Ground-state Properties

Abstract

In this chapter, we shall review some important ground-state properties that can be derived directly from the total energy calculated for different volumes, geometries and compositions. For a specific space group, chemical composition and magnetic structure, the equation of state is obtained from the total energy E(V) calculated as a function of volume (V). At each volume, the crystal structure should be fully relaxed with respect to the internal coordinates and unit cell shape. The negative volume derivative of E(V) gives the pressure,

$$ P(V) = - \frac{{\partial E(V)}} {{\partial V}}. $$

(6.1)

From the pressure-volume relation, we get the enthalpy H(P) = E (V (P))+ PV(P). The bulk modulus is defined from the volume derivative of the pressure as

$$ B(V) = - V\frac{{\partial P}} {{\partial V}} = V\frac{{\partial ^2 E(V)}} {{\partial V^2 }}. $$

(6.2)

The third order volume derivative of E(V) enters the expression of the pressure derivative of the bulk modulus (B′) or the Grüneisen constant¹. In order to minimize the numerical noise in P, B and B′, usually an analytic form is used to fit the ab initio energies versus volume. In Section 6.1, we shall describe the most commonly used equation of states: the Murnaghan, the Birch-Murnaghan and the Morse functions.

The Grüneisen constant (γ) describes the anharmonic effects in the vibrating lattice and it is given by γ = −f+B′/2. The best agreement with the experimental Grüneisen constant was obtained [123] for f = 1/2.