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 :math:`\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** :math:`C^{(m)}_k`: .. math:: 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 :math:`\langle r^k \rangle` are absorbed into the :math:`A_{k,m}` coefficients. In the Quanty templates this is the ``Akm`` table of ``{k, m, value}`` triples passed to ``NewOperator("CF", ...)``. Only **even** :math:`k` contribute within a shell (:math:`k = 0, 2, 4` for *d*; :math:`k = 0, 2, 4, 6` for *f*), and which :math:`\{k,m\}` terms survive — and with what coefficients — is fixed by the point group **and its orientation**. The :math:`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 :math:`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: https://www.quanty.org/physics_chemistry/point_groups 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. .. list-table:: :header-rows: 1 :widths: 12 14 74 * - 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 :math:`[\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 :math:`\sigma_v` lie along *x* and *y* (the equatorial ligands sit on *x* and *y*, so ``b1g`` is the :math:`d_{x^2-y^2}` orbital). The ``C2''`` axes / :math:`\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 :math:`\sigma_d` planes follow at 120°). * - ``D3h`` - `zx `__ - The three-fold ``C3`` axis is along *z*; the horizontal mirror :math:`\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 :math:`\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-\ :math:`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 :math:`+0.6\,\mathrm{10Dq}`, ``t2g`` at :math:`-0.4\,\mathrm{10Dq}`). * *f*: ``a2u`` + ``t1u`` + ``t2u`` (parameters ``Ea2u``, ``Et1u``, ``Et2u``). ``Td`` — Tetrahedral ~~~~~~~~~~~~~~~~~~~~ * *d*: ``e`` + ``t2`` (``e`` at :math:`-0.6\,\mathrm{10Dq}`, ``t2`` at :math:`+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 :math:`-\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-\ :math:`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 ---------- * Quanty point-group tables — https://www.quanty.org/physics_chemistry/point_groups * E. König and S. Kremer, *Ligand Field Energy Diagrams* (Plenum, 1977) — the ``C3v`` and trigonal conventions.