Orientational expansions, which are widely used in natural sciences, exist in angular and Cartesian forms. Although these expansions are orderwise equivalent, it is difficult to relate them in 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 coefficients for functions depending on one, two, and three angles in two and three spatial dimensions. These formulas 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.

Orientational expansions, i.e., expansions of the angular dependence of a function f(Ω), with Ω denoting an orientational variable, are used in many fields of natural sciences, such as liquid crystal physics,1–6 active matter physics,7–13 polymer physics,14 electrostatics,15,16 optics,17–19 geophysics,20 astrophysics and cosmology,21,22 general relativity,23,24 quantum mechanics,25 chemistry,26,27 engineering,28,29 machine learning,30 and medicine.31 Important examples for orientational expansions are the Fourier expansion for Ω = ϕ ∈ [0, 2π), the expansion in spherical harmonics for Ω = (θ, ϕ) ∈ [0, π] × [0, 2π), and the expansion in outer products of a normalized orientation vector û for Ω = û. The expansions can be classified in two main categories, which differ in the way the expansion coefficients transform under rotations: angular expansions (including the Fourier series and the spherical harmonics expansion) and Cartesian expansions (including expansions in symmetric traceless tensors and outer products of orientation vectors or rotation matrices32).

One of the main applications of such expansions is the description of the orientational order of liquid crystals. Here, we can use an orientation vector û depending on the angle ϕ in two spatial dimensions (2D) or on two angles θ and ϕ in three spatial dimensions (3D) to specify the orientation of a particle. A system of particles is then described using a distribution function f(û). The coefficients of the expansion of the function f(û) provide orientational order parameters. Of particular importance are the Cartesian order parameters of first and second order, given by the polarization P and the nematic tensor Q, respectively.33 They are used, e.g., to formulate field theories for liquid crystals.34,35 In addition, order parameters of third order are useful for the description of certain phase transitions.36 

In three spatial dimensions, the description of the orientational order using one orientation vector is only sufficient if the particles have an axis of continuous rotational symmetry (“uniaxial particles”37). The situation is more complex if one considers particles without such a symmetry (“biaxial particles”). Here, a description of their orientation requires two orientation vectors or, alternatively, three angles such as the Euler angles (θ, ϕ, χ) ∈ [0, π] × [0, 2π) × [0, 2π), which correspond to a rotation R that maps the laboratory-fixed Cartesian coordinate system onto a body-fixed one.38 Therefore, the definition of order parameters for biaxial particles requires more general orientational expansions.38–42 Based on liquid crystal terminology, we will refer to expansions for functions f(û) as “uniaxial” and to expansions for functions f(R) as “biaxial.” Note, however, that the applicability of our results is not restricted to liquid crystal physics but extends to all fields where such expansions are used.

Both angular and Cartesian expansions have their own advantages and disadvantages. Angular expansions allow us to make use of the mathematical properties of circular or spherical harmonics and have a smaller number of expansion coefficients, since they are all independent. Cartesian expansions, on the other hand, have a clearer geometrical interpretation and are more directly connected to computer simulations and experiments.42 A good example is the study of the orientational order of liquid crystals consisting of low-symmetry molecules. For theoretical studies, order parameters based on an expansion in Wigner D-matrices, which have a useful and well-known mathematical structure, are widely used.43–45 Experimentalists, on the other hand, frequently use the Saupé ordering matrix—a Cartesian order parameter—which is more practical in describing the outcomes of, e.g., nuclear magnetic resonance (NMR) experiments.46 This can constitute a difficulty in comparing the results of theoretical calculations with computer simulations and experiments.

The problem can be solved if the relations between the different types of expansions are known. It is very difficult to give such an expression in a general form.47 One can, however, explicitly calculate these relations for the lowest orders, which is sufficient for almost all practical applications. In this work, we provide tables containing all relevant relations. This includes coefficients of zeroth to third order for functions depending on one angle (2D, uniaxial), two angles (3D, uniaxial), and three angles (3D, biaxial). These relations constitute the main result of this article. The conversion rules can be used in any field of the natural sciences where orientational distributions are relevant. For example, they allow us to easily convert Cartesian data from a computer simulation or an NMR experiment on a liquid crystal into a form that can be compared with theoretical calculations that use angular functions. Moreover, these equations allow, e.g., to easily calculate the dipole vector corresponding to data that are given in the form of an expansion in spherical harmonics.

In addition, we give an overview over the mathematical structure of uniaxial and biaxial expansions, including definitions of all special functions that are involved. We also provide formulas for the expansion coefficients. Such an overview is very useful and difficult to find in the literature. It also clarifies the conventions used for obtaining the conversion formulas. Furthermore, we explain how these expansions allow us to define order parameters for liquid crystals.

This article is structured as follows: In Sec. II, we describe angular and Cartesian expansions with the relevant functions, coefficients, and order parameters for the uniaxial case. The biaxial expansions are described in Sec. III. We summarize this work and give an outlook in Sec. IV. The relations between the uniaxial expansion coefficients are listed in Sec. II C and those for the biaxial ones are listed in  Appendix A. A list of elements of the Wigner D-matrices can be found in  Appendix B.

We consider uniaxial particles in two and three spatial dimensions. The orientational distribution of the particles is described by a scalar orientation-dependent function f. In a 2D system, this function can be parameterized as f(ϕ) ≡ f(û) with the polar angle ϕ ∈ [0, 2π) and the orientational unit vector û(ϕ) = (cos(ϕ), sin(ϕ))T, where the superscript T denotes a transposition. When considering a 3D system, the function can be parameterized as f(θ, ϕ) ≡ f(û) with the spherical coordinates θ ∈ [0, π] and ϕ ∈ [0, 2π) as well as the orientational unit vector û(θ, ϕ) = (sin(θ) cos(ϕ), sin(θ) sin(ϕ), cos(θ))T.

The scalar orientation-dependent function f can be orthogonally expanded in terms of angular coordinates ϕ (in 2D) and θ and ϕ (in 3D). We follow Refs. 44 and 48.

1. Circular multipole expansion (2D)

In the case of two spatial dimensions, the angular multipole expansion is also called “circular multipole expansion” and is identical to the Fourier series expansion

f(ϕ)=k=fkeikϕ
(1)

with the imaginary unit i and the circular harmonics ei. The corresponding (Fourier) expansion coefficients are given by

fk=12πS1dΩf(ϕ)eikϕ,
(2)

where S1dΩ=02πdϕ denotes an angular integration over the unit circle S1. In general, these expansion coefficients are independent. If the coordinate system is rotated by an angle φ, the expansion coefficients fk change to

fk=fkeikφ.
(3)

2. Spherical multipole expansion (3D)

When there are three spatial dimensions, the angular multipole expansion is identical to the “spherical multipole expansion”

f(θ,ϕ)=l=0m=llflmYlm(θ,ϕ)
(4)

with the spherical harmonics

Ylm(θ,ϕ)=2l+14π(lm)!(l+m)!Plm(cos(θ))eimϕ
(5)

and the associated Legendre polynomials

Plm(x)=(1)m2ll!(1x2)m/2xl+m(x21)l.
(6)

The latter two functions are stated here explicitly to avoid confusion with other conventions. Now, the expansion coefficients are given by

flm=S2dΩf(θ,ϕ)Ylm(θ,ϕ),
(7)

where S2dΩ=0πdθsin(θ)02πdϕ is an angular integration over the unit sphere S2 and the star ⋆ denotes complex conjugation. As is the 2D case, the expansion coefficients are, in general, independent. Under passive49 rotations, the expansion coefficients transform according to

flm=n=llDmnl(ϖ)fln,
(8)

where the Dmnl(ϖ) are the Wigner D-matrices depending on the Euler angles ϖ (see Sec. III A).

The scalar orientation-dependent function f can also be orthogonally expanded in terms of the orientation vector û. This applies to both 2D and 3D. Our presentation of this expansion follows Ref. 48 (for 2D) and Ref. 44 (for 3D).

The so-called “Cartesian multipole expansion” is given by

f(û)=l=0i1,,il=1dfi1il(dD)ui1uil,
(9)

where d ∈ {2, 3} is the number of spatial dimensions and ui is the ith element of the orientation vector û=(u1,,ud)T. [For l = 0, the right-hand side of Eq. (9) gives a constant term f(dD).] For this expansion, the corresponding expansion coefficients are obtained as

fi1il(dD)=Al(dD)Sd1dΩf(û)Ti1il(dD)
(10)

with the prefactors

Al(2D)=2δ0lΩ2,  Al(3D)=2l+1Ω3
(11)

and the angular normalization factors

Ωd=Sd1dΩ=2π ford=24π ford=3.
(12)

The tensors Ti1il(dD) on the right-hand side of Eq. (10) equal the tensor Chebyshev polynomials of the first kind Ti1il(2D) for d = 2 and the tensor Legendre polynomials Ti1il(3D) for d = 3. They are given by

Ti1il(2D)=(1)ll!(l+δ0l)i1il(1ln(r))r=û,
(13)
Ti1il(3D)=(1)ll!i1il1rr=û
(14)

with r=r and the Euclidean norm ∥·∥. In Table I, the first four of these tensors for d = 2 and d = 3 are listed explicitly. The tensors Ti1il(dD) and also the Cartesian coefficient tensors (10) are symmetric and traceless for l > 1. When f(û) is real, the same applies to the Cartesian coefficient tensors fi1il(dD). In general, not more than 2 − δ0l (in 2D) and 2l + 1 (in 3D) elements of a Cartesian 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.,

fi1il(dD)=j1,,jl=1dRi1j1Riljlfj1jl(dD)
(15)

with the rotation matrix Rij (see Sec. III B). The rotation matrix for d = 2 is defined as

R(ϕ)=cos(ϕ)sin(ϕ)sin(ϕ)cos(ϕ),
(16)

and the rotation matrix for d = 3 is given by Eq. (100).

TABLE I.

Tensor Chebyshev polynomials of the first kind Ti1il(2D) and tensor Legendre polynomials Ti1il(3D) for different orders l, where (·)sym denotes the symmetrization of a tensor.

lTi1il(2D)Ti1il(3D)
ui1 ui1 
2ui1ui2δi1i2 12(3ui1ui2δi1i2) 
4ui1ui2ui33(ui1δi2i3)sym 12(5ui1ui2ui33(ui1δi2i3)sym) 
⋮ ⋮ ⋮ 
lTi1il(2D)Ti1il(3D)
ui1 ui1 
2ui1ui2δi1i2 12(3ui1ui2δi1i2) 
4ui1ui2ui33(ui1δi2i3)sym 12(5ui1ui2ui33(ui1δi2i3)sym) 
⋮ ⋮ ⋮ 
TABLE II.

Cartesian expansion coefficients fi1il(2D) and fi1il(3D) for different orders l.

lfi1il(2D)fi1il(3D)
12πS1dΩf(û) 14πS2dΩf(û) 
1πS1dΩf(û)ui1 34πS2dΩf(û)ui1 
1πS1dΩf(û)(2ui1ui2δi1i2) 58πS2dΩf(û)(3ui1ui2δi1i2) 
1πS1dΩf(û)(4ui1ui2ui3ui1δi2i3ui2δi3i1ui3δi1i2) 78πS2dΩf(û)(5ui1ui2ui3ui1δi2i3ui2δi3i1ui3δi1i2) 
⋮ ⋮ ⋮ 
lfi1il(2D)fi1il(3D)
12πS1dΩf(û) 14πS2dΩf(û) 
1πS1dΩf(û)ui1 34πS2dΩf(û)ui1 
1πS1dΩf(û)(2ui1ui2δi1i2) 58πS2dΩf(û)(3ui1ui2δi1i2) 
1πS1dΩf(û)(4ui1ui2ui3ui1δi2i3ui2δi3i1ui3δi1i2) 78πS2dΩf(û)(5ui1ui2ui3ui1δi2i3ui2δi3i1ui3δi1i2) 
⋮ ⋮ ⋮ 

An advantage of the Cartesian multipole expansion is that it is a direct expansion in the variable û, whereas the angular variables ϕ (in 2D) or θ and ϕ (in 3D) appear not directly but via exponential and trigonometric functions in the angular multipole expansion. On the other hand, the number of expansion coefficients is higher for the Cartesian multipole expansion although not more of the expansion coefficients can be independent. Despite the differences of both types of expansions, they are equivalent. Moreover, each order of one expansion is equivalent to the same order of the other expansion. This allows an orderwise mapping between both types of expansions and explains the maximal number of independent expansion coefficients for the Cartesian multipole expansion. In Sec. II C, explicit equations expressing the expansion coefficients of an angular multipole expansion in terms of the expansion coefficients of a Cartesian multipole expansion and vice versa are given for dimensionalities d = 2 and d = 3 and up to third order.

We now derive the conversion formulas (see also Ref. 44 for a geometrical discussion on this problem). The orientation vector in two spatial dimensions is given by û(ϕ) = (cos(ϕ), sin(ϕ))T. For computing the expansion coefficients fk in terms of the expansion coefficients fi1il(2D), we first insert Eq. (9) into Eq. (2) and obtain

fk=12πS1dΩf(ϕ)eikϕ=12πS1dΩl=0i1,,il=12fi1il(2D)ui1uileikϕ=l=0i1,,il=1212πS1dΩui1uileikϕfi1il(2D).
(17)

We can, thus, express fk as a linear combination of the coefficients fi1il(2D), with the prefactors being given by the expression in the larger parentheses. Fortunately, due to orderwise equivalence of both expansions, only terms of order l = |k| contribute to fk (for other orders, the sum of all terms is zero) such that we do not require an infinite sum. We thus find the conversion formula

fk=i1,,i|k|=1212πS1dΩui1ui|k|eikϕfi1i|k|(2D).
(18)

As an example, we find for f±1 the result

f±1=12πS1dΩu1eiϕf1(2D)+12πS1dΩu2eiϕf2(2D)=12πS1dΩcos(ϕ)eiϕf1(2D)+12πS1dΩsin(ϕ)eiϕf2(2D)=12(f1(2D)if2(2D)).
(19)

One has to take into account that these relations are not necessarily unique, since the Cartesian coefficients are not all independent. For example, we find at second order

f±2=12πS1dΩu1u1e2iϕf11(2D)+12πS1dΩu1u2e2iϕf12(2D)+12πS1dΩu2u1e2iϕf21(2D)+12πS1dΩu2u2e2iϕf22(2D)=12πS1dΩcos(ϕ)2e2iϕf11(2D)+12πS1dΩcos(ϕ)sin(ϕ)e2iϕf12(2D)+12πS1dΩsin(ϕ)cos(ϕ)e2iϕf21(2D)+12πS1dΩsin(ϕ)2e2iϕf22(2D)=14f11(2D)i4f12(2D)i4f21(2D)14f22(2D).
(20)

By symmetry and tracelessness, we also have

f21(2D)=f12(2D),
(21)
f22(2D)=f11(2D)
(22)

such that the result (20) can be written in a simpler way as

f±2=12(f11(2D)if12(2D)).
(23)

Similarly, for the 3D case with the orientation vector û(ϕ) = (sin(θ) cos(ϕ), sin(θ) sin(ϕ), cos(θ))T, we can express the flm in terms of fi1il(3D) by inserting Eq. (9) into Eq. (7). We find the conversion formula

flm=i1,,il=13S2dΩui1uilYlm(θ,ϕ)fi1il(3D),
(24)

where we have exploited the orderwise equivalence, which is proven rigorously in Ref. 42, to avoid an infinite sum. Finally, the fi1il(dD) can be expressed in terms of fk (in 2D) and flm (in 3D) by inserting Eq. (1) (in 2D) and Eq. (4) (in 3D) into Eq. (10) and evaluating the integrals. In 2D, we find the conversion formula

fi1il(2D)=k{l,l}Al(2D)S1dΩeikϕTi1il(2D)fk,
(25)

and in 3D, we find

fi1il(3D)=m=llAl(3D)S2dΩYlm(θ,ϕ)Ti1il(3D)flm.
(26)

These conversions are unique, since the coefficients fk and flm are independent.

The following equations allow us to convert the expansion coefficients of an angular and a Cartesian expansion, respectively, up to third order into each other. Higher-order relations can be derived using Eqs. (18) and (24)–(26).

  • Circular from Cartesian (2D):

f0=f(2D),
(27)
f±1=12(f1(2D)if2(2D)),
(28)
f±2=12(f11(2D)if12(2D)),
(29)
f±3=12(f111(2D)if112(2D)).
(30)
  • Cartesian from circular (2D):

f(2D)=f0,
(31)
f1(2D)=f1+f1,
(32)
f2(2D)=i(f1f1),
(33)
f11(2D)=f2+f2,
(34)
f12(2D)=f21(2D)=i(f2f2),
(35)
f22(2D)=f11(2D),
(36)
f111(2D)=f3+f3,
(37)
f112(2D)=f121(2D)=f211(2D)=i(f3f3),
(38)
f122(2D)=f212(2D)=f221(2D)=f111(2D),
(39)
f222(2D)=f211(2D).
(40)
  • Spherical from Cartesian (3D):

f00=2πf(3D),
(41)
f10=2π3f3(3D),
(42)
f1±1=2π3(f1(3D)+if2(3D)),
(43)
f20=2π5f33(3D),
(44)
f2±1=22π15(f13(3D)+if23(3D)),
(45)
f2±2=2π15(f11(3D)f22(3D)2if12(3D)),
(46)
f30=2π7f333(3D),
(47)
f3±1=3π7(f133(3D)+if233(3D)),
(48)
f3±2=6π35(f113(3D)f223(3D)2if123(3D)),
(49)
f3±3=π35(f111(3D)±3f122(3D)+3if112(3D)if222(3D)).
(50)
  • Cartesian from spherical (3D):

f(3D)=12πf00,
(51)
f1(3D)=1232π(f11f11),
(52)
f2(3D)=i1232π(f11+f11),
(53)
f3(3D)=123πf10,
(54)
f11(3D)=185π(2f206(f22+f22)),
(55)
f12(3D)=f21(3D)=i14152π(f22f22),
(56)
f13(3D)=f31(3D)=14152π(f21f21),
(57)
f22(3D)=185π(2f20+6(f22+f22)),
(58)
f23(3D)=f32(3D)=i14152π(f21+f21),
(59)
f33(3D)=f11(3D)f22(3D),
(60)
f111(3D)=187π(3(f31f31)5(f33f33)),
(61)
f112(3D)=f121(3D)=f211(3D)=i12421π(f31+f3115(f33+f33)),
(62)
f113(3D)=f131(3D)=f311(3D)=12442π(6f305(f32+f32)),
(63)
f122(3D)=f212(3D)=f221(3D)=12421π(f31f31+15(f33f33)),
(64)
f123(3D)=f132(3D)=f213(3D)=f231(3D)=f312(3D)=f321(3D)=i14356π(f32f32),
(65)
f133(3D)=f313(3D)=f331(3D)=f111(3D)f122(3D),
(66)
f222(3D)=i187π(3(f31+f31)+5(f33+f33)),
(67)
f223(3D)=f232(3D)=f322(3D)=12442π(6f30+5(f32+f32)),
(68)
f233(3D)=f323(3D)=f332(3D)=f211(3D)f222(3D),
(69)
f333(3D)=f311(3D)f322(3D).
(70)

The Cartesian coefficient tensors (10) can be used as order parameters in liquid-crystal theory. They correspond to the mean particle density (l = 0, monopole moment), polarization vector (l = 1, dipole moment), nematic tensor (l = 2, quadrupole moment), tetratic tensor (l = 3, octupole moment), and so on.34,36 Usually, only order parameters up to second order are considered.

In general, a liquid crystal consisting of particles that are (considered as) uniaxial is described microscopically using a probability distribution f(r,û) that depends on position r and orientation û. The vector û specifies the geometrical orientation of a particle or, for an active particle, the direction of self-propulsion.50,51 A Cartesian expansion in the form (9) up to second order gives

f(r,û)=ρ(r)+i=1dPi(r)ui+i,j=1dQij(r)uiuj
(71)

and for d = 2,

ρ(r)=12πS1dΩf(r,û),
(72)
Pi(r)=1πS1dΩf(r,û)ui,
(73)
Qij(r)=2πS1dΩf(r,û)uiuj12δij
(74)

and for d = 3

ρ(r)=14πS2dΩf(r,û),
(75)
Pi(r)=34πS2dΩf(r,û)ui,
(76)
Qij(r)=158πS2dΩf(r,û)uiuj13δij.
(77)

Now suppose that we have a system of N particles and that we can assign every particle a position rn and an orientation vector ûn = û(ϕn) for d = 2 and ûn = û(θn, ϕn) for d = 3 with n ∈ {1, …, N}. In this case, we can write the microscopic one-particle distribution function as

f(r,û)=i=1Nδ(rrn)δ(ûûn),
(78)

which gives the order parameters for d = 2

ρ(r)=12πn=1Nδ(rrn),
(79)
Pi(r)=1πn=1Nun,iδ(rrn),
(80)
Qij(r)=2πn=1Nun,iun,j12δijδ(rrn)
(81)

and for d = 3

ρ(r)=14πn=1Nδ(rrn),
(82)
Pi(r)=34πn=1Nun,iδ(rrn),
(83)
Qij(r)=158πn=1Nun,iun,j13δijδ(rrn)
(84)

with un,i=(ûn)i denoting the ith element of the vector ûn. These order parameters are, in the case of Eqs. (79) and (82) up to a prefactor, the usual microscopic definitions of the one-particle density 2(d1)πρ(r), polarization vector P(r), and nematic tensor Q(r). They can be used, e.g., to describe a system of active particles, where ûn denotes the direction of self-propulsion of the nth particle. In the case of passive apolar particles such as rods, where there is no physical difference between ûn and −ûn, one can incorporate the head–tail symmetry by replacing δ(ûûn) by (δ(ûûn) + δ(û + ûn))/2 in Eq. (78).

We consider biaxial particles in three spatial dimensions.52 The orientational distribution function of the particles can now be parameterized as f(θ, ϕ, χ) ≡ f(R) with the third Euler angle χ ∈ [0, 2π) and the rotation R ∈ SO(3) that can be represented by a rotation matrix Rij(θ, ϕ, χ).

For describing the orientation of a biaxial particle, we use the Euler angles (θ, ϕ, χ) ∈ [0, π] × [0, 2π) × [0, 2π) and introduce the vector ϖ=(θ,ϕ,χ)T as a shorthand notation. The Euler angles can be defined in various ways. For convenience, we use the popular definition of Gray and Gubbins.44 There, the angles θ and ϕ correspond to the usual angles of spherical coordinates, while the third angle χ ∈ [0, 2π) describes a rotation about the axis defined by θ and ϕ. The advantage of this convention is that, since the first two angles have the same definition as in the usual case of spherical coordinates, the order parameters for biaxial particles will contain the common definitions for uniaxial particles as a limiting case.

The Euler angles are also a way to specify a rotation in three spatial dimensions so that a function f(ϖ)f(θ,ϕ,χ) can be thought of as a function f(R) with R being a rotation represented by a 3 × 3 matrix. Thus, we essentially need to find a way for expanding a function that is defined on the rotation group SO(3). The Wigner D-matrices44 

Dmnl(ϖ)=eimϕdmnl(θ)einχ
(85)

with the Wigner d-matrices

dmnl(θ)=(l+m)!(lm)!(l+n)!(ln)!×kImnl(1)kcos(θ2)2l+mn2ksin(θ2)2km+n(l+mk)!(lnk)!k!(km+n)!
(86)

and −lm, nl are irreducible representations of the group SO(3). The set Imnl contains all integers that k can attain such that all factorial arguments in Eq. (86) are non-negative.53 See Ref. 53 for differential and integral representations of the functions dmnl(θ). The functions Dmnl(ϖ) are also referred to as Wigner D-functions,53 Wigner rotation matrices,54 and four-dimensional spherical harmonics.44 Note that the definition of the Wigner matrices used here assumes passive rotations.44,55

An expansion for functions defined on SO(3) can be performed using the Peter–Weyl theorem, which states (roughly) that for a compact topological group G, a square-integrable function fL2(G) can be expanded in terms of matrix elements of the irreducible representations of G.42 Since these representations are given by the Wigner D-matrices for the group SO(3), we can expand a function f(ϖ)L2(SO(3)) in the form42,44

f(ϖ)=l=0m=lln=llclmnDmnl(ϖ)
(87)

with the expansion coefficients

clmn=2l+18π2SO(3)dΩDmnl(ϖ)f(ϖ),
(88)

where the integral is defined as44 

SO(3)dΩ=02πdχ02πdϕ0πdθsin(θ).
(89)

The expansion coefficients (88) are, in general, independent such that there are (2l + 1)2 independent coefficients at order l.

Since the Wigner D-matrices are very useful—in addition to their role in liquid crystal physics discussed in this work, they have also found applications in machine learning56–58—we present here the elements of lowest order for the reader’s convenience,44,53

D000=1,
(90)
D111=eiϕ12(1+cos(θ))eiχ,
(91)
D101=eiϕsin(θ)2,
(92)
D111=eiϕ12(1cos(θ))eiχ,
(93)
D011=sin(θ)2eiχ,
(94)
D001=cos(θ),
(95)
D011=sin(θ)2eiχ,
(96)
D111=eiϕ12(1cos(θ))eiχ,
(97)
D101=eiϕsin(θ)2,
(98)
D111=eiϕ12(1+cos(θ))eiχ.
(99)

Explicit expressions for the elements of the Wigner D-matrices up to order l = 3 are given in  Appendix B.

We are now interested in a Cartesian expansion for a function depending on the Euler angles. For this expansion, various options are possible. Ehrentraut and Muschik described an expansion in higher-dimensional symmetric traceless tensors,41 which allows us to construct order parameters for biaxial particles, since one can map between the sphere in four dimensions S3 and the configuration space SO(3) of a biaxial particle.41,47 An orientational expansion of functions defined on S3 can be performed in terms of four-dimensional spherical harmonics or, equivalently, higher-dimensional orientation vectors.41 A Cartesian expansion in four-dimensional symmetric traceless tensors is discussed in Ref. 47.

We here describe an expansion derived by Turzi.42,59 It allows us to describe orientational order in terms of rotation matrices, which is geometrically more intuitive than using four-dimensional orientation vectors. Just as an angular expansion in terms of Wigner D-matrices can be obtained starting from the fact that they are representations of the rotation group SO(3), we can construct a Cartesian representation of this group to derive an expansion in outer products of a rotation matrix Rij. The rotation matrix can be defined in terms of the Euler angles as44 

R(ϖ)=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).
(100)

This matrix describes a successive rotation as R(θ, ϕ, χ) = Rz(ϕ)Ry(θ)Rz(χ) with the elementary rotation matrices

Ry(φ)=cos(φ)0sin(φ)010sin(φ)0cos(φ),
(101)
Rz(φ)=cos(φ)sin(φ)0sin(φ)cos(φ)0001.
(102)

We denote by (Ej(j),,Ej(j)) an orthonormal basis of the vector space of symmetric traceless tensors of rank j. As the Wigner D-matrices for the angular case, an irreducible Cartesian representation D̃(R) of the rotation group can be derived, which acts on a symmetric traceless tensor Ti1il as

(D̃(R)T)i1il=  j1,,jl=13  Ri1j1RiljlTj1jl.
(103)

Using the Peter–Weyl theorem, one can then expand a function fL2(SO(3)) in terms of the representation D̃(R), whose matrix elements form a complete orthogonal system.42 The resulting expansion is42 

f(R)=l=0i1,,il=13j1,,jl=13     ci1j1iljl(3D)Ri1j1Riljl
(104)

with the expansion coefficients

ci1j1iljl(3D)=2l+18π2SO(3)dΩα,β=lla1,,al=13b1,,bl=13f(R)×(Eα(l))a1al(Eβ(l))b1bl(Eα(l))i1il(Eβ(l))j1jl×Ra1b1Ralbl.
(105)

In practical applications, the expansion coefficients can be calculated more efficiently in other ways (see Ref. 42 for details). A list of the biaxial expansion coefficients up to order l = 3 can be found in Table III.

TABLE III.

Cartesian expansion coefficients ci1j1iljl(3D) for different orders l.42 

lci1j1iljl(3D)
18π2SO(3)dΩf(R) 
38π2SO(3)dΩf(R)Ri1j1 
58π2SO(3)dΩf(R)12Ri1j1Ri2j2+Ri1j2Ri2j113δi1i2δj1j2 
78π2SO(3)dΩf(R)16Ri1j1Ri2j2Ri3j3+Ri1j2Ri2j3Ri3j1+Ri1j3Ri2j1Ri3j2+Ri1j2Ri2j1Ri3j3 
 +Ri1j1Ri2j3Ri3j2+Ri1j3Ri2j2Ri3j1115Ri1j1δi2i3δj2j3+Ri2j2δi1i3δj1j3+Ri3j3δi1i2δj1j2 
 +Ri1j2δi2i3δj1j3+Ri1j3δi2i3δj1j2+Ri2j1δi1i3δj2j3+Ri2j3δi1i3δj1j2+Ri3j1δi1i2δj2j3+Ri3j2δi1i2δj1j3 
⋮ ⋮ 
lci1j1iljl(3D)
18π2SO(3)dΩf(R) 
38π2SO(3)dΩf(R)Ri1j1 
58π2SO(3)dΩf(R)12Ri1j1Ri2j2+Ri1j2Ri2j113δi1i2δj1j2 
78π2SO(3)dΩf(R)16Ri1j1Ri2j2Ri3j3+Ri1j2Ri2j3Ri3j1+Ri1j3Ri2j1Ri3j2+Ri1j2Ri2j1Ri3j3 
 +Ri1j1Ri2j3Ri3j2+Ri1j3Ri2j2Ri3j1115Ri1j1δi2i3δj2j3+Ri2j2δi1i3δj1j3+Ri3j3δi1i2δj1j2 
 +Ri1j2δi2i3δj1j3+Ri1j3δi2i3δj1j2+Ri2j1δi1i3δj2j3+Ri2j3δi1i3δj1j2+Ri3j1δi1i2δj2j3+Ri3j2δi1i2δj1j3 
⋮ ⋮ 

The expansion coefficients (105) are symmetric and traceless in the {ik} and {jk} separately for l > 1 (e.g., we have c1321(3D)=c2311(3D) but c1321(3D)c1231(3D) and i=13cikil(3D)=0 but i=13ciijl(3D)0), so the maximal number of independent expansion coefficients of a certain order l is (2l + 1)2 and thus identical to that of the expansion coefficients (88) for the same l. In consequence, angular and Cartesian expansions are also orderwise equivalent in the biaxial case.

In  Appendix A, explicit equations expressing the expansion coefficients of an angular multipole expansion in terms of the expansion coefficients of a Cartesian multipole expansion and vice versa are given up to third order for biaxial particles. These relations have been calculated using the same procedure as in the uniaxial case (see Sec. II B). The general conversion formulas read

clmn=i1,,il=13j1,,jl=13×2l+18π2SO(3)dΩDmnl(ϖ)Ri1j1Riljlci1j1iljl(3D)
(106)

and

ci1j1iljl(3D)=m=lln=ll2l+18π2α,β=lla1,,al=13b1,,bl=13×(Eα(l))a1al(Eβ(l))b1bl(Eα(l))i1il(Eβ(l))j1jl×SO(3)dΩDmnl(ϖ)Ra1b1Ralblclmn.
(107)

In the biaxial case, various definitions of orientational order parameters are possible and used in the literature. A very popular choice for l = 2 is the Saupé ordering matrix, which can be used to describe the outcomes of NMR measurements.38 Here, we state the order parameters in the form that follows from the Turzi expansion, as described in Sec. III B. We consider a system with a distribution function f(r,R). A Cartesian expansion up to second order gives

f(r,R)=ρ(r)+i,j=13Pij(r)Rij+     i,j,k,l=13Qijkl(r)RijRkl
(108)

with

ρ(r)=18π2SO(3)dΩf(r,R),
(109)
Pij(r)=38π2SO(3)dΩf(r,R)Rij,
(110)
Qijkl(r)=516π2SO(3)dΩf(r,R)RijRkl+RilRkj23δikδjl.
(111)

In particular, for the distribution function for a system of N biaxial particles with positions rn and orientations Rn with n = 1, …, N, given by

f(r,R)=n=1Nδ(rrn)δ(RRn),
(112)

we obtain the microscopic definitions

ρ(r)=18π2n=1Nδ(rrn),
(113)
Pij(r)=38π2n=1NRn,ijδ(rrn),
(114)
Qijkl(r)=516π2n=1NRn,ijRn,kl+Rn,ilRn,kj23δikδjlδ(rrn)
(115)

with Rn,ij=(Rn)ij denoting an element of the matrix Rn.

It is counterintuitive that, in the biaxial case, the first- and second-rank tensors giving polarization and nematic order from the uniaxial case have to be replaced by a second- and fourth-rank tensor, respectively. One can understand this in analogy to quantum mechanics: For uniaxial particles, various orientational states are degenerate such that one only needs to distinguish three rather than nine degrees of freedom at first order. This is similar as in the hydrogen atom, where various angular momentum eigenstates have the same energy and do not have to be distinguished in elementary treatments. In our case, the degeneracy is lifted by biaxiality.

This analogy is also helpful for understanding another aspect of orientational order, namely, the fact that it can occur in systems of active particles with spherical symmetry.33,50 These particles also have “orientational degeneracy,” which is lifted not by a geometrical anisotropy but by the preferred direction corresponding to the axis of self-propulsion. Thus, all types of orders discussed in Sec. II can also occur in these systems. In our comparison to the hydrogen atom, the activity has a similar role as the magnetic field in the Zeeman effect, which lifts the degeneracy of the energy levels of the hydrogen atom.60 (Actually, the analogy between active and quantum matter goes beyond this aspect. Like active orientational order, quantum orientational order requires no geometrical anisotropy of the individual particles and can occur even in systems of point particles.25)

In this article, we have discussed angular and Cartesian uniaxial and biaxial expansions in two and three spatial dimensions. We have given an overview over the relevant functions and definitions and derived formulas for conversions between the expansion coefficients of both types of expansions up to third order. These formulas allow us to relate the results of analytical calculations, computer simulations, and experiments based on different types of expansions, which makes it possible to combine their advantages by converting between them.

Possible continuations of this work include the consideration of angular and Cartesian expansions in higher-dimensional spaces, in particular, concerning the relations between four-dimensional symmetric traceless tensors and the biaxial expansions discussed here. Moreover, the formalism could be extended toward vector and higher-order tensor spherical harmonics.17,61–63

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

We are thankful to Jonas Lübken for helpful discussions. R.W. was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) (Grant No. WI 4170/3-1).

With the following equations, the expansion coefficients of an angular and Cartesian expansion, respectively, up to third order can be converted into each other. Relations for higher orders can be derived using Eqs. (106) and (107).

  • Angular from Cartesian:

c000=c(3D),
(A1)
c111=12(c11(3D)+c22(3D)+i(c12(3D)c21(3D))),
(A2)
c110=12(c13(3D)ic23(3D)),
(A3)
c111=12(c11(3D)+c22(3D)+i(c12(3D)+c21(3D))),
(A4)
c101=12(c31(3D)+ic32(3D)),
(A5)
c100=c33(3D),
(A6)
c101=12(c31(3D)ic32(3D)),
(A7)
c111=12(c11(3D)+c22(3D)i(c12(3D)+c21(3D))),
(A8)
c110=12(c13(3D)+ic23(3D)),
(A9)
c111=12(c11(3D)+c22(3D)+i(c21(3D)c12(3D))),
(A10)
c222=14(c1111(3D)+4c1122(3D)c1212(3D)c2121(3D)+c2222(3D)  +2i(c1112(3D)c1121(3D)+c1222(3D)c2122(3D))),
(A11)
c221=12(c1113(3D)+2c1223(3D)c2123(3D)+i(2c1123(3D)+c1213(3D)c2223(3D))),
(A12)
c220=1232(c1111(3D)+c1212(3D)c2121(3D)c2222(3D)2i(c1121(3D)+c1222(3D))),
(A13)
c221=12(c1113(3D)+2c1223(3D)+c2123(3D)+i(2c1123(3D)+c1213(3D)c2223(3D))),
(A14)
c222=14c1111(3D)4c1122(3D)c1212(3D)c2121(3D)+c2222(3D)  2ic1112(3D)+c1121(3D)c1222(3D)c2122(3D),
(A15)
c212=12(c1131(3D)c1232(3D)+2c2132(3D)+i(2c1132(3D)c2131(3D)+c2232(3D))),
(A16)
c211=c1133(3D)+c2233(3D)+i(c1233(3D)c2133(3D)),
(A17)
c210=32(c1131(3D)+c1232(3D)i(c2131(3D)+c2232(3D))),
(A18)
c211=c1133(3D)+c2233(3D)+i(c1233(3D)+c2133(3D)),
(A19)
c212=12(c1131(3D)c1232(3D)2c2132(3D)+i(2c1132(3D)c2131(3D)+c2232(3D))),
(A20)
c202=1232(c1111(3D)c1212(3D)+c2121(3D)c2222(3D)+2i(c1112(3D)+c2122(3D))),
(A21)
c201=32(c1113(3D)+c2123(3D)+i(c1213(3D)+c2223(3D))),
(A22)
c200=32(c1111(3D)+c1212(3D)+c2121(3D)+c2222(3D)),
(A23)
c201=32(c1113(3D)+c2123(3D)i(c1213(3D)+c2223(3D))),
(A24)
c202=1232(c1111(3D)c1212(3D)+c2121(3D)c2222(3D)2i(c1112(3D)+c2122(3D))),
(A25)
c212=12(c1131(3D)+c1232(3D)+2c2132(3D)+i(2c1132(3D)c2131(3D)+c2232(3D))),
(A26)
c211=c1133(3D)+c2233(3D)i(c1233(3D)+c2133(3D)),
(A27)
c210=32(c1131(3D)+c1232(3D)+i(c2131(3D)+c2232(3D))),
(A28)
c211=c1133(3D)+c2233(3D)+i(c2133(3D)c1233(3D)),
(A29)
c212=12c1131(3D)+c1232(3D)2c2132(3D)+i2c1132(3D)c2131(3D)+c2232(3D),
(A30)
c222=14c1111(3D)4c1122(3D)c1212(3D)c2121(3D)+c2222(3D)  +2ic1112(3D)+c1121(3D)c1222(3D)c2122(3D),
(A31)
c221=12c1113(3D)2c1223(3D)c2123(3D)+i2c1123(3D)+c1213(3D)c2223(3D),
(A32)
c220=1232c1111(3D)+c1212(3D)c2121(3D)c2222(3D)+2ic1121(3D)+c1222(3D),
(A33)
c221=12c1113(3D)2c1223(3D)+c2123(3D)+i2c1123(3D)+c1213(3D)c2223(3D),
(A34)
c222=14c1111(3D)+4c1122(3D)c1212(3D)c2121(3D)+c2222(3D)  2ic1112(3D)c1121(3D)+c1222(3D)c2122(3D),
(A35)
c333=18c111111(3D)+9c111122(3D)3c111212(3D)3c112121(3D)  +9c112222(3D)3c121222(3D)3c212122(3D)+c222222(3D)  +i3c111112(3D)3c111121(3D)+9c111222(3D)9c112122(3D)  c121212(3D)+3c122222(3D)+c212121(3D)3c212222(3D),
(A36)
c332=1432c111113(3D)+6c111223(3D)3c112123(3D)c121213(3D)  +3c122223(3D)2c212223(3D)+i3c111123(3D)  +2c111213(3D)6c112223(3D)+3c121223(3D)+c212123(3D)c222223(3D),
(A37)
c331=1815c111111(3D)+3c111122(3D)+c111212(3D)  3c112121(3D)3c112222(3D)+3c121222(3D)c212122(3D)  c222222(3D)+ic111112(3D)3c111121(3D)3c111222(3D)  3c112122(3D)+c121212(3D)3c122222(3D)+c212121(3D)+c212222(3D),
(A38)
c330=145c111113(3D)3c112123(3D)+c121213(3D)3c122223(3D)  +i3c111123(3D)3c121223(3D)+c212123(3D)+c222223(3D),
(A39)
c331=1815c111111(3D)3c111122(3D)+c111212(3D)3c112121(3D)  3c112222(3D)3c121222(3D)+c212122(3D)+c222222(3D)  +ic111112(3D)3c111121(3D)3c111222(3D)+3c112122(3D)  c121212(3D)+3c122222(3D)+c212121(3D)+c212222(3D),
(A40)
c332=1432c111113(3D)6c111223(3D)3c112123(3D)c121213(3D)  +3c122223(3D)+2c212223(3D)+i3c111123(3D)2c111213(3D)  +6c112223(3D)+3c121223(3D)+c212123(3D)c222223(3D),
(A41)
c333=18c111111(3D)+9c111122(3D)+3c111212(3D)+3c112121(3D)  9c112222(3D)3c121222(3D)3c212122(3D)+c222222(3D)  +i3c111112(3D)+3c111121(3D)9c111222(3D)9c112122(3D)  c121212(3D)+3c122222(3D)c212121(3D)+3c212222(3D),
(A42)
c323=1432c111131(3D)3c111232(3D)+6c112132(3D)  2c122232(3D)c212131(3D)+3c212232(3D)+i3c111132(3D)2c112131(3D)  +6c112232(3D)c121232(3D)3c212132(3D)+c222232(3D),
(A43)
c322=34c111133(3D)+4c112233(3D)c121233(3D)c212133(3D)+c222233(3D)  +2ic111233(3D)c112133(3D)+c122233(3D)c212233(3D),
(A44)
c321=3452c111131(3D)+c111232(3D)+2c112132(3D)+2c122232(3D)  c212131(3D)c212232(3D)+ic111132(3D)2c112131(3D)  2c112232(3D)+c121232(3D)c212132(3D)c222232(3D),
(A45)
c320=12152c111133(3D)+c121233(3D)c212133(3D)c222233(3D)  2ic112133(3D)+c122233(3D),
(A46)
c321=3452c111131(3D)+c111232(3D)2c112132(3D)2c122232(3D)  c212131(3D)c212232(3D)ic111132(3D)+2c112131(3D)  +2c112232(3D)+c121232(3D)c212132(3D)c222232(3D),
(A47)
c322=34c111133(3D)4c112233(3D)c121233(3D)c212133(3D)+c222233(3D)  2ic111233(3D)+c112133(3D)c122233(3D)c212233(3D),
(A48)
c323=1432c111131(3D)3c111232(3D)6c112132(3D)+2c122232(3D)  c212131(3D)+3c212232(3D)+i3c111132(3D)2c112131(3D)  +6c112232(3D)+c121232(3D)+3c212132(3D)c222232(3D),
(A49)
c313=1815c111111(3D)+3c111122(3D)3c111212(3D)  +c112121(3D)3c112222(3D)c121222(3D)+3c212122(3D)  c222222(3D)+i3c111112(3D)c111121(3D)+3c111222(3D)  +3c112122(3D)c121212(3D)c122222(3D)c212121(3D)+3c212222(3D),
(A50)
c312=3452c111113(3D)+2c111223(3D)+c112123(3D)c121213(3D)  c122223(3D)+2c212223(3D)+ic111123(3D)+2c111213(3D)  +2c112223(3D)+c121223(3D)c212123(3D)+c222223(3D),
(A51)
c311=158c111111(3D)+c111122(3D)+c111212(3D)+c112121(3D)  +c112222(3D)+c121222(3D)+c212122(3D)+c222222(3D)  +ic111112(3D)c111121(3D)c111222(3D)+c112122(3D)  +c121212(3D)+c122222(3D)c212121(3D)c212222(3D),
(A52)
c310=543c111113(3D)+c112123(3D)+c121213(3D)+c122223(3D)  ic111123(3D)+c121223(3D)+c212123(3D)+c222223(3D),
(A53)
c311=158c111111(3D)c111122(3D)+c111212(3D)+c112121(3D)  +c112222(3D)c121222(3D)c212122(3D)c222222(3D)  ic111112(3D)+c111121(3D)+c111222(3D)+c112122(3D)  +c121212(3D)+c122222(3D)+c212121(3D)+c212222(3D),
(A54)
c312=3452c111113(3D)2c111223(3D)+c112123(3D)c121213(3D)  c122223(3D)2c212223(3D)ic111123(3D)+2c111213(3D)  +2c112223(3D)c121223(3D)+c212123(3D)c222223(3D),
(A55)
c313=1815c111111(3D)3c111122(3D)3c111212(3D)+c112121(3D)  3c112222(3D)+c121222(3D)3c212122(3D)+c222222(3D)  +i3c111112(3D)c111121(3D)+3c111222(3D)3c112122(3D)  +c121212(3D)+c122222(3D)c212121(3D)+3c212222(3D),
(A56)
c303=145c111131(3D)3c111232(3D)+c212131(3D)3c212232(3D)  +i3c111132(3D)c121232(3D)+3c212132(3D)c222232(3D),
(A57)
c302=12152c111133(3D)c121233(3D)+c212133(3D)c222233(3D)  +2ic111233(3D)+c212233(3D),
(A58)
c301=543c111131(3D)+c111232(3D)+c212131(3D)+c212232(3D)  +ic111132(3D)+c121232(3D)+c212132(3D)+c222232(3D),
(A59)
c300=52(c111133(3D)+c121233(3D)+c212133(3D)+c222233(3D)),
(A60)
c301=543c111131(3D)+c111232(3D)+c212131(3D)+c212232(3D)  ic111132(3D)+c121232(3D)+c212132(3D)+c222232(3D),
(A61)
c302=12152c111133(3D)c121233(3D)+c212133(3D)c222233(3D)  2ic111233(3D)+c212233(3D),
(A62)
c303=145c111131(3D)3c111232(3D)+c212131(3D)3c212232(3D)  +i3c111132(3D)+c121232(3D)3c212132(3D)+c222232(3D),
(A63)
c313=1815c111111(3D)3c111122(3D)3c111212(3D)+c112121(3D)  3c112222(3D)+c121222(3D)3c212122(3D)+c222222(3D)  +i3c111112(3D)+c111121(3D)3c111222(3D)+3c112122(3D)  c121212(3D)c122222(3D)+c212121(3D)3c212222(3D),
(A64)
c312=3452c111113(3D)2c111223(3D)+c112123(3D)c121213(3D)  c122223(3D)2c212223(3D)+ic111123(3D)+2c111213(3D)  +2c112223(3D)c121223(3D)+c212123(3D)c222223(3D),
(A65)
c311=158c111111(3D)c111122(3D)+c111212(3D)+c112121(3D)  +c112222(3D)c121222(3D)c212122(3D)c222222(3D)  +ic111112(3D)+c111121(3D)+c111222(3D)+c112122(3D)  +c121212(3D)+c122222(3D)+c212121(3D)+c212222(3D),
(A66)
c310=543c111113(3D)+c112123(3D)+c121213(3D)+c122223(3D)  +ic111123(3D)+c121223(3D)+c212123(3D)+c222223(3D),
(A67)
c311=158c111111(3D)+c111122(3D)+c111212(3D)+c112121(3D)  +c112222(3D)+c121222(3D)+c212122(3D)+c222222(3D)  +ic111112(3D)+c111121(3D)+c111222(3D)c112122(3D)  c121212(3D)c122222(3D)+c212121(3D)+c212222(3D),
(A68)
c312=3452c111113(3D)+2c111223(3D)+c112123(3D)c121213(3D)  c122223(3D)+2c212223(3D)+ic111123(3D)2c111213(3D)  2c112223(3D)c121223(3D)+c212123(3D)c222223(3D),
(A69)
c313=1815c111111(3D)+3c111122(3D)3c111212(3D)  +c112121(3D)3c112222(3D)c121222(3D)+3c212122(3D)  c222222(3D)+i3c111112(3D)+c111121(3D)3c111222(3D)  3c112122(3D)+c121212(3D)+c122222(3D)+c212121(3D)3c212222(3D),
(A70)
c323=1432c111131(3D)3c111232(3D)6c112132(3D)+2c122232(3D)  c212131(3D)+3c212232(3D)+i3c111132(3D)+2c112131(3D)  6c112232(3D)c121232(3D)3c212132(3D)+c222232(3D),
(A71)
c322=34c111133(3D)4c112233(3D)c121233(3D)c212133(3D)+c222233(3D)  +2ic111233(3D)+c112133(3D)c122233(3D)c212233(3D),
(A72)
c321=3452c111131(3D)+c111232(3D)2c112132(3D)2c122232(3D)  c212131(3D)c212232(3D)+ic111132(3D)+2c112131(3D)  +2c112232(3D)+c121232(3D)c212132(3D)c222232(3D),
(A73)
c320=12152c111133(3D)+c121233(3D)c212133(3D)  c222233(3D)+2ic112133(3D)+c122233(3D),
(A74)
c321=3452c111131(3D)+c111232(3D)+2c112132(3D)+2c122232(3D)  c212131(3D)c212232(3D)+ic111132(3D)+2c112131(3D)  +2c112232(3D)c121232(3D)+c212132(3D)+c222232(3D),
(A75)
c322=34c111133(3D)+4c112233(3D)c121233(3D)c212133(3D)+c222233(3D)  2ic111233(3D)c112133(3D)+c122233(3D)c212233(3D),
(A76)
c323=1432c111131(3D)3c111232(3D)+6c112132(3D)2c122232(3D)  c212131(3D)+3c212232(3D)+i3c111132(3D)+2c112131(3D)  6c112232(3D)+c121232(3D)+3c212132(3D)c222232(3D),
(A77)
c333=18c111111(3D)+9c111122(3D)+3c111212(3D)+3c112121(3D)  9c112222(3D)3c121222(3D)3c212122(3D)+c222222(3D)  +i3c111112(3D)3c111121(3D)+9c111222(3D)+9c112122(3D)  +c121212(3D)3c122222(3D)+c212121(3D)3c212222(3D),
(A78)
c332=1432c111113(3D)6c111223(3D)3c112123(3D)c121213(3D)  +3c122223(3D)+2c212223(3D)+i3c111123(3D)+2c111213(3D)  6c112223(3D)3c121223(3D)c212123(3D)+c222223(3D),
(A79)
c331=1815c111111(3D)3c111122(3D)+c111212(3D)3c112121(3D)  3c112222(3D)3c121222(3D)+c212122(3D)+c222222(3D)  +ic111112(3D)+3c111121(3D)+3c111222(3D)3c112122(3D)  +c121212(3D)3c122222(3D)c212121(3D)c212222(3D),
(A80)
c330=145c111113(3D)3c112123(3D)+c121213(3D)3c122223(3D)  +i3c111123(3D)+3c121223(3D)c212123(3D)c222223(3D),
(A81)
c331=1815c111111(3D)+3c111122(3D)+c111212(3D)  3c112121(3D)3c112222(3D)+3c121222(3D)c212122(3D)  c222222(3D)ic111112(3D)3c111121(3D)3c111222(3D)  3c112122(3D)+c121212(3D)3c122222(3D)+c212121(3D)+c212222(3D),
(A82)
c332=1432c111113(3D)+6c111223(3D)3c112123(3D)  c121213(3D)+3c122223(3D)2c212223(3D)+i3c111123(3D)  2c111213(3D)+6c112223(3D)3c121223(3D)c212123(3D)+c222223(3D),
(A83)
c333=18c111111(3D)+9c111122(3D)3c111212(3D)3c112121(3D)  +9c112222(3D)3c121222(3D)3c212122(3D)+c222222(3D)  +i3c111112(3D)+3c111121(3D)9c111222(3D)  +9c112122(3D)+c121212(3D)3c122222(3D)c212121(3D)+3c212222(3D).
(A84)
  • Cartesian from angular:

c(3D)=c000,
(A85)
c11(3D)=12(c111c111c111+c111),
(A86)
c12(3D)=i2(c111+c111c111c111),
(A87)
c13(3D)=12(c110c110),
(A88)
c21(3D)=i2(c111c111+c111c111),
(A89)
c22(3D)=12(c111+c111+c111+c111),
(A90)
c23(3D)=i2(c110+c110),
(A91)
c31(3D)=12(c101c101),
(A92)
c32(3D)=i2(c101+c101),
(A93)
c33(3D)=c100,
(A94)
c1111(3D)=1122c2006c2026c2026c220+3c222  +3c2226c220+3c222+3c222,
(A95)
c1112(3D)=i126c2026c2023c222+3c222  3c222+3c222,
(A96)
c1113(3D)=1126c2016c2013c221+3c221  3c221+3c221,
(A97)
c1121(3D)=i126c2203c2223c2226c220  +3c222+3c222,
(A98)
c1122(3D)=14(c222c222c222+c222),
(A99)
c1123(3D)=i4(c221c221c221+c221),
(A100)
c1131(3D)=1126c2103c2123c2126c210  +3c212+3c212,
(A101)
c1132(3D)=i4(c212c212c212+c212),
(A102)
c1133(3D)=14(c211c211c211+c211),
(A103)
c1212(3D)=1122c200+6c202+6c2026c220  3c2223c2226c2203c2223c222,
(A104)
c1213(3D)=i12(6c201+6c2013c2213c2213c2213c221),
(A105)
c1222(3D)=i12(6c220+3c222+3c2226c2203c2223c222),
(A106)
c1223(3D)=14(c221c221+c221+c221),
(A107)
c1232(3D)=112(6c210+3c212+3c2126c2103c2123c212),
(A108)
c1233(3D)=i4(c211+c211c211c211),
(A109)
c2121(3D)=1122c2006c2026c202+6c220  3c2223c222+6c2203c2223c222,
(A110)
c2122(3D)=i126c2026c202+3c2223c222  +3c2223c222,
(A111)
c2123(3D)=112(6c2016c201+3c2213c221+3c2213c221),
(A112)
c2131(3D)=i126c2103c2123c212+6c210  3c2123c212,
(A113)
c2132(3D)=14(c212+c212c212+c212),
(A114)
c2133(3D)=i4(c211c211+c211c211),
(A115)
c2222(3D)=1122c200+6c202+6c202+6c220  +3c222+3c222+6c220+3c222+3c222,
(A116)
c2223(3D)=i12(6c201+6c201+3c221+3c221+3c221+3c221),
(A117)
c2232(3D)=i126c210+3c212+3c212+6c210+3c212  +3c212,
(A118)
c2233(3D)=14(c211+c211+c211+c211),
(A119)
c111111(3D)=1403c31115c3133c311+15c313  15c331+5c333+15c3315c333  3c311+15c313+3c311  15c313+15c3315c333  15c331+5c333,
(A120)
c111112(3D)=i1203c311315c313+3c311315c313  15c331+15c33315c331+15c3333c311  +315c3133c311+315c313  +15c33115c333+15c33115c333,
(A121)
c111113(3D)=112063c310+310c312+310c312  +65c33056c33256c332  +63c310310c312310c312  65c330+56c332+56c332,
(A122)
c111121(3D)=i1203c31115c3133c311+15c313  315c331+15c333+315c33115c333  +3c31115c3133c311  +15c313315c331+15c333  +315c33115c333,
(A123)
c111122(3D)=1120c31115c313+c31115c313  15c331+15c33315c331+15c333  +c31115c313+c31115c313  15c331+15c33315c331+15c333,
(A124)
c111123(3D)=i12023c31010c31210c312  65c330+56c332+56c332  +23c31010c31210c312  65c330+56c332+56c332,
(A125)
c111131(3D)=112063c301+65c303+63c301  65c303+310c32156c323  310c321+56c323+310c321  56c323310c321+56c323,
(A126)
c111132(3D)=i12023c30165c303+23c301  65c30310c321+56c323  10c321+56c32310c321  +56c32310c321+56c323,
(A127)
c111133(3D)=1606c30030c30230c30230c320  +5c322+5c32230c320+5c322+5c322,
(A128)
c111212(3D)=11203c311+315c3133c311  315c31315c33115c333  +15c331+15c3333c311  315c313+3c311+315c313  +15c331+15c33315c33115c333,
(A129)
c111213(3D)=i60235c31235c31253c332  +53c33235c312+35c312  +53c33253c332,
(A130)
c111222(3D)=i120c311+15c313c31115c313  15c33115c333+15c331+15c333  +c311+15c313c31115c313  15c33115c333+15c331+15c333,
(A131)
c111223(3D)=16025c3125c31253c332  +53c332+5c3125c312  53c332+53c332,
(A132)
c111232(3D)=112023c30165c303+23c301  +65c303+10c321+56c323  10c32156c323+10c321  +56c32310c32156c323,
(A133)
c111233(3D)=i6030c30230c3025c322+5c322  5c322+5c322,
(A134)
c112121(3D)=11203c31115c3133c311+15c313  +315c33115c333315c331+15c333  3c311+15c313+3c311  15c313315c331+15c333  +315c33115c333,
(A135)
c112122(3D)=i120c31115c313+c31115c313  +15c33115c333+15c33115c333  c311+15c313c311+15c313  15c331+15c33315c331+15c333,
(A136)
c112123(3D)=112023c310+10c312+10c312  65c330+56c332+56c332  +23c31010c31210c312  +65c33056c33256c332,
(A137)
c112131(3D)=i60235c32153c32335c321  +53c32335c321+53c323  +35c32153c323,
(A138)
c112132(3D)=16025c32153c323+5c321  53c3235c321+53c323  5c321+53c323,
(A139)
c112133(3D)=i6030c3205c3225c32230c320  +5c322+5c322,
(A140)
c112222(3D)=1120c311+15c313c31115c313  +15c331+15c33315c33115c333  c31115c313+c311+15c313  15c33115c333+15c331+15c333,
(A141)
c112223(3D)=i6025c3125c312+53c332  53c3325c312+5c312  53c332+53c332,
(A142)
c112232(3D)=i6025c321+53c3235c321  53c3235c32153c323  +5c321+53c323,
(A143)
c112233(3D)=112(c322c322c322+c322),
(A144)
c121212(3D)=i403c311+15c313+3c311+15c313  15c3315c33315c3315c333  3c31115c3133c31115c313  +15c331+5c333+15c331+5c333,
(A145)
c121213(3D)=112063c310310c312310c312  +65c330+56c332+56c332  +63c310+310c312+310c312  65c33056c33256c332,
(A146)
c121222(3D)=11203c311+15c313+3c311+15c313  315c33115c333315c33115c333  +3c311+15c313+3c311  +15c313315c33115c333  315c33115c333,
(A147)
c121223(3D)=i12023c310+10c312+10c312  65c33056c33256c332  +23c310+10c312+10c312  65c33056c33256c332,
(A148)
c121232(3D)=i12063c301+65c303+63c301  +65c303310c32156c323  310c32156c323310c321  56c323310c32156c323,
(A149)
c121233(3D)=1606c300+30c302+30c30230c320  5c3225c32230c3205c3225c322,
(A150)
c122222(3D)=i1203c311+15c313+3c311+15c313  +315c331+15c333+315c331+15c333  3c31115c3133c311  15c313315c33115c333  315c33115c333,
(A151)
c122223(3D)=112023c31010c31210c312  65c33056c33256c332  +23c310+10c312+10c312  +65c330+56c332+56c332,
(A152)
c122232(3D)=160235c321+53c323+35c321  +53c32335c32153c323  35c32153c323,
(A153)
c122233(3D)=i6030c320+5c322+5c32230c320  5c3225c322,
(A154)
c212121(3D)=i403c31115c3133c311+15c313  +15c3315c33315c331+5c333  +3c31115c3133c311  +15c313+15c3315c333  15c331+5c333,
(A155)
c212122(3D)=11203c311315c313+3c311  315c313+15c33115c333  +15c33115c333+3c311  315c313+3c311315c313  +15c33115c333+15c33115c333,
(A156)
c212123(3D)=i12063c310310c312310c312  +65c33056c33256c332  +63c310310c312310c312  +65c33056c33256c332,
(A157)
c212131(3D)=112063c301+65c303+63c301  65c303310c321+56c323  +310c32156c323310c321  +56c323+310c32156c323,
(A158)
c212132(3D)=i12023c30165c303+23c301  65c303+10c32156c323  +10c32156c323+10c321  56c323+10c32156c323,
(A159)
c212133(3D)=1606c30030c30230c302+30c320  5c3225c322+30c3205c3225c322,
(A160)
c212222(3D)=i1203c311+315c3133c311  315c313+15c331+15c333  15c33115c333+3c311  +315c3133c311315c313  +15c331+15c33315c33115c333,
(A161)
c212223(3D)=160235c31235c312+53c332  53c332+35c31235c312  +53c33253c332,
(A162)
c212232(3D)=112023c30165c303+23c301  +65c30310c32156c323  +10c321+56c32310c321  56c323+10c321+56c323,
(A163)
c212233(3D)=i6030c30230c302+5c3225c322  +5c3225c322,
(A164)
c222222(3D)=1403c311+15c313+3c311+15c313  +15c331+5c333+15c331+5c333  +3c311+15c313+3c311  +15c313+15c331+5c333  +15c331+5c333,
(A165)
c222223(3D)=i12063c310+310c312+310c312  +65c330+56c332+56c332  +63c310+310c312+310c312  +65c330+56c332+56c332,
(A166)
c222232(3D)=i12063c301+65c303+63c301  +65c303+310c321+56c323  +310c321+56c323+310c321  +56c323+310c321+56c323,
(A167)
c222233(3D)=1606c300+30c302+30c302+30c320  +5c322+5c322+30c320+5c322+5c322.
(A168)

The expansion coefficients that appear not explicitly in Eqs. (A1)(A168) follow from their symmetry properties

ci1j1i2j2(3D)=ci2j1i1j2(3D)=ci1j2i2j1(3D),
(A169)
ci1j1i2j2i3j3(3D)=ci2j1i1j2i3j3(3D)=ci3j1i2j2i1j3(3D)=ci1j1i3j2i2j3(3D)=ci1j2i2j1i3j3(3D)=ci1j3i2j2i3j1(3D)=ci1j1i2j3i3j2(3D)
(A170)

and tracelessness

c3j13j2(3D)=c1j11j2(3D)c2j12j2(3D),
(A171)
ci13i23(3D)=ci11i21(3D)ci12i22(3D),
(A172)
c3j13j2i3j3(3D)=c1j11j2i3j3(3D)c2j12j2i3j3(3D),
(A173)
c3j1i2j23j3(3D)=c1j1i2j21j3(3D)c2j1i2j22j3(3D),
(A174)
ci1j13j23j3(3D)=ci1j11j21j3(3D)ci1j12j22j3(3D),
(A175)
ci13i23i3j3(3D)=ci11i21i3j3(3D)ci12i22i3j3(3D),
(A176)
ci13i2j2i33(3D)=ci11i2j2i31(3D)ci12i2j2i32(3D),
(A177)
ci1j1i23i33(3D)=ci1j1i21i31(3D)ci1j1i22i32(3D).
(A178)

Finally, we list all elements of the Wigner D-matrices (85) with l ≤ 3,44,53

D000=1,
(B1)
D111=eiϕ12(1+cos(θ))eiχ,
(B2)
D101=eiϕsin(θ)2,
(B3)
D111=eiϕ12(1cos(θ))eiχ,
(B4)
D011=sin(θ)2eiχ,
(B5)
D001=cos(θ),
(B6)
D011=sin(θ)2eiχ,
(B7)
D111=eiϕ12(1cos(θ))eiχ,
(B8)
D101=eiϕsin(θ)2,
(B9)
D111=eiϕ12(1+cos(θ))eiχ,
(B10)
D222=e2iϕ14(1+cos(θ))2e2iχ,
(B11)
D212=e2iϕ12sin(θ)(1+cos(θ))eiχ,
(B12)
D202=e2iϕ1232sin(θ)2,
(B13)
D212=e2iϕ12sin(θ)(cos(θ)1)eiχ,
(B14)
D222=e2iϕ14(cos(θ)1)2e2iχ,
(B15)
D122=eiϕ12sin(θ)(1+cos(θ))e2iχ,
(B16)
D112=eiϕ12(1+cos(θ))(2cos(θ)1)eiχ,
(B17)
D102=eiϕ32sin(θ)cos(θ),
(B18)
D112=eiϕ12(1cos(θ))(1+2cos(θ))eiχ,
(B19)
D122=eiϕ12sin(θ)(cos(θ)1)e2iχ,
(B20)
D022=1232sin(θ)2e2iχ,
(B21)
D012=32sin(θ)cos(θ)eiχ,
(B22)
D002=12(3cos(θ)21),
(B23)
D012=32sin(θ)cos(θ)eiχ,
(B24)
D022=1232sin(θ)2e2iχ,
(B25)
D122=eiϕ12sin(θ)(cos(θ)1)e2iχ,
(B26)
D112=eiϕ12(1cos(θ))(1+2cos(θ))eiχ,
(B27)
D102=eiϕ32sin(θ)cos(θ),
(B28)
D112=eiϕ12(1+cos(θ))(2cos(θ)1)eiχ,
(B29)
D122=eiϕ12sin(θ)(1+cos(θ))e2iχ,
(B30)
D222=e2iϕ14(cos(θ)1)2e2iχ,
(B31)
D212=e2iϕ12sin(θ)(cos(θ)1)eiχ,
(B32)
D202=e2iϕ1232sin(θ)2,
(B33)
D212=e2iϕ12sin(θ)(1+cos(θ))eiχ,
(B34)
D222=e2iϕ14(1+cos(θ))2e2iχ,
(B35)
D333=e3iϕ18(1+cos(θ))3e3iχ,
(B36)
D323=e3iϕ1432sin(θ)(1+cos(θ))2e2iχ,
(B37)
D313=e3iϕ1815sin(θ)2(1+cos(θ))eiχ,
(B38)
D303=e3iϕ145sin(θ)3,
(B39)
D313=e3iϕ1815sin(θ)2(cos(θ)1)eiχ,
(B40)
D323=e3iϕ1432sin(θ)(cos(θ)1)2e2iχ,
(B41)
D333=e3iϕ18(cos(θ)1)3e3iχ,
(B42)
D233=e2iϕ1432sin(θ)(1+cos(θ))2e3iχ,
(B43)
D223=e2iϕ14(1+cos(θ))2(3cos(θ)2)e2iχ,
(B44)
D213=e2iϕ1452sin(θ)(1+cos(θ))(3cos(θ)1)eiχ,
(B45)
D203=e2iϕ12152sin(θ)2cos(θ),
(B46)
D213=e2iϕ1452sin(θ)(1cos(θ))(1+3cos(θ))eiχ,
(B47)
D223=e2iϕ14(cos(θ)1)2(2+3cos(θ))e2iχ,
(B48)
D233=e2iϕ1432sin(θ)(cos(θ)1)2e3iχ,
(B49)
D133=eiϕ1815sin(θ)2(1+cos(θ))e3iχ,
(B50)
D123=eiϕ1452sin(θ)(1+cos(θ))(3cos(θ)1)e2iχ,
(B51)
D113=eiϕ18(1+cos(θ))(15cos(θ)210cos(θ)1)eiχ,
(B52)
D103=eiϕ143sin(θ)(5cos(θ)21),
(B53)
D113=eiϕ18(1cos(θ))(15cos(θ)2+10cos(θ)1)eiχ,
(B54)
D123=eiϕ1452sin(θ)(1cos(θ))(1+3cos(θ))e2iχ,
(B55)
D133=eiϕ1815sin(θ)2(cos(θ)1)e3iχ,
(B56)
D033=145sin(θ)3e3iχ,
(B57)
D023=12152sin(θ)2cos(θ)e2iχ,
(B58)
D013=143sin(θ)(15cos(θ)2)eiχ,
(B59)
D003=12cos(θ)(5cos(θ)23),
(B60)
D013=143sin(θ)(5cos(θ)21)eiχ,
(B61)
D023=12152sin(θ)2cos(θ)e2iχ,
(B62)
D033=145sin(θ)3e3iχ,
(B63)
D133=eiϕ1815sin(θ)2(cos(θ)1)e3iχ,
(B64)
D123=eiϕ1452sin(θ)(cos(θ)1)(1+3cos(θ))e2iχ,
(B65)
D113=eiϕ18(1cos(θ))(15cos(θ)2+10cos(θ)1)eiχ,
(B66)
D103=eiϕ143sin(θ)(15cos(θ)2),
(B67)
D113=eiϕ18(1+cos(θ))(15cos(θ)210cos(θ)1)eiχ,
(B68)
D123=eiϕ1452sin(θ)(1+cos(θ))(3cos(θ)1)e2iχ,
(B69)
D133=eiϕ1815sin(θ)2(1+cos(θ))e3iχ,
(B70)
D233=e2iϕ1432sin(θ)(cos(θ)1)2e3iχ,
(B71)
D223=e2iϕ14(cos(θ)1)2(2+3cos(θ))e2iχ,
(B72)
D213=e2iϕ1452sin(θ)(cos(θ)1)(1+3cos(θ))eiχ,
(B73)
D203=e2iϕ12152sin(θ)2cos(θ),
(B74)
D213=e2iϕ1452sin(θ)(1+cos(θ))(3cos(θ)1)eiχ,
(B75)
D223=e2iϕ14(1+cos(θ))2(3cos(θ)2)e2iχ,
(B76)
D233=e2iϕ1432sin(θ)(1+cos(θ))2e3iχ,
(B77)
D333=e3iϕ18(cos(θ)1)3e3iχ,
(B78)
D323=e3iϕ1432sin(θ)(cos(θ)1)2e2iχ,
(B79)
D313=e3iϕ1815sin(θ)2(cos(θ)1)eiχ,
(B80)
D303=e3iϕ145sin(θ)3,
(B81)
D313=e3iϕ1815sin(θ)2(1+cos(θ))eiχ,
(B82)
D323=e3iϕ1432sin(θ)(1+cos(θ))2e2iχ,
(B83)
D333=e3iϕ18(1+cos(θ))3e3iχ.
(B84)
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