In 1912, Langmuir  immersed a hot tungsten wire in a hydrogen atmosphere, and found that above 3000 K, heat was carried away from the wire at a rate much higher than would be expected by convection alone. The abnormally high conductivity appears because hydrogen molecules dissociate into atoms at that temperature, absorbing heat which is later released when the atoms recombine. At higher temperatures (10000 K), hydrogen atoms separate into protons and electrons. Heavier atoms ionize at even higher temperatures, and Saha  proposed in 1920 that one could infer the temperature of stars from their relative concentration of ions.
For an equilibrium of ideal gases , we can write the dissociation fraction as a function of the gas temperature and density using the Saha equation (Appendix B, ):
Applying the Saha equation to the reactions and , we find that in dilute gases, dissociation is a gradual process, not an abrupt transition (Figure 4.15). These reactions are entropy driven -- the bond dissociation energy of is 50,000 K and the ionization potential of H is 150,000 K, yet dissociation and ionization occur at much lower temperatures, driven by the separation of one particle into two. Two particles take up more space than one, which means that by La Chatlier's principle, compressing a gas shifts the equilibrium toward association. Thus the temperature of dissociation and the temperature of ionization increase with increasing density.
This analysis shows that in dilute gases, dissociation and ionization are two separate events. At higher densities, however, it should become easier to ionize hydrogen, since as atoms are squeezed together, the band gap decreases. At extreme compressions (), hydrogen becomes metallic, and the electrons move freely as if in a uniform sea of background positive charge. This pressure ionization occurs even at absolute zero . There has been speculation that at intermediate densities, the temperature needed to ionize hydrogen decreases with temperature, and at some point matches the temperature required to dissociate hydrogen molecules . At such a plasma phase transition, hydrogen would simultaneously dissociate and ionize, and properties like pressure or conductivity could change abruptly with variations in temperature or pressure (depending on the order of the phase transition).
If a plasma phase transition existed, it could lead to the revision of astrophysical models  -- for example, giant planets like Jupiter have dense hydrogen near their core, and an abrupt phase change would change the way helium partitioned itself between molecular and metallic phases of hydrogen. Recently there has been a renewed interest in studying dense hydrogen, stemming from (1) the development of path-integral Monte Carlo methods to calculate hydrogen equations of state ab initio [45,50,46,47,48,49] (2) shock hugoniot experiments with gas guns , lasers [54,55], and exploding wires [52,53] able to access densities and temperatures near the postulated PPT (15000 K, bohr, according to a chemical model . We demonstrate that the electron force field gives results consistent with the most recent high-level theory  and shock Hugoniot experiments 
In our simulations, we placed hydrogen molecules (64 nuclei) in a cubic periodic box, set atom velocities randomly from a Boltzmann distribution, then integrated the dynamics equations of motion with fixed volume and energy and a time step of 0.01 fs. We set the electron mass to be the same as the proton mass, making our simulation a plausible model of deuterium. We calculated the instantaneous temperature as the total kinetic energy of nuclei and electrons divided by ; we computed electrostatic energies by the Ewald method; and we averaged thermodynamic data over 1 ps following a 200 fs equilibration period.
Holding density fixed ( = 2 bohr) and performing simulations at a range of temperatures, we observed a thermal transition from a molecular to an atomic fluid (Figure 4.16). Proton-proton pair distribution functions plotted as a function of temperature (Figure 4.17) show a gradual transition between molecular and atomic fluid extremes, with an intermediate point at K, which compares well with the phase transition temperature of 15300 K estimated from a chemical model . The pair distribution curves look similar to ones obtained using path-integral Monte Carlo , where an intermediate point (roughly estimated by looking at pair distribution curves by eye) occurs at 10000 K.
We can also calculate pressure using the virial expression
By looking at the distribution of electron sizes over the course of the simulation, we can estimate how many electrons become ionized. At low temperatures, we observe a Maxwell-Boltzmann-like distribution of electron sizes that broadens with increasing temperatures. At higher temperatures, we find that a small fraction of electrons escape and expand to be larger than the size of our periodic box; at that size, they no longer interact strongly with the rest of the system. Taking electrons with bohr to be ionized, we find that some ionization occurs for at 25000 K and for at 30000 K but not for . The ionization we observe is consistent with thermal ionization of hydrogen atoms in a dilute gas, where the electrons, not having a nucleus to associate with, expand to fill free space; in this regime, it is reasonable to expect the temperature required for ionization to increase with increasing density. However, further work needs to be done to determine whether metallic-like electrons appear at higher density and lower temperature. Metallic electrons in the electron force field would be characterized not by a large size, but by an increased mobility.
Experimentally, liquid deuterium can be compressed to near-metallic densities using shock waves generated by explosives, exploding wires, or lasers. In these experiments, the deuterium is compressed with a solid pusher; by measuring the position and acceleration of the pusher over a duration of nanoseconds, we deduce a density-pressure relation called a Hugoniot curve that is a characteristic of the material. From conservation of mass, energy, and momentum over the boundary of a shock wave, we know the internal energy, volume, and pressure must satisfy the Hugoniot relation
The Hugoniot curve measures how compressible liquid deuterium is to shock. In the last decade, there has been some controversy over compressibility, with laser driven experiments (Nova [54,55]) indicating a maximum compressibility of six times, and gas gun experiments indicating a lower compressibility of four times with a stiffer response. More recent experiments done with exploding wires ( [52,53]) support the stiffer response function. We would like to see what kind of Hugoniot our theory, which has only been parameterized to fit bond lengths of simple alkanes, would produce.
To estimate the Hugoniot curve using eFF, we carry out a series of simulations at fixed volume and different fixed temperatures; measure the pressure; plot the Hugoniot function versus pressure; and find by interpolation the pressure that makes the Hugoniot function zero. As a starting point, we compute a box of liquid hydrogen with
; we find
hartrees/atom and .
While PIMC shows a maximum density of 0.73 (compressibility of 4.3 times), eFF shows a maximum density of 0.84 (compressibility of 4.9 times). In contrast, the Nova laser Hugoniot shows a maximum compressibility of 6.0 times. eFF also shows a comparable rise in temperature to PIMC (ours is 1/3 less) over the course of its Hugoniot.
Another group has performed a WPMD simulation of hydrogen plasma, but used the earlier described Klakow potential, with an additional term added to capture electrostatic energy changes upon antisymmetrization. Their results agreed well with eFF at low temperatures, but deviated significantly at higher temperatures, with a higher compressibility (6.4 times) that better matches the Nova laser data. The same extra repulsion that prevents electrons from inappropriately coalescing in the eFF model may be serving to give a higher compressibility and better agreement with path-integral Monte Carlo over Pauli potentials based on Klakow's expression.
To summarize, the electron force field shows that at bohr, dissociation from molecular to atomic fluid is gradual, and the equation of state and proton-proton pair distribution functions are consistent with path-integral Monte Carlo calculations. We observed no evidence of a plasma phase transition at this density. The good agreement with Hugoniot curves obtained from gas gun and Z machine experiments, as well as those obtained from PIMC, further confirms that we are describing the thermal transition from molecules to atoms correctly.
We have examined ionization as well, but only by measuring the concentration of free (large) electrons; we find that at low densities, the temperature required to create large electrons in free space increases with increasing density, as we would expect from a Saha model. Pressure ionization creates metallic electrons that are smaller yet highly mobile. It would be interesting to measure electron mobility at higher densities to determine if eFF can model metallic phases of hydrogen properly, and to better characterize pressure induced transitions from molecular to metallic hydrogen.