A systematic method of handling a system composed of a number of particles and an intermediary Bose field has been developed. The main idea consists in linearizing the interaction by taking certain average values so that it may be amalgamated to the free Hamiltonian. Only the remaining fluctuational interaction is treated by perturbation theory. The effect of the averaged interaction appears in the form of two generalized Hartree fields associated with the particle and the Bose quantum, respectively. The self-consistent equations for these Hartree fields are derived from an analysis of the
S matrix, keeping close analogy to the renormalization procedure in quantum electrodynamics. The formalism includes such approximations as the Bohm-Pines, Hartree-Fock, and Thomas-Fermi theories as special cases. General criteria about the validity and plausibility of the picture involved in the present formulation are discussed from various points of view.
A methodology is introduced for the systematic treatment of electron-electron correlations in solids and other interacting quantum
N-particle systems. The method is developed within the framework of electronic structure theory (band theory) but, in contrast to conventional approaches, which are based on the single-particle picture, it is formulated within a many-particle picture in which
n particles in
d-dimensional phase space are treated as a single particle in a phase space of
nd dimensions. In this phase space, interparticle interactions appear as external potentials allowing the treatment of the system of particles through the use of single-particle methods, while at the same time allowing a systematic, direct, and nonperturbative treatment of interparticle interactions. The method makes use of the invariance of the Hamiltonian describing an interacting-particle system under partitioning into subsystems of
n particles. This treatment leads to exact results in the limit $n\ensuremath{\rightarrow}N$. Based on such partitioning, we propose a generalization of density functional theory and an appropriately defined local density approximation to treat the interactions between the
n-particle units in a system of
N> n particles. This approach yields
n-particle correlated densities and
n-particle states which can be used in an analysis of the electronic properties of materials, such as total energy, phase stability, electronic transport, and others. We use the formal construct of multiple-scattering theory to develop the method for the calculation of the two-particle electronic structure of a solid and the corresponding total energy of the ground state. We also illustrate some of the properties of the method in terms of a Hubbard model Hamiltonian on a linear ring. Various features of the method and further possible applications are presented in a discussion section.
A system of
N 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.
A finite many-particle system can have collective states for which the off-diagonal matrix elements of certain one-particle operators are of the same order of magnitude as the diagonal elements. In such cases it is suggested that the random-phase approximation is in need of generalization. Examples are the uniform translational motion of any system and the rotational motion of deformed nuclei. The generalization is suggested after a review and critical analysis of the Hartree-Fock approximation. The model single-particle wave functions of the latter are replaced by wave functions in a space labeled both by the particle variables and by the quantum numbers of the collective motion. These generalized amplitudes are defined field-theoretically, and a self-consistent scheme for their calculation is obtained from the equations of motion. In addition to the self-consistent potential defined in the enlarged space, the energies of the excited states also turn out to be given by a natural self-consistency requirement. The new calculational scheme is first applied to a systematic restudy of the random-phase approximation where the self-consistency requirement on the energies has previously been overlooked. As a first characteristic application we obtain without "pushing" the total mass of a system in uniform translation, and a reinterpretation of the Hartree-Fock average field.