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# IMIX

IMIX = 0 | 1 | 2 | 4
Default: IMIX = 4

Description: IMIX specifies the type of mixing.

• IMIX=0: no mixing.
${\displaystyle \rho _{{{\rm {mix}}}}=\rho _{{{\rm {out}}}}\,}$
• IMIX=1: Kerker mixing.[1]
The mixed density is given by
${\displaystyle \rho _{{{\rm {mix}}}}\left(G\right)=\rho _{{{\rm {in}}}}\left(G\right)+A{\frac {G^{2}}{G^{2}+B^{2}}}{\Bigl (}\rho _{{{\rm {out}}}}\left(G\right)-\rho _{{{\rm {in}}}}\left(G\right){\Bigr )}}$
with ${\displaystyle A}$ =AMIX and ${\displaystyle B}$ =BMIX
If BMIX is chosen to be very small, e.g. BMIX=0.0001, a simple straight mixing is obtained. Please mind, that BMIX=0 might cause floating point exceptions on some platforms.
• IMIX=2: A variant of the popular Tchebycheff mixing scheme.[2]
In our implementation a second order equation of motion is used, that reads:
${\displaystyle {\ddot {\rho }}_{{{\rm {in}}}}\left(G\right)=2*A{\frac {G^{2}}{G^{2}+B^{2}}}{\Bigl (}\rho _{{{\rm {out}}}}\left(G\right)-\rho _{{{\rm {in}}}}\left(G\right){\Bigr )}-\mu {\dot {\rho }}_{{{\rm {in}}}}\left(G\right)}$
with ${\displaystyle A}$ =AMIX, ${\displaystyle B}$ =BMIX, and ${\displaystyle \mu }$ =AMIN.
A simple velocity Verlet algorithm is used to integrate this equation, and the discretized equation reads (the index N now refers to the electronic iteration, F is the force acting on the charge):
${\displaystyle {\dot {\rho }}_{{N+1/2}}={\Bigl (}\left(1-\mu /2\right){\dot {\rho }}_{{N-1/2}}+2*F_{N}{\Bigr )}/\left(1+\mu /2\right)}$
where
${\displaystyle F\left(G\right)=A{\frac {G^{2}}{G^{2}+B^{2}}}{\Bigl (}\rho _{{{\rm {out}}}}\left(G\right)-\rho _{{{\rm {in}}}}\left(G\right){\Bigr )}}$
and
${\displaystyle \rho _{{N+1}}=\rho _{{N+1}}+{\dot {\rho }}_{{N+1/2}}}$ .
For BMIX≈0, no model for the dielectric matrix is used. It is easy to see, that for ${\displaystyle \mu =2}$ a simple straight mixing is obtained. Therefore, ${\displaystyle \mu =2}$ corresponds to maximal damping, and obviously ${\displaystyle \mu =0}$ implies no damping. Optimal parameters for ${\displaystyle \mu }$ and AMIX can be determined by converging first with the Pulay mixer (IMIX=4) to the groundstate. Then the eigenvalues of the charge dielectric matrix as given in the OUTCAR file must be inspected. Search for the last orrurance of
eigenvalues of (default mixing * dielectric matrix)

in the OUTCAR file. The optimal parameters are then given by:
 AMIX ${\displaystyle ={{\rm {AMIX}}}({{\rm {as\;used\;in\;Pulay\;run}}})*{{\rm {smallest\;eigenvalue}}}}$ AMIN ${\displaystyle =\mu =2{\sqrt {{{\rm {smallest\;eigenvalue}}}/{{\rm {largest\;eigenvalue}}}}}}$
• IMIX=4: Broyden's 2nd method,[3][4] or Pulay's mixing method[5] (depending on the choice of WC).
With the default settings, a Pulay mixer with an initial approximation for the charge dielectric function according to Kerker[1] is used
${\displaystyle A\times \min \left({\frac {G^{2}}{G^{2}+B^{2}}},A_{{{\rm {min}}}}\right)}$
where ${\displaystyle A}$ =AMIX, ${\displaystyle B}$ =BMIX, and ${\displaystyle A_{{{\rm {min}}}}}$ =AMIN.
A reasonable choice for AMIN is usually AMIN=0.4. AMIX depends very much on the system, for metals this parameter usually has to be rather small, e.g. AMIX= 0.02.
In the Broyden scheme, the functional form of the initial mixing matrix is determined by AMIX and BMIX (or alternatively specified by means of the INIMIX-tag). The metric used in the Broyden scheme is specified through MIXPRE.

## References

1. a b G. P. Kerker, Phys. Rev. B 23, 3082 (1981).
2. H. Akai and P.H. Dederichs, J. Phys. C 18 (1985).
3. S. Blügel, PhD Thesis, RWTH Aachen (1988).
4. D. D. Johnson, Phys. Rev. B38, 12807 (1988).
5. P. Pulay, Chem. Phys. Lett. 73, 393 (1980).