The Role of the Hohenberg Kohn Theorem in Density-Functional Theory

Transcription

1 The Role of the Hohenberg Kohn Theorem in Density-Functional Theory T. Helgaker, U. E. Ekström, S. Kvaal, E. Sagvolden, A. M. Teale,, E. Tellgren Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, Norway School of Chemistry, University of Nottingham, Nottingham, UK STCP X The 9th Congress of the nternational Society of Theoretical Chemical Physics Alerus Center, Grand Forks, North Dakota, USA 7 July 06 T. Helgaker (CTCC, University of Oslo) Role of Hohenberg Kohn Theorem in DFT 0 July 06 / 8

2 Hohenberg Kohn theorem The Hamiltonian of an electronic system can be written in the general form H(v) = T + W + v(r i ) i kinetic T and electron-repulsion W operators are the same for all N-electron systems the external potential v varies from system to system The ground-state energy for a given v is obtained by solving the Schrödinger equation: H(v)Ψ(x i ) = E(v)Ψ(x i ) a complicated problem: Ψ(x i ) depends on the coordinates x i = (r i, σ i ) of all electrons the associated ground-state density is much simpler, depending only on three coordinates: ρ(r) = Ψ(r, σ, x,... x N ) dσdx dx N The Hohenberg Kohn theorem (964) for ground-state densities Thus v(r) is (to within a constant) a unique functional of ρ(r); since, in turn, v(r) fixes H(v) we see that the full many-body ground state is a unique functional of ρ(r). ρ v ρ Ψ ρ Hohenberg Kohn mapping ground-state densities are said to be v-representable the proof is elementary (apart from some subtleties) the result is perhaps difficult to grasp intuitively T. Helgaker (CTCC, University of Oslo) Role of Hohenberg Kohn Theorem in DFT 0 July 06 / 8

3 Hohenberg Kohn universal density functional For a v-representable density ρ v ρ, the Hohenberg Kohn functional is defined as F (ρ) = E(v ρ) (v ρ ρ), (v ρ ρ) = v ρ(r)ρ(r) dr For each v v ρ that supports a ground state, HK established the strict inequality F (ρ) > E(v) (v ρ) The Hohenberg Kohn variation principle (964) now follows directly: E(v) = min ρ (F (ρ) + (v ρ)) caveat: only potentials v that support ground states are allowed caveat: only variations over v-representable densities ρ are allowed Under the same restrictions, the Lieb variation principle (983) also follows: F (ρ) = max v (E(v) (v ρ)) this alternative definition of F was not considered by Hohenberg and Kohn The v-representability problem of Hohenberg Kohn theory: the set of v-representable densities and the set of potentials with ground states are unknown we have no optimality conditions except by the HK mapping ρ v ρ it is our purpose to explore the optimality conditions in more detail T. Helgaker (CTCC, University of Oslo) Role of Hohenberg Kohn Theorem in DFT 0 July 06 3 / 8

4 Levy Lieb constrained-search functional The v-representability problem solved with the Levy Lieb constrained-search functional: F LL (ρ) = min Ψ T + W Ψ Ψ ρ F LL (ρ) well defined for each ρ 0 with a finite von Weizsäcker kinetic energy: ρ / (r) dr < + each such N-representable density comes from a wave function with finite kinetic energy We may now search over all N-representable densities in the HK variation principle: E(v) = inf ρ N (F LL (ρ) + (v ρ)) the set of N-representable densities N is explicitly known if v does not support a ground state, then E(v) is well defined as a greatest lower bound Regarding the optimality conditions, Levy writes (979): One can now confidently use existing Euler equations without being concerned about whether or not the functions in the immediate neighbourhood of the optimum functions are v-representable. δf LL (ρ) δρ(r) = v(r) + c, c R The Euler equations provide the desired HK mapping but is F LL differentiable? T. Helgaker (CTCC, University of Oslo) Role of Hohenberg Kohn Theorem in DFT 0 July 06 4 / 8

5 Discontinuity of universal density functional For a one-electron system, the universal density functional has a simple explicit form: F (ρ) = ρ / (r) dr one-electron kinetic energy A one-electron Gaussian density of unit exponent has a finite kinetic energy: ρ(r) = π 3/ exp ( r ), F (ρ) = 3/4 Let {ρ n} be a sequence that approaches ρ in the norm, lim ρ ρn p = 0, n + while developing increasingly rapid oscillations of increasingly small amplitude: The kinetic energy F (ρ n) is driven arbitrarily high in the sequence and F is not continuous: ) lim F (ρ n n) = + F (lim ρ n = 3/4 n The universal density functional is everywhere discontinuous and hence nondifferentiable P. E. Lammert, nt. J. Quantum Chem. 07, 943 (007) T. Helgaker (CTCC, University of Oslo) Role of Hohenberg Kohn Theorem in DFT 0 July 06 5 / 8

6 HK and Lieb variation principles extended to vector spaces Let us return to the Hohenberg Kohn and Lieb variation principles introduced above: E (v ) = minρ (F (ρ) + (v ρ)) Hohenberg Kohn F (ρ) = maxv (E (v ) (v ρ)) Lieb restricted to ground-state densities and to potentials with ground states These may be extended directly to full vector spaces for densities and potentials E (v ) = inf (F (ρ) + (v ρ)) Hohenberg Kohn F (ρ) = sup (E (v ) (v ρ)) Lieb (983) ρ X v X X = L3 L includes all N-representable densities X = L3/ + L includes all Coulomb potentials solutions are found only when v supports a ground state and ρ is v -representable, respectively if not, E (v ) and F (ρ) are well defined as greatest lower bound and least upper bound, respectively F is equal to FLL when the constrained search is extended to ensembles n the language of convex analysis, E and F are said to be conjugate functions E (concave) F (convex) such conjugate functions contain the same information, represented in different manners each property of one function is exactly reflected in some property of its conjugate function concavity and continuity of E guarantee the existence of F, making DFT possible T. Helgaker (CTCC, University of Oslo) Role of Hohenberg Kohn Theorem in DFT 0 July 06 6 / 8

7 E and F as conjugate functions The energy E and universal density functional F are conjugate functions: E(v) = inf ρ ( F (ρ) + (v ρ) ) F (ρ) = supv ( E(v) (v ρ) ) Assuming differentiability, we obtain the equivalent stationary conditions E (v) = ρ F (ρ) = v Hohenberg Kohn theorem hence E and F are each other s inverse functions: F = (E ) being invertible, E and F must be strictly monotonic functions we therefore have E < 0 (strictly concave) and F > 0 (strictly convex) Graphical illustration of conjugation: differentiate invert integrate E '(v) F(ρ) F '(ρ) - E(v) concavity of E and convexity of F are essential features of these functions but something is wrong: E(v) is negative for v > 0 and F (ρ) is finite for ρ < 0 in reality, neither E nor F is differentiable T. Helgaker (CTCC, University of Oslo) Role of Hohenberg Kohn Theorem in DFT 0 July 06 7 / 8

8 Concavity of the ground-state energy The ground-state energy is concave in the external potential E(v) = inf Ψ Ψ H(v) Ψ E Ψ 0 H(v ) Ψ 0 B C Ψ H(v ) Ψ A Ψ 0 H(v 0) Ψ 0 v v 0<0 v <0 v=0 The concavity of E(v) may be understood in the following two-step manner: from A to B, the energy increases linearly since H(v) is linear in v and Ψ 0 is fixed from B to C, the energy decreases as the wave function relaxes to the ground state Ψ T. Helgaker (CTCC, University of Oslo) Role of Hohenberg Kohn Theorem in DFT 0 July 06 8 / 8

9 Non-strict concavity of the ground-state energy The concavity of the energy E(v) follows from two circumstances: the linearity of H(v) changes the energy linearly from v 0 to v for fixed Ψ 0 the variation principle lowers the energy from Ψ 0 to Ψ for fixed v We have nonstrict concavity of E(v) when the energy changes linearly with v: the potentials differ by a scalar only (so that the wave function remains constant) E(v + c) = E(v) + Nc, (scalar c) for repulsive potentials (no ground state exists but energy is zero as greatest lower bound) E(v) = inf Ψ H(v) Ψ = 0 (repulsive v 0) Ψ E E E(v-) < 0 v0 v = v0+c v v-<0 v0=0 v+>0 v T. Helgaker (CTCC, University of Oslo) Role of Hohenberg Kohn Theorem in DFT 0 July 06 9 / 8

10 Hohenberg Kohn and Lieb variation principles As a result of nonstrict concavity of E, F becomes nondifferentiable and even discontinuous E '(v) F(ρ) F '(ρ) - E(v) Such behaviour is conveniently described in terms of subdifferentiation a subgradient of f at x is the slope of a supporting line at x the subdifferential f (x) is the set of all subgradients at x f (0) = [, ], f (0) = {0} Relationship to differentiation f is differentiable at x with derivative y if f (x) = {y} and f is continuous T. Helgaker (CTCC, University of Oslo) Role of Hohenberg Kohn Theorem in DFT 0 July 06 0 / 8

11 Optimality conditions of DFT Hohenberg Kohn and Lieb variation principles E(v) = inf ρ ( F (ρ) + (v ρ) ) F (ρ) = supv ( E(v) (v ρ) ) Hohenberg Kohn optimality conditions with the Lieb functional E(v) = F (ρ) + (v ρ) v F (ρ) ρ E(v) The subdifferential F provides the HK mapping from densities to potentials: {, ρ is not v-representable F (ρ) = { v + c c R}, ρ is v-representable (Hohenberg Kohn theorem) F is nonempty on a dense subset of all N-representable densities note: F is everywhere discontinuous and therefore nondifferentiable The subdifferential E provides the inverse mapping from potentials to densities:, v does not support a ground state E(v) = {ρ}, v supports a nondegenerate ground state { N i= λ i ρ i i λ i =, λ i 0}, v supports a N-degenerate ground state E is nonempty on a dense subset of all Coulomb potentials note: E is everywhere continuous and differentiable at nondegenerate ground-state densities There are no lost nor false densities with the Lieb functional S. Kvaal and T. Helgaker, JCP 43, 8406 (05) T. Helgaker (CTCC, University of Oslo) Role of Hohenberg Kohn Theorem in DFT 0 July 06 / 8

12 Moreau Yosida regularization of DFT The ground-state energy E(v) is concave but not strictly concave concave strictly concave We obtain a strictly concave energy E γ by subtracting a term proportional to v : E γ(v) = E(v) γ v, γ > 0 such strict concavity is sufficient to guarantee continuity of its conjugate function caveat: v must be square integrable (Coulomb potential in a box) We now introduce the density functional F γ in the usual manner, as the conjugate to E γ: ( F γ(ρ) = max Eγ(v) (v ρ) ) Lieb variation principle v E γ(v) = min ρ ( Fγ(ρ) + (v ρ) ) Hohenberg Kohn variation principle unlike F, the new density functional F γ is continuous and even differentiable δf γ(ρ) δρ(r) = v(r) µ Euler equation well defined This procedure is known as Moreau Yosida (MY) regularization S. Kvaal, U. Ekström, A. M. Teale, and T. Helgaker, JCP 40, 8A58 (04) T. Helgaker (CTCC, University of Oslo) Role of Hohenberg Kohn Theorem in DFT 0 July 06 / 8

13 llustration of Moreau Yosida regularization DFT with non-regularized ground-state energy E(v) derivative of F is not well defined (multi-valued) E '(v) F(ρ) F '(ρ) - E(v) MY DFT with regularized ground-state energy E γ(v) = E(v) /γ v, γ = 0. derivative of F is well defined (single-valued) Eγ'(v) Fγ(ρ) Fγ'(ρ) - Eγ(v) T. Helgaker (CTCC, University of Oslo) Role of Hohenberg Kohn Theorem in DFT 0 July 06 3 / 8

14 Ground-state energy in a magnetic field A magnetic field B is represented by a vector potential A such that B = A The resulting Hamiltonian H(v, A) is nonlinear in A (one electron, ignoring spin): H(v, A) = H 0 + v + A + A p Consequently, E(v, A) is not concave in A as illustrated for C 0 in a perpendicular field Tellgren, Helgaker and Soncini, PCCP, 5489 (009) T. Helgaker (CTCC, University of Oslo) Role of Hohenberg Kohn Theorem in DFT 0 July 06 4 / 8

15 Current-density-functional theory (CDFT) A simple reparameterization of the potentials restores linearity of H and concavity of E: H(u, A) = H 0 + u + A p, u = v + A, p = i the basic potentials of CDFT are u and A (Tellgren et al. PRA 86, (0)) Proceeding as for DFT, we arrive at the Hohenberg Kohn and Lieb variation principles: ( ) E(u, A) = inf F (ρ, κ) + (u ρ) + (A κ) ρ,κ ( ) F (ρ, κ) = sup E(u, A) (u ρ) (A κ) u,a the universal functional depends on the charge density and paramagnetic current density ρ(r) = Ψ (r)ψ(r) κ(r) = ReΨ (r)pψ(r) charge density paramagnetic current density these quantities are uniquely determined by the electronic structure F was obtained by Hohenberg Kohn-type arguments by Vignale and Rasolt (987) Why is the physical current density not the basic current variable in CDFT? j = κ + ρ A j is observable but not uniquely determined by the electronic structure, depending also on A (C)DFT variables should be the distillation of Ψ information needed for evaluation of interactions also, κ is the natural variable by convex conjugation T. Helgaker (CTCC, University of Oslo) Role of Hohenberg Kohn Theorem in DFT 0 July 06 5 / 8

16 Hohenberg Kohn theorem in CDFT? HK theorem in DFT: the ρ determines the potential v up to a constant Adding a uniform magnetic field to a spherical system, we obtain (ignoring spin) HB = H0 + BLz [H0, HB ] = 0 same ground state Ψ in Bmin B Bmax E Bmin 0 Bmax field the density (ρ, κ) are constant in the interval since Ψ is constant the potentials (u, A) change in the interval since B = A changes Hence, there is no HK theorem: the densities (ρ, κ) do not determine the potentials (u, A) von Barth and Hedin (97), Eschrig and Pickett (00), Capelle and Vignale (00,00) But the physical current density j = κ + ρa does change in the interval HK theorem for j is not ruled out but purported proofs are flawed (Tellgren et al. (0)) T. Helgaker (CTCC, University of Oslo) Role of Hohenberg Kohn Theorem in DFT 0 July 06 6 / 8

17 Densities from the RR and HK variation principles The universal density functional is not unique a minimal requirement is that the density functional gives the correct ground-state energy such functionals are said to be admissible (Ayres) E(v) = inf ρ (F 0 (ρ) + (v ρ)) = inf Ψ Ψ H(v) Ψ = inf Γ tr ΓH(v) optimality conditions for any admissible density functional E(v) = F 0 (ρ) + (v ρ) v F 0 (ρ) = ρ E(v) examples of admissible density functionals F LL (ρ) = min Ψ T + W Ψ, F Ψ ρ Are the minimizing densities in DFT always physical? DM(ρ) = min tr Γ (T + W ) = F (ρ) Γ ρ a general admissible density functional may give unphysical densities the constrained-search functionals in DFT recover precisely the corresponding physical densities: E(v) = Ψ ρ H(v) Ψ ρ v F LL (ρ) = ρ E(v) E(v) = tr Γ ρh(v) v F DM (ρ) ρ E(v) the same has not been shown for the VR constrained-search functional in CDFT S. Kvaal and T. Helgaker, JCP 43, 8406 (05) T. Helgaker (CTCC, University of Oslo) Role of Hohenberg Kohn Theorem in DFT 0 July 06 7 / 8

18 Conclusions Review: ground-state energy is concave and continuous but nondifferentiable universal density functional is convex but discontinuous and nondifferentiable subdifferentation replaces differentiation as the tool for studying variations Moreau Yosida regularization yields a continuous, differentiable density functional Hohenberg Kohn theorem in DFT follows from (almost) strictly concave energy there is no Hohenberg Kohn theorem for the paramagnetic current density in CDFT there are no false densities in DFT but possibly in CDFT Advertisement: Principles of Density-Functional Theory, Helgaker, Jørgensen, and Olsen (Wiley, 07) Acknowledgements: Ulf Ekström, Simen Kvaal, Espen Sagvolden, Andy Teale, Erik Tellgren ERC advanced grant ABACUS Norwegian Research Council for Centre of Excellence CTCC T. Helgaker (CTCC, University of Oslo) Role of Hohenberg Kohn Theorem in DFT 0 July 06 8 / 8