Abstract

The objective of this thesis is to compare the upper bound of the ground state energy of doubly ionized carbon (C⁺⁺) to the lower bound obtained by reduced density matrices.

To make a fair comparison, the number of orbitals used in both is the same, namely ten. The upper bound is obtained by use of the standard Rayleigh-Ritz Variational Principle which states ⟨Ψ|H|Ψ⟩ ≥ Eo(H) were Eo(H) is the exact ground state energy, H is the Hamiltonian, and Ψ is the trial wave function.

Abstract An improved ground state wavefunction for Helium for use in density matrix calculations has been obtained. This improvement was achieved by the introduction into the expansion for the singlet S-state wavefunction for Helium, of a term containing explicit angular dependence. This angular term was constructed so as to preserve angular momentum symmetry by being an eigenfunction of the two-particle, angular momentum operator with eigenvalue zero, as is required for S-states. The results give...

Abstract The solution to the problem of a one-electron ion under the influence of various generalized central-field potentials is considered. Solutions are obtained using a finite expansion of the perturbation wave functions in terms of associated Laguerre polynomials. Higher orders of perturbation theory are obtained without having to perform integrations over the continuum states, and the solution can be written explicitly to any order. This perturbation procedure is applied to the three...

Abstract A survey of the relative efficiency of various approximate methods for solving the non-relativistic Schroedinger Equation [sic] is made. In particular, a comparison is made of the results obtained for the problem of a simple harmonic oscillator perturbed by λx∝, (∝ =3,4 ) while using the Rayleigh-Ritz variational principle, Linear Variation of Functions, and the Reyleigh-Schroedinger [sic]...

Abstract

A recently developed theory enabling the construction of density matrices which are expressible in terms of a given spin geminal set and which are a nearly N-representable as possible is discussed.

Calculation of lower bounds to the ground state energy of the C⁺⁺ ion is performed utilizing reduced density matrices subject to the continuously variable N-representability constraint determined by the above mentioned theory.

Abstract The objective of this paper is to demonstrate the usefulness of density matrix techniques in the determination of the ground state energy of certain atomic systems. Use of the method, currently under development, results in a lower energy bound in contrast with the Rayleigh-Ritz method which yields an upper bound. The former technique is illustrated for the C⁺⁺ atomic system. It was found that the Pauli Principle was being violated even when the 2-particle density matrix is expanded in...

Abstract In Part I, a large parameter asymptotic expansion of a perturbed simple harmonic oscillator is presented. Perturbations of the form λx∝ are studied were λ is the coupling constant. The method is demonstrated by a general derviation of the wavefunctions to third order for any ∝ and quantum number n. Additonal expressions are presented for the ground states to tenth order in the energy. It is also shown that the "matching condition" for the basic and...

Abstract The non-relativistic energies have been determined for the singlet and triple S states (1¹S, 2¹S and 2³S) of the helium like two electron system. Calculations were based on a finite set of orbitals of the order 7. The set, which includes Laguerre polynomials, is expressed as a superposition of configurations built up from one-electron wave functions or orbitals and approximates the exact wave function. Application of the variational principle leads to a system of linear equations....