Relations between angular and Cartesian orientational expansions
aa r X i v : . [ m a t h - ph ] O c t Relations between angular and Cartesian orientational expansions
Michael te Vrugt
1, 2 and Raphael Wittkowski
1, 2, 3, ∗ Institut f¨ur Theoretische Physik, Westf¨alische Wilhelms-Universit¨at M¨unster, D-48149 M¨unster, Germany Center for Soft Nanoscience, Westf¨alische Wilhelms-Universit¨at M¨unster, D-48149 M¨unster, Germany Center for Nonlinear Science, Westf¨alische Wilhelms-Universit¨at M¨unster, D-48149 M¨unster, Germany
Orientational expansions, which are widely used in the natural sciences, exist in angular andCartesian form. Although these expansions are orderwise equivalent, it is difficult to relate themin practice. In this article, both types of expansions and their relations are explained in detail.We give explicit formulas for the conversion between angular and Cartesian expansion coefficientsfor functions depending on one, two, and three angles in two and three spatial dimensions. Theseformulas are useful, e.g., for comparing theoretical and experimental results in liquid crystal physics.The application of the expansions in the definition of orientational order parameters is also discussed.
I. INTRODUCTION
Orientational expansions, i.e., expansions of the angu-lar dependence of a function f (Ω), with Ω denoting anorientational variable, are used in many fields of the nat-ural sciences, such as liquid crystal physics [1–6], activematter physics [7–13], polymer physics [14], electrostat-ics [15, 16], optics [17–19], geophysics [20], astrophysicsand cosmology [21, 22], general relativity [23, 24], engi-neering [25, 26], chemistry [27, 28], and medicine [29].Important examples for orientational expansions are theFourier expansion for Ω = φ ∈ [0 , π ), the expansion inspherical harmonics for Ω = ( θ, φ ) ∈ [0 , π ] × [0 , π ), andthe expansion in outer products of a normalized orienta-tion vector ˆ u for Ω = ˆ u . The expansions can be classifiedin two main categories, which differ in the way the ex-pansion coefficients transform under rotations: angularexpansions (including the Fourier series and the sphericalharmonics expansion) and Cartesian expansions (includ-ing expansions in symmetric traceless tensors and outerproducts of orientation vectors or rotation matrices ).One of the main applications of such expansions is thedescription of the orientational order of liquid crystals.Here, we can use an orientation vector ˆ u depending onthe angle φ in two spatial dimensions (2D) or on twoangles θ and φ in three spatial dimensions (3D) to spec-ify the orientation of a particle. A system of particles isthen described using a distribution function f (ˆ u ). Thecoefficients of the expansion of the function f (ˆ u ) provideorientational order parameters. Of particular importanceare the Cartesian order parameters of first and second or-der, given by the polarization ~P and the nematic tensor Q , respectively [30]. They are used, e.g., to formulatefield theories for liquid crystals [31, 32]. Also order pa-rameters of third order are useful for the description ofcertain phase transitions [33]. ∗ Corresponding author: [email protected] By “rotation matrix”, we always denote a matrix of the form(16) or (36) that rotates a Cartesian vector. Wigner rotationmatrices, which correspond to the angular case, are referred toas “Wigner D-matrices”.
In three spatial dimensions, the description of the ori-entational order using one orientation vector is only suf-ficient if the particles have an axis of continuous ro-tational symmetry (“uniaxial particles” ). The situa-tion is more complex if one considers particles withoutsuch a symmetry (“biaxial particles”). Here, a descrip-tion of their orientation requires two orientation vectorsor, alternatively, three angles such as the Euler angles( θ, φ, χ ) ∈ [0 , π ] × [0 , π ) × [0 , π ), which correspond toa rotation R that maps the laboratory-fixed Cartesiancoordinate system onto a body-fixed one [34]. Therefore,the definition of order parameters for biaxial particlesrequires more general orientational expansions [34–38].Based on liquid crystal terminology, we will refer to ex-pansions for functions f (ˆ u ) as “uniaxial” and to expan-sions for functions f ( R ) as “biaxial”. Note, however, thatthe applicability of our results is not restricted to liquidcrystal physics, but extends to all fields where such ex-pansions are used.Both angular and Cartesian expansions have their ownadvantages and disadvantages. Angular expansions al-low to make use of the mathematical properties of cir-cular or spherical harmonics and have a smaller num-ber of expansion coefficients, since they are all indepen-dent. Cartesian expansions, on the other hand, have aclearer geometrical interpretation and are more directlyconnected to computer simulations and experiments [38].A good example is the study of the orientational orderof liquid crystals consisting of low-symmetry molecules.For theoretical studies, order parameters based on anexpansion in Wigner D-matrices, which have a usefuland well-known mathematical structure, are widely used[39, 40]. Experimentalists, on the other hand, frequentlyuse the Saup´e ordering matrix – a Cartesian order pa-rameter – which is more practical in describing outcomesof, e.g., nuclear magnetic resonance (NMR) experiments In this work, we use the terminology “uniaxial” and “biaxial”to distinguish particles with and without an axis of continuousrotational symmetry, respectively. Note that these adjectives canalso be used in a different way, referring to optical properties ofa system [34]. Moreover, we make the common approximationof assuming all particles to be rigid bodies. [41]. This can constitute a difficulty in comparing theresults of theoretical calculations with computer simula-tions and experiments.The problem can be solved if the relations between thedifferent types of expansions are known. It is very dif-ficult to give such an expression in a general form [42].One can, however, explicitly calculate these relations forthe lowest orders, which is sufficient for almost all prac-tical applications. In this work, we provide tables con-taining all relevant relations. This includes coefficientsof zeroth to third order for functions depending on oneangle (2D, uniaxial), two angles (3D, uniaxial), and threeangles (3D, biaxial). These relations, which can be foundin an Appendix, constitute the main result of this article.The conversion rules can be used in any field of the natu-ral sciences where orientational distributions are relevant.For example, they allow to easily convert Cartesian datafrom a computer simulation or an NMR experiment ona liquid crystal into a form that can be compared withtheoretical calculations that use angular functions. More-over, these equations allow, e.g., to easily calculate thedipole vector corresponding to data that are given in theform of an expansion in spherical harmonics.In addition, we give an overview over the mathemati-cal structure of uniaxial and biaxial expansions, includingdefinitions of all special functions that are involved. Wealso provide formulas for the expansion coefficients. Suchan overview is very useful and difficult to find in the liter-ature. It also clarifies the conventions used for obtainingthe conversion formulas. Furthermore, we explain howthese expansions allow to define order parameters for liq-uid crystals.This article is structured as follows: In Section II, wedescribe angular and Cartesian expansions with the rele-vant functions, coefficients, and order parameters for theuniaxial case. The biaxial expansions are described inSection III. We summarize the work and give an outlookin Section IV. The relations between the uniaxial expan-sion coefficients are listed in Appendix A and those forthe biaxial ones in Appendix B. A list of elements of theWigner D-matrices can be found in Appendix C.
II. UNIAXIAL EXPANSIONS
We consider uniaxial particles in two and three spatialdimensions. The orientational distribution of the parti-cles is described by a scalar orientation-dependent func-tion f . In a 2D system, this function can be parameter-ized as f ( φ ) ≡ f (ˆ u ) with the polar angle φ ∈ [0 , π ) andthe orientational unit vector ˆ u ( φ ) = (cos( φ ) , sin( φ )) T ,where the superscript T denotes a transposition. Whenconsidering a 3D system, the function can be parame-terized as f ( θ, φ ) ≡ f (ˆ u ) with the spherical coordinates θ ∈ [0 , π ] and φ ∈ [0 , π ) as well as the orientational unitvector ˆ u ( θ, φ ) = (sin( θ ) cos( φ ) , sin( θ ) sin( φ ) , cos( θ )) T . A. Angular multipole expansion
The scalar orientation-dependent function f can or-thogonally be expanded in terms of angular coordinates φ (in 2D) and θ and φ (in 3D). We follow Refs. [40, 43].
1. Circular multipole expansion (2D)
In the case of two spatial dimensions, the angular mul-tipole expansion is also called “circular multipole expan-sion” and identical to the Fourier series expansion f ( φ ) = ∞ X k = −∞ f k e i kφ (1)with the imaginary unit i and the circular harmonics e i kφ .The corresponding (Fourier) expansion coefficients aregiven by f k = 12 π Z S dΩ f ( φ ) e − i kφ , (2)where R S dΩ = R π d φ denotes an angular integration overthe unit circle S . In general, these expansion coefficientsare independent. If the coordinate system is rotated byan angle ϕ , the expansion coefficients f k change to f ′ k = f k e i kϕ . (3)
2. Spherical multipole expansion (3D)
When there are three spatial dimensions, the angularmultipole expansion is identical to the “spherical multi-pole expansion” f ( θ, φ ) = ∞ X l =0 l X m = − l f lm Y lm ( θ, φ ) (4)with the spherical harmonics Y lm ( θ, φ ) = s l + 14 π ( l − m )!( l + m )! P lm (cos( θ )) e i mφ (5)and the associated Legendre polynomials P lm ( x ) = ( − m l l ! (1 − x ) m/ ∂ l + mx ( x − l . (6)The latter two functions are stated here explicitly toavoid confusion with other conventions. Now, the ex-pansion coefficients are given by f lm = Z S dΩ f ( θ, φ ) Y ⋆lm ( θ, φ ) , (7)where R S dΩ = R π d θ sin( θ ) R π d φ is an angular integra-tion over the unit sphere S and the star ⋆ denotes com-plex conjugation. As is the 2D case, the expansion coef-ficients are in general independent. Under passive rota-tions, the expansion coefficients transform according to f lm = l X n = − l D lmn ( ~̟ ) f ′ ln , (8)where the D lmn ( ~̟ ) are the Wigner D-matrices dependingon the Euler angles ~̟ (see Section III A). B. Cartesian multipole expansion
The scalar orientation-dependent function f can alsoorthogonally be expanded in terms of the orientation vec-tor ˆ u . This applies to both 2D and 3D. Our presentationof this expansion follows Refs. [43] (for 2D) and [40] (for3D).The so-called “Cartesian multipole expansion” is givenby f (ˆ u ) = ∞ X l =0 d X i ,...,i l =1 f ( d D) i ··· i l u i · · · u i l , (9)where d ∈ { , } is the number of spatial dimensionsand u i is the i -th element of the orientation vectorˆ u = ( u , . . . , u d ) T . For this expansion, the correspondingexpansion coefficients are obtained as f ( d D) i ··· i l = A ( d D) l Z S d − dΩ f (ˆ u )T ( d D) i ··· i l (10)with the prefactors A (2D) l = 2 − δ l Ω , A (3D) l = 2 l + 1Ω (11)and the angular normalization factorsΩ d = Z S d − dΩ = ( π, for d = 2 , π, for d = 3 . (12)The tensors T ( d D) i ··· i l on the right-hand side of Eq. (10)equal the tensor Chebyshev polynomials of the first kind “Active” and “passive” here refer to two different conventionsused in the description of rotations, whereas elsewhere in thearticle we use this terminology to distinguish particles with andwithout self-propulsion, respectively. An active rotation corre-sponds to a rotation of a body in a fixed coordinate system,whereas in a passive rotation the coordinate axes are rotated.Switching between these conventions corresponds to turning aclockwise into a counterclockwise rotation [40], which is why itis important to clarify the convention. Throughout this article,we follow the conventions used by Gray and Gubbins [40] for thedescription of rotations. T (2D) i ··· i l for d = 2 and the tensor Legendre polynomialsT (3D) i ··· i l for d = 3. They are given byT (2D) i ··· i l = ( − l l ! ( l + δ l ) ∂ i · · · ∂ i l (1 − ln( r )) (cid:12)(cid:12)(cid:12)(cid:12) ~r =ˆ u , (13)T (3D) i ··· i l = ( − l l ! ∂ i · · · ∂ i l r (cid:12)(cid:12)(cid:12)(cid:12) ~r =ˆ u (14)with r = k ~r k and the Euclidean norm k·k . In Table I,the first four of these tensors for d = 2 and d = 3 arelisted explicitly. The tensors T ( d D) i ··· i l and therefore also l T (2D) i ··· i l T (3D) i ··· i l u i u i u i u i − δ i i (3 u i u i − δ i i )3 4 u i u i u i − u i δ i i ) sym 12 (5 u i u i u i − u i δ i i ) sym )... ... ...TABLE I. Tensor Chebyshev polynomials of the first kindT (2D) i ··· i l and tensor Legendre polynomials T (3D) i ··· i l for differentorders l , where ( · ) sym denotes the symmetrization of a tensor. the Cartesian coefficient tensors (10) are symmetric andtraceless for l >
1. When f (ˆ u ) is real, the same appliesto the Cartesian coefficient tensors f ( d D) i ··· i l . In general, notmore than 2 − δ l (in 2D) and 2 l + 1 (in 3D) elements of aCartesian coefficient tensor of order l can be independent.The first four Cartesian coefficient tensors for d = 2 and d = 3 are listed explicitly in Table II. Under rotations,the expansion coefficients transform as Cartesian tensors,i.e., f ( d D) i ··· i l = d X j ,...,j l =1 R i j · · · R i l j l f ( d D) ′ j ··· j l (15)with the rotation matrix R ij (see Section III B). The ro-tation matrix for d = 2 is defined as R ( φ ) = (cid:18) cos( φ ) − sin( φ )sin( φ ) cos( φ ) (cid:19) (16)and the rotation matrix for d = 3 is given by Eq. (36).An advantage of the Cartesian multipole expansion isthat it is a direct expansion in the variable ˆ u , whereasthe angular variables φ (in 2D) or θ and φ (in 3D) ap-pear not directly but via exponential and trigonometricfunctions in the angular multipole expansion. On theother hand, the number of expansion coefficients is higherfor the Cartesian multipole expansion although not moreof the expansion coefficients can be independent. De-spite of the differences of both types of expansions, theyare equivalent. Moreover, each order of one expansionis equivalent to the same order of the other expansion.This allows an orderwise mapping between both types ofexpansions and explains the maximal number of indepen-dent expansion coefficients for the Cartesian multipole l f (2D) i ··· i l f (3D) i ··· i l π R S dΩ f (ˆ u ) π R S dΩ f (ˆ u )1 π R S dΩ f (ˆ u ) u i π R S dΩ f (ˆ u ) u i π R S dΩ f (ˆ u )(2 u i u i − δ i i ) π R S dΩ f (ˆ u )(3 u i u i − δ i i )3 π R S dΩ f (ˆ u )(4 u i u i u i − u i δ i i − u i δ i i − u i δ i i ) π R S dΩ f (ˆ u )(5 u i u i u i − u i δ i i − u i δ i i − u i δ i i )... ... ...TABLE II. Cartesian expansion coefficients f (2D) i ··· i l and f (3D) i ··· i l for different orders l . expansion. In Appendix A, explicit equations expressingthe expansion coefficients of an angular multipole expan-sion in terms of the expansion coefficients of a Cartesianmultipole expansion and vice versa are given for dimen-sionalities d = 2 and d = 3 and up to third order.The equations for the expansion coefficients f k in termsof the expansion coefficients f (2D) i ··· i l have been obtained byinserting Eq. (9) into Eq. (2) and performing the angu-lar integration. Similarly, we have expressed the f lm interms of the f (3D) i ··· i l by inserting Eq. (9) into Eq. (7) andintegrating. Finally, the f ( d D) i ··· i l have been expressed interms of the f k (in 2D) and f lm (in 3D) by insertingEq. (1) (in 2D) and Eq. (4) (in 3D) into Eq. (10) andevaluating the integrals. C. Order parameters
The Cartesian coefficient tensors (10) can be used asorder parameters in liquid-crystal theory. They corre-spond to the mean particle density ( l = 0, monopolemoment), polarization vector ( l = 1, dipole moment), ne-matic tensor ( l = 2, quadrupole moment), tetratic tensor( l = 3, octupole moment), and so on [31, 33]. Usually,only order parameters up to second order are considered.In general, a liquid crystal consisting of particles thatare (considered as) uniaxial is described microscopicallyusing a probability distribution f ( ~r, ˆ u ) that depends onposition ~r and orientation ˆ u . The vector ˆ u specifies thegeometrical orientation of a particle or, for an active par-ticle, the direction of self-propulsion [44, 45]. A Cartesianexpansion in the form (9) up to second order gives f ( ~r, ˆ u ) = ρ ( ~r ) + d X i =1 P i ( ~r ) u i + d X i,j =1 Q ij ( ~r ) u i u j (17)with for d = 2 ρ ( ~r ) = 12 π Z S dΩ f ( ~r, ˆ u ) , (18) P i ( ~r ) = 1 π Z S dΩ f ( ~r, ˆ u ) u i , (19) Q ij ( ~r ) = 2 π Z S dΩ f ( ~r, ˆ u ) (cid:16) u i u j − δ ij (cid:17) (20) and for d = 3 ρ ( ~r ) = 14 π Z S dΩ f ( ~r, ˆ u ) , (21) P i ( ~r ) = 34 π Z S dΩ f ( ~r, ˆ u ) u i , (22) Q ij ( ~r ) = 158 π Z S dΩ f ( ~r, ˆ u ) (cid:16) u i u j − δ ij (cid:17) . (23)Now suppose that we have a system of N particles andthat we can assign every particle a position ~r n and an ori-entation vector ˆ u n = ˆ u ( φ n ) for d = 2 and ˆ u n = ˆ u ( θ n , φ n )for d = 3 with n ∈ { , . . . , N } . In this case, we can writethe microscopic one-particle distribution function as f ( ~r, ˆ u ) = N X i =1 δ ( ~r − ~r n ) δ (ˆ u − ˆ u n ) , (24)which gives the order parameters for d = 2 ρ ( ~r ) = 12 π N X n =1 δ ( ~r − ~r n ) , (25) P i ( ~r ) = 1 π N X n =1 u n,i δ ( ~r − ~r n ) , (26) Q ij ( ~r ) = 2 π N X n =1 (cid:16) u n,i u n,j − δ ij (cid:17) δ ( ~r − ~r n ) (27)and for d = 3 ρ ( ~r ) = 14 π N X n =1 δ ( ~r − ~r n ) , (28) P i ( ~r ) = 34 π N X n =1 u n,i δ ( ~r − ~r n ) , (29) Q ij ( ~r ) = 158 π N X n =1 (cid:16) u n,i u n,j − δ ij (cid:17) δ ( ~r − ~r n ) (30)with u n,i = (ˆ u n ) i denoting the i -th element of the vec-tor ˆ u n . These order parameters are, in the case of Eqs.(25) and (28) up to a prefactor, the usual microscopicdefinitions of the one-particle density 2( d − πρ ( ~r ), po-larization vector ~P ( ~r ), and nematic tensor Q ( ~r ). Theycan be used, e.g., to describe a system of active particleswhere ˆ u n denotes the direction of self-propulsion of the n -th particle. In the case of passive apolar particles suchas rods, where there is no physical difference between ˆ u n and − ˆ u n , one can incorporate the head-tail symmetryby replacing δ (ˆ u − ˆ u n ) by ( δ (ˆ u − ˆ u n ) + δ (ˆ u + ˆ u n )) / III. BIAXIAL EXPANSIONS
We consider biaxial particles in three spatial dimen-sions. The orientational distribution function of theparticles can now be parameterized as f ( θ, φ, χ ) ≡ f ( R )with the third Euler angle χ ∈ [0 , π ) and the rotation R ∈ SO(3) that can be represented by a rotation matrix R ij ( θ, φ, χ ). A. Angular multipole expansion
For describing the orientation of a biaxial particle, weuse the Euler angles ( θ, φ, χ ) ∈ [0 , π ] × [0 , π ) × [0 , π ) andintroduce the vector ~̟ = ( θ, φ, χ ) T as a shorthand no-tation. The Euler angles can be defined in various ways.For convenience, we use the popular definition of Grayand Gubbins [40]. There, the angles θ and φ correspondto the usual angles of spherical coordinates, while thethird angle χ ∈ [0 , π ) describes a rotation about the axisdefined by θ and φ . The advantage of this convention isthat, since the first two angles have the same definitionas in the usual case of spherical coordinates, the orderparameters for biaxial particles will contain the commondefinitions for uniaxial particles as a limiting case.The Euler angles are also a way to specify a rotationin three spatial dimensions, so that a function f ( ~̟ ) ≡ f ( θ, φ, χ ) can be thought of as a function f ( R ) with R being a rotation represented by a 3 × D lmn ( ~̟ ) = e − i mφ d lmn ( θ ) e − i nχ (31)with the Wigner d-matrices d lmn ( θ ) = p ( l + m )!( l − m )!( l + n )!( l − n )! X k ∈I lmn ( − k cos( θ ) l + m − n − k sin( θ ) k − m + n ( l + m − k )!( l − n − k )! k !( k − m + n )!(32)and − l ≤ m, n ≤ l are irreducible representations of thegroup SO(3). The set I lmn contains all integers that k can attain such that all factorial arguments in Eq. (32) In two spatial dimensions, one orientation vector is always suffi-cient, hence we do not need to consider the 2D case separatelyin this section. are nonnegative [46]. See Ref. [46] for differential and in-tegral representations of the functions d lmn ( θ ). The func-tions D lmn ( ~̟ ) are also referred to as Wigner D-functions[46], Wigner rotation matrices [47], and four-dimensionalspherical harmonics [40]. Note that the definition ofthe Wigner matrices used here assumes passive rotations[40, 48].An expansion for functions defined on SO(3) can beperformed using the Peter-Weyl theorem, which states(roughly) that for a compact topological group G , asquare-integrable function f ∈ L ( G ) can be expandedin terms of matrix elements of the irreducible represen-tations of G [38]. Since these representations are givenby the Wigner D-matrices for the group SO(3), we canexpand a function f ( ~̟ ) ∈ L (SO(3)) in the form [38, 40] f ( ~̟ ) = ∞ X l =0 l X m = − l l X n = − l c lmn D lmn ( ~̟ ) (33)with the expansion coefficients c lmn = 2 l + 18 π Z SO(3) dΩ D lmn ( ~̟ ) ⋆ f ( ~̟ ) , (34)where the integral is defined as [40] Z SO(3) dΩ = Z π d χ Z π d φ Z π d θ sin( θ ) . (35)The expansion coefficients (34) are in general indepen-dent, such that there are (2 l +1) independent coefficientsat order l .Explicit expressions for the elements of the Wigner D-matrices up to order l = 3 are given in Appendix C. B. Cartesian multipole expansion
We are now interested in a Cartesian expansion fora function depending on the Euler angles. For this ex-pansion, various options are possible. Ehrentraut andMuschik describe an expansion in higher-dimensionalsymmetric traceless tensors [37], which allows to con-struct order parameters for biaxial particles, since onecan map between the sphere in four dimensions S and the configuration space SO(3) of a biaxial particle[37, 42]. An orientational expansion of functions definedon S can be performed in terms of four-dimensionalspherical harmonics or, equivalently, higher-dimensionalorientation vectors [37]. A Cartesian expansion in four-dimensional symmetric traceless tensors is discussed inRef. [42].We here describe an expansion derived by Turzi [38].It allows to describe orientational order in terms of rota-tion matrices, which is geometrically more intuitive thanusing four-dimensional orientation vectors. Just as anangular expansion in terms of Wigner D-matrices can beobtained starting from the fact that they are represen-tations of the rotation group SO(3), we can construct aCartesian representation of this group to derive an ex-pansion in outer products of a rotation matrix R ij . The rotation matrix can be defined in terms of the Euler an-gles as [40] R ( ~̟ ) = cos( φ ) cos( θ ) cos( χ ) − sin( φ ) sin( χ ) − cos( φ ) cos( θ ) sin( χ ) − sin( φ ) cos( χ ) cos( φ ) sin( θ )sin( φ ) cos( θ ) cos( χ ) + cos( φ ) sin( χ ) − sin( φ ) cos( θ ) sin( χ ) + cos( φ ) cos( χ ) sin( φ ) sin( θ ) − sin( θ ) cos( χ ) sin( θ ) sin( χ ) cos( θ ) . (36)This matrix describes a successive rotation as R ( θ, φ, χ ) = R z ( φ ) R y ( θ ) R z ( χ ) with the elementaryrotation matrices R y ( ϕ ) = cos( ϕ ) 0 sin( ϕ )0 1 0 − sin( ϕ ) 0 cos( ϕ ) , (37) R z ( ϕ ) = cos( ϕ ) − sin( ϕ ) 0sin( ϕ ) cos( ϕ ) 00 0 1 . (38)We denote by ( E ( j ) − j , . . . , E ( j ) j ) an orthonormal basis ofthe vector space of symmetric traceless tensors of rank j . As the Wigner D-matrices for the angular case, anirreducible Cartesian representation ˜ D ( R ) of the rotationgroup can be derived, which acts on a symmetric tracelesstensor T i ··· i l as( ˜ D ( R ) T ) i ··· i l = X j ,...,j l =1 R i j · · · R i l j l T j ··· j l . (39)Using the Peter-Weyl theorem, one can then expand afunction f ∈ L (SO(3)) in terms of the representation˜ D ( R ), whose matrix elements form a complete orthogo-nal system [38]. The resulting expansion is [38] f ( R ) = ∞ X l =0 3 X i ,...,i l =1 3 X j ,...,j l =1 c (3D) i j ··· i l j l R i j · · · R i l j l (40)with the expansion coefficients c (3D) i j ··· i l j l = 2 l + 18 π Z SO(3) dΩ l X α,β = − l X a ,...,a l =1 3 X b ,...,b l =1 ( E ( l ) α ) a ··· a l ( E ( l ) β ) b ··· b l ( E ( l ) α ) i ··· i l ( E ( l ) β ) j ··· j l R a b · · · R a l b l . (41)In practical applications, the expansion coefficients canbe calculated more efficiently in other ways (see Ref. [38]for details). A list of the biaxial expansion coefficientsup to order l = 3 can be found in Table III.The expansion coefficients (41) are symmetric andtraceless in the { i k } and { j k } separately for l > c (3D)1321 = c (3D)2311 but c (3D)1321 = c (3D)1231 and P i =1 c (3D) ikil = 0but P i =1 c (3D) iijl = 0), so the maximal number of indepen-dent expansion coefficients of a certain order l is (2 l + 1) and thus identical to that of the expansion coefficients(34) for the same l . In consequence, angular and Carte-sian expansions are orderwise equivalent also in the bi-axial case.In Appendix B, explicit equations expressing the ex-pansion coefficients of an angular multipole expansion interms of the expansion coefficients of a Cartesian multi-pole expansion and vice versa are given up to third orderfor biaxial particles. These relations have been calcu-lated using the same procedure as in the uniaxial case(see Section II B). C. Order parameters
In the biaxial case, various definitions of orientationalorder parameters are possible and used in the literature.A very popular choice for l = 2 is the Saup´e orderingmatrix, which can be used to describe outcomes of NMRmeasurements [34]. Here, we state the order parame-ters in the form that follows from the Turzi expansionfor consistency with the previous section. We consider asystem with a distribution function f ( ~r, R ). A Cartesianexpansion up to second order gives f ( ~r, R ) = ρ ( ~r ) + X i,j =1 P ij ( ~r ) R ij + X i,j,k,l =1 Q ijkl ( ~r ) R ij R kl (42)with ρ ( ~r ) = 18 π Z SO(3) dΩ f ( ~r, R ) , (43) P ij ( ~r ) = 38 π Z SO(3) dΩ f ( ~r, R ) R ij , (44) Q ijkl ( ~r ) = 516 π Z SO(3) dΩ f ( ~r, R ) (cid:16) R ij R kl + R il R kj − δ ik δ jl (cid:17) . (45)In particular, for the distribution function for a systemof N biaxial particles with positions ~r n and orientations R n with n = 1 , . . . , N , given by f ( ~r, R ) = N X n =1 δ ( ~r − ~r n ) δ ( R − R n ) , (46) l c (3D) i j ...i l j l π R SO(3) dΩ f ( R )1 π R SO(3) dΩ f ( R ) R i j π R SO(3) dΩ f ( R )( ( R i j R i j + R i j R i j ) − δ i i δ j j )3 π R SO(3) dΩ f ( R )( ( R i j R i j R i j + R i j R i j R i j + R i j R i j R i j + R i j R i j R i j + R i j R i j R i j + R i j R i j R i j ) − ( R i j δ i i δ j j + R i j δ i i δ j j + R i j δ i i δ j j + R i j δ i i δ j j + R i j δ i i δ j j + R i j δ i i δ j j + R i j δ i i δ j j + R i j δ i i δ j j + R i j δ i i δ j j ))... ...TABLE III. Cartesian expansion coefficients c (3D) i j ...i l j l for different orders l [38]. we get the microscopic definitions ρ ( ~r ) = 18 π N X n =1 δ ( ~r − ~r n ) , (47) P ij ( ~r ) = 38 π N X n =1 R n,ij δ ( ~r − ~r n ) , (48) Q ijkl ( ~r ) = 516 π N X n =1 (cid:16) R n,ij R n,kl + R n,il R n,kj − δ ik δ jl (cid:17) δ ( ~r − ~r n ) (49)with R n,ij = ( R n ) ij denoting an element of the matrix R n .It is counterintuitive that, in the biaxial case, the first-and second-rank tensors giving polarization and nematicorder from the uniaxial case have to be replaced by asecond- and fourth-rank tensor, respectively. One canunderstand this in analogy to quantum mechanics: Foruniaxial particles, various orientational states are degen-erate, such that one only needs to distinguish three ratherthan nine degrees of freedom at first order. This is similaras in the hydrogen atom, where various angular momen-tum eigenstates have the same energy and do not have tobe distinguished in elementary treatments. In our case,the degeneracy is lifted by biaxiality.There is also an interesting analogy between active andquantum systems. Passive spherical particles have no dis-tinguishable orientations and thus no orientational order.However, active spherical particles have a preferred direc-tion (the axis of self-propulsion), such that the orienta-tional degeneracy is lifted. In consequence, systems ofactive spherical particles can exhibit orientational order.Staying in the comparison to the hydrogen atom, activ-ity has a similar role as the magnetic field in the Zeemaneffect, which lifts the degeneracy of the energy levels ofthe hydrogen atom [49]. IV. CONCLUSIONS
In this article, we have discussed angular and Cartesianuniaxial and biaxial expansions in two and three spatialdimensions. We have given an overview over the relevant functions and definitions, and derived formulas for con-versions between the expansion coefficients of both typesof expansions up to third order. These formulas allowto relate the results of analytical calculations, computersimulations, and experiments based on different types ofexpansions, which makes it possible to combine their ad-vantages by converting between them.Possible continuations of this work include the consid-eration of angular and Cartesian expansions in higher-dimensional spaces, in particular concerning the rela-tions between four-dimensional symmetric traceless ten-sors and the biaxial expansions discussed here. More-over, the formalism could be extended towards vectorand higher-order tensor spherical harmonics [17, 50–52]or quantum state multipoles [39, 53].
ACKNOWLEDGEMENTS
We thank Jonas L¨ubken for helpful discussions.R.W. is funded by the Deutsche Forschungsgemeinschaft(DFG, German Research Foundation) – WI 4170/3-1.
Appendix A: Relation of angular and Cartesianexpansion coefficients for the uniaxial case
The following equations allow to convert the expan-sion coefficients of an angular and Cartesian expansion,respectively, up to third order into each other. Higher-order relations can be derived using Eqs. (1), (2), (4),(7), (9), and (10). • Circular from Cartesian (2D): f = f (2D) , (A1) f ± = 12 ( f (2D)1 ∓ i f (2D)2 ) , (A2) f ± = 12 ( f (2D)11 ∓ i f (2D)12 ) , (A3) f ± = 12 ( f (2D)111 ∓ i f (2D)112 ) (A4) • Cartesian from circular (2D): f (2D) = f , (A5) f (2D)1 = f + f − , (A6) f (2D)2 = i( f − f − ) , (A7) f (2D)11 = f + f − , (A8) f (2D)12 = f (2D)21 = i( f − f − ) , (A9) f (2D)22 = − f (2D)11 , (A10) f (2D)111 = f + f − , (A11) f (2D)112 = f (2D)121 = f (2D)211 = i( f − f − ) , (A12) f (2D)122 = f (2D)212 = f (2D)221 = − f (2D)111 , (A13) f (2D)222 = − f (2D)211 (A14) • Spherical from Cartesian (3D): f = 2 √ πf (3D) , (A15) f = 2 r π f (3D)3 , (A16) f ± = r π ∓ f (3D)1 + i f (3D)2 ) , (A17) f = 2 r π f (3D)33 , (A18) f ± = 2 r π
15 ( ∓ f (3D)13 + i f (3D)23 ) , (A19) f ± = r π
15 ( f (3D)11 − f (3D)22 ∓ f (3D)12 ) , (A20) f = 2 r π f (3D)333 , (A21) f ± = r π ∓ f (3D)133 + i f (3D)233 ) , (A22) f ± = r π
35 ( f (3D)113 − f (3D)223 ∓ f (3D)123 ) , (A23) f ± = r π
35 ( ∓ f (3D)111 ± f (3D)122 + 3i f (3D)112 − i f (3D)222 )(A24) • Cartesian from spherical (3D): f (3D) = 12 √ π f , (A25) f (3D)1 = − r π ( f − f − ) , (A26) f (3D)2 = − i 12 r π ( f + f − ) , (A27) f (3D)3 = 12 r π f , (A28) f (3D)11 = − r π (2 f − √ f + f − )) , (A29) f (3D)12 = f (3D)21 = i 14 r π ( f − f − ) , (A30) f (3D)13 = f (3D)31 = − r π ( f − f − ) , (A31) f (3D)22 = − r π (2 f + √ f + f − )) , (A32) f (3D)23 = f (3D)32 = − i 14 r π ( f + f − ) , (A33) f (3D)33 = − f (3D)11 − f (3D)22 , (A34) f (3D)111 = 18 r π ( √ f − f − ) − √ f − f − )) , (A35) f (3D)112 = f (3D)121 = f (3D)211 = i 124 r π ( f + f − − √ f + f − )) , (A36) f (3D)113 = f (3D)131 = f (3D)311 = − r π ( √ f − √ f + f − )) , (A37) f (3D)122 = f (3D)212 = f (3D)221 = 124 r π ( f − f − + √ f − f − )) , (A38) f (3D)123 = f (3D)132 = f (3D)213 = f (3D)231 = f (3D)312 = f (3D)321 = i 14 r π ( f − f − ) , (A39) f (3D)133 = f (3D)313 = f (3D)331 = − f (3D)111 − f (3D)122 , (A40) f (3D)222 = i 18 r π ( √ f + f − ) + √ f + f − )) , (A41) f (3D)223 = f (3D)232 = f (3D)322 = − r π ( √ f + √ f + f − )) , (A42) f (3D)233 = f (3D)323 = f (3D)332 = − f (3D)211 − f (3D)222 , (A43) f (3D)333 = − f (3D)311 − f (3D)322 (A44) Appendix B: Relation of angular and Cartesianexpansion coefficients for the biaxial case
With the following equations, the expansion coeffi-cients of an angular and Cartesian expansion, respec-tively, up to third order can be converted into each other.Relations for higher orders can be derived using Eqs. (33),(34), (40), and (41). • Angular from Cartesian: c = c (3D) , (B1) c − − = 12 ( c (3D)11 + c (3D)22 + i( c (3D)12 − c (3D)21 )) , (B2) c − = 1 √ c (3D)13 − i c (3D)23 ) , (B3) c − = 12 ( − c (3D)11 + c (3D)22 + i( c (3D)12 + c (3D)21 )) , (B4) c − = 1 √ c (3D)31 + i c (3D)32 ) , (B5) c = c (3D)33 , (B6) c = − √ c (3D)31 − i c (3D)32 ) , (B7) c − = 12 ( − c (3D)11 + c (3D)22 − i( c (3D)12 + c (3D)21 )) , (B8) c = − √ c (3D)13 + i c (3D)23 ) , (B9) c = 12 ( c (3D)11 + c (3D)22 + i( c (3D)21 − c (3D)12 )) , (B10) c − − = 14 ( c (3D)1111 + 4 c (3D)1122 − c (3D)1212 − c (3D)2121 + c (3D)2222 + 2i( c (3D)1112 − c (3D)1121 + c (3D)1222 − c (3D)2122 )) , (B11) c − − = 12 ( c (3D)1113 + 2 c (3D)1223 − c (3D)2123 + i( − c (3D)1123 + c (3D)1213 − c (3D)2223 )) , (B12) c − = − r
32 ( c (3D)1111 + c (3D)1212 − c (3D)2121 − c (3D)2222 − c (3D)1121 + c (3D)1222 )) , (B13) c − = 12 ( − c (3D)1113 + 2 c (3D)1223 + c (3D)2123 + i(2 c (3D)1123 + c (3D)1213 − c (3D)2223 )) , (B14) c − = 14 ( c (3D)1111 − c (3D)1122 − c (3D)1212 − c (3D)2121 + c (3D)2222 − c (3D)1112 + c (3D)1121 − c (3D)1222 − c (3D)2122 )) , (B15) c − − = 12 ( c (3D)1131 − c (3D)1232 + 2 c (3D)2132 + i(2 c (3D)1132 − c (3D)2131 + c (3D)2232 )) , (B16) c − − = c (3D)1133 + c (3D)2233 + i( c (3D)1233 − c (3D)2133 ) , (B17) c − = − r
32 ( c (3D)1131 + c (3D)1232 − i( c (3D)2131 + c (3D)2232 )) , (B18) c − = − c (3D)1133 + c (3D)2233 + i( c (3D)1233 + c (3D)2133 ) , (B19) c − = 12 ( c (3D)1131 − c (3D)1232 − c (3D)2132 + i( − c (3D)1132 − c (3D)2131 + c (3D)2232 )) , (B20) c − = − r
32 ( c (3D)1111 − c (3D)1212 + c (3D)2121 − c (3D)2222 + 2i( c (3D)1112 + c (3D)2122 )) , (B21) c − = − r
32 ( c (3D)1113 + c (3D)2123 + i( c (3D)1213 + c (3D)2223 )) , (B22) c = 32 ( c (3D)1111 + c (3D)1212 + c (3D)2121 + c (3D)2222 ) , (B23) c = r
32 ( c (3D)1113 + c (3D)2123 − i( c (3D)1213 + c (3D)2223 )) , (B24) c = − r
32 ( c (3D)1111 − c (3D)1212 + c (3D)2121 − c (3D)2222 − c (3D)1112 + c (3D)2122 )) , (B25) c − = 12 ( − c (3D)1131 + c (3D)1232 + 2 c (3D)2132 + i( − c (3D)1132 − c (3D)2131 + c (3D)2232 )) , (B26) c − = − c (3D)1133 + c (3D)2233 − i( c (3D)1233 + c (3D)2133 ) , (B27) c = r
32 ( c (3D)1131 + c (3D)1232 + i( c (3D)2131 + c (3D)2232 )) , (B28) c = c (3D)1133 + c (3D)2233 + i( c (3D)2133 − c (3D)1233 ) , (B29) c = 12 ( − c (3D)1131 + c (3D)1232 − c (3D)2132 + i(2 c (3D)1132 − c (3D)2131 + c (3D)2232 )) , (B30) c − = 14 ( c (3D)1111 − c (3D)1122 − c (3D)1212 − c (3D)2121 + c (3D)2222 + 2i( c (3D)1112 + c (3D)1121 − c (3D)1222 − c (3D)2122 )) , (B31) c − = 12 ( c (3D)1113 − c (3D)1223 − c (3D)2123 + i(2 c (3D)1123 + c (3D)1213 − c (3D)2223 )) , (B32) c = − r
32 ( c (3D)1111 + c (3D)1212 − c (3D)2121 − c (3D)2222 + 2i( c (3D)1121 + c (3D)1222 )) , (B33) c = 12 ( − c (3D)1113 − c (3D)1223 + c (3D)2123 + i( − c (3D)1123 + c (3D)1213 − c (3D)2223 )) , (B34) c = 14 ( c (3D)1111 + 4 c (3D)1122 − c (3D)1212 − c (3D)2121 + c (3D)2222 − c (3D)1112 − c (3D)1121 + c (3D)1222 − c (3D)2122 )) , (B35) c − − = 18 ( c (3D)111111 + 9 c (3D)111122 − c (3D)111212 − c (3D)112121 + 9 c (3D)112222 − c (3D)121222 − c (3D)212122 + c (3D)222222 + i(3 c (3D)111112 − c (3D)111121 + 9 c (3D)111222 − c (3D)112122 − c (3D)121212 + 3 c (3D)122222 + c (3D)212121 − c (3D)212222 )) , (B36) c − − = 14 r
32 ( c (3D)111113 + 6 c (3D)111223 − c (3D)112123 − c (3D)121213 + 3 c (3D)122223 − c (3D)212223 + i( − c (3D)111123 + 2 c (3D)111213 − c (3D)112223 + 3 c (3D)121223 + c (3D)212123 − c (3D)222223 )) , (B37) c − − = − √ c (3D)111111 + 3 c (3D)111122 + c (3D)111212 − c (3D)112121 − c (3D)112222 + 3 c (3D)121222 − c (3D)212122 − c (3D)222222 + i( c (3D)111112 − c (3D)111121 − c (3D)111222 − c (3D)112122 + c (3D)121212 − c (3D)122222 + c (3D)212121 + c (3D)212222 )) , (B38) c − = − √ c (3D)111113 − c (3D)112123 + c (3D)121213 − c (3D)122223 + i( − c (3D)111123 − c (3D)121223 + c (3D)212123 + c (3D)222223 )) , (B39)0 c − = 18 √ c (3D)111111 − c (3D)111122 + c (3D)111212 − c (3D)112121 − c (3D)112222 − c (3D)121222 + c (3D)212122 + c (3D)222222 + i( − c (3D)111112 − c (3D)111121 − c (3D)111222 + 3 c (3D)112122 − c (3D)121212 + 3 c (3D)122222 + c (3D)212121 + c (3D)212222 )) , (B40) c − = 14 r
32 ( c (3D)111113 − c (3D)111223 − c (3D)112123 − c (3D)121213 + 3 c (3D)122223 + 2 c (3D)212223 + i( − c (3D)111123 − c (3D)111213 + 6 c (3D)112223 + 3 c (3D)121223 + c (3D)212123 − c (3D)222223 )) , (B41) c − = 18 ( − c (3D)111111 + 9 c (3D)111122 + 3 c (3D)111212 + 3 c (3D)112121 − c (3D)112222 − c (3D)121222 − c (3D)212122 + c (3D)222222 + i(3 c (3D)111112 + 3 c (3D)111121 − c (3D)111222 − c (3D)112122 − c (3D)121212 + 3 c (3D)122222 − c (3D)212121 + 3 c (3D)212222 )) , (B42) c − − = 14 r
32 ( c (3D)111131 − c (3D)111232 + 6 c (3D)112132 − c (3D)122232 − c (3D)212131 + 3 c (3D)212232 + i(3 c (3D)111132 − c (3D)112131 + 6 c (3D)112232 − c (3D)121232 − c (3D)212132 + c (3D)222232 )) , (B43) c − − = 34 ( c (3D)111133 + 4 c (3D)112233 − c (3D)121233 − c (3D)212133 + c (3D)222233 + 2i( c (3D)111233 − c (3D)112133 + c (3D)122233 − c (3D)212233 )) , (B44) c − − = − r
52 ( c (3D)111131 + c (3D)111232 + 2 c (3D)112132 + 2 c (3D)122232 − c (3D)212131 − c (3D)212232 + i( c (3D)111132 − c (3D)112131 − c (3D)112232 + c (3D)121232 − c (3D)212132 − c (3D)222232 )) , (B45) c − = − r
152 ( c (3D)111133 + c (3D)121233 − c (3D)212133 − c (3D)222233 − c (3D)112133 + c (3D)122233 )) , (B46) c − = 34 r
52 ( c (3D)111131 + c (3D)111232 − c (3D)112132 − c (3D)122232 − c (3D)212131 − c (3D)212232 − i( c (3D)111132 + 2 c (3D)112131 + 2 c (3D)112232 + c (3D)121232 − c (3D)212132 − c (3D)222232 )) , (B47) c − = 34 ( c (3D)111133 − c (3D)112233 − c (3D)121233 − c (3D)212133 + c (3D)222233 − c (3D)111233 + c (3D)112133 − c (3D)122233 − c (3D)212233 )) , (B48) c − = − r
32 ( c (3D)111131 − c (3D)111232 − c (3D)112132 + 2 c (3D)122232 − c (3D)212131 + 3 c (3D)212232 + i( − c (3D)111132 − c (3D)112131 + 6 c (3D)112232 + c (3D)121232 + 3 c (3D)212132 − c (3D)222232 )) , (B49) c − − = − √ c (3D)111111 + 3 c (3D)111122 − c (3D)111212 + c (3D)112121 − c (3D)112222 − c (3D)121222 + 3 c (3D)212122 − c (3D)222222 + i(3 c (3D)111112 − c (3D)111121 + 3 c (3D)111222 + 3 c (3D)112122 − c (3D)121212 − c (3D)122222 − c (3D)212121 + 3 c (3D)212222 )) , (B50) c − − = − r
52 ( c (3D)111113 + 2 c (3D)111223 + c (3D)112123 − c (3D)121213 − c (3D)122223 + 2 c (3D)212223 + i( − c (3D)111123 + 2 c (3D)111213 + 2 c (3D)112223 + c (3D)121223 − c (3D)212123 + c (3D)222223 )) , (B51) c − − = 158 ( c (3D)111111 + c (3D)111122 + c (3D)111212 + c (3D)112121 + c (3D)112222 + c (3D)121222 + c (3D)212122 + c (3D)222222 + i( c (3D)111112 − c (3D)111121 − c (3D)111222 + c (3D)112122 + c (3D)121212 + c (3D)122222 − c (3D)212121 − c (3D)212222 )) , (B52) c − = 54 √ c (3D)111113 + c (3D)112123 + c (3D)121213 + c (3D)122223 − i( c (3D)111123 + c (3D)121223 + c (3D)212123 + c (3D)222223 )) , (B53) c − = −
158 ( c (3D)111111 − c (3D)111122 + c (3D)111212 + c (3D)112121 + c (3D)112222 − c (3D)121222 − c (3D)212122 − c (3D)222222 − i( c (3D)111112 + c (3D)111121 + c (3D)111222 + c (3D)112122 + c (3D)121212 + c (3D)122222 + c (3D)212121 + c (3D)212222 )) , (B54) c − = − r
52 ( c (3D)111113 − c (3D)111223 + c (3D)112123 − c (3D)121213 − c (3D)122223 − c (3D)212223 − i( c (3D)111123 + 2 c (3D)111213 + 2 c (3D)112223 − c (3D)121223 + c (3D)212123 − c (3D)222223 )) , (B55) c − = 18 √ c (3D)111111 − c (3D)111122 − c (3D)111212 + c (3D)112121 − c (3D)112222 + c (3D)121222 − c (3D)212122 + c (3D)222222 + i( − c (3D)111112 − c (3D)111121 + 3 c (3D)111222 − c (3D)112122 + c (3D)121212 + c (3D)122222 − c (3D)212121 + 3 c (3D)212222 )) , (B56) c − = − √ c (3D)111131 − c (3D)111232 + c (3D)212131 − c (3D)212232 + i(3 c (3D)111132 − c (3D)121232 + 3 c (3D)212132 − c (3D)222232 )) , (B57) c − = − r
152 ( c (3D)111133 − c (3D)121233 + c (3D)212133 − c (3D)222233 + 2i( c (3D)111233 + c (3D)212233 )) , (B58)1 c − = 54 √ c (3D)111131 + c (3D)111232 + c (3D)212131 + c (3D)212232 + i( c (3D)111132 + c (3D)121232 + c (3D)212132 + c (3D)222232 )) , (B59) c = 52 ( c (3D)111133 + c (3D)121233 + c (3D)212133 + c (3D)222233 ) , (B60) c = − √ c (3D)111131 + c (3D)111232 + c (3D)212131 + c (3D)212232 − i( c (3D)111132 + c (3D)121232 + c (3D)212132 + c (3D)222232 )) , (B61) c = − r
152 ( c (3D)111133 − c (3D)121233 + c (3D)212133 − c (3D)222233 − c (3D)111233 + c (3D)212233 )) , (B62) c = 14 √ c (3D)111131 − c (3D)111232 + c (3D)212131 − c (3D)212232 + i( − c (3D)111132 + c (3D)121232 − c (3D)212132 + c (3D)222232 )) , (B63) c − = 18 √ c (3D)111111 − c (3D)111122 − c (3D)111212 + c (3D)112121 − c (3D)112222 + c (3D)121222 − c (3D)212122 + c (3D)222222 + i(3 c (3D)111112 + c (3D)111121 − c (3D)111222 + 3 c (3D)112122 − c (3D)121212 − c (3D)122222 + c (3D)212121 − c (3D)212222 )) , (B64) c − = 34 r
52 ( c (3D)111113 − c (3D)111223 + c (3D)112123 − c (3D)121213 − c (3D)122223 − c (3D)212223 + i( c (3D)111123 + 2 c (3D)111213 + 2 c (3D)112223 − c (3D)121223 + c (3D)212123 − c (3D)222223 )) , (B65) c − = −
158 ( c (3D)111111 − c (3D)111122 + c (3D)111212 + c (3D)112121 + c (3D)112222 − c (3D)121222 − c (3D)212122 − c (3D)222222 + i( c (3D)111112 + c (3D)111121 + c (3D)111222 + c (3D)112122 + c (3D)121212 + c (3D)122222 + c (3D)212121 + c (3D)212222 )) , (B66) c = − √ c (3D)111113 + c (3D)112123 + c (3D)121213 + c (3D)122223 + i( c (3D)111123 + c (3D)121223 + c (3D)212123 + c (3D)222223 )) , (B67) c = 158 ( c (3D)111111 + c (3D)111122 + c (3D)111212 + c (3D)112121 + c (3D)112222 + c (3D)121222 + c (3D)212122 + c (3D)222222 + i( − c (3D)111112 + c (3D)111121 + c (3D)111222 − c (3D)112122 − c (3D)121212 − c (3D)122222 + c (3D)212121 + c (3D)212222 )) , (B68) c = 34 r
52 ( c (3D)111113 + 2 c (3D)111223 + c (3D)112123 − c (3D)121213 − c (3D)122223 + 2 c (3D)212223 + i( c (3D)111123 − c (3D)111213 − c (3D)112223 − c (3D)121223 + c (3D)212123 − c (3D)222223 )) , (B69) c = − √ c (3D)111111 + 3 c (3D)111122 − c (3D)111212 + c (3D)112121 − c (3D)112222 − c (3D)121222 + 3 c (3D)212122 − c (3D)222222 + i( − c (3D)111112 + c (3D)111121 − c (3D)111222 − c (3D)112122 + c (3D)121212 + c (3D)122222 + c (3D)212121 − c (3D)212222 )) , (B70) c − = 14 r
32 ( c (3D)111131 − c (3D)111232 − c (3D)112132 + 2 c (3D)122232 − c (3D)212131 + 3 c (3D)212232 + i(3 c (3D)111132 + 2 c (3D)112131 − c (3D)112232 − c (3D)121232 − c (3D)212132 + c (3D)222232 )) , (B71) c − = 34 ( c (3D)111133 − c (3D)112233 − c (3D)121233 − c (3D)212133 + c (3D)222233 + 2i( c (3D)111233 + c (3D)112133 − c (3D)122233 − c (3D)212233 )) , (B72) c − = − r
52 ( c (3D)111131 + c (3D)111232 − c (3D)112132 − c (3D)122232 − c (3D)212131 − c (3D)212232 + i( c (3D)111132 + 2 c (3D)112131 + 2 c (3D)112232 + c (3D)121232 − c (3D)212132 − c (3D)222232 )) , (B73) c = − r
152 ( c (3D)111133 + c (3D)121233 − c (3D)212133 − c (3D)222233 + 2i( c (3D)112133 + c (3D)122233 )) , (B74) c = 34 r
52 ( c (3D)111131 + c (3D)111232 + 2 c (3D)112132 + 2 c (3D)122232 − c (3D)212131 − c (3D)212232 + i( − c (3D)111132 + 2 c (3D)112131 + 2 c (3D)112232 − c (3D)121232 + c (3D)212132 + c (3D)222232 )) , (B75) c = 34 ( c (3D)111133 + 4 c (3D)112233 − c (3D)121233 − c (3D)212133 + c (3D)222233 − c (3D)111233 − c (3D)112133 + c (3D)122233 − c (3D)212233 )) , (B76) c = − r
32 ( c (3D)111131 − c (3D)111232 + 6 c (3D)112132 − c (3D)122232 − c (3D)212131 + 3 c (3D)212232 + i( − c (3D)111132 + 2 c (3D)112131 − c (3D)112232 + c (3D)121232 + 3 c (3D)212132 − c (3D)222232 )) , (B77) c − = 18 ( − c (3D)111111 + 9 c (3D)111122 + 3 c (3D)111212 + 3 c (3D)112121 − c (3D)112222 − c (3D)121222 − c (3D)212122 + c (3D)222222 + i( − c (3D)111112 − c (3D)111121 + 9 c (3D)111222 + 9 c (3D)112122 + c (3D)121212 − c (3D)122222 + c (3D)212121 − c (3D)212222 )) , (B78)2 c − = − r
32 ( c (3D)111113 − c (3D)111223 − c (3D)112123 − c (3D)121213 + 3 c (3D)122223 + 2 c (3D)212223 + i(3 c (3D)111123 + 2 c (3D)111213 − c (3D)112223 − c (3D)121223 − c (3D)212123 + c (3D)222223 )) , (B79) c − = 18 √ c (3D)111111 − c (3D)111122 + c (3D)111212 − c (3D)112121 − c (3D)112222 − c (3D)121222 + c (3D)212122 + c (3D)222222 + i( c (3D)111112 + 3 c (3D)111121 + 3 c (3D)111222 − c (3D)112122 + c (3D)121212 − c (3D)122222 − c (3D)212121 − c (3D)212222 )) , (B80) c = 14 √ c (3D)111113 − c (3D)112123 + c (3D)121213 − c (3D)122223 + i(3 c (3D)111123 + 3 c (3D)121223 − c (3D)212123 − c (3D)222223 )) , (B81) c = − √ c (3D)111111 + 3 c (3D)111122 + c (3D)111212 − c (3D)112121 − c (3D)112222 + 3 c (3D)121222 − c (3D)212122 − c (3D)222222 − i( c (3D)111112 − c (3D)111121 − c (3D)111222 − c (3D)112122 + c (3D)121212 − c (3D)122222 + c (3D)212121 + c (3D)212222 )) , (B82) c = − r
32 ( c (3D)111113 + 6 c (3D)111223 − c (3D)112123 − c (3D)121213 + 3 c (3D)122223 − c (3D)212223 + i(3 c (3D)111123 − c (3D)111213 + 6 c (3D)112223 − c (3D)121223 − c (3D)212123 + c (3D)222223 )) , (B83) c = 18 ( c (3D)111111 + 9 c (3D)111122 − c (3D)111212 − c (3D)112121 + 9 c (3D)112222 − c (3D)121222 − c (3D)212122 + c (3D)222222 + i( − c (3D)111112 + 3 c (3D)111121 − c (3D)111222 + 9 c (3D)112122 + c (3D)121212 − c (3D)122222 − c (3D)212121 + 3 c (3D)212222 )) (B84) • Cartesian from angular: c (3D) = c , (B85) c (3D)11 = 12 ( c − c − − c − + c − − ) , (B86) c (3D)12 = i2 ( c + c − − c − − c − − ) , (B87) c (3D)13 = − √ c − c − ) , (B88) c (3D)21 = − i2 ( c − c − + c − − c − − ) , (B89) c (3D)22 = 12 ( c + c − + c − + c − − ) , (B90) c (3D)23 = i √ c + c − ) , (B91) c (3D)31 = − √ c − c − ) , (B92) c (3D)32 = − i √ c + c − ) , (B93) c (3D)33 = c , (B94) c (3D)1111 = 112 (2 c − √ c − √ c − − √ c + 3 c + 3 c − − √ c − + 3 c − + 3 c − − ) , (B95) c (3D)1112 = − i12 ( √ c − √ c − − c + 3 c − − c − + 3 c − − ) , (B96) c (3D)1113 = 112 ( √ c − √ c − − c + 3 c − − c − + 3 c − − ) , (B97) c (3D)1121 = i12 ( √ c − c − c − − √ c − + 3 c − + 3 c − − ) , (B98) c (3D)1122 = 14 ( c − c − − c − + c − − ) , (B99) c (3D)1123 = i4 ( c − c − − c − + c − − ) , (B100) c (3D)1131 = 112 ( √ c − c − c − − √ c − + 3 c − + 3 c − − ) , (B101) c (3D)1132 = − i4 ( c − c − − c − + c − − ) , (B102) c (3D)1133 = 14 ( c − c − − c − + c − − ) , (B103) c (3D)1212 = 112 (2 c + √ c + √ c − − √ c − c − c − − √ c − − c − − c − − ) , (B104) c (3D)1213 = i12 ( √ c + √ c − − c − c − − c − − c − − ) , (B105) c (3D)1222 = i12 ( √ c + 3 c + 3 c − − √ c − − c − − c − − ) , (B106) c (3D)1223 = 14 ( − c − c − + c − + c − − ) , (B107) c (3D)1232 = 112 ( √ c + 3 c + 3 c − − √ c − − c − − c − − ) , (B108) c (3D)1233 = i4 ( c + c − − c − − c − − ) , (B109) c (3D)2121 = 112 (2 c − √ c − √ c − + √ c − c − c − + √ c − − c − − c − − ) , (B110) c (3D)2122 = − i12 ( √ c − √ c − + 3 c − c − + 3 c − − c − − ) , (B111)3 c (3D)2123 = 112 ( √ c − √ c − + 3 c − c − + 3 c − − c − − ) , (B112) c (3D)2131 = − i12 ( √ c − c − c − + √ c − − c − − c − − ) , (B113) c (3D)2132 = 14 ( − c + c − − c − + c − − ) , (B114) c (3D)2133 = − i4 ( c − c − + c − − c − − ) , (B115) c (3D)2222 = 112 (2 c + √ c + √ c − + √ c + 3 c + 3 c − + √ c − + 3 c − + 3 c − − ) , (B116) c (3D)2223 = i12 ( √ c + √ c − + 3 c + 3 c − + 3 c − + 3 c − − ) , (B117) c (3D)2232 = − i12 ( √ c + 3 c + 3 c − + √ c − + 3 c − + 3 c − − ) , (B118) c (3D)2233 = 14 ( c + c − + c − + c − − ) , (B119) c (3D)111111 = 140 (3 c − √ c − c − + √ c − − √ c + 5 c + √ c − − c − − c − + √ c − + 3 c − − − √ c − − + √ c − − c − − √ c − − + 5 c − − ) , (B120) c (3D)111112 = i120 (3 c − √ c + 3 c − − √ c − − √ c + 15 c − √ c − + 15 c − − c − + 3 √ c − − c − − + 3 √ c − − + √ c − − c − + √ c − − − c − − ) , (B121) c (3D)111113 = 1120 ( − √ c + 3 √ c + 3 √ c − + 6 √ c − √ c − √ c − + 6 √ c − − √ c − − √ c − − − √ c − + 5 √ c − + 5 √ c − − ) , (B122) c (3D)111121 = − i120 (3 c − √ c − c − + √ c − − √ c + 15 c + 3 √ c − − c − + 3 c − − √ c − − c − − + √ c − − − √ c − + 15 c − + 3 √ c − − − c − − ) , (B123) c (3D)111122 = 1120 ( c − √ c + c − − √ c − − √ c + 15 c − √ c − + 15 c − + c − − √ c − + c − − − √ c − − − √ c − + 15 c − − √ c − − + 15 c − − ) , (B124) c (3D)111123 = i120 (2 √ c − √ c − √ c − − √ c + 5 √ c + 5 √ c − + 2 √ c − − √ c − − √ c − − − √ c − + 5 √ c − + 5 √ c − − ) , (B125) c (3D)111131 = 1120 ( − √ c + 6 √ c + 6 √ c − − √ c − + 3 √ c − √ c − √ c − + 5 √ c − + 3 √ c − − √ c − − √ c − − + 5 √ c − − ) , (B126) c (3D)111132 = − i120 (2 √ c − √ c + 2 √ c − − √ c − − √ c + 5 √ c − √ c − + 5 √ c − − √ c − + 5 √ c − − √ c − − + 5 √ c − − ) , (B127) c (3D)111133 = 160 (6 c − √ c − √ c − − √ c + 5 c + 5 c − − √ c − + 5 c − + 5 c − − ) , (B128) c (3D)111212 = 1120 (3 c + 3 √ c − c − − √ c − − √ c − c + √ c − + 15 c − − c − − √ c − + 3 c − − + 3 √ c − − + √ c − + 15 c − − √ c − − − c − − ) , (B129) c (3D)111213 = i60 √ √ c − √ c − − √ c + 5 √ c − − √ c − + 3 √ c − − + 5 √ c − − √ c − − ) , (B130) c (3D)111222 = − i120 ( c + √ c − c − − √ c − − √ c − c + √ c − + 15 c − + c − + √ c − − c − − − √ c − − − √ c − − c − + √ c − − + 15 c − − ) , (B131)4 c (3D)111223 = 160 √ √ c − √ c − − √ c + 5 √ c − + √ c − − √ c − − − √ c − + 5 √ c − − ) , (B132) c (3D)111232 = 1120 ( − √ c − √ c + 2 √ c − + 6 √ c − + √ c + 5 √ c − √ c − − √ c − + √ c − + 5 √ c − − √ c − − − √ c − − ) , (B133) c (3D)111233 = − i60 ( √ c − √ c − − c + 5 c − − c − + 5 c − − ) , (B134) c (3D)112121 = 1120 (3 c − √ c − c − + √ c − + 3 √ c − c − √ c − + 15 c − − c − + √ c − + 3 c − − − √ c − − − √ c − + 15 c − + 3 √ c − − − c − − ) , (B135) c (3D)112122 = i120 ( c − √ c + c − − √ c − + √ c − c + √ c − − c − − c − + √ c − − c − − + √ c − − − √ c − + 15 c − − √ c − − + 15 c − − ) , (B136) c (3D)112123 = 1120 ( − √ c + √ c + √ c − − √ c + 5 √ c + 5 √ c − + 2 √ c − − √ c − − √ c − − + 6 √ c − − √ c − − √ c − − ) , (B137) c (3D)112131 = − i60 √ √ c − √ c − √ c − + 5 √ c − − √ c − + 5 √ c − + 3 √ c − − − √ c − − ) , (B138) c (3D)112132 = 160 √ √ c − √ c + √ c − − √ c − − √ c − + 5 √ c − − √ c − − + 5 √ c − − ) , (B139) c (3D)112133 = i60 ( √ c − c − c − − √ c − + 5 c − + 5 c − − ) , (B140) c (3D)112222 = 1120 ( c + √ c − c − − √ c − + √ c + 15 c − √ c − − c − − c − − √ c − + c − − + √ c − − − √ c − − c − + √ c − − + 15 c − − ) , (B141) c (3D)112223 = i60 √ √ c − √ c − + 5 √ c − √ c − − √ c − + √ c − − − √ c − + 5 √ c − − ) , (B142) c (3D)112232 = − i60 √ √ c + 5 √ c − √ c − − √ c − − √ c − − √ c − + √ c − − + 5 √ c − − ) , (B143) c (3D)112233 = 112 ( c − c − − c − + c − − ) , (B144) c (3D)121212 = i40 (3 c + √ c + 3 c − + √ c − − √ c − c − √ c − − c − − c − − √ c − − c − − − √ c − − + √ c − + 5 c − + √ c − − + 5 c − − ) , (B145) c (3D)121213 = 1120 ( − √ c − √ c − √ c − + 6 √ c + 5 √ c + 5 √ c − + 6 √ c − + 3 √ c − + 3 √ c − − − √ c − − √ c − − √ c − − ) , (B146) c (3D)121222 = 1120 (3 c + √ c + 3 c − + √ c − − √ c − c − √ c − − c − + 3 c − + √ c − + 3 c − − + √ c − − − √ c − − c − − √ c − − − c − − ) , (B147) c (3D)121223 = i120 (2 √ c + √ c + √ c − − √ c − √ c − √ c − + 2 √ c − + √ c − + √ c − − − √ c − − √ c − − √ c − − ) , (B148) c (3D)121232 = − i120 (6 √ c + 6 √ c + 6 √ c − + 6 √ c − − √ c − √ c − √ c − − √ c − − √ c − − √ c − − √ c − − − √ c − − ) , (B149)5 c (3D)121233 = 160 (6 c + √ c + √ c − − √ c − c − c − − √ c − − c − − c − − ) , (B150) c (3D)122222 = i120 (3 c + √ c + 3 c − + √ c − + 3 √ c + 15 c + 3 √ c − + 15 c − − c − − √ c − − c − − − √ c − − − √ c − − c − − √ c − − − c − − ) , (B151) c (3D)122223 = 1120 ( − √ c − √ c − √ c − − √ c − √ c − √ c − + 2 √ c − + √ c − + √ c − − + 6 √ c − + 5 √ c − + 5 √ c − − ) , (B152) c (3D)122232 = 160 √ √ c + 5 √ c + 3 √ c − + 5 √ c − − √ c − − √ c − − √ c − − − √ c − − ) , (B153) c (3D)122233 = i60 ( √ c + 5 c + 5 c − − √ c − − c − − c − − ) , (B154) c (3D)212121 = − i40 (3 c − √ c − c − + √ c − + √ c − c − √ c − + 5 c − + 3 c − − √ c − − c − − + √ c − − + √ c − − c − − √ c − − + 5 c − − ) , (B155) c (3D)212122 = 1120 (3 c − √ c + 3 c − − √ c − + √ c − c + √ c − − c − + 3 c − − √ c − + 3 c − − − √ c − − + √ c − − c − + √ c − − − c − − ) , (B156) c (3D)212123 = i120 (6 √ c − √ c − √ c − + 6 √ c − √ c − √ c − + 6 √ c − − √ c − − √ c − − + 6 √ c − − √ c − − √ c − − ) , (B157) c (3D)212131 = 1120 ( − √ c + 6 √ c + 6 √ c − − √ c − − √ c + 5 √ c + 3 √ c − − √ c − − √ c − + 5 √ c − + 3 √ c − − − √ c − − ) , (B158) c (3D)212132 = − i120 (2 √ c − √ c + 2 √ c − − √ c − + √ c − √ c + √ c − − √ c − + √ c − − √ c − + √ c − − − √ c − − ) , (B159) c (3D)212133 = 160 (6 c − √ c − √ c − + √ c − c − c − + √ c − − c − − c − − ) , (B160) c (3D)212222 = − i120 (3 c + 3 √ c − c − − √ c − + √ c + 15 c − √ c − − c − + 3 c − + 3 √ c − − c − − − √ c − − + √ c − + 15 c − − √ c − − − c − − ) , (B161) c (3D)212223 = 160 √ √ c − √ c − + 5 √ c − √ c − + 3 √ c − − √ c − − + 5 √ c − − √ c − − ) , (B162) c (3D)212232 = 1120 ( − √ c − √ c + 2 √ c − + 6 √ c − − √ c − √ c + √ c − + 5 √ c − − √ c − − √ c − + √ c − − + 5 √ c − − ) , (B163) c (3D)212233 = − i60 ( √ c − √ c − + 5 c − c − + 5 c − − c − − ) , (B164) c (3D)222222 = 140 (3 c + √ c + 3 c − + √ c − + √ c + 5 c + √ c − + 5 c − + 3 c − + √ c − + 3 c − − + √ c − − + √ c − + 5 c − + √ c − − + 5 c − − ) , (B165) c (3D)222223 = i120 (6 √ c + 3 √ c + 3 √ c − + 6 √ c + 5 √ c + 5 √ c − + 6 √ c − + 3 √ c − + 3 √ c − − + 6 √ c − + 5 √ c − + 5 √ c − − ) , (B166) c (3D)222232 = − i120 (6 √ c + 6 √ c + 6 √ c − + 6 √ c − + 3 √ c + 5 √ c + 3 √ c − + 5 √ c − + 3 √ c − + 5 √ c − + 3 √ c − − + 5 √ c − − ) , (B167)6 c (3D)222233 = 160 (6 c + √ c + √ c − + √ c + 5 c + 5 c − + √ c − + 5 c − + 5 c − − ) (B168)The expansion coefficients that appear not explicitly inthe previous table follow from their symmetry properties c (3D) i j i j = c (3D) i j i j = c (3D) i j i j , (B169) c (3D) i j i j i j = c (3D) i j i j i j = c (3D) i j i j i j = c (3D) i j i j i j = c (3D) i j i j i j = c (3D) i j i j i j = c (3D) i j i j i j (B170)and tracelessness c (3D)3 j j = − c (3D)1 j j − c (3D)2 j j , (B171) c (3D) i i = − c (3D) i i − c (3D) i i , (B172) c (3D)3 j j i j = − c (3D)1 j j i j − c (3D)2 j j i j , (B173) c (3D)3 j i j j = − c (3D)1 j i j j − c (3D)2 j i j j , (B174) c (3D) i j j j = − c (3D) i j j j − c (3D) i j j j , (B175) c (3D) i i i j = − c (3D) i i i j − c (3D) i i i j , (B176) c (3D) i i j i = − c (3D) i i j i − c (3D) i i j i , (B177) c (3D) i j i i = − c (3D) i j i i − c (3D) i j i i . (B178) Appendix C: Elements of Wigner D-matrices
Finally, we list all elements of the Wigner D-matrices(31) with l ≤ D = 1 , (C1) D − − = e i φ
12 (1 + cos( θ )) e i χ , (C2) D − = e i φ sin( θ ) √ , (C3) D − = e i φ
12 (1 − cos( θ )) e − i χ , (C4) D − = − sin( θ ) √ e i χ , (C5) D = cos( θ ) , (C6) D = sin( θ ) √ e − i χ , (C7) D − = e − i φ
12 (1 − cos( θ )) e i χ , (C8) D = − e − i φ sin( θ ) √ , (C9) D = e − i φ
12 (1 + cos( θ )) e − i χ , (C10) D − − = e φ
14 (1 + cos( θ )) e χ , (C11) D − − = e φ
12 sin( θ )(1 + cos( θ )) e i χ , (C12) D − = e φ r
32 sin( θ ) , (C13) D − = − e φ
12 sin( θ )(cos( θ ) − e − i χ , (C14) D − = e φ
14 (cos( θ ) − e − χ , (C15) D − − = − e i φ
12 sin( θ )(1 + cos( θ )) e χ , (C16) D − − = e i φ
12 (1 + cos( θ ))(2 cos( θ ) − e i χ , (C17) D − = e i φ r
32 sin( θ ) cos( θ ) , (C18) D − = e i φ
12 (1 − cos( θ ))(1 + 2 cos( θ )) e − i χ , (C19) D − = − e i φ
12 sin( θ )(cos( θ ) − e − χ , (C20) D − = 12 r
32 sin( θ ) e χ , (C21) D − = − r
32 sin( θ ) cos( θ ) e i χ , (C22) D = 12 (3 cos( θ ) − , (C23) D = r
32 sin( θ ) cos( θ ) e − i χ , (C24) D = 12 r
32 sin( θ ) e − χ , (C25) D − = e − i φ
12 sin( θ )(cos( θ ) − e χ , (C26) D − = e − i φ
12 (1 − cos( θ ))(1 + 2 cos( θ )) e i χ , (C27) D = − e − i φ r
32 sin( θ ) cos( θ ) , (C28) D = e − i φ
12 (1 + cos( θ ))(2 cos( θ ) − e − i χ , (C29) D = e − i φ
12 sin( θ )(1 + cos( θ )) e − χ , (C30) D − = e − φ
14 (cos( θ ) − e χ , (C31) D − = e − φ
12 sin( θ )(cos( θ ) − e i χ , (C32) D = e − φ r
32 sin( θ ) , (C33) D = − e − φ
12 sin( θ )(1 + cos( θ )) e − i χ , (C34) D = e − φ
14 (1 + cos( θ )) e − χ , (C35) D − − = e φ
18 (1 + cos( θ )) e χ , (C36) D − − = e φ r
32 sin( θ )(1 + cos( θ )) e χ , (C37)7 D − − = e φ √
15 sin( θ ) (1 + cos( θ )) e i χ , (C38) D − = e φ √ θ ) , (C39) D − = − e φ √
15 sin( θ ) (cos( θ ) − e − i χ , (C40) D − = e φ r
32 sin( θ )(cos( θ ) − e − χ , (C41) D − = − e φ
18 (cos( θ ) − e − χ , (C42) D − − = − e φ r
32 sin( θ )(1 + cos( θ )) e χ , (C43) D − − = e φ
14 (1 + cos( θ )) (3 cos( θ ) − e χ , (C44) D − − = e φ r
52 sin( θ )(1 + cos( θ ))(3 cos( θ ) − e i χ , (C45) D − = e φ r
152 sin( θ ) cos( θ ) , (C46) D − = e φ r
52 sin( θ )(1 − cos( θ ))(1 + 3 cos( θ )) e − i χ , (C47) D − = e φ
14 (cos( θ ) − (2 + 3 cos( θ )) e − χ , (C48) D − = e φ r
32 sin( θ )(cos( θ ) − e − χ , (C49) D − − = e i φ √
15 sin( θ ) (1 + cos( θ )) e χ , (C50) D − − = − e i φ r
52 sin( θ )(1 + cos( θ ))(3 cos( θ ) − e χ , (C51) D − − = e i φ
18 (1 + cos( θ ))(15 cos( θ ) −
10 cos( θ ) − e i χ , (C52) D − = e i φ √ θ )(5 cos( θ ) − , (C53) D − = e i φ
18 (1 − cos( θ ))(15 cos( θ ) + 10 cos( θ ) − e − i χ , (C54) D − = e i φ r
52 sin( θ )(1 − cos( θ ))(1 + 3 cos( θ )) e − χ , (C55) D − = − e i φ √
15 sin( θ ) (cos( θ ) − e − χ , (C56) D − = − √ θ ) e χ , (C57) D − = 12 r
152 sin( θ ) cos( θ ) e χ , (C58) D − = 14 √ θ )(1 − θ ) ) e i χ , (C59) D = 12 cos( θ )(5 cos( θ ) − , (C60) D = 14 √ θ )(5 cos( θ ) − e − i χ , (C61) D = 12 r
152 sin( θ ) cos( θ ) e − χ , (C62) D = 14 √ θ ) e − χ , (C63) D − = − e − i φ √
15 sin( θ ) (cos( θ ) − e χ , (C64) D − = e − i φ r
52 sin( θ )(cos( θ ) − θ )) e χ , (C65) D − = e − i φ
18 (1 − cos( θ ))(15 cos( θ ) + 10 cos( θ ) − e i χ , (C66) D = e − i φ √ θ )(1 − θ ) ) , (C67) D = e − i φ
18 (1 + cos( θ ))(15 cos( θ ) −
10 cos( θ ) − e − i χ , (C68) D = e − i φ r
52 sin( θ )(1 + cos( θ ))(3 cos( θ ) − e − χ , (C69) D = e − i φ √
15 sin( θ ) (1 + cos( θ )) e − χ , (C70) D − = − e − φ r
32 sin( θ )(cos( θ ) − e χ , (C71) D − = e − φ
14 (cos( θ ) − (2 + 3 cos( θ )) e χ , (C72) D − = e − φ r
52 sin( θ )(cos( θ ) − θ )) e i χ , (C73) D = e − φ r
152 sin( θ ) cos( θ ) , (C74) D = − e − φ r
52 sin( θ )(1 + cos( θ ))(3 cos( θ ) − e − i χ , (C75) D = e − φ
14 (1 + cos( θ )) (3 cos( θ ) − e − χ , (C76) D = e − φ r
32 sin( θ )(1 + cos( θ )) e − χ , (C77) D − = − e − φ
18 (cos( θ ) − e χ , (C78) D − = − e − φ r
32 sin( θ )(cos( θ ) − e χ , (C79) D − = − e − φ √
15 sin( θ ) (cos( θ ) − e i χ , (C80)8 D = − e − φ √ θ ) , (C81) D = e − φ √
15 sin( θ ) (1 + cos( θ )) e − i χ , (C82) D = − e − φ r
32 sin( θ )(1 + cos( θ )) e − χ , (C83) D = e − φ
18 (1 + cos( θ )) e − χ (C84) [1] Y. Yuan, A. Martinez, B. Senyuk, M. Tasinkevych, andI. I. Smalyukh, “Chiral liquid crystal colloids,” NatureMaterials , 71–79 (2018).[2] J. Stenhammar, R. Wittkowski, D. Marenduzzo, andM. E. Cates, “Light-induced self-assembly of active rec-tification devices,” Science Advances , e1501850 (2016).[3] R. Wittmann, M. Marechal, and K. Mecke, “Fundamen-tal measure theory for non-spherical hard particles: pre-dicting liquid crystal properties from the particle shape,”Journal of Physics: Condensed Matter , 244003 (2016).[4] R. Wittkowski, H. L¨owen, and H. R. Brand, “Derivationof a three-dimensional phase-field-crystal model for liquidcrystals from density functional theory,” Physical ReviewE , 031708 (2010).[5] R. Wittkowski, H. L¨owen, and H. R. Brand, “Polar liq-uid crystals in two spatial dimensions: the bridge frommicroscopic to macroscopic modeling,” Physical ReviewE , 061706 (2011).[6] R. Wittkowski, H. L¨owen, and H. R. Brand, “Micro-scopic and macroscopic theories for the dynamics of polarliquid crystals,” Physical Review E , 041708 (2011).[7] D. Pearce, “Activity driven orientational order in ac-tive nematic liquid crystals on an anisotropic substrate,”Physical Review Letters , 227801 (2019).[8] R. Mueller, J. M. Yeomans, and A. Doostmohammadi,“Emergence of active nematic behavior in monolayersof isotropic cells,” Physical Review Letters , 048004(2019).[9] R. Hartmann, P. K. Singh, P. Pearce, R. Mok, B. Song,F. D´ıaz-Pascual, J. Dunkel, and K. Drescher, “Emer-gence of three-dimensional order and structure in grow-ing biofilms,” Nature Physics , 251–256 (2019).[10] A. Doostmohammadi, J. Ign´es-Mullol, J. M. Yeomans,and F. Sagu´es, “Active nematics,” Nature Communica-tions , 1–13 (2018).[11] P. W. Ellis, D. J. G. Pearce, Y. Chang, G. Gold-sztein, L. Giomi, and A. Fernandez-Nieves, “Curvature-induced defect unbinding and dynamics in active nematictoroids,” Nature Physics , 85–90 (2018).[12] S. Praetorius, A. Voigt, R. Wittkowski, and H. L¨owen,“Active crystals on a sphere,” Physical Review E ,052615 (2018).[13] L. Ophaus, S. V. Gurevich, and U. Thiele, “Resting andtraveling localized states in an active phase-field-crystalmodel,” Physical Review E , 022608 (2018).[14] Z. Wang, C. N. Lam, W. Chen, W. Wang, J. Liu, Y. Liu,L. Porcar, C. B. Stanley, Z. Zhao, K. Hong, et al. , “Fin-gerprinting molecular relaxation in deformed polymers,”Physical Review X , 031003 (2017).[15] M. Swart, P. T. van Duijnen, and J. G. Snijders, “Acharge analysis derived from an atomic multipole expan-sion,” Journal of Computational Chemistry , 79–88(2001). [16] J. Applequist, “Traceless Cartesian tensor forms forspherical harmonic functions: new theorems and appli-cations to electrostatics of dielectric media,” Journal ofPhysics A: Mathematical and General , 4303–4330(1989).[17] R. Grinter and G. A. Jones, “Interpreting angular mo-mentum transfer between electromagnetic multipoles us-ing vector spherical harmonics,” Optics Letters , 367–370 (2018).[18] T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R.Heckenberg, “Multipole expansion of strongly focussedlaser beams,” Journal of Quantitative Spectroscopy andRadiative Transfer , 1005–1017 (2003).[19] C. J. R. Sheppard and P. T¨or¨ok, “Efficient calculationof electromagnetic diffraction in optical systems using amultipole expansion,” Journal of Modern Optics , 803–818 (1997).[20] M. A. Wieczorek and M. Meschede, “Shtools: tools forworking with spherical harmonics,” Geochemistry, Geo-physics, Geosystems , 2574–2592 (2018).[21] N. Bartolo, A. Kehagias, M. Liguori, A. Riotto, M. Shi-raishi, and V. Tansella, “Detecting higher spin fieldsthrough statistical anisotropy in the CMB and galaxypower spectra,” Physical Review D , 023503 (2018).[22] M. Kamionkowski, “Circular polarization in a sphericalbasis,” Physical Review D , 123529 (2018).[23] P. Kundu, “Multipole expansion of stationary asymptot-ically flat vacuum metrics in general relativity,” Journalof Mathematical Physics , 1236–1242 (1981).[24] X. Zhang, “Multipole expansions of the general-relativistic gravitational field of the external universe,”Physical Review D , 991–1004 (1986).[25] H. Zuo, P. N. Samarasinghe, T. D. Abhayapala, andG. Dickins, “Spatial sound intensity vectors in sphericalharmonic domain,” Journal of the Acoustical Society ofAmerica , EL149–EL155 (2019).[26] F. Brinkmann and S. Weinzierl, “Comparison of head-related transfer functions pre-processing techniques forspherical harmonics decomposition,” in Audio Engineer-ing Society Conference: 2018 AES International Confer-ence on Audio for Virtual and Augmented Reality (AudioEngineering Society, 2018).[27] C. Bannwarth, S. Ehlert, and S. Grimme, “GFN2-xTB–an accurate and broadly parametrized self-consistenttight-binding quantum chemical method with multipoleelectrostatics and density-dependent dispersion contribu-tions,” Journal of Chemical Theory and Computation ,1652–1671 (2019).[28] R. Singh, R. Adhikari, and M. E. Cates, “Competingchemical and hydrodynamic interactions in autophoreticcolloidal suspensions,” Journal of Chemical Physics ,044901 (2019). [29] Y. Lu, Z. Yu, X. Zhan, and R. Thompson, “Scatter cor-rection with a deterministic integral spherical harmon-ics method in computed tomography,” in Medical Imag-ing 2019: Physics of Medical Imaging , Vol. 10948 (In-ternational Society for Optics and Photonics, 2019) p.109485L.[30] R. Wittkowski, J. Stenhammar, and M. E. Cates,“Nonequilibrium dynamics of mixtures of active andpassive colloidal particles,” New Journal of Physics ,105003 (2017).[31] P.-G. de Gennes and J. Prost, The Physics of LiquidCrystals , 2nd ed., International Series of Monographs onPhysics, Vol. 83 (Oxford University Press, Oxford, 1995).[32] H. Emmerich, H. L¨owen, R. Wittkowski, T. Gruhn, G. I.T´oth, G. Tegze, and L. Gr´an´asy, “Phase-field-crystalmodels for condensed matter dynamics on atomic lengthand diffusive time scales: an overview,” Advances inPhysics , 665–743 (2012).[33] T. C. Lubensky and L. Radzihovsky, “Theory of bent-core liquid-crystal phases and phase transitions,” Physi-cal Review E , 031704 (2002).[34] R. Rosso, “Orientational order parameters in biaxial ne-matics: polymorphic notation,” Liquid Crystals , 737–748 (2007).[35] A. Matsuyama, S. Arikawa, M. Wada, and N. Fukutomi,“Uniaxial and biaxial nematic phases of banana-shapedmolecules and the effects of an external field,” LiquidCrystals , 1–14 (2019).[36] G. R. Luckhurst and T. J. Sluckin, Biaxial Nematic Liq-uid Crystals: Theory, Simulation and Experiment , 1st ed.(John Wiley & Sons, Chichester, 2015).[37] H. Ehrentraut and W. Muschik, “On symmetric irre-ducible tensors in d-dimensions,” ARI - An InternationalJournal for Physical and Engineering Sciences , 149–159 (1998).[38] S. S. Turzi, “On the Cartesian definition of orientationalorder parameters,” Journal of Mathematical Physics ,053517 (2011).[39] K. Blum, Density matrix theory and applications ,Springer Series on Atomic, Optical, and Plasma Physics,Vol. 64 (Springer, Berlin, 2012).[40] C. G. Gray and K. E. Gubbins,
Theory of Molecular Flu-ids: Fundamentals , 1st ed., International Series of Mono- graphs on Chemistry 9, Vol. 1 (Oxford University Press,Oxford, 1984).[41] Q. Teng,
Structural Biology: Practical NMR Applications (Springer, New York, 2007).[42] C. G. Joslin and C. G. Gray, “Multipole expansions infour dimensions,” Journal of Physics A: Mathematicaland General , 1313–1331 (1984).[43] C. G. Joslin and C. G. Gray, “Multipole expansions intwo dimensions,” Molecular Physics , 329–345 (1983).[44] J. Bickmann and R. Wittkowski, “Predictive local fieldtheory for interacting active Brownian spheres in two spa-tial dimensions,” preprint, arXiv:1909.03369 (2019).[45] J. Bickmann and R. Wittkowski, “Predictive local fieldtheory for interacting spherical active Brownian particlesin three spatial dimensions,” in preparation (2019).[46] D. A. Varshalovich, A. N. Moskalev, and V. K. Kher-sonskii, Quantum theory of angular momentum (WorldScientific, Singapore, 1988).[47] N. Tajima, “Analytical formula for numerical evaluationsof the Wigner rotation matrices at high spins,” PhysicalReview C , 014320 (2015).[48] P. P. Man, “Wigner active and passive rotation matricesapplied to NMR tensor,” Concepts in Magnetic Reso-nance Part A , e21385 (2016).[49] C. J. Foot, Atomic Physics , Oxford Master Series inPhysics (Oxford University Press, New York, 2005).[50] W. Freeden and M. Schreiner,
Spherical Functions ofMathematical Geosciences – A Scalar, Vectorial, andTensorial Setup , Advances in Geophysical and Environ-mental Mechanics and Mathematics (Springer, Berlin,2009).[51] M. Fengler and W. Freeden, “A nonlinear Galerkinscheme involving vector and tensor spherical harmonicsfor solving the incompressible Navier-Stokes equation onthe sphere,” SIAM Journal on Scientific Computing ,967–994 (2005).[52] W. Freeden, T. Gervens, and M. Schreiner, “Ten-sor spherical harmonics and tensor spherical splines,”Manuscripta Geodaetica , 80–100 (1994).[53] P. de la Hoz, G. Bj¨ork, A. B. Klimov, G. Leuchs, andL. S´anchez-Soto, “Unpolarized states and hidden polar-ization,” Physical Review A90