diff --git a/ffprime/electrostatics/spherical.py b/ffprime/electrostatics/spherical.py
new file mode 100644
index 0000000..fd7ee60
--- /dev/null
+++ b/ffprime/electrostatics/spherical.py
@@ -0,0 +1,112 @@
+import numpy as np
+
+
+def dipole_cartesian_to_spherical(p: np.ndarray) -> np.ndarray:
+    """
+    Convert a Cartesian dipole moment to real spherical (solid harmonic) form.
+
+    Parameters
+    ----------
+    p : np.ndarray, shape (3,)
+        Cartesian dipole moment [px, py, pz] in atomic units (e * bohr).
+
+    Returns
+    -------
+    np.ndarray, shape (3,)
+        Spherical components [Q_10, Q_11c, Q_11s].
+
+    Notes
+    -----
+    Convention follows Stone, A.J. "Theory of Intermolecular Forces", 2nd ed.
+    Q_10 = pz,  Q_11c = px,  Q_11s = py
+    """
+    px, py, pz = p
+    return np.array([pz, px, py])
+
+
+def dipole_spherical_to_cartesian(q: np.ndarray) -> np.ndarray:
+    """
+    Convert real spherical dipole components back to Cartesian form.
+
+    Parameters
+    ----------
+    q : np.ndarray, shape (3,)
+        Spherical components [Q_10, Q_11c, Q_11s].
+
+    Returns
+    -------
+    np.ndarray, shape (3,)
+        Cartesian dipole moment [px, py, pz] in atomic units (e * bohr).
+    """
+    Q_10, Q_11c, Q_11s = q
+    return np.array([Q_11c, Q_11s, Q_10])
+
+
+def quadrupole_cartesian_to_spherical(theta: np.ndarray) -> np.ndarray:
+    """
+    Convert a traceless Cartesian quadrupole tensor to real spherical form.
+
+    Parameters
+    ----------
+    theta : np.ndarray, shape (3, 3)
+        Traceless symmetric quadrupole tensor in atomic units (e * bohr^2).
+        Must satisfy np.trace(theta) == 0.
+
+    Returns
+    -------
+    np.ndarray, shape (5,)
+        Spherical components [Q_20, Q_21c, Q_21s, Q_22c, Q_22s].
+
+    Raises
+    ------
+    ValueError
+        If the input tensor is not traceless within numerical tolerance.
+
+    Notes
+    -----
+    Convention follows Stone, A.J. "Theory of Intermolecular Forces", 2nd ed.
+    Q_20  = Θzz
+    Q_21c = Θxz
+    Q_21s = Θyz
+    Q_22c = (Θxx - Θyy) / 2
+    Q_22s = Θxy
+    """
+    if not np.isclose(np.trace(theta), 0.0, atol=1e-10):
+        raise ValueError(
+            f"Quadrupole tensor must be traceless, got trace = {np.trace(theta):.6e}"
+        )
+
+    Q_20  = theta[2, 2]
+    Q_21c = theta[0, 2]
+    Q_21s = theta[1, 2]
+    Q_22c = (theta[0, 0] - theta[1, 1]) / 2.0
+    Q_22s = theta[0, 1]
+
+    return np.array([Q_20, Q_21c, Q_21s, Q_22c, Q_22s])
+
+
+def quadrupole_spherical_to_cartesian(q: np.ndarray) -> np.ndarray:
+    """
+    Convert real spherical quadrupole components back to Cartesian tensor form.
+
+    Parameters
+    ----------
+    q : np.ndarray, shape (5,)
+        Spherical components [Q_20, Q_21c, Q_21s, Q_22c, Q_22s].
+
+    Returns
+    -------
+    np.ndarray, shape (3, 3)
+        Traceless symmetric quadrupole tensor in atomic units (e * bohr^2).
+    """
+    Q_20, Q_21c, Q_21s, Q_22c, Q_22s = q
+
+    theta = np.zeros((3, 3))
+    theta[2, 2] = Q_20
+    theta[0, 0] = Q_22c - Q_20 / 2.0
+    theta[1, 1] = -Q_22c - Q_20 / 2.0
+    theta[0, 2] = theta[2, 0] = Q_21c
+    theta[1, 2] = theta[2, 1] = Q_21s
+    theta[0, 1] = theta[1, 0] = Q_22s
+
+    return theta
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diff --git a/tests/test_spherical.py b/tests/test_spherical.py
new file mode 100644
index 0000000..18cb98e
--- /dev/null
+++ b/tests/test_spherical.py
@@ -0,0 +1,57 @@
+import numpy as np
+import pytest
+from ffprime.electrostatics.spherical import (
+    dipole_cartesian_to_spherical,
+    dipole_spherical_to_cartesian,
+    quadrupole_cartesian_to_spherical,
+    quadrupole_spherical_to_cartesian,
+)
+
+
+def test_dipole_z_aligned():
+    """Pure z-dipole maps entirely to Q_10, others zero."""
+    p = np.array([0.0, 0.0, 1.0])
+    sph = dipole_cartesian_to_spherical(p)
+    assert np.isclose(sph[0], 1.0)
+    assert np.isclose(sph[1], 0.0)
+    assert np.isclose(sph[2], 0.0)
+
+
+def test_dipole_roundtrip():
+    """Cartesian -> spherical -> Cartesian recovers original."""
+    p = np.array([1.5, -2.0, 3.0])
+    assert np.allclose(
+        dipole_spherical_to_cartesian(dipole_cartesian_to_spherical(p)), p
+    )
+
+
+def test_quadrupole_traceless_check():
+    """Non-traceless tensor raises ValueError."""
+    bad_tensor = np.eye(3)
+    with pytest.raises(ValueError, match="traceless"):
+        quadrupole_cartesian_to_spherical(bad_tensor)
+
+
+def test_quadrupole_z_axial():
+    """Axially symmetric quadrupole along z has only Q_20 nonzero."""
+    theta = np.array([
+        [-0.5,  0.0,  0.0],
+        [ 0.0, -0.5,  0.0],
+        [ 0.0,  0.0,  1.0]
+    ])
+    sph = quadrupole_cartesian_to_spherical(theta)
+    assert np.isclose(sph[0], 1.0)
+    assert np.allclose(sph[1:], 0.0)
+
+
+def test_quadrupole_roundtrip():
+    """Cartesian -> spherical -> Cartesian recovers original tensor."""
+    theta = np.array([
+        [-0.5,  0.3,  0.1],
+        [ 0.3, -0.5,  0.2],
+        [ 0.1,  0.2,  1.0]
+    ])
+    recovered = quadrupole_spherical_to_cartesian(
+        quadrupole_cartesian_to_spherical(theta)
+    )
+    assert np.allclose(recovered, theta)
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