Notes on open questions within density functional theory (existence of a derivative of the Lieb functional in a restricted sense and non-interacting v-representability)
aa r X i v : . [ qu a n t - ph ] F e b On the non-interacting v -representabilityconjecture within density functional theory J´er´emie MessudFebruary 9, 2021
Density functional theory together with the Kohn-Sham (KS) scheme rep-resent an efficient framework to recover the ground state density and energyof a many-body quantum system from an auxiliary “non-interacting” system(one-body with a local potential that is a density functional). However, to fullyachieve establishing the KS scheme, a general proof of the validity of the so-called “non-interacting v -representability” conjecture is still required. In thisarticle, we propose such a proof. We reduce the demonstration to proving (1)that the KS potential is differentiable for all non-interacting v -representabledensities and (2) that its derivative is bounded. Then, we demonstrate points(1) and (2) applying static linear response to the non-interacting system.
1. Introduction
Density functional theory (DFT) (Hohenberg and Kohn, 1964; Parr and Yang, 1989; Dreizler and Gross,1990; Kohn, 1999; Engel and Dreizler, 2011) has become over the last decades a widely usedtheoretical tool for the description and analysis of particles properties in quantum sys-tems. The essence of DFT is the Hohenberg-Kohn (HK) theorem (Hohenberg and Kohn,1964), ensuring that ground state many-particle systems in an “external” potential v canbe completely characterized by the ground state particle density (Parr and Yang, 1989;Dreizler and Gross, 1990; Engel and Dreizler, 2011). Indeed, the HK theorem implies thatany ground state observable of such a system can be written as a density functional, in par-ticular the energy of the system; hence the possibility of substituting the involved (many-body and non-local) “exchange-correlation” contribution to the energy by a functional of thesimple particle density. The form of this exchange-correlation energy functional may be verycomplicated, but the proof of its existence leads to the appealing idea of replacing the origi-nal interacting (or many-body) system equations by a non-interacting (or one-body) systemequations that exactly reproduce the ground state density and energy of the interacting sys-tem. The corresponding reformulation has been proposed by Kohn and Sham (1965) and iscalled the Kohn-Sham (KS) scheme. All quantum effects, including the exchange-correlationones, are there described through a local potential, called the KS potential, defined as thefunctional derivative of the energy functional with respect to the density (Kohn and Sham,1965; Parr and Yang, 1989; Dreizler and Gross, 1990; Engel and Dreizler, 2011). The oc-currence in the KS scheme of this one-body local potential that accounts for all exchange-correlation effects eventually makes the scheme computationally very efficient to describesystems with a sizeable number of particles. 1nfortunately, DFT states existence theorems but does not give a clue on the en-ergy functional form. A whole branch of DFT research is dedicated to set up approxi-mate parameterized functionals. These are usually based on the local density approxima-tion (LDA), which is widely used to describe, for instance, many-electron systems in amolecule (Parr and Yang, 1989; Dreizler and Gross, 1990; Engel and Dreizler, 2011).Another branch of DFT research is dedicated to bringing rigorous foundations to theformalism. In particular, as DFT involves functional derivatives of the energy functional(to build the KS scheme...), the functional differentiability of the exact energy functionalmust be built on solid ground. This implies, among other things, that the energy functionalmust be defined on sufficiently dense sets of admissible densities and the notion of a normto make sense. The original HK definition of the energy functional (Hohenberg and Kohn,1964; Kohn and Sham, 1965) is not sufficient to achieve this goal, so various extensionshave been proposed. The Levy-Lieb formulation (Levy, 1982; Lieb, 1983) extends the en-ergy functional to be defined on all “interacting pure state v -representable” densities, i.e. alldensities that arise from non-degenerate ground states of a many-body or interacting systemwith an external potential v . However, the differentiability of this functional has not been es-tablished. Lieb (1983) proposed another formulation which further extends the energy func-tional to be defined on all “interacting (ensemble) v -representable” densities, i.e. all densi-ties that arise from ground states of an interacting system (degenerate or not). The Liebfunctional is the only one that so far that has been demonstrated to satisfy the good proper-ties regarding differentiability (Lieb, 1983; Englisch and Englisch, 1984a,b). Thus, any for-mal consideration on DFT must be based on this functional. Considering the Lieb functionalof the interacting system, here called interacting Lieb functional, and further introducing aLieb functional for the non-interacting system, here called non-interacting Lieb functional,the KS equations can be derived on solid ground (Englisch and Englisch, 1984a,b). A re-maining difficulty is that the interacting and non-interacting Lieb functionals domains arenot the same. The interacting Lieb functional domain consists of the set of interacting v -representable densities, whereas the non-interacting Lieb functional domain consists of theset of non-interacting v -representable densities (which will be defined later). Fortunately,it has been demonstrated that for every interacting v -representable density there exists anarbitrarily close non-interacting v -representable density (Englisch and Englisch, 1984a,b),allowing one to theoretically approximate with arbitrary precision a ground state densityof an interacting system with that of a non-interacting or KS system. A comprehensivereview of these aspects has been presented by van Leeuwen (2003) and Engel and Dreizler(2011).However, a final question remained open, called the “non-interacting v -representabilityquestion”: are the sets of interacting v -representable densities and non-interacting v -representabledensities equal? A positive answer would achieve to establish the KS scheme, implying thatit is possible to reproduce by a KS scheme any interacting v -representable density (this in apractically friendly way, i.e. with a KS potential that would not change very rapidly, oscil-late... approaching the limit). A recent attempt at a proof has been done by Gonis (2019) inthe non-degenerate non-interacting system case, using the Levy-Lieb functional. However,as already mentioned, the differentiability of the Levy-Lieb functional has not been proven.A proof using the Lieb functional and extending to degenerate non-interacting systems isstill to be done, which represents the goal of this article. The first part of the article isdedicated to a review of the Levy functionals and the KS scheme. The second part is dedi-cated to a rigorous demonstration that a positive answer can be given to the non-interacting v -representability question. To that end, we explain how we can reduce the demonstrationto proving (1) that the KS potential is differentiable for all non-interacting v -representabledensities and (2) that its derivative is bounded. Then, we prove points (1) and (2) applyingstatic linear response to the non-interacting system, achieving the demonstration.2 . Notations, Lieb functional and Kohn-Sham scheme(reminders) We consider a stationary quantum system of N interacting identical particles, called inter-acting system, whose Hamiltonian isˆ H v = ˆ T + ˆ W + ˆ V [ v ] . (1)ˆ T denotes the kinetic energy operator and ˆ W denotes the particles interaction operator.ˆ V [ v ] = R R v ( r )ˆ n ( r ) dr denotes a potential operator defined through an “external” potential v ( r ), and the diagonal part of the particle density operator,ˆ n ( r ) = N X i =1 δ ( r − r i ) . (2) r i , indexed by i ∈ { , ..., N } , denotes the position of a particle. Any following reasoningconsider ( r, r i ) ∈ R × R , and fixed N and ˆ W .A Banach space is a (possibly infinite dimensional) vector space with a norm (Rudin,1991; Brezis, 2020). We consider the L p ( p ≥
1) Banach spaces defined by L p ( R M ) = n f ( x ) (cid:12)(cid:12)(cid:12) x ∈ R M , || f || p < ∞ o with norm || f || p = (cid:16) Z R M | f ( x ) | dx (cid:17) p . (3)When p = ∞ , || f || ∞ = sup r ∈ R | f ( r ) | which is called the uniform norm. The Banach spaceof admissible potentials v for the Hamiltonian in eq. (1) is (Lieb, 1983; van Leeuwen, 2003;Engel and Dreizler, 2011) V L = n v = v + v (cid:12)(cid:12)(cid:12) v ∈ L ( R ) , v ∈ L ∞ ( R ) o (4)with norm || v || V L = inf v ∈ L ( R ) ,v ∈ L ∞ ( R ) v + v = v || v || + || v || ∞ , where potentials that differ by an additive constant are identified.We also consider the Sobolev space H ( R M ) = n f (cid:12)(cid:12)(cid:12) f ∈ L ( R M ) , ∇ f ∈ ~ L ( R M ) o with norm || f || H = (cid:16) Z R M ( | f ( x ) | + |∇ f ( x ) | ) dx (cid:17) . (5)In the following, we will be interested in the particle density defined through eq. (2) by n ( r ) = h ψ | ˆ n ( r ) | ψ i , (6)where | ψ i can represent any eigenstate of ˆ H v (and particularly the ground state thatwill interest us further). More generally, | ψ i can be any state so that h r , .., r N | ψ i ∈ H ( R N ). A general (convex) set of pertinent densities is (Lieb, 1983; van Leeuwen, 2003;Engel and Dreizler, 2011) S = n n (cid:12)(cid:12)(cid:12) n ( r ) ≥ , Z R n ( r ) dr = N, √ n ∈ H ( R ) o . (7)The || . || p =1 and || . || p =3 norms, here denoted by || . || p =1 , , can be associated to S as S ⊂ N L where N L is the Banach space : N L = L ( R ) ∩ L ( R ) . (8) The topological dual N ∗ L of N L is “represented” by V L , eq. (4) (Brezis (2020) chapter IV). Similarly, || . || V L “represents” the dual norm of || . || p =1 , (Brezis (2020) chapter I, Rudin (1991) chapter IV). N and ˆ W ), herecalled the “interacting Lieb functional” ∀ n ∈ S : F L [ n ] = inf ˆ D ∈ D ( n ) Tr ˆ D ( ˆ T + ˆ W ) , (9)where Tr denotes the trace operator and D ( n ) denotes the set of N -particle density matricesthat yield a gives density n D ( n ) = n ˆ D (cid:12)(cid:12)(cid:12) ˆ D = X k d k | ψ k ih ψ k | , d ∗ k = d k ≥ , X k d k = 1 , (10) h ψ k | ψ l i = δ kl , h r , ..., r N | ψ k i ∈ H ( R N ) , n ( r ) = X k d k h ψ k | ˆ n ( r ) | ψ k i o . A corresponding “interacting energy functional” is defined by ∀ n ∈ S : E v [ n ] = F L [ n ] + Z R v ( r ) n ( r ) dr. (11) Theorem 2.1 (Demonstrated by Lieb (1983)) . The minimum of the interacting energyfunctional, E [ v ] = inf n ∈ S E v [ n ] , (12) equals the ground state energy of the Hamiltonian in eq. (1) for any external potential v ∈ V L that yields a ground state. In the non-degenerate case, i.e. when a unique ground states is associated to ˆ H v , there is aone-to-one correspondence between v and the corresponding minimizing (ground state) den-sity. In the degenerate case, however, the many minimizing (ground states) densities relatedto the degenerate ground states of ˆ H v are identified; this provides a one-to-one correspon-dence between v and the equivalence class of these ground state densities (Parr and Yang,1989; Dreizler and Gross, 1990; Engel and Dreizler, 2011). Then, the potential v can beconsidered as a functional of the ground state density and conversely. F L [ n ], eq. (9), isconvex with respect to n (Lieb, 1983), which is a desirable property from a formal point ofview, among other things regarding differentiability. Theorem 2.2 (Demonstrated by Lieb (1983) and Englisch and Englisch (1984a,b)) . F L [ n ] is differentiable only in a subset B ⊂ S . ∀ n ∈ B : ∂F L [ n ] ∂n ( r ) = − v ( r ) , (13) where v ∈ V L is the potential that generates n ∈ B and B is the set of interacting (ensem-ble) v -representable densities defined by B = n n (cid:12)(cid:12)(cid:12) n ( r ) = q X i =1 c i h ψ i | ˆ n ( r ) | ψ i i , c ∗ i = c i ≥ , q X i =1 c i = 1 , (14) | ψ i i belongs to the set of ground states of ˆ H v for some v o . The set B includes the interacting pure state v -representable densities ( q = 1 restriction).The q > .2. Non-interacting system We now consider a stationary system of N identical particles whose Hamiltonian isˆ H s = ˆ T + ˆ V [ v s ] , (15)where ˆ V s [ v s ] = R R v s ( r )ˆ n ( r ) dr denotes a potential operator defined through a potential v s ( r ). Compared to the interacting system Hamiltonian in eq. (1), the non-interactingsystem Hamiltonian in eq. (15) omits the particles interaction operator. We denote the“non-interacting Lieb functional” by ∀ n ∈ S : T L [ n ] = inf ˆ D ∈ D ( n ) Tr ˆ D ˆ T , (16)where the particles interaction term is omitted compared to the interacting Lieb functionalin eq. (9). From a straightforward adaptation of theorem 2.2, T L [ n ] is differentiable only ina subset B s ⊂ S (Englisch and Englisch, 1984a,b), ∀ n s ∈ B s : ∂T L [ n s ] ∂n s ( r ) = − v s ( r ) , (17)where v s ∈ V L is the potential that generates n s ∈ B s , called the KS potential, and B s isthe set of non-interacting (ensemble) v -representable densities defined by B s = n n s (cid:12)(cid:12)(cid:12) n s ( r ) = q s X i =1 c s i h ψ s i | ˆ n ( r ) | ψ s i i , c ∗ s i = c s i ≥ , q s X i =1 c s i = 1 , (18) | ψ s i i belongs to the set of ground states of ˆ H s for some v s o . The set B s includes the non-interacting pure state v -representable densities set denoted by A s ( q s = 1 restriction). The q s > § v s and the corresponding minimizing density n s . In the degenerate case, there is a one-to-onecorrespondence between v s and the equivalence class of ground state densities obtained fromthe degenerate ground states of ˆ H s . Then, the potential v s can be considered as a functionalof the density n s , i.e. v s [ n s ], and conversely (Parr and Yang, 1989; Dreizler and Gross, 1990;Engel and Dreizler, 2011).From eqs. (13) and (17), we notice that the interacting and non-interacting Lieb func-tionals domains, respectively B and B s , are not the same. Let us first consider the caseof an interacting system ground state density n ∈ B ∩ B s . The main outcome of previousconsiderations is that such densities can be reproduced by a non-interacting system usingin eq. (15) the potential v s [ n ] computed by eq. (17) with n s = n , i.e. through ∀ k ∈ { , ..., q s } : (cid:16) ˆ T + ˆ V [ v s [ n ]] (cid:17) | ψ s k i = E s | ψ s k i (19) n ( r ) = q s X k =1 λ k n k ( r ) , n k ( r ) = h ψ s k | ˆ n ( r ) | ψ s k i , λ ∗ k = λ k ≥ , q s X k =1 λ k = 1 , where q s degenerate ground states | ψ s k i with energy E s are possibly considered for thenon-interacting system when the computed potential v s [ n ] leads to such a situation. When v s [ n ] does not lead to ground state degeneracy, we have q s = 1. The {| ψ s k i} can be chosento be orthonormal eigenstates, for instance Slater determinants in the case of a fermionicsystem (Dreizler and Gross, 1990; Engel and Dreizler, 2011). When the non-interactingsystem is degenerate, at least one of the many minimizing densities P q s k =1 λ k h ψ s i | ˆ n ( r ) | ψ s i i λ k . The equivalenceclass of these degenerate densities is thus identified to this element.Now, what about an interacting system ground state density n ∈ B \ B s ? van Leeuwen(2003) and Engel and Dreizler (2011) detail in a comprehensive way the steps leading tothe following important theorem: Theorem 2.3 (Demonstrated by Englisch and Englisch (1984a,b)) . ∀ n ∈ B , ∀ ǫ > , (20) ∃ n ǫ ∈ B s : || n ǫ − n || p =1 , ≤ ǫ.n ǫ is a ground state density of the non-interacting system with potential v s [ n ǫ ] = − ∂T L [ n ǫ ] ∂n ǫ ( r ) . As a consequence of the first part of Theorem 2.3, the set B s , eq. (18), is dense in theset B , eq. (14) (and conversely). We thus havelim ǫ → n ǫ = n. (21)However, as there for now is no proof that the sets B and B s are equal, two possibilitiescan occur regarding this limit: • If B = B s , the limit for n ∈ B \ B s cannot be reached by n ǫ ∈ B s , and conversely. Anon-interacting system that reproduces n with an arbitrary precision can still theo-retically be defined, but this does not mean there exists a practically friendly way toachieve it. Indeed, v s [ n ǫ ] could then possibly change very rapidly (oscillate or evendiverge) to make n ǫ slightly closer to n . • If B = B s , the limit is always reached, and conversely.The aim of this paper is to decide which of these two possibilities is true, representing thestill open “non-interacting v -representability question” (van Leeuwen, 2003; Engel and Dreizler,2011). Before diving into this topic, we conclude the first part of this article by giving fewreminders regarding the KS scheme. Eq. (19) forms a basis to establish the KS scheme (Kohn and Sham, 1965; Parr and Yang,1989; Dreizler and Gross, 1990; Engel and Dreizler, 2011). Considering v s as a known func-tional of n ∈ B ∩ B s , we could self-consistently resolve eq. (19) to recover the ground statedensity n of the corresponding interacting system. In practice, as the exact v s [ n ] functionalform is not known (DFT only states existence theorems), a parameterized functional formis to be defined.A first step is usually done rewriting the interacting energy functional in eq. (11) as (Parr and Yang,1989; Dreizler and Gross, 1990; Engel and Dreizler, 2011) ∀ n ∈ S : E v [ n ] = T L [ n ] + E H [ n ] + E XC [ n ] + Z R v ( r ) n ( r ) dr, (22)where E H [ n ] is the Hartree energy, and E XC [ n ] is the exchange-correlation energy definedby ∀ n ∈ S : E XC [ n ] = F L [ n ] − T L [ n ] − E H [ n ] . We deduce, using eqs. (13), (17) and (22), that the KS potential can be split into threecontributions ∀ n ∈ B ∩ B s : v s [ n ]( r ) = v H [ n ]( r ) + v XC [ n ]( r ) + v ( r ) , (23)6here v H [ n ] is the Hartree potential, v the external potential (the same than the one of theinteracting system) and v XC [ n ]( r ) = ∂E XC [ n ] /∂n ( r ) is called the “exchange-correlationpotential” (Parr and Yang, 1989; Dreizler and Gross, 1990; Engel and Dreizler, 2011).This represents only a rewriting of the interacting energy functional, but the nice pointis that it implies defining a parameterized functional form for the E XC [ n ] term only.This is advantageous as this term usually represents a “correction”, i.e. is usually quitesmall compared to the other terms in eq. (22). An approximate parameterization of E XC [ n ] is the LDA one, widely used for instance to describe many-electron systems ina molecule (Parr and Yang, 1989; Dreizler and Gross, 1990; Engel and Dreizler, 2011).Finally, we remind that if the answer to the non-interacting v -representability question is B = B s , then B ∩ B s = B so that a KS scheme with the potential in eq. (23) can alwaysbe built to reproduce exactly the ground state density of any interacting system. We nowdemonstrate that this is the case, to achieve to establish the KS scheme.
3. On non-interacting v -representability Proposition 3.1 (Non-interacting v -representability conjecture) . B = B s . Remark 3.2 (Non-interacting v -representability condition) . The non-interacting v -representabilityconjecture is true if ∀ n ∈ B , ∃ v n ∈ V L , (24) ∀ ǫ > , ∃ α ( ǫ ) n satisfying lim ε → α ( ε ) n = 0 , ∀ n ǫ ∈ B ( ǫ,n ) s : (cid:12)(cid:12) v s [ n ǫ ]( r ) − v n ( r ) (cid:12)(cid:12) ≤ α ( ǫ ) n ( r ) , where we considered the following set (defined by a given n ∈ B and a given ǫ value) B ( ǫ,n ) s = n n ǫ ∈ B s : || n ǫ − n || p =1 , ≤ ǫ o . (25) B ( ǫ,n ) s represents the subset of densities in B s that are “ ǫ -close” to a given density n ∈ B from the || . || p =1 , norm point of view . Eq. (24) would straightforwardly imply lim ǫ → v s [ n ǫ ] = v n . (26) Justification . If Proposition 3.1 is true, then any n ∈ B would be exactly reproductible bya non-interacting system with a potential v s [ n ]. This is equivalent to say, using the notationsin Theorem 2.3, that: (A) v s [ n ǫ ] should converge for ǫ → v n ∈ V L insome smooth way (i.e. that v s [ n ǫ ] should not change rapidly, oscillate... approaching thelimit) (van Leeuwen (2003), section 15) and (B) v s [ n ǫ ] should reach this v n element (andnot only tend to it without reaching it). Satisfying eq. (24) would explicitly ensure point(A) and implicitly ensure point (B). Indeed, as v s [ n ǫ ] and v n belong to the same space V L ,it would always be possible to infinitesimally “tweak” a v n that satisfies eq. (24) so that thelimit is reached in eq. (26). Thus, proving eq. (24) would not only imply that v s [ n ǫ ] wouldconverge towards some element in V L in a smooth way, but also that it is always possibleto select this element so that the limit is reached. Equivalently, the density domain limitin eq. (21) would always be reached, implying through eq. (20) that B = B s . (cid:4) Eq. (24) represent a practical way to demonstrate the non-interacting v -representabilityconjecture, probably more practical than working with eq. (20). Indeed, in eq. (20) n ǫ and n do not belong to the same set, whereas in eq. (24) v s [ n ǫ ] and v n belong to the same space. Using this set, the first relation in eq. (20) can be reformulated: ∀ n ∈ B , ∀ ǫ > B ( ǫ,n ) s = ∅ . .2. Non-interacting v -representability, v s [ n s ] differentiability and derivativeboundedness In previous section, we reduced the proof of the non-interacting v -representability conjectureto the proof of eq. (24). In current section, we reduce the proof of eq. (24) to demonstratingthat v s [ n s ] is differentiable ∀ n s ∈ B s and that its derivative is bounded. This makes sensephysically: if in eq. (24) v s [ n ǫ ] is continuous and varies sufficiently smoothly with respectto n ǫ ∈ B s when n ǫ approaches some limit n ∈ B , it should be possible at some point tocontinuously extend v s [ n ǫ ] at n ∈ B . This should lead to the definition of a an element v n ∈ V L towards which v s [ n ǫ ] converges smoothly for a n ǫ sufficiently close to n from the || . || p =1 , norm point of view. Theorem 3.3 (Non-interacting v -representability and differentiability) . If v s [ n s ]( r ) is dif-ferentiable ∀ n s ∈ B s and its derivative is bounded, then the non-interacting v -representabilityconjecture (Proposition 3.1) is true. Proof . Theorem 2.3 ensures that B s is dense in B . We moreover do the hypothesis that v s [ n s ] is differentiable for all n s ∈ B s and that its derivative is bounded. Mathematicalanalysis theorems (Rudin, 1976) then ensure the existence of a unique continuous extensionof v s [ n s ] at n ∈ B , thus the existence of the limit in eq. (26), which is sufficient for ourconsideration. However, let us give complementary insight by explicitly demonstrating thateq. (24) is true, building expressions for the v n and α ( ǫ ) n terms.We consider a given density n ∈ B and a given ǫ > B ( ǫ,n ) s setthrough eq. (25), and denote by ˜ n ǫ any particular density in B ( ǫ,n ) s . Eqs. (20) and (25)imply lim ǫ → ˜ n ǫ = n . Because we consider that v s [ n s ] is differentiable ∀ n s ∈ B s , we candevelop v s [ n ǫ ] in Taylor series around ˜ n ǫ . We have, for all n ǫ ∈ B ( ǫ,n ) s : v s [ n ǫ ]( r ) = v s [˜ n ǫ ]( r ) + Z R ∂v s [˜ n ǫ ]( r ) ∂ ˜ n ǫ ( r ′ ) (cid:0) n ǫ ( r ′ ) − ˜ n ǫ ( r ′ ) (cid:1) dr ′ + δv ( ǫ ) s [ n ǫ , ˜ n ǫ ]( r ) , (27)where δv ( ǫ ) s [ n ǫ , ˜ n ǫ ] denotes the remainder term satisfying lim ǫ → δv ( ǫ ) s [ n ǫ , ˜ n ǫ ] = 0. We con-sider an element v n, ˜ n ǫ ∈ V L that equals v s [˜ n ǫ ] if n = ˜ n ǫ , and continuously extends v s [˜ n ǫ ] if n ∈ B by: v n, ˜ n ǫ ( r ) = v s [˜ n ǫ ]( r ) + Z R ∂v s [˜ n ǫ ]( r ) ∂ ˜ n ǫ ( r ′ ) (cid:0) n ( r ′ ) − ˜ n ǫ ( r ′ ) (cid:1) dr ′ + δv ( ǫ ) n, ˜ n ǫ ( r ) , (28)where δv ( ǫ ) n, ˜ n ǫ denotes the “smallest correction” so that v n, ˜ n ǫ belongs to V L , which satisfieslim ǫ → δv ( ǫ ) n, ˜ n ǫ = 0. This definition of the correction term makes v n, ˜ n ǫ dependent on the choiceof ˜ n ǫ , hence the subscript “ n, ˜ n ǫ ”, contrarywise to v s [ n ǫ ] in eq. (27) that is independentof the choice of ˜ n ǫ because the Taylor series remainder term compensates for this choice.Eq. (28) represents the simplest way to build a potential in V L that is a functional of thedensity n in B from the knowledge of a potential v s [˜ n ǫ ] that is a functional of a density ˜ n ǫ (sufficiently close to n ) in B s .Combining eqs. (27) and (28), we obtain for all n ǫ ∈ B ( ǫ,n ) s v s [ n ǫ ]( r ) − v n, ˜ n ǫ ( r ) = Z R ∂v s [˜ n ǫ ]( r ) ∂ ˜ n ǫ ( r ′ ) (cid:0) n ǫ ( r ′ ) − n ( r ′ ) (cid:1) dr ′ + ∆ v ( ǫ ) n [ n ǫ , ˜ n ǫ ]( r ) , (29)where ∆ v ( ǫ ) n [ n ǫ , ˜ n ǫ ] = δv ( ǫ ) s [ n ǫ , ˜ n ǫ ] − δv ( ǫ ) n, ˜ n ǫ satisfies lim ǫ → ∆ v ( ǫ ) n [ n ǫ , ˜ n ǫ ] = 0 (because of8forementioned properties). We compute (cid:12)(cid:12) v s [ n ǫ ]( r ) − v n, ˜ n ǫ ( r ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z R ∂v s [˜ n ǫ ]( r ) ∂ ˜ n ǫ ( r ′ ) (cid:0) n ǫ ( r ′ ) − n ( r ′ ) (cid:1) dr ′ + ∆ v ( ǫ ) n [ n ǫ , ˜ n ǫ ]( r ) (cid:12)(cid:12)(cid:12) (30) ≤ (cid:12)(cid:12)(cid:12) Z R ∂v s [˜ n ǫ ]( r ) ∂ ˜ n ǫ ( r ′ ) (cid:0) n ǫ ( r ′ ) − n ( r ′ ) (cid:1) dr ′ (cid:12)(cid:12)(cid:12) + | ∆ v ( ǫ ) n [ n ǫ , ˜ n ǫ ]( r ) |≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂v s [˜ n ǫ ]( r ) ∂ ˜ n ǫ (cid:0) n ǫ − n (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p =1 + | ∆ v ( ǫ ) n [ n ǫ , ˜ n ǫ ]( r ) |≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂v s [˜ n ǫ ]( r ) ∂ ˜ n ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q = ∞ × || n ǫ − n || p =1 + | ∆ v ( ǫ ) n [ n ǫ , ˜ n ǫ ]( r ) |≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂v s [˜ n ǫ ]( r ) ∂ ˜ n ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q = ∞ × ǫ + | ∆ v ( ǫ ) n [ n ǫ , ˜ n ǫ ]( r ) | , where we used the uniform norm definition, the H¨older inequality (Brezis, 2020; Rudin,1991) for the penultimate line and eq. (25) for the last line. We define β ( ǫ ) n [ n ǫ , ˜ n ǫ ]( r ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂v s [˜ n ǫ ]( r ) ∂ ˜ n ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q = ∞ × ǫ + | ∆ v ( ǫ ) n [ n ǫ , ˜ n ǫ ]( r ) | . (31)Because we consider that the derivative of v s [ n s ] is bounded, ∃ K ∈ R + , ∀ n s ∈ B s : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂v s [ n s ]( r ) ∂n s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q = ∞ < K, (32)we have lim ǫ → β ( ǫ ) n [ n ǫ , ˜ n ǫ ]( r ) = 0 . (33)Gathering these results, we have demonstrated that if v s [ n s ] is differentiable ∀ n s ∈ B s andits derivative bounded, we have ∀ n ∈ B , ∀ ǫ > , (defining B ( ǫ,n ) s thus allowing to choose a ˜ n ǫ ) (34) ∃ v n, ˜ n ǫ ∈ V L , ∀ n ǫ ∈ B ( ǫ,n ) s , ∃ β ( ǫ ) n [ n ǫ , ˜ n ǫ ] satisfying lim ε → β ( ε ) n [ n ǫ , ˜ n ǫ ] = 0 : (cid:12)(cid:12) v s [ n ǫ ]( r ) − v n, ˜ n ǫ ( r ) (cid:12)(cid:12) ≤ β ( ǫ ) n [ n ǫ , ˜ n ǫ ]( r ) . Eq. (34) shares some similarity with eq. (24), the major differences being that in eq. (34) v n, ˜ n ǫ depends on the choice for ˜ n ǫ , and β ( ǫ ) n [ n ǫ , ˜ n ǫ ] depends on n ǫ and the choice for ˜ n ǫ .If we can find a way to remove these dependencies, then eq. (34) would naturally convertinto eq. (24) and the proof would be done. For the β ( ǫ ) n [ n ǫ , ˜ n ǫ ] term, this is achieved byconsidering α ( ǫ ) n ( r ) = max ( n ǫ , ˜ n ǫ ) ∈ B ( ǫ,n ) s × B ( ǫ,n ) s β ( ǫ ) n [ n ǫ , ˜ n ǫ ]( r ) (satisfies lim ε → α ( ε ) n ( r ) = 0) . (35)For the v n, ˜ n ǫ term, this can be achieved only if the following limit exists: v n = lim ǫ → v n, ˜ n ǫ . (36)Again, as B s is dense in B , and v s [ n s ] is differentiable and its derivative is bounded, math-ematical analysis theorems (Rudin, 1976) ensure the existence of a continuous extension of v s [˜ n ǫ ] at n ∈ B . Such a continuous extension being unique, it is given by eq. (28) at thelimit when ǫ approaches 0. As a consequence, the limit in eq. (36) exists.Thus, for a given density n ∈ B , we showed how to define an element v n ∈ V L , eqs. (28)and (36), that is compliant with eq. (24). For a given ǫ > α ( ǫ ) , eqs. (31) and (35), that is compliant with eq. (24). This concludes the proof. (cid:4) To achieve the proof of the non-interacting v -representability conjecture, it remains todemonstrate that v s [ n s ] is differentiable and that its derivative is bounded. This is the goalof the rest of the article. 9 .3. Differentiability of v s [ n s ] (proof) Definition 3.4 (Differential) . A differential is defined considering a Banach space, heredenoted by B ( R ) , with the corresponding norm denoted by || . || B . For a functional G whosedomain is B ( R ) , the differential evaluated at some n ∈ B ( R ) is denoted by dG n ( r ) . If thedifferential exists, it is uniquely defined by ∀ h ∈ B ( R ) : G [ n + h ] − G [ n ] = Z R dG n ( r ) h ( r ) dr + o ( || h || B ) , (37) where “ little- o ” corresponds to the Landau notation. dG n ( r ) is then equal to the functionalderivative ∂G [ n ] /∂n ( r ) . To demonstrate the existence of the differential of v s [ n s ] evaluated at n s ∈ B s , we mustdemonstrate that v s [ n s ] satisfies a relation similar to eq. (37). Before working on this point,for pedagogy, we first discuss the existence of the differential of n s [ v s ] evaluated at v s ∈ V L .Indeed, the potential v s can be considered as a functional of the density n s and, conversely,the density n s can be considered as a functional of the potential v s , remind § n s [ v s ] in the non-degenerate case Theorem 3.5 (Differentiability of n s [ v s ] – restricted case) . n s [ v s ] is differentiable for all v s ∈ V nondegen L , V nondegen L denoting the subset of V L that leads to non-degenerate solutionsfor the non-interacting systems and whose first order potentials perturbations also lead tonon-degenerate solutions. We then have n s [ v s ] ∈ A s , where A s denotes the non-interactingpure state v -representable densities set, that has been defined right after eq. (18). Proof (inspired from van Leeuwen (2003)) . Reminding that V L represents a Banachspace for the norm || . || V L ( § dn s vs ( r ′ ) ( r ) thatsatisfies for sufficiently small δw ∈ V L : n s [ v s + δw ]( r ) − n s [ v s ]( r ) = Z R dn s vs ( r ′ ) ( r ) δw ( r ′ ) dr ′ + o ( || δw || V L ) . (38)Then, from considerations of § dn s vs ( r ′ ) ( r ) would equal the differential of n s [ v s ]( r )evaluated at v s ( r ′ ), i.e. ∂n s [ v s ]( r ) /∂v s ( r ′ ).We consider a potential v s ∈ V nondegen L . Then, n s [ v s ] ∈ A s and, for sufficiently smallperturbations δw , n s [ v s + δw ] ∈ A s .We now adapt the considerations of section 7 of van Leeuwen (2003) to the non-interactingsystems and approach the question of differentiability from a slightly different point of view.The static linear response of the Schr¨odinger equation gives in the non-degenerate case thefollowing relationship between a perturbation δw ∈ V L that represents a first order changein v s and the corresponding first order change δm in n s : δm [ v s , δw ]( r ) = Z R χ s [ v s ]( r ′ , r ) δw ( r ′ ) dr ′ , (39)where χ s [ v s ] represents the “static density response function” defined in Appendix A.1. If(as usual) we identify potential perturbations defined up to an additive constant, we candeduce from eq. (39) that δm is a unique functional of v s and δw (van Leeuwen, 2003).The obtained δm [ v s , δw ]( r ) satisfies R R δm [ v s , δw ]( r ) dr = 0, see Appendix A.1 for details, In the general case, dG n ( r ) can possibly be a linear operator acting on h ( r ) (like a derivative...) or adistribution. This is not the case here. N = R R n ( r ) dr . Defining the new Banach space δ S = n δn (cid:12)(cid:12)(cid:12) Z R δn ( r ) dr = 0 , p | δn | ∈ H ( R ) o with norm || . || p =1 , , (40)we can demonstrate that any δm [ v s , δw ] obtained from eq. (39) belongs to δ S .We now would like to approximate n s [ v s + δw ] ∈ A s using δm [ v s , δw ] to verify if wecan recover a form like eq. (38). The difficulty underlined by van Leeuwen (2003) is that n s [ v s ]+ δm [ v s , δw ] / ∈ A s , i.e. does not in general represent a non-degenerate non-interactingground state density for the potential v s + δw . Nevertheless, n s [ v s ] + δm [ v s , δw ] / ∈ A s approximates n s [ v s + δw ] ∈ A s to first order. Thus, we can introduce a “corrective” secondorder term ∆ m ( δw ) ∈ δ S satisfying lim || δw || VL → ∆ m ( δw ) ( r ) / || δw || V L = 0, defined so that n s [ v s + δw ] = n s [ v s ] + δm [ v s , δw ] + ∆ m ( δw ) ∈ A s . We then obtain, using eq. (39) n s [ v s + δw ]( r ) − n s [ v s ]( r ) = Z R χ s [ v s ]( r, r ′ ) δw ( r ′ ) dr ′ − ∆ m ( δw ) ( r ) , (41)which has the form of eq. (38). We thus can deduce that the differential exists and is givenby ∀ v s ∈ V nondegen L : dn s vs ( r ′ ) ( r ) = ∂n s [ v s ]( r ) ∂v s ( r ′ ) = χ s [ v s ]( r ′ , r ) . (42) (cid:4) v s [ n s ] in the non-degenerate case Now, we would like to demonstrate the “reverse”, i.e. the differentiability of v s [ n s ] for all n s ∈ A s , reminding that we can consider n s [ v s ] (as done in previous section) as well as v s [ n s ] (as done in the current section). Theorem 3.6 (Differentiability of v s [ n s ] - restricted case) . v s [ n s ] is differentiable for all n s ∈ A s . Proof (inspired from van Leeuwen (2003)) . We verify is we can recover a form likeeq. (37) for v s [ n s ] when n s ∈ A s . As eq. (39) is invertible (identifying potential perturba-tions defined up to an additive constant), see section 8 of van Leeuwen (2003) for details,we obtain ∀ δm ∈ δ S : δw [ n s , δm ]( r ) = Z R χ − s [ v s [ n s ]]( r, r ′ ) δm ( r ′ ) dr ′ , (43)where χ − s [ v s [ n s ]] is defined through Z R χ − s [ v s [ n s ]]( r, r ′ ) χ s [ v s [ n s ]]( r ′ , r ′′ ) dr ′ = δ ( r − r ′′ ) . (44) δm ∈ δ S must be considered in eq. (43) for the inverse to make sense. The resulting δw [ n s , δm ] lives in V L .Again, as n s + δm does not belong to A s for any δm ∈ δ S , we introduce the “smallestcorrection” ∆ m ( δw [ n s + δm ]) ∈ δ S so that n s + δm +∆ m ( δw [ n s ,δm ]) ∈ A s . The correction termsatisfies lim || δm || p =1 , → ∆ m ( δw [ n s + δm ]) ( r ) / || δm || p =1 , = 0 and necessarilly imply v s (cid:2) n s + We can prove that δ S is a Banach space, i.e. has a vector space structure, unlike S , eq. (7). This isbecause linear combinations of density perturbations in δ S still integrate to 0. m + ∆ m ( δw [ n s ,δm ]) (cid:3) = v s (cid:2) n s (cid:3) + δw (cid:2) n s , δm (cid:3) + o ( || δm || p =1 , ). Using eq. (43), we obtain forall δm ∈ δ S : v s (cid:2) n s + δm + ∆ m ( δw [ n s ,δm ]) (cid:3) ( r ) − v s (cid:2) n s (cid:3) ( r ) = Z R χ − s [ v s [ n s ]]( r, r ′ ) δm ( r ′ ) dr ′ + o ( || δm || p =1 , ) . (45)We have not yet obtained a form similar to the differential definition, eq. (37). Indeed: • n s lives in a set, A s , that is not a vector space. Even if A s is provided with a norm, || . || p =1 , , it thus does not represent a Banach space. However, we can consider any n s ∈ A s and n s + δm + ∆ m ( δw [ n s ,δm ]) ∈ A s as elements of the Banach space N L provided with the || . || p =1 , norm, defined in § A s is a subset of N L . Also, the δm and ∆ m ( δw [ n s ,δm ]) terms live in the Banach space δ S provided with the || . || p =1 , norm. Thus, for any element δm in the Banach space δ S , n s + δm + ∆ m ( δw [ n s ,δm ]) belongs to another Banach space S provided with the same norm. This is sufficientto consider that eq. (45) is compliant with the differential definition from the domainspoint of view. • Another difference is that a correction term ∆ m ( δw [ n s ,δm ]) appears inside the first v s term in eq. (45). However, the correction term becomes negligible compared to δm at the limit where || δm || p =1 , →
0, so that we recover a relation similar to eq. (37)for ǫ sufficiently small.This allows to establish the relation between eq. (45) and the differential definition, eq. (37),and thus deduce dv s ns ( r ′ ) ( r ) = ∂v s [ n s ]( r ) ∂n s ( r ′ ) = χ − s [ v s [ n s ]]( r, r ′ ) . (46) (cid:4) v s [ n s ] in the general (possibly degenerate) case Theorem 3.7 (Differentiability of v s [ n s ]) . Theorem 3.6, can be extended to all n s ∈ B s . Inthe degenerate non-interacting system case, i.e. when n s ∈ B s \ A s , at least one element ofthe equivalence class of the degenerate ground state densities is differentiable; the derivativeof this element is chosen to represent the differential of v s [ n s ] . Proof (together with Appendix A.2.2) . If an equation like eq. (43) can be establishedalso in the degenerate case, i.e. for n s ∈ B s , then the considerations of § δw ∈ V L that represents a first order change in v s and the corresponding firstorder change δm ∈ δ S in n s : δm [ v s , δw ]( r ) = Z R χ s [ v s ]( r ′ , r ) δw ( r ′ ) dr ′ + Z R Z R ξ s [ v s , δw ]( r, r ′′ , r ′ ) δw ( r ′′ ) δw ( r ′ ) dr ′′ dr ′ χ s [ v s ]( r ′ , r ) = q s X k =1 λ k × χ s k [ v s ]( r ′ , r ) ξ s [ v s , δw ]( r ′′ , r, r ′ ) = q s X k =1 λ k × ξ s k [ v s , δw ]( r ′′ , r, r ′ ) δm [ v s , δw ]( r ) = q s X k =1 λ k × δm k [ v s , δw ]( r ) , (47)12here all terms that appear in this equation are detailed in Appendix A.2.1. The importantpoints are, using similar notations than the ones in eq. (19), that λ k denotes the weightsassociated to each degenerate ground states | ψ s k i of ˆ H v , chosen to be orthonormal eigen-states (for instance Slater determinants in the case of fermions), and that q s denotes thedegeneracy. Each δm k [ v s , δw ]( r ) represents a first order change in the k th eigenstate density n s k ( r ) = h ψ s k | ˆ n ( r ) | ψ s k i , so that δm [ v s , δw ]( r ) represents a first order change in the totaldensity n ( r ) = P q s k =1 λ k n k ( r ).One difficulty with the degenerate case, eq. (47), is that δm does not depend linearly onthe perturbation δw , unlike in the non-degenerate case, eq. (39). A second difficulty is thatan explicit form for the inverse of eq. (47) is missing. A third difficulty is that ξ s [ v s , δw ]depends on δw as explained in Appendix A.2.1. All this prevents to straightforwardlygeneralize the considerations of § λ k that always nullifies the ξ s [ v s , δw ] term in eq. (47).This particular choice is λ k = 1 /q s , implying that each degenerate eigenstate | ψ s k i is equallyrepresented in n . Thus, this choice allows to recover an equation of the form of eq. (43). Theresulting δm [ v s , δw ] satisfies R R δm [ v s , δw ]( r ) dr = 0, see Appendix A.2.2, which is physical.As the obtained equation is invertible (identifying potential perturbations defined up to anadditive constant), we then straightforwardly can generalize the differential considerationsof § λ k = 1 /q s specific choice, we canextend the differentiability of v s [ n s ] to n s ∈ B s .Now, does this choice make sense? As explained in § (cid:4) v s [ n s ] is bounded (proof) It now remains to show that the derivative of v s [ n s ] as defined in Theorem 3.7 is boundedfor n s ∈ B s to achieve the proof of the non-interacting v -representability conjecture. Theorem 3.8 (Boundedness of the derivative of v s [ n s ]) . The derivative of v s [ n s ] as definedin Theorem 3.7 is bounded: ∃ K ∈ R + , ∀ n s ∈ B s : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂v s [ n s ]( r ) ∂n s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q = ∞ < K. (48) Proof (together with Appendix A.2.3) . Equivalently to eq. (48), we demonstrate ∃ K ∈ R + , ∀ n s ∈ B s , ∀ r ′ ∈ R : (cid:12)(cid:12)(cid:12) ∂v s [ n s ]( r ) ∂n s ( r ′ ) (cid:12)(cid:12)(cid:12) < K. (49)Appendix A.2.3 shows that the kernel − χ s [ v s [ n s ]] is real and symmetric, and has real posi-tive eigenvalues α j such that ∃ ǫ > α j > ǫ . As a consequence, the kernel − χ − s [ v s [ n s ]] hasreal eigenvalues 1 /α j such that 1 /α j < /ǫ , thus has eigenvalues bounded by K = 1 /ǫ ∈ R + .This implies that the representation of − χ − s [ v s [ n s ]] in the eigenvector basis is bounded,so as its representations in any other basis. The derivative of v s [ n s ] as defined in Theo-rem 3.7 being equal to χ − s [ v s [ n s ]], we deduce that (cid:12)(cid:12) ∂v s [ n s ]( r ) /∂n s ( r ′ ) (cid:12)(cid:12) = (cid:12)(cid:12) χ − s [ v s [ n s ]]( r, r ′ ) (cid:12)(cid:12) is bounded. (cid:4) . Conclusion We have demonstrated the validity of the non-interacting v -representability conjecture,using the Lieb functional and considering possibly degenerate non-interacting systems. Tothat end, we firstly reduced the proof of the non-interacting v -representability conjectureto proving (1) that the KS potential is differentiable for all non-interacting v -representabledensities and (2) that its derivative is bounded. Then, we demonstrated points (1) and (2)applying static linear response considerations to the non-interacting systems. A subtletyregarding point (1) occurs in the degenerate non-interacting system case, which requires tointroduce an equivalence class for degenerate ground state densities. We demonstrated thatat least one element of this class is differentiable, the derivative of this element representingthe “derivative of the equivalence class”. This is sufficient to conclude on the non-interacting v -representability conjecture and complete to establish that a KS scheme can theoreticallybe set up to exactly reproduce any interacting v -representable density (with a KS potentialthat would not change very rapidly, oscillate... approaching the limit).We conclude by mentioning that the considerations in this article may be slightly adaptedto give a solid ground to the stationary “internal DFT formalism” (Messud et al., 2009;Messud, 2011), that deals with self-bound many-particles systems (like nucleons in a nu-cleus). Details are left for a future work. 14 ppendixA. Static linear response for a non-interacting system A.1. Non-degenerate case reminders
We use the notations of § H s , eq. (15). The static linear response of the correspond-ing Schr¨odinger equation gives the relationship in eq. (39), as detailed in section 7 ofvan Leeuwen (2003). The “static density response function” is defined by (“c.c.” denotesthe complex conjugate) χ s [ v s ]( r ′ , r ) = − ∞ X i =2 h ψ (1) s | ˆ n ( r ′ ) | ψ ( i ) s ih ψ ( i ) s | ˆ n ( r ) | ψ (1) s i E s i − E s + c.c.. (A.1) {| ψ ( i ) s i , i ≥ } denotes the set of orthonormal eigenstates of ˆ H s (for instance Slater determi-nants in the case of fermions), which are obviously functionals of the potential v s (implicitin the following). The ground state is | ψ (1) s i , which equals | ψ s i in the notations of §
2. Theground state density is n ( r ) = h ψ (1) s | ˆ n ( r ) | ψ (1) s i . E s i = h ψ ( i ) s | ˆ H s | ψ ( i ) s i denotes the energyassociated to the eigenstate i , E s being the ground state energy.Because h ψ ( k ) s | ψ ( l ) s i = δ kl , we deduce from eq. (A.1) that R R χ s [ v s ]( r ′ , r ) dr = 0. Thus,the δm [ v s , δw ]( r ) term in eq. (39) satisfies R R δm [ v s , δw ]( r ) dr = 0. A.2. Degenerate case
A.2.1. Reminders
We consider a degenerate non-interacting system described by the Hamiltonian ˆ H s , eq. (15). {| ψ ( i ) s i , i ≥ } still denotes the set of orthonormal eigenstates of ˆ H s (for instance Slaterdeterminants in the case of fermions). However, the first eigenstates are now degenerate for i = 1 , ..., q s . Using the letter k to index these eigenstates (for coherency with the notationsof § ∀ k ∈ { , ..., q s } : h ψ ( k ) s | ˆ H s | ψ ( k ) s i = E s . The density associated to theeigenstate k is n s k ( r ) = h ψ ( k ) s | ˆ n ( r ) | ψ ( k ) s i . As explained in § v s and the class of ground state densities n n s (cid:12)(cid:12)(cid:12) n s = q s X k =1 λ k × n s k ( r ) , λ ∗ k = λ k ≥ , q s X k =1 λ k = 1 o . (A.2)As detailed in section 11 of van Leeuwen (2003), the perturbation theory applied to thecorresponding Schr¨odinger equation gives for k ∈ { , ..., q s } : δm k [ v s , δw ]( r ) = Z R χ s k [ v s ]( r ′ , r ) δw ( r ′ ) dr ′ + Z R Z R ξ s k [ v s , δw ]( r, r ′′ , r ′ ) δw ( r ′′ ) δw ( r ′ ) dr ′′ dr ′ , (A.3)where χ s k [ v s ]( r ′ , r ) = − ∞ X i = q s +1 h ψ ( k ) s | ˆ n ( r ′ ) | ψ ( i ) s ih ψ ( i ) s | ˆ n ( r ) | ψ ( k ) s i E s i − E s + c.c. (A.4) ξ s k [ v s , δw ]( r, r ′′ , r ′ ) = q s X l =1 l = k ∞ X i = q +1 h ψ ( k ) s | ˆ n ( r ) | ψ ( l ) s ih ψ ( l ) s | ˆ n ( r ′′ ) | ψ ( i ) s ih ψ ( i ) s | ˆ n ( r ′ ) | ψ ( k ) s i ( E ′ s k − E ′ s l )( E s i − E s ) + c.c.. ∀ k ∈ { , ..., q s } : E s k ( ǫδw ) = h ψ ( k ) s [ v s + ǫδw ] | ˆ H s + ǫδw | ψ ( k ) s [ v s + ǫδw ] i , whichsatisfies E s k = lim ǫ → E s k ( ǫδw ), E ′ s k is defined by E ′ s k = lim ǫ → ∂E s k ( ǫδw ) /∂ǫ . E s k isindependent on the choice of δw (the ground state energy remains the same whatever thepath used to approach it), i.e. is a functional of v s only. E ′ s k is however dependent on thechoice of δw (the variation of the ground state energy can depend on the path used), i.e.is a functional of v s and δw . Thus, ξ s k is also a functional of v s and δw .Because h ψ ( k ) s | ψ ( l ) s i = δ kl , we deduce from eq. (A.4) that R R χ s k [ v s ]( r ′ , r ) dr = R R ξ s k [ v s ]( r, r ′′ , r ′ ) dr =0. As a consequence, the δm k [ v s , δw ] in eq. (A.3) satisfies R R δm k [ v s , δw ]( r ) dr = 0. Astronger property that can be demonstrated is that each of the two terms that compose δm k [ v s , δw ] in eq. (A.3) separately integrate to 0.From eq. (A.3), we deduce that any degenerate ground state density first order pertur-bation δm [ v s , δw ]( r ) = P q s k =1 λ k × δm k [ v s , δw ]( r ) satisfies eq. (47). A.2.2. Recovering a linear dependency (proof)
We demonstrate that there exists a particular choice for the λ k that always nullifies the ξ s [ v s ] term in eq. (47) or equivalently the ξ s k [ v s ] terms in eq. (A.3). Defining N kl ( r ) = h ψ ( k ) s | ˆ n ( r ) | ψ ( l ) s i ⇒ N lk ( r ) = N ∗ kl ( r ) (A.5) W li = Z R h ψ ( l ) s | ˆ n ( r ′′ ) | ψ ( i ) s i δw ( r ′′ ) dr ′′ ⇒ W il = W ∗ li , we compute Z R Z R ξ s [ v s , δw ]( r, r ′′ , r ′ ) δw ( r ′′ ) δw ( r ′ ) dr ′′ dr ′ (A.6)= q s X k =1 λ k q s X l =1 l = k ∞ X i = q s +1 N kl ( r ) W li W ik ( E ′ s k − E ′ s l )( E s i − E s ) + c.c. = ∞ X i = q s +1 E s i − E s q s X k =1 q s X l =1 l = k λ k N kl ( r ) W li W ik E ′ s k − E ′ s l + c.c. = ∞ X i = q s +1 E s i − E s q s X k =1 q s X l>k (cid:16) λ k N kl ( r ) W li W ik E ′ s k − E ′ s l + λ l N lk ( r ) W ki W il E ′ s l − E ′ s k (cid:17) + c.c. = ∞ X i = q s +1 E s i − E s q s X k =1 q s X l>k ( λ k − λ l ) N kl ( r ) W li W ik E ′ s k − E ′ s l + c.c.. If we do the particular choice of equal λ k ’s, which implies λ k = 1 /q s , eq. (A.6) alwaysequals to zero. This choice allows to neglect the second term in eq. (A.3) and thus ineq. (47). Because of aforementioned properties, the resulting δm [ v s , δw ] would still satisfy R R δm k [ v s , δw ]( r ) dr = 0, which makes sense. A.2.3. Bounded eigenvalues of χ − s [ v s [ n s ]] (proof) − χ s [ v s [ n s ]] represents a real and symmetric kernel. It has real eigenvalues α j related toorthonormal eigenvectors f j ( r ), satisfying (eq. (A.4) is used to deduce the second line) − Z R χ s [ v s [ n s ]]( r, r ′ ) f j ( r ′ ) dr ′ = α j f j ( r ) (A.7) α j = 2 q s X k =1 ∞ X i = q s +1 λ k (cid:12)(cid:12) R R h ψ ( k ) s | ˆ n ( r ) | ψ ( i ) s i f j ( r ) dr (cid:12)(cid:12) E s i − E s .
16e must have α j > α j would imply a constant f j ( r ), which is not possiblebecause of the orthonormalization constraint on the f j ( r ). We recover that χ s [ v s [ n s ]] isinvertible. Using eq. (44), we have − Z R χ − s [ v s [ n s ]]( r ′ , r ) f j ( r ) dr = 1 α j f j ( r ′ ) , (A.8)i.e. the real and symmetric kernel − χ − s [ v s [ n s ]] has real positive eigenvalues 1 /α j related tothe orthonormal eigenvectors f j ( r ). We now study if the 1 /α j are bounded, which wouldbe the case if ∃ ǫ > α j > ǫ ⇒ α j < ǫ . (A.9)Eq. (A.9) would be satisfied if the α j cannot become arbitrarily close to 0; i.e., fromeq. (A.7), if the R R h ψ ( k ) s | ˆ n ( r ) | ψ ( i ) s i f j ( r ) dr cannot become arbitrarily close to 0 for k ∈{ , ..., q s } and i ≥ q s + 1; i.e. if the f j ( r ) cannot become arbitrarily close to a constant. Thelatter is prevented again by the orthonormalization constraint, which achieves to demon-strate that the 1 /α j are bounded. 17 eferences Brezis, H. (2020).
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