Excited-State Density-Functional Theory Revisited: on the Uniqueness, Existence, and Construction of the Density-to-Potential Mapping
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J un Excited-State Density-Functional Theory Revisited: on the Uniqueness, Existence,and Construction of the Density-to-Potential Mapping
Prasanjit Samal, ∗ Subrata Jana, and Sourabh S. Chauhan School of Physical Sciences, National Institute of Science Education & Research, Bhubaneswar 751005, INDIA. (Dated: September 20, 2018)The generalized constrained search formalism is used to address the issues concerning density-to-potential mapping for excited states in time-independent density-functional theory. The multiplicityof potentials for any given density and the uniqueness in density-to-potential mapping are explainedwithin the framework of unified constrained search formalism for excited-states due to G¨orling,Levy-Nagy, Samal-Harbola and Ayers-Levy. The extensions of Samal-Harbola criteria and it’s linkto the generalized constrained search formalism are revealed in the context of existence and uniqueconstruction of the density-to-potential mapping. The close connections between the proposed cri-teria and the generalized adiabatic connection are further elaborated so as to keep the desiredmapping intact at the strictly correlated regime. Exemplification of the unified constrained searchformalism is done through model systems in order to demonstrate that the seemingly contradic-tory results reported so far are neither the true confirmation of lack of Hohenberg-Kohn theoremnor valid representation of violation of Gunnarsson-Lundqvist theorem for excited states. Hencethe misleading interpretation of subtle differences between the ground and excited state densityfunctional formalism are exemplified.
I. INTRODUCTION
Since its advent, density-functional theory (DFT) [1–10] is routinely applied for calculating the electronic,magnetic, spectroscopic and thermodynamic propertiesof atoms, molecules and materials in ground and excitedstates. In the last couple of decades, studying excited-states employing DFT has become the main research in-terest [8–48]. Thus one of the most natural approach todo excited-state DFT is to adopt the time-independentdensity functional formalism [23, 33, 41, 49] in whichthe individual excited-state energies are determined fromthe stationary states of the energy density functional.However, the question is whether there exists any suchfunctional(s) for excited states analogous to the ground-state. Not only energy functionals but also the mostfundamental and essential requirement for excited-statedensity functional theory ( e DFT) is to establish the one-to-one mapping similar to the Hohenberg-Kohn theoremwhich is the main intent of the present work. Althoughthe issue of density ρ ( ~r ) to potential ˆ v ( ~r ) mapping forexcited states has been addressed in the past [50–55],but the question still remains unanswered. So the cur-rent work will answer the critiques of density-to-potentialmapping based on the generalized/unified constrainedsearch(CS) due to Perdew-Levy(PL) [21], G¨orling [24–26], Levy-Nagy(LN) [27–29], Samal-Harbola(SH) [56–59] ∗ E-mail: [email protected] and Ayers-Levy-Nagy [60–62].In the present work, we will critically analyse and makefurtherance to the e DFT ideas proposed by Samal andHarbola [41, 58]. According to it, (i) the CS approach canbe extended to excited-state in the light of the stationarystate formalism of G¨orling [24–26] and variational e DFTformalism by Levy-Nagy [27–29]; (ii) within the varia-tional e DFT formalism, the construction of the Kohn-Sham(KS) system by comparing only the ground-statedensity is insufficient and can’t explain the existence ofmultiple potentials; (iii) the density-to-potential map-ping in e DFT can be achieved through the following crite-ria: compare the ground states of the true and KS systemenergetically such that it can account for the most closeresemblance of the densities in a least square sense. SHshowed it by comparing the expectation value of the orig-inal ground-state KS Hamiltonian (obtained using theHarbola-Sahni [63] exact exchange potential) with thatof the alternative KS systems. Finally, the kinetic energyof true and KS system need to be kept closest. This isalso another way of comparing the ground states basedon the differential virial theorem(DVT) [64]; (iv) the CSapproach is capable of generating all the potentials for agiven excited state density and at the same time uniquelyestablishes the density-to-potential mapping.The work is organized as follows. In Sec.II, the gen-eralized/unified CS e DFT will be briefly discussed fromthe prospective of density-to-potential mapping. It willbe shown that there exist multitude of potentials for agiven density. In Sec.III, furtherance of SH e DFT will bepresented in order to show the existence and unique con-struction of the desired density-to-potential mapping. Inthis, we will show, how the proposed e DFT is also con-sistent with the generalized adiabatic connection(GAC)KS formalism [11, 24–26, 69–77] and in principle appli-cable to (non-)coulombic densities. In Sec.IV, we willshow the existence of multiple potentials for given groundor lowest excited states can never be ruled out evenwithin Li. et al.[50] demonstration of Gunnarsson andLundqvist(GL) theorem [11, 12]. However, based on thetheories presented in Sec.II & III, these seemingly con-tradictory results will be explained in order to justify thenon-violation of Hohenberg-Kohn(HK) [1] and GL the-orems. Thus the density-to-potential mapping will bedemonstrated within [50] approach by making use of theunified e DFT for the two model systems (i.e. 1 D quan-tum harmonic oscillator(QHO) with finite boundary andinfinite well external potentials). For completeness, inSec. V, same set of model systems will be used to ex-emplify density-to-potential mapping based on the CSformalism [78]. Finally, we will provide firm footing todensity-to-potential mapping based on the proposed cri-teria of e DFT.
II. UNIFIED CONSTRAINED-SEARCHFORMULATION OF eDFT
Although in principle the ground-state CS formalism[3–7] has all the information about the excited-states,the desired density-to-potential mapping for individualexcited-states are not so trivial and straightforward. Todo so, series of attempts being made based on the originalCS approach [21–29, 48, 56–62]. In the recent past, theform of functional for ground state (both for degenerateand non-degenerate) has been extended [57, 58, 65–68] tostudy the excited states. Now we will briefly describe howthe generalized CS formalism explains the existence ofmultiple potentials for any given fermionic density with-out hindering the density-to-potential mapping.Let’s consider N fermions trapped in a local externalpotential ˆ v ext ( ~r ), described by the Hamiltonianˆ H [ˆ v ; N ] = ˆ T + ˆ V ee + N X i =1 ˆ v ext ( ~r i ) , (1)where ˆ T and ˆ V ee are the kinetic and electron-electroninteraction operators with the corresponding stationary states are given byˆ H [ˆ v ( ~r ) , N ]Ψ k ( ~r ) = E k [ˆ v ( ~r ) , N ]Ψ k ( ~r ) , (2)where ˆ v ext ( ~r ) ≡ ˆ v ( ~r ). In Eq.(2), Ψ k ( ~r ) ≡ Ψ k [ˆ v ( ~r ) , N ]are the pure state v − representable stationary quan-tum states i.e. it is coming from the solution of theSchr¨odinger equation. But for N − representable densi-ties (i.e. R ρ ( ~r ) d~r = N ) and therefore wavefunctions (i.e. R Ψ[ N ] d~r = N ), similar to the HK universal functionalthere exists an analogous functional which is stationaryw.r.t all the variations that do not change the density(i.e. δ Ψ → ρ ) and is given by Q S [ ρ ; N ] = δ Ψ[ N ] → ρ ( ~r ) h Ψ | ˆ T + ˆ V ee | Ψ i . (3)Now according to the Perdew and Levy extremum prin-ciple [21] and generalized CS formalism [25, 58, 60], theenergy of the k th excited state is given by E k = E [ ρ k ; N ] = Q S [ ρ k ; N ] + Z ρ k ( ~r ) v ext ( ~r ) d~r. (4)In Eq.(4), the minimization occurs only over G¨orling’sstationary-state functional Q S [ ρ k ] and the correspondingwavefunctions are given byΨ Sk = Ψ S [ ρ k , N ] = arg min Ψ[ N ] → ρ k h Ψ[ N ] | ˆ T + ˆ V ee | Ψ[ N ] i . (5)On the other hand, in the LN [27–29] variational con-strained minimization approach for excited-states leadsto the k th stationary state energy E k [ ρ, ρ ] = min ρ [ˆ v ] → N n Z ρ ( ~r ) v ext ( ~r ) d~r + F [ ρ, ρ ] o = Z ρ k ( ~r ) v ext ( ~r ) d~r + F [ ρ k , ρ ] , (6)where ρ is the ground state density of the system un-der consideration. The LN energy density functional dif-fers from the HKS ground-state and the stationary state e DFT functional due to the bifunctional F [ ρ, ρ ], whichis defined by F k [ ρ, ρ ] = min Ψ[ N ] → ρ, h Ψ[ N ] | Ψ j [ˆ v ; N ] i =0 ,j The CS formulation described in the previous sectionimplies that the content of the excited state function-als Q S [ ρ e ] and F [ ρ e , ρ ] differs from the HK universalfunctional F [ ρ ] except their stationarity with respect tovariation in the external potential. Actually, only in thecase of ground-state, all the three functionals are identi-cal to one another and in general there exists a close linkbetween G¨orling Q S [ ρ e ] and Levy-Nagy F [ ρ e , ρ ] [60]. Soin the unified e DFT formalism, for a given excited-stateeigendensity ρ e ( ~r ), both Q S [ ρ e ] and F [ ρ, ˆ v ext ] are station-ary about the corresponding ˆ v ext which also holds for thedesired excited-state Ψ Sk ≡ Ψ LNk [41, 58, 60]. Now due tothe presence of orthogonality constraint in F [ ρ, ˆ v ext ], sev-eral choices for the set of low lying states can be madeto which Ψ LNk will be orthogonal and for each choice,there may exists a generalized potential function ˆ w ext .So some extra deciding factors are required for settingup the ρ ⇐⇒ ˆ v mapping which is the intent of the cur-rent section.Now resorting back to the work of Samal-Harbola [58],we would also like to re-emphasis that the direct or in-direct comparison of ground states are not sufficient toestablish the ρ ( ~r ) ⇐⇒ ˆ v ext ( ~r ) mapping or to constructthe KS system for excited-states [56]. Given the discus-sions on unified CS e DFT in the previous section, we nowpresent a consistent approach to address the density-to-potential mapping issues. Fundamentally rigorous andcrucial tenets of the proposed eDFT are: ( i ) There ex-ist ways for mapping an excited-state density ρ e ( ~r ) to thecorresponding many-electron wavefunction Ψ( ~r ) which inturn maps to the external potential ˆ v ext ( ~r ) through the ρ -stationary wavefunctions [25, 58, 60]. In this, the wave-function depends upon the ground-state density ρ im-plicitly. ( ii ) The KS system is to be defined througha comparison of the kinetic energy, ground-state den-sity and variation of the energy w.r.t. symmetry of theexcited-states.The claim is, unified CS approach can provide the map-ping from an excited-state density ρ e ( ~r ) to many-bodywavefunction. Stationary state formalism [25, 58] pro-vides a straightforward method of mapping ρ e ( ~r ) ⇐⇒ ˆ v ext ( ~r ), just by making sure whether h Ψ k | ˆ T + ˆ V ee | Ψ k i isstationary or not, subject to the condition that Ψ k gives ρ e . But [25, 41, 58, 60] shows that different Ψ k ( ~r )s cor-respond to potentials ˆ v kext ( ~r ). The same problem alsopervades through the variational e DFT approach as pro-posed by LN [27, 28, 58]. Thus unified CS gives, manydifferent wavefunctions Ψ k ( ~r ) and the corresponding ex-ternal potential ˆ v kext ( ~r ) can be associated with a givendensity. Now if in addition to the excited-state density wealso have the ground-state information ρ , then ˆ v ext ( ~r )can be uniquely determined out of all possible multiplepotentials ˆ v kext ( ~r ). Hence with the knowledge of ρ , itis quite trivial to select a particular Ψ that belongs toa given [ ρ e , ρ ] combination by comparing ˆ v kext ( ~r ) withthe actual ˆ v ext ( ~r ). Alternatively, one can think of it asfinding Ψ variationally for a [ ρ e , ˆ v ext ] combination. Itsbecause the knowledge of ρ and ˆ v ext is equivalent. Nowwith the above information, the bifunctional F [ ρ e , ρ ] canbe redefined as F [ ρ e , ρ ] = h Ψ[ ρ e , ρ ] | ˆ T + ˆ V ee | Ψ[ ρ e , ρ ] i . (14)The above theoretical formulation is similar to that ofLN [27] but avoids the orthogonality constraint imposedby LN formalism. This is because, the densities for dif-ferent excited states for a given ground-state density ρ (that corresponds to a unique external potential ˆ v ext ) canbe found in following manner: take a density and searchfor Ψ that makes h Ψ | ˆ T + ˆ V ee | Ψ i stationary and simulta-neously make sure whether the corresponding potentialˆ w ext (cid:16) i.e. w ext = − δF [ ρ,ρ ] δρ (cid:12)(cid:12)(cid:12) ρ = ρ e (cid:17) resembles the given ρ ( or ˆ v ext ); if not, search for another density and re-peat the procedure until the correct ρ is found. Thus itis clear that excited state orbitals Ψ are now functionalof [ ρ e , ρ ]. So the correct density ρ is excited state den-sity of the potential and the Ψ obtained in this method isalso excited state wavefunction corresponding to that po-tential and density. After finding the correct density ρ e ,make a variation over it so that ( ρ e → ρ e + δρ ) and again perform the CS to find Ψ[ ρ e + δρ ; ρ ]. In this case, choosethat ( ˆ w ext + δ ˆ w ext ) which converges to ˆ v ext as δρ → T s [ ρ e , ρ ] and using it to further define theexchange-correlation functional as E xc [ ρ e , ρ ] = F [ ρ e , ρ ] − E H [ ρ e ] − T s [ ρ e , ρ ] , (15)solves the purpose. So the Euler equation for the excited-state densities becomes v ext = µ − n δT s [ ρ e , ρ ] δρ ( ~r ) + V H [ ρ e ] + δE xc [ ρ e , ρ ] δρ ( ~r ) o , (16)which is equivalent to solving the KS equation (cid:26) − ∇ + ˆ v s ( ~r ) (cid:27) Ψ i ( ~r ) = ε i Ψ i ( ~r ) , (17)where v s ( ~r ) = v ext ( ~r )+ δ n F [ ρ, ρ ] − T s [ ρ, ρ ] o δρ ( ~r ) (cid:12)(cid:12)(cid:12) ρ ( ~r )= ρ e [ v ext ( ~r )] . (18)In ground state DFT, one can easily find the T s [ ρ ] byminimizing the kinetic energy for a given density; here T s [ ρ ] for a given density is obtained by occupying thelowest energy orbitals for a non-interacting system. Butin e DFT, to define T s [ ρ e , ρ ] is not easy, as for theexcited-states it is not clear which orbitals to occupy fora given density. Particularly because a density can begenerated by many different configurations of the non-interacting systems. Levy-Nagy select one of these sys-tems by comparing the ground-state density correspond-ing to the excited-state non-interacting system with thetrue ground-state density. However, LN criterion is notsatisfactory as pointed out by Samal and Harbola [56].The reason of this discrepancy is due to the inconsis-tency of the ground-state density of an excited state KSsystem with the true ground-state density. The ground-state density corresponding to the excited-state KS sys-tem is not same as the ground-state density of the truesystem. This means the desired state is not associatedwith ˆ v ext ( ~r ), rather it comes from a different local poten-tial ˆ v ′ ext ( ~r ). To settle this inconsistency, KS system mustbe so chosen that it is energetically very close to the orig-inal system and it can be ensured through the followingcriterion. Criterion I: the non-interacting kinetic energy T s [ ρ e , ρ ] obtained through the CS need to be very closeto the actual T [ ρ e , ρ ], where T s [ ρ e , ρ ] and T [ ρ e , ρ ] aredefined as T s [ ρ e , ρ ] = min Φ → ρ e h Φ | ˆ T + ˆ V ee = 0 | {z } | Φ i T [ ρ e , ρ ] = min Ψ → ρ e h Ψ | ˆ T + ˆ V ee | Ψ i . (19)So defining ∆ T = T − T s smallest not only ensures thatDFT exchange-correlation energy remains closer to theconventional quantum mechanical exchange-correlationenergy but also keeps the structure of the KS potentialappropriate for the desired excited-state which is shownbelow. Based on the DVT [64], it can be argued howfor a given density ρ e one can have different exchange -correlation ˆ v xc and external ˆ v ext potentials. According toDVT, the exact expression for the gradient of the exter-nal potential (for interacting system) for a given excited-state density ρ e is − ∇ ˆ v ext = − ρ e ( ~r ) ∇∇ ρ e ( ~r ) + 1 ρ e ( ~r ) ~Z ( ~r ; Γ ( ~r ; ~r ′ ))+ 2 ρ e ( ~r ) Z [ ∇ ˆ u ( ~r, ~r ′ )] Γ ( ~r, ~r ′ ) d~r ′ , (20)where ˆ u = | ~r − ~r ′ | . This equation represents an exact rela-tion between the gradient of the external potential ˆ v ext ,the e − e interaction potential ˆ u ( ~r, ~r ′ ) and the densitymatrices ρ ( ~r ), Γ ( ~r ; ~r ′ ) and Γ ( ~r, ~r ′ ). The vector field ~Z in Eq.(20)is related to the kinetic-energy density tensorvia Z α [ ~r ; Γ ( ~r ; ~r ′ )] = h (cid:16) ∂ ∂r ′ α ∂r ′′ β + ∂ ∂r ′ β ∂r ′′ α (cid:17) Γ ( ~r ′ ; ~r ′′ ) i ~r ′ = ~r ′′ = ~r (21)So, ~Z can be called a ”local” functional of Γ . Similarly,for KS potential Eq.(20) reduces to ∇ ˆ v KS = − ρ e ( ~r ) ∇∇ ρ e ( ~r ) + 1 ρ e ( ~r ) ~Z KS ( ~r ; Γ ( ~r ; ~r ′ )) . (22) As a given ground-state density ρ fixes the external po-tential uniquely via HK theorem, which implies that ρ , Γ and Γ are also fixed from Eq.(20). Since the den-sity matrices generated by some eigenfunction Ψ of theHamiltonian ˆ H . So the fixed pair of excited-state andground-state density i.e. [ ρ e , ρ ] may be arising from dif-ferent configurations − different configurations can bethought of as arising from different external potential ordifferent exchange-correlation potential and this is due tothe different Γ and Γ for a fixed ρ e . Suppose a givendensity ρ e is generated through an i th KS system, then ∇ ˆ v i KS = − ρ e ( ~r ) ∇∇ ρ e ( ~r ) + 1 ρ e ( ~r ) ~Z i KS ( ~r ; Γ i ( ~r ; ~r ′ )) . (23)If the density is generated through a j th external poten-tial then − ∇ ˆ v j ext = − ρ e ( ~r ) ∇∇ ρ ( ~r ) + 1 ρ ( ~r ) ~Z j ( ~r ; Γ j ( ~r ; ~r ′ ))+ 2 ρ e ( ~r ) Z [ ∇ u ( ~r, ~r ′ )] Γ j ( ~r, ~r ′ ) d~r ′ . (24)As a matter of which − ∇ ˆ v xc = ~Z KS ( ~r ; Γ ( ~r ; ~r ′ )) − ~Z ( ~r ; Γ ( ~r ; ~r ′ )) ρ e ( ~r ) + R [ ∇ ˆ u ( ~r, ~r ′ )][ ρ e ( ~r ) ρ e ( ~r ′ ) − Γ ( ~r, ~r ′ )] d~r ′ ρ e ( ~r ) (25)becomes − ∇ ˆ v ijxc = ~Z i KS − ~Z j ρ ( ~r ) + ~ε j xc , (26)where ~ε j xc is the field due to the Fermi-Coulomb hole ofthe j th system [ Γ j ] . So the kinetic energy differencebetween the true system and KS system is given by∆ T = 12 Z ~r. n ~Z KS (cid:16) ~r ; [ Γ ] (cid:17) − ~Z (cid:16) ~r ; [ Γ ] (cid:17)o d~r. (27)This difference should be kept the smallest for the trueKS system so that it gives the KS system consistent withthe original system. As a matter of which, we concludethat one way to establish the ρ e ⇐⇒ ˆ v ext mapping viathe LN formalism [27–29] is: if among the several poten-tials − which have the same excited -state density, onecan choose the correct KS potential by comparing theground-state density i.e. keep that KS-potential whoseground-state density resembles with the true ground-state density. Keeping the ground-state density close weactually keep the external potential fixed via HK theo-rem. Thus LN criterion is exact for non-interacting sys-tem as there is no interaction, so the ground-state densitymatch perfectly.This proposal of LN for ρ e ⇐⇒ ˆ v ext mapping was car-ried by Samal and Harbola [58] but they argued in aslightly different way. They proposed that both for in-teracting and non-interacting case among all the mul-tiple potentials, choose the correct KS potential whoseground-state density differ from the exact ground-state“most closely by least-square sense” which is done in thefollowing manner. If ρ ( ~r ) is the exact ground state den-sity and ˜ ρ ( ~r ) is that of the KS system (OR the alternatepotentials ˆ w ext ) then SH proposition can be further im-proved intuitively. Criterion II: the mean square distancebetween ρ ( ~r ) and ˜ ρ ( ~r ) should remain very close to zero .Thus∆[ ρ ( ~r ) , ˜ ρ ( ~r )] = min v [ ˜ ρ ,ρ e ] (cid:26)Z ∞ | [ ρ ( ~r ) − ˜ ρ ( ~r )] | d~r (cid:27) ≥ , (28)where the integration is carried out in the Sobolev space.This criterion is more appropriate in the context of ρ e ⇐⇒ ˆ v ext than the one proposed by [58, 61]. Thecriterion as given in Eq.(28) will be fully satisfied if onemakes use of the excited state functionals [41, 57, 66–68].Otherwise it may fail in certain situations as pointed outby SH [58].Instead of sticking to the Criterion I & II , one can evengo beyond the same through Criterion III: compare theground states of the true and alternate systems energet-ically . It can be done in the following manner in orderto select the KS system for a given density. The alterna-tive approach is to compare the ground-state expectationvalue of the KS system and the true system, instead ofcomparing their ground-state densities and kinetic ener-gies. The procedure for comparing ground-state energylevel is the following. First solve the exact DFT equa-tion (say Harbola-Sahni [63] etc) for ground-state of thetrue system and obtain the ground-state of KS Hamil-tonian H . If the expectation value of the ground stateHamiltonian of the true system is h H i true and that of theKS system is h H i KS , then one need to choose that KSsystem whose h H i KS ≃ h H i true . These criteria are wellconnected to the GAC-KS [11, 24–26, 69–77] as discussedbelow.Since GAC-KS in principle helps for the self-consistenttreatment of excited states and could be considered as aplausible extension of HK theorem to the same. So nowthe furtherance of the propositions made by SH [58] asdiscussed previously will be justified within the GAC-KS.Indeed, relying on the principles of GAC-KS, unified CSformalism along with the SH criteria can also establish the density-to-potential mapping at the strictly corre-lated regime which will be shown below. In GAC, the λ dependent Hamiltonian which is also used in the PLextremum principle is given byˆ H λ [ˆ v, N ] = ˆ T + λ ˆ V ee + N X i =1 ˆ v ( ~r i ) , (29)with the corresponding equation of stateˆ H λ [ˆ v, N ]Ψ λ [ˆ v, N ] = E λ [ˆ v, N ]Ψ λ [ˆ v, N ] , (30)where λ is the coupling constant with 0 ≤ λ ≤ v ( ~r ), is independent of λ . Analogous to the Levy-Lieb CS functionals, the GAC for the conjugate densityfunctionals F λ [ ρ ] (density fixed AC) and E λ [ˆ v ] (potentialfixed AC) are given by F λ =1 [ ρ ] = F λ =0 [ ρ ] + Z dF λ [ ρ ] dλ dλ , (31) E λ =1 [ˆ v ] = E λ =0 [ˆ v ] + Z dE λ [ˆ v ] dλ dλ . (32)Similar to Eq.(31) and (32), the excited-state function-als T λ [ ρ, ρ ], Q Sλ [ ρ ], F λ [ ρ, ρ ] and E λ [ ρ, ρ ] can be de-fined. Upon finding these e DFT functionals, one candefine the GAC by starting at a ρ stationary wavefunc-tion for λ = 1 and then by gradually turning off ( λ = 0)the electron-electron interaction. Thus the ρ -stationarywavefunctions for 0 ≤ λ ≤ e DFT.Since the ρ -stationary wave functions for a given ρ arenumerable and the adiabatic connections do not overlapwith each other, states Φ i of non-interacting model sys-tems equals to the ρ -stationary wave functions at λ = 0(i.e.Φ i = Ψ Sλ =0 [ ρ ]) and can be assigned to real electronicstates Ψ j = Ψ[ ρ, ν, α = 1] [25]. These assigned modelstates are the eigenstates of the GAC-KS formalism. Asdiscussed above, they are eigenstates of a Hamiltonianoperator with local multiplicative potential. In this way,the GAC will define the path of going from a non- inter-acting system to an interacting system via a ρ − stationarypath. Although for each of the interacting system, onecan still end up with multiple non-interacting KS system.But with the criteria discussed previously it’s possible toselect the appropriate ones. So once the ρ ⇐⇒ ˆ v ext forthe interacting system is fixed, it do carries over to theKS system via GAC and vice versa. This shows howthe proposed unified CS formalism not only establishesthe density-to-potential mapping concretely but also con-structs the KS system successfully. In the following sec-tions we will exemplify what we have proposed so farthrough two model systems. This will be done in orderaddress the critiques about density-to-potential mappingin e DFT. IV. eDFT BEYOND THE HK AND GLTHEOREM The issue of non-uniqueness in the density-to-potentialmapping is also persuaded [50] in the context of GL theo-rem [11, 12]. In [50], it has been demonstrated for higherexcited states of the considered 1 D model system thereis no equivalence of the GL/HK theorem. But the crit-ical analysis of [50] presented in this section will out-line how the multiplicity of potentials still can’t be ruledout even in the case of ground as well as lowest excitedstates. So one need to go beyond [50] approach in orderto address the validity of HK & GL theorem for suchstate. In fact, relying on the principles of unified e DFTapproach [25, 41, 58, 60] as discussed in Sec. II alongwith the proposed criteria of Sec. III, it will be shownhere why the claim made in [50] lacks merit to addressthe excited-state density-to-potential mapping. To vali-date the density-to-potential mapping (i.e. the analogueof HK/GL theorem) in [50] proposed approach, we willconsider as test cases: the examples of the 1 D QHO withfinite boundary and then the infinite potential well.For clarity in understanding let’s first briefly discussthe theoretical formulation of [50]. The Schr¨odingerequation of two non-interacting fermions subjected to lo-cal one dimensional potentials v ( x ) and w ( x ) s.t. v ( x ) = w ( x ) + C , where C is a constant are given by h − d dx + v ( x ) i Φ i ( x ) = ε i Φ i ( x ) , (33) h − d dx + w ( x ) i Ψ i ( x ) = λ i Ψ i ( x ) . (34)Suppose that the eigenfunctions of the local potential w ( x ) generates the ground/excited-state eigendensity of v ( x ) as one of it’s eigendensity but with some arbitraryconfiguration which is either same or different from theoriginal one. Then one possible way of achieving thisis: the wavefunctions Ψ( x ) of the potential w ( x ) can beassociated to the wavefunctions Φ( x ) of the potential v ( x ) via the following unitary transformation i.e. (cid:18) Ψ k ( x )Ψ l ( x ) (cid:19) = (cid:18) cos θ ( x ) sin θ ( x ) − sin θ ( x ) cos θ ( x ) (cid:19) (cid:18) Φ i ( x )Φ j ( x ) (cid:19) = (cid:18) Φ i ( x ) cos θ ( x ) + Φ j ( x ) sin θ ( x ) − Φ i ( x ) sin θ ( x ) + Φ j ( x ) cos θ ( x ) (cid:19) , (35)As a matter of which the density preserving constraintwill be satisfied and the ground/excited state density oftwo potentials remain invariant i.e. ρ ( x ) = | Φ i ( x ) | + | Φ j ( x ) | = | Ψ k ( x ) | + | Ψ l ( x ) | . (36)Now the potentials can be obtained from the Eqs.(33)and (34) by inverting the same v ( x ) = ε i + ¨Φ i ( x )2Φ i ( x ) = ε j + ¨Φ j ( x )2Φ j ( x ) (37) w ( x ) = λ k + ¨Ψ k ( x )2Ψ k ( x ) = λ l + ¨Ψ l ( x )2Ψ l ( x ) . (38)Also from Eqs.(33) and (34), the difference between anytwo eigenvalues ∆ and ∆ ′ corresponding to the potentials v ( x ) and w ( x ) are given by∆ = ε j − ε i = 12Φ i ( x )Φ j ( x ) ddx [Φ j ( x ) ˙Φ i ( x ) − Φ i ( x ) ˙Φ j ( x )] , (39)∆ ′ = λ k − λ l = 12Ψ k ( x )Ψ l ( x ) ddx [Ψ l ( x ) ˙Ψ k ( x ) − Ψ k ( x ) ˙Ψ l ( x )] . (40)Now by plugging the values Ψ k ( x ) and Ψ l ( x ) fromEq.(35) back in Eq.(40), the rotation θ ( x ) can be ob-tained from the following ddx [ ˙ θ ( x ) { Φ i ( x ) + Φ j ( x ) } + { Φ j ( x ) ˙Φ i ( x ) − Φ i ( x ) ˙Φ j ( x ) } ]= ∆ ′ [2Φ i ( x )Φ j ( x cos 2 θ ( x ) + { Φ j ( x ) − Φ i ( x ) } sin 2 θ ( x )](41)or ρ ( x )¨ θ ( x )+ ˙ ρ ( x ) ˙ θ ( x )+ f (Φ i ( x ) , Φ j ( x ) , ∆ , ∆ ′ , θ ) = 0 , (42)where f =2∆Φ i ( x )Φ j ( x ) − ∆ ′ [Φ i ( x )Φ j ( x ) cos 2 θ ( x )+ { Φ j ( x ) − Φ i ( x ) } sin 2 θ ( x )] . (43)The Eq.(42) is the central equation of [50] theoreticalframework which need to be solved numerically withproper initial conditions in order to obtain the alternatepotential w ( x ) for any given density and eigenvalue dif-ferences. In this work, the adopted numerical procedureto solve the above mentioned differential equation is verymuch accurate even at the boundary where obtaining ap-propriate structure and behavior of the multiple poten-tials and the corresponding wavefunctions are importantand crucial. A. Results: D Quantum Harmonic Oscillator The first model system for demonstrating the density-to-potential mapping is the 1 D QHO defined by v ( x ) = 12 ω x , where − l ≤ x ≤ l . (44)So the wavefunctions and energy eigenvalues of the n th eigenstate are given byΦ n ( x ) = (cid:16) ωπ (cid:17) √ n n ! H n ( √ ωx ) exp( − ωx , (45) ε n = ( n + 12 ) ω , (46)where n = 0 , , ..... (atomic units are adopted i.e. ~ = 1 and m e = 1) -1-0.500.51 ρ [ V ] , Ψ [ V ] V [ ρ ] Δ '=10.0 x -20020406080100 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 FIG. 1. Upper panel: Shows (red color) the ground statedensity of the 1D QHO and the corresponding transformedwavefunctions Ψ k (blue) and Ψ l (green) for ∆ ′ = 10 . 0. Lowerpanel: Shows the alternate potential associated with abovewavefunctions and density. B. Fermions in The Ground State Now consider two non-interacting fermions occupyingthe ground-state of the QHO i.e. n = 0 = m . So theeigenvalue difference for this state ∆ = ε − ε = 0 andthe density is given by ρ ( x ) = 2 (cid:16) ωπ (cid:17) exp( − ωx ) . (47) -1-0.500.51 -20020406080100 -2 -1 0 1 2 ρ [ V ] , Ψ [ V ] V [ ρ ] Δ '=46.0 x FIG. 2. The figure caption is same as Fig.1 but with ∆ ′ =46 . Thus the corresponding equation for rotation θ ( x ) can beobtained from the Eq.(42) and is given by ρ ( x )¨ θ ( x )+ ˙ ρ ( x ) ˙ θ ( x ) − ∆ ′ [2 (cid:16) ωπ (cid:17) exp( − ωx ) cos 2 θ ( x )] = 0 . (48)Now Eq.(48) has to be solved with proper initial condi-tions. The initial conditions can be fixed by taking intoconsideration the symmetry of the differential Eq.(48)and the normalization condition of the wavefunction.From Eq.(48) it is clear that dθdx | ( x =0) = 0 as both Φ( x )and ρ ( x ) are symmetric about x = 0. Now another con-dition is that Ψ k ( x ) and Ψ l ( x ) must also be normalized.So if we plot the renormalization R Z l − l | Ψ k,l ( x ) | dx − R = 0 (49)as a function of θ ( x = 0), then the points where R = 0corresponds to the normalization of Ψ k ( x ) and Ψ l ( x ) [50]and it will provide the initial condition on θ ( x = 0). Af-ter finding θ ( x ), the transformed set of normalized wave-functions Ψ k ( x ) and Ψ l ( x ) is being obtained. Again us-ing these wavefunctions the potential w ( x ) can be deter-mined from the Eq.(38). In Fig.1 and Fig.2, we haveshown two different potentials which are obtained forthe eigenvalue differences ∆ ′ = 10 . ′ = 46 . ρ QHO = ρ [ v ( x ) = v QHO ( x ) , N =2] now corresponds to some arbitrary excite-state hav-ing density ρ e [ w ( x ) = v QHO ( x ) , N = 2]. As a mat-ter of which, for the fixed ρ QHO and ∆ ′ , the systemgets transformed to some other system w ( x ) for which Q S [ ρ e [ w ( x )] = ρ QHO ] and/or F [ ρ e [ w ( x )] = ρ QHO , ˜ ρ ]will be stationary. The corresponding stationary statesare basically the transformed wavefunctions which aregiven by Eq.(34) Ψ Sl [ w ( x )] = Ψ l [ ρ e , ˜ ρ ]. In this ˜ ρ isthe ground state density of the newly generated poten-tial w ( x ) and ˜ ρ = ρ QHO . Now from the proposed cri-teria it follows that ∆ T = 0 , ∆[ ρ ( x ) , ˜ ρ ( x )] > v ( x ) althoughthere exist several multiple potentials w ( x ). This resultis consistent with the generalized/unified CS formalism[25, 41, 58, 60]. -0.4-0.200.20.40.6 -50510152025303540-3 -2 -1 0 1 2 3 x ρ [ V ] , Ψ [ V ] V [ ρ ] Δ ’(cid:0)(cid:1)(cid:2)(cid:3)(cid:4) FIG. 3. The figure caption is same as Fig.1 but for the lowestexcited state density being produced with ∆ ′ = 15 . C. Fermions in The Lowest Excited State As the second example, we consider the lowest excited-state of the QHO. So the two non-interacting fermions arenow occupying the n = 0 and m = 1 state. For this case, ε = ω , ε = ω and ∆ = ε − ε = ω with the density ρ ( x ) = (cid:16) ωπ (cid:17) exp( − ωx )(1 + 2 ωx ) , (50) -0.6-0.4-0.200.20.40.6 -50510152025303540-3 -2 -1 0 1 2 3 Δ'=35.0 ρ [ V ] , Ψ [ V ] V [ ρ ] x FIG. 4. The figure caption is same as Fig.3 but with ∆ ′ =35 . and the corresponding equation for rotation θ ( x ) is givenby ρ ( x )¨ θ ( x ) + ˙ ρ ( x ) ˙ θ ( x ) + 2 ωx (cid:18) ω π (cid:19) exp( − ωx ) − ∆ ′ [2 x (cid:18) ω π (cid:19) exp( − ωx ) cos 2 θ ( x ) + ωπ exp( − ωx ) { ωx − } sin 2 θ ( x )] = 0 . (51)Since in this case Φ ( x ) is symmetric, Φ ( x ) is antisym-metric, so ρ ( x ) symmetric around x = 0. Thus Eq.(51)implies that θ ( x ) should be symmetric at x = 0. Theinitial conditions on dθdx | ( x =0) is obtained from the behav-ior of the renormalization R as a function of dθdx | ( x =0) .Following the same procedure as before, in this case alsowe have obtained different potentials for the fixed lowestexcited state density which are shown in the Fig.3 andFig.4. These two alternative potentials and the trans-formed wavefunctions correspond to two different eigen-value differences ∆ ′ = 16 . ′ = 35 . 0. As describedin the ground state case, in this case also the structureof the potential is different from the original 1D QHOas the potential should follow the structure of the wave-functions. However, according to the unified CS e DFT,the results are never due to the violation of GL theorem.This is because the ground and lowest excited states ofthe newly found potential are quite different from that ofthe QHO. So following similar argument as in the previ-ous case, now the lowest excited-state density of the QHO0corresponds to some different eigendensity of the multi-ple potentials. Thus the multitude of potentials poses noissues for the validity of the GL theorem. -0.8-0.6-0.4-0.200.20.40.6 -5051015 -3 -2 -1 0 1 2 3 Δ '=8.0 ρ [ V ] , Ψ [ V ] V [ ρ ] x FIG. 5. The figure caption is same as Fig.1 but for one of thehigher excited state density being produced with ∆ ′ = 8 . -0.6-0.4-0.200.20.40.60.8 -50510152025303540 -3 -2 -1 0 1 2 3 Δ '=30.0 ρ [ V ] , Ψ [ V ] V [ ρ ] x FIG. 6. The figure caption is same as Fig.5 but with ∆ ′ =30 . -2-1.5-1-0.500.511.52 -300-200-1000100200300400500600 0 0.2 0.4 0.6 0.8 1 Δ '=200.0 (cid:5) [ V ] V [ (cid:6) ] x (cid:7) ( x ) x FIG. 7. Upper panel: Shows the alternate wavefunctions Ψ k and Ψ l (green & red) resulting the ground state density of1D potential well for ∆ ′ = 200 . 0. Lower panel: Shows thealternate potentials (green & red) and the density (magenta)associated with above wavefunctions. D. Fermions in Higher Excited States Here we consider one of the higher excited-state of 1DQHO (i.e. two non-interacting fermions are in the n = 0and m = 2 states). For this case, the eigenvalue differenceis ∆ = ε − ε = 2 ω and the density corresponding to itis given by ρ ( x ) = (cid:16) ωπ (cid:17) exp( − ωx ) { − ωx ) } . (52)Similarly, the corresponding equation for rotation θ ( x ) isgiven by ρ ( x )¨ θ ( x ) + ˙ θ ( x ) ˙ θ ( x ) + 4 ω (cid:16) ω π (cid:17) (2 ωx − 1) exp( − ωx ) − ∆ ′ [ (cid:16) ω π (cid:17) (2 ωx − 1) exp( − ωx ) cos 2 θ ( x )+ (cid:16) ωπ (cid:17) exp( − ωx ) { 12 (2 ωx − − } sin 2 θ ( x )] = 0 . (53)Now by solving Eq.(53) for rotation θ ( x ) in analogouswith the ground-state of the QHO and after taking careof the normalization of the transformed wavefunctions,the potential w ( x ) is obtained for ∆ ′ = 8 . , . 0. The1 -2-1.5-1-0.500.511.52 -200-1000100200300400500 0 0.2 0.4 0.6 0.8 1 ρ ( x ) x x ψ [ V ] V [ ρ ] Δ '=600.0 FIG. 8. The figure caption is same as Fig.7 but with ∆ ′ =600 . potentials along with the wavefunctions are shown in Fig.5 & Fig.6. Similar to ground and lowest excited-state,here too the given density is a different eigendensity ofthe new potentials. If it would have the same eigenden-sity of w ( x ) then w ( x ) should have been identical to the v QHO ( x ). But it is not the case. That’s why the gener-ated potentials are completely different from the QHO. E. Results: D Infinite Potential Well As our second case study, we consider the model sys-tem same as that reported in [50] (i.e. particles aretrapped inside an 1 D infinite potential well). For aninfinite potential well with length varying from 0 to 1,the n th eigenfunction Φ n ( x ) and the energy eigenvalue ε n are given byΦ n ( x ) = √ nπx ) ; ε n = n π , (54)where n = 1 , , .... . The density ρ ( x ) corresponding tothe two potentials v ( x ) and w ( x ) is given by Eq.(36). -2-1.5-1-0.500.511.52 -2000200400600 0 0.2 0.4 0.6 0.8 1 Δ '=1000.0 ψ [ V ] (cid:8) ( x ) x x V [ (cid:9) ] FIG. 9. The figure caption is same as Fig.7 but with ∆ ′ =1000 . F. Fermions in The Ground State For two spinless non-interacting particles in n = 1 = m states, the energies of two states and the difference are ε = π ε ; ∆ = ε − ε = 0 . (55)The density corresponding to these states is ρ ( x ) = 4[sin ( πx )] , (56)and the equation corresponding to Eq.(42) for the rota-tion θ ( x ) is ρ ( x )¨ θ ( x ) + ˙ ρ ( x ) ˙ θ ( x ) − ∆ ′ [4 sin πx cos 2 θ ( x )] = 0 . (57)Since Φ ( x ) is symmetric and ρ ( x ) is symmetric about x = . Thus Eq.(57) indicates that θ ( x ) should be sym-metric such that ˙ θ ( ) = 0. With this initial conditionand choosing any value of ∆ ′ one can solve for θ ( x ) andsubsequently obtain the Ψ k s. Now using these Ψ k s,the alternate potentials w ( x ) will be obtained by usingEq. (38). Since the transformed wavefunction Ψ k ( x )must also be normalized. This condition will be fulfilledby choosing the appropriate value of θ ( ) at which theΨ k ( x ) should be normalized. Once Ψ k ( x ) is normalizedthen Ψ l ( x ) will also be normalized. Again by adopting2 -2-1.5-1-0.500.511.52 -50050100150200250300350400 0 0.2 0.4 0.6 0.8 1 Δ '=200.0 ρ ( x ) xx V [ ρ ] ψ [ V ] FIG. 10. The figure caption is same as Fig.7 but for the lowestexcited state density being produced with ∆ ′ = 200 . the same procedure as that described in the case of 1 D QHO, the alternative multiple potentials are obtained bymaking use of the following renormalization R condition Z | Ψ k ( x ) | dx − R = 0 . (58)All the wavefunctions, densities and multiple potentialsare shown in the Figs.(7 to ′ = 200 . , . . D infinitewell. Although the density remains to be the same in allthe cases. But its not the ground state eigendensity ofthe multiple potentials. So this poses no issue for the HKtheorem. G. Fermions in The Lowest Excited State Now consider two fermions occupying the n = 1 , m =2 (i.e. the lowest excited-state) eigenstates of the infinitepotential well. Here too we have obtained several mul-tiple potentials unlike [50]. For this excited-state, theenergy eigenvalues are ε = π , ε = 2 π with ∆ = π .Hence the density arising from these two states is given -1.5-1-0.500.511.52 -50050100150200250300 0 0.2 0.4 0.6 0.8 1 Δ '=600.0 ρ ( x ) x V [ ρ ] Ψ [ V ] x FIG. 11. The figure caption is same as Fig.10 but with ∆ ′ =600 . by ρ ( x ) = 2[sin ( πx ) + sin (2 πx )] . (59)Similar to the previous examples, the equation for therotation θ ( x ) is the following ρ ( x )¨ θ ( x ) + ˙ ρ ( x ) ˙ θ ( x ) + 6 π sin( πx ) sin(2 πx ) − ∆ ′ [4 sin( πx ) sin(2 πx ) cos 2 θ ( x )+2 { sin (2 πx ) − sin ( πx ) } sin 2 θ ( x )] = 0 . (60)Here Φ ( x ) is symmetric, Φ ( x ) is antisymmetric and ρ ( x ) symmetric about x = . Thus Eq.(60) predicts that θ ( x ) is antisymmetric such that θ ( ) = 0. In this casealso normalization of both Ψ k ( x ) and Ψ l ( x ) are takencare and the proper R (renormalization) values are ob-tained w.r.t. dθdx ( ). Quite interestingly, in this casealso we have successfully generated multiple potentialsfor ∆ ′ = 200 . , . . 0. This is where [50] failedto explain the validity of GL theorem. As expected, thepotential follows the wavefunctions pattern. This is ob-vious at the boundary where the wavefunctions are per-fectly vanishing, the potential shoots up to a very largepositive value. The potentials along with wavefunctionsare shown in the Figs.(10 to ρ ( x ) xx V [ ρ ] ψ [ V ] -50050100150200250300 0 0.2 0.4 0.6 0.8 1 -1-0.500.511.52 Δ '=1000.0 FIG. 12. The figure caption is same as Fig.10 but with ∆ ′ =1000 . H. Fermions in Higher Excited States Now to complete our exploration on 1 D well, we haveconsidered here the second excited-state of it. This isthe only excited-state for which [50] reported multipleexternal potentials for various eigenvalue differences. Wetoo generated multiple potentials and the correspondingwavefunctions for ∆ ′ = 200 . , . . to -2-1.5-1-0.500.511.52 -1000100200300400500 0 0.2 0.4 0.6 0.8 1 Ψ [ V ] Δ '=200.0 ρ ( x ) x x V [ ρ ] FIG. 13. The figure caption is same as Fig.7 but for one of thehigher excited state density being produced with ∆ ′ = 200 . tion. But in getting the potential structure whether byinverting the Schr¨odinger equation or CS method solelydepends on the wavefunction behavior in a given domain.So better access of the wavefunction’s behavior will bydefault lead to reliable potential structure. V. RESULTS WITHIN THE CS FORMALISM In this section, we will discuss the results in connec-tion with the density-to-potential mapping based on theCS-formalism discussed earlier. According to it, thereexist multiple potentials for a given ground or excitedstate (eigen)density. But for the case of excited statedensity, when it is produced as some different excited-state of these multiple potentials (except the actual one)the corresponding ground-states are completely differentfrom that of the original system. Similarly, one can pro-duce potentials whose ground-state density may be sameas the excited-state density of the original system. Theresults we have obtained for the systems of our study arefully consistent with the unified CS e DFT. The Zhao-Parr [78] CS method is being used to show the multiplic-ity of potentials for a given density.To begin the CS exemplification (shown in Fig.16), letsconsider f our non-interacting particles in an 1 D poten-4 -2-1.5-1-0.500.511.52 -1000100200300400500600 0 0.2 0.4 0.6 0.8 1 Δ '=600.0 Ψ [ V ] ρ ( x ) xx V [ ρ ] FIG. 14. The figure caption is same as Fig.13 but with ∆ ′ =600 . tial well, where two fermions are in n = 1 state and onefermion each in n = 2 and n = 4 state. As a result, thisgives some excited state density ρ e ( x ) associated with theabove configuration which is shown in the Fig.16(a) andis given by ρ e ( x ) = ρ V e ( x ) = 2 | Ψ ( x ) | + | Ψ ( x ) | + | Ψ ( x ) | , (61)where Ψ i ( x ) s are the wavefunctions of the 1 D poten-tial well. In all our results shown in the figures (16)to (22), we have adopted notation ρ ( n i ( f j )), where n i denotes the quantum number of the eigenfunctions ofthe potential V or V i ( i = 1 , , , 4) and f j , the occu-pation. Using CS [78] the excited state density ρ e ( x )given by Eq.(61) is produced through another alterna-tive potential V (say) whose n = 1 state is occupiedwith 2 fermions (i.e. f = 2) and n = 2 , n = 3 with onefermion each (i.e. f = 1 = f ). Now the ground statedensity of the potential V is different from that of the V (i.e. particle in an infinite potential well) which is givenby ˜ ρ (1)0 (Fig.16a). As per our formalism, there can bemany such multiple potentials having the given densityas it’s eigendensity associated with some combination ofeigenfunctions. So it is possible that one can also ob-tain second alternative potential V (say) whose ground-state density is same as the above excited state density( ρ e ( x )) of the original system ( V ). In this way, we have -2-1.5-1-0.500.511.52 -2000200400600800 0 0.2 0.4 0.6 0.8 1 V [ ρ ] Ψ [ V ] x ρ ( x ) x Δ '=1000.0 FIG. 15. The figure caption is same as Fig.13 but with ∆ ′ =1000 . studied six such excited states of the 1 D potential well(Figs.16 to 21) and for each case we are able to producesymmetrically different multiple potentials for fix densi-ties. Also in each case, we have produced the alternativepotential whose ground-state density is nothing but thegiven excited-state density of the original configuration(i.e. 1 D potential well).As our final case study, we have considered the excited-states of the 1 D QHO. This is also an interesting modelsystem like the potential well. The results for this case,are shown in Fig.22. Now consider the Fig.22a, in thiscase we have produced three symmetrically different al-ternative potentials V , V and V (shown in Fig.22b)whose ground-states densities (i.e. ρ (1)0 ( x ) , ρ (2)0 ( x ) and ρ (3)0 ( x ))are same as the different excited-states densi-ties (i.e. ρ (1) e ( x ) , ρ (2) e ( x ) and ρ (3) e ( x )) of the QHO po-tential V ( x ). Here ρ (1) e ( x ) corresponds to the config-uration [ n = 0( f = 1) , n = 3( f = 1)]. Similarly, ρ (2) e ( x ) and ρ (3) e ( x ) are arising from the excited-stateconfigurations [ n = 1( f = 1) , n = 2( f = 1)] and[ n = 2( f = 1) , n = 3( f = 1)] respectively. In Fig.22(d),we have produced a different potential V whose excited-state density corresponding to the configuration [ n =0( f = 1) , n = 2( f = 1)] is same as the excited-statedensity ρ e ( x ) ([ n = 0( f = 1) , n = 3( f = 1)]) of the5 (a) (b) (c) (d) FIG. 16. (a) ρ e [ n (2) , n (1) , n (1)] is the excited-state den-sity of 1 D potential well with ground-state ρ . ˜ ρ (1)0 isthe ground-state density of potential V whose excited-stateconfiguration [ n (2) , n (1) , n (1)] results the same ρ e . (b) V [ ρ e ] is the potential whose ground-state configuration re-sults the same ρ e of (a) and is shown along with V [˜ ρ (1)0 ].(c) ρ e [ n (2) , n (1) , n (1)] is the excited-state density of 1 D potential well with ground-state ρ and produced in an al-ternative configuration [ n (2) , n (1) , n (1)] ( V [˜ ρ (1)0 ]) besidesthe ground-state configuration leading to V [ ρ e ]. (d) Showsall the alternative potentials of (c). original 1 D QHO potential. Although we have producedso many potentials, but our criteria will only select theoriginal potentials (i.e. the infinite potential well in theprevious and QHO in the current study) for any given(i.e. either ground or excited-state) density. Thus estab-lishes the excited-state ρ ( x ) ⇐⇒ ˆ v ( x ) mapping uniquely. VI. DISCUSSIONS Now the conceptually basic questions of e DFT: whatare the consequences as well as similarities and differ-ences between the results of the CS formalism and thatobtained in connection to the HK/GL theorem? Sec-ondly, whether there arisen any critical scenario which isinconsistent with the HK and /or GL theorem(s)? This isbecause several multiple potentials are obtained for non-interacting fermions in the ground as well as lowest ex-cited state. Not only that, [50, 55] have also claimed thatfor higher excited-states there is no analogue of HK theo-rem. So the seemingly contradictory results may give rise (a) (a) (b) (c) (d) FIG. 17. (a) ρ e [ n (1) , n (2) , n (1)] is the excited-state densityof 1 D potential well with ground-state ρ . ˜ ρ (1)0 and ˜ ρ (2)0 arethe ground -state densities of V and V whose excited-stateconfigurations [ n (2) , n (1) , n (1)] and [ n (2) , n (1) , n (1)]results the same ρ e . (b) V [ ρ e ] is the potential whoseground-state configuration gives the same ρ e of (a) and isshown along with V , V . (c) ρ e [ n (2) , n (1) , n (1)] is theexcited-state density produced in alternative configuration[ n (2) , n (1) , n (1)] ( V [˜ ρ (1)0 ]), besides the ground-state con-figuration leading to V [ ρ e ]. (d) Shows all the alternativepotentials in (c). to the wrong conclusion about the validity of HK/GL the-orem and non-existence of density-to- potential mappingfor excited-states. However, the generalized/unified CSformalism overrules all these claims by showing that theground-state density of a given symmetry (potential) canbe the excited-state density of differing symmetry (poten-tial). Now this excited-state will have a correspondingground state which will be obviously quite different fromthe ground-state of the original system. As a matter ofwhich there will exist a different potential according toHK theorem. This is also true for the excited -state den-sity of the actual system: when it becomes either theground-state or some arbitrary excited -state density ofanother potential. So the unified CS formalism justifiesthe non-violation of HK/GL theorem for such states.Now based on the unified/generalized CS e DFT, onecan very nicely interpret ours as well as [50] results. Ac-tually by keeping the excited/ground state density fix viaa unitary transformation never guarantee the symmetriesof the states involve will remain intact. This is becauseby changing the ∆ ′ value and keeping either ground or6 (a) (b) (c) (d) FIG. 18. (a) ρ e [ n (1) , n (2) , n (1)] is the excited-state den-sity of 1 D infinite potential well with ground-state ρ .˜ ρ (1)0 and ˜ ρ (2)0 are the ground-state densities of V and V ,whose excited-state configurations [ n (2) , n (1) , n (1)] and[ n (2) , n (1) , n (1)] results the same ρ e . (b) V is the po-tential whose ground-state density is same as ρ e of (a) andis shown along with V , V . (c) ρ e [ n (1) , n (2) , n (1)] is theexcited-state density produced via the alternative configura-tions [ n (2) , n (1) , n (1)] ( V [˜ ρ (1)0 ]) besides the ground-stateconfiguration leading to V [ ρ e ]. (d) Shows all the alternativepotentials of (c). the excited state density fix, we are forcing the systemto change itself accordingly without hindering only thefixed density constraint. Since ∆ ′ is not fixed. So inprinciple one can make several choices for ∆ ′ and foreach choice, the system will converge to different poten-tials (systems/configurations) which can give the desireddensity of ground/ excited-state of the original system(potential/configurations) as one of it’s eigendensity. Ac-tually, the converged potentials are those for which theG¨orling and LN functionals are stationary and minimumrespectively. So everything is again automatically fitsinto realm of generalized CS formalism and nothing reallycontradicting or posing issues for the e DFT formulationsprovided by [24–29, 41, 58, 60, 61]. Also the transformedquantum states leading to multitude of potential for agiven density are energetically far off from the actual sys-tem and even the ground-states are also very different.Thus the generalized CS formalism proposed in this workalong with the SH criteria can be considered as the mostessential steps for establishing the ρ ( ~r ) ⇐⇒ ˆ v ext ( ~r ) whichfurther elaborated below. (a) (b) (c) (d) FIG. 19. (a) ρ e [ n (1) , n (1) , n (2)] is the excited-statedensity of 1 D infinite potential well with ground-state ρ . ˜ ρ (1)0 , ˜ ρ (2)0 and ˜ ρ (3)0 are the ground-state densi-ties of V , V and V whose excited-state configurations[ n (2) , n (1) , n (1)],[ n (2) , n (1) , n (1)] and [ n (2) , n (2)] re-sults the same ρ e . (b) V is the potential whose ground-state density is same as ρ e of (a) and is shown along with V , V and V . (c) ρ e [ n (1) , n (1) , n (2)] is the excited-state density produced in the alternative configurations[ n (2) , n (1) , n (1)] ( V [˜ ρ (1)0 ]), [ n (2) , n (1) , n (1)] ( V [˜ ρ (2)0 ])and [ n (2) , n (1) , n (1)] ( V [˜ ρ (3)0 ]) besides the ground-stateconfiguration leading to V [ ρ e ]. (d) Shows all the alternativepotentials of (c). Now the question is out of these existing multiple po-tentials in association with a fix density and ∆ ′ , whichpotential in principle should be picked in view of the ρ ( x ) ⇐⇒ ˆ v ( x )? The criteria of selecting the exact poten-tial out of all possibilities have already been discussed inSec.III. First of all it is quite obvious from the Figs.(1 to 6) and from Fig.7 to Fig.15 that the ground-state den-sities of the generated alternate potentials are differentfrom that of the original potential. This is also trueeven for the results of the CS formalism as shown in theFigs.(16 to (a) (b) (c) (d) FIG. 20. (a) ρ e [ n (1) , n (1) , n (2)] is the excited-state den-sity of 1 D infinite potential well with ground-state ρ .˜ ρ (1)0 and ˜ ρ (2)0 are the ground state densities of V and V , whose excited-state configurations [ n (2) , n (1) , n (1)]and [ n (2) , n (1) , n (1)] results the same ρ e . (b) V is the potential whose ground state density is sameas ρ e of (a) and is shown along with V , V . (c) ρ e [ n (2) , n (2)] is the excited-state density produced in al-ternative configurations [ n (2) , n (1) , n (1)] ( V [˜ ρ (1)0 ]) and[ n (2) , n (1) , n (1)] ( V [˜ ρ (2)0 ]) besides the ground-state con-figuration leading to V [ ρ e ]. (d) Shows all the alternativepotentials of (c). Additionally the kinetic energies of the two systems needto be kept closest, which we have pointed out on the ba-sis of DVT. So in all the non-interacting model systemsreported here, ∆ T should have been zero. But the drasti-cally differing structures of the transformed and originalwavefunctions are nothing but the manifestation of non-vanishing difference of kinetic energies and thus leadingto the multiple potentials. Furthermore, the most signif-icant differences between the symmetries of the old andnew systems implies that principally there exist discrep-ancies in the expectation values of the Hamiltonian w.r.t.the ground-states of various multiple potentials. This iswhat trivially follows from the reported results. Hence,the proposed criteria uniquely maps a given density ofthe 1 D QHO/infinite well to a potential which is noth-ing but the 1 D QHO/infinite well and discards rest ofthe multiple potentials. (a) (b) (c) (d) FIG. 21. (a) ρ e [ n (2) , n (2)] is the excited-state densityof 1 D infinite potential well with ground-state ρ . ˜ ρ (1)0 and ˜ ρ (2)0 are the ground-state densities of V and V ,whose excited-state configurations [ n (2) , n (1) , n (1)] and[ n (2) , n (2)] results the same ρ e . (b) V is the potentialwhose ground-state density is same as ρ e of (a) and isshown along with V , V . (c) ρ e [ n (1) , n (1) , n (1) , n (1)]is the excited-state density produced in alterna-tive configurations [ n (2) , n (1) , n (1)] ( V [˜ ρ (1)0 ]) and[ n (2) , n (1) , n (1)] ( V [˜ ρ (2)0 ]) besides the ground-state con-figuration leading to V [ ρ e ]. (d) Shows all the alternativepotentials of (c). VII. SUMMARY AND CONCLUDINGREMARKS In this work, we have tried to obtain a consistent the-ory for e DFT based on the stationary state, variationaland GAC formalism of modern DFT. We have provideda unified and general approach for dealing with excited-states which follows from previous attempts made byPerdew-Levy, G¨orling, Levy-Nagy-Ayers and in particu-lar the work of Samal-Harbola in the recent past. In thiscurrent attempt, we have answered the questions raisedabout the validity of HK and GL theorems to excited-states. We have settled the issues by explaining whythere exist multiple potentials not only for higher excitedstates but also for the ground as well as lowest excitedstate of given symmetry. In fact, the existing e DFT for-malism allows the above possibility and at the same timekeeps the uniqueness of density-to-potential mapping in-tact. So we have established in a rigorous fundamentalfooting the non-violation of the HK and GL theorem.8 (a) (b) (c) (d) FIG. 22. (a) ρ (1) e [ n = 0 , n = 3](both half-filled) , ρ (2) e [ n =1 , n = 2] (both half filled) and ρ (3) e [ n = 2 , n = 3](both halffilled) are the excited-state densities of the potential V pro-duced as the ground state density of the potentials V , V and V . (b) Shows all the four potentials V , V , V and V of(a). (c) ρ e [ n = 0 , n = 3] (both half filled) is the excited-statedensity of the potential V produced in an alternative excitedstate configuration [ n = 0 , n = 2] ( V ). (d) Shows both thepotentials of (c). Actually, the generalized CS approach gives us a strongbasis in choosing a potential out of several multiple po-tentials for a fixed ground/excited state density. In ourpropositions, we have strictly defined the bi-density func-tionals for a fix pair of ground and excited-state densitiesin order to establish the density-to-potential mapping. Not only that, the theory also gives us a clear definitionof excited-state KS systems through the comparison ofkinetic and exchange-correlation energies w.r.t. the truesystem. It does takes care the stationarity and orthog-onality of the quantum states. So everything fits quitenaturally into the realm of modern DFT.To conclude, we have demonstrated density-to-potential mapping for non-interacting fermions. For in-teracting case the GAC can be used to formulate all thetheoretical and numerical contents in a similar way. Weare working along this direction for strictly correlatedfermions and the results will be reported in future. Fi-nally, our conclusion is that nothing really reveals themanifestation of the failure or violation of the basic the-orems and existing principles of modern DFT irrespectiveof the states under consideration. 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