Wigner Matrix Based Convolution Algorithm (WMCA)

• WMCA is a formalism that redefines LCAO matrix elements by leveraging convolution operations with Wigner-matrix rotations, overcoming angular momentum limitations.
• It integrates multipole expansion of full crystal potentials with spherical Bessel transforms, generalizing standard Slater–Koster methods to high angular momentum.
• The algorithm retains two-center computational efficiency while enabling on-the-fly evaluation of matrix elements, promising improved scalability for ab initio LCAO methods.

Searching arXiv for the cited WMCA paper and closely related Wigner-matrix convolution work. arXiv search query: "Wigner Matrix Based Convolution Algorithm" arXiv search query: "conviqt Wigner matrices convolution reduced Wigner matrices" Wigner Matrix Based Convolution Algorithm (WMCA) is a formalism for evaluating matrix elements in the linear combination of atomic orbitals (LCAO) method by replacing low-angular-momentum, tabulated Slater–Koster machinery with a universal rotation-and-convolution framework based on Wigner matrices. In the formulation introduced for matrix elements in an LCAO basis, the full crystal potential is expanded into multipoles, each contribution is reduced to a two-center-like convolution between localized angular-momentum channels, the coordinate system is rotated with Wigner $D$ matrices, and the remaining radial part is evaluated through spherical-Bessel-transform convolutions. The resulting construction is intended to retain the efficiency associated with two-center integrals while extending the formalism to arbitrary angular momentum and to multipole-expanded full crystal potentials (Sterling, 6 Oct 2025).

1. Conceptual setting and motivation

The algorithm is motivated by the observation that, in LCAO methods, the dominant cost can lie in constructing matrix elements rather than in diagonalization. The relevant quantities include the overlap $S_{\mu\nu}$, kinetic-energy $T_{\mu\nu}$, and potential $V_{\mu\nu}$ matrix elements. Tight binding in the two-center approximation (2CA) is efficient because each matrix element is reduced to a two-center integral (2CI) that can be parameterized and rotated into a special coordinate system, but the standard Slater–Koster formulae are limited to $l\leq 3$, and the usable tabulated formulas are correspondingly limited. The difficulty becomes acute when one wants $d$, $f$, or higher orbitals, a full crystal potential rather than a simplified two-center potential, or an algorithm suitable for ab-initio LCAO methods (Sterling, 6 Oct 2025).

WMCA addresses this by replacing manual angular tables with a programmatic Wigner-matrix treatment. The central move is to exploit the fact that overlap, kinetic, and potential matrix elements can all be written as convolutions of localized functions. This suggests a generalization of the 2CA logic: retain the reduction to symmetry-adapted two-center-like objects, but compute the angular dependence through Wigner rotations and the radial dependence through transform-based convolution. A plausible implication is that WMCA should be viewed less as a new approximation than as a reorganization of the matrix-element problem into a form that scales more naturally to high angular momentum.

2. LCAO formulation and convolution structure

In the construction described for WMCA, the basis functions are Bloch sums of localized pseudoatomic orbitals,

$$\varphi_{\vec k \mu}(\vec r) = c_{\vec k \mu} \sum_{\vec R} e^{i\vec k \cdot (\vec R + \vec \tau_\mu)} \phi_\mu(\vec r - (\vec R + \vec \tau_\mu)),$$

with

$$\phi_\mu(\vec r)=X_{lm}(\Omega_{\vec r})\,\chi_\mu(r),$$

where $X_{lm}$ is a real spherical harmonic and $S_{\mu\nu}$0 is a localized radial function. For two sites separated by $S_{\mu\nu}$1, the matrix elements are written as

$S_{\mu\nu}$2

$S_{\mu\nu}$3

$S_{\mu\nu}$4

The paper’s key observation is that all of these have convolution form (Sterling, 6 Oct 2025).

The treatment of the potential is especially important. Rather than imposing the 2CA assumption of a purely spherical, atom-centered potential, the full crystal potential is expanded around each atom $S_{\mu\nu}$5 in multipoles,

$S_{\mu\nu}$6

with

$S_{\mu\nu}$7

This represents the potential as radial functions $S_{\mu\nu}$8 times spherical harmonics, so each potential channel carries a definite angular momentum. Matrix elements thereby become sums over angular-momentum-resolved two-center-like contributions rather than a single spherical term (Sterling, 6 Oct 2025).

3. Rotation to the internuclear axis and Wigner-matrix reduction

For a generic convolution

$S_{\mu\nu}$9

WMCA expands each function in spherical harmonics,

$T_{\mu\nu}$0

Each angular channel then yields a two-center integral

$T_{\mu\nu}$1

The central rotational step is to choose coordinates such that

$T_{\mu\nu}$2

Under this rotation, the real spherical harmonics transform as

$T_{\mu\nu}$3

where $T_{\mu\nu}$4 is the real-basis Wigner $T_{\mu\nu}$5 matrix. Azimuthal symmetry in the rotated frame implies that only a single magnetic quantum number $T_{\mu\nu}$6 survives, and the integral takes the form

$T_{\mu\nu}$7

with

$T_{\mu\nu}$8

This is the WMCA analogue of the Slater–Koster decomposition: the geometric dependence is isolated into Wigner rotations, while the remaining quantity is a symmetry-reduced radial–angular integral (Sterling, 6 Oct 2025).

The paper’s universal Wigner-matrix algorithm is based on the factorization

$T_{\mu\nu}$9

so that

$V_{\mu\nu}$0

The little-$V_{\mu\nu}$1 matrix is obtained by diagonalizing $V_{\mu\nu}$2,

$V_{\mu\nu}$3

which gives

$V_{\mu\nu}$4

with $V_{\mu\nu}$5 diagonal. The real-basis matrix follows from

$V_{\mu\nu}$6

and hence

$V_{\mu\nu}$7

This is the mechanism by which the same procedure applies to $V_{\mu\nu}$8, $V_{\mu\nu}$9, $l\leq 3$0, or higher $l\leq 3$1 without hand-derived angular tables (Sterling, 6 Oct 2025).

4. Convolution theorem, spherical Bessel transforms, and the generalized Slater–Koster relation

Once the rotation has reduced the problem to $l\leq 3$2-integrals, WMCA computes them in Fourier space using the convolution theorem. Defining the spherical Bessel transform of a channel by

$l\leq 3$3

with

$l\leq 3$4

and similarly for $l\leq 3$5, one obtains

$l\leq 3$6

where

$l\leq 3$7

The angular selection rules are enforced through Gaunt coefficients: $l\leq 3$8 The algorithm therefore separates into channel decomposition, Wigner rotation, radial convolution via spherical Bessel transforms, and summation over the allowed $l\leq 3$9 values (Sterling, 6 Oct 2025).

For low angular momentum, the paper shows that the formalism reproduces the usual Slater–Koster relations. In particular,

$d$0

and for $d$1-$d$2 terms,

$d$3

$d$4

$d$5

with

$d$6

Accordingly, WMCA is best understood as a generalized Slater–Koster theory in which tabulated direction-cosine algebra is replaced by on-the-fly Wigner-matrix evaluation (Sterling, 6 Oct 2025).

5. Full-potential matrix elements and demonstrated applications

After multipole expansion, the full-potential matrix element becomes

$d$7

Using the product decomposition

$d$8

each term is rewritten as a two-center-like integral in which the bra-side angular momentum is $d$9, potentially larger than the original orbital $f$0. The paper gives

$f$1

with

$f$2

The convolution-theorem form is

$f$3

where

$f$4

and

$f$5

Thus, full-potential matrix elements are reduced to radial Bessel transforms together with Wigner and Gaunt algebra (Sterling, 6 Oct 2025).

The reported application is silicon with a model local pseudopotential,

$f$6

using the empirical reciprocal-space form

$f$7

The basis in the main text is a minimal spherical Gaussian pseudoatomic basis with $f$8 and $f$9 functions. The paper reports that the multipole expansion of the potential converges with increasing $$\varphi_{\vec k \mu}(\vec r) = c_{\vec k \mu} \sum_{\vec R} e^{i\vec k \cdot (\vec R + \vec \tau_\mu)} \phi_\mu(\vec r - (\vec R + \vec \tau_\mu)),$$0, that the band structure from WMCA converges to the grid-based LCAO result, and that with $$\varphi_{\vec k \mu}(\vec r) = c_{\vec k \mu} \sum_{\vec R} e^{i\vec k \cdot (\vec R + \vec \tau_\mu)} \phi_\mu(\vec r - (\vec R + \vec \tau_\mu)),$$1 the WMCA band structure is essentially indistinguishable from the converged reference. It further states that accuracy improves roughly exponentially with $$\varphi_{\vec k \mu}(\vec r) = c_{\vec k \mu} \sum_{\vec R} e^{i\vec k \cdot (\vec R + \vec \tau_\mu)} \phi_\mu(\vec r - (\vec R + \vec \tau_\mu)),$$2, though not perfectly uniformly because some angular channels are symmetry forbidden or weak. The rotation-coefficient machinery was also tested on silicon, LaFeAsO, and thorium with $$\varphi_{\vec k \mu}(\vec r) = c_{\vec k \mu} \sum_{\vec R} e^{i\vec k \cdot (\vec R + \vec \tau_\mu)} \phi_\mu(\vec r - (\vec R + \vec \tau_\mu)),$$3 orbitals, with perfect agreement reported against reference tight-binding/Slater–Koster results when WMCA is used to compute the rotation coefficients (Sterling, 6 Oct 2025).

6. Computational characteristics, limitations, and related developments

The method is described as potentially $$\varphi_{\vec k \mu}(\vec r) = c_{\vec k \mu} \sum_{\vec R} e^{i\vec k \cdot (\vec R + \vec \tau_\mu)} \phi_\mu(\vec r - (\vec R + \vec \tau_\mu)),$$4 with respect to system size because the orbitals are local, only nearby overlaps matter, only a fixed number of matrix elements per atom need to be evaluated within a cutoff sphere, and the angular-momentum cutoffs are system-independent. The expensive part is identified as the spherical Bessel transforms, although these are described as manageable and parallelizable. At the same time, several limitations are explicit: the multipole expansion is truncated at $$\varphi_{\vec k \mu}(\vec r) = c_{\vec k \mu} \sum_{\vec R} e^{i\vec k \cdot (\vec R + \vec \tau_\mu)} \phi_\mu(\vec r - (\vec R + \vec \tau_\mu)),$$5, the demonstrated implementation uses a grid/plane-wave representation of the potential and is therefore not $$\varphi_{\vec k \mu}(\vec r) = c_{\vec k \mu} \sum_{\vec R} e^{i\vec k \cdot (\vec R + \vec \tau_\mu)} \phi_\mu(\vec r - (\vec R + \vec \tau_\mu)),$$6 in system size for large cells, a fully linear-scaling potential-generation algorithm without grids is still needed for full $$\varphi_{\vec k \mu}(\vec r) = c_{\vec k \mu} \sum_{\vec R} e^{i\vec k \cdot (\vec R + \vec \tau_\mu)} \phi_\mu(\vec r - (\vec R + \vec \tau_\mu)),$$7 behavior, the paper focuses on matrix elements rather than total energies or forces, and the method assumes a local potential and localized basis functions. It is also stated that, given a suitable method for generating a local ab-initio Kohn–Sham potential, the algorithm is applicable to fully ab-initio LCAO methods, but that implementation is “forthcoming” (Sterling, 6 Oct 2025).

A common misconception is to treat WMCA as merely a higher-$$\varphi_{\vec k \mu}(\vec r) = c_{\vec k \mu} \sum_{\vec R} e^{i\vec k \cdot (\vec R + \vec \tau_\mu)} \phi_\mu(\vec r - (\vec R + \vec \tau_\mu)),$$8 extension of existing Slater–Koster tables. The formal development indicates a broader scope: the algorithm is intended not only for arbitrary angular momentum in the basis, but also for matrix elements generated by a multipole-expanded full crystal potential. Another misconception is to treat the present formulation as already constituting a complete ab-initio electronic-structure workflow; the stated scope is narrower, centering on matrix-element evaluation.

Related work clarifies the placement of WMCA within a larger computational landscape. In spherical $$\varphi_{\vec k \mu}(\vec r) = c_{\vec k \mu} \sum_{\vec R} e^{i\vec k \cdot (\vec R + \vec \tau_\mu)} \phi_\mu(\vec r - (\vec R + \vec \tau_\mu)),$$9 beam–sky convolution, reduced Wigner matrix elements $$\phi_\mu(\vec r)=X_{lm}(\Omega_{\vec r})\,\chi_\mu(r),$$0 have been computed fast and exactly and embedded in the massively parallel conviqt code, where Wigner matrices are the rotation matrices of spherical harmonics on $$\phi_\mu(\vec r)=X_{lm}(\Omega_{\vec r})\,\chi_\mu(r),$$1 and the main computational kernel is the harmonic-space convolution

$$\phi_\mu(\vec r)=X_{lm}(\Omega_{\vec r})\,\chi_\mu(r),$$2

That work is distinct in application and formulation, but it exemplifies the same broad pattern of combining Wigner-matrix evaluation with convolution structure for computational gain (Prezeau et al., 2010). A different but conceptually related development appears in phase-space quantum mechanics, where Wigner-function dynamics with hard boundaries are reformulated as a convolution of free-particle Wigner dynamics with a geometry-dependent kernel. This also uses convolution to encode otherwise awkward structure, but it concerns the Wigner function rather than Wigner rotation matrices (Seidov, 2023).

Within LCAO theory, WMCA is therefore most precisely characterized as a generalized Slater–Koster framework in which multipole-expanded full potentials, Wigner-matrix rotations, Gaunt-coefficient angular reduction, and spherical-Bessel-transform convolutions are combined into a unified method for matrix-element construction at arbitrary angular momentum (Sterling, 6 Oct 2025).