This text is inspired from the following article:

Analysis of Equation of States for the Suitability at High Pressure: MgO as an Example

Definition

In physics and thermodynamics, an equation of state is a thermodynamic equation relating state variables which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature (PVT), or internal energy.^[1] Equations of state are useful in describing the properties of fluids, mixtures of fluids, solids, and the interior of stars. ( Reference: Wikipedia)

In ab initio calculations in solids we will be interested only to equations relating volume to pressure. The famous equations are those of Murnagham and Birch-Murnagham but there are also other equations which good results and we can also put an EOS for some materials or for all materials.

EOSs used in solid calculations

When putting a new EOS, the equation must satisft the Stacey criterion as follows:

(i) For small compression almost every EOS gives the results within the experimental uncertainty, as laboratory measured P-V data are often subjected to pressure calibration errors [2].

(ii) Almost every EOS, new or old, can be fitted to a given range of P-V data by adjusting the values of the zero pressure, isothermal bulk modulus B0 and its pressure derivatives [2].

Reference:Analysis of Equation of States for the Suitability at High Pressure: MgO as an Example

Here we present some EOSs

Parsafar-Mason EOS

On the basis of the first-principle calculations, using the augmented-plane-wave (APW) method and quantum statistical model Hama and Suito [35] revealed that the Parsafar-Mason EOS becomes less successful at high compressions (V/V0 > 0.65 ). In addition to this, Shanker and Kushwah [36] pointed out that the Parsafar and Mason EOS gives P as a fourth degree expression in V/V0 ; therefore, the determination of the higher-derivative properties such as the bulk modulus and its pressure derivatives becomes less convenient.

Shanker-Kushwah EOS

Shanker and Kushwah [36] have expanded PV2 in powers of [1-/(V/V0] up to the quadratic term and found that the equation works well for materials having pressure derivative of the isothermal bulk modulus less than four. Recently, Kholiya [8] has expanded pressure in powers of density up to the quadratic term and achieved the EOS as