Caesar Harmonic Calculation Library¶

Purpose of Module
Theory
Performing Calculations
Source Code

Purpose of Module ¶

The Caesar harmonic calculation library aims to provide an efficient method for calculating vibrational properties under the harmonic approximation [Hoja_ea1]. This can be done using a range of electronic structure codes, using the Caesar electronic structure interface.

[Hoja_ea1]

(1, 2, 3) First-principles modelling of molecular crystals: structures and stabilities, temperature and pressure. https://doi.org/10.1002/wcms.1294

Theory ¶

The Harmonic Approximation ¶

The nuclear potential energy surface $V$ is a function of the $3n$ -dimensional nuclear coordinate $\mathbf{r}$ . Under the harmonic approximation [Hoja_ea1], this function is approximated as quadratic in the difference between $\mathbf{r}$ and the value of $\mathbf{r}$ for the undisplaced structure, $\mathbf{r}^{(0)}$ . Formally, this is

$V(\mathbf{r}) = (\mathbf{r}-\mathbf{r}^{(0)})\cdot H \cdot(\mathbf{r}-\mathbf{r}^{(0)})$

where $H$ is the Hessian matrix.

By making a coordinate transform from $\mathbf{r}=\sum_ir_i\hat{\mathbf{r}}_i$ to $\mathbf{r}=\sum_ju_j\hat{\mathbf{u}}_j$ , the Hessian matrix can be diagonalised, to give

$V(\mathbf{u}) = \sum_j\frac{1}{2}N\omega_j^2u_j^2$

where each $u_j$ is a normal mode of the system, each $\omega_j$ is the corresponding frequency of that mode, and $N$ is the size of the supercell (defined as the ratio of the volume of the supercell to the volume of the primitive cell).

Provided that every value of $\omega_j$ is real, the simple form of $V$ means that the Hamiltonian can be diagonalised analytically, and so the free energy and other properties can be calculated analytically [Hoja_ea1].

Calculating the Hessian Matrix ¶

Caesar calculates the Hessian matrix using a finite difference method. This involves calculating the forces on the atoms in atomic configurations where the atomic displacement $\mathbf{r}-\mathbf{r}^{(0)}$ is small. Forces must be calculated by an electronic structure code, via the Caesar electronic structure interface.

Caesar minimises the total computational cost of these electronic structure calculations by exploiting crystal symmetry, as calculated by the spglib crystal symmetry library, and by using the non-diagonal supercell method [Lloyd-Williams_Monserrat].

[Lloyd-Williams_Monserrat]

Lattice dynamics and electron-phonon coupling calculations using nondiagonal supercells. https://doi.org/10.1103/PhysRevB.92.184301

Supercells ¶

Most vibrations in crystals cause atoms to move in ways which break translational symmetry. This means that it is not sufficient to calculate vibrational properties in the primitive unit cell alone. Instead, properties must be calculated in a supercell containing multiple copies of the primitive unit cell. Properties should be calculated in a number of different supercells, and the size of the supercell should be increased until the results of the calculations converge.

Performing Calculations ¶

Running the Caesar Harmonic Calculation Library is a four-stage process.

Firstly, caesar setup_harmonic parses the input data, calls spglib to calculate the crystal symmetries, and generates a directory structure containing directories in which all the necessary electronic structure calculations must be run.

Secondly, caesar run_harmonic performs the electronic structure calculations, using the Caesar electronic structure interface. There is no connection between the separate electronic structure calculations, so they can be run sequentially, in parallel, or across multiple computers as desired.

Thirdly, caesar calculate_normal_modes uses the results of the electronic structure calculations to calculate the Hessian matrix and the normal modes of the crystal.

Finally, caesar calculate_harmonic_observables calculates the vibrational properties of the crystal under the harmonic approximation.

The stages are separated so that the potentially computationally costly run_harmonic can be run on a separate computer or computers as needed, and so that calculate_harmonic_observables can be run repeatedly to calculate different observables as required.

The calculated properties are written to a harmonic_observables directory. These can be visualised using the various caesar plot_ utilities.

Each stage of the calculation has its own helptext, which can be accessed through the caesar executable by calling caesar --help.

Source Code ¶

The source code for the Caesar harmonic library is available from the src/harmonic directory of the Caesar repository

Caesar Harmonic Calculation Library¶

Purpose of Module¶

Theory¶

The Harmonic Approximation¶

Calculating the Hessian Matrix¶

Supercells¶

Performing Calculations¶

Source Code¶