# Coordination chemistry in molecular symmetry adapted spin space (mSASS)

R. Matthias Geilhufe<EMAIL_ADDRESS>Department of Physics,
Chalmers University of Technology, Kemigården 1, 412 58 Göteborg, Sweden
Jeffrey D. Rinehart<EMAIL_ADDRESS>Department of Chemistry and
Biochemistry, University of California, San Diego, La Jolla, California, USA

###### Abstract

Many areas of chemistry are devoted to the challenge of understanding,
predicting, and controlling the behavior of strongly localized electrons.
Examples include molecular magnetism and luminescence, color centers in
crystals, photochemistry and quantum sensing to name but a few. Over the
years, an amalgam of powerful quantum chemistry methods, simple intuitive
models, and phenomenological parameterizations have been developed, providing
increasingly complex and specialized methodologies. Even with increasing
specialization, a pervasive challenge remains that is surprisingly universal -
the simultaneous description of continuous symmetries (e.g. spin and orbital
angular momenta) and discrete symmetries (e.g. crystal field). Modeling and
predicting behavior in these complex systems is increasingly important for
metal ions of unusual or technologically relevant behavior. Additionally,
development and adoption of broad-scope models with physically-meaningful
parameters carries the potential to facilitate interdisciplinary collaboration
and large-scale meta analysis. Here, we propose a generalized algorithmic
approach, the molecular symmetry adapted spin space (mSASS), to localized
electronic structure via descent directly from fermionic (spin) rather than
bosonic (orbital) symmetry. We derive the Hamiltonian in symmetry-constrained
matrix form with an exact account of free parameters and several usage
examples. Although preliminary in its implementation, a fundamental benefit of
this approach is the treatment of spatial and spin-orbit symmetries without
the need for perturbative approximations. In general, the mSASS Hamiltonian is
large but finite and can be diagonalized numerically with high efficiency,
providing a basis for conceptual models of electronic structure that naturally
incorporates spin while leveraging the intuition and efficiency benefits of
crystallographic symmetry. For the generation of the mSASS Hamiltonian, we
provide an implementation into the Mathematica Software Package, GTPack.

##  I Introduction

Chemistry flourishes when synthetic efforts are driven by intuitive physical
models that link structure to function. An iconic example from molecular
chemistry is the control of the d-orbital manifold of transition metals via
ligand coordination environment. In many cases, a surprising level of insight
can be gained through the heuristic assumptions of a simplified symmetry and a
basic electrostatic model (i.e. Crystal Field Theory; CFT) [1, 2, 3, 4]. The
limitations of CFT are often cited as evidence for involvement of ligand
orbitals and thus the necessity of an expanded basis allowing for mixing with
ligand orbitals (Ligand Field Theory; LFT) [5]. Indeed, benchmark structure-
function relationships such as the spectrochemical series and nephelauxetic
effect [6] require LFT insight [7]. Rightly so, the influence of ligand
orbitals is considered foundational knowledge for many chemists. Knowledge of
orbital moment and the spin-orbit interaction, however, is much less
pervasive, despite being vital to understanding of electronic structure,
especially for the d-orbital manifold of many materials of high physical and
technological interest. Even though CFT/LFT models do not explicitly treat
angular momentum, basic qualitative predictions can be made in simple cases
with only the Pauli exclusion principle as a guide (e.g. spin-only models).
Caution in this respect is imperative, however, as the underlying physics of
electronic spin is absent here, limiting insight in both scope and scale.

Recognition of this issue goes back to the advent of quantum mechanics, and as
disciplines became more specialized, approaches to the problem developed a
dizzying array of formalisms. A common historic theme was that full spin
quantum mechanical modeling or fitting was too computationally-intensive for
practical use, and, in the end, unnecessary for empirical descriptions of most
phenomena. For instance, exclusions of energy states outside of the range of
interest vastly reduces the computational cost. Such simplifications trade
physically meaningful operators for phenomenologically parameterized tensors
to compensate for the simplified basis. Often referred to as spin
Hamiltonians, [8, 9, 10, 11], such models are particularly effective in
efficiently reproducing magnetometry and magnetic spectroscopy data of
transition metal complexes when the ground state lacks orbital angular
momentum. Developed largely in the context of Electron Paramagnetic Resonance
(EPR) [12], spin Hamiltonians are characterized by a lack of orbital operators
with compensation via a phenomenological symmetry-restricted Hamiltonian
acting directly on functions of the spin operators. There are a number of
common methods for perturbations involving the orbital operators where the
symmetry of the local crystal field is expanded into operator equivalents
following methods of Stevens [13, 14, 15] or Buckmaster-Smith-Thornley [16],
for example. This approach has proven powerful and efficient, in particular in
situations of well-defined symmetry and low anisotropy.

While effective in fitting complex spectral data, there are reasons to
consider alternative approaches to the crystal field expansion. The first of
these reasons being the lack of intrinsic physical meaning to the parameters.
Without the ability to generate falsifiable hypotheses for predicted spin
state structure from coordination environment, progress towards new,
overarching models or technological goals is drastically hindered.
Additionally, many of the most exciting systems of current interest are
incompatible with the spin Hamiltonian model. In cases of low-lying excited
states and significant orbital moment, for example, employing a spin
Hamiltonian will provide little insight. As a final point, somewhat counter-
intuitively, a more complete incorporation of the underlying spin and orbital
moment can actually reduce the overall complexity despite an expansion of the
Hilbert space. The spin symmetry lost in the truncated basis can often be
leveraged by modern computational algorithms to considerable effect. In this
work, we present a method for deploying point group representations of spin-
symmetry called the molecular symmetry adapted spin space (mSASS) method.
mSASS relies solely on symmetry constraints. By working in the full spin
rotational group and subgroups thereof we are able to maintain the meaningful
angular momentum operators with crystal field and spin orbit interactions
arising via mixing terms that emerge from the symmetry restrictions.

##  II Method Overview

In the following section, we provide theoretical background outlining the
molecular symmetry adapted spin space approach. To efficiently evaluate the
symmetry constraints imposed on the crystal field Hamiltonian, we make use of
the Mathematica group theory package GTPack [17, 18]. In particular, we extend
the functionality of GTPack to calculate representation matrices of O(3) and
SU(2), as described throughout the main text and in the appendix. All examples
presented here can be found in the Mathematica notebook format (Supporting
information).

Figure 1: Outline of the molecular symmetry adapted spin space method.

###  II.1 Spinor basis

The molecular symmetry adapted spin space (mSASS) method is constructed in the
spinor basis (Fig. 1). For total angular momentum $J$, a $(2J+1)$-dimensional
basis is spanned by the functions $\left|Jm\right>$, with $m=-J,\dots,J$. For
multiple values of $J$, the basis is extended accordingly,
$\left\\{\left|J_{1},J_{1}\right>,\left|J_{1},J_{1}-1\right>,\dots,\left|J_{1},-J_{1}\right>,\left|J_{2},J_{2}\right>,\left|J_{2},J_{2}-1\right>,\dots,\left|J_{2},-J_{2}\right>\right\\}$.

###  II.2 Multi-electron basis

Many-particle basis functions are constructed in terms of direct products of
single particle wave functions. For electrons, the wave function needs to
satisfy fermionic statistics, i.e., being odd under particle-exchange. This
constraint is satisfied by taking the alternating square of two
representations, instead of the ordinary direct product [19].

In the following we introduce the characters of $SU(2)$ representations and
the construction of the alternating square. We start from rotation of angle
$\phi$ about the arbitrary axis $\mathbf{n}$, $D^{J}(\mathbf{n},\phi)$,
represented as

$D^{J}(\mathbf{n},\phi)=D^{J}(\mathbf{n}^{\prime},-\theta)D^{J}(\mathbf{e}_{z},\phi)D^{J}(\mathbf{n}^{\prime},\theta),$
(1)

with $\mathbf{n}^{\prime}$ and $\theta$ being the respective axis and angle to
transform the initial axis $\mathbf{n}$ to the $\mathbf{e}_{z}$ axis. Due to
the cyclic permutation rule for the trace operation we obtain for the
characters $\chi^{J}(\mathbf{n},\phi)$ and $\chi^{J}(\mathbf{e}_{z},\phi)$,
$\chi^{J}(\mathbf{n},\phi)=\operatorname{Tr}\left[D^{J}(\mathbf{n}^{\prime},-\theta)D^{J}(\mathbf{e}_{z},\phi)D^{J}(\mathbf{n}^{\prime},\theta)\right]=\operatorname{Tr}\left[D^{J}(\mathbf{n}^{\prime},\theta)D^{J}(\mathbf{n}^{\prime},-\theta)D^{J}(\mathbf{e}_{z},\phi)\right]=\operatorname{Tr}\left[D^{J}(\mathbf{e}_{z},\phi)\right]=\chi^{J}(\mathbf{e}_{z},\phi)$.
Hence, without loss of generality, we focus on rotations about the $z$-axis in
the following.

Since,
$D^{J}(\mathbf{e}_{z},\phi)=\exp\left(-\mathrm{i}\hat{J}_{z}\phi\right)$ and
$\left<J,m|D^{J}(\mathbf{e}_{z},\phi)|J^{\prime}m^{\prime}\right>=\delta_{mm^{\prime}}\delta_{JJ^{\prime}}\exp\left(-\mathrm{i}m^{\prime}\phi\right)$
we obtain for the character

$\chi^{J}(\mathbf{e}_{z},\phi)=\sum_{m=-J}^{J}e^{-\mathrm{i}m\phi}.$ (2)

We continue by considering the direct product of two $SU(2)$ representations,
where we assume $J_{1}>J_{2}$. By calculating the product
$\chi^{J_{1}}(\mathbf{e}_{z},\phi)\chi^{J_{2}}(\mathbf{e}_{z},\phi)=\chi^{J_{1}+J_{2}}(\mathbf{e}_{z},\phi)+\chi^{J_{1}+J_{2}-1}(\mathbf{e}_{z},\phi)+\dots+\chi^{J_{1}-J_{2}}(\mathbf{e}_{z},\phi)$
we obtain

$D^{J_{1}}\otimes D^{J_{2}}\simeq D^{J_{1}+J_{2}}\oplus
D^{J_{1}+J_{2}-1}\oplus\dots\oplus D^{J_{1}-J_{2}}.$ (3)

For $J_{1}=J_{2}$, the representation matrices operate on the same finite
vector space $\mathcal{V}^{J}$. The basis of the direct product space
$\mathcal{V}^{J}\otimes\mathcal{V}^{J}$ can be constructed from products
$v_{i}v_{j}$, where $v_{i},v_{j}\in\mathcal{V}^{J}$ are a basis of
$\mathcal{V}^{J}$. This direct product can be decomposed into a symmetric
(bosonic) and an antisymmetric (fermionic) part,

$\mathcal{V}^{J}\otimes\mathcal{V}^{J}=S\left(\mathcal{V}^{J}\otimes\mathcal{V}^{J}\right)\oplus
A\left(\mathcal{V}^{J}\otimes\mathcal{V}^{J}\right).$ (4)

The symmetric and antisymmetric parts of the decomposition are also known as
the symmetric and antisymmetric square of the representation. Correspondingly,
one obtains for the direct product representation $D^{J}\otimes D^{J}\simeq
D^{S}\otimes D^{A}$ with $\chi^{J}(g)^{2}=\chi^{S}(g)+\chi^{A}(g)$. The
respective characters for the symmetric and the antisymmetric square are [19],

$\displaystyle\chi^{S}(g)$
$\displaystyle=\frac{1}{2}\left(\chi^{J}(g)^{2}+\chi^{J}\left(g^{2}\right)\right),$
(5) $\displaystyle\chi^{A}(g)$
$\displaystyle=\frac{1}{2}\left(\chi^{J}(g)^{2}-\chi^{J}\left(g^{2}\right)\right),$
(6)

where $g\in\mathcal{G}$ is a symmetry element of the group $\mathcal{G}$.

Finally, we give the decomposition of the antisymmetric square in terms of
representations of $SU(2)$,

$A\left(D^{J}\otimes D^{J}\right)\simeq D^{2J-1}\oplus D^{2J-3}\oplus\dots.$
(7)

The above equation can be obtained from applying (2) to equation (6).

As an example, we consider $J=\nicefrac{{1}}{{2}}$ and the well known direct
product

$D^{\nicefrac{{1}}{{2}}}\otimes D^{\nicefrac{{1}}{{2}}}\simeq D^{1}\oplus
D^{0}.$ (8)

From equation (7) it follows that the antisymmetric part of the direct product
is given by the spin-singlet $D^{0}$,

$A\left(D^{\nicefrac{{1}}{{2}}}\otimes D^{\nicefrac{{1}}{{2}}}\right)\simeq
D^{0}.$ (9)

The respective results for all direct products involving single values of $J$,
relevant for $s-f$ electrons with and without spin orbit coupling are given in
Table 1.

Table 1: The antisymmetric squares for relevant direct products arising for
$s-f$ electrons with and without spin-orbit coupling

$\displaystyle A\left(D^{\nicefrac{{1}}{{2}}}\otimes
D^{\nicefrac{{1}}{{2}}}\right)$ $\displaystyle\simeq D^{0}$ $\displaystyle
A\left(D^{1}\otimes D^{1}\right)$ $\displaystyle\simeq D^{1}$ $\displaystyle
A\left(D^{\nicefrac{{3}}{{2}}}\otimes D^{\nicefrac{{3}}{{2}}}\right)$
$\displaystyle\simeq D^{2}\oplus D^{0}$ $\displaystyle A\left(D^{2}\otimes
D^{2}\right)$ $\displaystyle\simeq D^{3}\oplus D^{1}$ $\displaystyle
A\left(D^{\nicefrac{{5}}{{2}}}\otimes D^{\nicefrac{{5}}{{2}}}\right)$
$\displaystyle\simeq D^{4}\oplus D^{2}\oplus D^{0}$ $\displaystyle
A\left(D^{3}\otimes D^{3}\right)$ $\displaystyle\simeq D^{5}\oplus D^{3}\oplus
D^{1}$ $\displaystyle A\left(D^{\nicefrac{{7}}{{2}}}\otimes
D^{\nicefrac{{7}}{{2}}}\right)$ $\displaystyle\simeq D^{6}\oplus D^{4}\oplus
D^{2}\oplus D^{0}$

###  II.3 Construction of the mSASS Hamiltonian

Consider a system with a basis belonging to several total angular momenta
$J_{i}$, where each $J_{i}$ corresponds to a $(2J_{i}+1)$-dimensional subspace
with basis function $u^{J_{i}}_{m}$ ($J_{i}\geq m\geq-J_{i}$). Furthermore,
consider a crystal field with symmetry group $\mathcal{G}$. By definition, the
transformation of $u^{J_{i}}_{m}$ under a symmetry element $g\in\mathcal{G}$
is given by

$gu^{J_{i}}_{m}=\sum_{m^{\prime}=-J_{i}}^{J_{i}}D_{m^{\prime}m}^{J_{i}}(g)u^{J_{i}}_{m^{\prime}},$
(10)

with $D_{m^{\prime}m}^{J_{i}}(g)$ denoting the matrix of the irreducible
representation of SO(3) corresponding to $J_{i}$. In general,
$D_{m^{\prime}m}^{J_{i}}(g)$ form a reducible representation in
$\mathcal{G}\subset SO(3)$.

The total dimension of a Hamiltonian spanned by
$(u^{J_{1}}_{J_{i}},u^{J_{1}}_{J_{i}-1},\dots u^{J_{2}}_{J_{2}},\dots)$ is
given by

$d=\sum_{i}(2J_{i}+1).$ (11)

The corresponding Hamiltonian is a $d\times d$-dimensional Hermitian matrix
111We note that a Hermitian matrix was historically chosen to enforce real
eigenvalues of the Hamiltonian. This condition can be lifted towards
$\mathcal{P}\mathcal{T}$-symmetric matrices [35] which bring a new twist to
quantum chemistry. Also, non-Hermitian matrices are meaningful, in the context
of open quantum systems [29]..

$\tilde{H}=\left(\begin{array}[]{cccc}h_{11}&h_{12}&\dots&h_{1d}\\\
h^{*}_{12}&h_{22}&\dots&h_{2d}\\\ \vdots&\vdots&\ddots&\vdots\\\
h^{*}_{1d}&h^{*}_{2d}&\dots&h_{dd}\end{array}\right)$ (12)

This matrix has to be invariant under all symmetry elements of $\mathcal{G}$,
imposing

$\tilde{H}=\sum_{g\in\mathcal{G}}\tilde{D}(g)\tilde{H}\tilde{D}(g)^{-1},$ (13)

where $\tilde{D}(g)$ denotes the super representation of $g$ incorporating all
matrices $\tilde{D}^{J_{i}}(g)$ as follows,

$\tilde{D}(g)=\left(\begin{array}[]{cccc}\tilde{D}^{J_{1}}(g)&0&\dots&0\\\
0&\tilde{D}^{J_{2}}(g)&\dots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\dots&\tilde{D}^{J_{N}}(g)\end{array}\right).$ (14)

While $\tilde{H}$ is diagonal in blocks belonging to different $J_{i}$,
imposing (13) for $\mathcal{G}\subset SO(3)$ separates $\tilde{H}$ into blocks
belonging to different irreducible representations of $\mathcal{G}$. The
diagonal formulation of $\tilde{H}$ in terms of $J_{i}$ represents the
unrestricted rotational symmetry of the chosen angular momentum operators. If
multiple types of angular momenta are used (e.g. $J=S+L$) intra-$J_{i}$
coupling terms naturally incorporate any associated interactions via symmetry
(e.g. the spin-orbit interaction). When the further constraint of
$\mathcal{G}\subset SO(3)$ is included, (13) inter-$J_{i}$ terms arise
naturally from the symmetry of $\mathcal{G}$ to account for the new crystal
field interactions. Due to the symmetry constraints of real molecules,
$D^{J_{i}}$ can now be reduced in $\mathcal{G}$ to give

$D^{J_{i}}\sim n_{1}\Gamma^{1}\oplus n_{2}\Gamma^{2}\oplus\dots\oplus
n_{n}\Gamma^{n},$ (15)

where $\Gamma^{i}$ are the $n$ nonequivalent irreducible representations of
$\mathcal{G}$. Hence, in the most general form, the mSASS Hamiltonian
$\tilde{H}$ is given by

$\tilde{H}=\left(\begin{array}[]{ccccccc}\tilde{H}_{\Gamma^{1,1}}&\tilde{h}_{1}&\dots&0&\dots&\tilde{\Delta}_{1}&\dots\\\
\tilde{h}^{\dagger}_{1}&\tilde{H}_{\Gamma^{1,2}}&\dots&0&\dots&\tilde{\Delta}_{2}&\dots\\\
\vdots&\vdots&\ddots&\vdots&&\vdots&\\\
0&0&\dots&\tilde{H}_{\Gamma^{i,\alpha}}&\dots&0&\dots\\\
\vdots&\vdots&&\vdots&\ddots&\vdots&\\\
\tilde{\Delta}^{\dagger}_{1}&\tilde{\Delta}^{\dagger}_{1}&\dots&0&\dots&\tilde{H}_{\Gamma^{1,\beta}}&\dots\\\
\vdots&\vdots&&\vdots&&\vdots&\end{array}\right)$ (16)

Here $\tilde{h}_{i}$ ($\tilde{\Delta}_{i}$) are intra-hybridization (inter-
hybridization) coupling terms belonging to equivalent irreducible
representations $\Gamma^{i}$ of $\mathcal{G}$, but the same (different) $J$.
The implementation of this formalism is shown below in several cases where
pre-existing intuition will be helpful. In this work, we do not yet apply the
method in complex or quantitative methods, instead focusing on establishing
the groundwork for further development and implementation. As an aid in this
process, the mSASS Hamiltonian $\tilde{H}$ can be deduced from evaluating
equation (13) using the Mathematica Software Package, GTPack. [17, 18]

##  III Examples

To evaluate equation (13) for specific point group symmetries
$g\in\mathcal{G}$, we implemented a new module GTAngularMomenumRep into the
Mathematica group theory package GTPack [17, 18], calculating the $SU(2)$ and
$SO(3)$ representation matrices $D^{J}(g)$. Full details of the implementation
are given in the appendix and software documentation. Briefly, the starting
point for implementation of orbital symmetry is often $O(3)$, the full roto-
reflection group of $\mathbb{R}^{3}$ represented by orthonormal $3\times 3$
matrices of $\det=\pm 1$). $O(3)$ is the direct product of SO(3) and the group
$S_{I}=\left\\{E,I\right\\}$ ($E$ the identity, $I$ the inversion),
$O(3)=SO(3)\times S_{I}$. The irreducible representations of O(3) come as even
and odd variants. The physically relevant representations are even (odd) under
inversion for even (odd) angular momenta $l$. Hence, for $d$-electrons, the
corresponding irreducible representation of O(3) would be $D^{2}_{g}$.

###  III.1 The $d^{1}$ configuration in an octahedral crystal field

To introduce the methodology, We will build up the textbook example of a
single $d$-electron in an octahedral, $O$, crystal field where spin is not a
factor outside of the Pauli exclusion principle. In Section III.B, we will add
detail through the introduction of spinor representations, constructing the
fermionic picture in the presence of simultaneous symmetry restrictions of the
spin-orbit and crystal field interactions.

Considering only the orbital moment, the five-fold degenerate $d$-electron
wave function in $O$-symmetry splits according to

$D^{2}\simeq E\oplus T_{2}.$ (17)

Note that we choose $O$ for our analysis of the restrictions of the crystal
field rather than the inversion symmetric $O_{h}$. The pure rotational
octahedral group, $O$, provides a more straightforward example, as well as one
of more general utility due to the ready extension to the tetrahedral case,
$T$.

In terms of real (tesseral) spherical harmonics
$Y^{l}_{m}=\left<\mathbf{r}|l,m\right>$, the basis of the $5$-dimensional
subspace belonging to $l=2$ is given by

$\left\\{\left|2,2\right>,\left|2,1\right>,\left|2,0\right>,\left|2,-1\right>,\left|2,-2\right>\right\\}.$
(18)

In this basis, the mSASS Hamiltonian matrix is diagonal, giving rise to three-
fold degenerate ($T_{2}$) and two-fold degenerate ($E$) electronic
configurations,

$\tilde{H}=\left(\begin{array}[]{ccccc}\epsilon_{E}&0&0&0&0\\\
0&\epsilon_{T_{2}}&0&0&0\\\ 0&0&\epsilon_{E}&0&0\\\
0&0&0&\epsilon_{T_{2}}&0\\\ 0&0&0&0&\epsilon_{T_{2}}\end{array}\right).$ (19)

To refer to the general form of the mSASS Hamiltonian in (16), we note that
the matrices $\tilde{H}_{E}$ and $\tilde{H}_{T_{2}}$ are diagonal, i.e.,
$\tilde{H}_{E}=\operatorname{diag}\left(\epsilon_{E},\epsilon_{E}\right)$ and
$\tilde{H}_{T_{2}}=\operatorname{diag}\left(\epsilon_{T_{2}},\epsilon_{T_{2}},\epsilon_{T_{2}}\right)$.
As each representation only occurs once, the coupling matrices vanish,
$\tilde{h}_{i}=\tilde{\Delta}_{i}=0$. The eigenfunctions of the mSASS
Hamiltonian for a $d^{1}$ configuration in an octahedral field can be
constructed to transform as

$\displaystyle\Psi_{T_{2};1}$
$\displaystyle\sim\left<\mathbf{r}|2,1\right>\sim xz,$ (20)
$\displaystyle\Psi_{T_{2};2}$
$\displaystyle\sim\left<\mathbf{r}|2,-1\right>\sim yz,$ (21)
$\displaystyle\Psi_{T_{2};3}$
$\displaystyle\sim\left<\mathbf{r}|2,-2\right>\sim xy,$ (22)
$\displaystyle\Psi_{E;1}$ $\displaystyle\sim\left<\mathbf{r}|2,0\right>\sim
z^{2}-r^{2},$ (23) $\displaystyle\Psi_{E;2}$
$\displaystyle\sim\left<\mathbf{r}|2,2\right>\sim x^{2}-y^{2}.$ (24)

Note that in the case of a single electron, these eigenfunctions have the
mathematical form of the commonly used Cartesian d-orbital set. It is
important, however, not to conflate the solutions to (19) (corresponding to
possible configurations available to a $d^{1}$ electron count) with the
concept of orbital energy levels. The corresponding Mathematica code can be
found in the supplemental material.

Throughout this work we focus mainly on the cubic symmetry $O$ as a
representative example. Without modification, however, the algorithm used for
$O$ may be applied to any point group symmetry. For example, the degeneracy of
possible configurations for $d^{1}$ may be lowered from $O$ to $D_{4}$ by an
axial Jahn-Teller distortion, leading to symmetry lowering according to:

$\displaystyle E$ $\displaystyle\simeq A_{1}\oplus B_{1},$ (25) $\displaystyle
T_{2}$ $\displaystyle\simeq E+B_{2}.$ (26)

In agreement with this decomposition, the mSASS Hamiltonian (16) is of the
form

$\tilde{H}=\left(\begin{array}[]{ccccc}\epsilon_{B_{1}}&0&0&0&0\\\
0&\epsilon_{E}&0&0&0\\\ 0&0&\epsilon_{A_{1}}&0&0\\\ 0&0&0&\epsilon_{E}&0\\\
0&0&0&0&\epsilon_{B_{2}}\end{array}\right).$ (27)

The corresponding eigenfunctions of the mSASS Hamiltonian for the $d^{1}$
configuration in square symmetry transform as follows

$\displaystyle\Psi_{E;1}$ $\displaystyle\sim\left<\mathbf{r}|2,1\right>\sim
xz,$ (28) $\displaystyle\Psi_{E;2}$
$\displaystyle\sim\left<\mathbf{r}|2,-1\right>\sim yz,$ (29)
$\displaystyle\Psi_{B_{2}}$ $\displaystyle\sim\left<\mathbf{r}|2,-2\right>\sim
xy,$ (30) $\displaystyle\Psi_{A_{1}}$
$\displaystyle\sim\left<\mathbf{r}|2,0\right>\sim z^{2}-r^{2},$ (31)
$\displaystyle\Psi_{B_{1}}$ $\displaystyle\sim\left<\mathbf{r}|2,2\right>\sim
x^{2}-y^{2}.$ (32)

A schematic of the hierarchical symmetry lowering of the $d$-state by $O$ and
$D_{4}$ crystal fields is shown in Fig. 3.

Figure 2: Splitting of a single $d$-electron in octahedral and square
(bipyramidal) geometries, with symmetry groups $O$ and $D_{4}$. The axes
represent relative energy changes vs distortion coordinates for the two
symmetry transformations.

###  III.2 The $d^{1}$ configuration in an octahedral crystal field with spin-
orbit coupling

Orbital symmetry has been arguably the greatest driver of intuitive models for
chemical synthesis, properties, and reactivity in the era of modern quantum
theory; spin-symmetry, on the other hand, has had only minor influence. While
the reasons for this are complex and beyond the scope of this work, in many
ways the triage of spin can be attributed to the search for configuration
subspaces accessible by contemporary computational methods. These subspaces,
though vastly reduced in dimensionality, lack the overarching symmetry
restrictions of the original space. When applied to systems with non-trivial
spin interactions, the reduced space may still effectively emulate the
properties of the local region of the full space assuming the subspace remains
largely orthogonal to the full configuration space (i.e. cases where
interactions arising from spin comprise only very minor perturbations on the
energy). In the framework of (16), $\tilde{\Delta}_{i}\approx 0$.

Because of the limitations of the restricted subspace to minor perturbations,
the spin-orbit interaction is typically discussed in terms of two limits. In
the first limit, the spin-orbit interaction dominates the crystal field,
causing a splitting of the $d$-level into two levels with angular momenta
$j=\nicefrac{{3}}{{2}}$ and $j=\nicefrac{{5}}{{2}}$. These levels then split
in the presence of the crystal field. Continuing from our example of $O$
symmetry, the splitting follows

$\displaystyle D^{\nicefrac{{3}}{{2}}}$ $\displaystyle\simeq
F_{\nicefrac{{3}}{{2}}},$ (33) $\displaystyle D^{\nicefrac{{5}}{{2}}}$
$\displaystyle\simeq E_{\nicefrac{{5}}{{2}}}\oplus F_{\nicefrac{{3}}{{2}}}.$
(34)

The far more common case in valence electrons of chemical importance occurs
when the crystal field dominates. Here, one can discuss the splitting of the
$E$ and $T_{2}$ levels discussed previously, due to spin orbit interaction. In
the case of the group $O$, these representations split as follows

$\displaystyle E$ $\displaystyle\simeq F_{\nicefrac{{3}}{{2}}},$ (35)
$\displaystyle T_{2}$ $\displaystyle\simeq E_{\nicefrac{{5}}{{2}}}\oplus
F_{\nicefrac{{3}}{{2}}}.$ (36)

Figure 3: Splitting of a single $d$-electron in octahedral field with symmetry
group $O$ and spin-orbit interaction. In the presence of both, the
$j=\nicefrac{{3}}{{2}}$ and $j=\nicefrac{{5}}{{2}}$ multiplets, states
belonging to the irreducible representation $F_{\nicefrac{{3}}{{2}}}$ mix, an
effect which is usually neglected in the strong spin-orbit limit.

In general, however, the two levels transforming as $F_{\nicefrac{{3}}{{2}}}$
interact. To obtain the corresponding mSASS Hamiltonian, we constrain the
problem to the 10-dimensional subspace containing spinor functions belonging
to the angular momenta $j_{1}=\nicefrac{{3}}{{2}}$ and
$j_{2}=\nicefrac{{5}}{{2}}$,

$\left\\{\left|\nicefrac{{3}}{{2}},\nicefrac{{3}}{{2}}\right>,\dots\left|\nicefrac{{3}}{{2}},-\nicefrac{{3}}{{2}}\right>,\left|\nicefrac{{5}}{{2}},\nicefrac{{5}}{{2}}\right>,\dots,\left|\nicefrac{{5}}{{2}},-\nicefrac{{5}}{{2}}\right>\right\\}.$
(37)

The corresponding super representation (14) is given by the Clebsch-Gordan sum

$D=D^{\nicefrac{{3}}{{2}}}\oplus D^{\nicefrac{{5}}{{2}}}.$ (38)

As there are two equivalent representations ($F_{\nicefrac{{3}}{{2}}}$) and
one additional representation ($E_{\nicefrac{{5}}{{2}}}$), we expect four free
parameters in our theory, three determining the three levels, plus one
parameter determining the interaction between the levels belonging to the
equivalent representation $F_{\nicefrac{{3}}{{2}}}$. Evaluating (13), our
algorithm gives the following mSASS Hamiltonian matrix for $d^{1}$
configurations under the full spin-orbit symmetry, with the parameters
$\lambda_{1},\dots,\lambda_{4}$,

$\tilde{H}=\left(\begin{array}[]{cccccccccc}\lambda_{1}&0&0&0&0&\lambda_{4}&0&0&0&\sqrt{5}\lambda_{4}\\\
0&\lambda_{1}&0&0&0&0&-\sqrt{6}\lambda_{4}&0&0&0\\\
0&0&\lambda_{1}&0&0&0&0&\sqrt{6}\lambda_{4}&0&0\\\
0&0&0&\lambda_{1}&-\sqrt{5}\lambda_{4}&0&0&0&-\lambda_{4}&0\\\
0&0&0&-\sqrt{5}\lambda_{4}^{*}&\lambda_{2}&0&0&0&\lambda_{3}&0\\\
\lambda_{4}^{*}&0&0&0&0&\lambda_{2}-\frac{4\lambda_{3}}{\sqrt{5}}&0&0&0&\lambda_{3}\\\
0&-\sqrt{6}\lambda_{4}^{*}&0&0&0&0&\lambda_{2}+\frac{\lambda_{3}}{\sqrt{5}}&0&0&0\\\
0&0&\sqrt{6}\lambda_{4}^{*}&0&0&0&0&\lambda_{2}+\frac{\lambda_{3}}{\sqrt{5}}&0&0\\\
0&0&0&-\lambda_{4}^{*}&\lambda_{3}&0&0&0&\lambda_{2}-\frac{4\lambda_{3}}{\sqrt{5}}&0\\\
\sqrt{5}\lambda_{4}^{*}&0&0&0&0&\lambda_{3}&0&0&0&\lambda_{2}\\\
\end{array}\right).$ (39)

By diagonalizing the matrix, one of the four parameters can be seen to
represent the absolute energy scale and set to zero or an arbitrary offset.
For real-valued parameters, the resulting energies for each term can be
expressed as follows,

$\displaystyle\epsilon_{E_{\nicefrac{{5}}{{2}}}}$
$\displaystyle=\lambda_{2}-\sqrt{5}\lambda_{3}$ (40)
$\displaystyle\epsilon_{F_{\nicefrac{{3}}{{2}}},1}$ $\displaystyle=a-b,$ (41)
$\displaystyle\epsilon_{F_{\nicefrac{{3}}{{2}}},2}$ $\displaystyle=a+b,$ (42)

with

$\displaystyle a$
$\displaystyle=\frac{1}{2}\left(\lambda_{1}+\lambda_{2}+\frac{1}{\sqrt{5}}\lambda_{3}\right),$
(43) $\displaystyle b$
$\displaystyle=\sqrt{a^{2}-\lambda_{1}\left(a-\lambda_{1}\right)+6\lambda_{4}^{2}}.$
(44)

The corresponding eigenvectors are parameter dependent and are expressed as a
linear combination of the basis functions given in (37). As the eigenvectors
transform as irreducible representations retaining the full spinor symmetry,
they can be used to form a symmetry-restricted model of the full configuration
space as a function of the full parameter space. Such models can provide
useful insight on the often opaque intermediate configuration space away from
the strong crystal field or spin-orbit limits. Additionally, the use of the
symmetry restricted full configuration space allows for a more physically-
meaningful model for fitting spin-dependent data. This flows from the fact
that a given physical measurement often represents only a fraction of the
energy range needed to parameterize a complex electronic structure. If the
model is constructed for the energy range of the measurement instead of the
energy range of the physical parameters, the relationship between the fitted
parameters and the “true” ones is ill-defined and generally not useful for
comparison across different systems.

###  III.3 Continuous symmetry parameterization of multielectronic
configurations

In the last example, we provide a simple extension to the multielectron case:
a $p^{4}$ configuration within an octahedral field. This case extends the
coordination chemistry examples used so far and serves as a demonstration of
the versatility offered by taking a wider view of electronic structure
problems. Configurations of several $p$-electrons occur e.g., in color centers
or vacancies, and play an important role in $d^{0}$-magnetism [21, 22] and NV
centers [23, 24, 25, 26]. For simplicity, we focus on two electrons in the
$p_{\nicefrac{{3}}{{2}}}$-state, assuming that the $p_{\nicefrac{{1}}{{2}}}$
state is fully occupied and sufficiently separated in energy from
$p_{\nicefrac{{3}}{{2}}}$. The corresponding alternating square from Tab. 1 is

$A\left(D^{\nicefrac{{3}}{{2}}}\otimes D^{\nicefrac{{3}}{{2}}}\right)\simeq
D^{2}\oplus D^{0}.$ (45)

As before, we choose a basis in terms of angular momentum states. However,
here we need to consider 2-electron wave functions,

$\left|J,M_{J};j_{1}j_{2}\right>=\left\\{\left|2,2;\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>,\dots,\\{\left|2,-2;\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>,\left|0,0;\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>\right\\}.$
(46)

which are constructed using Clebsch-Gordan coefficients and the addition of
angular momenta,

$\displaystyle\left|2,2;\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\left|\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>\left|\nicefrac{{3}}{{2}}\nicefrac{{1}}{{2}}\right>-\left|\nicefrac{{3}}{{2}}\nicefrac{{1}}{{2}}\right>\left|\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>\right),$
(47) $\displaystyle\left|2,1;\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\left|\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>\left|\nicefrac{{3}}{{2}}-\nicefrac{{1}}{{2}}\right>-\left|\nicefrac{{3}}{{2}}-\nicefrac{{1}}{{2}}\right>\left|\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>\right),$
(48) $\displaystyle\left|2,0;\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>$
$\displaystyle=\frac{1}{2}\left(\left|\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>\left|\nicefrac{{3}}{{2}}-\nicefrac{{3}}{{2}}\right>+\left|\nicefrac{{3}}{{2}}\nicefrac{{1}}{{2}}\right>\left|\nicefrac{{3}}{{2}}-\nicefrac{{1}}{{2}}\right>\right.$
$\displaystyle-\left.\left|\nicefrac{{3}}{{2}}-\nicefrac{{3}}{{2}}\right>\left|\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>-\left|\nicefrac{{3}}{{2}}-\nicefrac{{1}}{{2}}\right>\left|\nicefrac{{3}}{{2}}\nicefrac{{1}}{{2}}\right>\right),$
(49) $\displaystyle\left|2,-1;\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\left|\nicefrac{{3}}{{2}}\nicefrac{{1}}{{2}}\right>\left|\nicefrac{{3}}{{2}}-\nicefrac{{3}}{{2}}\right>-\left|\nicefrac{{3}}{{2}}-\nicefrac{{3}}{{2}}\right>\left|\nicefrac{{3}}{{2}}\nicefrac{{1}}{{2}}\right>\right),$
(50) $\displaystyle\left|2,-2;\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\left|\nicefrac{{3}}{{2}}-\nicefrac{{1}}{{2}}\right>\left|\nicefrac{{3}}{{2}}-\nicefrac{{3}}{{2}}\right>\right.$
$\displaystyle\qquad-\left.\left|\nicefrac{{3}}{{2}}-\nicefrac{{3}}{{2}}\right>\left|\nicefrac{{3}}{{2}}-\nicefrac{{1}}{{2}}\right>\right),$
(51) $\displaystyle\left|0,0;\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>$
$\displaystyle=\frac{1}{2}\left(\left|\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>\left|\nicefrac{{3}}{{2}}-\nicefrac{{3}}{{2}}\right>-\left|\nicefrac{{3}}{{2}}\nicefrac{{1}}{{2}}\right>\left|\nicefrac{{3}}{{2}}-\nicefrac{{1}}{{2}}\right>\right.$
$\displaystyle-\left.\left|\nicefrac{{3}}{{2}}-\nicefrac{{3}}{{2}}\right>\left|\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>+\left|\nicefrac{{3}}{{2}}-\nicefrac{{1}}{{2}}\right>\left|\nicefrac{{3}}{{2}}\nicefrac{{1}}{{2}}\right>\right).$
(52)

In $O$ symmetry, the representations $D^{2}$ and $D^{0}$ are decomposed as
before,

$D^{2}\simeq E\oplus T_{2},\qquad D^{0}\simeq A_{1}.$ (53)

In contrast to the single $d$-electron Hamiltonian in (19), we choose a
complex valued representation according to our basis in (46). Applying
equation (13) as before, we obtain a mSASS Hamiltonian with 3 free parameters,

$\tilde{H}=\left(\begin{array}[]{cccccc}\lambda_{1}&0&0&0&\lambda_{3}&0\\\
0&\lambda_{1}-\lambda_{3}&0&0&0&0\\\ 0&0&\lambda_{1}+\lambda_{3}&0&0&0\\\
0&0&0&\lambda_{1}-\lambda_{3}&0&0\\\ \lambda_{3}&0&0&0&\lambda_{1}&0\\\
0&0&0&0&0&\lambda_{2}\end{array}\right).$ (54)

Figure 4: Splitting of energy levels for two $p$-electrons in an octahedral
field.

The eigenvalues of this Hamiltonian are the level energies

$\displaystyle\epsilon_{T_{2}}$ $\displaystyle=\lambda_{1}-\lambda_{3},$ (55)
$\displaystyle\epsilon_{E}$ $\displaystyle=\lambda_{1}+\lambda_{3},$ (56)
$\displaystyle\epsilon_{A_{1}}$ $\displaystyle=\lambda_{2}.$ (57)

The eigenvectors of the Hamiltonian in terms of the respective two-electron
basis functions are given by

$\displaystyle\Psi_{T_{2};1}$
$\displaystyle\sim\left|2,1;\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>,$
(58) $\displaystyle\Psi_{T_{2};2}$
$\displaystyle\sim\frac{1}{\sqrt{2}}\left(\left|2,-2;\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>-\left|2,2;\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>\right),$
(59) $\displaystyle\Psi_{T_{2};3}$
$\displaystyle\sim\left|2,-1;\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>,$
(60) $\displaystyle\Psi_{E;1}$
$\displaystyle\sim\left|2,0;\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>,$
(61) $\displaystyle\Psi_{E;2}$
$\displaystyle\sim\frac{1}{\sqrt{2}}\left(\left|2,-2;\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>+\left|2,2;\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>\right),$
(62) $\displaystyle\Psi_{A_{1}}$
$\displaystyle\sim\left|0,0;\nicefrac{{3}}{{2}}\nicefrac{{3}}{{2}}\right>.$
(63)

As can be seen, the $T_{2}$ state corresponds to a spin-triplet with $J=1$,
whereas the levels $E$ and $A_{1}$ give rise to spin singlets with $J=0$. The
result is summarized in Fig. 4.

##  IV Conclusion

Ideally, computational modeling, synthesis, and physical characterization of
spin physics in molecular species would all fall under an overarching,
universal framework robust enough to allow scientists to communicate through
everything from back-of-the-envelope to supercomputer computations.

The molecular symmetry adapted spin space (mSASS) method allows for a
generalization of the crystal field theory for calculating atomic spectra,
taking into account the mixing of levels with different angular momenta, when
exposed to a symmetry breaking molecular environment. The mSASS Hamiltonian is
constructed by, first, fixing a subspace consisting of several angular
momentum multiplets belonging to a central atom or cluster, and, second,
imposing symmetry constraints of the molecular environment. We outlined the
approach on three simple examples, the $d^{1}$ configuration in octahedral and
square symmetry, the $d^{1}$ configuration in octahedral symmetry with spin-
orbit interaction, and, the $p^{2}$ configuration in an octahedral field.

By way of introduction, we have provided a case for the need, theoretical
basis, and simple implementation of the method, leaving detailed exploration
of specific applications and fitting to experiments for subsequent work. These
works will include, for example, symmetry-adapted splitting of states in a
magnetic field.

We close by providing one more advantage of the mSASS method, compared to
conventional crystal field theory. Note, that evaluating (13) is generic. By
lifting the Hermiticity of the mSASS Hamiltonian, the method can be
generalized towards open quantum systems, i.e., quantum systems interacting
with a thermal bath [27, 28, 29]. In such systems, electrons occupying certain
levels decay in time, which is a natural description for excited local
electrons. A non-Hermitian mSASS Hamiltonian contains the respective decay
channels in terms of generally complex parameters. As such, it might play a
powerful role, e.g., in optimizing the coherence time of molecular qubits [30]
by symmetry principles.

## Author Contributions

All authors contributed equally. RMG implemented the necessary algorithms into
the Mathematica group theory package GTPack.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

We thank Wolfram Hergert for helpful discussions. RMG acknowledges support
from Chalmers University of Technology. JDR acknowledges the National Science
Foundation Division of Chemistry CHE-2154830.

## Appendix

### Irreducible representation of SU(2)

In the process of the present work, a new module GTAngularMomenumRep was
implemented into the Mathematica group theory package GTPack [17, 18],
calculating the $SU(2)$ and $SO(3)$ representation matrices. We follow the
approach derived by Altmann [31]. Note that the $SU(2)$ matrices represent the
$j=\nicefrac{{1}}{{2}}$ representation of $SU(2)$. We obtain all other
representations by constructing corresponding tensor products as described
below. As $SU(2)$ is the double cover of $SO(3)$, all $SO(3)$ representations
are obtained in terms of integer valued $j$. The $j=\nicefrac{{1}}{{2}}$
representation is spanned by two basis functions $\mu_{-\nicefrac{{1}}{{2}}}$
and $\mu_{\nicefrac{{1}}{{2}}}$, such that

$\hat{g}\mu_{i}=\sum_{j=-\nicefrac{{1}}{{2}}}^{\nicefrac{{1}}{{2}}}\tilde{R}_{ji}(g)\mu_{j}.$
(64)

The basis can be used to construct tensors
$\mu_{ijk\dots}=\mu_{i}\mu_{j}\mu_{k}\dots$, transforming as

$\hat{g}\mu_{ijk\dots}=\sum_{IJK\dots}\tilde{R}_{Ii}(g)\tilde{R}_{Jj}(g)\tilde{R}_{Kk}(g)\dots\mu_{ijk\dots}.$
(65)

We constrain ourselves to focus on the totally symmetric tensors having the
property $\mu_{i\dots mn\dots}=\mu_{i\dots nm\dots}$, i.e., they are symmetric
with respect to pairwise permutation of indices. Hence, we can order indices
such that all values $-\nicefrac{{1}}{{2}}$ are listed on the left and all
$\nicefrac{{1}}{{2}}$ on the right. This construction can be abbreviated by a
function $v^{j}_{m}$ as follows

$v_{m}^{j}=\mu_{\underbrace{-\nicefrac{{1}}{{2}}-\nicefrac{{1}}{{2}}\dots}_{j-m\text{~{}times}}\underbrace{\nicefrac{{1}}{{2}}\,\nicefrac{{1}}{{2}}\dots}_{j+m\text{~{}times}}}=\mu^{j-m}_{-\nicefrac{{1}}{{2}}}\mu^{j+m}_{\nicefrac{{1}}{{2}}}.$
(66)

Acting on $v_{m}^{j}$ with the angular momentum operator $\hat{J}_{z}$ gives

$\hat{J}_{z}v_{m}^{j}=\left(-(j-m)\frac{1}{2}+(j+m)\frac{1}{2}\right)v_{m}^{j}=mv_{m}^{j}.$
(67)

Note that $\hat{J}_{z}$ acts like a differential operator, i.e.,
$\hat{J}_{z}\mu_{m_{1}}\mu_{m_{2}}=\left(\hat{J}_{z}\mu_{m_{1}}\right)\mu_{m_{2}}+\mu_{m_{1}}\left(\hat{J}_{z}\mu_{m_{2}}\right)$.
Normalizing $v_{m}^{j}$ leads to the final form

$v_{m}^{j}=\left((j+m)!(j-m)!\right)^{-\nicefrac{{1}}{{2}}}\mu^{j-m}_{-\nicefrac{{1}}{{2}}}\mu^{j+m}_{\nicefrac{{1}}{{2}}}.$
(68)

The representation matrices for higher $j$ can now be deduced by acting on the
normalized $v_{m}^{j}$. A general transformation using SU(2) matrices for
$j=\nicefrac{{1}}{{2}}$ can be written as

$\left(\begin{array}[]{c}\mu_{-\nicefrac{{1}}{{2}}}\\\
\mu_{\nicefrac{{1}}{{2}}}\end{array}\right)\left(\begin{array}[]{cc}a&b\\\
-b^{*}&a^{*}\end{array}\right)=\left(\begin{array}[]{c}a\mu_{-\nicefrac{{1}}{{2}}}-b*\mu_{\nicefrac{{1}}{{2}}}\\\
b\mu_{-\nicefrac{{1}}{{2}}}+a^{*}\mu_{\nicefrac{{1}}{{2}}}\end{array}\right).$
(69)

Using (68) we therefore obtain

$\hat{g}v_{m}^{j}=\left((j+m)!(j-m)!\right)^{-\nicefrac{{1}}{{2}}}\left(a\mu_{-\nicefrac{{1}}{{2}}}-b*\mu_{\nicefrac{{1}}{{2}}}\right)^{j-m}\times\\\
\times\left(b\mu_{-\nicefrac{{1}}{{2}}}+a^{*}\mu_{\nicefrac{{1}}{{2}}}\right)^{j+m}_{\nicefrac{{1}}{{2}}}.$
(70)

This expression can be expanded using the binomial theorem, giving

$\hat{g}v_{m}^{j}=\left((j+m)!(j-m)!\right)^{-\nicefrac{{1}}{{2}}}\times\\\
\times\sum_{k=0}^{j+m}\sum_{\kappa=0}^{j-m}\frac{(j+m)!(j-m)!}{(j+m-k)!(j-m-\kappa)!k!\kappa!}\times\\\
\times
a^{j+m-k}(a^{*})^{\kappa}b^{j-m-\kappa}(b^{*})^{k}\mu^{2j-k-\kappa}_{-\nicefrac{{1}}{{2}}}\mu^{k+\kappa}_{\nicefrac{{1}}{{2}}}.$
(71)

Rewriting the indices as $2j-k-\kappa=j+m^{\prime}$ and
$k+\kappa=j-m^{\prime}$ and noting that $j-k\leq m^{\prime}\leq m-k$, leads to
the final equation

$\hat{g}v_{m}^{j}\\\
=c^{j}_{mm^{\prime}}\sum_{m^{\prime}=j-k}^{m-k}\sum_{k=0}^{j+m}a^{j+m-k}(-b^{*})^{k}b^{m^{\prime}-m+k}(a^{*})^{j-m^{\prime}-k}v_{m^{\prime}}^{j},$
(72)

with

$c^{j}_{mm^{\prime}}=\left((j+m)!(j-m)!(j+m^{\prime})!(j-m^{\prime})!\right)^{\nicefrac{{1}}{{2}}}.$
(73)

Equation (72) is used to determine the representation matrices
$\tilde{D}^{j}(g)$, which satisfy

$\hat{g}v_{m}^{j}=\sum_{m^{\prime}=-j}^{j}D_{m^{\prime}m}^{j}(g)v_{m^{\prime}}^{j}.$
(74)

The limits where either $a=0$ or $b=0$ can be derived straightforwardly from
(71) without using the binomial series.

### Character Tables

The character tables are calculated using a Burnside algorithm, implemented in
GTPack. We choose the Mulliken notation for the irreducible representations
[32, 33], as tabulated by Altmann and Herzig [34]. Representations under the
separation are double group representations (see [18] for more information).

Table 2: Top: character table of $O$. Bottom: character table of $D_{4}$.

| E | $3C_{2}$ | $8C_{3}$ | $6C_{4}$ | $6C_{2}^{\prime}$
---|---|---|---|---|---
$A_{1}$ | 1 | 1 | 1 | 1 | 1
$A_{2}$ | 1 | 1 | 1 | -1 | -1
$E$ | 2 | 2 | -1 | 0 | 0
$T_{1}$ | 3 | -1 | 0 | 1 | -1
$T_{2}$ | 3 | -1 | 0 | -1 | 1
$E_{\nicefrac{{1}}{{2}}}$ | 2 | 0 | 1 | $\sqrt{2}$ | 0
$E_{\nicefrac{{5}}{{2}}}$ | 2 | 0 | 1 | $-\sqrt{2}$ | 0
$F_{\nicefrac{{3}}{{2}}}$ | 4 | 0 | -1 | 0 | 0

E $2C_{4}$ $C_{2}$ $2C_{2}^{\prime}$ $2C_{2}^{\prime\prime}$ $A_{1}$ 1 1 1 1 1
$A_{2}$ 1 1 1 -1 -1 $B_{1}$ 1 -1 1 1 -1 $B_{2}$ 1 -1 1 -1 1 $E$ 2 0 -2 0 0
$E_{\nicefrac{{1}}{{2}}}$ 2 $\sqrt{2}$ 0 0 0 $E_{\nicefrac{{3}}{{2}}}$ 2
-$\sqrt{2}$ 0 0 0

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