Theor Chem Account (2007) 118:557–561 DOI 10.1007/s00214-007-0367-6 REGULAR ARTICLE A re-statement of the Hohenberg–Kohn theorem and its extension to finite subspaces Ramiro Pino · Olivier Bokanowski · Eduardo V. Ludeña · Roberto López Boada Received: 18 December 2006 / Accepted: 19 January 2007 / Published online: 27 June 2007 © Springer-Verlag 2007 Abstract Bearing in mind the insight into the Hohenberg– Kohn theorem for Coulomb systems provided recently by Kryachko (Int J Quantum Chem 103:818, 2005), we present a re-statement of this theorem through an elaboration on Lieb’s proof as well as an extension of this theorem to finite subspaces. Keywords Hohenberg–Kohn theorem · DFT · Finite subspaces 1 Introduction Density functional theory, DFT [1–12] has become a basic tool in contemporary quantum chemistry [13–15] but, as shown some decades ago by Lieb [16] and more recently Contribution to the Serafin Fraga Memorial Issue. R. Pino North American Scientific, Nomos Radiation Oncology, 200 W Kensinger Dr., Cranberry Township, PA 16066, USA e-mail: ramiropino@yahoo.com O. Bokanowski Lab. J.-L. Lions, Université Pierre et Marie Curie, 175 R. Chevralet, 75013 Paris, France e-mail: boka@math.jussieu.fr E. V. Ludeña (B) Química, IVIC, Apdo. 21827, Caracas 1020-A, Venezuela e-mail: eludena@ivic.ve R. L. Boada Miami-Dade College Wolfson Campus, Department of Natural Sciences, Health and Wellness, 300 NE 2nd Ave., Miami, FL 33132, USA e-mail: rlopezbo@mdc.edu by several authors [17–23] due to its subtleties, this theory cannot be considered as yet to be entirely elaborated. The Hohenberg–Kohn theorem [24,25] has played a fundamental role in the development of DFT. In a recent work, however, Kryachko [26] has pointed out that the usual reductio ad absurdum proof of this theorem is unsatisfactory since the would-be-refuted assumptions on the one-electron density and the assumption on the external potential evince incompatibilities with the Kato cusp condition. Nevertheless, as shown by Kryachko [26], application of the Kato cusp conditions actually leads to a satisfactory proof of this theorem. In the present work, within the context of Kryachko’s analysis, we advance an alternative proof of the Hohenberg–Kohn theorem, which is based on the rigorous examination of the original formulation of this theorem made by Lieb [16], a number of years ago. In Lieb’s proof, it is required that the N -particle wavefunction Ψ not vanish in a set of positive measure. This condition, however, cannot be easily fulfilled. In order to avoid this difficulty we present below an essentially algebraic proof of the Hohenberg–Kohn theorem which dispenses with the latter condition. In addition, we propose an extension of the present reformulation of the Hohenberg–Kohn theorem to the case of finite subspaces. This finite subspace problem has been treated in a restricted sense by Epstein and Rosenthal [27] and by Katriel et al. [28,29] and in a general sense by Harriman [30]. More recently, Görling and Ernzerhof have reexamined this problem in relation to the linear response method to determine Kohn–Sham orbitals (and, purportedly, Kohn–Sham wavefunctions; strictly speaking, it is not possible to attach a rigorous meaning to Kohn–Sham wavefunctions as through the application of the variational principle there only result Kohn–Sham single-particle equations and their corresponding single-particle orbitals) from electron densities [31]. 123 558 Theor Chem Account (2007) 118:557–561 In order to set the proper background for our discussion, we review in Sect. 1 both the original proof given by Hohenberg and Kohn, in the context of Kryachko’s work, as well as Lieb’s reformulation. In Sect. 2 we discuss the modifications introduced in our present proof. In Sect. 3, we consider the conditions that must be fulfilled in order that this theorem be extended to finite subspaces. 2 The original Hohenberg–Kohn proof and Lieb’s reformulation Let us consider a system formed by N -electrons interacting with a positive background through an “external" potential V (r1 , . . . , r N ) = N
(1) v(ri ). i=1 The many-electron Hamiltonian for such a system is v = H o + V  H (2) o is defined by where H o = − 1 H 2
∇r2i + N −1
N
i=1 j=i+1 1 . |ri − r j | (3) It is assumed that the selected class {v(r)} of single-particle external potentials is such that it possesses a ground-state wavefunction {Ψov }. The one-electron density ρov (r) associated with Ψov is defined by   v 3 ρo (r1 ) = N d r2 · · · d 3 r N |Ψov (r1 , . . . , r N )|2 . (4) For such systems, the Hohenberg–Kohn theorem states that there exists a one to one correspondence between an external potential v(r) and the exact ground-state density ρov (r). The original proof of this theorem [24] is carried out by reductio ad absurdum. Consider two potentials v(r) and v ′ (r) differing by more than a constant. The exact groundv state wavefunctions for the corresponding Hamiltonians H v ′ ) are assumed to be different (actually, these assump(or H tions immediately evoke the Kato theorem and show the way to a proof that dispenses with the reductio ad absurdum argument) and for this reason the following strict variational inequalities hold: ′ ′ v |Ψov  > Ψov | H v |Ψov  ≡ E ov Ψov | H (5) v ′ |Ψov ′  ≡ E ov ′ . v ′ |Ψov  > Ψov ′ | H Ψov | H (6) and Adding these inequalities and carrying out the integration over all coordinates but one, one obtains  ′ (7) d 3 r(v ′ (r) − v(r))(ρ0v (r) − ρ0v (r)) < 0 . 123 Because Eq. (7) is a strict inequality, a contradiction ensues (0 < 0) when it is assumed that different potentials yield the same one-particle density. Thus, it follows that there is a one to one correspondence between the exact ground-state oneparticle densities and their corresponding external potentials. In the present notation, Lieb’s statement of this theorem (Theorem 3.2 of Ref. [16]) is the following: suppose Ψov ′ (respectively, Ψov ) is a ground state for v (respectively, v ′ ) ′ and v = v ′ + constant. Then ρ0v (r) = ρ0v (r). Lieb’s proof ′ starts from the suppositions that ρ0v (r) = ρ0v (r) = ρ0 and ′ Ψov = Ψov because they satisfy different Schrödinger equations, and proceeds as in the original proof showing that this leads to a contradiction. As it was mentioned above, the argument for writing the strict inequalities [Eqs. (5) and (6)] in Hohenberg–Kohn’s paper [24] is based on the assump′ tion that Ψov and Ψov satisfy different Schrödinger equations, ′ namely, that Ψov = Ψov . The fact that the space of single particle potentials is not specified in the original Hohenberg–Kohn proof was remedied in Lieb’s proof [16] by selecting this space as Y = L 3/2 (R3 ) + L ∞ (R3 ) (where f (x) ∈ L m if d x | f (x)|m < m if f ∈ L m and it is integrable in any bounded ∞. f ∈ L loc set; f ∈ H 1 if f, ∇ f ∈ L 2 ) and by demanding that v(r) ∈ Y . This choice—which follows from the requirement that  ρ 1/2 ∈ H 1 (R3 )—guarantees that the integral d 3 rρ(r)v(r) (in fact, the essentially self-adjoint character of the Hamiltonian [32]) is well defined. An important difference arises, however, from the fact that Lieb notes that in order to prove the statement that Ψov and ′ Ψov satisfy different Schrödinger equations it is necessary to show that the equivalence V (r1 , . . . , r N )Ψ (r1 , . . . , r N ) = V ′ (r1 , . . . , r N )Ψ (r1 , . . . , r N ) implies that v(r) = v ′ (r). Fulfillment of this condition requires that the Ψov corresponding to the external potential v ∈ Y not vanish on a set of positive measure. As has been indicated by Lieb [16] (p. 255), the unique continuation theorem may be invoked to guarantee that Ψov does not vanish in an open set. However, this theorem strictly holds only for v ∈ L 3loc although it is believed to hold also for v ∈ Y . But let us mention that there are subtle problems related to the space to which a single particle potential belongs and to its relation to the wavefunction. Thus, for example, as shown by Englisch and Englisch [33], for a one particle case there exists a non-vanishing density ρ (or equivalently, a non-vanishing wavefunction given as Ψ = ρ 1/2 ) which does not arise from any v, in the sense that for a v = ρ −1/2 ∇ 2 ρ 1/2 , −∇ 2 + v cannot be defined as a semibounded operator. Precisely in order to avoid these difficulties, we advance an algebraic proof of the Hohenberg–Kohn theorem where these issues are avoided. Theor Chem Account (2007) 118:557–561 559 3 A re-statement of the Hohenberg–Kohn theorem The present proof is essentially based on Lieb’s version of the HK theorem (Theorem 3.2 and Remark (ii) in p. 255 of Ref. [16]). But as mentioned above, in order to avoid some mathematical complications, we have, however, removed the ′ assumption that Ψov = Ψov , i.e., we consider the case where ′ v = v ′ + constant but Ψov = Ψov (Case I of Kryachko [26]) and have added the condition on the ground state wavefunco be tion that it vanishes at most on a zero-measure set. Let H the Hamiltonian of an electronic Coulomb system without o is not external potential [cf. Eq. (3)]. In fact, the form of H very important, as the proof is essentially algebraic. Let us v given by Eq. (2). consider the many-electron Hamiltonian H We denote Y as in the above Section. We assume that ρov is v if there exists a ground-state the ground-state density of H v  wavefunction Ψo of Hv . We denote by E ov the corresponding eigenvalue. Theorem 1 (Hohenberg–Kohn) Let v, v ′ be in Y . Let ρov v and ρov ′ a ground state denbe a ground state density of H v ′ . We assume that the ground state wavefunction sity of H v  Ψo of Hv vanishes at most on a Lebesgue’s zero-measure set ′ of R3N . Suppose that ρov = ρov . Then almost everywhere in the Lebesgue’s measure sense (a.e.) ′ v(r) − v ′ (r) = (E ov − E ov )/N . (8) Proof We essentially make explicit what was implicit in ′ Lieb’s proof [16]. Let us introduce the notation ∆E = E ov − N v ′ E o , ∆v = v − v and ∆V = i=1 ∆v(ri ). We have then v = H v ′ − ∆V and H  v |Ψov  ≤ Ψov ′ | H v |Ψov ′  = E ov ′ − ρov ′ ∆v. (9) E ov = Ψov | H where the equal sign must be included as we are not assum′ ing that for v = v ′ + constant the condition Ψov = Ψov holds.  So we get a ≥ 0 where a = ∆E − ρo ∆v, and ρo = ρov = ′ ρov . Reversing v and v ′ we get similarly a ≤ 0. So a = 0 and this implies also that all the preceding inequalities are ′ v |Ψov ′  in fact equalities. In particular, we have E ov = Ψov | H ′ ′ v : H v Ψov = E ov Ψov ′ . In so Ψov is also a ground state of H v Ψov = E o Ψov v ′ Ψov = E ov ′ Ψov . Using also H the same way: H v = ∆V , by subtraction we obtain v ′ − H and H ∆V Ψov = ∆E Ψov . (10) or, equivalently, (∆V − ∆E)Ψov = 0. (11) Since we have by assumption that Ψ vanishes at most on a set of zero measure (we take it to be a nodeless ground state wavefunction) it follows from Eq. (11) that ∆V = ∆E almost everywhere for (r1 , . . . , r N ) ∈ R3N , except for a set of zero measure. Then setting r1 = · · · = r N = r we obtain N ∆v(r) = ∆E (see also Harriman’s comments in p. 641 and in the Appendix of Ref. [30]). The present argument is rigorous provided v is continuous; otherwise, the proof can be completed using Lemma 1 proved in the Appendix. ⊔ ⊓ 4 The Hohenberg–Kohn theorem in finite subspaces We first state a Hohenberg–Kohn theorem that holds in subspaces which are not necessarily finite-dimensional. Theorem 2 (Infinite-dimensional subspaces) Let v, v ′ be in Y . Let F be some subspace of the antisymmetric N -particle v ′ ) such that F be v and H Hilbert space (in the domains of H   v F ⊂ F and ′ stable under the action of Hv and Hv , i.e., ( H v  Hv ′ F ⊂ F). Take ρo a ground state density of the restriction v | F and ρov ′ a ground state density of H v ′ | F . Again, assume H that the ground state wavefunction vanishes at most on a set ′ of zero measure. Suppose that ρov = ρov . Then ′ v(r) − v ′ (r) = (E 0v − E 0v )/N . (12) Proof It is carried out along the same steps as in Theorem 1, ′ except for the fact that Ψov and Ψov must be in F in order to apply the variational principle and obtain a = 0, and, hence, ′ v |Ψov ′  implying that Ψov ′ is a ground state of E ov = Ψov | H v | F . ⊔ ⊓ H We see, therefore, that it is possible to extend the HK formulation of Density Functional Theory to a subspace F as long as the conditions of stability of Theorem 2 are satisfied. However, as shown in Theorem 3 below, it is not possible, in general, to satisfy the assumptions of Theorem 2. First v ′ (F) ⊂ F then by taking the v (F) ⊂ F and H note that if H difference we obtain ∆V (F) ⊂ F. We recall also that the  associated to a scalar potential V is defined by operator V  (V (Ψ ))(x) := V (x)Ψ (x). Theorem 3 (Finite-dimensional subspaces) Let F be a finite-dimensional subspace of L 2 (Rn ) (n ≥ 1). We suppose that F = V ect(u 1 (x), .. . , u M (x)) where the (u i (x)) is an orthonormal set (i.e., u i u ∗j = δi j ) and such that M 2 n i=1 |u i (x)| > 0 a.e. for x ∈ R . Let V (x) be real-valued potential, and continuous. Then (F) ⊂ F) ⇒ (V (x) = const on Rn ). (V We note that this theorem also holds with weaker assumptions, such as, for instance, F ⊂ L 1loc (Rn ) (the space of 1 (Rn ) [i.e. locally integrable functions on Rn ), and V ∈ Hloc 2 n V, ∇V ∈ L loc (R )]. Proof We first remark that V behaves on F as an M × M matrix since it is a linear operator. So there exists M = (m i j ) such that 123 560 Theor Chem Account (2007) 118:557–561 V (x)u i (x) =
m i j u j (x). (13) j=1,...,M Since u j is orthonormal, we have  dxu i (x)∗ V (x)u j (x) mi j = 6 Appendix Rn using (13). Since V is real we obtain m i j = m ∗ji and thus M is an hermitian matrix. So, we can diagonalize M in an orthonormal basis: there exists a unitary matrix P (P † P = P P † = Id ) and a diagonal matrix D = diag(λ1 , . . . , λ M ) such that M = P † D P. Let us write u(x) = (u 1 (x), . . . , u M (x)). Then (13) reads V (x)u = Mu. So, it follows that V (x)Pu = P V (x)u = PMu = P P † D Pu = D Pu. Hence if we define ψ(x) = Pu and denote its components as (ψ0 (x), . . . , ψ M (x)), we obtain: V (x)ψi (x) = λi ψi (x), i = 1, . . . , M. (14) We have simply diagonalized V (x) in an orthonormal basis M 2 2 set. Then let us notice that i=1 |ψi (x)| = ||ψ|| =  M 2 2 ||u|| = i=1 |u i (x)| since P is unitary. Obviously this quantity is non-negative and thus we have a.e. x ∈ R n the existence of an i ∈ {1, . . . , M} such that ψi (x) = 0. From (14) we obtain V (x) = λi for this x. This implies finally that the range of V is included in the finite set {λ1 , . . . , λ M }. For 1 this means that V a regular V (x) such as continuous or Hloc ⊔ is a constant, which concludes the proof of Theorem 3. ⊓ A consequence of Theorem 3 is that, in general, it is not possible to fulfill the stability conditions of Theorem 2 when F is finite dimensional, except if we suppose that V (x) and V ′ (x) are constants as then they would trivially satisfy the main conclusion of Theorem 2, namely, ∆V = const. Let us mention that this result is in agreement with the conclusion of Görling and Ernzerhof for local potentials in finite subspaces [see Eq. (A9) and the discussion below in Ref. [31]]. But in the infinite dimensional case Theorem 3 does not hold and thus Theorem 2 becomes interesting. As an example, let F = L 2 (R3 ) and v(x) = 1/(1 + |x|2 ). Then we have obviously v(F) ⊂ F (since v(x) ≤ 1) but v(x) is not constant. 5 Conclusions The main contribution of this article is to provide an algebraic proof for the Hohenberg–Kohn theorem that allows us to discuss in a very simple way the extensions of this theorem to both infinite-dimensional and finite-dimensional subspaces. In the former case, such an extension is possible as long as the v . In the latter case, subspace is stable under the action of H ′ when the external potentials V and V , or their one-particle components v and v ′ are constants. 123 Acknowledgments E.V.L. would like to express his gratitude to FONACIT of Venezuela, for its support of the present work through Project G-97000741. Lemma 1 Suppose that v(r1 ) + · · · + v(r N ) = 0, a.e. x = (r1 , . . . , r N ) ∈ R3N . (15) Then v(r1 ) = 0 a.e. r1 ∈ R3 . Proof First note that we cannot (a priori) take r1 = r2 = · · · = r N because {(r1 , . . . , r N ) ∈ R3N , r1 = r2 = · · · = r N } is a set of zero measure in R3N . To bypass this difficulty, we consider a real-valued  continuous function ρ(r ) > 0, 3 , such that ρ(r )dr = 1, and denote ρ ǫ (x) = defined on R   1 ρ xǫ . We multiply Eq. (15) by ρ ǫ (x1 −r1 ) · · · ρ ǫ (x N −r N ) ǫ3 and then integrate over (r1 , . . . , r N ) ∈ R3N . We obtain (16) v ǫ (r1 ) + · · · + v ǫ (r N ) = 0, a.e.,  where v ǫ (x) = R3 v(y)ρ ǫ (x − y)dy (convolution product). Then it is well known [34] that v ǫ is a continuous function and thus Eq. (16) holds everywhere and not only almost everywhere. Then we can take r1 = · · · = r N = r and conclude that v ǫ (r ) = 0 for all r . On the other hand it is also well known that, as ǫ → 0+ v ǫ (r ) → v(r ) for a.e. r ∈ R3 (eventually for some subsequence v ǫn extracted from v ǫ , [34]). ⊔ ⊓ Hence we conclude that v(r ) = 0 a.e. r ∈ R3 . References 1. Parr RG, Yang W (1989) Density functional theory of atoms and molecules. Oxford University Press, Oxford 2. Dreizler RM, Gross EKU (1990) Density functional theory. Springer, Berlin 3. Kryachko ES, Ludeña EV (1990) Energy density functional theory of many electron systems. Kluwer, Dordrecht 4. March NH (1992) Electron density theory of atoms and molecules. Academic, New York 5. Cioslowski J (ed) (2000) Many-electron densities and reduced density matrices. Kluwer Academic/Plenum Publishers, New York 6. Gross EKU, Dreizler RM (eds) (1995) Density functional theory. NATO ASI Series, vol B337. Plenum, New York 7. Seminario JM, Politzer P (eds) (1995) Modern density functional theory: a tool for chemistry. Elsevier, Amsterdam 8. Dobson JF, Vignale G, Das MP, Electronic density functional theory. Recent progress and new directions. Plenum Press, New York 9. Chong DP (ed) (1995) Recent advances in density functional methods. World Scientific, Singapore 10. Geerlings P, de Proft F, Langenaeker W (eds) (1999) Density functional theory. A bridge between chemistry and physics. VUB University Press, Brussels 11. Nalewajski RF (ed) (1996) Density functional theory In: Topics in current chemistry, vols 180–183. Springer, Berlin Theor Chem Account (2007) 118:557–561 12. Sen KD (ed) (2002) Reviews in modern quantum chemistry: a celebration of the contribution of Robert G. Parr. World Scientific, Singapore 13. Koch W, Holthausen MC, A chemist’s guide to density functional theory. Wiley-VCH, Weinheim 14. Kohanoff J, Gidopoulos NI (2003) In: Handbook of molecular physics and quantum chemistry, vol. 2. Molecular electronic structure. Wiley, Chichester 15. Scuseria GE, Staroverov VN (2005) Ch. 12 In: Dykstra CE, Frenking G, Kim KS, Scuseria GE (eds) Theory and application of computational chemistry: the first 40 years (A volume of technical and historical perspectives). Elsevier, Amsterdam 16. Lieb EH (1983) Int J Quantum Chem 24:243 17. Eschrig H (1996) The fundamentals of density functional theory. Teubner, Sttutgart. Section 6.3 18. van Leeuwen R (2003) Adv Quantum Chem 43:24 19. Ludeña EV (2004) J Mol Struct (Theochem) 709:25 20. Ayers PW, Levy M (2005) J Chem Sci 117:507 561 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. Görling A (2005) J Chem Phys 123:062203 Ayers PW, Golden S, Levy M (2006) J Chem Phys 124:054101 Kutzelnigg W (2006) J Mol Struct (Theochem) 768:163 Hohenberg P, Kohn W (1964) Phys Rev 136B:864 Hohenberg P, Kohn W, Sham LJ (1990) Adv Quantum Chem 21:7 Kryachko ES (2005) Int J Quantum Chem 103:818 Epstein ST, Rosenthal CM (1976) J Chem Phys 64:247 Katriel J, Appellof CJ, Davidson ER (1981) Int J Quantum Chem 19:293 Meron E, Katriel J (1977) Phys Lett 61A:19 Harriman JE (1983) Phys Rev A 27:632 Görling A, Ernzerhof M (1995) Phys Rev A 51:4501 Reed M, Simon B (1975) Methods of modern mathematical physics II. Academic, New York Englisch H, Englisch R (1983) Physica 121A:253 Rudin W (1987) Real and complex analysis 3rd edn. McGraw– Hill, New York 123