.. module:: ase.constraints
   :synopsis: Constraining some degrees of freedom

===========
Constraints
===========

When performing minimizations or dynamics one may wish to keep some
degrees of freedom in the system fixed. One way of doing this is by
attaching constraint object(s) directly to the atoms object.

Important: setting constraints will freeze the corresponding atom positions.
Changing such atom positions can be achieved:

- by directly setting the :attr:`~ase.Atoms.positions` attribute
  (see example of setting :ref:`atoms_special_attributes`),

- alternatively, by removing the constraints first::

    del atoms.constraints

  or::

    atoms.set_constraint()

  and using the :meth:`~ase.Atoms.set_positions` method.


The FixAtoms class
==================

This class is used for fixing some of the atoms.

.. class:: FixAtoms(indices=None, mask=None)

You must supply either the indices of the atoms that should be fixed
or a mask. The mask is a list of booleans, one for each atom, being true
if the atoms should be kept fixed.

For example, to fix the positions of all the Cu atoms in a simulation
with the indices keyword:

>>> from ase.constraints import FixAtoms
>>> c = FixAtoms(indices=[atom.index for atom in atoms if atom.symbol == 'Cu'])
>>> atoms.set_constraint(c)

or with the mask keyword:

>>> c = FixAtoms(mask=[atom.symbol == 'Cu' for atom in atoms])
>>> atoms.set_constraint(c)


The FixBondLength class
=======================

This class is used to fix the distance between two atoms specified by
their indices (*a1* and *a2*)

.. class:: FixBondLength(a1, a2)

Example of use::

  >>> c = FixBondLength(0, 1)
  >>> atoms.set_constraint(c)

In this example the distance between the atoms
with indices 0 and 1 will be fixed in all following dynamics and/or
minimizations performed on the *atoms* object.

This constraint is useful for finding minimum energy barriers for
reactions where the path can be described well by a single bond
length (see the :ref:`mep2` tutorial).

Important: If fixing multiple bond lengths, use the FixBondLengths class
below, particularly if the same atom is fixed to multiple partners.

.. _FixBondLengths:

The FixBondLengths class
========================

RATTLE-type holonomic constraints. More than one bond length can be fixed by
using this class. Especially for cases in which more than one bond length
constraint is applied on the same atom. It is done by specifying the indices
of the two atoms forming the bond in pairs.

.. class:: FixBondLengths(pairs)

Example of use::

  >>> c = FixBondLengths([[0, 1], [0, 2]])
    >>> atoms.set_constraint(c)

    Here the distances between atoms with indices 0 and 1 and atoms with
    indices 0 and 2 will be fixed. The constraint is for the same purpose
    as the FixBondLength class.

The FixLinearTriatomic class
============================

This class is used to keep the geometry of linear triatomic molecules
rigid in geometry optimizations or molecular dynamics runs. Rigidness
of linear triatomic molecules is impossible to attain by constraining
all interatomic distances using :class:`FixBondLength`, as this won't
remove an adequate number of degrees of freedom. To overcome this,
:class:`FixLinearTriatomic` fixes the distance between the outer atoms
with RATTLE and applies a linear vectorial constraint to the central
atom using the RATTLE-constrained positions of the outer atoms (read
more about the method here: G. Ciccotti, M. Ferrario, J.-P. Ryckaert,
Molecular Physics 47, 1253 (1982)).

When setting these constraints one has to specify a list of triples
of atomic indices, each triple representing a specific triatomic molecule.

.. autoclass:: FixLinearTriatomic

The example below shows how to fix the geometry of two carbon dioxide
molecules::

    >>> from ase.build import molecule
    >>> from ase.constraints import FixLinearTriatomic
    >>> atoms = molecule('CO2')
    >>> dimer = atoms + atoms.copy()
    >>> c = FixLinearTriatomic(triples=[(1, 0, 2), (4, 3, 5)])
    >>> dimer.set_constraint(c)

.. note::
    When specifying a triple of indices, the second element must correspond
    to the index of the central atom.

The FixedLine class
===================

.. autoclass:: FixedLine


The FixedPlane class
====================

.. autoclass:: FixedPlane

Example of use: :ref:`constraints diffusion tutorial`.


The FixedMode class
===================

.. autoclass:: FixedMode

A mode is a list of vectors specifying a direction for each atom. It often
comes from :meth:`ase.vibrations.Vibrations.get_mode`.


The FixCom class
===================

.. autoclass:: FixCom

Example of use::

  >>> from ase.constraints import FixCom
  >>> c = FixCom()
  >>> atoms.set_constraint(c)

The FixSubsetCom class
======================

.. autoclass:: FixSubsetCom

The Hookean class
=================


This class of constraints, based on Hooke's Law, is generally used to
conserve molecular identity in optimization schemes and can be used in three
different ways. In the first, it applies a Hookean restorative force between
two atoms if the distance between them exceeds a threshold. This is useful to
maintain the identity of molecules in quenched molecular dynamics, without
changing the degrees of freedom or violating conservation of energy. When the
distance between the two atoms is less than the threshold length, this
constraint is completely inactive.

.. autoclass:: Hookean

The below example tethers atoms at indices 3 and 4 together::

  >>> c = Hookean(a1=3, a2=4, rt=1.79, k=5.)
  >>> atoms.set_constraint(c)

Alternatively, this constraint can tether a single atom to a point in space,
for example to prevent the top layer of a slab from subliming during a
high-temperature MD simulation. An example of tethering atom at index 3 to its
original position:

>>> from ase.constraints import Hookean
>>> c = Hookean(a1=3, a2=atoms[3].position, rt=0.94, k=2.)
>>> atoms.set_constraint(c)

Reasonable values of the threshold (rt) and spring constant (k) for some
common bonds are below.

.. list-table::

  * - Bond
    - rt (Angstroms)
    - k (eV Angstrom^-2)
  * - O-H
    - 1.40
    - 5
  * - C-O
    - 1.79
    - 5
  * - C-H
    - 1.59
    - 7
  * - C=O
    - 1.58
    - 10
  * - Pt sublimation
    - 0.94
    - 2
  * - Cu sublimation
    - 0.97
    - 2

A third way this constraint can be applied is to apply a restorative force if an atom crosses a plane in space. For example::

  >>> c = Hookean(a1=3, a2=(0, 0, 1, -7), k=10.)
  >>> atoms.set_constraint(c)

This will apply a restorative force on atom 3 in the downward direction of magnitude k * (atom.z - 7) if the atom's vertical position exceeds 7 Angstroms. In other words, if the atom crosses to the (positive) normal side of the plane, the force is applied and directed towards the plane. (The same plane with the normal direction pointing in the -z direction would be given by (0, 0, -1, 7).)

For an example of use, see the :ref:`mhtutorial` tutorial.

.. note::

  In previous versions of ASE, this was known as the BondSpring constraint.


The ExternalForce class
=======================

This class can be used to simulate a constant external force
(e.g. the force of atomic force microscope).
One can set the absolute value of the force *f_ext* (in eV/Ang) and two
atom indices *a1* and *a2* to define on which atoms the force should act.
If the sign of the force is positive, the two atoms will be pulled apart.
The external forces which acts on both atoms are parallel to the
connecting line of the two atoms.

.. class:: ExternalForce(a1, a2, f_ext)

Example of use::

  >>> form ase.constraints import ExternalForce
  >>> c = ExternalForce(0, 1, 0.5)
  >>> atoms.set_constraint(c)

One can combine this constraint with :class:`FixBondLength` but one has to
consider the correct ordering when setting both constraints.
:class:`ExternalForce` must come first in the list as shown in the following
example.

  >>> from ase.constraints import ExternalForce, FixBondLength
  >>> c1 = ExternalForce(0, 1, 0.5)
  >>> c2 = FixBondLength(1, 2)
  >>> atoms.set_constraint([c1, c2])


The FixInternals class
======================

This class allows to fix an arbitrary number of bond lengths, angles
and dihedral angles as well as linear combinations of bond lengths
('bondcombos').
A fixed linear combination of bond lengths fulfils
:math:`\sum_i \mathrm{coef}_i \times \mathrm{bond_length}_i = \mathrm{constant}`.
The defined constraints are satisfied self consistently.
To define the constraints one needs to specify the
atoms object on which the constraint works (needed for atomic
masses), a list of bond, angle and dihedral constraints.
Those constraint definitions are always list objects containing
the value to be set and a list of atomic indices.
For the linear combination of bond lengths the list of atomic
indices is a list of bond definitions with coeficients
([[a1, a2, coef],[a3, a4, coef],]).
The usage of mic is supported by providing the keyword argument `mic=True`.
Using mic slows the algorithm and is probably not necessary in most cases.
The epsilon value specifies the accuracy to which the constraints are
fulfilled.
Please specify angles and dihedrals in degrees using the keywords angles_deg
and dihedrals_deg.

.. autoclass:: FixInternals

    .. automethod:: get_bondcombo


Example of use::

  >>> bond1 = [1.20, [1, 2]]
  >>> angle_indices1 = [2, 3, 4]
  >>> dihedral_indices1 = [2, 3, 4, 5]
  >>> bondcombo_indices1 = [[6, 7, 1.0], [8, 9, -1.0]]
  >>> angle1 = [atoms.get_angle(*angle_indices1), angle_indices1]
  >>> dihedral1 = [atoms.get_dihedral(*dihedral_indices1), dihedral_indices1]
  >>> bondcombo1 = [0.0, bondcombo_indices1]
  >>> c = FixInternals(bonds=[bond1], angles_deg=[angle1],
  ...                  dihedrals_deg=[dihedral1], bondcombos=[bondcombo1])
  >>> atoms.set_constraint(c)

This example defines a bond, an angle and a dihedral angle constraint
to be fixed at the same time
at which also the linear combination of bond lengths
:math:`1.0 * \text{bond}_{6-7} -1.0 * \text{bond}_{8-9}`
is fixed to the value of 0.0 Ångstrom.


Combining constraints
=====================

It is possible to supply several constraints on an atoms object. For
example one may wish to keep the distance between two nitrogen atoms
fixed while relaxing it on a fixed ruthenium surface::

  >>> pos = [[0.00000, 0.00000,  9.17625],
  ...        [0.00000, 0.00000, 10.27625],
  ...        [1.37715, 0.79510,  5.00000],
  ...        [0.00000, 3.18039,  5.00000],
  ...        [0.00000, 0.00000,  7.17625],
  ...        [1.37715, 2.38529,  7.17625]]
  >>> unitcell = [5.5086, 4.7706, 15.27625]

  >>> atoms = Atoms(positions=pos,
  ...               symbols='N2Ru4',
  ...               cell=unitcell,
  ...               pbc=[True,True,False])

  >>> fa = FixAtoms(mask=[a.symbol == 'Ru' for a in atoms])
  >>> fb = FixBondLength(0, 1)
  >>> atoms.set_constraint([fa, fb])

When applying more than one constraint they are passed as a list in
the :meth:`~ase.Atoms.set_constraint` method, and they will be applied
one after the other.

Important: If wanting to fix the length of more than one bond in the
simulation, do not supply a list of :class:`FixBondLength`
instances; instead, use a single instance of
:class:`FixBondLengths`.


Making your own constraint class
================================

A constraint class must have these two methods:

.. method:: adjust_positions(oldpositions, newpositions)

   Adjust the *newpositions* array inplace.

.. method:: adjust_forces(positions, forces)

   Adjust the *forces* array inplace.


A simple example::

  import numpy as np
  class MyConstraint:
      """Constrain an atom to move along a given direction only."""
      def __init__(self, a, direction):
          self.a = a
          self.dir = direction / sqrt(np.dot(direction, direction))

      def adjust_positions(self, atoms, newpositions):
          step = newpositions[self.a] - atoms.positions[self.a]
          step = np.dot(step, self.dir)
          newpositions[self.a] = atoms.positions[self.a] + step * self.dir

      def adjust_forces(self, atoms, forces):
          forces[self.a] = self.dir * np.dot(forces[self.a], self.dir)

A constraint can optionally have two additional methods, which
will be ignored if missing:

.. method:: adjust_momenta(atoms, momenta)

   Adjust the *momenta* array inplace.

.. method:: adjust_potential_energy(atoms, energy)

   Provide the difference in the *potential energy* due to the constraint.
   (Note that inplace adjustment is not possible for energy, which is a
   float.)

.. module:: ase.spacegroup.symmetrize

The FixSymmetry class
=====================

.. autoclass:: ase.constraints.FixSymmetry

The following are some utility functions to prepare
symmetrized configurations and to check symmetry.

.. autofunction:: ase.spacegroup.symmetrize.refine_symmetry

.. autofunction:: ase.spacegroup.symmetrize.check_symmetry

Here is an example of using these tools to demonstrate the difference between
minimising a perturbed bcc Al cell with and without symmetry-preservation.
Since bcc is unstable with respect to fcc with a Lennard Jones model, the
unsymmetrised case relaxes to fcc, while the constraint keeps the original
symmetry.

.. literalinclude:: fix_symmetry_example.py