utils¶

def cell_to_cellpar(
    cell: Tuple[Vector3D, Vector3D, Vector3D], radians: bool = False
) -> List[float]:
    """Returns the cell parameters `[a, b, c, alpha, beta, gamma]`.

    Angles are in degrees unless `radian=True` is used.

    Note:
        Based on [ASE code](https://wiki.fysik.dtu.dk/ase/_modules/ase/geometry/cell.html#cell_to_cellpar).

    Parameters:
        cell: A Cartesian 3x3 cell. This equates to the
            [`lattice_vectors`][optimade.models.structures.StructureResourceAttributes.lattice_vectors] attribute.
        radians: Use radians instead of degrees (default) for angles.

    Returns:
        The unit cell parameters as a `list` of `float` values.

    """
    if globals().get("np", None) is None:
        warn(NUMPY_NOT_FOUND, AdapterPackageNotFound)
        return None

    cell = np.asarray(cell)

    lengths = [np.linalg.norm(vector) for vector in cell]
    angles = []
    for i in range(3):
        j = i - 1
        k = i - 2
        outer_product = lengths[j] * lengths[k]
        if outer_product > 1e-16:
            x_vector = np.dot(cell[j], cell[k]) / outer_product
            angle = 180.0 / np.pi * np.arccos(x_vector)
        else:
            angle = 90.0
        angles.append(angle)
    if radians:
        angles = [angle * np.pi / 180 for angle in angles]
    return np.array(lengths + angles)

def cellpar_to_cell(
    cellpar: List[float],
    ab_normal: Tuple[int, int, int] = (0, 0, 1),
    a_direction: Tuple[int, int, int] = None,
) -> List[Vector3D]:
    """Return a 3x3 cell matrix from `cellpar=[a,b,c,alpha,beta,gamma]`.

    Angles must be in degrees.

    The returned cell is orientated such that a and b
    are normal to `ab_normal` and a is parallel to the projection of
    `a_direction` in the a-b plane.

    Default `a_direction` is (1,0,0), unless this is parallel to
    `ab_normal`, in which case default `a_direction` is (0,0,1).

    The returned cell has the vectors va, vb and vc along the rows. The
    cell will be oriented such that va and vb are normal to `ab_normal`
    and va will be along the projection of `a_direction` onto the a-b
    plane.

    Example:
        >>> cell = cellpar_to_cell([1, 2, 4, 10, 20, 30], (0, 1, 1), (1, 2, 3))
        >>> np.round(cell, 3)
        array([[ 0.816, -0.408,  0.408],
            [ 1.992, -0.13 ,  0.13 ],
            [ 3.859, -0.745,  0.745]])

    Note:
        Direct copy of [ASE code](https://wiki.fysik.dtu.dk/ase/_modules/ase/geometry/cell.html#cellpar_to_cell).

    Parameters:
        cellpar: The unit cell parameters as a `list` of `float` values.

            **Note**: The angles must be given in degrees.
        ab_normal: Unit vector normal to the ab-plane.
        a_direction: Unit vector defining the a-direction (default: `(1, 0, 0)`).

    Returns:
        A Cartesian 3x3 cell.

        This should equate to the
        [`lattice_vectors`][optimade.models.structures.StructureResourceAttributes.lattice_vectors] attribute.

    """
    if globals().get("np", None) is None:
        warn(NUMPY_NOT_FOUND, AdapterPackageNotFound)
        return None

    if a_direction is None:
        if np.linalg.norm(np.cross(ab_normal, (1, 0, 0))) < 1e-5:
            a_direction = (0, 0, 1)
        else:
            a_direction = (1, 0, 0)

    # Define rotated X,Y,Z-system, with Z along ab_normal and X along
    # the projection of a_direction onto the normal plane of Z.
    a_direction_array = np.array(a_direction)
    Z = unit_vector(ab_normal)
    X = unit_vector(a_direction_array - np.dot(a_direction_array, Z) * Z)
    Y = np.cross(Z, X)

    # Express va, vb and vc in the X,Y,Z-system
    alpha, beta, gamma = 90.0, 90.0, 90.0
    if isinstance(cellpar, (int, float)):
        a = b = c = cellpar
    elif len(cellpar) == 1:
        a = b = c = cellpar[0]
    elif len(cellpar) == 3:
        a, b, c = cellpar
    else:
        a, b, c, alpha, beta, gamma = cellpar

    # Handle orthorhombic cells separately to avoid rounding errors
    eps = 2 * np.spacing(90.0, dtype=np.float64)  # around 1.4e-14
    # alpha
    if abs(abs(alpha) - 90) < eps:
        cos_alpha = 0.0
    else:
        cos_alpha = np.cos(alpha * np.pi / 180.0)
    # beta
    if abs(abs(beta) - 90) < eps:
        cos_beta = 0.0
    else:
        cos_beta = np.cos(beta * np.pi / 180.0)
    # gamma
    if abs(gamma - 90) < eps:
        cos_gamma = 0.0
        sin_gamma = 1.0
    elif abs(gamma + 90) < eps:
        cos_gamma = 0.0
        sin_gamma = -1.0
    else:
        cos_gamma = np.cos(gamma * np.pi / 180.0)
        sin_gamma = np.sin(gamma * np.pi / 180.0)

    # Build the cell vectors
    va = a * np.array([1, 0, 0])
    vb = b * np.array([cos_gamma, sin_gamma, 0])
    cx = cos_beta
    cy = (cos_alpha - cos_beta * cos_gamma) / sin_gamma
    cz_sqr = 1.0 - cx * cx - cy * cy
    assert cz_sqr >= 0
    cz = np.sqrt(cz_sqr)
    vc = c * np.array([cx, cy, cz])

    # Convert to the Cartesian x,y,z-system
    abc = np.vstack((va, vb, vc))
    T = np.vstack((X, Y, Z))
    cell = np.dot(abc, T)

    return cell

def fractional_coordinates(
    cell: Tuple[Vector3D, Vector3D, Vector3D], cartesian_positions: List[Vector3D]
) -> List[Vector3D]:
    """Returns fractional coordinates and wraps coordinates to `[0,1[`.

    Note:
        Based on [ASE code](https://wiki.fysik.dtu.dk/ase/_modules/ase/atoms.html#Atoms.get_scaled_positions).

    Parameters:
        cell: A Cartesian 3x3 cell. This equates to the
            [`lattice_vectors`][optimade.models.structures.StructureResourceAttributes.lattice_vectors] attribute.
        cartesian_positions: A list of cartesian atomic positions. This equates to the
            [`cartesian_site_positions`][optimade.models.structures.StructureResourceAttributes.cartesian_site_positions]
            attribute.

    Returns:
        A list of fractional coordinates for the atomic positions.

    """
    if globals().get("np", None) is None:
        warn(NUMPY_NOT_FOUND, AdapterPackageNotFound)
        return None

    cell = np.asarray(cell)
    cartesian_positions = np.asarray(cartesian_positions)

    fractional = np.linalg.solve(cell.T, cartesian_positions.T).T

    # Expecting a bulk 3D structure here, note, this may change in the future.
    # See `ase.atoms:Atoms.get_scaled_positions()` for ideas on how to handle lower dimensional structures.
    # Furthermore, according to ASE we need to modulo 1.0 twice.
    # This seems to be due to small floats % 1.0 becomes 1.0, hence twice makes it 0.0.
    for i in range(3):
        fractional[:, i] %= 1.0
        fractional[:, i] %= 1.0

    return [tuple(position) for position in fractional]

def pad_cell(
    lattice_vectors: Tuple[Vector3D, Vector3D, Vector3D], padding: float = None
) -> Tuple[Tuple[Vector3D, Vector3D, Vector3D], bool]:
    """Turn any `null`/`None` values into a `float` in given `tuple` of
    [`lattice_vectors`][optimade.models.structures.StructureResourceAttributes.lattice_vectors].

    Parameters:
        lattice_vectors: A 3x3 cartesian cell. This is the
            [`lattice_vectors`][optimade.models.structures.StructureResourceAttributes.lattice_vectors]
            attribute.
        padding: A value with which `null` or `None` values should be replaced.

    Returns:
        The possibly redacted/padded `lattice_vectors` and a `bool` declaring whether or not
        the value has been redacted/padded or not, i.e., whether it contained `null` or `None`
        values.

    """
    return _pad_iter_of_iters(
        iterable=lattice_vectors,
        padding=padding,
        outer=tuple,
        inner=tuple,
    )

def scaled_cell(
    cell: Tuple[Vector3D, Vector3D, Vector3D]
) -> Tuple[Vector3D, Vector3D, Vector3D]:
    """Return a scaled 3x3 cell from cartesian 3x3 cell (`lattice_vectors`).

    This is based on PDB's method of calculating SCALE from CRYST data.
    For more info, see [this site](https://www.wwpdb.org/documentation/file-format-content/format33/sect8.html#SCALEn).

    Parameters:
        cell: A Cartesian 3x3 cell. This equates to the
            [`lattice_vectors`][optimade.models.structures.StructureResourceAttributes.lattice_vectors] attribute.

    Returns:
        A scaled 3x3 cell.

    """
    if globals().get("np", None) is None:
        warn(NUMPY_NOT_FOUND, AdapterPackageNotFound)
        return None

    cell = np.asarray(cell)

    volume = np.dot(cell[0], np.cross(cell[1], cell[2]))
    scale = []
    for i in range(3):
        vector = np.cross(cell[(i + 1) % 3], cell[(i + 2) % 3]) / volume
        scale.append(tuple(vector))
    return tuple(scale)

def unit_vector(x: Vector3D) -> Vector3D:
    """Return a unit vector in the same direction as `x`.

    Parameters:
        x: A three-dimensional vector.

    Returns:
        A unit vector in the same direction as `x`.

    """
    if globals().get("np", None) is None:
        warn(NUMPY_NOT_FOUND, AdapterPackageNotFound)
        return None

    y = np.array(x, dtype="float")
    return y / np.linalg.norm(y)