E.H. Lieb and B. Simon. The Hartree-Fock theory for Coulomb systems. Comm. Math. Phys., 53(3):185--194, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....equation (1. 2) by simpler (usually non linear) ones in 3 dimensions (Thomas Fermi theory and Hartree Fock theory) The relation of these approximations with the N electron Schr odinger equation has been analysed in detail for big atoms, see e.g. Lieb and Simon [10] Lieb [9] and Lieb and Simon [11]. One important observation and motivation for the development of these and other approximation schemes was the insight that in order to calculate the energy E or one and two electron operator expectation values there is no need for the full wave function (x 1 ; x 2 ; xN ) but only for ....

Lieb, E. H., and Simon, B. The Hartree-Fock Theory for Coulomb Systems. Comm. Math. Phys. 53, 3 (1977), 185-194.

....energy . This trial projection can be constructed such that it has an energy arbitrarily close to the original N dimensional projection. Therefore we also have that (N; Z) inf = fl; fl = fl; Trfl N : 29) This Hartree Fock minimization problem was studied by Lieb and Simon in [13]. They proved the following about the existence of minimizers. 3.4. THEOREM (Existence of HF minimizers) If N is a positive integer such that N Z 1 then there exists an N dimensional projection fl minimizing the functional E in (26) i.e. E (N; Z) E ) is a minimum. In the ....

....(Bound on the ionization energy) The ionization energy of a neutral atom E (Z; Z) is bounded by a universal constant (in particular, independent of Z) This theorem is proved in Sect. 13 on page 73. The variational equations (Euler Lagrange equations) for the minimizer was also given in [13]. Since the Hartree Fock variational equations shall be used later in this work, we shall derive them in Theorem 3.11 below. We first note that the Hartree Fock functional E may be extended from projections (i.e. density matrices with fl = fl) to all density matrices. If Tr [ Gamma Deltafl ....

E. H. Lieb & B. Simon, The Hartree-Fock theory for Coulomb systems, Commun. Math. Phys., 53, 185--194, (1977).

....in the case of the smeared nuclei brings no additional difficulty and the proofs are even easier in that case (see [11] In the Hartree Fock setting, the existence of a minimizer for neutral molecules for the standard Hartree Fock model (2.7) 2.8) has been proved by E. H. Lieb and B. Simon in [33] and by P. L. Lions in [37] Moreover, the equivalence between the standard Hartree Fock model (2.7) 2.8) and the Hartree Fock model stated in terms of density matrices (2.3) 2.5) without restricting the minimization to projectors) is due to E.H. Lieb [30] Lieb s proof has been simplified ....

E. H. Lieb & B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53, 1977, pp. 185--194.

....energy, and only depends on the density ae. The second term, the so called exchange energy E ex , takes into account the Pauli principle (a purely quantum effect) but a priori does not depend only on the density. The existence of a minimum for (15) which is a difficult problem when posed on R 3 [LS2], PLL] can be more easily proved here since Omega is compact. This minimization gives (after a unitary transformation on the orbitals) the so called Hartree Fock equations Gamma Delta i V ext (x) i ( 1 jxj ae) i (V ex i ) x) ffl i i (21) where (ffl i ) are the eigenvalues and ....

E.H. Lieb and B. Simon "The Hartree-Fock theory for Coulomb systems" Comm. Math. Phys. 53 (1977) 185 - 194

....The story began twenty years ago from a rather theoretical standpoint with the fundamental contributions of E.H. Lieb, B. Simon, H. Brezis and coworkers, and continued with the works of P L. Lions. A rapid list of the most significant articles in this field should include at least the following [22, 23, 24] (For a complete list, we refer to [5, 11, 12] To this day, it sounds reasonable to claim that most of the molecular models of Quantum Chemistry are now well understood mathematically and have been carefully analyzed 1 .The focus has now turned either towards the side of the study of the ....

E.H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems,Com- mun. Math. Phys. 53 (1977) 185-194

....The Hartree model was historically introduced by Hartree in [19] It is a well known fact that, for any subset of R 3 , this minimization problem is attained by at least one vector ( 1 ; Delta Delta Delta ; jj ) with i 0 for every 1 i jj (see the works by E.H. Lieb and B. Simon in [33] and by P. L. Lions in [35] In the smeared nuclei case, the energy functional of the Hartree model reads as follows E m;H ( 1 ; Delta Delta Delta ; jj ) jj X i=1 Z R 3 jr i j 2 Gamma 1 2 D(j i j 2 ; j i j 2 ) Gamma Z R 3 V m ae 1 2 D(ae; ae) 22) and ....

E. H. Lieb & B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53, 1977, pp. 185-194.

....Euler Lagrange equations have a form similar to (1.19) with # instead of H 0 in the expression of H # . The physically interesting states correspond to # 1 # . # # N 0, and the ground state minimizes EHF on #, which implies that # 1 , # N are the N first eigenvalues of H # (see [30]) In the DF model, the physically interesting states correspond to 0 # k 1 : a positive energy inferior to the rest mass of the electron. The definition of a ground state is less clear : the DF functional has no minimum on #. This fact is at the origin of serious di#culties in the numerical ....

.... (see [22] In [35] Mittleman derived the DF equations with self consistent projector (1.21) from a variational procedure applied to a QED Hamiltonian in Fock space, followed by the standard Hartree Fock approximation. Important existence results are known on the HF equations. Lieb and Simon [30] proved the existence of a ground state of EHF on #, provided N Z 1, where Z = P m i=1 Z i . P.L. Lions [33] proved the existence of infinitely many excited states if N # Z. Using inequality (1.7) one can easily extend the results of [30, 33] to the projected equations (1.20) assuming ....

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E. H. Lieb, B. Simon. The Hartree-Fock theory for Coulomb systems. Comm. Math. Phys., 53 (1977) p. 185-194.

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E. H. Lieb and B. Simon, The Hartree--Fock theory for Coulomb systems, Comm. Math. Phys. 53 (1977) 185--194.

....known but seemingly was not. It applies to repulsive two body potentials (as in the real world of electrons with Coulomb interaction) and states two things. The first is that the N one particle states are precisely the energetically lowest eigenvectors of the HF operator. This fact was stated in [LS], and the proof was sketched in [LE4] While the N HF orbitals are distinct eigenvectors of the HF operator, it is not obvious, a priori, that they are the lowest ones; indeed, this might not be true when the interactions are attractive. The second part is surprising, for its conclusion runs ....

....clearly the case that the infimum in (2c.9) is attained since the set of admissible denisity matrices is a compact subset of a finite dimensional space. In case of the Publ. in J.Stat.Phys. 76:3 90, 1994 atomic Hamiltonian it is also true that the infimum is attained. This result was proved in [LS], where it was assumed that ff = 0, but this follows from Theorem 2.11 below) A 1 pdm for which the infimum (2c.9) is attained defines a HF ground state. To define the finite temperature HF Gibbs state we must introduce the entropy of a quasi free state: S ( Gamma) Gamma 1 2 Tr[ Gamma ln ....

E.H. Lieb and B. Simon. The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys., 53:185--194, 1977.

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E.H. Lieb and B. Simon. The Hartree-Fock theory for Coulomb systems. Comm. Math. Phys., 53(3):185--194, 1977.

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