a r X i v :c o n d -m a t /0610322v 1 [c o n d -m a t .m t r l -s c i ] 11 O c t 2006
Degenerate ground states and nonunique potentials:breakdown and restoration of
density functionals
K.Capelle,1C.A.Ullrich,2and G.Vignale 2
1
Departamento de F´ısica e Inform´a tica,Instituto de F´ısica de S˜a o Carlos,Universidade de S˜a o Paulo,Caixa Postal 369,13560-970S˜a o Carlos,SP,Brazil
2
Department of Physics and Astronomy,University of Missouri-Columbia,Columbia,Missouri 65211,USA
(Dated:February 6,2008)
The Hohenberg-Kohn (HK)theorem is one of the most fundamental theorems of quantum mechan-ic
s,and constitutes the basis for the very successful density-functional approach to inhomogeneous interacting many-particle systems.Here we show that in formulations of density-functional theory (DFT)that employ more than one density variable,applied to systems with a degenerate ground state,there is a subtle loophole in the HK theorem,as all mappings between densities,wave functions and potentials can break down.Two weaker theorems which we prove here,the joint-degeneracy theorem and the internal-energy theorem ,restore the internal,total and exchange-correlation en-ergy functionals to the extent needed in applications of DFT to atomic,molecular and solid-state physics and quantum chemistry.The joint-degeneracy theorem constrains the nature of possible degeneracies in general many-body systems.
Introduction.Quantum mechanics is bad on the as-sumption that all information that one can,in principle,extract from a system in a pure state at zero temperature is contained in its wave function.In nonrelativistic quan-tum mechanics the wave function obeys Schr¨o dinger’s
equation,1
which implies a powerful variational principle according to which the ground-state wave function min-imizes the expectation value of the Hamiltonian.This variational principle was ud by Hohenberg an
d Kohn (HK)2to show that the entire information contained in the wave function is also contained in the system’s ground-state particle density n (r ).
HK established the existence of two mappings,
v (r )1
⇐⇒Ψ(r 1,...r N )2
⇐⇒n (r ),
二年级小作文(1)
where the first guarantees that the single-particle poten-tial is a unique functional of the wave function,v [Ψ],and the cond implies that the ground-state wave function is a unique functional of the ground-state density,Ψ[n ].Taken together,both mappings are encapsulated in the single statement that the single-particle potential is a unique density functional v [n ].In this formulation,the HK theorem forms the basis of the spectacularly success-ful approach to many-body physics,electronic-structure theory and quantum chemistry that became known as density-functional theory (DFT).3,4,5
Mapping 2was originally proven by contradiction 2and later by constrained arch.6Note that,in spite of occa-sional statements to the contrary in the literature,nei-ther proof directly proves the combined mapping,and thus the existence of the functional v [n ].This requires additionally mapping 1,which in the ca of density-only DFT is proven by inverting Schr¨o dinger’s equation 4,7
ˆV
=
i
v (r i )=E k −
(ˆT
+ˆU )Ψk (r 1,...r N )
2
cal density-functional theory,one must define the total-energy functional E v[n],the universal internal-
energy functional F[n],and the exchange-correlation energy functional E xc[n].The Kohn-Sham formulation addition-ally requires the noninteracting kinetic-energy functional T s[n].21years after proving the original HK theorem, Kohn14showed how the functionals can be defined even if degeneracy renders the original proof ineffective.4Since all degenerate wave functions by definition yield the same ground-state energy E,one can directly define the func-tional
F[n]:=E− d3r n(r)v[n](r)=E−V[n].(3)
Conventionally,this functional is defined as F[n]= T[n]+U[n],but the information that the kinetic energy T and interaction energy U are density functionals is only available if the cond mapping,Ψ[n],holds,and cannot be taken for granted in the prence of degeneracy.By contrast,the alternative definition above only requires the mapping v[n]to establish the existence of the uni-versal internal-energy functional F[n].Similarly,in the noninteracting KS system one defines
T s[n]:=E s− d3r n(r)v s[n](r)=E s−V s[n](4)
where v s(r)is the effective KS potential and E s the ground-state energy of the KS ,the sum of the KS eigenvalues.The F and T s functionals defined in this way can be ud to establish a density variational principle for E v[n]and to define the exchange-correlation energy E xc[n]=F[n]−T s
[n]−E H[n],where E H is the Hartree energy.4
Thus tamed,degeneracy actually becomes helpful in
further strengthening the foundations of DFT:on a lat-tice,any density can be written as a linear combination of densities arising from enmbles of degenerate ground states of a local potential,thus solving the discretized v−reprentability problem.15
Breakdown of mappings:Nonuniqueness.It is known16at least since1983that mapping1,v[Ψ],breaks down iffinite basis ts are ud to reprent the wave functions.Harriman16gives both general arguments and an explicit example illustrating this breakdown.This breakdown of mapping1is the only one occuring already in the charge-density-only formulation of DFT,and it is manifestly an artifact of the u of afinite basis t.
In multi-density DFTs,such as SDFT and CDFT, the mapping between the t of effective potentials and the t of ground-state densities can break down even in the complete basis-t limit,becau inversion of Schr¨o dinger’s equation does not establish a unique rela-tion between the t of densities and the t of conjugate potentials.This is the so-called nonuniqueness problem of SDFT(and CDFT and others).Following an early obrvation of the problem by von Barth and Hedin,
10 the problem has been shown to be fundamental and per-vasive in recent work by Eschrig and Pickett17and by two of us,18who provided explicit examples of differ-ent SDFT potentials sharing the same ground-state wave function.Ref.18propod a classification of nonunique-ness into systematic(arising from the existence of cer-tain constants of motion)and accidental(arising from special features of the ground state).In both cas,the nonuniqueness is associated with the external potential. Since the mappingΨ[n]remains intact,and internal-energy functionals can be defined exclusively in terms of wave functions,
F[n]= Ψ[n]|ˆT+ˆU|Ψ[n] ,(5) the functionals E v[n]=F[n]+V[n],T s[n]= Φ[n]|ˆT|Φ[n] ,and E xc[n]=F[n]−E H[n]−T s[n]still exist.
Additional examples of both systematic and accidental nonuniqueness were found in CDFT19and in SDFT on a lattice.20The extent to which nonuniqueness of the potentials affects various types of applications of multi-density DFTs,as well as possible remedies,are discusd in Refs.19,20,21.
Breakdown of mappings:Nonuniqueness and degener-acy.We have just en that in the prence of nonunique-ness the mapping v[Ψ]breaks down,whereas in the pres-ence of degeneracy the mappingΨ[n]breaks down.In-terestingly,a crucial fact has been overlooked in the standard analysis o
f either degeneracy or nonunique-ness:The complications can occur simultaneously!If a system with a degenerate ground state is treated with SDFT or any other formulation of DFT suffering from a nonuniqueness problem,none of the mappings holds,and no conventional HK theorem exists.In fact,it isΨ[n] that was ud above to define F[n],T s[n]and E xc[n]in the abnce of v[Ψ](nonuniqueness),while v[Ψ]guaran-teed the existence of v[n]and thus of F[n],T s[n]and E xc[n]in the abnce ofΨ[n](degeneracy).If bothΨ[n] and v[Ψ]break down,it ems nothing is left.The break-down of both mappings is illustrated in Fig. 1.Three simple examples are given below.
Ourfirst example is an extension of the ca of the noninteracting Li atom13to collinear SDFT.If the spin degree of freedom is included,each of the four degenerate states13is additionally twofold degenerate with respect to S z.The t of external potentials B=0,v=3/r thus has an8-fold degenerate ground-state manifold. The Slater determinants formed from the configurations 1s22p+↑and1s22p−↑have the same charge and spin densities.Again,we e that in the prence of a degener-ate ground state the densities do not uniquely determine the wave function.Differently from above,in SDFT we can now also consider the alternative t of external po-tentials B′=const=0,v=3/r.The spin-only mag-neticfield B simply splits the ground-state manifold into two,one comprising the four spin-up configurations,the other the four spin-down configurations.The new grou
nd state will be in the spin-up manifold,where the configu-rations1s22p+↑and1s22p−↑remain and still yield the same densities.From the point of view of the mapping
属木的行业between densities and potentials,this is simply the by now well known17,18,19,20,21nonuniqueness of the poten-tials of SDFT with respect to a weak collinear magnetic field.The full situation,however,is now one in which the densities do not determine the wave functions but only a (ground-state)manifold of them,and some members of the manifolds are ground states in more than one t of external potentials.The functionalsΨ[n],V[Ψ]and V[n]thus do not exist.
Consider next an interacting atom in an S=1,L=1 state.Concrete examples are6C and14Si(with term 3P0)and8O and16S(with term3P2).In the t of ex-
ternal potentials B=0,v=Z/r the ground state of such systems is(2L+1)(2S+1)=9-fold degenerate.Let’s de-note the members of this manifold asΨL z,S z.Several of the,such asΨ1,1andΨ−1,1have the same charge and spin densities.Hence,we have another situation in which the densities do not determine the wave functions but only the manifold.Now consider the same system in ex-ternal potentials B′=const=0and v=Z/r.The statesΨ1,1,Ψ0,1andΨ−1,1remain degenerate ground states in this new t of potentials,and the density and spin density of thefirst and the last are still the same as
for B′=0.Hence,as in other examples of nonunique-ness,knowledge of this state alone does not determine the external potentials.Upon combining both obrvations wefind that to a given t of ground-state densities(n,m) there may correspond more than one degenerate wave function(all in external potentials B=0,v=Z/r),and all of the wave functions are also degenerate ground-states of the different t of external potentials(B′,v). Again,the functionalsΨ[n],V[Ψ]and V[n]do not exist. Lastly,we discuss a modification of the one-electron example by von Barth and Hedin.10Consider a sin-gle electron in the prence of an external4-potential wαβ(r)=V(r)δαβ−[B(r)· σ]αβ,where σis the vec-tor of Pauli matrices.Let wαβ(r)be uniform along one spatial direction(say,x),with periodic boundary con-ditions along that direction parated by a distance L (which is topologically equivalent to confining the elec-tron on a ring).The two-fold degenerate ground state of the Hamiltonian Hαβ=−¯h2
with the constrained-arch formulation of DFT,6which defines F[n]:=minΨ→n Ψ|ˆT+ˆU|Ψ ,although in the prence of degeneracy this definition cannot be ud to defineΨ[n].
Since the noninteracting kinetic energy T s is the inter-nal energy of the Kohn-Sham system,it is also a well-defined density functional,and E xc[n]=F[n]−E H[n]−T s[n]can be constructed as usual.Finally,for a given external potential,the functional E v[n]then obviously also exists.
Conclusions.We have shown both by general argu-ments and by specific examples that in the ca of degen-eracy in multi-density DFTs all three mappings,Ψ[n], V[ψ]and V[n],and thus the entire body of informa-tion usually considered the content of the HK theorem,break down.The weaker joint-degeneracy and internal-energy theorems,however,still allow the definition of the internal-energy functional F[n],and thus also the func-tionals T s[n],E v[n]and E xc[n].Practical DFT,which assumes the existence of the functionals,is thus largely unaffected by the breakdown of the various mappings. However,we stress that we have only proven existence of the functionals,not their differentiability.In fact,in the prence of nonuniqueness,all the functionals are expected to display derivative discontinuities.
K.C.was supported by FAPESP and CNPq. C.A.U. acknowledges support by DOE Grant No.DE-FG02-05ER46213,NSF Grant No.DMR-0553485and Rearch Corporation.GV acknowledges support from NSF Grant No.DMR-0313681.
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虚拟团队
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什么运动最减肥
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帖的读音
5
Local Potentials Ground State
Wavefunctions
Densities
FIG.1:(color online)Schematic illustration of the break-down of mappings that occurs in the prence of degeneracy (black)and the additional complication pod by nonunique-ness(red).V and n stand generically for ts of conjugate po-tentials and ,v↑,v↓and n↑,
n↓in SDFT).Large ovals are ts of functions,medium-size ovals collect degener-ate wave functions,and small ovals enclo degenerate wave functions that give ri to the same density.
办公室规章制度
V
V
n
n'
n
n'
方舟神器位置
V
V
FIG.2:(color online)The joint-degeneracy theorem(ca I) and a situation excluded by it(ca II).Colored ovals enclo degenerate wave functions coming from a local potential.
