Spherical harmonics

Research: Science & Education

332 Journal of Chemical Education • Vol. 75 No. 3 March 1998 • JChemEd.chem.wisc.edu

Most chemistry undergraduates encounter surface spheri-cal harmonics and their squares for the first time in quan-tum chemistry courses, interpreted as angular wave functionsand angular probability densities, respectively. The teachingof modern chemistry leans more heavily on quantum theorythan on any other single pillar. Thus, answers to students’questions like “What do these functions represent?” or “Howdo these functions look?” cannot be avoided and constructivevisualization aids must be developed. Besides, surface spheri-cal harmonics occur not only in quantum chemistry but alsoin a large variety of physicochemical problems dealing withspherical symmetry or spherical surfaces.

In trying to meet visualization needs and difficulties of thesefunctions one could use a good quality illustration, a suit-able computer graphic, a model made from a solid material,or one’s own drawing. Nevertheless, problems can arise withthese possibilities also. In many recent quantum chemistrytextbooks the graphical representation of harmonics is treatedas not worthy to be mentioned. The insufficiently detailedand inaccurate portrayals of the harmonics and their squares,as well as the use of various definitions and terminologies,can be a source of confusion for students. There are a fewadvanced texts that nicely explain two-dimensional representa-tion of surface spherical harmonics and their squares (1–8),but the cross sections used are not always good indicationsof the overall shapes of the functions. However, many stu-dents find it difficult to appreciate the directional propertiesof the functions even using the cross sections. Some nice andsophisticated computer-generated three-dimensional polarplots of the surface spherical harmonics and their squares havebeen appeared in last few years (9, 10). When computergraphics is used, familiarity with mathematical concepts, com-puter programing, and graphics is necessary. Accurate handdrawings require fairly detailed knowledge of calculus,geometry, and drawing. Moreover, the harmonics are mod-erately difficult functions to draw. A simple remedy to allthese problems could be to give students exercises and home-work problems dealing with the harmonics, having themmake their own computer-generated or hand drawings.

This paper focuses on some important concepts as theresult of a literature survey regarding the definition of sur-face spherical harmonics and spherical coordinates, and theoccurrence of the harmonics and their squares in variousfields. It is assumed that the student has become familiar withthis subject before any serious approach to visualization ofthe functions. In a companion paper to this one, a drawingsystem clear to undergraduates, based on normal projections,will be given (Kiralj, R., manuscript in preparation). Calcu-lated coordinates will be used for accurate three-dimensional

Three-Dimensional Representation of Surface SphericalHarmonics and Their Squares Using Normal Projections:Some Comments on the Functions

Preparing for an Undergraduate Exercise

Rudolf KiraljRudjer Bo skovic Institute, P.O.B. 1016, HR-10 001 Zagreb, Croatia

Figure 1. Spherical polar coordinate system (top), right-handedthree-dimensional (rectangular) Cartesian system (middle), and theirsuperposition (bottom), with an illustrative example. Top: the pointP is the intersection of the coordinate sphere (white; r = 2.5 arbi-trary units) with the cone (gray; θ = 40°) and the half-plane (gray-ish; ϕ = 70°). Middle: the point P is the place of intersection of thethree planes (x = 0.550, y = 1.510, z = 1.915). Bottom: the geo-metrical relationship between the polar (r,θ,ϕ) and the Cartesian(x,y, z ) coordinates as the superposition result.


JChemEd.chem.wisc.edu • Vol. 75 No. 3 March 1998 • Journal of Chemical Education 333

hand drawings of some surface spherical harmonics and theirsquares in spherical coordinates, and the worked exampleswill be explained in detail.

Some Basic Concepts

Spherical Polar Coordinate SystemThe spherical polar coordinate system is one of the most

frequently used coordinate systems in three-dimensional Eu-clidean space. The spherical system could be defined and ex-plained from several points of view—which should be madeclear to undergraduates. Three definitions of spherical systemare explained here: geometrical, algebraic, and physical.

The Geometrical DefinitionSpherical polar coordinate system is completely defined

by two intersecting axes at right angles, one horizontal andanother vertical (polar axis), with the origin or pole O at thepoint of the intersection (Fig. 1, top). Spherical coordinatesof a point P (the length r of the radius vector, the colatitudeθ, and the azimuth ϕ) are specified by intersection of threecoordinate surfaces: the concentric sphere about the origin(r = const), the right circular cone with apex at the originand axis along the polar axis (θ = const), and the half-planefrom the polar axis (ϕ = const) such that 0 ≤ r ≤ ∞, 0 ≤ θ ≤ π,0 ≤ ϕ ≤ 2π. The spherical system with this (geometrical) defi-nition is very useful in pictorial representation of various phe-nomena, and it occurs sometimes in chemistry (11).

The Algebraic DefinitionMany problems in physics and chemistry are treated in

both Cartesian and spherical coordinate systems. The three-dimensional (rectangular) Cartesian system (Fig. 1, middle)is defined by three mutually perpendicular axes X, Y, Z in-tersecting in the point at the origin O. The Cartesian coor-dinates x, y, z of a point P are specified by intersection ofthree planes: the planes perpendicular to the X (x = const), Y(y = const), and Z (z = const) axes. The x, y, z coordinatesare positions of the πx, πy, πz planes from the coordinate planes(YZ, XZ, XY) such that { ∞ ≤ x, y, z ≤ + ∞. If the Cartesiansystem is superimposed on the spherical system in the waythat the origin, the Z axis, and the XZ plane of the Cartesiansystem coincide with the pole, the polar axis, and the initialmeridian plane of the spherical system, respectively (Fig. 1,bottom), the coordinate transformations (12–14) are x = r sinθ cosϕ y = r sinθ sinϕ z = r cosθ

r = x 2 + y 2 + z 2 ; ϕ = tan{1 y

x ; θ = cos{1 z

x 2 + y 2 + z 2(1)

This is the easiest way to transform spherical coordinates toCartesian and vice versa. This definition of the spherical sys-tem (algebraic) enables very accurate drawings, since coordi-nates are calculated and then plotted. More complicated ob-jects can be drawn this way, rather than in the geometricalapproach to the spherical system.

The Physical DefinitionIn crystallography (15) and crystal physics (16), in as-

tronomy (17), in geography and cartography (18) and someother fields, the spherical system is defined by real physicalobjects (crystal, celestial objects, earth): the radius r is alwaysconstant or undetermined, and the sphere is divided into

hemispheres. Such a spherical system is not as appropriatefor spherical harmonics representation as the previous twoapproaches.

Spherical HarmonicsThe Classical Definitions

Many problems of classical physics and mathematics(12–14, 19–22) involve the solution of a partial differentialequation of the second order known as Laplace’s equation:1

=2ψ = 0 ; =2 = ∂2

∂x 2+ ∂2

∂y 2+ ∂2

∂z 2(2)

=2 is the Laplace’s operator, Laplacian (given in Cartesiancoordinates in eq 2), and ψ is a harmonic or harmonic func-tion. ψ is a spherical harmonic when eq 2 is solved in spheri-cal coordinates. Whenever eq 2 is solved by the separationof variables in spherical coordinates (spherical form of =2 isin Table 1), particular solutions occur, among which a solidspherical harmonic of the first kind T 2 is one of the most gen-eral solutions:3

T r, θ, ϕ

= ra

lal P l cos θ + al mcos mϕ + bl msin mϕ Pl m cos θΣ

m =1

l

Σl=1

∞(3)

where al , alm , bl m are arbitrary constants, and Pl (cos θ) andPl m (cos θ) are Legendre’s polynomial of degree (rank) l andassociated Legendre’s function of degree l and order m, re-spectively, both of the first kind:

Pl x = 1

2 ll!⋅ d l

dxlx 2 – 1

l

P l m x = 1 – x 2m/ 2 d m

d x m⋅ Pl x

x = cos θ

(4)

where l and m are positive integers (including zero values)satisfying m < l + 1.

The solution of eq 2 used in many physicochemicalproblems must satisfy certain boundary conditions:4 ψ isperiodic in ϕ so ψ(ϕ) = ψ(ϕ + 2π), it is a function of cos θbut not of θ, and many problems require bl m= i al m (i is thesquare root of {1).

Here are the special cases of the T harmonic (12–14,19–22):

1. A solid spherical harmonic of degree l, of the first kindTl (T with summation only over m);

2. A solid spherical harmonic of degree l and order m, ofthe first kind Tlm (it is without any summation, and isgiven in Table 1 as linear combination of two elemen-tary solutions);

3. A surface spherical harmonic of degree l, of the firstkind Yl (derived from Tl for r = 1);



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Fψ = Eψ F = H + G

L2 = {Λ2 Lz = {i ∂∂ϕ

L2ψ = l l + 1 hψ

Lzψ = m hψ

L2ψ = L L + 1 hψ

Lzψ = M hψ

J2 = {Λ2

J2ψ = 2 I Eh2

ψ

Rnl r = Cnl e{ ρ/2ρlLn+ l

2 l+1 ρ ρ = 2Z rn a0

En = {µe4

8h2ε02

Z2

n2

ψnlm r,θ,ϕ = Rnl r Ylm θ,ϕEnl

ψlm θ,ϕ = Ylm θ,ϕ

L2 = l l + 1 h Lz = mh

ψLM θ,ϕ = YLM θ,ϕ

L2 = L L + 1 h Lz = Mh

ψjm θ,ϕ = Yjm θ,ϕ

Ej = j j + 1 h2

2I

ψklm r,θ,ϕ = Ckljl k r Ylm θ,ϕ

ψnlm r,θ,ϕ = Cnl α rne{ α r 2 / 2 ap α r

pΣp

Ylm θ,ϕ

En = h2ω n + 32

ψnlm r,θ,ϕ = Rnl r Ylm θ,ϕ e{ iEnlh / t

Enl

ψnlm r,θ,ϕ = Rnl r Ylm θ,ϕEnl

ψnlm r,θ,ϕ = Rnl r Ylm θ,ϕ

=2 = 1r2

∂∂r r2 ∂

∂r + Λ2

=2ψ = 0

=2ψ + 4πρ = 0

=2ψ – 1α2

∂ψ∂t

= 0

=2ψ – 1ν2

∂2ψ

∂t2= 0

=2ψ + k2ψ = 0

=2ψ +2µ

h2E – 1

2mω2r2 ψ = 0

=2ψ +

2µh

U(r)h

– i ∂∂t

ψ = 0

=2ψ +2µ

h2E – U(r) ψ = 0

=2ψ +

2µ

h2E + Z e2

4πε0rψ = 0

ψklm r,θ,ϕ, t = Ckljl k r Ylm θ,ϕ e{ ikν t

Λ2 = 1sinθ

∂∂θ sinθ ∂

∂θ + 1sin2θ

∂2

∂ϕ2

ψnlm r,θ,ϕ = A rn + B r{ n–1 Ylm θ,ϕ

the spherical Green function expression

ψklmλ r,θ,ϕ, t = Cklλjl k r Ylm θ,ϕ e{ λ2α2t



4. A tesseral harmonic of degree l and order m, of thefirst kind

Yl m θ,ϕ = Nl m Pl m cos θ Φm ϕ

Nl m =2l + 1

2π⋅

l – m !

l + m !

(5)

where Nlm is a normalizing constant and Φm(ϕ) =cos(mϕ)or sin(mϕ);

5. A surface spherical harmonic of degree l and order m , ofthe first kind Ylm (derived from Yl with blm = ialm and with-out any summation, and taking Φm(ϕ) = (2π){1/2 e imϕ);

6. A sectoral harmonic of degree l , of the first kind Yll(this is a tesseral harmonic with l = m);

7. A surface zonal harmonic of degree l , of the first kind(this is another name for the associated Legendre’sfunction);

8. A solid zonal harmonic of degree l, of the first kind(for r ≠ 1 this is the previous function times r l ).

The term spherical harmonic is commonly used for themost general solutions as well as for any special case. Alsothe term spherical is dropped from solid and surface sphericalharmonics. The best way to avoid this confusion is to look atthe exact definition of the harmonic functions.

The Quantum Mechanical DefinitionSpherical harmonics in quantum mechanics (2, 3, 8, 23–

39) are defined by equations

Yl m θ ,ϕ = Θl m θ Φm ϕ

Θl m θ = pl mPl m cos θ Nl m = 2l + 12π

⋅l – m !

l + m !

Φm ϕ = 12π

⋅ e i m ϕ

(6)

where |m| is used instead of m in the associated Legendre’sfunction (eq 4), and m is an integer satisfying |m| < l + 1. Astandard phase factor plm is not uniquely defined in the lit-erature (31), but usually it is (+1) (2, 3, 8, 23, 26, 29) or({1)(m+|m|)/2 (25, 30, 34–38). Some authors use Plm(cos θ) orΘlm(θ) with their own phase factors and pl m = (+1) (32, 33):

Pl,{m cos θ = {1m

Pl, m cos θ

or

Θl,{ m θ = {1m Θl m θ

(7)

The relation between the Ylm(θ, ϕ) and its complex con-jugate Y *

l m(θ, ϕ) is

Yl,{m θ ,ϕ = Yl m* θ ,ϕ or Yl,{m θ ,ϕ = {1

mYl m

* θ ,ϕ (8)

The first case in eq 8 is valid when plm = (+1) in eq 6,and the second one when pl m = ({1)( m+|m| )/2 in eq 6 or pl m =

(+1) in eq 7. Real spherical harmonics are surface sphericalharmonics with m = 0, tesseral harmonics, and real linearcombinations of the Yl m(θ,ϕ) functions. The term real spheri-cal harmonics is also referred to ylm(p) functions (40, 41) de-fined by

yl0 θ,ϕ = Yl0 θ,ϕ

yl m+ θ,ϕ = 12

Yl m θ,ϕ + Yl,{m θ,ϕ

yl m{θ,ϕ =

{i

2Yl m θ,ϕ – Yl,{m θ,ϕ

(9)

where the Ylm(θ,ϕ) harmonic has a phase factor plm = (+1).Real spherical harmonics represent directional dependenceof many multivariable functions F(ν1, ν2, … , νn, θ, ϕ). Whenν1 = const1, ν2 = const2, …, νn = constn, the functions be-come F(θ,ϕ) containing one or more single, double, or mul-tiple summations including the Yl m(θ,ϕ). The function F(θ,ϕ)can be represented as a contour graph on the surface of asphere of arbitrary radius (42, 43), since the Yl m(θ,ϕ) har-monic itself is a function of the two coordinates (θ,ϕ) onsuch surface.

From a pure mathematical point of view, every Y spheri-cal harmonic is a solution of partial differential equation ofthe second order

Λ2Y + l (l + 1)Y = 0 (10)

where Λ is the Legendre’s operator (Legendrian), the angularpart of the =2 operator (it is given in Table 1). In fact, eq 10is the angular part of eq 2 as the result of the separation ofvariables in spherical coordinates

1. in magnetic resonance spectroscopy: Y2m (θ,ϕ) in posi-tional vectors of atomic nuclei (44)

2. in quantum chemistry: in atomic orbitals—hydrogen-like orbitals (2–8, 26, 29, 34, 35, 37, 38, 45 ), Slater’sand Gaussian orbitals (36, 46), Xα orbitals (47), nu-merical Hartree–Fock orbitals (48), and others.

Functions as Summations or Integrals ContainingSurface Spherical Harmonics

Many angular dependent functions contain the Ylm(θ,ϕ)functions in the form of linear combinations such as

tesseral and real surface spherical harmonics (eqs 6–9)cubic and other symmetry-adapted spherical harmonics(10, 40, 49)electrostatic potentials in the crystal field theory (50)angular parts of some hybrid orbitals (51)Hirschfeld’s cosn(θk) angular functions in the multipolerefinement formalism (52), etc.

single, double, or multiple summations such as

general solutions of some differential equations (repre-sented in Table 1)wave functions in scattering theory, containing theYl0(θ, ϕ) harmonics (53 )angular functions in the multipole refinement and chargedensity studies (41, 54 )Patterson (55) and orientation distribution functions (43)used in crystallography, etc.



and single, double, or multiple integrals, as for example mul-tipole moments and fields (56, 57).

Occurrence of the Double Products and Squares ofSurface Spherical Harmonics

Functions Containing the double Products or Squaresof Surface Spherical Harmonics

There are many functions containing the squares|Ylm(θ,ϕ)|2 = Y*

lm(θ, ϕ)Ylm(θ,ϕ) or the double productsY*

l1m1(θ,ϕ)Yl2m2

(θ,ϕ). For example, the squares represent an-gular one-electron probability densities (2, 6, 29, 8) and an-gular distributions of multipole radiation (57). The doubleproduct is, for example, transitional charge density or chargedistribution from the multipole (58), or is just an angulardouble-electron wave function (59).

Occurrence of Surface Spherical Harmonics

Occurrence in Partial Differential EquationsSome partial differential equations of the second order

are given in Table 1. Whenever such an equation is solvedby the separation of variables in spherical coordinates, themost elementary solution is a product of an angular func-tion (usually a Yl m harmonic) times a radial and other func-tions. More general solutions include such products in linearcombinations or summations over l, m, and other integers.Various physical quantities are represented by the ψ function,as can be seen from Table 1. Five equations on the top ofTable 1 are well known in mathematics, as special functionsand solutions of some partial differential equations of the sec-ond order (19–21, 60), and in classical physics in regard topotential fields, heat, wave, and diffusion phenomena (20–23, 61), and especially gravitational (62–64) and electrostat-ics (65) problems.

The Schrödinger equation occurs in many quantum me-chanics problems, as, for example, one-particle problems (aharmonic oscillator [56, 66], a particle on a sphere [37] andin a spherical box [67], a free particle [7, 68], angular mo-mentum of a single particle [6, 27, 29, 35, 39, 56, 69]); two-particle problems (hydrogen-like atoms [2–8, 26, 29, 34, 35,37, 38, 45 ],5 a rigid rotator [4, 28, 70], rotation of a di-atomic molecule [4, 34, 71]; and many-particle problems(many-electron atoms in self-consistent field calculations[48], nuclear models [72, 73 ], angular momentum of a com-posite system [27, 74 ], rotation of a polyatomic linear mol-ecule [75 ], quantum-size effects in colloidal chemistry [76]).

Functions in Terms of Surface Spherical HarmonicsSurface spherical harmonics occur in simple definitions

or expressions for many functions F(ν1 , ν2, …, νn, θ, ϕ)—for example, other kinds of harmonic functions:6 solid spheri-cal harmonics (eq 3), four- and higher-dimensional surfacespherical harmonics (62, 64, 77 ), tensor and vector surfacespherical harmonics (61, 63, 64, 78), spherical harmonics withspin (spinors) (79, 80), generalized spherical harmonics (81 );in magnetic resonance spectroscopy: Y2m(θ,ϕ) in positionalvectors of atomic nuclei (44 ); in quantum chemistry: inatomic orbitals—hydrogen-like orbitals (2–8, 26, 29, 34, 35,37, 38, 45), Slater’s and Gaussian orbitals (36, 46 ), Xα or-bitals (47 ), numerical Hartree–Fock orbitals (48), and others.

Functions as Summations or Integrals ContainingSurface Spherical Harmonics

Very common functions defined as summations and in-tegrals containing the squares |Yl m(θ,ϕ)|2 or the productsY *l1 m1

(θ,ϕ) Yl 2 m 2(θ,ϕ) are electrostatic potential, 1/r operator,

and various electron–electron interaction integrals (82).

Concluding Remarks

There are many fields in which surface spherical harmon-ics or their squares occur and are used to define and calculatevarious quantities, or are simply solutions of some differentialequations solved in spherical coordinates. In some fields suchas quantum chemistry, there is a great need for exact graphicalrepresentation of these functions, especially in spherical coordi-nates. The first step for undergraduates when introduced to thesefunctions is to understand what they are and to learn necessarymathematical concepts for further considerations. The nexttask is to become familiar with the rules of graphical repre-sentation of the harmonics and their squares, then to try tomake accurate hand-drawings, and finally to visualize the exactsize, shape, and symmetry of these functions.

Acknowledgments

Thanks to Márcia M. C. Ferreira and Sanja Tomic forreading this manuscript and giving me constructive com-ments and suggestions to improve it.

Notes

1. Laplace introduced this equation into the theory of potentialin 1782. A brief historical sketch of this subject with references can befound in some monographs (19 ).

2. The character ψ is not so commonly used for various sphericalharmonics as T, Y, P.

3. Since the general solution of the Laplace’s equation cannot befound, the solution defined by eq 3 or any other most general solutionis always less general than the general solution.

4. Boundary condition means a function (solution) that satisfiescertain conditions (properties of the function or its derivative) in certainpoints or regions of its variables, to give a unique solution to a problem.

5. Three simplifications are understood here: relativistic effectsare ignored, an ideal ground state is assumed, and the state is in absenceof external fields. In a real situation, the surface spherical harmonicsthat are angular parts of atomic orbitals are significantly changed ow-ing to the relativistic effects (83 ), external fields (84, 85 ) and elec-tronic transitions (85 ).

6. Both the surface and the solid spherical harmonics could beclassified according to the number of their variables (two-, three-, four-,or many-variable harmonics; or analogously, three-, four-, five-, andhigher-dimensional harmonics), or kind of their variables (scalar, vector,or tensor harmonics). The surface spherical harmonics that are subjectof this paper (eqs 5–9) are three-dimensional (or two-variable) and scalarfunctions.

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