On Csanyi's and Arias' Functional for Ground States Energy of Multi-Particle Fermion Systems: Asymptotics
We show that Csanyi's and Arias' energy functional of the reduced one-particle density matrix is bounded from below by the Müller functional and bounded from above by the Hartree-Fock functional. We use this fact to derive an asymptotic expansion of …
Authors: Heinz Siedentop
ON CS ´ ANYI’S AND ARIAS’ FUNCTIONAL F OR GR OUND ST A TES ENER GY OF MUL TI-P AR TICLE FERMION SYSTEMS: ASYMPTOTICS HEINZ SIEDENTOP Abstract. W e show that Cs´ anyi’s and Arias’ energy functional of the one- particle reduced density matrix is bounded from b elow by the M ¨ uller functional and b ounded from ab ov e by the Hartree-F ock functional. W e use this fact to derive an asymptotic expansion of the ground state energy of this functional which agrees with the quantum energy to third order. 1. Introduction Cs´ an yi and Arias [2] in troduced an energy functional, hereafter referred to as the CA functional of the one-particle reduced densit y matrix γ which they call“corrected Hartree-F o ck functional”. T o define it we first define the Hartree-F o ck functional (1) E HF ( γ ) := tr ( 1 2 p 2 − V ) γ + 1 2 Z R 3 d x Z R 3 d y ρ γ ( x ) ρ γ ( y ) | x − y | | {z } =: D [ ρ γ ] − 1 2 Z Γ d x Z Γ d y | γ ( x, y ) | 2 | x − y | | {z } =: X [ γ ] where ρ γ is the particle densit y of γ , i.e., ρ γ ( x ) := P n P q σ =1 λ n | ξ n ( x ) | 2 with the notation x = ( x , σ ) ∈ Γ := R 3 × C q and R Γ d x := R R 3 d x P q σ =1 . Here λ 1 , λ 2 , ... and ξ 1 , ξ 2 , ... are the eigenv alues and eigenfunctions of γ , i.e., γ = P n λ n | ξ n ⟩⟨ ξ n | . More- o v er, q is the n um b er of spin states p er particle, i.e., for non-relativistic electrons q = 2. Ev en tually , T ( p ) is the kinetic energy written in terms of the momentum p := − i ∇ and V is the external p oten tial. As we will see in the proof, the exact form of the kinetic energy T and of the p otential V is not relev ant for our main result. How ever, for definiteness we assume T ( p ) = p 2 / 2 and V ( x ) = Z/ | x | whic h is essential for the asymptotic expansion (18). Using this notation the CA functional is (2) E CA ( γ ) := E HF ( γ ) − X [ p γ (1 − γ )] . Cs´ an yi and Arias found the functional E CA b y approximating the tw o-particle densit y matrix quadratically keeping some quantum statistical properties, some symmetries, and normalization. The only other functional that they found under Date : Marc h 15, 2026. 2020 Mathematics Subje ct Classification. 81-V45, 81-V55, 81-V74. Key words and phr ases. Exchange-correlation energies, 1-p dm functionals. This work was partially supp orted by the Deutsche F orschungsgemeinsc haft via the TRR 352 – Pro ject-ID 470903074 and EXC-2111-390814868. 1 2 H. SIEDENTOP those requirements is the M ¨ uller functional (in CA’s diction the “corrected Hartree functional”): (3) E M ( γ ) := tr (( T ( p ) − V ( x )) γ ) + D [ ρ γ ] − X [ √ γ ] . W e consider these functionals on certain subsets of the set S 1 ( L 2 (Γ : C )) of trace class op erators on square integrable complex v alued functions on Γ, namely Q := { γ ∈ S 1 ( L 2 (Γ : C )) 0 ≤ γ ≤ 1 , T ( p ) γ ∈ S 1 ( L 2 (Γ : C ) } , (4) Q N := { γ ∈ Q tr γ ≤ N } (5) where N ∈ N is the n um b er of electrons, i.e., the set of 1-p dms with finite particle n um b er and finite kinetic energy , and, p ossibly , with particle num ber not exceeding N . Hartree-F o ck theory is w ell established in non-relativistic quantum mechanics. It is an upper bound on the true quan tum energy and agrees energetically for hea vy atoms with the underlying non-relativistic quan tum mec hanics up to the subleading order in the atomic n umber (see Lieb and Simon [16] for the leading order, Sieden top and W eik ard [22, 19, 20, 21] (upp er and lo w er b ound) and Hughes [10, 11] (low er b ound) for the leading correction, and Bach [1] and F efferman and Seco [6, 7, 3, 8, 4] for the subleading order). Moreo ver agreements of the densities are shown on the scale Z − 1 3 in [16] and on smaller scales b y Ian tc henk o et al. [13], Ian tchenk o [12] and b y Iantc henk o et al. [14] (densit y matrix). The same accuracy of the energy has b een shown for the M ¨ uller functional [17, 18]. It is conjectured a low er b ound on the quantum energy . How ever, this w as only sho wn for tw o electrons (F rank et al. [9] based on an inequality of Wigner and Y anase [23]. Although the CA functional has b een already successfully b enchmark ed n umer- ically by Jara-Cortes et al. [15, T able 1], an analytic study is lac king. Our goal is to fill this gap and to show how the CA functional fits in the picture from an analytical point of view. In particular w e will sho w that it is actually sandwiched in b et w een the Hartree-F o ck and the M ¨ uller functional. Establishing this result will ha v e the consequence that the CA ground state energy of atoms of atomic num b er Z agrees with the true quantum energy up subsubleading order in Z . W e start with some preparatory facts. Lemma 1. Assume µ, λ ∈ [0 , 1] . Then (6) λµ + p λ (1 − λ ) µ (1 − µ ) ≤ p λµ. Pr o of. Applying the Sch warz inequality to the sum on the left side of (6) giv e (7) λµ + p λ (1 − λ ) µ (1 − µ ) ≤ p λ 2 + λ (1 − λ ) p µ 2 + µ (1 − µ ) = p λµ whic h is iden tical with the right side of (6). □ W e also remind of the formula (F efferman and de la Llav e [5, p. 124]) (8) 1 | x − y | = 1 π Z ∞ 0 d r r 5 Z R 3 d z B r, z ( x ) B r, z ( x ) where B r, z is the c haracteristic function of the unit ball centered around z , i.e., (9) B r, z ( x ) := ( 1 | x − z | < r 0 | x − z | ≥ r . CS ´ ANYI AND ARIAS 3 W riting (10) E M ( Z ) := inf E M ( Q Z ) for the neutral atomic M¨ uller ground state energy our basic result is Theorem 1. Assume γ ∈ Q . Then (11) E M ( γ ) ≤ E CA ( γ ) ≤ E HF ( γ ) . Pr o of. The upp er b ound on E CA ( γ ) is immediate from the definition of E CA . The remaining low er b ound is equiv alent to showing (12) X [ γ ] + X [ p γ (1 − γ )] ≤ X [ √ γ ] . First note that (13) 0 ≤ Z Γ d x Z Γ d y B r, z ( x ) ξ m ( x ) ξ n ( x ) ξ m ( y ) ξ n ( y ) B r, z ( y ) and therefore, c ho osing λ = λ m and µ = λ n in (6), (14) 0 ≤ p λ m λ n − λ m λ n − p λ m (1 − λ m ) λ n (1 − λ n ) × Z Γ d x Z Γ d y B r, z ( x ) ξ m ( x ) ξ n ( x ) ξ m ( y ) ξ n ( y ) B r, z ( y ) . W e divide the inequality by 2 π r 5 and in tegrate o v er r and z and use F efferman’s and de la Llav e’s formula (8) for the Coulomb k ernel and get (15) 1 2 Z Γ d x Z Γ d y | P m λ m ξ m ( x ) ξ m ( y ) | 2 + | P m p λ m (1 − λ m ) ξ m ( x ) ξ m ( y ) | 2 | x − y | ≤ 1 2 Z Γ d x Z Γ d y | P m √ λ m ξ m ( x ) ξ m ( y ) | 2 | x − y | whic h is (12) written in terms of the eigenfunctions and eigenv alues of γ . □ Theorem 1 has an immediate consequence on the expansion of ground state energy E Q ( Z ) := inf σ ( H Z,Z ) of the neutral atomic Hamiltonian (in the N -electron space V N n =1 L 2 (Γ : C )) (16) H N ,Z := N X n =1 p 2 n 2 − Z | x n | + X 1 ≤ m
Original Paper
Loading high-quality paper...
Comments & Academic Discussion
Loading comments...
Leave a Comment