Density Functional Theory (I)

Background on Solid State Physics

Thermodynamics Background

Volume or Pressure as the Most Fundamental Variable

Let \(\Omega\) denote the volume of a unit cell. We have \begin{align*} E &= E(\Omega), \\ P &= -\dv{E}{\Omega}, \\ B &= -\Omega \dv{P}{\Omega} = \Omega \dv[2]{E}{\Omega}, \end{align*} where \(B\) denotes the bulk modulus.

To predict the equilibrium volume \(\Omega^0\), i.e. where \(E\) attains its minimum, we may either

• calculate \(E\) for several values of \(\Omega\), and fit to an analytic form, e.g. the Murnaghan equation; or
• calculate \(P\) from a generalization of the virial theorem, and \(B\) from response functions.