Requesting Vibrational Properties

CASTEP allows you to prepare calculations that provide vibrational frequencies and eigenvectors at a selected group of q-vectors in the Brillouin zone. CASTEP provides two approaches to calculating these data:

There are two predefined modes of calculation: the phonon dispersion spectrum along high symmetry directions and the phonon density of states spectrum, which is a prerequisite for the calculation of thermodynamic properties.

Phonon calculations take into account existing fixed atom constraints, regardless of the overall Task setting. Such fixed atoms are excluded from the calculation of vibrational properties, which corresponds to the "partial Hessian" approach. Constrained lattice dynamics approach is applied for both linear response and finite displacement approach.

Select Fix fractional position on the Atom tab of the Edit Constraints dialog, accessible from the Modify menu.

Phonon Frequencies from Linear Response Calculations

The linear response scheme treats atomic displacements as perturbations. In addition, it evaluates an electric field response to calculate a non-analytical correction to longitudinal optical (LO) phonon frequencies at the Γ-point. Longitudinal and transverse modes at the Γ-point are identical (no LO-TO splitting), unless you apply this correction.

The linear response scheme allows the calculation of phonon spectra in any commensurate or incommensurate q-vector of the reciprocal space. This provides full flexibility in the choice of q-vectors to use for calculations. Nevertheless, linear response calculations at a general q-vector are very time consuming and often unnecessary as a real space dynamical matrix that defines vibrational properties is often a short range property. You can use a coarse Monkhorst-Pack mesh to calculate this real space dynamical matrix and then use it to obtain the phonon spectrum at any given q-vector in the Brillouin zone. Thus, there are two major modes of linear response calculations: direct and using interpolation.

Limitation of the Linear Response Implementation in CASTEP

This version of CASTEP supports linear response calculations of phonons or finite displacement calculations that require LO-TO splitting correction only for norm-conserving pseudopotentials. For more information, see Setting up pseudopotentials.

Phonon Frequencies from Finite Displacements Calculations

CASTEP provides an alternative method of calculating phonon frequencies that is based on the finite displacements method (Montanari and Harrison, 2002). This technique is based on the numerical differentiation of forces on atoms calculated for a number of unit cells with atomic displacements. CASTEP determines the optimum number of such displacements automatically, based on the crystal symmetry. Displacements with a very small magnitude give results that are similar to the linear response frequencies. An advantage of the finite displacements method is that it is also possible to investigate phonon anharmonicity by increasing the amplitude of the displacements. A further important advantage is that you can use efficient ultrasoft potentials to obtain vibrational frequencies.

The formalism is based on evaluation of the dynamical matrix at a (coarse) commensurate grid of q-points at the cost of creating a supercell. Then the dynamical matrix is re-interpolated onto a requested group of q-points. There are two available approaches to dynamical matrix calculation:

Limitations of the Finite Displacements Implementation in CASTEP

There is a limitation with the implementation of the finite displacements method in CASTEP related to the treatment of the non-analytical term responsible for the LO-TO splitting in insulators. This term is calculated using linear response with an electric field as a perturbation. Therefore, the computational limitation applies to finite displacements calculations if you request LO-TO splitting.

An LO-TO calculation imposes the restriction that applies when you use the linear response technique. You might want to proceed with finite displacement calculations without LO-TO splitting if this restriction affects your study. This case only reliably produces TO modes when there is LO-TO degeneracy in the results.

Requesting Phonon Dispersion

Calculating the phonon dispersion produces phonon frequencies and eigenvectors along high symmetry directions in the Brillouin zone. The standard path for each lattice type is taken from Bradley and Cracknell (1972). You can modify the path using the Brillouin Zone Path dialog. The Quality in the Dispersion section on the CASTEP Phonon Properties Setup dialog controls the density of points along the path, which affects the appearance of the resulting chart.

To calculate phonon dispersion

  1. Choose Modules | CASTEP | Calculation from the menu bar to display the CASTEP Calculation dialog.
  2. Select the Properties tab.
  3. Select Phonons in the list of properties, then select Dispersion below the list.
  4. If required, select Calculate LO-TO splitting.
  5. Select the technique to use to calculate the phonon frequencies as the Method.
  6. Click More... to display the CASTEP Phonon Properties Options dialog.
  7. For linear response calculations, specify the convergence criterion for the force constants in Convergence tolerance.

    If you do not want to carry out a direct calculation, select Use interpolation and specify the q-point separation in q-vector grid spacing for interpolation.

    For finite displacement calculations, specify the cutoff radius to use to construct the appropriate supercell in Supercell defined by cutoff radius.

  8. Optionally, adjust the density of q-vectors along the high symmetry directions used for the phonon dispersion calculation using the Quality control in the Dispersion section.
  9. If you want to inspect or modify the Brillouin zone path used for the phonon dispersion calculation, click Path...to display the Brillouin Zone Path dialog and alter the path segments accordingly.
  10. Optionally, use the partial Hessian approach to accelerate the calculation. Apply fixed atom constraints to the atoms that are not essential to the calculation; for example, the atoms in a molecule on surface calculations.

Requesting Phonon Density of States

Calculating the phonon density of states produces phonon frequencies and eigenvectors on the regular Monkhorst-Pack mesh of q-vectors. This information is required to produce either the total or projected (partial) phonon density of states and, thus, to calculate thermodynamic properties. The Quality setting in the Density of states section on the CASTEP Phonon Properties Options dialog controls the quality of the q-vector.

When you request phonon density of states CASTEP also calculates atomic displacement parameters, allowing assignment of temperature factors during analysis.

To calculate the density of states

  1. Choose Modules | CASTEP | Calculation from the menu bar to display the CASTEP Calculation dialog.
  2. Select the Properties tab.
  3. Select Phonons in the list of properties, then select Density of states below the list.
  4. If required, select Calculate LO-TO splitting.
  5. Select the technique to use to calculate the phonon frequencies as the Method.
  6. Click More... to display the CASTEP Phonon Properties Options dialog.
  7. For linear response calculations, specify the convergence criterion for the electronic eigenvalues in Convergence tolerance.

    If you do not want to carry out a direct calculation, select Use interpolation and specify the q-point separation in q-vector grid spacing for interpolation.

    For finite displacement calculations, specify the cutoff radius to use to construct the appropriate supercell in Supercell defined by cutoff radius.

    Optionally, use the Quality control in the Density of states section to adjust the density of Monkhorst-Pack q-vectors used for the phonon density of states calculation.
  8. If you want to inspect or modify the setup of the Monkhorst-Pack grid, click More... to display the Phonon Density of States Options dialog and alter the settings accordingly.
  9. Optionally, use the partial Hessian approach accelerate the calculation. Apply fixed atom constraints to the atoms that are not essential to the calculation; for example, the substrate atoms in a molecule on surface calculations.

If you choose quality gamma frequencies, then CASTEP only calculates eigenvectors at the Γ-point, which determines most of the experimental data on IR or Raman spectra. This can accelerate finite displacement calculations tremendously as no supercell is constructed for this case. However, the accuracy of thermodynamic properties obtained with this single q-vector is very poor as it does not take into account any acoustic degrees of freedom.

Calculating Vibrational Frequencies of an Isolated Molecule

You can use CASTEP to calculate the vibrational frequencies of isolated molecules as well as the phonon dispersion of solids. Use only the Γ-point for electronic calculations and request the Γ-point when performing either phonon dispersion or phonon density of states calculations. In addition, the all bands/EDFT electronic minimizer is known to be more efficient for studies of isolated molecules in supercell geometry than the density mixing minimizer. Therefore, change the Electronic minimizer to All Bands/EDFT on the SCF tab of the CASTEP Electronic Options dialog when performing such calculations.

Calculations for "molecule in a box" systems that do not require geometry optimization can be sped up if the molecular symmetry is utilized. Use the Find Symmetry tool to find and apply the symmetry of the molecule to the supercell.

Do not calculate LO-TO splitting for a "molecule in a box" system. There is no physical meaning to this term when studying molecular, rather than crystalline, vibrations.

Guidelines for the Successful Calculation of Vibrational Properties Using CASTEP

  1. Parameters for geometry optimization

    Starting from a well-converged geometry for the input structure leads to inaccurate results of a phonon calculation. Well-converged forces ensure a true ground state for the geometry optimization.

    In general, using a Quality setting of Ultra-fine for the geometry optimization on the Setup tab of the CASTEP Calculation dialog ensures accurate forces. The safest procedure is to investigate the convergence of the forces as a function of calculation parameters; primarily the k-point sampling. Carry out a series of single-point energy calculations using increasingly dense k-point meshes. Monitor the convergence of the forces until they converge to within the force tolerance criterion.

  2. Geometry optimization of the input structure

    Once you have generated reasonably well-converged forces, you can carry out the geometry optimization. Use the same force convergence tolerance as for the force convergence test. A well-converged geometry optimization is of little value if the forces are inaccurate because of convergence errors.

    You may want to perform a fixed-cell or variable-cell geometry optimization, depending on your goals for the calculation. The lattice-dynamical problem is well-defined in either case, but the pressure or stress differs. Also monitor the convergence of the stress before carrying out a variable-cell optimization (though the behavior is usually similar to the convergence of forces).

  3. Use of consistent electronic settings

    To be a valid lattice dynamics calculation, the phonon run must use the same specification as the geometry optimization. If you want to change anything of significance, such as the cutoff, XC functional, or k-point sampling, you must carry out a fresh geometry optimization using these new settings before starting the lattice dynamics run.

  4. Checking the convergence of the results

    Unfortunately, following the above guidelines does not guarantee that your final phonon frequencies converge to any given tolerance. The only way to determine that this is the case is to verify the convergence behavior of the final properties. This requires a subtle judgment of whether you are measuring a 'primary' change resulting from the convergence of force constants or a 'secondary' one consequent on the slightly different geometry.

    As noted in the literature, and from experience, phonon frequencies are more sensitive to k-point convergence than geometries, assuming tolerance criteria of a few cm-1 and a few hundredths of an angstrom. Acoustic phonons near the Γ-point are the most sensitive, as they depend on near-cancellation between large force constant terms in the dynamical matrix. Dielectric properties and Born charges tend to be more sensitive than phonons, on a fractional proportion tolerance.

    The ultimate convergence test would require geometry optimization performed for a denser k-point grid, followed by a new phonon calculation - this is the only way to determine whether the k-point sampling has an effect on the accuracy of the calculated vibrational frequencies.

Phonon spectra in metallic compounds are very sensitive to the details of the Fermi surface. To obtain reasonable spectra using the linear response method for metals, the k-point sampling must be extremely dense: much denser than the Fine mesh. Consider specifying a value of less than 0.02 for the Separation parameter on the k-points tab of the CASTEP Electronic Options dialog. A clear sign of insufficient quality of k-point sampling is the presence of imaginary acoustic modes near the Γ-point.

See Also:

Phonon density of states
Phonons
Linear response
Setting up a calculation on an isolated molecule
CASTEP Calculation dialog