# Polarizable models

In polarizable force fields the charge distributions in molecules and
materials respond to their electrostatic environments. Polarizable
systems can be simulated in LAMMPS using three methods:

-   the fluctuating charge method, implemented in the [QEQ](fix_qeq)
    package,
-   the adiabatic core-shell method, implemented in the
    [CORESHELL](Howto_coreshell) package,
-   the thermalized Drude dipole method, implemented in the
    [DRUDE](Howto_drude) package.

The fluctuating charge method calculates instantaneous charges on
interacting atoms based on the electronegativity equalization principle.
It is implemented in the [fix qeq](fix_qeq) which is available in
several variants. It is a relatively efficient technique since no
additional particles are introduced. This method allows for charge
transfer between molecules or atom groups. However, because the charges
are located at the interaction sites, off-plane components of
polarization cannot be represented in planar molecules or atom groups.

The two other methods share the same basic idea: polarizable atoms are
split into one core atom and one satellite particle (called shell or
Drude particle) attached to it by a harmonic spring. Both atoms bear a
charge and they represent collectively an induced electric dipole. These
techniques are computationally more expensive than the QEq method
because of additional particles and bonds. These two charge-on-spring
methods differ in certain features, with the core-shell model being
normally used for ionic/crystalline materials, whereas the so-called
Drude model is normally used for molecular systems and fluid states.

The core-shell model is applicable to crystalline materials where the
high symmetry around each site leads to stable trajectories of the
core-shell pairs. However, bonded atoms in molecules can be so close
that a core would interact too strongly or even capture the Drude
particle of a neighbor. The Drude dipole model is relatively more
complex in order to remedy this and other issues. Specifically, the
Drude model includes specific thermostatting of the core-Drude pairs and
short-range damping of the induced dipoles.

The three polarization methods can be implemented through a
self-consistent calculation of charges or induced dipoles at each
timestep. In the fluctuating charge scheme this is done by the matrix
inversion method in [fix qeq/point](fix_qeq), but for core-shell or
Drude-dipoles the relaxed-dipoles technique would require an slow
iterative procedure. These self-consistent solutions yield accurate
trajectories since the additional degrees of freedom representing
polarization are massless. An alternative is to attribute a mass to the
additional degrees of freedom and perform time integration using an
extended Lagrangian technique. For the fluctuating charge scheme this is
done by [fix qeq/dynamic](fix_qeq), and for the charge-on-spring models
by the methods outlined in the next two sections. The assignment of
masses to the additional degrees of freedom can lead to unphysical
trajectories if care is not exerted in choosing the parameters of the
polarizable models and the simulation conditions.

In the core-shell model the vibration of the shells is kept faster than
the ionic vibrations to mimic the fast response of the polarizable
electrons. But in molecular systems thermalizing the core-Drude pairs at
temperatures comparable to the rest of the simulation leads to several
problems (kinetic energy transfer, too short a timestep, etc.) In order
to avoid these problems the relative motion of the Drude particles with
respect to their cores is kept \"cold\" so the vibration of the
core-Drude pairs is very slow, approaching the self-consistent regime.
In both models the temperature is regulated using the velocities of the
center of mass of core+shell (or Drude) pairs, but in the Drude model
the actual relative core-Drude particle motion is thermostatted
separately as well.