Perturbation Expansion of the Ground State Energy of Solid Helium

Abstract

The ground state energy of helium is expanded in powers of a non-singular effective potential, i.e., the reaction matrix, which is defined by a self-consistent equation of an interacting pair of atoms in the solid. Two alternative schemes are given by taking advantage of arbitrariness in introducing the reaction matrix. One, in the first approxima­ tion, leads to the formula of ground state energy proposed by the authors previously. The other is a modified scheme which results in a vanishing second order term. In both schemes the expansion of the energy is explicitly written up to the third order in the reaction matrix. Solid helium of both isotopes distinguishes itself from other crystals as the solid due to its large zero-point oscillation about the equilibrium posi­ tion of the atoms. This characteristic is due to the small atomic mass and the weak attraction between atoms at a distance. However, at a smaller distance the interatomic _r:>otential contains the hard-core of a radius as large as the zero­ point oscillation itself. Therefore, construction of a theory of helium from first principles encounters difficulty in treating the large quantum oscillation and the hard-core interaction at the same time. It is this difficulty that causes a serious failure both to the conventional approach of lattice dynamics and to the Hartree approach, in explaining the observed value of the cohesive energy of helium. In 1966, the present authors proposed a theory of helium 1 ) (hereafter referred to as I) which copes with this difficulty, and calculated the ground state energy numerically. However, their basic equations relied upon an intuitive argument. In this paper, we derive the equations from the total Hamiltonian and give an appropriate form of perturbation expansion for the ground state energy. Needless to say, it is meaningless to expand any quantity in terms of the original two-body potential v due to its hard-core behavior. Usually we replace v by the two-body reaction matrix K with the help of diagram technique. In this paper, however, we will develop a different approach ju eliminating the