One question that sometimes comes up regarding DFT computation (typically at zero pressure) is whether one can safely neglect the effect of pressure at ambient conditions. Here’s a simple back-of-the-envelope calculation that shows why it’s OK to neglect pressure under normal circumstances.

## The effect of pressure on absolute energy

Under constant temperature and pressure conditions, the relevant thermodynamic potential is the Gibbs Free Energy, defined as:

where **G** is the Gibbs free energy, **U** is internal energy, **P** is pressure, **V** is volume, **T** is temperature and **S** is entropy. Since we’re doing a back-of-the-envelope calculation and want to single out pressure effects, let’s conduct our analysis at zero temperature.^{[1]} Also let’s normalize extrinsic quantities per atom; the “per atom” versions will be denoted by lowercase letters:

Clearly, the difference between **g** at finite pressure and zero pressure is the **Pv** term:

The value of **P** under ambient conditions is 100 kPa (10^{5} Pa). For **v** (the volume per atom), let’s plug in values for Si, which has a 40 Angstrom^{3} unit cell containing 2 atoms,^{[2]} so **v** ~ (20)*10^{-30} m^{3}. So:

10^{-5} eV/atom is very small. For comparison, energy differences between different crystal structures are on the order of 10^{-2} eV/atom. The effect of ambient pressure on absolute energy is about *1000 times smaller* than the quantities we care about! Note that the only real assumption in this analysis – other than zero temperature – was **v**, and this will not vary by too much between different compounds.

## The effect of pressure on relative energies between phases

What matters physically is not absolute energy but relative energies between compounds. In particular, at high pressure compounds with smaller **v** (more dense) will have lower **g** and thus be preferred.

We can approximate the difference in **g** between two compounds due to pressure as:

All we need to evaluate this expression numerically are number densities for two different phases. Let’s choose two phases of Si – cubic ground state (**v _{1}** ~20 *10

^{-30}m

^{3}) and high pressure (

**v**~14*10

_{2}^{-30}m

^{3}).

^{[2]}Then:

Again, the effect of ambient pressure is several orders of magnitude smaller than what we care about.

Of course, one could always crank up the pressure – a lot. For example, the high-pressure phase of Si is calculated to be about 0.3 eV/atom higher in energy than the ground state (according to DFT-GGA),^{4} making it quite unstable under ambient conditions. However, according to our calculation above, if we crank up the pressure to about 100,000 times ambient conditions,^{[3]} the effect of pressure would be just enough to overcome the calculated energy difference. Nature agrees – Si is known to transition to the high-pressure form at 112,000 times pressure compared to ambient conditions.^{[4]} So pressure can certainly have an effect in extreme conditions.

## The effect of pressure on gases

The assumption that V is about the same order of magnitude for all compounds breaks down for gases. Fortunately, the product PV for a gas can be evaluated using the ideal gas law:

Interestingly, the effect of *pressure* on the Gibbs free energy depends strongly on the *temperature*. Note that in contrast to solids, the effect of pressure on gases is large enough that it should be added to the calculations even in normal situations.

## References

[1] Note that pressure can have an effect on **S**; a fundamental thermodynamic relation equates and (or the product of the volume and coefficient of thermal expansion).

[2] I’m using experimental volumes for **v** at ambient pressure; they’re generally very close to DFT values. Data is from the Materials Project. Ground state (cubic) is mp-149, high pressure (beta-Sn) is mp-92.

[3] Note: to keep the math simple and easy for everyone to follow in their head, I was a bit sloppy with the rounding. If you follow the math more closely you would predict a transition closer to 75,000 times ambient pressure.

[4] Reported in “Phases of Silicon at High Pressure” by Hu and Spain.