We suggest to use the Newton iteration method for constructing a (locally unique) solution of the atomic and nuclear Hartree-Fock equations for an arbitrary number of particles. Our proposal is based on a theorem by Kantorovic and rests on the following points: 1) the two-body potential must satisfy a boundedness condition; 2) the zero-order approximation, used to start the iteration sequence, must satisfy certain conditions, to be proved numerically. Condition 1) holds, for instance, for all local potentials, defined by a bounded function and for a class of nonlocal potentials; it does not hold for local potentials, behaving as 1/r near the origin.

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Solution of the Hartree-Fock equations

Hartree-Fock theory with constrained deformation

Differential properties of the energy functional on the set of Slater determinants are defined according to a new differential calculus on complex Banach spaces previously considered. Such properties permit us to derive, with a new and rigorous procedure, the Hartree–Fock equations and to define an iterative method of the gradient type to minimize this functional on the above set. For a large class of one‐body and two‐body potentials, including Coulombic potentials, we show that our gradient method yields a decreasing sequence of values of the energy functional and a sequence of approximate solutions of the Hartree–Fock equations. The convergence of the latter sequence to a solution of these equations is proved in any finite‐dimensional subspace.

On the extremum problem of the energy functional on the set of Slater determinants

Consistent symmetries in a rigorous approach to the Hartree-Fock problem

The problem of existence and uniqueness of solutions of the Hartree-Fock equations is studied by means of the Lyusternik-Schrinlmann category theory In the case of a box with all the states with negative energies the existence and uniqueness of the solutions is discussed

Limitations and validity of the Hartree-Fock method

A rigorous iteration method for the solution of nonlinear equations in Banach spaces, due to Isaac Newton and Kantorovic, is reported. We consider its application to three nonlinear problems of atomic and nuclear physics, in finite-dimensional spaces: the ordinary, the constrained and a generalized Hartree-Fock problem. We have proved the existence of a solution of the Hartree-Fock equations and its uniqueness in a definite small region of the functional space, for the nucleus16O. This method converges faster than the usual Hartree algorithm and displays some further advantages.

A rigorous method for solving nonlinear equations: Hartree-Fock and related problems

The homogeneous Bethe-Salpeter equation (BSE) of a (1/2)$^+$ bound system, that has both fermionic and bosonic degrees of freedom, that we call a {\em mock nucleon}, is studied in Minkowski space, in order to analyse the chiral limit in covariant gauges. After adopting an interaction kernel built with a one-particle exchange, the $\chi$-BSE is numerically solved by means of the Nakanishi integral representation and light-front projection. Noteworthy, the chiral limit induces a scale-invariance of the model and consequently generates a wealth of striking features: i) it reduces the number of non trivial Nakanishi weight functions to only one; ii) the form of the surviving weight function has a factorized dependence on the two relevant variables, compact and non-compact one; iii) the coupling constant becomes an explicit function of the real exponent governing the power-law fall-off of the non trivial Nakanishi weight function. The thorough investigation at large transverse-momentum of light-front Bethe-Salpeter amplitudes, obtained with massive constituents, provides a confirmation of the expected universal power-law fall-off, with exponents predicted by our non-perturbative framework. Finally, one can shed light on the exponents that govern the approach to the upper extremum of the longitudinal-momentum fraction distribution function of the {\em mock nucleon}, when the coupling constant varies.

The chiral limit of a fermion-scalar $(1/2)^+$ system in covariant gauges

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A rigorous method for solving nonlinear equations: Hartree-Fock and related problems

The chiral limit of a fermion-scalar $(1/2)^+$ system in covariant gauges