Abstract Expressions for moments of spectral density distributions of a many‐electron Hamiltonian defined in a finite‐dimensional, antisymmetric, and spin‐adapted model space (as, e.g., a full configuration interaction space) are derived. The moments are expressed in terms of combinations of two‐electron integrals corresponding to a symmetric (a two‐electron singlet) and antisymmetric (two‐electron triplet) two‐electron wave functions. A diagrammatic approach based on Hugenholtz‐type diagrams and leading to a simple and universal classification scheme of the terms appearing in the expression for a specific moment is proposed. © 1995 John Wiley & Sons, Inc.
Spectra of several types of finite Heisenberg lattices have been studied using methods of statistical spectroscopy. In particular, spectral density distributions, discrete spectra generated from the spectral density distribution moments, and the distributions of spacings between the neighboring energy levels have been analyzed. The densities have been shown to be nearly Gaussian and the spectra generated from their several-moment Gram-Charlier approximations are in a good agreement with the exact ones, except for several extreme energy levels. Spacings between the neighboring levels exhibit clustering of the energy levels stronger than that given by the Poisson distribution, showing an essential influence of the point-group symmetries of the lattices.
The density matrix for a many-electron system has been examined. A formalism has been arrived at which facilitates its evaluation in terms of a basis corresponding to a discrete spectrum of eigenvalues. This allows a representation to be employed associated with a suitably chosen approximation to the Hamiltonian. Thereby, reasonably accurate estimates of the properties of many-electron systems may be anticipated for low orders of approximation.In developing the formalism, attention was focused upon a means for approximating the quantity
exp(zH), where H is the many-particle Hamiltonian and
z is a complex number. It was found possible to represent this quantity exactly as an infinite product of exponential factors, each of which depends upon a portion of the Hamiltonian alone. The relation of this result to statistical mechanical applications is indicated. An approximation procedure is described in terms of which various finite products of exponential factors approach the desired quantity in the limit of indefinitely increasing numbers of factors; this limit is approached in a manner that permits any given approximation to contain all the terms of the previous approximation.The first order of approximation was employed to approximate the many-particle density matrix. The resulting theory was applied to the helium-like systems: He,
Li+,
Be++. The energies of the ground
1S state of helium and the three lowest states of
Li+ and
Be++ were calculated. The agreement with experimental term values is reasonably satisfactory, the discrepancy between calculations and experiments not exceeding one percent. The triplet-singlet splitting for
Li+ was in the order observed; the magnitude calculated was about one-half that observed.The relation between the present theory and the Thomas-Fermi theory is discussed. It is pointed out that the former is the analog of the latter when the density matrix is expressed in other than eigenfunctions of momentum. The quasi-classical character of the approximation is discussed.