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The Journal of Chemical Physics

Many-body interaction free energy to infinite order. II. Collective behavior

T. B. MacRury, Bruno Linder

Published Jun 15, 1973• 3 citations
Publisher

Abstract

The closed-form expressions for the infinite order free energy, derived in the preceding paper [J. Chem. Phys. 58, 5388 (1973), Paper I] are recast in several different forms which focus on the collective behavior of the many-body system. In particular, it is shown that the present formulation is equivalent to the treatments based on collective modes of motion and to those developed on the basis of time-dependent Hartree theory.

DOI
10.1063/1.1679158

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The closed-form expressions for the infinite order free energy, derived in the preceding paper [J. Chem. Phys. 58, 5388 (1973), Paper I] are recast in several different forms which focus on the collective behavior of the many-body system. In particular, it is shown that the present formulation is equivalent to the treatments based on collective modes of motion and to those developed on the basis of time-dependent Hartree theory.
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A microscopic theory of collective and independent-particle motion in many-fermion system is developed in the framework of the classical theory. The basic idea is an extension of the conventional time-dependent Hartree-Fock method with the use of fermion coherent state representation. Equation to determine collective path and constraints to govern the collective and the independent-particle variables are given. Hamiltonian and other physical quantities given with the original fermion variables are rewritten to the forms expressed in terms of the collective and the independent-particle variables.
Some model Hamiltonians are proposed for quantum-mechanical many-body systems with pair forces. In the case of an infinite system in thermal equilibrium, they lead to temperature-domain propagator expansions which are expressible by closed, formally exact equations. The expansions are identical with infinite subclasses of terms from the propagator expansion for the true many-body problem. The two principal models introduced correspond, respectively, to ring and ladder summations from the true propagator expansion, but augmented by infinite classes of self-energy corrections. The latter are expected to yield damping of single-particle excitations. The eigenvalues of the ring and ladder model Hamiltonians are real, and they are bounded from below if the pair potential obeys certain conditions. The models are formulated for fermions, bosons, and distinguishable particles. In addition to the ring and ladder models, two simpler types are discussed, one of which yields the Hartree-Fock approximation to the true problem. A novel feature of all the model Hamiltonians (except the Hartree-Fock) is that they contain an infinite number of parameters whose phases are fixed by random choices. Explicit closed expressions are obtained for the Helmholtz free energy of all the models in the classical limit.
The phonon is discussed as an example of a collective mode which restores a symmetry property (in this case, translational invariance) to a system whose Hamiltonian is invariant under the symmetry operation, but in whose ground state the symmetry is broken. The crystal lattice is first studied within the time-independent Hartree approximation. It is then shown that allowing small time-dependent changes in the Hartree field generates an equation for the normal modes of vibration. The three k=0 modes with $\ensuremath{\omega}=0$ are shown to represent uniform translations of the solid, as expected. The $\mathbf{k}\ensuremath{\ne}0$ modes are analyzed to extract those three which may be identified as one-phonon modes, and the contribution of these modes to the free energy is computed. The recent theory of Brout is found to be equivalent to the Hartree approximation with the further assumption that the atoms are infinitely heavy. No restrictions are made here on the interatomic potential other than that a hard core is absent. The reduction of the present theory to well-known results in the case of harmonic forces is demonstrated. An extension to include hard cores, analogous to the Brueckner theory, is discussed.