Correlation energy of the uniform electron gas from an interpolation between high- and low-density limits

We show that known or knowable information about the high-$({r}_{s}\ensuremath{\rightarrow}0)$ and low-density $({r}_{s}\ensuremath{\rightarrow}\ensuremath{\infty})$ asymptotes can be used to predict the correlation energy per electron, ${e}_{c}({r}_{s},\ensuremath{\zeta})$, of the three-dimensional uniform gas over the whole range of the density parameter $(0\ensuremath{\le}{r}_{s}<\ensuremath{\infty})$ and relative spin polarization $(0\ensuremath{\le}|\ensuremath{\zeta}|\ensuremath{\le}1)$, without quantum Monte Carlo or other input. For $\ensuremath{\zeta}=0$, the high-density limit through order ${r}_{s}$ is known exactly from many-body perturbation theory, and for all $\ensuremath{\zeta}$, the low-density limit through order $1/{r}_{s}^{2}$ is known accurately from a simple, intuitive, and accurate model. We propose a single interpolation formula with the expected analytic structure to all orders in both limits, and use it to predict ${e}_{c}({r}_{s},0)$ in excellent agreement with quantum Monte Carlo data. For $|\ensuremath{\zeta}|>0$, we derive the $\ensuremath{\zeta}$ dependence of the coefficient ${a}_{1}(\ensuremath{\zeta})$ of the ${r}_{s}\text{ }\text{ln}\text{ }{r}_{s}$ term, previously known only for $|\ensuremath{\zeta}|=0$ and 1. For ${b}_{1}(\ensuremath{\zeta})$, the coefficient of the ${r}_{s}$ term (not yet derived for $\ensuremath{\zeta}\ensuremath{\ne}0$), we approximately extend the known ${b}_{1}(0)$ by using a simplification of the available quantum Monte Carlo information that replaces the second-order transition over $50<{r}_{s}<100$ by a sudden transition to full spin polarization at ${r}_{s}=75$.