Ze Chen

## 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.

2021/1/23 0:29:30

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