Unitary Group Approach to the Many-Electron Correlation Problem

Abstract

The fascination with symmetry and the multitude of its manifestations in nature ean be traced back to the earliest times of human civilization. Its importance as a basic unifying principle for our understanding of the physical universe can hardly be overestimated. Initially it was of course the most conspicuous geometrical symmetry, with its unquestionable aesthetic appeal, that enabled scientists to systematically classify erystal structures and unravel the eomplexities of molecular spectra. More reeently, especially with the advent of quantum mechanics, an entirely new universe of very subtle symmetries was opened up and the intrinsic elegance and mathematical beauty of fundamental physical la ws of nature was revealed, manifesting itself in countless phenomena ranging from the discovery of antiparticles to our understanding of the periodic table and ehemical bonding. Nowadays, these limits are further expanded by probing various dynamical and gauge symmetries, quasi-symmetries and so-called supersymmetries.While classical geometrie al symmetries deal with at most three-dimensional objeets and are easily comprehended and appreciated by our senses. various symmetries that are relevant in the submicroscopic world characterize objects and concepts that can only be defined in highly-dimensional abstract spaces. Even here, however, there is a fundamental difference between present-day classical symmetries that are characterized by the so-called invariance, symmetry or degeneracy group of the Hamiltonian of the studied system (see, for example, Hamermesh, 1962;McWeeny, 1963;Chen, 1989; and references therein) and the rather recently introduced dynamical groups or spectrum generating algebras (Bohm et al., 1988) that are required to contain all the bound states of the system within a single irreducible representation (e.g., SO(4,2) or SO(2,

On the use of Brillouin-Wigner perturbation theory for many-body systems

“…This feature guarantees the existence of a natural gap and thereby rapid convergence of the perturbation series'. Hubač and his co-workers [23] have explored the use of Brillouin-Wigner perturbation theory in solving the equations of coupled cluster theory [34,36]. By adopting an exponential expansion for the wave operator they obtain the Brillouin-Wigner coupled cluster theory [25,26] which is entirely equivalent to other manybody formulations for the case of a single reference function since the perturbation expansion is summed through all orders.…”

Spin-dependent unitary group approach to the Pauli–Breit Hamiltonian. II. First order energy level shifts due to spin–spin interaction

“…13, the Pauli-Breit Hamiltonian is expressible in the second quantized formalism as a sum of terms (8) where Ho is the usual spin-independent molecular Hamiltonian (5), the remaining terms constituting the relativistic corrections. The second term is a spin-independent oneelectron operator and thus is expressible as a linear combination of orbital U(n) generators (2) Finally the spin-spin interaction is expressible in terms of the (two-electron) orbital replacement operators (6) and the spin-orbital replacement operators (17) It is worth noting that the (two-electron) operators of Eqs.…”

“…To illustrate the main effects of the spin-spin interaction on a predominantly spin-conserved system, here we consider the problem, within the UGA framework, of calculating the eigenvalues of the spin-dependent operator (48) Here Ho is the usual spin-independent molecular Hamiltonian (5) and Hss is the spin-spin interaction. Since the changes in energy due to this spin-dependent interaction are expected to be small for many systems, owing to its relativistic nature, we may apply standard RayleighSchrodinger perturbation theory (for degenerate states).…”

An explicit basis of lowering operators for irreducible representations of unitary groups

“…Elliott's SU(3) model of the nucleus provides a bridge between the standard and collective models [2][3][4], and various low-dimensional unitary groups have been used in particle physics [5]. More general unitary groups arise in the many-body problem [6,7], quantum chemistry [8,9], and in quantum computation [10,11].…”