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Equivariance

These documents develop the transformation language used by ELFES. The section covers group actions and representations, spatial and spin rotations, angular coupling, real-space fields, and LCAO operators.

This discussion is limited to identifying the symmetry and equivariance requirements on the relevant physical quantities and expressing them in terms of irreducible representations; it does not address how to construct models that satisfy these requirements.

The subject sits between the physical definitions in Electronic Structure and their eventual use in models and software:

  • electronic-structure objects are taken as already defined;
  • their spatial transformation laws and angular content belong here;
  • array ordering and software-specific angular conventions are implementation concerns, not definitions of the mathematical objects.

Spatial and spin indices are treated as distinct transformation types. Their rotations are independent without spin–orbit coupling and become joint when spin is coupled to the spatial frame. Time reversal remains a separate antiunitary symmetry.

Documents

Document Scope
Symmetry and Groups Group actions, representations, changes of basis, equivariant maps, and the relation between \(\mathrm{E}(3)\) and the angular \(\mathrm{O}(3)\) action.
Representations of SO(3) and O(3) Irreducible representations of \(\mathrm{SO}(3)\) and \(\mathrm{O}(3)\), Wigner \(\mathcal D\) matrices, spherical tensors, irrep features, parity, and multiplicity.
Spherical Harmonics Complex and real spherical harmonics, spherical coefficients and rotations, parity, and changes of angular basis.
Tensor Product Representations Tensor-product representations, coupled and uncoupled bases, Clebsch–Gordan coefficients, selection rules, multiplicity, and operator spaces.
Spatial and Spin Symmetries Independent spatial and spin rotations, the Pauli scalar/vector decomposition, and the joint action selected by spin–orbit coupling.
Real-Space Fields A unified treatment of scalar, spatial-vector, and spin-vector-valued fields and their centered irrep features.
LCAO Operators Spatial coupling of the two AO indices and the product or joint transformation of Pauli components.

The same framework applies to other electronic-structure objects once their spatial transformation laws have been specified.

Conventions Used in This Section

A representation is a group action on a vector space; a Wigner matrix is the coordinate form of that action after a basis has been selected. The same distinction applies to an abstract Clebsch–Gordan map and its numerical coefficients.

Bra–ket notation is used for abstract states and operators in the \(\mathrm{SO}(3)\), spherical-harmonic, and tensor-product chapters. A ket such as \(|\psi\rangle\) does not select a basis; basis dependence enters only when states such as \(|\ell m\rangle\) and their numerical components are chosen.

Unless stated otherwise:

  • transformations are active;
  • vectors of components are columns;
  • Cartesian space is right-handed;
  • \(R\in\mathrm{SO}(3)\) denotes a proper rotation;
  • \(Q\in\mathrm{O}(3)\) may also include a reflection or inversion;
  • real and complex representation spaces are both allowed.

The spherical-harmonic chapter gives a common complex and real reference. ELFES adopts the Wikipedia real spherical harmonic convention; software-specific conversions and their evidence are implementation concerns documented separately.

Background Sources