# 10. DFPT

Many materials properties can be formulated as response of their perturbed ground-state to various perturbations. For example, optical properties characterize interaction of materials with electromagnetic field perturbations. These materials properties (response functions) can be calculated (in most cases) as derivatives of total energy with respect to perturbation factors. For example, in case of atomic displacements perturbation, 1st, 2nd, and 3rd order derivatives determine the atomic forces, harmonic, and anharmonic lattice dynamics, respectively.

Traditionally, these derivatives can be calculated by means finite difference (direct, also frozen phonon) method, i.e. total energy is directly calculated for several values of perturbation parameter, and derivatives are evaluated by differences in corresponding total energy values. Although, finite difference method might be useful for certain types of problems, it has major drawbacks in treating important perturbations such as atomic displacements. For example it cannot handle phonons with wavevectors incommensurate with systems unitcell, or requires very large supercells to account for phonon with wavevectors which span several unitcells, hence substantially increasing the computation cost.

Density functional perturbation theory (DFPT) provides an efficient framework to calculate various orders of total energy derivatives and addresses the shortcomings of finite difference method [BGCG01, GC97]. It also opens the door to treating perturbations which might be very difficult to fit into finite difference framework. In short, DFPT provides necessary formalisms to obtain perturbed wavefunctions, self-consistent potential and electronic density for a broad range of perturbations, which are combined with ground-state quantities to calculate response functions.

Current DFPT implementation in RESCU can calculate first order perturbed quantities
with respect to perturbation parameter by solving the corresponding **Sternheimer**
equations, and up to third order derivatives of total energy by using
**“2n+1”** theorem. It supports real-space basis set with norm conserving pseudo potentials, and
following perturbations are covered:

Static electric field

Atomic displacements

The overall workflow of a DFPT calculation is consisted of the following three main steps:

First, ground-state of the system is obtain by a self-consistent DFT calculation.

Second, for a given perturbation, DFPT equations are self-consistently solved to obtain perturbed wavefunctions, electronic density, and self-consistent potential.

Third, various response functions are calculated in a nonself-consistent manner using the results of previous two steps.

In the following sections, we demonstrate workflow of different DFPT calculations in more details. Certain response functions, e.g. dielectric response, are not well-defined in metallic system where there is no electronic band-gap. Hence, details of DFPT calculations for metals are covered in Metals.

- 10.1. Preparing the ground state data
- 10.2. Ion-clamped dielectric tensor
- 10.3. Dynamical matrix and Born effective charges
- 10.4. Phonon band structure
- 10.5. Phonon density of states
- 10.6. Infrared region optical properties
- 10.7. The Raman spectrum and non-linear optical susceptibility
- 10.8. Metals
- 10.9. Electron-phonon interaction

Stefano Baroni, Stefano De Gironcoli, Andrea Dal Corso, and Paolo Giannozzi. Phonons and related crystal properties from density-functional perturbation theory. Reviews of Modern Physics 73.2 (2001), p. 515

Xavier Gonze and Changyol Lee. Dynamical matrices, Born effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory. Physical Review B 55 10355 (1997)