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# LEPSILON

LEPSILON = .TRUE. | .FALSE.
Default: LEPSILON = .FALSE.

Description: LEPSILON=.TRUE. determines the static dielectric matrix, ion-clamped piezoelectric tensor and the Born effective charges using density functional perturbation theory.

Determines the static ion-clamped dielectric matrix using density functional perturbation theory. The dielectric matrix is calculated with and without local field effects. Usually local field effects are determined on the Hartree level, i.e. including changes of the Hartree potential. To include microscopic changes of the exchange correlation potential the tag LRPA=.FALSE. must be set. The method is explained in detail in paper by Gajdoš et al.,[1] and closely follows the original work of Baroni and Resta.[2] A summation over empty conduction band states is not required, instead the usual expressions in perturbation theory (LOPTICS=.TRUE.),

${\displaystyle |\nabla _{\mathbf {k} }|{\tilde {u}}_{n{\mathbf {k} }}\rangle =\sum _{n'\neq n}{\frac {|{\tilde {u}}_{n'{\mathbf {k} }}\rangle \langle {\tilde {u}}_{n'{\mathbf {k} }}|{\frac {\partial \,({\mathbf {H} }({\mathbf {k} })-\epsilon _{n{\mathbf {k} }}{\mathbf {S} }({\mathbf {k} }))}{\partial {\mathbf {k} }}}|{\tilde {u}}_{n{\mathbf {k} }}\rangle }{\epsilon _{n{\mathbf {k} }}-\epsilon _{n'{\mathbf {k} }}}},}$

are rewritten as linear Sternheimer equations:

${\displaystyle \left({\mathbf {H} }({\mathbf {k} })-\epsilon _{n{\mathbf {k} }}{\mathbf {S} }({\mathbf {k} })\right)|\nabla _{\mathbf {k} }{\tilde {u}}_{n{\mathbf {k} }}\rangle =-{\frac {\partial \,({\mathbf {H} }({\mathbf {k} })-\epsilon _{n{\mathbf {k} }}{\mathbf {S} }({\mathbf {k} }))}{\partial {\mathbf {k} }}}|{\tilde {u}}_{n{\mathbf {k} }}\rangle .}$

The solution of this equation involves similar iterative techniques as the conventional selfconsistency cycles. Hence, for each element of the dielectric matrix several lines will be written to the stdout and OSZICAR. These possess a similar structure as for conventional selfconsistent or non-selfconsistent calculations (a residual minimization scheme is used to solve the linear equation, other schemes such as Davidson do not apply to a linear equation):

       N       E              dE             d eps       ncg     rms          rms(c)
RMM:   1    -0.14800E+01   -0.85101E-01   -0.72835E+00   220   0.907E+00    0.146E+00
RMM:   2    -0.14248E+01    0.55195E-01   -0.27994E-01   221   0.449E+00    0.719E-01
RMM:   3    -0.13949E+01    0.29864E-01   -0.10673E-01   240   0.322E+00    0.131E-01
RMM:   4    -0.13949E+01    0.13883E-04   -0.31511E-03   242   0.600E-01    0.336E-02
RMM:   5    -0.13949E+01    0.28357E-04   -0.25757E-04   228   0.177E-01    0.126E-02


It is important to note that exact values for the dielectric matrix are obtained even if only valence band states are calculated. Hence this method does not require to increase the NBANDS parameter. The final values for the static dielectric matrix can be found in the OUTCAR file after the lines

 MICROSCOPIC STATIC DIELECTRIC TENSOR (excluding local field effects)


and

 MACROSCOPIC STATIC DIELECTRIC TENSOR (including local field effects in DFT)


The values found after MACROSCOPIC STATIC DIELECTRIC TENSOR (excluding local field effects) should match exactly to the zero frequency values ω→0 determined by the method selected using LOPTICS=.TRUE.. This offers a convenient way to determine how many empty bands are required for LOPTICS=.TRUE.. Simply execute VASP using LEPSILON=.TRUE. in order to determine the exact values for the dielectric constants. Next, switch to LOPTICS=.TRUE. and increase the number of conduction bands until the same values are obtained as using density functional perturbation theory.

Note that the routine also parses and uses the value supplied in the LNABLA tag. Furthermore, the routine calculates the Born effective charge tensor (dynamical charges) and electronic contribution to the the piezoelectric tensor , and prints them after

 BORN EFFECTIVE CHARGES (in e, cummulative output)


and

 PIEZOELECTRIC TENSOR  for field in x, y, z        (C/m^2)


if LRPA=.FALSE. is set (the calculated tensors are not sensible in the random phase approximation LRPA=.TRUE.).

Pros compared to LOPTICS=.TRUE.
• no conduction bands required.
• local field effects included on the RPA and DFT level (see ACFDT/RPA calculations).

Cons compared to LOPTICS=.TRUE.
• presently only static properties available.
• requires a relatively time consuming iterative process.
• does not support HF or hybrid functionals, whereas LOPTICS=.TRUE. and the GW routines do.

We do not recommend to select LOPTICS=.TRUE. and LEPSILON=.TRUE. in a single run (although it might work in some versions). Density functional perturbation theory LEPSILON=.TRUE. does not require to increase NBANDS and is, in fact, much slower if NBANDS is increased, whereas the summation over emtpy conduction band states requires a large number of such states.