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Ground state energy fluctuations in nuclear matter II

Arturo De Pace, Herman Feshbach, Alfredo Molinari

Published Jan 1, 1999• 3 citations
OpenAlex

Abstract

Improvements are performed on a recently proposed statistical theory of the mean field of a many-fermion system. The dependence of the predictions of the theory upon its two basic ingredients, namely the Hartree-Fock energy and the average energy of the two particle–two hole excitations, is explored. Typeset using REVTEX 1 I.

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Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Mar 8, 1940•2 citations
The general form of the energy of the ground state of a many-fermion system is shown to be exactly of the form proposed by Brueckner and Bethe, without approximation. In a variational treatment, if the trial wave function is picked containing only pair correlations, together with all possible unlinked pairs, it is described by a two-body excitation matrix 〈m1m2∣A∣p1p2〉〈{m}_{1}{m}_{2}|A|{p}_{1}{p}_{2}〉〈m1​m2​∣A∣p1​p2​〉. Variation of this matrix in the Ritz-Rayleigh principle yields a set of integral equations of the scattering type for the matrix AAA. Hole-state energies are given self-consistently in terms of the matrix AAA, but particle-state energies are Hartree-Fock energies. This may be corrected for by widely enlarging the class of terms admitted into the wave function. If the approximation is then made of omitting a class of terms, defined as cross-linked clusters in $〈\ensuremath{\psi}|H|\ensuremath{\psi}〉$, the particle-state energies are easily renormalized. Variation then leads to an infinite hierarchy of integral equations.
A new theory of treating the ground-state correlations and the correspond-ing renormalization effects is proposed. In the theory, two-particle-two-hole and four-particle-four-hole excitations are taken up as the important parts in the ground-state correlations. The correlated true ground state is given through multiplication of the unitary operator to the Hartree-Fock ground state. The unitary operator is determined from the condition of minimizing the energy of the correlated true ground state. New single-particle operators are defined through the unitary transformation of the Hartree-Fock single-particle operators in the second-quantization formalism. The transformed Hamiltonian represented by the new single-particle operators is established to display the renormalization effects in the single-particle energies and the two-body matrix elements, and further it is proved that in the Hamiltonian the two-body interactions inducing the ground-state correlati6ns are reduced to small and negligible values. With this Hamiltonian, we are led to a simple
A system of NNN antisymmetric particles, moving under the influence of a fixed potential and their mutual many-particle interactions, is investigated in the ordinary Hartree-Fock scheme, having the total wave function approximated by a single Slater determinant. It is shown that all the density matrices of various orders, the wave function, and the entire physical situation depends only on a fundamental invariant $\ensuremath{\rho}({x}_{1}, {x}_{2})$, which is identical with the first-order density matrix. The Hartree-Fock equations are expressed in terms of this quantity.The Hartree-Fock equations are also solved by expanding the eigenfunctions in a given complete set, and applications to the MO-LCAO theory of the electronic structure of molecules, and crystals are given. It is shown that, in this scheme, the entire physical situation depends on a charge- and bond-order matrix $R(\ensuremath{\nu}\ensuremath{\mu})$ with respect to the ordinary atomic spin-orbitals involved. The Hartree-Fock equations for this matrix are investigated.Finally, the ionized and excited states are investigated, and it is shown that the Hartree-Fock scheme has a high degree of physical visuality also in case of many-particle interactions. The excitation energy of the system is the difference (${{\ensuremath{\omega}}_{j}}^{\ensuremath{'}}\ensuremath{-}{\ensuremath{\omega}}_{i}$) between two "spin-orbital energies," being eigenvalues to the effective Hamiltonians associated with the two states under consideration.
The ground-state wave function and energy of a finite system of interacting fermions are expanded in terms of multiple-particle excitations on an uncorrelated zero-order state. The resulting set of coupled equations constitutes a systematic variational generalization of Hartree-Fock theory. Comparison is made with many-body perturbation theory and it is shown that to any order the theory incorporates an infinite number of perturbation theory terms. Solutions of the equations for ground-state atomic systems are discussed and related to previous work using many-body perturbation theory. It is shown that the sums of perturbation terms necessary for convergence are automatically included in the equations for two-particle excitations. Application of the equations to open-shell atoms is described.
Abstract The calculation of the energies of heavy nuclei presents a many-body problem with a very large number of particles. For the treatment of this problem the statistical method seems to be indispensable. But the statistical method, as applied hitherto in nuclear physics, has some serious drawbacks. It proceeds from the Hartree approximation, each particle being supposed to move in the average field of the other particles. With this approximation the total wave function can be represented by a determinant of independent wave functions. The calculation of the energy then leads to expressions containing the wave functions only in the form of densities or mixed densities ρ(r) = Σiϕ*i (r) ϕi(r); ρ(r, r') = Σ i+jϕ*i (r) ϕ*j (r') ϕi(r') ϕj(r), (1.1) where the summation extends over all independent wave functions.