# Longwave

This page gives hints on how to compute spatial dispersion properties (e.g. flexoelectric tensor or dynamical quadrupoles) with the longwave DFPT driver of the ABINIT package.

## Introduction¶

In condensed-matter physics, spatial dispersion refers to the dependence of many material
properties on the wavevector **q** at which they are probed, or equivalently on the gradients of the
external field (electric, magnetic, strain…) and/or the response in real space. Remarkable
examples of such gradient effects include the natural optical rotation, [Belinicher1980] whereby some crystals
rotate the polarization plane of the transmitted light, or the flexoelectric tensor, [Zubko2013] which
describes the polarization response to a gradient of applied strain.

Since ABINIT v9.0.2, the calculation of several spatial dispersion quantities is accessible
via the longwave driver. The implementation, detailed in [Romero2020], follows the formalism developed in
[Royo2019] that adapts the classic longwave method of Born and Huang [Born1954] with the modern tools of
the DFPT. Technically, the driver computes analytical third-order energy derivatives (readily converted to physical
spatial dispersion quantities) with respect to two of the *standard*
perturbations (atomic displacements, electric field and strain) and to the wavevector **q**.

At present (April 30, 2021 ), ABINIT enables the calculation of four spatial dispersion tensors required in order to build all the contributions to the bulk flexoelectric tensor following the prescriptions exposed in [Stengel2013]. These include, the clamped-ion flexoelectric tensor (a purely electronic contribution) and the first real-space moment of three other tensors (entering the mixed and lattice mediated contributions): the polarization response to an atomic displacement, the interatomic force constants and the piezoelectric force-response tensor. The implementation also provides access, as a by-product of the flexoelectric formalism, to the dynamical quadrupoles, which can be considered as the spatial-dispersion counterpart of the Born effective charges (dynamical dipoles). After execution, the longwave routines generate a third-order derivative database that is subsequently used by ANADDB either to compute and print the different contributions to the flexoelectric tensor, or to consider quadrupolar fields within the Fourier interpolation procedure of the dynamical matrix [Royo2020].

The underlying theory of the long-wave DFPT approach [Royo2019] has been developed for its application on time-reversal symmetric insulating crystals only. Therefore, the usage of the longwave driver is restricted to materials of this kind. An extension of the theoretical framework and the ABINIT implementation to magnetic insulators and/or metals will be hopefully pursued in the future.

Regarding the flexoelectric tensor that ABINIT provides, a few remarks are in order. First, recall that
this is the **bulk** flexoelectric tensor and that a surface counterpart is still missing in order to obtain
the total flexoelectric tensor of a system.[Stengel2016] Even though the implementation can be applied
to slabs or low-dimensional systems (such as 2D materials), via a supercell approach, the outcome of such a
calculation will not directly
produce the total (i.e., bulk+surface) flexoelectric tensor, neither the surface contribution can be readily estimated
from it. How to obtain the total flexoelectric tensor within the current capabilities of ABINIT will be the
subject of a future publication.

The interested user must be likewise aware of the physical ambiguities existing in the definition of the bulk flexoelectric tensor which inherently affect the longwave driver. One of them precludes its usage to obtain the flexoelectric tensor of non-centrosymmetric (i.e., piezoelectric) materials (see section VII.c of [Stengel2013]). The other one is related with the dependence of the bulk flexoelectric coefficients on the choice of an arbitrary reference energy. The average electrostatic potential has been taken as the energy reference within the longwave driver. Nonetheless, as illustrated in section IV.d of [Stengel2016a], other choices might lead to quantitatively and qualitatively different outcomes. On the other hand, such ambiguity is well known to disappear when the surface-specific part is accounted for, as done e.g. in [Stengel2014].

The longwave implementation is still under heavy development. To date it requires the use of norm-conserving pseudopotentials without XC nonlinear core corrections and is limited to LDA XC functionals. The use of spherical harmonics for the nonlocal projectors is mandatory through the option useylm=1.

The following steps are required to perform a longwave DFPT calculation of the bulk flexoelectric tensor (see, e.g., tests lw[1] to lw[3]):

- Perform ground-state calculation.
- Perform ddk and d2_dkdk response function calculations.
- Perform response function calculations at
**q**=Γ to atomic displacements, electric field and strain perturbations, including the option prepalw=1. - Perform a longwave DFTP calculation of third-order energy derivatives (optdriver=10 and lw_flexo=1).
- Use MRGDDB to merge 1
^{st}, 2^{nd}and 3^{rd}order DDB files. - Run ANADDB with flexoflag=1.

The following steps are required to perform a phonon dispersion calculation including quadrupolar fields in the nonanalytical part of the dynamical matrix (see, e.g., tests lw[4] to lw[6]):

- Perform ground-state calculation.
- Perform ddk and d2_dkdk response function calculations.
- Perform response function calculations at
**q**=Γ to atomic displacements and electric field, including the option prepalw=1. - Perform a longwave DFTP calculation of third-order energy derivatives (optdriver=10 and lw_qdrpl=1).
- Perform response function calculations to atomic displacements at finite
**q**(coarse grid). - Use MRGDDB to merge 2
^{nd}and 3^{rd}order DDB files. - Run a phonon dispersion calculation of ANADDB including dipquad=1 and/or quadquad=1.

## Related Input Variables¶

*compulsory:*

- lw_flexo LongWave calculation of FLEXOelectricity related spatial dispersion tensors
- lw_qdrpl LongWave calculation of dynamical QuaDRuPoLes tensor
- prepalw PREPAre LongWave calculation

*basic:*

- dipquad DIPole-QUADdrupole interaction
- flexoflag FLEXOelectric tensor FLAG
- quadquad QUADdrupole-QUADdrupole interaction

## Selected Input Files¶

*tutorespfn:*

- tests/tutorespfn/Input/tlw_1.abi
- tests/tutorespfn/Input/tlw_2.abi
- tests/tutorespfn/Input/tlw_4.abi
- tests/tutorespfn/Input/tlw_5.abi
- tests/tutorespfn/Input/tlw_6.abi
- tests/tutorespfn/Input/tlw_7.abi

## Tutorials¶

A tutorial is in preparation.