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[1/2U(rᵢⱼ)]; (ii) paired-phonon factors ρ_→kρ_-→k to all powers, (iii) multiple phonon factors ρ_→k⋯ρ_→1 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 ρ_→k 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 ⁴He at the equilibrium density, (1/N)δE~-0.7 ^∘{}K; for the hypothetical boson-type ³He system at ρ=0.0164 ^-3, $(1/N){δ}E{~}{-}0.3^∘{}K (HNC) or {-}0.5^∘{}K (PY)$. 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 ${10}²$ - ${10}³$ particles.