Hartree-Fock Excited States Mathieu LEWIN - PowerPoint PPT Presentation

Hartree-Fock Excited States Mathieu LEWIN mathieu.lewin@math.cnrs.fr (CNRS & Universit´ e de Paris-Dauphine) Conference on “Mathematical challenges in classical & quantum statistical mechanics” Venice, August 2017 Mathieu LEWIN (CNRS / Paris-Dauphine) HF Excited States 1 / 13

HF Excited States ◮ Hartree-Fock theory: simplest nonlinear approx. of fermionic N -particle ground state problem not always efficient (correlation) � Kohn-Sham / DFT recently discovery: can be efficient for excited states Mathieu LEWIN (CNRS / Paris-Dauphine) HF Excited States 2 / 13

N -particle Schr¨ odinger operator N -particle fermionic Hamiltonian N � � H V ( N ) = − ∆ x j + V ( x j ) + w ( x j − x k ) j =1 1 ≤ j < k ≤ N V , w infinitesimally relatively − ∆–form bounded in R d Ground state energy E V ( N ) = min Spec � N H V ( N ) Ψ , H V ( N )Ψ � � � � = inf 1 L 2 ( R d ) 1 H 1 ( R d ) Ψ ∈ � N � Ψ � =1 Bottom of essential spectrum Σ V ( N ) = min Ess Spec � N � H V ( N ) � � Ψ n , H V ( N )Ψ n � = inf lim inf 1 L 2 ( R d ) n →∞ Ψ n ⇀ 0 � Ψ n � =1 Mathieu LEWIN (CNRS / Paris-Dauphine) HF Excited States 3 / 13

HVZ Theorem Excited state energies � Ψ , H V ( N )Ψ � λ V k ( N ) = inf max V⊂ � N 1 H 1 ( R d ) Ψ ∈V � Ψ � =1 dim( V )= k is the k th eigenvalue of H V ( N ), counted with multiplicity, or = Σ V ( N ). Theorem (HVZ) Σ V ( N ) = min � E V ( N − k ) + E 0 ( k ) , k = 1 , ..., N � (Hunziker ’66, Van Winter ’64, Zhislin ’60) Mathieu LEWIN (CNRS / Paris-Dauphine) HF Excited States 4 / 13

Atoms & Molecules ◮ Atoms & Molecules (Born-Oppenheimer): M z m w ( x ) = 1 � V ( x ) = − | x − R m | , | x | m =1 Since w ≥ 0, E 0 ( k ) = 0, hence Σ V ( N ) = E V ( N − 1) M = 3, N = 10 z 1 = z 2 = 1, z 3 = 8 Theorem (Spectrum of atoms & molecules) ◮ If N < � M k ( N ) < Σ V ( N ) for all k ≥ 1 . m =1 z m + 1 then λ V (Zhislin ’60, Zhislin-Sigalov ’65) ◮ If N ≥ � M k 0 ( N ) = Σ V ( N ) for some k 0 ≥ 1 . m =1 z m + 1 then λ V (Yafaev ’76, Vugalter-Zhislin ’77, Sigal ’82) ◮ If N ≫ 1 (e.g. N ≥ 2 � M m =1 z m + 1 ), then k 0 = 1 . (Lieb ’84, Nam ’12, Ruskai ’82, Sigal ’82-84, Lieb-Sigal-Simon-Thirring ’88, Seco-Sigal-Solovej ’90, Fefferman-Seco ’90, Lenzmann-Lewin ’13) Mathieu LEWIN (CNRS / Paris-Dauphine) HF Excited States 5 / 13

Curse of dimensionality  i ∂ � N �  ∂ t Ψ( t , x 1 , ..., x N ) � � − ∆ x j + V ( t , x j )+ w ( x j − x k ) Ψ( t , x 1 , ..., x N ) =  λ Ψ( x 1 , ..., x N ) j =1 1 ≤ j < k ≤ N N ∼ 10 3 in small macromolecules N ∼ 10 57 in neutron star N = 10 electrons in water molecule (short segments of DNA) “the mathematical theory of a large part of physics and the whole of chemistry is thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed” Dirac (1929) Mathieu LEWIN (CNRS / Paris-Dauphine) HF Excited States 6 / 13

Hartree-Fock theory Hartree-Fock state 1 Ψ = ϕ 1 ∧ · · · ∧ ϕ N = √ det( ϕ j ( x k )) N ! where ϕ j ∈ L 2 ( R d , R ) and � ϕ j , ϕ k � = δ jk ◮ Restrict N -particle energy to manifold M = { Ψ = ϕ 1 ∧ · · · ∧ ϕ N } N ˆ Ψ , H V ( N )Ψ R 3 |∇ ϕ j | 2 + V | ϕ j | 2 � � � = j =1 � N N N 2 � � � + 1 ¨ | ϕ k ( y ) | 2 − � | ϕ j ( x ) | 2 � � � � R 6 w ( x − y ) ϕ j ( x ) ϕ j ( y ) dx dy � � 2 � � j =1 k =1 j =1 N ˆ ¨ R 3 |∇ ϕ j | 2 + V | ϕ j | 2 + R 6 w ( x − y ) | ϕ j ∧ ϕ k ( x , y ) | 2 dx dy � � = j =1 1 ≤ j < k ≤ N h Ψ ϕ j = µ j ϕ j , j = 1 , ..., N � � j =1 | ϕ j | 2 ∗ w � � − ∆ + V + � N f − � N h Ψ f := ( ϕ j f ) ∗ w ϕ j j =1 Mathieu LEWIN (CNRS / Paris-Dauphine) HF Excited States 7 / 13

Hartree-Fock ground states Hartree-Fock ground state energy Ψ , H V ( N )Ψ ≥ E V ( N ) � � E V HF ( N ) = inf Ψ ∈M � Ψ � =1 Theorem (Existence of HF ground states) Let V , w be infinitesimally − ∆ –form bounded in R d . The following are equivalent: (i) All the minimizing sequences { Ψ n } ⊂ M for E V HF ( N ) have a convergent subsequence in H 1 ( R dN ) HF ( N − k ) + E 0 (ii) E V HF ( N ) < E V HF ( k ) for all k = 1 , ... N (Friesecke ’03, Lewin ’11) Rmk. Sort of nonlinear HVZ . Very important that HF = restriction of H V ( N ) M � Atoms and molecules: existence for N < z m + 1 (Lieb-Simon ’77, Lions ’87) m =1 Mathieu LEWIN (CNRS / Paris-Dauphine) HF Excited States 8 / 13

Weyl ≡ Palais-Smale condition Exercise Linear problem with E V ( N ) < Σ V ( N ). Any minimizing sequence { Ψ n } for E V ( N ) is precompact 1 � Ψ n , H V ( N )Ψ n � → c < Σ V ( N ) ∃ non-compact sequences { Ψ n } such that 2 If ( H V ( N ) − c )Ψ n → 0 (Weyl) with c < Σ V ( N ), then { Ψ n } is precompact 3 Proof of 3) Extract subsequence such that Ψ n ⇀ Ψ Passing to weak limits gives ( H V ( N ) − c )Ψ = 0 � Ψ n , H V ( N )Ψ n � � Ψ , H V ( N )Ψ � � (Ψ n − Ψ) , H V ( N )(Ψ n − Ψ) � c ← = + + o (1) � �� � � �� � ≥ Σ V ( N )(1 −� Ψ � 2 )+ o (1) c � Ψ � 2 Mathieu LEWIN (CNRS / Paris-Dauphine) HF Excited States 9 / 13

Weyl ≡ Palais-Smale condition II Theorem: HF Palais-Smale condition (Lewin ’17) Assume w ≥ 0 and E V HF ( N ) < E V HF ( N − 1). Let Ψ n = ϕ 1 , n ∧ · · · ∧ ϕ N , n ∈ M with � � � � Ψ n , H V ( N )Ψ n E V HF ( N ) , E V • → c ∈ HF ( N − 1) , [ ∂ M E V (Ψ n ) → 0] • h Ψ n ϕ j , n − µ j , n ϕ j , n → 0 in H − 1 ( R d ), ∀ j = 1 , ..., N , then { Ψ n } is precompact in H 1 ( R dN ) and converges strongly, after extraction of a subsequence, to Ψ = ϕ 1 ∧ · · · ∧ ϕ N ∈ M which is a Hartree-Fock critical point. Mathieu LEWIN (CNRS / Paris-Dauphine) HF Excited States 10 / 13

HF Excited states Theorem: HF Excited States (Lewin ’17) For atoms and molecules with N < � M m =1 z m + 1, the HF energy has infinitely many critical points { Ψ ( k ) } k ≥ 1 on M with energies � Ψ ( k ) , H V ( N )Ψ ( k ) � λ V k ( N ) ≤ λ V < E V HF , k ( N ) = HF ( N − 1) , k ≥ 1 such that k →∞ λ V HF , k ( N ) = E V lim HF ( N − 1) Rmk. Lions ’87 also constructed infinitely many HF critical point, but with � Ψ ( k ) , H V ( N )Ψ ( k ) � energies → 0 ( ≃ “embedded eigenvalues”) ◮ Lions worked in one-particle space, his method applies to other HF-like theories ◮ I work in N -particle space, the method uses that HF = restriction linear problem on M Mathieu LEWIN (CNRS / Paris-Dauphine) HF Excited States 11 / 13

Critical Point Theory Nonlinear minimax method � Ψ , H V ( N )Ψ � λ V ≤ E V HF , k ( N ) := inf sup HF ( N − 1) f : S k − 1 →M Ψ ∈ f ( S k − 1 ) continuous and odd generalizes usual Courant-Fischer / Rayleigh-Ritz linear minimax λ V k ( N ) =same formula on whole sphere instead of M one can use instead Krasnoselskii index, homology classes, etc Palais-Smale at minimax level = ⇒ ∃ critical point Palais-Smale does not hold for energies < 0, Lions uses Morse index bounds to get compactness (Ambrosetti-Rabinowitz ’73, Berestycki-Lions ’83, Rabinowitz ’86,...) Mathieu LEWIN (CNRS / Paris-Dauphine) HF Excited States 12 / 13

Proof of Palais-Smale property Lemma (Geometric limits of HF states) If M ∋ Ψ n ⇀ Ψ , then � � n →∞ E V (Ψ n ) ≥ � 1 − � Ψ � 2 � + E V (Ψ) . HF ( N − k ) + E 0 E V lim inf min HF ( k ) k =1 ,..., N (Friesecke ’03, Lewin ’11) Main fact: the (geometric) localization of a pure HF state is a convex combination of HF pure states Lemma (Energy of weak limit of Palais-Smale sequence) Assume w ≥ 0 . If M ∋ Ψ n ⇀ Ψ with E V (Ψ n ) → c and ∂ M E V (Ψ n ) → 0 , then E V (Ψ) ≥ c � Ψ � 2 Mathieu LEWIN (CNRS / Paris-Dauphine) HF Excited States 13 / 13