers, “Discontinuities in components of the energy with respect to geometric perturbations” (unpublished). 70 D. R. Bates, K. Ledsham, and A. L. Stewart, Philos. Trans. R. Soc. London, Ser. A 246, 215 (1953). 71 T. H. Dunning, J. Chem. Phys. 90, 1007 (1989). 72 M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M. Dupuis, and J. A. Montgomery, J. Comput. Chem. 14, 1347 (1993). 73 M. S. Gordon and M. W. Schmidt, in Theory and Applications of Computational Chemistry: The First Forty Years, edited by C. E. Dykstra et al. (Elsevier, Amsterdam, 2005), p. 1167. 25 S.

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Kinetic and electron-electron energies for convex sums of ground state densities with degeneracies and fractional electron number.

Properties of exact density functionals provide useful constraints for the development of new approximate functionals. This paper focuses on convex su...
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