# 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:

## The FixAtoms class¶

This class is used for fixing some of the atoms.

class ase.constraints.FixAtoms(indices=None, mask=None)[source]

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)


>>> 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 ase.constraints.FixBondLength(a1, a2)[source]

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 Dissociation of a molecule using the NEB method tutorial).

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

## 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 ase.constraints.FixBondLengths(pairs)[source]

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 FixBondLength, as this won’t remove an adequate number of degrees of freedom. To overcome this, 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.

class ase.constraints.FixLinearTriatomic(triples)[source]

Holonomic constraints for rigid linear triatomic molecules.

Apply RATTLE-type bond constraints between outer atoms n and m and linear vectorial constraints to the position of central atoms o to fix the geometry of linear triatomic molecules of the type:

n–o–m

Parameters:

triples: list

Indices of the atoms forming the linear molecules to constrain as triples. Sequence should be (n, o, m) or (m, o, n).

When using these constraints in molecular dynamics or structure optimizations, atomic forces need to be redistributed within a triple. The function redistribute_forces_optimization implements the redistribution of forces for structure optimization, while the function redistribute_forces_md implements the redistribution for molecular dynamics.

References:

Ciccotti et al. Molecular Physics 47 (1982) https://doi.org/10.1080/00268978200100942

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¶

class ase.constraints.FixedLine(a, direction)[source]

Constrain an atom index a to move on a given line only.

The line is defined by its vector direction.

## The FixedPlane class¶

class ase.constraints.FixedPlane(a, direction)[source]

Constrain an atom index a to move in a given plane only.

The plane is defined by its normal vector direction.

Example of use: Surface diffusion energy barriers using ASE constraints.

## The FixedMode class¶

class ase.constraints.FixedMode(mode)[source]

Constrain atoms to move along directions orthogonal to a given mode only.

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

## The FixCom class¶

class ase.constraints.FixCom[source]

Constraint class for fixing the center of mass.

References

https://pubs.acs.org/doi/abs/10.1021/jp9722824

Example of use:

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


## 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.

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.

 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 Constrained minima hopping (global optimization) 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 ase.constraints.ExternalForce(a1, a2, f_ext)[source]

Example of use:

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


One can combine this constraint with FixBondLength but one has to consider the correct ordering when setting both constraints. 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 these. A fixed linear combination of bond lengths fulfils $$\sum_i \text{coef}_i \times \text{bond_length}_i = \text{constant}$$. Fixed linear combinations of angles and dihedrals are defined similarly. 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],]). Linear combinations of angles and dihedrals are defined similarly with the corresponding number of atom indices. 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.

class ase.constraints.FixInternals(bonds=None, angles=None, dihedrals=None, angles_deg=None, dihedrals_deg=None, bondcombos=None, anglecombos=None, dihedralcombos=None, mic=False, epsilon=1e-07)[source]

Constraint object for fixing multiple internal coordinates.

Allows fixing bonds, angles, and dihedrals. Please provide angular units in degrees using angles_deg and dihedrals_deg.

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]]
>>> anglecombo_indices1 = [[10, 11, 12, 1.0], [13, 14, 15, 1.0]]
>>> dihedralcombo_indices1 = [[16, 17, 18, 19, 1.0], [20, 21, 22, 23, 1.0]]
>>> angle1 = [atoms.get_angle(*angle_indices1), angle_indices1]
>>> dihedral1 = [atoms.get_dihedral(*dihedral_indices1), dihedral_indices1]
>>> bondcombo1 = [0.0, bondcombo_indices1]
>>> anglecombo1 = [90.0, bondcombo_indices1]
>>> dihedralcombo1 = [90.0, bondcombo_indices1]
>>> c = FixInternals(bonds=[bond1], angles_deg=[angle1],
...                  dihedrals_deg=[dihedral1], bondcombos=[bondcombo1],
...                  anglecombos=[anglecombo1],
...                  dihedralcombos=[dihedralcombo1])
>>> 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 $$1.0 * \text{bond}_{6-7} -1.0 * \text{bond}_{8-9}$$ is fixed to the value of 0.0 Ångstrom. In addition, a linear combination of two angles and a linear combination of two dihedrals is fixed to the value of 90.0 degrees.

## 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 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 FixBondLength instances; instead, use a single instance of FixBondLengths.

## Making your own constraint class¶

A constraint class must have these two methods:

ase.constraints.adjust_positions(oldpositions, newpositions)

ase.constraints.adjust_forces(positions, forces)

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))

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

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:

ase.constraints.adjust_momenta(atoms, momenta)

ase.constraints.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.)

## The Filter class¶

Constraints can also be applied via filters, which acts as a wrapper around an atoms object. A typical use case will look like this:

 -------       --------       ----------
|       |     |        |     |          |
| Atoms |<----| Filter |<----| Dynamics |
|       |     |        |     |          |
-------       --------       ----------


and in Python this would be:

>>> atoms = Atoms(...)
>>> filter = Filter(atoms, ...)
>>> dyn = Dynamics(filter, ...)


This class hides some of the atoms in an Atoms object.

class ase.constraints.Filter(atoms, indices=None, mask=None)[source]

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

Example of use:

>>> from ase import Atoms, Filter
>>> atoms=Atoms(positions=[[ 0    , 0    , 0],
...                        [ 0.773, 0.600, 0],
...                        [-0.773, 0.600, 0]],
...             symbols='OH2')
>>> f1 = Filter(atoms, indices=[1, 2])
>>> f2 = Filter(atoms, mask=[0, 1, 1])
>>> f3 = Filter(atoms, mask=[a.Z == 1 for a in atoms])
>>> f1.get_positions()
[[ 0.773  0.6    0.   ]
[-0.773  0.6    0.   ]]


In all three filters only the hydrogen atoms are made visible. When asking for the positions only the positions of the hydrogen atoms are returned.

## The UnitCellFilter class¶

The unit cell filter is for optimizing positions and unit cell simultaneously. Note that ExpCellFilter will probably perform better.

class ase.constraints.UnitCellFilter(atoms, mask=None, cell_factor=None, hydrostatic_strain=False, constant_volume=False, scalar_pressure=0.0)[source]

Modify the supercell and the atom positions.

Create a filter that returns the atomic forces and unit cell stresses together, so they can simultaneously be minimized.

The first argument, atoms, is the atoms object. The optional second argument, mask, is a list of booleans, indicating which of the six independent components of the strain are relaxed.

• True = relax to zero

• False = fixed, ignore this component

Degrees of freedom are the positions in the original undeformed cell, plus the deformation tensor (extra 3 “atoms”). This gives forces consistent with numerical derivatives of the potential energy with respect to the cell degreees of freedom.

For full details see:

E. B. Tadmor, G. S. Smith, N. Bernstein, and E. Kaxiras, Phys. Rev. B 59, 235 (1999)

You can still use constraints on the atoms, e.g. FixAtoms, to control the relaxation of the atoms.

>>> # this should be equivalent to the StrainFilter
>>> atoms = Atoms(...)
>>> atoms.set_constraint(FixAtoms(mask=[True for atom in atoms]))
>>> ucf = UnitCellFilter(atoms)


You should not attach this UnitCellFilter object to a trajectory. Instead, create a trajectory for the atoms, and attach it to an optimizer like this:

>>> atoms = Atoms(...)
>>> ucf = UnitCellFilter(atoms)
>>> qn = QuasiNewton(ucf)
>>> traj = Trajectory('TiO2.traj', 'w', atoms)
>>> qn.attach(traj)
>>> qn.run(fmax=0.05)


• 0.05 eV/A^3 = 8 GPA

• 0.003 eV/A^3 = 0.48 GPa

• 0.0006 eV/A^3 = 0.096 GPa

• 0.0003 eV/A^3 = 0.048 GPa

• 0.0001 eV/A^3 = 0.02 GPa

cell_factor: float (default float(len(atoms)))

Factor by which deformation gradient is multiplied to put it on the same scale as the positions when assembling the combined position/cell vector. The stress contribution to the forces is scaled down by the same factor. This can be thought of as a very simple preconditioners. Default is number of atoms which gives approximately the correct scaling.

hydrostatic_strain: bool (default False)

Constrain the cell by only allowing hydrostatic deformation. The virial tensor is replaced by np.diag([np.trace(virial)]*3).

constant_volume: bool (default False)

Project out the diagonal elements of the virial tensor to allow relaxations at constant volume, e.g. for mapping out an energy-volume curve. Note: this only approximately conserves the volume and breaks energy/force consistency so can only be used with optimizers that do require do a line minimisation (e.g. FIRE).

scalar_pressure: float (default 0.0)

Applied pressure to use for enthalpy pV term. As above, this breaks energy/force consistency.

## The StrainFilter class¶

The strain filter is for optimizing the unit cell while keeping scaled positions fixed.

class ase.constraints.StrainFilter(atoms, mask=None, include_ideal_gas=False)[source]

Modify the supercell while keeping the scaled positions fixed.

Presents the strain of the supercell as the generalized positions, and the global stress tensor (times the volume) as the generalized force.

This filter can be used to relax the unit cell until the stress is zero. If MDMin is used for this, the timestep (dt) to be used depends on the system size. 0.01/x where x is a typical dimension seems like a good choice.

The stress and strain are presented as 6-vectors, the order of the components follow the standard engingeering practice: xx, yy, zz, yz, xz, xy.

Create a filter applying a homogeneous strain to a list of atoms.

The first argument, atoms, is the atoms object.

The optional second argument, mask, is a list of six booleans, indicating which of the six independent components of the strain that are allowed to become non-zero. It defaults to [1,1,1,1,1,1].

## The ExpCellFilter class¶

The exponential cell filter is an improved UnitCellFilter which is parameter free.

class ase.constraints.ExpCellFilter(atoms, mask=None, cell_factor=None, hydrostatic_strain=False, constant_volume=False, scalar_pressure=0.0)[source]

Modify the supercell and the atom positions.

Create a filter that returns the atomic forces and unit cell stresses together, so they can simultaneously be minimized.

The first argument, atoms, is the atoms object. The optional second argument, mask, is a list of booleans, indicating which of the six independent components of the strain are relaxed.

• True = relax to zero

• False = fixed, ignore this component

Degrees of freedom are the positions in the original undeformed cell, plus the log of the deformation tensor (extra 3 “atoms”). This gives forces consistent with numerical derivatives of the potential energy with respect to the cell degrees of freedom.

For full details see:

E. B. Tadmor, G. S. Smith, N. Bernstein, and E. Kaxiras, Phys. Rev. B 59, 235 (1999)

You can still use constraints on the atoms, e.g. FixAtoms, to control the relaxation of the atoms.

>>> # this should be equivalent to the StrainFilter
>>> atoms = Atoms(...)
>>> atoms.set_constraint(FixAtoms(mask=[True for atom in atoms]))
>>> ecf = ExpCellFilter(atoms)


You should not attach this ExpCellFilter object to a trajectory. Instead, create a trajectory for the atoms, and attach it to an optimizer like this:

>>> atoms = Atoms(...)
>>> ecf = ExpCellFilter(atoms)
>>> qn = QuasiNewton(ecf)
>>> traj = Trajectory('TiO2.traj', 'w', atoms)
>>> qn.attach(traj)
>>> qn.run(fmax=0.05)


• 0.05 eV/A^3 = 8 GPA

• 0.003 eV/A^3 = 0.48 GPa

• 0.0006 eV/A^3 = 0.096 GPa

• 0.0003 eV/A^3 = 0.048 GPa

• 0.0001 eV/A^3 = 0.02 GPa

cell_factor: (DEPRECATED)

Retained for backwards compatibility, but no longer used.

hydrostatic_strain: bool (default False)

Constrain the cell by only allowing hydrostatic deformation. The virial tensor is replaced by np.diag([np.trace(virial)]*3).

constant_volume: bool (default False)

Project out the diagonal elements of the virial tensor to allow relaxations at constant volume, e.g. for mapping out an energy-volume curve.

scalar_pressure: float (default 0.0)

Applied pressure to use for enthalpy pV term. As above, this breaks energy/force consistency.

Implementation details:

The implementation is based on that of Christoph Ortner in JuLIP.jl: https://github.com/libAtoms/JuLIP.jl/blob/expcell/src/Constraints.jl#L244

We decompose the deformation gradient as

F = exp(U) F0 x = F * F0^{-1} z = exp(U) z

If we write the energy as a function of U we can transform the stress associated with a perturbation V into a derivative using a linear map V -> L(U, V).

phi( exp(U+tV) (z+tv) ) ~ phi’(x) . (exp(U) v) + phi’(x) .

( L(U, V) exp(-U) exp(U) z )

where

nabla E(U)V = [S exp(-U)’]L(U,V)

= L’(U, S exp(-U)’) : V = L(U’, S exp(-U)’) : V = L(U, S exp(-U)) : V (provided U = U’)

where the : operator represents double contraction, i.e. A:B = trace(A’B), and

F = deformation tensor - 3x3 matrix F0 = reference deformation tensor - 3x3 matrix, np.eye(3) here U = cell degrees of freedom used here - 3x3 matrix V = perturbation to cell DoFs - 3x3 matrix v = perturbation to position DoFs x = atomic positions in deformed cell z = atomic positions in original cell phi = potential energy S = stress tensor [3x3 matrix] L(U, V) = directional derivative of exp at U in direction V, i.e d/dt exp(U + t V)|_{t=0} = L(U, V)

This means we can write

d/dt E(U + t V)|_{t=0} = L(U, S exp (-U)) : V

and therefore the contribution to the gradient of the energy is

nabla E(U) / nabla U_ij = [L(U, S exp(-U))]_ij

## The FixSymmetry class¶

class ase.spacegroup.symmetrize.FixSymmetry(atoms, symprec=0.01, adjust_positions=True, adjust_cell=True, verbose=False)[source]

Constraint to preserve spacegroup symmetry during optimisation.

Requires spglib package to be available.

The module also provides some utility functions to prepare symmetrized configurations and to check symmetry.

ase.spacegroup.symmetrize.refine_symmetry(atoms, symprec=0.01, verbose=False)[source]

Refine symmetry of an Atoms object

Parameters
• - input Atoms object (atoms) –

• - symmetry precicion (symprec) –

• - if True (verbose) –

• out symmetry information before and after (print) –

Returns

Return type

spglib dataset

ase.spacegroup.symmetrize.check_symmetry(atoms, symprec=1e-06, verbose=False)[source]

Check symmetry of $$atoms$$ with precision $$symprec$$ using $$spglib$$

Prints a summary and returns result of $$spglib.get_symmetry_dataset()$$

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