A max-min principle for the ground state of the Dirac-Fock functional
9
0
0
Texte intégral
(2) ne structure constant is small enough, the energy of the \ground state" of the Dirac-Fock model can be obtained by a nonlinear max-min principle.. AMS Subject Classi
(3) cation: 49 S 05, 35 J 60, 35 P 30, 35 Q 75, 81 Q 05, 81 V 70, 81 V 45, 81 V 55.. Key Words : Relativistic quantum mechanics, nonrelativistic limit, quan-. tum chemistry, ground state, nonlinear eigenvalue problems, Dirac-Fock equations, Hartree-Fock equations, variational methods, critical points, strongly inde
(4) nite functionals, bifurcations from the essential spectrum.. 1. The Dirac-Fock model. In [6] we proved that solutions of Dirac-Fock equations converge, in a certain sense, towards solutions of the Hartree-Fock equations when the speed of light tends to in
(5) nity. This limiting process allowed us to de
(6) ne a notion of ground state for the Dirac-Fock equations, valid when the
(7) ne structure constant is small enough. In the present paper we show, as a consequence of our results in [6], that the energy of this ground state can be de
(8) ned as a max-min level. We hope that this new characterization will Membre de l'Institut universitaire de France. Partially supported by the A.C.I. blanche \Modeles mathematiques pour la chimie quantique atomique et moleculaire".. 1.
(9) help to bridge the gap between the Dirac-Fock equations and the DiracFock-Bogoliubov model derived from Q.E.D. (see [4, 1, 2]). First of all, we choose units for which m = c = h = 1, where m is the mass of the electron, c the speed of light and h is Planck's constant. In these units, the free Dirac Hamiltonian can be written as (1). !. H0 = ?i r +
(10) ; !. where
(11) = 10I ?01I , k = 0 0k (k = 1; 2; 3) and the k are the k well known Pauli matrices. The operator H0 acts on 4-spinors, i.e. functions from IR3 to CI4 , and it is self-adjoint in L2 (IR3 ; CI4 ), with domain H 1 (IR3 ; CI4 ) and form-domain H 1=2 (IR3 ; CI4 ). Its spectrum is (?1; ?1] [ [1; +1). Let us denote E := H 1=2 (IR3 ; CI4 ). Since. (H0 ) = (?1; ?1] [ [1; +1) ; the Hilbert space E can be split as. E = E+ E? ; where E := 0 E , and 0 := IR (H0 ). The projectors 0 have a simple ( ) = ^ ( ) ^( ), with expression in the Fourier domain : d 0 0 b 0 ( ). (2). !. := 21 1ICI4 p +
(12) 2 : 1 + j j. Let us consider a system of N electrons coupled to a
(13) xed nuclear density Z, where Z > 0 is the total number of protons and is a probability measure de
(14) ned on IR3 . Note that in the particular case of m point-like nuclei, m X each one having atomic number Zi at a
(15) xed location xi ; Z = Zi x and. Z=. m X i=1. i=1. Zi .. i. In our system of units, the Dirac-Fock equations (
(16) rst introduced in [16]). 2.
(17) for such a molecule are given by (DF). 8 > > > > > > > < > > > > > > > :. 1. 1. k := H0 k ? Z ( jxj ) k + ( jxj ) k Z ? R (jx;x ?y) yjk (y) dy = "k k (k = 1; :::N );. H. IR3. GramL2 = 1IN (i:e. R. . IR3 k l. = kl ; 1 k; l N):. Here, > 0 is a dimensionless constant called the
(18) ne structure con1 , but in this paper will be a stant. Its physical value is close to 137 \small" parameter. Moreover, = ( 1 ; ; N ) ; each k is a 4-spinor in H 1=2 (IR3 ; CI4 ) (by bootstrap, k is also in any W 1;p(IR3 ) space, 1 p < 3=2), and N N X X (2) (x) := k (x) k (y) : k (x) k (x); R (x; y) := k=1. k=1. We have denoted the complex line vector whose components are the conjugates of those of a complex (column) vector , and 2 is the inner 1. product of two complex (column) vectors 1 , 2 . The nn matrix Gram 2 is de
(19) ned by the usual formulas L. (3). (GramL2 ) :=. Z. kl. IR3. . k (x) l (x) dx :. Finally, "1 ::: "N are eigenvalues of H : Each one represents the energy of one of the electrons, in the mean
(20) eld created by the molecule. For physical reasons, we impose 0 < "k < 1 : Note that the scalars "k can also be seen as Lagrange multipliers. Indeed, the Dirac-Fock equations are the Euler-Lagrange equations of the Dirac-Fock energy functional. E ( ) =. N Z X k=1. IR3. k H0 k ? Z. . jx1j k . k . (x) (y) ? tr R (x; y)R (y; x) dxdy + 2 3 3 jx ? yj IR IR R under the constraints IR3 k l = kl : ZZ. The de
(21) nition of a ground state is not easy : the DF functional is not bounded from below. This fact is at the origin of serious diculties in the 3.
(22) numerical implementation, as well as the interpretation, of the DF equations (see [4] and the references therein). One way to deal with this problem, is to restrict the energy functional to the space (+ E )N , + being the projector on the space of positive states of a perturbed Dirac operator H0 +. [3, 14, 12, 15]. The associated Euler-Lagrange equations are the \projected" Dirac-Fock equations (4) + H + k = k k : The diculty here is in the choice of the projector. Note that, in the case k > 0, the \unprojected" (DF) equations can be written (5). + H + k = k k :. Here, + is the \self-consistent" projector on the positive spectral space of H . Numerical computations with various choices of projectors, and comparisons with experimental data, show that this self-consistent projector is the best choice (see e.g. [7, 11]). In fact, when + in (4) is not equal to + , the eigenvalues "k and the total energy E ( ) are underestimated (see [9] for an theoretical explanation based on perturbation theory). In [12] Mittleman derived the DF equations with \self-consistent projector" (5), from a variational procedure applied to a QED Hamiltonian in Fock space, followed by the standard Hartree-Fock approximation. More precisely, let + = (0;1) (H0 + ), let be a minimizer of the DF energy in the projected space (+ E )N under normalization constraints, and let E ( ) := E ( ). Mittleman showed that the stationarity of E ( ) with respect to. implies that + is the self-consistent projector associated to . His arguments are formal. One of our motivations in the present work, is to present a rigorous max-min method related to Mittelman's ideas. In [5] and [13], under some assumptions on , N and Z , an in
(23) nite sequence ( j )j 0 of solutions of (DF) was found. The assumptions in [13] cover all existing atoms. In [6], we were able to prove that, for small enough, the
(24) rst solution 0 can be viewed as an electronic ground state for the Dirac-Fock equations in the following sense: it minimizes the DiracFock energy among all electronic con
(25) gurations which are orthogonal to the \Dirac sea".. Theorem 1 [6]. Fix N; Z with N < Z + 1 and take suciently small. Then there exists a solution 0 of the (DF) equations, such that (6). E ( 0) = minfE ( ) ; GramL2 = 1IN ; ? = 0 g 4.
(26) where ? = (?1;0) (H ) is the negative spectral projector of the mean-
(27) eld operator H , and ? := (? 1 ; ; ? N ) :. The constraint ? = 0 has a physical meaning. Indeed, according to Dirac's original ideas, the vacuum consists of in
(28) nitely many electrons which completely
(29) ll up the negative space of H : these electrons form the \Dirac sea". So, by the Pauli exclusion principle, additional electronic states should be in the positive space of the mean-
(30) eld Hamiltonian H :. 2. A new \max-min" principle. First of all, note that the Dirac-Fock energy only depends on the N dimensional space spanned by . So, in the sequel, we consider that E is de
(31) ned on the Grassmannian manifold. G := fV subspace of H 1=2 (IR3 ; CI4 ) ; dimCI(V ) = N g : Similarly, since the mean-
(32) eld operator H only depends on V = Span( ), we shall denote it H V . The associated positive and negative energy projectors will be denoted +V and ?V . Recalling that E := H 1=2 (IR3 ; CI4 ); E := 0 (E ), let Then the set. G+ := fV 2 G ; V E + g :. U := fV 2 G ; dimCI(+0(V )) = N g. is an open neighborhood of G+ in G. Moreover, it has the structure of a
(33) ber bundle over G+, with canonical projection : V 2 U ! +0 (V ) 2 G+ : Note that U is isomorphic to the vector bundle [. V +2G+. fV +g LCI(V +; E ? ) :. This is due to the fact that any V 2 U is the graph of a unique linear map LV : +0 (V ) ! E ? : Now, let ? := fs 2 C 1 (G+ ; U ) ; s = IdG+ g : In terms of di erential geometry, ? is the set of smooth sections of U over G+ . Note that the canonical injection s0 : G+ ! U belongs to ?. 5.
(34) More generally, let F be a Hilbert subspace of E such that the restriction F of +0 to F is an isomorphism between F and E + . Then the map. sF : V + 2 G+ ! ?F 1 (V +) 2 U belongs to ?. The next lemma gives another important example of a smooth section over G+.. Lemma 2 . Fix N; Z with N < Z +1. Take > 0 small enough. Consider the set. S~ := fV 2 G : ?V (V ) = 0g : Then S~ U , and the map V 2 S~ ! +0 (V ) 2 G+ has a unique inverse s~ : G+ ! S~. The map s~ belongs to ? :. Proof of Lemma 2. Denoting KV := 1 (H V ? H0), we
(35) nd, as in the proof. of Lemma 1 of [8], +0 ? ?V = . Z +1 0. h. dz H02 + z 2. i?1 . H0 KV H V ? z 2 KV. h. 2. H V + z2. i?1. :. Since KV is independent of we
(36) nd, by estimates similar to those in [8] (proof of Lemma 3), that the map V 2 G ! 1 (+ ? ? ) 2 L(E ). 0. V. is bounded in C 1 , independently of . As a consequence,. V 2 G ! ?V 2 L(E ). is C 1 -close to the constant map V ! ?0 , for small. Then Lemma 2 directly follows from the implicit functions theorem. Note that when goes to 0, s~ converges in C 1 to the canonical injection s0 .. tu. We now de
(37) ne the max-min level. e0 := sup V 2inf E (V ) : s(G+ ) s2?. Our main result is the following:. 6.
(38) Theorem 3 . Let N < Z + 1. Take > 0 small enough. Then e0 = inf~ E (V ) = E ( 0 ) V 2S. with 0 de
(39) ned in Theorem 1, and S~ de
(40) ned in Lemma 2.. Proof of Theorem 3. Lemma 2 implies that inf~ E (V ) = inf + E (V ) e0 : V 2 s~(G ) V2S. Now, if we denote V 0 := Span( 0) and V 0;+ := (V 0 ) = +0 (V 0 ), it follows directly from [6] (proof of Lemma 9, Step 1), that (7). E (V 0 ) =. sup. W 2G + 0 (W )=V 0 +. E (W ):. ;. As a consequence, for any s 2 ?,. E (V 0 ) E (s(V 0;+)) ; since +0 (s(V 0;+ )) = s(V 0;+ ) = V 0;+ . We thus have. E (V 0 ) sup E (s(V 0;+)) e0 : s2?. But if we replace, in Theorem 1, the variable by the more intrinsic variable V = Span( ), we get inf~ E (V ) = E (V 0 ) : So Theorem 3 is proved.. V 2S. tu. Remark : One could try the following simpler \max-min" principle. Let F be the family of all Hilbert subspaces F of E such that the restriction F. of +0 to F is an isomorphism between F and E + . This family contains, in particular, the positive spectral subspaces associated to the perturbed Dirac operators H0 + , for \not too large". Let. (8). c00 := sup inf E (V ) : F 2F V 2G V F. 7.
(41) We have already seen that the formula s(V + ) := ?F 1 (V + ) ; V + 2 G+ ; de
(42) nes a section sF 2 ?, for all F 2 F . As a consequence, c00 c0 . For N = 1, the mean-
(43) eld Hamiltonian H V is simply the Dirac operator with
(44) xed potential ? Z jx1j , so it does not depend on V . Let F~ be its positive spectral subspace. Then the de
(45) nition of S~ (see Lemma 2) reduces to S~ = fV 2 G ; V F~ g : So Theorem 3 reduces to. c0 = inf E (V ) : V 2G V F~. When is small enough, F~ belongs to the class F , and the above formula implies that c0 c00 . So, in the case N = 1, c00 = c0 , and this energy level is the
(46) rst positive eigenvalue 1 of H0 ? Z jx1j . A formula similar to c00 = 1 has already been obtained in an unpublished work by Barbaroux and Siedentop (personal communication), in the more general context of the Dirac-Fock-Bogoliubov model, but with the same restriction N = 1. In the case N 2, it would be interesting to show that c00 = c0 . Unfortunately, we do not know how to prove this, even when is very small.. References [1] V. Bach, J.M. Barbaroux, B. Hel er, H. Siedentop. Stability of matter for the Hartree-Fock functional of the relativistic electronpositron
(47) eld. Doc. Math. 3 (1998), p. 353-364. [2] V. Bach, J.M. Barbaroux, B. Hel er, H. Siedentop. On the stability of the relativistic electron-positron
(48) eld. Comm. Math. Phys. 201(2) (1999), p. 445-460. [3] G. E. Brown, D.G. Ravenhall. On the interaction of two electrons. Proc. Roy. Soc. London. A208 (1951), p. 552-559. [4] P. Chaix, D. Iracane. The Bogoliubov-Dirac-Fock formalism. J. Phys. At. Mol. Opt. Phys. 22 (1989), p. 3791-3814. [5] M.J. Esteban, E. Sere. Solutions for the Dirac-Fock equations for atoms and molecules. Comm. Math. Phys. 203 (1999), p. 499-530. 8.
(49) [6] M.J. Esteban, E. Sere. Nonrelativistic limit of the Dirac-Fock equations. A.H.P. 2 (2001), p. 941-961. [7] I.P. Grant. Relativistic Calculation of Atomic Structures. Adv. Phys. 19 (1970), p. 747-811. [8] M. Griesemer, R.T. Lewis, H. Siedentop. A minimax principle for eigenvalues in spectral gaps : Dirac operators with Coulomb potentials. Doc. Math. 4 (1999), p. 275-283. [9] J.L. Heully, I. Lindgren, E. Lindroh, A.M. Martensson-Pendrill. Comment on relativistic wave equations and negative-energy states. Phys. Rev. A 33 (1986), p. 4426-4429. [10] Y.-K. Kim. Relativistic self-consistent
(50) eld theory for closed-shell atoms. Phys. Rev. 154 (1967), p. 17-39. [11] I. Lindgren, A. Rosen. Relativistic self-consistent
(51) eld calculations. Case Stud. At. Phys. 4 (1974), p. 93-149. [12] M.H. Mittleman. Theory of relativistic e ects on atoms: Con
(52) guration-space Hamiltonian. Phys. Rev. A 24(3) (1981), p. 1167-1175. [13] E. Paturel. Solutions of the Dirac equations without projector. A.H.P. 1 (2000), p. 1123-1157. [14] J. Sucher. Foundations of the relativistic theory of many-particle atoms. Phys. Rev. A 22 (2) (1980), p. 348-362. [15] J. Sucher. Relativistic many-electron Hamiltonians. Phys. Scrypta 36 (1987), p. 271-281. [16] B. Swirles. The relativistic self-consistent
(53) eld. Proc. Roy. Soc. A 152 (1935), p. 625-649.. 9.
(54)
Documents relatifs