aa r X i v : . [ qu a n t - ph ] A ug Hohenberg-Kohn Theorem for Coulomb Type Systems ∗ Aihui Zhou † Abstract
Density functional theory (DFT) has become a basic tool for the study of electronicstructure of matter, in which the Hohenberg-Kohn theorem plays a fundamental role inthe development of DFT. Unfortunately, the existing proofs are incomplete even incor-rect; besides, the statement of the Hohenberg-Kohn theorem for many-electron Coulombsystems is not perfect. In this paper, we shall restate the Hohenberg-Kohn theorem forCoulomb type systems and present a rigorous proof by using the Fundamental Theoremof Algebra.
Keywords:
Coulomb system, density functional theory, electronic structure, Fundamen-tal Theorem of Algebra, Hohenberg-Kohn theorem.
The modern formulation of density functional theory (DFT) originated in the work of Hohen-berg and Kohn [4], on which based the other classic work in this field by Kohn and Sham[7], the Kohn-Sham equation, has become a basic mathematical model of much of present-daymethods for treating electrons in atoms, molecules, condensed matter, and man-made struc-tures [1, 2, 5, 12, 13]. Although it is quite profound, DFT is not entirely elaborated yet (c.f.,e.g., [11, 14, 16] and references cited therein). Since the relevant assumptions are incompatiblewith the Kato cusp condition, Kryachko has pointed out that the usual reductio ad absurdum proof of the original Hohenberg-Kohn theorem is unsatisfactory [8, 9]. Note that Kato theo-rem [6] tells the electron-nucleus cusp conditions at nucleus positions only, we are not able touniquely determine the electron density from the cusp conditions, though the electron densityuniquely determines the external Coulombic potential. Consequently, there is a gap in theproof of the theorem by using the Kato theorem in [9]. We note that Lieb has tried to examinethe theorem rigorously [11]. But Lieb’s proof required that the N -particle wavefunction doesnot vanish in a set of positive measure that is unclear in a real system (c.f. [14]). We referto [3, 8, 10, 11, 14, 16] and references cited therein for discussions on the Hohenberg-Kohntheorem. Indeed, the statement of the original Hohenberg-Kohn theorem for many-electronCoulomb systems is not perfect (see Section 2).In this paper, we shall state the Hohenberg-Kohn theorem for Coulomb type systems pre-cisely (see Theorem 2.2) and present a rigorous proof by using the Fundamental Theorem ofAlgebra (see Section 4). ∗ This work was partially supported by the National Science Foundation of China under grants 10871198and 10971059, the Funds for Creative Research Groups of China under grant 11021101, and the National BasicResearch Program of China under grant 2011CB309703. † LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Math-ematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China ([email protected]). Hohenberg-Kohn theorem
We see that the approach of Hohenberg and Kohn is to formulate DFT as an exact theoryof many-body systems. The formulation applies to any system of interacting particles in anexternal potential v , including any problem of electrons and fixed nuclei, where the Hamiltoniancan be written as H = − N X i =1 ~ m e ∇ x i + N X i =1 v ( x i ) + 12 N X i,j =1 ,i = j e | x i − x j | , (2.1)where ~ is Planck’s constant divided by 2 π, m e is the mass of the electron, { x i : i = 1 , · · · , N } are the variables that describe the electron positions, and e is the electronic charge. For anelectronic Coulomb system, v ( x ) ≡ v { Zj } , { r j } ( x ) = − M X j =1 Z j e | x − r j | (2.2)is determined by { Z j : j = 1 , , · · · , M } , which are the valence charges of the nuclei, and { r j : j = 1 , , · · · , M } , which are the positions of the nuclei. The energy of the system can beexpressed by E = (Ψ , H Ψ) = (Ψ , ( T + V ee )Ψ) + Z R v ( x ) ρ ( x ) , (2.3)where T = − N X i =1 ~ m e ∇ x i is the kinetic energy operator, V ee = 12 N X i,j =1 ,i = j e | x i − x j | is the electron-electron repulsion energy operator, and ρ ( x ) ≡ ρ Ψ ( x ) = N X σ ,σ , ··· ,σ N Z R N − | Ψ(( x, σ ) , ( x , σ ) , · · · , ( x N , σ N )) | dx · · · dx N (2.4)is the single-particle density.Let H = T + V ee and v be a single-particle potential in function space V ≡ L / ( R ) + L ∞ ( R ) . The total Hamiltonian is H v = H + V , where V = N X i =1 v ( x i ) . The associated ground state energy E ( v ) is defined to be E ( v ) ≡ E ( v, N ) = inf { (Ψ , H v Ψ) : Ψ ∈ W N } , (2.5)2here W N = { Ψ ∈ H ( R N ) : X σ ,σ , ··· ,σ N Z R N | Ψ | = 1 } . Note that there may or may not be a minimizer ψ in W N , and if there is one it may not beunique [11]. Thus, we should introduce a set of minimizers G v ≡ G v,N = arg inf { (Ψ , H v Ψ) : Ψ ∈ W N } . Any Ψ in G v is called a ground state of (2.5). Define V N = { v ∈ V : G v,N = ∅} and D N = { ρ Ψ : Ψ ∈ G v for some v ∈ V N } . If Ψ ∈ G v , then H v Ψ = E ( v )Ψ (2.6)in the distributional sense.The original Hohnberg-Kohn theorem (see page B865 of [4]) states that the external po-tential v “is a unique functional of” the electronic density in the ground state,“apart from atrivial additive constant.” In our notation, Lieb’s statement of this theorem may be written asthe following (see Theorem 3.2 of [11]): Theorem 2.1.
Suppose Ψ v ∈ G v and Ψ v ′ ∈ G v ′ . If v = v ′ + constant , then ρ Ψ v = ρ Ψ v ′ . Consider Coulomb potential set V C = − M X j =1 Z j e | x − r j | : Z j ∈ R , r j ∈ R ( j = 1 , , · · · , M ); M = 1 , , · · · . Obviously, V C ( V .It will be shown by the Fundamental Theorem of Algebra that for any v, v ′ ∈ V C , v = v ′ + constant for any constant if v = v ′ (see Section 4 for details), which means that therequirement v = v ′ + constant in the Hohnberg-Kohn theorem or Theorem 2.1 is superfluousfor electronic Coulomb system. More precisely, v = v ′ + constant in the Hohnberg-Kohntheorem should be replaced by v = v ′ . Therefore the Hohenberg-Kohn theorem or Theorem2.1 for Coulomb type systems should be restated as follows: Theorem 2.2.
The map: v ∈ V C −→ ρ v ∈ D N is one-to-one, where ρ v = ρ Ψ with Ψ ∈ G v . To prove Theorem 2.2, we need some lemmas.
Lemma 3.1.
Let n ≥ . If t j ∈ R ( j = 1 , , · · · , n ) and δ = n X j =1 t j , hen there exist non-zero polynomials { H n,j ( s , s , · · · , s n ) : j = 0 , , , · · · , n − } with realcoefficients satisfying n − X j =0 H n,j ( t , t , · · · , t n ) δ n − − j ) = 0 , (3.1) where H n,j ( s , s , · · · , s n )( j = 1 , , · · · , n − ) are homogeneous: H n,j ( λs , λs , · · · , λs n ) = λ j H n,j ( s , s , · · · , s n ) , ∀ λ ∈ R , j = 1 , , · · · , n − ,H n, ( s , s , · · · , s n ) = 1 , and H n, n − ( s , s , · · · , s n ) is a monic polynomial of degree n − .Proof. We prove the conclusion by induction on n . First, for n = 2, t + t = δ , which implies t + 2 t t + t = δ and hence t + t − t t − δ ( t + t ) + δ = 0 . Namely, (3.1) is true for n = 2.For the induction step, suppose (3.1) is true for n . If n X j =1 t j + t n +1 = δ, then n X j =1 t j = δ − t n +1 . By the induction hypothesis, we have that there exist non-zero polynomials { H n,j ( s , s , · · · , s n ) : j = 0 , , , · · · , n − } with real coefficients satisfying H n,j ( s , s , · · · , s n )( j = 1 , , · · · , n − )are homogeneous, H n, n − ( s , s , · · · , s n ) is a monic polynomial of degree 2 n − , and n − X j =0 H n,j ( t , t , · · · , t n )( δ − t n +1 ) n − − j ) = 0 . (3.2)Applying Newton binomial theory, we then get that H n,n ( t , t , · · · , t n ) + δ n + t n n +1 + n − − X j =1 H n,j ( t , t , · · · , t n ) n − − j X l =0 (cid:18) n − j l (cid:19) δ n − j − l t ln +1 + n − − X l =1 (cid:18) n l (cid:19) δ n − l t ln +1 = δt n +1 n − − X j =1 H n,j ( t , t , · · · , t n ) n − − j X l =1 (cid:18) n − j l − (cid:19) δ n − j − l t l − n +1 + δt n +1 2 n − X l =1 (cid:18) n l − (cid:19) δ n − l t l − n +1 . n is replaced by n + 1. This completes the proof.The following conclusion results from the proof of Theorem 1 of [14] (c.f. also [11]): Lemma 3.2.
Given v, v ′ ∈ V . Let ρ v = ρ Ψ v and ρ v ′ = ρ Ψ v ′ with Ψ v ∈ G v and Ψ v ′ ∈ G v ′ . If ρ v = ρ v ′ , then N X i =1 ( v ′ − v )( x i ) − ( E ( v ′ ) − E ( v )) ! Ψ v = 0 . (3.3) Proof.
For completion, we present a proof here, which essentially comes from the proof ofTheorem 1 of [14]. We see that E ( v ) = (Ψ v , H v Ψ v ) ≤ (Ψ v ′ , H v Ψ v ′ )= E ( v ′ ) − Z R ρ v ′ ( v ′ − v ) . Similarly, E ( v ′ ) ≤ E ( v ) − Z R ρ v ( v − v ′ ) . Thus we obtain that if ρ v = ρ v ′ , then E ( v ′ ) = E ( v ) − Z R ρ v ( v − v ′ ) , or Z R ρ v ( v − v ′ ) = E ( v ) − E ( v ′ ) , which leads to E ( v ) = (Ψ v ′ , H v Ψ v ′ ). Therefore Ψ v ′ ∈ G v and H v Ψ v ′ = E ( v )Ψ v ′ . By a similar argument, we have H v ′ Ψ v = E ( v ′ )Ψ v . Since H v Ψ v = E ( v )Ψ v , we arrive at (3.3). This completes the proof.Due to the Fundamental Theorem of Algebra (c.f., e.g., [15]), every non-zero single-variablepolynomial with real or complex coefficients has exactly as many real or complex zeroes as itsdegree, if each zero is counted up to its multiplicity. Hence we have a multivariate version ofthe Fundamental Theorem of Algebra as follows: Lemma 3.3.
The Lebesgue’s measure of the set of zeroes of any non-zero multivariate poly-nomial with real coefficients is zero. Proof
In this section, we prove Theorem 2.2, the precise and new statement of Hohenberg-Kohntheorem.
Proof.
Let v, v ′ ∈ V C . Choose Ψ v ∈ G v and Ψ v ′ ∈ G v ′ such that ρ v = ρ Ψ v and ρ v ′ = ρ Ψ v ′ . Itis sufficient to prove that v = v ′ if ρ v = ρ v ′ .Note that there exist m ≥ , r j ∈ R and α j ∈ R ( j = 1 , , · · · , m ) such that( v ′ − v )( x ) = m X j =1 α j | x − r j | . Suppose v ′ = v , we have α j = 0 for some j ∈ { , , · · · , m } . As a result, if equation N X i =1 ( v ′ − v )( x i ) = E ( v ′ ) − E ( v ) (4.1)holds, then there exist non-zero polynomials { H n,j ( s , s , · · · , s n ) : j = 0 , , , · · · , n − } withreal coefficients satisfying the conclusion of Lemma 3.1 with n = mN and δ = α j | x − r j | ,t = E ( v ′ ) − E ( v ) , { t l : l = 2 , , · · · , n } = (cid:26) − α j | x i − r j | : i = 1 , , · · · , N ; j = 1 , , · · · , m (cid:27) \ {− δ } . Therefore there exists a non-zero multivariate polynomial P ( s , s , · · · , s n ) with real coefficientssuch that P ( | x − r | , · · · , | x i − r j | , · · · , | x N − r m | ) = 0 ,x i ∈ R \ { r j : j = 1 , , · · · , m } , i = 1 , , · · · , N (4.2)and the set of the solutions ( x , x , · · · , x N ) of (4.1) are a subset of zeroes of (4.2) and henceof zero measure in R N by Lemma 3.3. Since N X i =1 ( v ′ − v )( x i ) − ( E ( v ′ ) − E ( v ))is continuous over domain { ( x , x , · · · , x N ) : x i ∈ R \ { r j : j = 1 , , · · · , m } , i = 1 , , · · · , N } as a function of ( x , x , · · · , x N ) ∈ R N , we must have Ψ v = 0 almost every where in R N from(3.3), which is a contradiction to X σ ,σ , ··· ,σ N Z R N | Ψ v | = 1 . This completes the proof.
Acknowledgements.
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