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Flexibility

Linker ring rotation and isotropic scaling. Used by CageBuilder.rotors(), scaled() and distance_scaling(): cage_isomer_builder.utils.flexibility.

Framework flexibility for descriptors: linker ring rotation and isotropic pore expansion/contraction.

Ring rotation

A rotor is an aromatic ring of a linker that can turn about the axis through its two backbone atoms: the ring atoms bonded (outside the ring) to the rest of the framework, e.g. C1 and C4 of a 1,4-phenylene. Everything attached to the ring except through those two backbone bonds (its H atoms, R sites, substituents) turns with it; the backbone itself does not move, so every bond length and every bond to the node is preserved exactly.

A backbone atom is a ring atom whose exocyclic heavy neighbour leads to a metal, to another ring, or out of the linker; an exocyclic neighbour that only leads to a terminal substituent (NH2, CH3, ...) is not backbone. Rings with other than two backbone atoms (e.g. a tritopic core) are not rotors.

:func:rotation_samples gives every slot's position at each angle, ready for :func:cage_isomer_builder.utils.ensemble.ensemble_descriptor: slots on one ring move together; different rings move independently. Angle weights may come from an energy profile (e.g. a UFF4MOF torsion scan, via :func:cage_isomer_builder.utils.ensemble.boltzmann_weights); the default is uniform.

Pore scaling

:func:scale_structure moves every building block rigidly so its centroid's distance from the pore centre is multiplied by a factor (a periodic cell is scaled instead, keeping each block's internal geometry). Internal geometry is untouched, so the result shows how each FG-FG distance responds to an isotropic expansion or contraction; :func:distance_scaling gives the slope d(distance)/d(factor) of every pair.

Rotor dataclass 

Rotor(ring: tuple, axis_atoms: tuple, moving: tuple, linker: int = -1)

Attributes:

Name Type Description
ring tuple of int
axis_atoms (int, int)

The two backbone ring atoms; the rotation axis passes through them.
moving tuple of int

Every atom that turns (ring atoms other than the axis atoms, and everything hanging off the ring).
linker int

Index of the linker (in the list given to :func:find_rotors), or -1.

Examples:

>>> from cage_isomer_builder import example_data
>>> from cage_isomer_builder.cage import Tri4Di6CageBuilder
>>> cage = Tri4Di6CageBuilder(node=example_data.path("uio66_tri_node"),
...                           linker=example_data.path("bdc"), scale_multiplier=0.225)
>>> _ = cage.build()
>>> _ = cage.optimise(include_charges=False, logfile=None)
>>> _ = cage.functionalise()
>>> atoms, adjacency = cage.to_ase(), cage.adjacency()
>>> linkers, nodes = cage.building_blocks()
>>> from cage_isomer_builder.utils.flexibility import find_rotors
>>> rotor = find_rotors(atoms, linkers, adjacency)[0]
>>> len(rotor.ring), len(rotor.axis_atoms), len(rotor.moving)   # ring C, axis C, moving atoms
(6, 2, 8)

find_rotors 

find_rotors(atoms, linker_atom_indices=None, adjacency=None)

Rotatable aromatic rings (see the module docstring).

Parameters:

Name Type Description Default
atoms Atoms
required
linker_atom_indices list of list of int

Atoms of each linker; only rings inside a linker are returned, and a neighbour outside it counts as backbone. Default: every ring.

 None

Examples:

>>> from cage_isomer_builder import example_data
>>> from cage_isomer_builder.cage import Tri4Di6CageBuilder
>>> cage = Tri4Di6CageBuilder(node=example_data.path("uio66_tri_node"),
...                           linker=example_data.path("bdc"), scale_multiplier=0.225)
>>> _ = cage.build()
>>> _ = cage.optimise(include_charges=False, logfile=None)
>>> _ = cage.functionalise()
>>> atoms, adjacency = cage.to_ase(), cage.adjacency()
>>> linkers, nodes = cage.building_blocks()
>>> from cage_isomer_builder.utils.flexibility import find_rotors
>>> len(find_rotors(atoms, linkers, adjacency))       # one phenylene ring per linker
6
Source code in cage_isomer_builder/utils/flexibility.py 
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def find_rotors(atoms, linker_atom_indices=None, adjacency=None):
    """
    Rotatable aromatic rings (see the module docstring).

    Parameters
    ----------
    atoms : ase.Atoms
    linker_atom_indices : list of list of int, optional
        Atoms of each linker; only rings inside a linker are returned, and a
        neighbour outside it counts as backbone. Default: every ring.

    Examples
    --------
    >>> from cage_isomer_builder import example_data
    >>> from cage_isomer_builder.cage import Tri4Di6CageBuilder
    >>> cage = Tri4Di6CageBuilder(node=example_data.path("uio66_tri_node"),
    ...                           linker=example_data.path("bdc"), scale_multiplier=0.225)
    >>> _ = cage.build()
    >>> _ = cage.optimise(include_charges=False, logfile=None)
    >>> _ = cage.functionalise()
    >>> atoms, adjacency = cage.to_ase(), cage.adjacency()
    >>> linkers, nodes = cage.building_blocks()
    >>> from cage_isomer_builder.utils.flexibility import find_rotors
    >>> len(find_rotors(atoms, linkers, adjacency))       # one phenylene ring per linker
    6
    """
    if adjacency is None:
        adjacency = bond_graph(atoms)
    numbers = atoms.numbers
    rings = aromatic_rings(atoms, adjacency)
    ring_atoms = {a for r in rings for a in r}
    owner = {}
    if linker_atom_indices is not None:
        owner = {a: k for k, lk in enumerate(linker_atom_indices) for a in lk}
    rotors = []
    for ring in rings:
        lk = owner.get(ring[0], -1)
        if linker_atom_indices is not None and lk < 0:
            continue
        members = set(ring)
        backbone = []
        for a in ring:
            for b in adjacency[a]:
                if b in members or numbers[b] <= 1:
                    continue
                side = _side(adjacency, b, members)
                leads_out = (any(is_metal(numbers[k]) for k in side)
                             or any(k in ring_atoms for k in side)
                             or (linker_atom_indices is not None
                                 and any(owner.get(k, -1) != lk for k in side)))
                if leads_out:
                    backbone.append((a, b))
        if len({a for a, _ in backbone}) != 2:
            continue
        (a1, b1), (a2, b2) = sorted({a: (a, b) for a, b in backbone}.values())
        # The two axis atoms split the ring into two arcs: collect both,
        # with everything hanging off them.
        moving = set()
        for start in ring:
            if start not in (a1, a2) and start not in moving:
                moving |= _side(adjacency, start, {a1, a2, b1, b2})
        # A ring whose "substituents" reach a metal or another building block
        # (e.g. an ortho group that coordinates the node) cannot turn without
        # dragging the framework along: not a rotor.
        if any(is_metal(numbers[k]) for k in moving) or (
                linker_atom_indices is not None
                and any(owner.get(k, -1) != lk for k in moving)):
            continue
        rotors.append(Rotor(tuple(ring), (a1, a2), tuple(sorted(moving)), lk))
    return rotors

rotor_axis 

rotor_axis(atoms, rotor)

Origin (first axis atom) and unit axis of a rotor.

Examples:

>>> from cage_isomer_builder import example_data
>>> from cage_isomer_builder.cage import Tri4Di6CageBuilder
>>> cage = Tri4Di6CageBuilder(node=example_data.path("uio66_tri_node"),
...                           linker=example_data.path("bdc"), scale_multiplier=0.225)
>>> _ = cage.build()
>>> _ = cage.optimise(include_charges=False, logfile=None)
>>> _ = cage.functionalise()
>>> atoms, adjacency = cage.to_ase(), cage.adjacency()
>>> linkers, nodes = cage.building_blocks()
>>> import numpy as np
>>> from cage_isomer_builder.utils.flexibility import find_rotors, rotor_axis
>>> origin, axis = rotor_axis(atoms, find_rotors(atoms, linkers, adjacency)[0])
>>> round(float(np.linalg.norm(axis)), 6)
1.0
Source code in cage_isomer_builder/utils/flexibility.py 
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def rotor_axis(atoms, rotor):
    """Origin (first axis atom) and unit axis of a rotor.

    Examples
    --------
    >>> from cage_isomer_builder import example_data
    >>> from cage_isomer_builder.cage import Tri4Di6CageBuilder
    >>> cage = Tri4Di6CageBuilder(node=example_data.path("uio66_tri_node"),
    ...                           linker=example_data.path("bdc"), scale_multiplier=0.225)
    >>> _ = cage.build()
    >>> _ = cage.optimise(include_charges=False, logfile=None)
    >>> _ = cage.functionalise()
    >>> atoms, adjacency = cage.to_ase(), cage.adjacency()
    >>> linkers, nodes = cage.building_blocks()
    >>> import numpy as np
    >>> from cage_isomer_builder.utils.flexibility import find_rotors, rotor_axis
    >>> origin, axis = rotor_axis(atoms, find_rotors(atoms, linkers, adjacency)[0])
    >>> round(float(np.linalg.norm(axis)), 6)
    1.0
    """
    p0 = atoms.positions[rotor.axis_atoms[0]]
    d = atoms.positions[rotor.axis_atoms[1]] - p0
    if atoms.pbc.any():
        d, _ = find_mic(d, atoms.cell, atoms.pbc)
    return p0, d / np.linalg.norm(d)

rotate_rotor 

rotate_rotor(atoms, rotor, angle_deg)

Copy of atoms with one ring turned by angle_deg.

Examples:

>>> from cage_isomer_builder import example_data
>>> from cage_isomer_builder.cage import Tri4Di6CageBuilder
>>> cage = Tri4Di6CageBuilder(node=example_data.path("uio66_tri_node"),
...                           linker=example_data.path("bdc"), scale_multiplier=0.225)
>>> _ = cage.build()
>>> _ = cage.optimise(include_charges=False, logfile=None)
>>> _ = cage.functionalise()
>>> atoms, adjacency = cage.to_ase(), cage.adjacency()
>>> linkers, nodes = cage.building_blocks()
>>> from cage_isomer_builder.utils.flexibility import find_rotors, rotate_rotor
>>> rotor = find_rotors(atoms, linkers, adjacency)[0]
>>> turned = rotate_rotor(atoms, rotor, 30.0)
>>> bonds = [(i, j) for i in adjacency for j in adjacency[i] if i < j]
>>> bool(max(abs(atoms.get_distance(i, j) - turned.get_distance(i, j)) for i, j in bonds) < 1e-9)
True
Source code in cage_isomer_builder/utils/flexibility.py 
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def rotate_rotor(atoms, rotor, angle_deg):
    """Copy of ``atoms`` with one ring turned by ``angle_deg``.

    Examples
    --------
    >>> from cage_isomer_builder import example_data
    >>> from cage_isomer_builder.cage import Tri4Di6CageBuilder
    >>> cage = Tri4Di6CageBuilder(node=example_data.path("uio66_tri_node"),
    ...                           linker=example_data.path("bdc"), scale_multiplier=0.225)
    >>> _ = cage.build()
    >>> _ = cage.optimise(include_charges=False, logfile=None)
    >>> _ = cage.functionalise()
    >>> atoms, adjacency = cage.to_ase(), cage.adjacency()
    >>> linkers, nodes = cage.building_blocks()
    >>> from cage_isomer_builder.utils.flexibility import find_rotors, rotate_rotor
    >>> rotor = find_rotors(atoms, linkers, adjacency)[0]
    >>> turned = rotate_rotor(atoms, rotor, 30.0)
    >>> bonds = [(i, j) for i in adjacency for j in adjacency[i] if i < j]
    >>> bool(max(abs(atoms.get_distance(i, j) - turned.get_distance(i, j)) for i, j in bonds) < 1e-9)
    True
    """
    out = atoms.copy()
    origin, axis = rotor_axis(atoms, rotor)
    pts = _unwrapped(atoms, rotor.moving, origin)
    out.positions[list(rotor.moving)] = _rotate(pts, origin, axis, np.radians(angle_deg))
    return out

rotation_samples 

rotation_samples(atoms, rotors, slot_indices, angles, centre=None, threshold=0.5, adjacency=None)

Slot geometry at every rotation angle.

Parameters:

Name Type Description Default
rotors list of Rotor
required
slot_indices sequence of int

Site atoms in slot order.

 required
angles sequence of float

Rotation angles in degrees (applied to every ring independently).

 required
centre array - like

Pore centre, to re-classify endo/exo at each angle.

 None

Returns:

Name Type Description
positions (ndarray, shape(n_slots, n_angles, 3))
vectors (ndarray, shape(n_slots, n_angles, 3))

Unit ring-C -> site direction.
orientations np.ndarray of str, shape (n_slots, n_angles)
rotor_of_slot np.ndarray of int

Rotor of each slot (-1: the slot is on no rotor and stays put).

Examples:

>>> from cage_isomer_builder import example_data
>>> from cage_isomer_builder.cage import Tri4Di6CageBuilder
>>> cage = Tri4Di6CageBuilder(node=example_data.path("uio66_tri_node"),
...                           linker=example_data.path("bdc"), scale_multiplier=0.225)
>>> _ = cage.build()
>>> _ = cage.optimise(include_charges=False, logfile=None)
>>> _ = cage.functionalise()
>>> atoms, adjacency = cage.to_ase(), cage.adjacency()
>>> linkers, nodes = cage.building_blocks()
>>> from cage_isomer_builder.utils.flexibility import find_rotors, rotation_samples
>>> rotors = find_rotors(atoms, linkers, adjacency)
>>> pos, vec, orientation, rotor_of_slot = rotation_samples(
...     atoms, rotors, cage._fg_anchor_indices, [0, 90, 180, 270],
...     centre=cage.pore_centre(), adjacency=adjacency)
>>> pos.shape, sorted(set(rotor_of_slot.tolist()))      # 24 slots x 4 angles
((24, 4, 3), [0, 1, 2, 3, 4, 5])
Source code in cage_isomer_builder/utils/flexibility.py 
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def rotation_samples(atoms, rotors, slot_indices, angles, centre=None, threshold=0.5,
                     adjacency=None):
    """
    Slot geometry at every rotation angle.

    Parameters
    ----------
    rotors : list of Rotor
    slot_indices : sequence of int
        Site atoms in slot order.
    angles : sequence of float
        Rotation angles in degrees (applied to every ring independently).
    centre : array-like, optional
        Pore centre, to re-classify endo/exo at each angle.

    Returns
    -------
    positions : np.ndarray, shape (n_slots, n_angles, 3)
    vectors : np.ndarray, shape (n_slots, n_angles, 3)
        Unit ring-C -> site direction.
    orientations : np.ndarray of str, shape (n_slots, n_angles)
    rotor_of_slot : np.ndarray of int
        Rotor of each slot (-1: the slot is on no rotor and stays put).

    Examples
    --------
    >>> from cage_isomer_builder import example_data
    >>> from cage_isomer_builder.cage import Tri4Di6CageBuilder
    >>> cage = Tri4Di6CageBuilder(node=example_data.path("uio66_tri_node"),
    ...                           linker=example_data.path("bdc"), scale_multiplier=0.225)
    >>> _ = cage.build()
    >>> _ = cage.optimise(include_charges=False, logfile=None)
    >>> _ = cage.functionalise()
    >>> atoms, adjacency = cage.to_ase(), cage.adjacency()
    >>> linkers, nodes = cage.building_blocks()
    >>> from cage_isomer_builder.utils.flexibility import find_rotors, rotation_samples
    >>> rotors = find_rotors(atoms, linkers, adjacency)
    >>> pos, vec, orientation, rotor_of_slot = rotation_samples(
    ...     atoms, rotors, cage._fg_anchor_indices, [0, 90, 180, 270],
    ...     centre=cage.pore_centre(), adjacency=adjacency)
    >>> pos.shape, sorted(set(rotor_of_slot.tolist()))      # 24 slots x 4 angles
    ((24, 4, 3), [0, 1, 2, 3, 4, 5])
    """
    if adjacency is None:
        adjacency = bond_graph(atoms)
    slot_indices = [int(s) for s in slot_indices]
    angles = np.radians(np.asarray(angles, dtype=float))
    n, m = len(slot_indices), len(angles)
    positions = np.repeat(atoms.positions[slot_indices][:, None, :], m, axis=1)
    carbons = [[j for j in adjacency[s] if atoms.numbers[j] > 1][0] for s in slot_indices]
    carbon_pos = np.repeat(atoms.positions[carbons][:, None, :], m, axis=1)
    rotor_of_slot = np.full(n, -1)
    for r, rotor in enumerate(rotors):
        origin, axis = rotor_axis(atoms, rotor)
        moving = set(rotor.moving)
        for k, (s, c) in enumerate(zip(slot_indices, carbons)):
            if s not in moving:
                continue
            rotor_of_slot[k] = r
            p_s = _unwrapped(atoms, [s], origin)[0]
            p_c = _unwrapped(atoms, [c], origin)[0]
            for t, ang in enumerate(angles):
                positions[k, t] = _rotate(p_s[None], origin, axis, ang)[0]
                carbon_pos[k, t] = (_rotate(p_c[None], origin, axis, ang)[0]
                                    if c in moving else p_c)
    vectors = positions - carbon_pos
    vectors /= np.linalg.norm(vectors, axis=2, keepdims=True)
    orientations = np.full((n, m), "", dtype="<U16")
    if centre is not None:
        inward = np.asarray(centre)[None, None, :] - carbon_pos
        inward /= np.linalg.norm(inward, axis=2, keepdims=True)
        cos = np.einsum("ijk,ijk->ij", vectors, inward)
        orientations = np.vectorize(lambda x: orientation_class(x, threshold))(cos).astype("<U16")
    return positions, vectors, orientations, rotor_of_slot

scale_structure 

scale_structure(atoms, groups, factor, centre=None)

Copy of atoms with every building block moved rigidly so its centroid-to-centre distance is multiplied by factor.

Parameters:

Name Type Description Default
groups list of list of int

Atoms of each building block (nodes and linkers). Atoms in no group stay where they are, so every atom should be in some group.

 required
centre array - like

Default: centroid of all atoms (finite structures). For periodic structures the cell is scaled by factor and every block keeps its fractional centroid.

 None

Examples:

>>> from cage_isomer_builder import example_data
>>> from cage_isomer_builder.cage import Tri4Di6CageBuilder
>>> cage = Tri4Di6CageBuilder(node=example_data.path("uio66_tri_node"),
...                           linker=example_data.path("bdc"), scale_multiplier=0.225)
>>> _ = cage.build()
>>> _ = cage.optimise(include_charges=False, logfile=None)
>>> _ = cage.functionalise()
>>> atoms, adjacency = cage.to_ase(), cage.adjacency()
>>> linkers, nodes = cage.building_blocks()
>>> import numpy as np
>>> from cage_isomer_builder.utils.flexibility import scale_structure
>>> centre = cage.pore_centre()
>>> bigger = scale_structure(atoms, linkers + nodes, 1.1, centre)
>>> node = nodes[0]
>>> r0 = np.linalg.norm(atoms.positions[node].mean(axis=0) - centre)
>>> r1 = np.linalg.norm(bigger.positions[node].mean(axis=0) - centre)
>>> round(float(r1 / r0), 6)                          # each block moves out by 10 %
1.1
Source code in cage_isomer_builder/utils/flexibility.py 
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def scale_structure(atoms, groups, factor, centre=None):
    """
    Copy of ``atoms`` with every building block moved rigidly so its
    centroid-to-centre distance is multiplied by ``factor``.

    Parameters
    ----------
    groups : list of list of int
        Atoms of each building block (nodes and linkers). Atoms in no group
        stay where they are, so every atom should be in some group.
    centre : array-like, optional
        Default: centroid of all atoms (finite structures). For periodic
        structures the cell is scaled by ``factor`` and every block keeps
        its fractional centroid.

    Examples
    --------
    >>> from cage_isomer_builder import example_data
    >>> from cage_isomer_builder.cage import Tri4Di6CageBuilder
    >>> cage = Tri4Di6CageBuilder(node=example_data.path("uio66_tri_node"),
    ...                           linker=example_data.path("bdc"), scale_multiplier=0.225)
    >>> _ = cage.build()
    >>> _ = cage.optimise(include_charges=False, logfile=None)
    >>> _ = cage.functionalise()
    >>> atoms, adjacency = cage.to_ase(), cage.adjacency()
    >>> linkers, nodes = cage.building_blocks()
    >>> import numpy as np
    >>> from cage_isomer_builder.utils.flexibility import scale_structure
    >>> centre = cage.pore_centre()
    >>> bigger = scale_structure(atoms, linkers + nodes, 1.1, centre)
    >>> node = nodes[0]
    >>> r0 = np.linalg.norm(atoms.positions[node].mean(axis=0) - centre)
    >>> r1 = np.linalg.norm(bigger.positions[node].mean(axis=0) - centre)
    >>> round(float(r1 / r0), 6)                          # each block moves out by 10 %
    1.1
    """
    out = atoms.copy()
    pos = atoms.positions.copy()
    if atoms.pbc.any():
        new_cell = atoms.cell[:] * factor
        for g in groups:
            g = list(g)
            local = _unwrapped(atoms, g, pos[g[0]])
            com = local.mean(axis=0)
            frac = np.linalg.solve(atoms.cell[:].T, com)
            new_com = new_cell.T @ frac
            pos[g] = local - com + new_com
        out.set_cell(new_cell, scale_atoms=False)
        out.positions = pos
        out.wrap()
        return out
    if centre is None:
        centre = pos.mean(axis=0)
    for g in groups:
        g = list(g)
        com = pos[g].mean(axis=0)
        pos[g] += (factor - 1.0) * (com - centre)
    out.positions = pos
    return out

distance_scaling 

distance_scaling(atoms, groups, slot_indices, factors=(0.95, 1.0, 1.05), centre=None)

How every slot-slot distance changes with isotropic scaling.

Returns:

Name Type Description
pairs (ndarray, shape(n_pairs, 2))

Slot pairs (i < j).
distances (ndarray, shape(n_factors, n_pairs))
slopes (ndarray, shape(n_pairs))

Least-squares d(distance)/d(factor). A pair whose two slots sit on building blocks that move apart rigidly has slope close to the distance between those blocks' centroids; two slots on one block have slope 0.

Examples:

>>> from cage_isomer_builder import example_data
>>> from cage_isomer_builder.cage import Tri4Di6CageBuilder
>>> cage = Tri4Di6CageBuilder(node=example_data.path("uio66_tri_node"),
...                           linker=example_data.path("bdc"), scale_multiplier=0.225)
>>> _ = cage.build()
>>> _ = cage.optimise(include_charges=False, logfile=None)
>>> _ = cage.functionalise()
>>> atoms, adjacency = cage.to_ase(), cage.adjacency()
>>> linkers, nodes = cage.building_blocks()
>>> from cage_isomer_builder.utils.flexibility import distance_scaling
>>> pairs, distances, slopes = distance_scaling(
...     atoms, linkers + nodes, cage._fg_anchor_indices, (0.95, 1.0, 1.05), cage.pore_centre())
>>> distances.shape, bool((slopes >= -1e-9).all())    # 3 factors x 276 slot pairs
((3, 276), True)
Source code in cage_isomer_builder/utils/flexibility.py 
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def distance_scaling(atoms, groups, slot_indices, factors=(0.95, 1.0, 1.05), centre=None):
    """
    How every slot-slot distance changes with isotropic scaling.

    Returns
    -------
    pairs : np.ndarray, shape (n_pairs, 2)
        Slot pairs (i < j).
    distances : np.ndarray, shape (n_factors, n_pairs)
    slopes : np.ndarray, shape (n_pairs,)
        Least-squares d(distance)/d(factor). A pair whose two slots sit on
        building blocks that move apart rigidly has slope close to the
        distance between those blocks' centroids; two slots on one block
        have slope 0.

    Examples
    --------
    >>> from cage_isomer_builder import example_data
    >>> from cage_isomer_builder.cage import Tri4Di6CageBuilder
    >>> cage = Tri4Di6CageBuilder(node=example_data.path("uio66_tri_node"),
    ...                           linker=example_data.path("bdc"), scale_multiplier=0.225)
    >>> _ = cage.build()
    >>> _ = cage.optimise(include_charges=False, logfile=None)
    >>> _ = cage.functionalise()
    >>> atoms, adjacency = cage.to_ase(), cage.adjacency()
    >>> linkers, nodes = cage.building_blocks()
    >>> from cage_isomer_builder.utils.flexibility import distance_scaling
    >>> pairs, distances, slopes = distance_scaling(
    ...     atoms, linkers + nodes, cage._fg_anchor_indices, (0.95, 1.0, 1.05), cage.pore_centre())
    >>> distances.shape, bool((slopes >= -1e-9).all())    # 3 factors x 276 slot pairs
    ((3, 276), True)
    """
    slot_indices = [int(s) for s in slot_indices]
    i, j = np.triu_indices(len(slot_indices), k=1)
    rows = []
    for f in factors:
        scaled = scale_structure(atoms, groups, f, centre)
        p = scaled.positions[slot_indices]
        d = p[j] - p[i]
        if scaled.pbc.any():
            d, _ = find_mic(d, scaled.cell, scaled.pbc)
        rows.append(np.linalg.norm(d, axis=1))
    distances = np.array(rows)
    f = np.asarray(factors, dtype=float)
    slopes = np.polyfit(f, distances, 1)[0] if len(f) > 1 else np.zeros(len(i))
    return np.stack([i, j], axis=1), distances, slopes