Symmetry, the Crystal Field, and the Choice of Axes#

Where Do the x, y, z Axes Come From?#

A free ion is spherically symmetric. Every orbital of a given angular momentum \(\ell\) (the five d orbitals, the seven f orbitals) is degenerate, and the energy cannot depend on direction — there is simply no preferred axis. In that situation talking about “the z axis” is meaningless.

Axes appear the moment the ion is placed in a non-spherical environment. The surrounding ligands (or the crystalline lattice) lower the symmetry from the full rotation group to one of the molecular point groups (Oh, Td, D4h …). Two things happen at once:

  • the orbital degeneracy is partially lifted — the level splits into sets that transform as the irreducible representations (irreps) of the point group;

  • the splitting is defined relative to the symmetry elements of that group — its rotation axes and mirror planes.

To turn this into numbers a calculation has to pin a Cartesian frame onto those symmetry elements: the z axis is put along the principal rotation axis, x and y along secondary axes or mirror planes, and so on. That frame is a convention — several are equally valid, related by rotations — but the numerical values of the crystal-field parameters are only meaningful once it is fixed. This page documents the convention Crispy uses for every implemented geometry.

The Crystal-Field Potential#

Crispy builds the one-electron crystal-field term as a potential expanded on renormalized (Racah) spherical harmonics \(C^{(m)}_k\):

\[V(\theta,\phi) = \sum_{k,m} A_{k,m}\, C^{(m)}_k(\theta,\phi), \qquad C^{(m)}_k = \sqrt{\tfrac{4\pi}{2k+1}}\, Y^{(m)}_k ,\]

where the radial integrals \(\langle r^k \rangle\) are absorbed into the \(A_{k,m}\) coefficients.

In the Quanty templates this is the Akm table of {k, m, value} triples passed to NewOperator("CF", ...). Only even \(k\) contribute within a shell (\(k = 0, 2, 4\) for d; \(k = 0, 2, 4, 6\) for f), and which \(\{k,m\}\) terms survive — and with what coefficients — is fixed by the point group and its orientation. The \(k=0\) (monopole) term is a constant shift of all levels and is dropped (or, equivalently, the irrep energies are referenced to their degeneracy-weighted average).

The \(A_{k,m}\) coefficients used in Crispy are taken directly from the Quanty point-group tables, which are the source of truth for both the allowed terms and the orientation conventions:

Each point group there offers several orientations (sub-pages such as .../d3d/orientation_zy); the symmetry-operation vectors listed on a page state exactly how the rotation axes and mirror planes point in the xyz frame.

Orientation Convention for Each Implemented Geometry#

The table below gives, for every symmetry implemented in Crispy, the Quanty orientation that the templates reproduce and how the Cartesian axes are tied to the symmetry elements.

Symmetry

Quanty setting

Alignment of the xyz axes with the symmetry elements

Oh

xyz

The three four-fold (C4) axes lie along x, y and z — the octahedral ligands sit on the Cartesian axes. The C3 axes run along the cube body-diagonals \([\pm1,\pm1,\pm1]\).

Td

xyz

The tetrahedron is inscribed in a cube whose edges lie along x, y, z. The S4/C2 axes are along x, y, z and the four C3 axes along the cube body-diagonals. (The d field is exactly the negative of the Oh one.)

D4h

zxy

The four-fold C4 principal axis is along z; the C2' axes and the vertical mirror planes \(\sigma_v\) lie along x and y (the equatorial ligands sit on x and y, so b1g is the \(d_{x^2-y^2}\) orbital). The C2'' axes / \(\sigma_d\) planes are at 45°.

D3d

zy

The three-fold C3 axis is along z; one basal C2 axis is along y (the other two and the \(\sigma_d\) planes follow at 120°).

D3h

zx

The three-fold C3 axis is along z; the horizontal mirror \(\sigma_h\) is the xy plane; one C2' axis lies along x.

C3v

D3d “zy”

The three-fold C3 axis is along z; a vertical mirror plane \(\sigma_v\) contains the y axis. The C3v pages of Quanty are unpopulated, so Crispy uses the König & Kremer convention, which is the D3d zy setting with the inversion centre removed (the d–d block is identical because it is built from even-\(k\) harmonics).

What Is Implemented for Each Geometry#

The orbital splittings and the parameters exposed in the interface are listed below for both the d and f blocks. Energies are quoted relative to the barycenter.

Oh — Octahedral#

  • d: eg + t2g separated by the single parameter 10Dq (eg at \(+0.6\,\mathrm{10Dq}\), t2g at \(-0.4\,\mathrm{10Dq}\)).

  • f: a2u + t1u + t2u (parameters Ea2u, Et1u, Et2u).

Td — Tetrahedral#

  • d: e + t2 (e at \(-0.6\,\mathrm{10Dq}\), t2 at \(+0.4\,\mathrm{10Dq}\)) — the inverted Oh field.

  • f: a2 + t1 + t2 (parameters Ea2, Et1, Et2).

D4h — Tetragonal#

  • d: a1g + b1g + b2g + eg, parametrized by Dq, Ds, Dt (a1g = 6Dq − 2Ds − 6Dt, b1g = 6Dq + 2Ds − Dt, b2g = −4Dq + 2Ds − Dt, eg = −4Dq − Ds + 4Dt).

  • f: a2u + b1u + b2u + 2 eu; the two eu sets mix through the off-diagonal Meu (parameters Ea2u, Eb1u, Eb2u, Eeu1, Eeu2, Meu).

D3d — Trigonal (With Inversion)#

  • d: a1g + 2 eg; the two eg sets (descended from the cubic eg and t2g) mix through Meg (parameters Ea1g, Eegσ, Eegπ, Meg).

  • f: a1u + 2 a2u + 2 eu; the a2u pair mixes through Ma2u and the eu pair through Meu (parameters Ea1u, Ea2uA, Ea2uB, Eeu1, Eeu2, Ma2u, Meu).

D3h — Trigonal (With a Horizontal Mirror)#

  • d: a1' + e' + e'', parametrized by Dmu and Dnu (a1' = −2Dμ − 6Dν, e' = 2Dμ − Dν, e'' = −Dμ + 4Dν).

  • f: a1' + a2' + a2'' + e' + e'' (no off-diagonal mixing — every irrep appears once; parameters Ea1p, Ea2p, Ea2pp, Eep, Eepp).

C3v — Trigonal (No Inversion)#

  • d: a1 + e + e, parametrized by Dq, Dσ, Dτ. The two e sets share an irrep and mix, so the Hamiltonian is not diagonal in the irrep basis (the off-diagonal element is \(-\tfrac{\sqrt2}{3}(3D\sigma - 5D\tau)\); see König & Kremer, p. 56).

  • f: 2 a1 + a2 + 2 e; the two a1 sets mix through Ma1 and the two e sets through Me (parameters Ea2, Ea1A, Ea1B, Ee1, Ee2, Ma1, Me).

Note

Because only even-\(k\) harmonics enter the on-site crystal field, two point groups that differ only by an inversion centre (for example D3d and C3v, or Oh and Td for the d–d coupling apart from the sign) share the same Akm structure. This is why C3v can be taken over from the populated D3d Quanty page.

References#