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.
I. INTRODUCTION
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 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.
II. UNIAXIAL EXPANSIONS
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.
A. Angular multipole expansion
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
with the imaginary unit i and the circular harmonics eikϕ. The corresponding (Fourier) expansion coefficients are given by
where 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
2. Spherical multipole expansion (3D)
When there are three spatial dimensions, the angular multipole expansion is identical to the “spherical multipole expansion”
with the spherical harmonics
and the associated Legendre polynomials
The latter two functions are stated here explicitly to avoid confusion with other conventions. Now, the expansion coefficients are given by
where 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
where the are the Wigner D-matrices depending on the Euler angles (see Sec. III A).
B. Cartesian multipole expansion
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
where d ∈ {2, 3} is the number of spatial dimensions and ui is the ith element of the orientation vector . [For l = 0, the right-hand side of Eq. (9) gives a constant term .] For this expansion, the corresponding expansion coefficients are obtained as
with the prefactors
and the angular normalization factors
The tensors on the right-hand side of Eq. (10) equal the tensor Chebyshev polynomials of the first kind for d = 2 and the tensor Legendre polynomials for d = 3. They are given by
with and the Euclidean norm ∥·∥. In Table I, the first four of these tensors for d = 2 and d = 3 are listed explicitly. The tensors 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 . 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.,
with the rotation matrix Rij (see Sec. III B). The rotation matrix for d = 2 is defined as
and the rotation matrix for d = 3 is given by Eq. (100).
Tensor Chebyshev polynomials of the first kind and tensor Legendre polynomials for different orders l, where (·)sym denotes the symmetrization of a tensor.
| l . | . | . |
|---|---|---|
| 0 | 1 | 1 |
| 1 | ||
| 2 | ||
| 3 | ||
| ⋮ | ⋮ | ⋮ |
| l . | . | . |
|---|---|---|
| 0 | 1 | 1 |
| 1 | ||
| 2 | ||
| 3 | ||
| ⋮ | ⋮ | ⋮ |
Cartesian expansion coefficients and for different orders l.
| l . | . | . |
|---|---|---|
| 0 | ||
| 1 | ||
| 2 | ||
| 3 | ||
| ⋮ | ⋮ | ⋮ |
| l . | . | . |
|---|---|---|
| 0 | ||
| 1 | ||
| 2 | ||
| 3 | ||
| ⋮ | ⋮ | ⋮ |
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 , we first insert Eq. (9) into Eq. (2) and obtain
We can, thus, express fk as a linear combination of the coefficients , 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
As an example, we find for f±1 the result
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
By symmetry and tracelessness, we also have
such that the result (20) can be written in a simpler way as
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 by inserting Eq. (9) into Eq. (7). We find the conversion formula
where we have exploited the orderwise equivalence, which is proven rigorously in Ref. 42, to avoid an infinite sum. Finally, the 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
and in 3D, we find
These conversions are unique, since the coefficients fk and flm are independent.
C. Relation of angular and Cartesian expansion coefficients for the uniaxial case
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):
Cartesian from circular (2D):
Spherical from Cartesian (3D):
Cartesian from spherical (3D):
D. Order parameters
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 that depends on position 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
and for d = 2,
and for d = 3
Now suppose that we have a system of N particles and that we can assign every particle a position 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
which gives the order parameters for d = 2
and for d = 3
with 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 , polarization vector , and nematic tensor . 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(û − ûn) by (δ(û − ûn) + δ(û + ûn))/2 in Eq. (78).
III. BIAXIAL EXPANSIONS
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).
A. Angular multipole expansion
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 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 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
with the Wigner d-matrices
and −l ≤ m, n ≤ l are irreducible representations of the group SO(3). The set 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 . The functions 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 f ∈ L2(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 in the form42,44
with the expansion coefficients
where the integral is defined as44
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
Explicit expressions for the elements of the Wigner D-matrices up to order l = 3 are given in Appendix B.
B. Cartesian multipole expansion
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
This matrix describes a successive rotation as R(θ, ϕ, χ) = Rz(ϕ)Ry(θ)Rz(χ) with the elementary rotation matrices
We denote by 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 of the rotation group can be derived, which acts on a symmetric traceless tensor as
Using the Peter–Weyl theorem, one can then expand a function f ∈ L2(SO(3)) in terms of the representation , whose matrix elements form a complete orthogonal system.42 The resulting expansion is42
with the expansion coefficients
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.
Cartesian expansion coefficients for different orders l.42
| l . | . |
|---|---|
| 0 | |
| 1 | |
| 2 | |
| 3 | |
| ⋮ | ⋮ |
| l . | . |
|---|---|
| 0 | |
| 1 | |
| 2 | |
| 3 | |
| ⋮ | ⋮ |
The expansion coefficients (105) are symmetric and traceless in the {ik} and {jk} separately for l > 1 (e.g., we have but and but ), 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
and
C. Order parameters
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 . A Cartesian expansion up to second order gives
with
In particular, for the distribution function for a system of N biaxial particles with positions and orientations Rn with n = 1, …, N, given by
we obtain the microscopic definitions
with 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)
IV. CONCLUSIONS
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 AVAILABILITY
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
ACKNOWLEDGMENTS
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).
APPENDIX A: RELATION OF ANGULAR AND CARTESIAN EXPANSION COEFFICIENTS FOR THE BIAXIAL CASE
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:
Cartesian from angular:
The expansion coefficients that appear not explicitly in Eqs. (A1)–(A168) follow from their symmetry properties
and tracelessness
APPENDIX B: ELEMENTS OF WIGNER D-MATRICES
REFERENCES
In this work, we use the terminology “uniaxial” and “biaxial” to distinguish particles with and without an axis of continuous rotational symmetry, respectively. Note that these adjectives can also be used in a different way, referring to the optical properties of a system.38 Moreover, we make the common approximation of assuming all particles to be rigid bodies.
“Active” and “passive” here refer to two different conventions used in the description of rotations, whereas elsewhere in the article we use this terminology to distinguish particles with and without self-propulsion, respectively. An active rotation corresponds 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 a clockwise rotation into a counterclockwise rotation,44 which is why it is important to clarify the convention. Throughout this article, we follow the conventions used by Gray and Gubbins44 for the description of rotations.
In two spatial dimensions, one orientation vector is always sufficient; hence, we do not need to consider the 2D case separately in this section.
