Hohenberg-Kohn Theorems in Electrostatic and Uniform Magnetostatic Fields
aa r X i v : . [ c ond - m a t . s t r- e l ] A ug Hohenberg-Kohn Theorems in Electrostatic and UniformMagnetostatic Fields
Xiao-Yin Pan
Department of Physics, Ningbo University, Ningbo 315211, China
Viraht Sahni
Brooklyn College and The Graduate School of the City University of New York,365 Fifth Avenue, New York, NY 10016 (Dated: July 11, 2018)
Abstract
The Hohenberg-Kohn (HK) theorems of bijectivity between the external scalar potential andthe gauge invariant nondegenerate ground state density, and the consequent Euler variationalprinciple for the density, are proved for arbitrary electrostatic field and the constraint of fixedelectron number. The HK theorems are generalized for spinless electrons to the added presenceof an external uniform magnetostatic field by introducing the new constraint of fixed canonicalorbital angular momentum. Thereby a bijective relationship between the external scalar andvector potentials, and the gauge invariant nondegenerate ground state density and physical currentdensity, is proved. A corresponding Euler variational principle in terms of these densities is alsodeveloped. These theorems are further generalized to electrons with spin by imposing the addedconstraint of fixed canonical orbital and spin angular momentum. The proofs differ from theoriginal HK proof, and explicitly account for the many-to-one relationship between the potentialsand the nondegenerate ground state wave function. . INTRODUCTION The Hohenberg-Kohn (HK) theorems [1] constitute a fundamental advance in quantummechanics. As a consequence they have furthered our understanding of the electronic struc-ture of matter: atoms, molecules, solids, clusters, surfaces, lower dimensional electronicsystems such as heterostructures, quantum dots, graphene, etc. Matter, according to HK, isdescribed as a system of N electrons in an external electrostatic field E ( r ) = − ∇ v ( r ). Thefirst HK theorem defines the concept of a basic variable of quantum mechanics. Knowledgeof this gauge invariant property – the nondegenerate ground state density ρ ( r ) – is of two-foldsignificance: (a) It determines the Schr¨odinger theory wave functions Ψ of the system, bothground and excited state; (b)
As the wave function Ψ is now proved to be a functional of thebasic variable, it constitutes together with the second HK theorem – the energy variationalprinciple for arbitrary variations of the density – the basis of theories of electronic structuresuch as of Hohenberg-Kohn [1], Kohn-Sham [2], and quantal density functional theory [3, 4].The theorems are valid for arbitrary confining potential v ( r ) and electron number N , butare derived [5] for the constraint of fixed N . In this paper we generalize the HK theoremsfor spinless electrons to the added presence of an external uniform magnetostatic field. Asthe presence of the magnetic field constitutes a new degree of freedom, we introduce thefurther natural constraint of fixed canonical orbital angular momentum. Thereby we provethat the basic variables in quantum mechanics in a uniform magnetic field are the gaugeinvariant nondegenerate ground state density ρ ( r ) and physical current density j ( r ). Thesetheorems are then further generalized to electrons with spin by imposing the constraints of fixed canonical orbital and spin angular momentum.The generalization is motivated by the considerable recent interest in yrast states whichare states of lowest energy for fixed angular momentum. These states have been studiedexperimentally and theoretically for both bosons and fermions, e.g. rotating trapped Bose-Einstein condensates [6], and harmonically trapped electrons in the presence of a uniformperpendicular magnetic field [7]. The theorems derived are applicable to all experimentationwith a uniform magnetic field such as the magneto-caloric effect [8], the Zeeman effect,cyclotron resonance, magnetoresistance, the de-Haas-van Alphen effect, the Hall effect, thequantum Hall effect, the Meissner effect, nuclear magnetic resonance, etc.The manner by which a basic variable is so defined is via the proof of the first HK2heorem for v -representable densities. To explain this, and to contrast the present proofswith the HK proof, we first briefly describe the HK arguments. The HK theorems areproved for a nondegenerate ground state. Particularizing to electrons without any loss ofgenerality, the Hamiltonian ˆ H in atomic units (charge of electron − e ; | e | = ~ = m = 1)is ˆ H = P k p k + P ′ k,ℓ / | r k − r ℓ | + P k v ( r k ), where the terms correspond to the kineticˆ T (with momentum ˆ p k = − i ∇ r k ), the electron-interaction ˆ W , and external potential ˆ V operators, respectively. The Schr¨odinger equation is ˆ H ( R )Ψ( X ) = E Ψ( X ), where Ψ( X ) , E are the eigenfunctions and eigenenergies, with R = r , . . . , r N ; X = x , . . . , x N ; x = r σ beingthe spatial and spin coordinates of the electron. The energy E is the expectation E = < Ψ( X ) | ˆ H ( R ) | Ψ( X ) > . In the first HK theorem it is initially proved (Map C) that there is aone-to-one relationship between the external potential v ( r ) and the nondegenerate ground-state wave function Ψ( X ). Employing this relationship , it is then proved (Map D) thatthere is a one-to-one relationship between the wave function Ψ( X ) and the correspondingnondegenerate ground state density ρ ( r ). Thus, knowledge of ρ ( r ) determines v ( r ) to withina constant. Since for a fixed electron number N , the kinetic ˆ T and electron-interactionpotential ˆ W energy operators are known, so is the system Hamiltonian. Solution of thecorresponding Schr¨odinger equation then leads to the wave functions Ψ of the system. Itis the one-to-one relationship between the external potential and the gauge invariant densitythat defines the latter as a basic variable . As the wave function Ψ, and hence energy E v [ ρ ]are functionals of the density ρ ( r ), the variational Euler equation for the density with fixed v ( r ) follows subject to the constraint of known electron number N (see Table 1). (The lowestnondegenerate [9, 10] excited state density ρ e ( r ) of a given symmetry different from that ofthe ground state is also a basic variable.)In the added presence of an external magnetostatic field B ( r ) = ∇ × A ( r ), where A ( r )is the vector potential, the Hamiltonian when the interaction of the field is only with theorbital angular momentum isˆ H = 12 X k (cid:20) ˆ p k + 1 c A ( r k ) (cid:21) + ˆ W + ˆ V . (1)When the interaction of the magnetic field is with both the orbital and spin angular mo-mentum, the Hamiltonian isˆ H = 12 X k (cid:20) ˆ p k + 1 c A ( r k ) (cid:21) + ˆ W + ˆ V + 1 c X k B ( r k ) · s k , (2)3here s is the electron spin angular momentum vector. In deriving the Hamiltonians of Eqs.(1) and (2), we have hewed to the philosophy [11] that the only ‘fundamental’ interactionsare those that can be generated by the substitution ˆ p → ˆ p + c A . (This then defines thephysical momentum operator in the presence of a magnetic field, and thereby the physicalcurrent density j ( r ).) In non-relativistic quantum mechanics, the Hamiltonian of Eq. (2) isderived [11] by Schr¨odinger-Pauli theory for spin particles via the kinetic energy operator σ · ( p + A ) σ · ( p + A ), where σ is the Pauli matrix, and s = σ . The spin magneticmoment generated in this way has the correct gyromagnetic ratio g = 2.It would appear that one could prove a one-to-one relationship between the gauge invari-ant properties { ρ ( r ) , j ( r ) } and the external potentials { v ( r ) , A ( r ) } along the lines of the HKproof. However, no such proof is possible as the relationship between the external potentials { v ( r ) , A ( r ) } and the non-degenerate ground state wave function Ψ( X ) can be many-to-one [12] and even infinite-to-one [13]. Hence, in these cases, there is no equivalent of Map C,and therefore the original HK path is not possible. The proof that { ρ ( r ) , j ( r ) } are the basicvariables must then differ from the original HK proof. Furthermore, the proof must accountfor the many-to-one relationship between { v ( r ) , A ( r ) } and Ψ( X ).In the literature [2, 12, 14], the proofs of what properties constitute the basic variables arenot rigorous in the HK sense of the one-to-one relationship between the basic variables andthe external potentials { v, A } . Further, they do not account for the many-to-one relationshipbetween { v, A } and Ψ. Additionally, the system angular momentum is not considered. Thechoice of the basic variables is arrived at solely on the basis of a Map D-type proof betweenthese assumed properties and the nondegenerate ground state Ψ, thereby the claim thatΨ is a functional of these properties. In these proofs, the existence of a bijective MapC is implicitly assumed, [15, 16] (see also last reference of 12). For example, in spin-DFT [2, 12, 14] for which the Hamiltonian is that of Eq. (2) with the field componentof the momentum absent, the basic variables are assumed to be { ρ ( r ) , m ( r ) } , where m ( r )is the magnetization density. In current-DFT [14], corresponding to the Hamiltonian ofEq. (1), the basic variables are assumed to be ρ ( r ) and the gauge variant paramagneticcurrent density j p ( r ). For the Hamiltonian of Eq. (2), the basic variables are assumedto be { ρ ( r ) , m ( r ) , j p ( r ) } or { ρ ( r ) , m ( r ) , j p ( r ) , j p, m ( r ) } where j p, m ( r ) are the gauge variantparamagnetic currents of each component of the magnetization density. Subsequently, aMap D proof is provided. Additionally, with the basic variables now assumed known, a4ercus-Levy-Lieb (PLL)-type proof [17, 18] can then be formulated [19]. More recently, wegave a derivation [15, 20] which purported to prove that { ρ ( r ) , j ( r ) } were the basic variablesbut the proof was in error [21]. Subsequently, we proved [22] for the Hamiltonian of Eq. (1)that for the significant subset of systems [13, 23] for which the ground state wave function Ψis real, the basic variables are { ρ ( r ) , j ( r ) } . Our proof of bijectivity between { ρ ( r ) , j ( r ) } and { v ( r ) , A ( r ) } explicitly accounts for the many-to-one { v ( r ) , A ( r ) } to Ψ relationship. Thisproof then constitutes a special case of the more general proof for Ψ complex presented inthis work.Here we extend the HK theorems to systems of electrons in external electrostatic E ( r ) = − ∇ v ( r ) and magnetostatic B ( r ) = ∇ × A ( r ) fields with known electron num-ber N and angular momentum J . The proofs are for a uniform magnetostatic field, andfor Hamiltonians in which the interaction of the magnetic field is (i) solely with the orbitalangular momentum ( J = L ), and (ii) with both the orbital and spin angular momentum( J = [ L and S ]). We prove, in the rigorous HK sense, that for fixed N and J the ba-sic variables are the gauge invariant nondegenerate ground state density ρ ( r ) and physical current density j ( r ). In other words, knowledge of { ρ ( r ) , j ( r ) } determines the potentials { v ( r ) , A ( r ) } to within a constant and the gradient of a scalar function, respectively. Hence,with the Hamiltonians known, solution of the respective Schr¨odinger and Schr¨odinger-Pauliequations lead to the wave functions of each system. The proof is for ( v, A )-representable { ρ ( r ) , j ( r ) } . The extension to the Percus-Levy-Lieb (PLL) [17, 18] constrained-search pathfor N -representable and degenerate states readily follows. As the wave function Ψ is afunctional of { ρ ( r ) , j ( r ) } , theories of electronic structure based on { ρ ( r ) , j ( r ) } as the basicvariables can then be formulated. II. PROOF OF GENERALIZED HOHENBERG-KOHN THEOREMS
To accentuate the role of the density ρ ( r ) and physical current density j ( r ), we rewritethe Hamiltonians of Eqs. (1) and (2) in terms of operators representative of these gaugeinvariant properties. The Hamiltonians can then be written, respectively, asˆ H = ˆ T + ˆ W + ˆ V A , (3)5nd ˆ H = ˆ T + ˆ W + ˆ V A − Z ˆ m ( r ) · B ( r ) d r , (4)where the total external potential operator ˆ V A isˆ V A = ˆ V + 1 c Z ˆ j ( r ) · A ( r ) d r − c Z ˆ ρ ( r ) A ( r ) d r , (5)and the corresponding energy expectations E = < Ψ( X ) | ˆ H | Ψ( X ) > as E = T + E ee + V A , (6)and E = T + E ee + V A − Z m ( r ) · B ( r ) d r , (7)where the total external potential energy V A is V A = < Ψ( X ) | ˆ V A | Ψ( X ) = Z ρ ( r ) v ( r ) d r + 1 c Z j ( r ) · A ( r ) d r − c Z ρ ( r ) A ( r ) d r , (8)and where T and E ee are the kinetic and electron-interaction energy expectations. In theabove equations, the physical current density j ( r ) is defined in terms of the physical mo-mentum operator (ˆ p + c A ) as j ( r ) = N ℜ X σ Z Ψ ⋆ ( r σ, X N − ) (cid:18) ˆ p + 1 c A ( r ) (cid:19) Ψ( r σ, X N − ) d X N − , (9)or equivalently as the expectation of the current density operator ˆ j ( r ): j ( r ) = < Ψ( X ) | ˆ j ( r ) | Ψ( X ) > (10)where ˆ j ( r ) = ˆ j p ( r ) + ˆ j d ( r ) , (11)with the paramagnetic ˆ j p ( r ) and diamagnetic ˆ j d ( r ) operator components defined, respec-tively, as ˆ j p ( r ) = 12 X k (cid:2) ˆ p k δ ( r k − r ) + δ ( r k − r )ˆ p k (cid:3) , (12)and ˆ j d ( r ) = ˆ ρ ( r ) A ( r ) /c, (13)with the density operator ˆ ρ ( r ) beingˆ ρ ( r ) = X k δ ( r k − r ) . (14)6he magnetization density m ( r ) is the expectation m ( r ) = < Ψ( X ) | ˆ m ( r ) | Ψ( X ) >, (15)with the local magnetization density operator ˆ m ( r ) defined asˆ m ( r ) = − c X k s k δ ( r k − r ) . (16)(The current density operator ˆ j ( r ) can also be defined in terms of the Hamiltonian ˆ H asˆ j ( r ) = c∂ ˆ H/∂ A . This confirms that for both the Hamiltonians of Eqs. (3) and (4), thephysical current density is the orbital current density.)We first present the proof of bijectivity between { ρ ( r ) , j ( r ) } and { v ( r ) , A ( r ) } for spinlesselectrons corresponding to the Hamiltonian of Eq. (1) or (3) for fixed electron number N and angular momentum L . The proof is by reductio ad absurdum . Let us consider twodifferent physical systems { v, A } and { v ′ , A ′ } that generate different nondegenerate groundstate wave functions Ψ and Ψ ′ . We assume the gauges of the unprimed and primed systemsto be the same. Let us further assume that these systems lead to the same nondegenerateground state { ρ ( r ) , j ( r ) } . We prove this cannot be the case. From the variational principlefor the energy for a nondegenerate ground state, one obtains the inequality E = < Ψ | ˆ H | Ψ > < < Ψ ′ | ˆ H | Ψ ′ > . (17)Now < Ψ ′ | ˆ H | Ψ ′ > = < Ψ ′ | ˆ T + ˆ W + ˆ V ′ + 1 c Z ˆ j ′ ( r ) · A ′ ( r ) d r − c Z ˆ ρ ( r ) A ′ ( r ) d r | Ψ ′ > + < Ψ ′ | ˆ V − ˆ V ′ | Ψ ′ > + 1 c < Ψ ′ | Z [ˆ j ( r ) · A ( r ) − ˆ j ′ ( r ) · A ′ ( r )] d r | Ψ ′ > − c < Ψ ′ | Z ˆ ρ ( r )[ A ( r ) − A ′ ( r )] d r | Ψ ′ > . (18)Employing the above assumptions, and following the same steps as in [22], one obtains theinequality E + E ′ < E + E ′ + Z (cid:2) j ′ p ( r ) − j p ( r ) (cid:3) · (cid:2) A ( r ) − A ′ ( r ) (cid:3) d r , (19)where E ′ = < Ψ ′ | ˆ H ′ | Ψ ′ > . 7s the majority of the experimental and consequent theoretical work is performed foruniform magnetic fields, our proof too is for such fields.Consider next the third term on the right hand side of Eq. (19). With B ( r ) = B ˆ i z , B ′ ( r ) = B ′ ˆ i z , and the symmetric gauge A ( r ) = B × r , A ′ ( r ) = B ′ × r , this term may bewritten as I = 12 ∆ B · Z r × (cid:20) j ′ p − j p ( r ) (cid:21) d r , (20)where ∆ B = ( B − B ′ )ˆ i z . First consider the integral I = Z r × j p ( r ) d r (21)= − i X k Z d X Z d r Ψ ⋆ ( X ) (cid:2) r × ∇ r k δ ( r − r k ) + δ ( r − r k ) r × ∇ r k (cid:3) Ψ( X ) . (22)Next consider the second integral of I of Eq. (22): I = 12 Z d X Ψ ⋆ ( X ) (cid:0) X k r k × ˆ p k (cid:1) Ψ( X ) (23)= 12 Z d X Ψ ⋆ ( X ) X k ˆ L k Ψ( X ) = 12 L , (24)where ˆ L k = r k × ˆ p k is the canonical orbital angular momentum operator, with ˆ p the canonicalmomentum operator (ˆ p = ˆ p kinetic + ˆ p field = m v + qc A ), and L the total canonical orbitalangular momentum defined by Eq. (24). Note that the canonical angular momentum isgauge variant.The first integral of I of Eq. (22) is I = − i X k Z d X Z d r Ψ ⋆ ( X ) ǫ αβγ ∂∂r kγ (cid:0) r β δ ( r − r k )Ψ( X ) (cid:1) . (25)On integrating the inner integral by parts and dropping the surface term, one obtains I = − i X k Z d X (cid:2) − ǫ αβγ Z d r ∂ Ψ ⋆ ( X ) ∂r kγ r β δ ( r − r k )Ψ( X ) (cid:3) (26)= − i X k Z d X (cid:2) − ǫ αβγ ∂ Ψ ⋆ ( X ) ∂r kγ r kβ Ψ( X ) (cid:3) . (27)On integrating by parts again, one obtains I = − i X k ǫ αβγ Z d X Ψ ⋆ ( X ) ∂∂r kγ (cid:0) r kβ Ψ( X ) (cid:1) (28)= − i X k Z d X Ψ ⋆ ( X ) ( r k × ∇ r k )Ψ( X ) = 12 L (29)8ence, the integral I of Eq. (20) is I = 12 ∆ B · ( L ′ − L ) . (30)If one imposes the condition that the total canonical orbital angular momentum is fixed sothat L = L ′ , then the integral I vanishes so that the third term on the right hand side ofEq. (19) vanishes.For states with fixed orbital angular momentum L , Eq. (19) then reduces to the contra-diction E + E ′ < E + E ′ . (31)What this means is that the original assumption that Ψ and Ψ ′ differ is erroneous, and thatthere can exist a { v, A } and a { v ′ , A ′ } with the same nondegenerate ground state wavefunction. The fact that Ψ = Ψ ′ means that ρ ( r ) | Ψ = ρ ′ ( r ) | Ψ ′ . However, the correspondingphysical current densities are not the same: j ( r ) | Ψ = j ′ ( r ) | Ψ ′ , because j d ( r ) | Ψ = j ′ d | Ψ ′ if onehews with the original assumption that A ( r ) is different from A ′ ( r ). This proves that theassumption that there exists a different { v ′ , A ′ } (with the same N and L ) that leads to thesame { ρ, j } as that due to { v, A } is incorrect. This step takes into account the fact thatthere could exist many { v, A } that lead to the same nondegenerate ground state Ψ. Hence,there exists only one { v, A } for fixed N and L that leads to a nondegenerate ground state { ρ, j } . The one-to-one relationship between { ρ, j } and { v, A } is therefore proved for the casewhen the interaction of the magnetic field is solely with the orbital angular momentum.With { ρ ( r ) , j ( r ) } as the basic variables, the wave function Ψ is a functional of theseproperties. By a density and physical current density preserving unitary transformation[4, 15, 24] it can be shown that the wave function must also be a functional of a gauge function α ( R ). This ensures that the wave function when written as a functional: Ψ = Ψ[ ρ, j , α ] isgauge variant. However, as the physical system remains unchanged for different gaugefunctions, the choice of vanishing gauge function is valid.As the ground state energy is a functional of the basic variables: E = E v, A [ ρ, j ], a varia-tional principle for E v, A [ ρ, j ] exists for arbitrary variations of ( v, A )-representable densities { ρ ( r ) , j ( r ) } . The corresponding Euler equations for ρ ( r ) and j ( r ) follow, and these must besolved self-consistently with the constraints R ρ ( r ) d r = N , R r × ( j ( r ) − c ρ ( r ) A ( r ) d r = L and ∇ · j ( r ) = 0. Implicit in this variational principle, as in all such energy variational9rinciples, is that the external potentials remain fixed throughout the variation . (See TableI.) We next consider electrons with spin corresponding to the Hamiltonian of Eq. (2) or (4).In this case, with the same assumptions made regarding the two different physical systems { v, A ; ψ } and { v ′ , A ′ ; ψ ′ } leading to the same { ρ ( r ) , j ( r ) as before, the inequality of Eq. (19)is replaced by E + E ′ < E + E ′ + Z (cid:2) j ′ p ( r ) − j p ( r ) (cid:3) · (cid:2) A ( r ) − A ′ ( r ) (cid:3) d r − Z (cid:2) m ′ ( r ) − m ( r ) (cid:3) · (cid:2) B ( r ) − B ′ ( r ) (cid:3) d r . (32)The third term on the right hand side vanishes if one imposes the constraint that the orbitalangular momentum L of the unprimed and primed systems are the same. Hence, nextconsider the last term of Eq. (32). For a uniform magnetic field with B ( r ) = B ˆ i z and B ′ ( r ) = B ˆ i z , we have Z m ( r ) · B ( r ) d r = B Z m z ( r ) d r , (33)where [19] m z ( r ) = − c (cid:2) ρ α ( r ) − ρ β ( r ) (cid:3) , (34)with ρ α ( r ) , ρ β ( r ) being the spin-up and spin-down spin densities. The last term of theinequality is then Z (cid:2) m ′ ( r ) − m ( r ) (cid:3) · ∆ B ( r ) d r = − c ∆ B Z (cid:2) { ρ ′ α ( r ) − ρ ′ β ( r ) } − { ρ α ( r ) − ρ β ( r ) } (cid:3) d r , (35)with ∆ B = B − B ′ . If the z -component of the total spin angular momentum S z for theunprimed and primed systems are the same, the corresponding spin densities are the same.The last term of Eq. (35) thus vanishes leading once again to the contradiction E + E ′ In conclusion, we have generalized the HK theorems to the added presence of a uniformmagnetic field. We have considered the cases of the interaction of the magnetic field withthe orbital angular momentum as well as when the interaction is with both the orbital andspin angular momentum. In this work we have proved a one-to-one relationship between theexternal potentials { v ( r ) , A ( r ) } and the nondegenerate ground state densities { ρ ( r ) , j ( r ) } .The proof differs from that of the original HK theorem, and explicitly accounts for the many-to-one relationship between the potentials { v ( r ) , A ( r ) } and the nondegenerate ground statewave function Ψ. To account for the presence of the magnetic field, which constitutes anadded degree of freedom, one must then impose a further constraint beyond that of fixedelectron number N as in the original HK theorems. For the Hamiltonian corresponding tospinless electrons, the added constraint is that of fixed canonical orbital angular momentum L . For that corresponding to electrons with spin, the constraints imposed are those of fixedcanonical orbital L and spin S angular momentum. (The gauge employed for the canonicalangular momentum L can be chosen to be the same as that employed for the Hamiltonian.)It is the further constraint on the angular momentum that makes a rigorous HK-type proofof bijectivity between the gauge invariant basic variables and the external scalar and vectorpotentials possible. Additionally, the HK-type proofs are possible because the Hamiltoniansconsidered are rigorously derived from the tenets of nonrelativistic quantum mechanics.With the knowledge that the basic variables are { ρ ( r ) , j ( r ) } , a variational principlefor the energy functional E v, A [ ρ, j ] for arbitrary variations of ( v, A )-representable densi-ties { ρ ( r ) , j ( r ) } is then developed for each Hamiltonian considered. The constraints on thecorresponding Euler equations are those of fixed electron number and angular momentum,and the satisfaction of the equation of continuity.Again, knowing what the basic variables are, it is possible to map the interacting systemdefined by the Hamiltonians of Eqs. (1) and (2) to one of noninteracting fermions with thesame ρ ( r ) , j ( r ), and J . Such a mapping has been derived within QDFT [26]. The theoryhas been applied to map an interacting system [13] of two electrons in a magnetic fieldand a harmonic trap v ( r ) = ω r for which the ground state wave function is Ψ( r , r ) = C (1 + r ) e − ( r r ) , where r = | r − r | and C = 1 /π (3 + √ π ), to one of noninteractingfermions with the same { ρ ( r ) , j ( r ) } . This example corresponds to the special case of zero12ngular momentum. However, the QDFT mapping for finite angular momentum is straightforward. For other recent work see [27, 28]. The conclusions in the latter are based on theassumption of existence of a HK theorem but one without the requirement of the constrainton the angular momentum.X.-P. was supported by the National Natural Sciences Foundation of China, Grant No.11275100, and the K.C. Wong Magna Foundation of Ningbo University. The work of V.S.was supported in part by the Research Foundation of the City University of New York. [1] P. Hohenberg and W. Kohn, Phys. Rev. , B864 (1964).[2] W. Kohn and L. J. Sham, Phys. Rev. , A1133 (1965).[3] V. Sahni, Quantal Density Functional Theory , Springer-Verlag, Berlin, Heidelberg (2004).[4] V. Sahni, Quantal Density Functional Theory II: Approximation Methods and Applications ,Springer-Verlag, Berlin, Heidelberg (2010).[5] X.-Y. Pan and V. Sahni, J. Chem. Phys. , 164116 (2010).[6] A. Fetter, Rev. Mod. Phys. , 647 (2009); N. K. Wilkin et al, Phys. Rev. , 2265 (1998);D. A. Butts and D. S. Rokhsar, Nature (London) , 327 (1999); G. F. Bertsch and T.Papenbrock, Phys. Rev. Lett. , 5412 (1999); R. A. Smith and N. K. Wilkin, Phys. Rev. A , 061602(R) (2000); G. M. Kavoulakis et al, Phys. Rev. A , 063605 (2000); M. Linn et al,Phys. Rev. A , 023602 (2001); E. Kamanishi et al, J. Phys: Conference Series , 012030(2014).[7] H. Saarikovski et al, Rev. Mod. Phys. , 2785 (2010); S. A. Trugman and S. Kivelson, Phys.Rev. B , 5280 (1985); N. K. Wilkin et al, Phys. Rev. Lett. , 2265 (1998); B. Mottelson,Phys. Rev. Lett. , 2695 (1999); E. Anisimovas et al, Phys. Rev. B , 195334 (2004).[8] J.B. Staunton et al, Phys. Rev. B (R), 060404 (2013); K.G. Sandeman, Scr. Mater. , 566(2012).[9] O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B , 4274 (1976).[10] Y. -Q. Li et al, Phys. Rev. A , 032517 (2012).[11] J. J. Sakurai, Advanced Quantum Mechanics , Addison-Wesley, Reading, MA, (1967).[12] U. von Barth and L. Hedin, J. Phys. C ,1629 (1972); K. Capelle and G. Vignale, Phys. Rev.Lett. , 5546 (2001); H. Eschrig and W. E. Pickett, Solid State Commun. , 123 (2001); . Capelle and G. Vignale, Phys. Rev. B , 113106 (2002); A. Laestadius and M. Benedicks,Int. J. Quantum Chem. , 782 (2014).[13] M. Taut, J. Phys. A: Math. Gen. , 1045 (1994); , 4723 (1994); M. Taut and H. Eschrig,Z. Phys. Chem. , 631 (2010).[14] M. M. Pant and A. K. Rajagopal, Solid State Commun. , 1157 (1972); G. Vignale and M.Rasolt, Phys. Rev. Lett. , 2360 (1987); Phys. Rev. B. , 10685 (1988); G. Vignale et al,Adv. Quantum Chem. , 235 (1990); G. Diener, J. Phys. Cond. Matter , 9417 (1991); K.Capelle and E. K. U. Gross, Phys. Rev. Lett. , 1872 (1997); W. Kohn et al, Int. J. QuantumChem. , 20 (2004); S. Rohra and A. G¨orling, Phys. Rev. Lett. , 013005 (2006); W. Yanget al, Phys. Rev. Lett. , 146404 (2004); P. W. Ayers and W. Yang, J. Chem. Phys. ,224108 (2006); T. Heaton-Burgess et al, Phys. Rev. Lett. , 036403 (2007).[15] X.-Y. Pan and V. Sahni, Int. J. Quantum Chem. , 2833 (2010); J. Phys. Chem. Solids, , 630 (2012); G. Vignale et al, Int. J. Quantum Chem. , 1422 (2013); X.-Y. Pan and V.Sahni, Int. J. Quantum Chem. , 424 (2013).[16] M. Taut et al, Phys. Rev. A , 022517 (2009).[17] V. Sahni and X.-Y. Pan, Phys. Rev. A , 052502 (2012).[18] J. Percus, Int. J. Quantum Chem.13, (1978); M. Levy, Proc. Natl. Acad. Sci. USA ,6062 (1979); E. Lieb, Int. J. Quantum Chem. , 243 (1983); M. Levy, Int. J. Quantum Chem. , 3140 (2010).[19] J. P. Perdew and A. Zunger, Phys. Rev. B , 5048 (1981); R.G. Parr and W. Yang, DensityFunctional Theory of Atoms and Molecules , Oxford Uniiversity Press, New York (1989); R.M.Dreizler and E.K.U. Gross, Density Functional Theory , Springer-Verlag, Berlin (1990).[20] X.-Y. Pan and V. Sahni, Phys. Rev. A , 042502 (2012).[21] E. I. Tellgren et al, Phys. Rev. A , 062506 (2012).[22] X.-Y. Pan and V. Sahni, Int. J. Quantum Chem. , 233 (2014).[23] S. M. Reimann and M. Manninen, Rev. Mod. Phys. , 1283 (2002); J.-L. Zhu, et al, Phys.Rev. B. , 045324 (2003); P.-F. Loos and P. M. W. Gill, Phys. Rev. Lett. , 083002 (2012).[24] X.-Y. Pan and V. Sahni, Int. J. Quantum Chem. , 2756 (2008).[25] L. D. Landau and E. M. Lifshitz, Quantum Mechanics , Pergamon Press (1965).[26] T. Yang, X. -Y. Pan, and V. Sahni, Phys. Rev. A , 042518 (2011).[27] E. H. Lieb and R. S. Schrader, Phys. Rev. A , 032516 (2013). 28] A. Laestadius and M. Benedicks. Phys. Rev. A , 032508 (2015). heory Hohenberg-Kohn DFT Generalized HK DFT Parameters characterizingground state Electron Number N Electron Number N Angular momentum L Relationship betweenpotentials and wave function One-to-one between v ( r ) and Ψ Many-to-one between { v ( r ) , A ( r ) } and ΨProperties characterizingground state Electron density ρ ( r ) Electron density ρ ( r )Physical current density j ( r )Bijectivity theorem For fixed Nρ ( r ) ↔ v ( r ) For fixed N and L { ρ ( r ) , j ( r ) } ↔ { v ( r ) , A ( r ) } Wave functionand Energy functionals Ψ = Ψ[ ρ, α ]For fixed v : E = E v [ ρ ] Ψ = Ψ[ ρ, j , α ]For fixed { v, A } : E = E v, A [ ρ, j ]Euler equationsand constraints Variational principle forfixed v and known N : δE v [ ρ ] δρ = 0 R ρ ( r ) d r = N Variational principle forfixed { v, A } and known N, L : δE v, A [ ρ, j ] δρ (cid:12)(cid:12)(cid:12)(cid:12) j = 0 δE v, A [ ρ, j ] δ j (cid:12)(cid:12)(cid:12)(cid:12) ρ = 0 R ρ ( r ) d r = N R r × ( j ( r ) − c ρ ( r ) A ( r )) d r = L ∇ · j ( r ) = 0TABLE I: Comparison of Hohenberg-Kohn and Generalized Hohenberg-Kohn theories.) = 0TABLE I: Comparison of Hohenberg-Kohn and Generalized Hohenberg-Kohn theories.