Paired-phonon analysis for the ground state and low excited states of liquid helium

The paired-phonon analysis operates in the function space generated by product functions compounded from (i) a starting trial function ψ of the Bijl-Dingle-Jastrow-type (BDJ) (a product of two-particle correlation factors exp[12U(rij)]; (ii) paired-phonon factors ρk→ρ-k→ to all powers, (iii) multiple phonon factors ρk→ρ1→ to all powers, with neglect of all matrix elements representing processes in which phonons coalesce, split, or scatter. Results in the present study include (i) a simpler and more general derivation of the fundamental relations; (ii) proof that the improved ground-state trial function ψ^ generated by the analysis is still in the BDJ function space [with U(r) replaced by U(r)+δU(r)]; (iii) a formula expressing δU(r) in terms of S(k), the starting liquid-structure function, and w(k), the residual interaction function; (iv) a convenient representation of the phonon factor ρk→ as a linear combination of phonon creation and annihilation operators; (v) explicit statement of the relation between the optimization condition w(k)0 and the variational extremum property of the expectation value of H in the BDJ-type function space; (vi) usable approximate procedures for evaluating the residual interaction function w(k) based on the hypernetted-chain (HNC) and Percus-Yevick (PY) relations; and (vii) numerical evaluation of w(k), the energy shift δE, and the improved liquid-structure function S(k) using ψ's computed by Massey and Woo as starting functions. For He4 at the equilibrium density, (1N)δE∼-0.7 °K; for the hypothetical boson-type He3 system at ρ=0.0164 -3, (1N)δE∼-0.3°K(HNC) or-0.5°K (PY). In the discussion, emphasis is placed on the practical possibility of accurate numerical evaluation of the interaction function ω(k) by the method of molecular dynamics applied to systems containing 102 - 103 particles.