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Physical Review

Exact-Energy Self-Consistent Field

Robert A. Kromhout

Published Jul 1, 1957• 6 citations
Publisher

Abstract

A self-consistent field Hamiltonian is constructed, for arbitrary pair interactions, which yields (in principle) an exact value for the energy of a chosen state of a system of $N$ particles, as well as estimates for other states. The method for extending the formalism to interactions involving arbitrary numbers of particles is indicated and is straightforward. The method is carried out within the bound-state framework, and while motivated by the recent work of Bethe, results in a Hamiltonian somewhat different from his. An argument is presented which indicates that the independent-particle eigenfunctions of this Hamiltonian are "good" representations of the original $N$-body system. A discussion of the practicality of the computations is included.

DOI
10.1103/physrev.107.215

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A self-consistent field Hamiltonian is constructed, for arbitrary pair interactions, which yields (in principle) an exact value for the energy of a chosen state of a system of NNN particles, as well as estimates for other states. The method for extending the formalism to interactions involving arbitrary numbers of particles is indicated and is straightforward. The method is carried out within the bound-state framework, and while motivated by the recent work of Bethe, results in a Hamiltonian somewhat different from his. An argument is presented which indicates that the independent-particle eigenfunctions of this Hamiltonian are "good" representations of the original NNN-body system. A discussion of the practicality of the computations is included.
A generalized self-consistent field (SCF) theory of many-particle systems is developed by modifying the usual functional relationship between energy and wave function. A parameter $\ensuremath{\xi}$, $0\ensuremath{\le}\ensuremath{\xi}\ensuremath{\le}1$, is thus introduced into the energy so that for $\ensuremath{\xi}=0$, the particles interact only with a self-consistent field, for $\ensuremath{\xi}=1$ the particles interact dynamically among themselves with the correct field, and for intermediate values of $\ensuremath{\xi}$ both SCF and dynamical interactions occur. A systematic development of the theory is given for time-independent and time-dependent probelms, for finite temperatures, and for both uniform and nonuniform systems. The derivative with respect to $\ensuremath{\xi}$ of the total energy, of free energy, is expressed in terms of the dielectric function and an improved version of the dielectric formulation of the many-body problem thereby obtained. A brief discussion of the advantages of the method, possible applications, and further generalizations or extensions is included.
A procedure is proposed to obtain the exact self-energy in the many-body problem. This algorithm is based on the formal analysis to reach the exact theory by repeated applications of an operator F to an arbitrarily chosen input self-energy represented as a functional of the dressed Green's function. The operator F is so defined that the microscopic conservation law for particle number is satisfied. The rigorous self-energy is obtained by the solution of an eigenfunction of F. Particular attention is paid to the relation between the present procedure and the Baym-Kadanoff framework of conserving approximations. By simplifying the procedure in F with use of the generalized Ward identity, we suggest a practical method to implement this algorithm rather easily in actual systems. In order to suggest future directions to improve on this practical method, the recently developed mean-field theory for the Hubbard model in the limit of high spatial dimensions is also discussed in the context of our theory.
A constrained energy extremal principle for natural orbitals and natural spin orbitals is demonstrated. The extremal principle is used to derive a one-electron Hamiltonian for the natural spin orbitals of a system of NNN electrons. The natural spin-orbital Hamiltonian depends on the one- and two-body density matrices, and is thus of the self-consistent-field type. Whether or not the natural spin-orbital Hamiltonian will be of any practical significance in the determination of exact or accurate natural spin orbitals depends upon finding a practical solution to the NNN-representability problem.
The independent-particle model in the theory of many-particle systems is studied by means of the self-consistent-field (SCF) idea. After a review of the characteristic features of the Hartree and Hartree-Fock schemes, the extension of the SCF method developed by Brueckner is further refined by introducing the exact reaction operator containing all correlation effects. This operator is here simply defined by means of the partitioning technique, and, if the SCF potentials are derived from this operator, one obtains a formalism which is completely analogous to the Hartree scheme but which still renders the exact energy and the exact wave function. An elementary derivation of the linked-cluster theorem is given, and finally the inclusion of various symmetry properties is discussed.
The total energy and electron addition and removal spectra can, in principle, be obtained exactly from the one-body Green's function (GF). In practice, the GF is obtained from an approximate self-energy. In the framework of many-body perturbation theory, we derive different expressions that are based on an approximate self-energy, but that yield nevertheless, in principle, the exact exchange-correlation contribution to the total energy for any interaction strength. Response functions play a crucial role, which explains why, for example, ingredients from time-dependent density functional theory can be used to build these approximate self-energies. We show that the key requirement for obtaining exact results is the consistent combination of ingredients. Also when further approximations are made, as it is necessary in practice, this consistency remains the key to obtain good results. All findings are illustrated using the exactly solvable symmetric Hubbard dimer.