A density functional representation of quantum chemistry. III. Rigorous realization of the program in lattice space

Abstract

AbstractsA mathematically rigorous reformulation of molecular quantum mechanics in terms of the particle density operator and a canonically conjugated phase field is given. Using a momentum cutoff, it is shown that the usual molecular Hamiltonian can be expressed in terms of the particle density operator and a rigorously defined phase operator. It is shown that this Hamiltonian converges strongly to the cutoff-free Hamiltonian. In spite of the fact that this Hamiltonian is of second order in the phase operators, all hitherto published expressions are not correct. Unfortunately, the correct formulation destroys the intuitive appeal of using the particle density operator as a coordinate for the many-body problems of quantum chemistry. Unless somebody provides an essential new and clever idea, we propose to resist the fascination of a local quantum field theory of molecular matter in terms of the particle density operator.On utilise I'opCrateur de densit6 de particules et son conjuguC, l'operateur de densit6 de phase, comme variables dynamiques fondamentales pour reformuler la mdcanique quantique molCculaire en toute rigueur mathtmatique. Par limitation d'impulsion l'hahituel Hamiltonien moliculaire peut Ctre exprim6 exactement B l'aide d'un opCrateur de densit6 de charge et d'un opkrateur de phase dtfinis rigoureusement. On dCmontre que cet opdrateur converge dans la topologie forte vers l'Hamiltonien sans limitation d'impulsion. Bien que 1'Hamiltonien ne contienne l'op6rateur de phase qu'en second ordre, toutes les formulations antCrieures se revitlent n'Ctre pas correctes. Malheureusement, la formulation exacte du problbme enlkve a une telle entreprise une grande partie de son intCrCt. A notre avis on ne peut espCrer que la reformulation de la mCcanique quantique molCculaire B l'aide de l'opkrateur de densitC de particules puisse Ctre d'une utilitd quelquonque, avant que n'apparaissent des idCes substantiellement neuves et pourvues d'intelligence.

Energy Density Functional Theory of Many-Electron Systems

“…is a very difficult one (Primas, 1967;Primas and Schleicher, 1975;Schleicher and Primas, 1975). In order to illustrate some of its complications it is instructive to carry out a detailed comparative analysis between the real t(r) == t( [-y(r; r'll; r) and its local density forms tn· ([p(r)]; T) and tTF ([p(r) ) since the latter varies in an infinite interval, including positive and negative values and produces all the local peculiarities of the quantum kinetic energy (for example, see Figure 4.20, we plot t(l), tw and tTF for the Hartree-Fock ground-state of the neon atom.…”

“…According to Smith et al (1979), "it gives no direct guidance to the construction of the density functional which, in its dependency on the density, in principle can be and most likely is much more complex than anything". It is clear that the Hohenberg-Kohn energy density functional is valid, or well-defined only at the ground-state electron density, and, for that reason, it implicitly absorbs the statistics of the N -electron system under study and the form of its Hamiltonian, and in that sense, it becomes valid as the expectation nlue of the Hamiltonian at the exact level (~Iikolas and Tomasek, 1977;Primas, 1967;Berrondo and Goscinski, 1975;Schleicher and Primas, 1975;Kryachko, 1980aKryachko, , 1984Parr, 1983;l\Iaynau et al, 1983). The first attempts in this direction have been made by Theophilou (1972) and Osaka (1974a,b) who have developed a formal theory for deriving Ev[Po(i)] for an inhomoE;eneouselectron gas.…”