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 $\psi$ of the Bijl-Dingle-Jastrow-type (BDJ) (a product of two-particle correlation factors $\exp \left[\frac{1}{2}U\right(r_{\mathrm{ij}}\left)\right]$; (ii) paired-phonon factors ${\rho }_{\overrightarrow{\mathrm{k}}}{\rho }_{-\overrightarrow{\mathrm{k}}}$ to all powers, (iii) multiple phonon factors ${\rho }_{\overrightarrow{\mathrm{k}}}\cdots {\rho }_{\overrightarrow{1}}$ 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 $\hat{\psi }$ generated by the analysis is still in the BDJ function space [with $U\left(r\right)\mathrm{}$ replaced by $U\left(r\right)+\delta U\left(r\right)\mathrm{}$]; (iii) a formula expressing $\delta U\left(r\right)\mathrm{}$ in terms of $S\left(k\right)\mathrm{}$, the starting liquid-structure function, and $w\left(k\right)\mathrm{}$, the residual interaction function; (iv) a convenient representation of the phonon factor ${\rho }_{\overrightarrow{\mathrm{k}}}$ as a linear combination of phonon creation and annihilation operators; (v) explicit statement of the relation between the optimization condition $w\left(k\right)\mathrm{}\equiv 0$ 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 $w\left(k\right)\mathrm{}$ based on the hypernetted-chain (HNC) and Percus-Yevick (PY) relations; and (vii) numerical evaluation of $w\left(k\right)\mathrm{}$, the energy shift $\delta E$, and the improved liquid-structure function $\hat{S}\mathrm{}\left(k\right)\mathrm{}$ using $\psi \text{'}\mathrm{s}$ computed by Massey and Woo as starting functions. For ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ at the equilibrium density, $\left(\frac{1}{N}\right)\delta E\sim -0.7\mathrm{}$ °K; for the hypothetical boson-type ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ system at $\rho =0.0164\mathrm{}$ ${\mathrm{\AA }}^{-3\mathrm{}}$, $\left(\frac{1}{N}\right)\delta E\sim -0.3\mathrm{}°\mathrm{K} \left(\mathrm{HNC}\right) or -0.5\mathrm{}°\mathrm{K} \left(\mathrm{PY}\right)$. In the discussion, emphasis is placed on the practical possibility of accurate numerical evaluation of the interaction function $\omega \mathrm{}\left(k\right)\mathrm{}$ by the method of molecular dynamics applied to systems containing ${10}^2$ - ${10}^3$ particles.