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Progress of Theoretical Physics

Many-Body Variation Theory. II

K. Sawada

Published May 1, 1970• 14 citations
Publisher

Abstract

The equation-of-motion method is shown to be related to the variational method for free energy. In the special case, the variational Hamiltonian does not appear in the formulation and the equation-of-motion method provides a variational answer.
DOI
10.1143/ptp.43.1199

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An equation-of-motion approach is used to develop the relativistic many-body theory of atomic transitions. The relativistic equations of motion for transition matrices are formulated with the use of techniques of quantum-field theory. To reduce the equations of motion to a tractable form which is appropriate for numerical calculations, a graphical method to resolve the complication arising from the antisymmetrization and angular-momentum coupling is employed. The relativistic equation-of-motion method allows an ab initio treatment of correlation and relativistic effects in both closed- and open-shell many-body systems. A special case of the present formulation reduces to the relativistic random-phase approximation.
An approach is formulated to the problem of obtaining approximate solutions to many-body quantum mechanics. The starting point is the representation of quantum mechanics as Hamiltonian mechanics on a symplectic manifold (phase space). It is shown that Dirac's variation of an action integral provides a natural mechanism for constraining the dynamics to symplectic submanifolds and gives rise to a hierarchy of approximate many-body theories of which Hartree-Fock, random-phase approximation, time-dependent Hartree-Fock, and the double commutator equations of motion formalism are special cases.
Recently [W. Kutzelnigg and D. Mukherjee, Chem. Phys. Lett. 317, 567 (2000); D. Mukherjee and W. Kutzelnigg, J. Chem. Phys 114, 2047 (2001)] the irreducible k-particle Brillouin conditions IBCk and the irreducible k-particle contracted Schrödinger equations ICSEk were derived. These permit the definition of a hierarchy of k-particle approximations for the direct calculation of the cumulants λk of the k-particle density matrices. Now, the spin-free form of these conditions, appropriate for a spin-free Hamiltonian, is given. This is particularly useful for open-shell states. The definition of the cumulants of the reduced densities has to be generalized for these anyway, making use of irreducible tensor operators with respect to SU2. There are two alternative definitions of spin-free cumulants, of which the one in terms of spin-free reduced density matrices appears to be preferable. Alternatively to the straight spin-free formulation, we also present a theory in terms of spin-free operators adapted to the symmetric group. Partial trace relations that relate the elements of the cumulants of different particle rank are derived. There are partial trace relations for “exchange elements,” which are determined by the total spin quantum number S. From these relations the individual exchange elements of Λ2 can be obtained in special cases. This allows a simple formulation of the stationarity conditions for open-shell states of any spin multiplicity.