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Data from recent laser-shock experiments, density-functional theory (DFT)with molecular-dynamics (MD), and path-integral Monte Carlo (PIMC) simulationson carbon are compared with predictions from the neutral-pseudo-atom (NPA)+hyper-netted-chain (HNC) approach for carbon, a complex liquid in thewarm-dense matter regime. The NPA results are in good agreement, not only withhigh-density regimes that have been studies via PIMC, but even at low densitiesand low temperatures where transient covalent bonding dominates ioniccorrelations. Thus the `pre-peak' due to the C-C bond at $\sim$1.4-1.6 \AA$\,$and other features found in the pair-distribution function from DFT+MDsimulations at 0.86 eV and 3.7 g/cm$^3$ etc., are recovered accurately in theNPA+HNC calculations. Such C-C bonding peaks have not been captured viaaverage-atom ion-sphere (IS) models. Evidence for an unusual liquid $\to$ vaporand metal$\to$ semi-metal transition occurring simultaneously is presented.Here a strongly correlated metallic-liquid with transient C-C bonds, i.e.,carbon at density $\sim$ 1.0 g/cm$^3$ and mean ionization $Z=4$ transitsabruptly to a disordered mono-atomic vapour at 7 eV, with $Z\simeq$ 3. Othercases where $Z$ drops abruptly are also noted. The nature of $Z$, itsdiscontinuities, and the role of exchange correlation, are reviewed. Thelimitations of IS models in capturing the physics of transient covalent bondingin warm dense matter are discussed.

Theory of complex fluids in the warm-dense-matter regime, and application to an unusual phase-transitions in liquid carbon