Caesar Anharmonic Calculation Library¶

Purpose of Module
Theory
Performing Calculations
Source Code

Purpose of Module ¶

The Caesar anharmonic calculation library aims to provide an efficient method for calculating vibrational properties beyond the harmonic approximation; under the vibrational self-consistent harmonic approximation (VSCHA) [Errea_ea1] or using vibrational self-consistent field theory (VSCF) [Christiansen1].

[Errea_ea1]

(1, 2) Anharmonic free energies and phonon dispersions from the stochastic self-consistent harmomic approximation: Application to platinum and palladium hydrides. https://doi.org/10.1103/PhysRevB.89.064302

[Christiansen1]

Vibrational structure theory: new vibrational wave function methods for calculation of anharmonic vibrational energies and vibrational contributions to molecular properties. https://doi.org/10.1039/B618764A

Theory ¶

Fitting the Potential Energy Surface ¶

Caesar models the nuclear potential energy surface (PES) using a truncated Taylor expansion in normal-mode coordinates. Constructing and fitting this model happens over several steps:

Firstly, a set of symmetry-invariant basis functions are generated, using the crystal symmetries as calculated by spglib.

Secondly, a set of nuclear coordinates $\mathbf{r}$ are generated at which the PES will be sampled.

Thirdly, electronic structure calculations are performed at each coordinate, using the Caesar electronic structure interface.

Finally, the results of the electronic structure calculations, including calculated energies, forces and other information, are used to calculate the basis function coefficients.

As with the harmonic calculation, the anharmonic calculation uses the non-diagonal supercell method [Lloyd-Williams_Monserrat1] to reduce the total computational cost of the electronic structure calculations where possible.

[Lloyd-Williams_Monserrat1]

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

The Vibrational Self-Consistent Harmonic Approximation ¶

VSCHA approximates the eigenstates of the system as those which diagonalise an effective harmonic potential $V^\text{effective}$ . The effective harmonic potential $V^\text{effective}$ implemented by Caesar has the same functional form and normal modes $\{u_j\}$ as the harmonic potential $V^\text{harmonic}$ , but has a different set of frequencies $\{\omega_j\}$ .

The frequencies $\{\omega_j\}$ are calculated as those which minimise the free energy of the anharmonic PES with respect to the VSCHA eigenstates [Errea_ea1].

Vibrational Self-consistent Field Theory ¶

Traditional VSCF separates the PES $V(\mathbf{u})$ into a sum of single-mode effective potentials $\{V_j(u_j)\}$ , each of which is the expectation of $V$ with respect to all modes other than $u_j$ . The Hamiltonian corresponding to each mode is then constructed in the VSCHA eigenbasis, and this is diagonalised to give the single-mode VSCF eigenstates $\{|\psi_{jk}>\}$ . This process can be written as two equations,

$(T+V_j)|\psi_{jk}> = E_{jk}|\psi_{jk}>$

and

$V_j = <V>_{j'\neq j}$

These equations are solved self-consistently, using a Pulay scheme [Pulay].

The VSCF method implemented by Caesar differs from traditional VSCF methods in that rather than separating the PES single-mode potentials, the PES is instead separated into single-subspace potentials, where each subspace contains a complete set of modes whose frequencies are degenerate as a result of symmetry. This implementation of VSCF is symmetry invariant, unlike the single-mode methods.

[Pulay]

Convergence acceleration of iterative sequences. The case of scf iteration.https://doi.org/10.1016/0009-2614(80)80396-4

Performing Calculations ¶

Prior to performing anharmonic calculations, a harmonic calculation must be performed. This can be done using the Caesar Harmonic Calculation Library, or the Hessian matrix of the undisplaced structure can be read using the Caesar electronic structure interface.

Like running the Caesar Harmonic Calculation Library, running the Caesar Anharmonic Calculation Library is a four-stage process.

Firstly, caesar setup_anharmonic parses the input data and reads the output of the harmonic calculation. It then generates a directory structure containing directories in which all the necessary electronic structure calculations must be run.

Secondly, caesar run_anharmonic 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_potential uses the results of the electronic structure calculations to fit the anharmonic potential.

Finally, caesar calculate_anharmonic_observables calculates the vibrational properties of the crystal under VSCHA and VSCF.

The calculated properties are written to an anharmonic_observables directory. These can be visualised using the various caesar_plot_ utilities.

As with the harmonic stages, each anharmonic stage has its own helptext, which can be accessed through the caesar executable by calling caesar --help.

Source Code ¶

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

Caesar Anharmonic Calculation Library¶

Purpose of Module¶

Theory¶

Fitting the Potential Energy Surface¶

The Vibrational Self-Consistent Harmonic Approximation¶

Vibrational Self-consistent Field Theory¶

Performing Calculations¶

Source Code¶