For an
n-fermion system in which each particle can occupy any one of
k states, an "irreducible
b-body operator" is defined as an operator belonging to [
2b, ${1}^{k\ensuremath{-}2b}$] of
SUk. It is shown that such an operator can always be written as a
b-body operator multiplied by a function of
n. A measure of the failure of a basis to diagonalize the Hamiltonian is constructed. If the Hamiltonian is analyzed into irreducible
b-body parts, the measure of error for
n particles can be calculated from two-particle parameters. An upper limit on the error of a group theoretically defined basis can be found without having to calculate any
n-particle interaction matrix elements. A measure of the magnitude of an interaction is defined, and shown to depend differently on
n for irreducible 0-body, 1-body, and 2-body interactions. The effect of the Pauli principle on the formation of shell-model potentials is discussed from this point of view.
Abstract An explicitly connected commutator expansion for the average value of an observable in the coupled‐cluster theory is derived. Specifically, it is shown that the expectation value of an operator for the state Ψ, related to the Fermi vacuum Φ by the exponential Ansatz ψ = eT Φ, is expressed as a finite commutator series containing the cluster operator T and an auxiliary operator S, defined by a linear equation involving again a finite commutator series in T. The above result is applied to derive the explicitly connected commutator form of the order‐by‐order many‐body perturbation theory (MBPT) expansion for the expectation values and density matrices. We also show how the commutator expansion derived by us can be used as a basis for size‐extensive infinite‐order summation techniques. An operator technique of eliminating the nonlocal, “off‐energy shell” denominators from MBPT expressions is proposed and applied to obtain compact commutator formulas for the expectation values of one‐ and two‐electron operators through the fourth and third order, respectively, and for the correlation energy through the fifth order of perturbation theory. © 1993 John Wiley & Sons, Inc.
A theory of strongly interacting Fermi systems of a few particles is developed. At high excitation energies (a few times the single-particle level spacing) these systems are characterized by an extreme degree of complexity due to strong mixing of the shell-model-based many-particle basis states by the residual two-body interaction. This regime can be described as many-body quantum chaos. Practically, it occurs when the excitation energy of the system is greater than a few single-particle level spacings near the Fermi energy. Physical examples of such systems are compound nuclei, heavy open shell atoms (e.g. rare earths) and multicharged ions, molecules, clusters and quantum dots in solids. The main quantity of the theory is the strength function which describes spreading of the eigenstates over many-particle basis states (determinants) constructed using the shell-model orbital basis. A nonlinear equation for the strength function is derived, which enables one to describe the eigenstates without diagonalization of the Hamiltonian matrix. We show how to use this approach to calculate mean orbital occupation numbers and matrix elements between chaotic eigenstates and introduce typically statistical variables such as temperature in an isolated microscopic Fermi system of a few particles.
A theory of strongly interacting Fermi systems of a few particles is developed.At high excitation energies (a few times the single-particle level spacing) these systems are characterized by an extreme degree of complexity due to strong mixing of the shell-model-based many-particle basis states by the residual twobody interaction.This regime can be described as many-body quantum chaos.Practically, it occurs when the excitation energy of the system is greater than a few single-particle level spacings near the Fermi energy.Physical examples of such systems are compound nuclei, heavy open shell atoms (e.g.rare earths) and multicharged ions, molecules, clusters and quantum dots in solids.The main quantity of the theory is the strength function which describes spreading of the eigenstates over many-particle basis states (determinants) constructed using the shell-model orbital basis.A nonlinear equation for the strength function is derived, which enables one to describe the eigenstates without diagonalization of the Hamiltonian matrix.We show how to use this approach to calculate mean orbital occupation numbers and matrix elements between chaotic eigenstates and introduce typically statistical variables such as temperature in an isolated microscopic Fermi system of a few particles.