Three-Dimensional Representation of Surface Spherical Harmonics

Research: Science & Education

Three-Dimensional Representation of Surface Spherical Harmonics and Their Squares Using Normal Projections: Some Comments on the Functions Preparing for an Undergraduate Exercise Rudolf Kiralj Rudjer Boˇs ković Institute, P.O.B. 1016, HR-10 001 Zagreb, Croatia

Most chemistry undergraduates encounter surface spherical harmonics and their squares for the first time in quantum chemistry courses, interpreted as angular wave functions and angular probability densities, respectively. The teaching of modern chemistry leans more heavily on quantum theory than on any other single pillar. Thus, answers to students’ questions like “What do these functions represent?” or “How do these functions look?” cannot be avoided and constructive visualization aids must be developed. Besides, surface spherical harmonics occur not only in quantum chemistry but also in a large variety of physicochemical problems dealing with spherical symmetry or spherical surfaces. In trying to meet visualization needs and difficulties of these functions one could use a good quality illustration, a suitable computer graphic, a model made from a solid material, or one’s own drawing. Nevertheless, problems can arise with these possibilities also. In many recent quantum chemistry textbooks the graphical representation of harmonics is treated as not worthy to be mentioned. The insufficiently detailed and 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 few advanced texts that nicely explain two-dimensional representation of surface spherical harmonics and their squares (1–8), but the cross sections used are not always good indications of the overall shapes of the functions. However, many students find it difficult to appreciate the directional properties of the functions even using the cross sections. Some nice and sophisticated computer-generated three-dimensional polar plots of the surface spherical harmonics and their squares have been appeared in last few years (9, 10). When computer graphics is used, familiarity with mathematical concepts, computer programing, and graphics is necessary. Accurate hand drawings require fairly detailed knowledge of calculus, geometry, and drawing. Moreover, the harmonics are moderately difficult functions to draw. A simple remedy to all these problems could be to give students exercises and homework problems dealing with the harmonics, having them make their own computer-generated or hand drawings. This paper focuses on some important concepts as the result of a literature survey regarding the definition of surface spherical harmonics and spherical coordinates, and the occurrence of the harmonics and their squares in various fields. It is assumed that the student has become familiar with this subject before any serious approach to visualization of the functions. In a companion paper to this one, a drawing system clear to undergraduates, based on normal projections, will be given (Kiralj, R., manuscript in preparation). Calculated coordinates will be used for accurate three-dimensional 332

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

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


hand drawings of some surface spherical harmonics and their squares in spherical coordinates, and the worked examples will be explained in detail. Some Basic Concepts

Spherical Polar Coordinate System The spherical polar coordinate system is one of the most frequently used coordinate systems in three-dimensional Euclidean space. The spherical system could be defined and explained from several points of view—which should be made clear to undergraduates. Three definitions of spherical system are explained here: geometrical, algebraic, and physical. The Geometrical Definition Spherical polar coordinate system is completely defined by two intersecting axes at right angles, one horizontal and another vertical (polar axis), with the origin or pole O at the point of the intersection (Fig. 1, top). Spherical coordinates of a point P (the length r of the radius vector, the colatitude θ , and the azimuth ϕ) are specified by intersection of three coordinate surfaces: the concentric sphere about the origin (r = const), the right circular cone with apex at the origin and axis along the polar axis (θ = const), and the half-plane from the polar axis (ϕ = const) such that 0 ≤ r ≤ ∞, 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π. The spherical system with this (geometrical) definition is very useful in pictorial representation of various phenomena, and it occurs sometimes in chemistry (11). The Algebraic Definition Many problems in physics and chemistry are treated in both Cartesian and spherical coordinate systems. The threedimensional (rectangular) Cartesian system (Fig. 1, middle) is defined by three mutually perpendicular axes X, Y, Z intersecting in the point at the origin O. The Cartesian coordinates x, y, z of a point P are specified by intersection of three planes: the planes perpendicular to the X (x = const), Y (y = const), and Z (z = const) axes. The x, y, z coordinates are positions of the πx, πy, πz planes from the coordinate planes (YZ, XZ, XY) such that ᎑ ∞ ≤ x, y, z ≤ + ∞. If the Cartesian system is superimposed on the spherical system in the way that the origin, the Z axis, and the XZ plane of the Cartesian system coincide with the pole, the polar axis, and the initial meridian plane of the spherical system, respectively (Fig. 1, bottom), the coordinate transformations (12–14) are x = r sin θ cosϕ r=

y = r sinθ sinϕ ᎑1

x 2 + y 2 + z 2 ; ϕ = tan

y ᎑1 θ x ; = cos

z = r cosθ z x 2+y2+z2

(1)

This is the easiest way to transform spherical coordinates to Cartesian and vice versa. This definition of the spherical system (algebraic) enables very accurate drawings, since coordinates are calculated and then plotted. More complicated objects can be drawn this way, rather than in the geometrical approach to the spherical system. The Physical Definition In crystallography (15) and crystal physics (16), in astronomy (17), in geography and cartography (18) and some other fields, the spherical system is defined by real physical objects (crystal, celestial objects, earth): the radius r is always constant or undetermined, and the sphere is divided into

hemispheres. Such a spherical system is not as appropriate for spherical harmonics representation as the previous two approaches.

Spherical Harmonics The Classical Definitions Many problems of classical physics and mathematics (12–14, 19–22) involve the solution of a partial differential equation of the second order known as Laplace’s equation: 1 ⵜ2ψ = 0 ;

ⵜ2 =

∂2 ∂x 2

∂2

+

∂y 2

+

∂2

(2)

∂z 2

ⵜ2 is the Laplace’s operator, Laplacian (given in Cartesian coordinates in eq 2), and ψ is a harmonic or harmonic function. ψ is a spherical harmonic when eq 2 is solved in spherical coordinates. Whenever eq 2 is solved by the separation of variables in spherical coordinates (spherical form of ⵜ 2 is in Table 1), particular solutions occur, among which a solid spherical harmonic of the first kind T 2 is one of the most general solutions: 3 T r, θ, ϕ ∞

=Σ l=1

r a

(3) l

a l P l cos θ +

l

Σ

m =1

a l mcos m ϕ + b l msin m ϕ P l m cos θ

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

d Pl x = 1 ⋅ x2–1 l l 2 l! dx Pl m x = 1 – x 2

m/ 2

d

m

dx

m

l

⋅ Pl x

(4)

x = cos θ where l and m are positive integers (including zero values) satisfying m < l + 1. The solution of eq 2 used in many physicochemical problems must satisfy certain boundary conditions:4 ψ is periodic in ϕ so ψ( ϕ) = ψ (ϕ + 2π), it is a function of cos θ but not of θ , and many problems require bl m= i al m (i is the square 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 kind T l (T with summation only over m); 2. A solid spherical harmonic of degree l and order m, of the first kind Tlm (it is without any summation, and is given in Table 1 as linear combination of two elementary solutions); 3. A surface spherical harmonic of degree l, of the first kind Yl (derived from Tl for r = 1);

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

333


Table 1. Some Differential Equations of the Second Order Whose Solutions contain the Surface Spherical Harmonics Ylm (␪, ␸) Name of Equation EQUATIONS WITH LAPLACEAN ⵜ2

Equation

Solution

ⵜ2 = 12 ∂ r 2 ∂ + Λ2 ∂r r ∂r

∂2 Λ2 = 1 ∂ sin θ ∂ + 12 2 ∂θ sin θ ∂θ sin θ ∂ϕ

Laplace's equation

ⵜ2ψ = 0

ψnlm r, θ, ϕ = A r + B r

Poisson's equation

ⵜ2ψ + 4πρ = 0

the spherical Green function expression

Diffusion equation

∂ψ ⵜ2ψ – 12 =0 α ∂t

ψklmλ r, θ, ϕ, t = C klλj l k r Y lm θ, ϕ e ᎑ λ α t

Wave equation

∂2ψ ⵜ2ψ – 12 2 = 0 ν ∂t

ψklm r, θ, ϕ, t = C kl j l k r Y lm θ, ϕ e ᎑ ikν t

Helmholtz's equation

ⵜ2ψ + k ψ = 0

n

᎑ n–1

Y lm θ, ϕ

2 2

ψklm r, θ, ϕ = C kl j l k r Y lm θ, ϕ

2

Schrödinger equations (nonrelativistic, absence of external fields); H ψ = E ψ

for an isotropic spherical oscillator

for a 2-particle problem time-dependent

for a 2-particle problem time-independent

for a hydrogen-like atom

ⵜ2ψ +

ⵜ2ψ +

ⵜ2ψ + ⵜ2ψ +

2µ h

2

E – 1 mω2r 2 ψ = 0 2

n

ψnlm r, θ, ϕ = C nl α r e ᎑

αr

2

2

2µ U(r) –i ∂ ψ=0 h h ∂t

ψnlm r, θ, ϕ = R nl r Y lm θ, ϕ e ᎑ iE nl h / t

2µ

ψnlm r, θ, ϕ = R nl r Y lm θ, ϕ

h

2

2µ h

2

E – U(r) ψ = 0

E nl

E nl 2

E+

Ze ψ=0 4πε0r

ψnlm r, θ, ϕ = R nl r Y lm θ, ϕ 2 l+1

En = ᎑

µe 4 Z 2 2 2 8h ε2 n

ρ = 2Z r n a0

0

for a single electron in an atom (SCF) Fψ = Eψ

F =H +G

ψnlm r, θ, ϕ = R nl r Y lm θ, ϕ E nl

EQUATIONS WITH LEGENDRIAN Λ2 2

L = ᎑Λ2

L z = ᎑i ∂ ∂ϕ

L ψ = l l + 1 hψ

ψlm θ, ϕ = Y lm θ, ϕ

L z ψ = m hψ

L =l l +1 h

2

for a single particle

for a composite system

334

L z = mh

ψLM θ, ϕ = Y LM θ, ϕ

L z ψ = M hψ

L =L L +1 h

2

J ψ = 2 I 2E ψ h 2

2

L z = Mh

2

Equations for some molecular rotations: J = ᎑Λ

for rotation of a rigid rotor, a diatomic and a polyatomic linear molecule

2

L ψ = L L + 1 hψ

2

Σp a p α r p Y lm θ, ϕ

En = h ω n + 3 2

R nl r = C nl e ᎑ ρ/2ρl L n+ l ρ

Angular momentum equations:

/2

ψjm θ, ϕ = Y jm θ, ϕ 2

Ej = j j + 1

h 2I


Research: Science & Education 4. A tesseral harmonic of degree l and order m, of the first kind

(+1) in eq 7. Real spherical harmonics are surface spherical harmonics with m = 0, tesseral harmonics, and real linear combinations of the Yl m( θ, ϕ) functions. The term real spherical harmonics is also referred to ylm(p) functions (40, 41) defined by

Y l m θ,ϕ = N l m P l m cos θ Φm ϕ (5)

2l + 1 l – m ! ⋅ 2π l+m !

Nl m =

5.

6. 7.

8.

y l0 θ, ϕ = Y l0 θ, ϕ

where Nlm is a normalizing constant and Φm(ϕ) = cos(m ϕ) or sin(m ϕ); A surface spherical harmonic of degree l and order m, of the first kind Ylm (derived from Yl with blm = i alm and without any summation, and taking Φm(ϕ) = (2π)᎑1/ 2 e imϕ); A sectoral harmonic of degree l, of the first kind Yll (this is a tesseral harmonic with l = m); A surface zonal harmonic of degree l, of the first kind (this is another name for the associated Legendre’s function); A solid zonal harmonic of degree l, of the first kind (for r ≠ 1 this is the previous function times r l ).

y l m+ θ, ϕ = 1 Y l m θ, ϕ + Y l,᎑ m θ, ϕ 2 y l m᎑ θ, ϕ =

The term spherical harmonic is commonly used for the most general solutions as well as for any special case. Also the term spherical is dropped from solid and surface spherical harmonics. The best way to avoid this confusion is to look at the exact definition of the harmonic functions. The Quantum Mechanical Definition Spherical harmonics in quantum mechanics (2, 3, 8, 23– 39) are defined by equations

Φm ϕ =

m

cos θ

l– m ! 2l + 1 ⋅ 2π l+ m !

Nl m =

(6)

1 ⋅ e imϕ 2π

where |m| is used instead of m in the associated Legendre’s function (eq 4), and m is an integer satisfying |m| < l + 1. A standard phase factor plm is not uniquely defined in the literature (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 ) or Θ lm(θ ) with their own phase factors and pl m = (+1) (32, 33):

P l,᎑m cos θ = ᎑1

m

P l,

m

cos θ

or Θl,᎑

m

2

Y l m θ, ϕ – Y l,᎑ m θ, ϕ

where the Ylm( θ, ϕ) harmonic has a phase factor plm = (+1). Real spherical harmonics represent directional dependence of many multivariable functions F(ν1, ν 2, … , ν n, θ, ϕ). When ν 1 = const1, ν 2 = const2, …, νn = constn, the functions become F(θ,ϕ) containing one or more single, double, or multiple summations including the Yl m(θ, ϕ). The function F(θ, ϕ) can be represented as a contour graph on the surface of a sphere of arbitrary radius (42, 43), since the Yl m(θ , ϕ) harmonic itself is a function of the two coordinates (θ , ϕ) on such surface. From a pure mathematical point of view, every Y spherical harmonic is a solution of partial differential equation of the second order Λ2Y + l (l + 1)Y = 0

Y l m θ , ϕ = Θl m θ Φm ϕ

Θl m θ = p l mP l

᎑i

(9)

θ = ᎑1

m

(7)

Θl m θ

where Λ is the Legendre’s operator (Legendrian), the angular part of the ⵜ 2 operator (it is given in Table 1). In fact, eq 10 is the angular part of eq 2 as the result of the separation of variables in spherical coordinates 1. in magnetic resonance spectroscopy: Y2m (θ,ϕ) in positional vectors of atomic nuclei (44) 2. in quantum chemistry: in atomic orbitals—hydrogenlike orbitals (2–8, 26, 29, 34, 35, 37, 38, 45 ), Slater’s and Gaussian orbitals (36, 46), X α orbitals (47), numerical Hartree–Fock orbitals (48), and others.

Functions as Summations or Integrals Containing Surface Spherical Harmonics Many angular dependent functions contain the Y lm(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(θ k) angular functions in the multipole refinement formalism ( 52), etc.

single, double, or multiple summations such as

The relation between the Ylm( θ, ϕ) and its complex conjugate Y *l m(θ , ϕ) is Y l,᎑m θ , ϕ = Y l*m θ , ϕ or Y l,᎑m θ , ϕ = ᎑1

(10)

m

Y l*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 =

general solutions of some differential equations (represented in Table 1) wave functions in scattering theory, containing the Yl0(θ , ϕ) harmonics (53 ) angular functions in the multipole refinement and charge density studies (41, 54 ) Patterson (55) and orientation distribution functions (43) used in crystallography, etc.


335


and single, double, or multiple integrals, as for example multipole moments and fields (56, 57). Occurrence of the Double Products and Squares of Surface Spherical Harmonics

Functions as Summations or Integrals Containing Surface Spherical Harmonics Very common functions defined as summations and integrals containing the squares |Yl m( θ, ϕ)|2 or the products Y *l m (θ , ϕ) Yl m (θ , ϕ) are electrostatic potential, 1/r operator, and various electron–electron interaction integrals (82). 1

Functions Containing the double Products or Squares of Surface Spherical Harmonics There are many functions containing the squares |Y lm(θ,ϕ)| 2 = Y *lm(θ, ϕ)Y lm(θ,ϕ) or the double products Y*l m (θ,ϕ)Yl m (θ,ϕ). For example, the squares represent angular one-electron probability densities (2, 6, 29, 8) and angular distributions of multipole radiation (57). The double product is, for example, transitional charge density or charge distribution from the multipole (58), or is just an angular double-electron wave function (59). 1

1

2

2

Occurrence of Surface Spherical Harmonics

Occurrence in Partial Differential Equations Some partial differential equations of the second order are given in Table 1. Whenever such an equation is solved by the separation of variables in spherical coordinates, the most elementary solution is a product of an angular function (usually a Yl m harmonic) times a radial and other functions. More general solutions include such products in linear combinations 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 of Table 1 are well known in mathematics, as special functions and solutions of some partial differential equations of the second order (19–21, 60), and in classical physics in regard to potential fields, heat, wave, and diffusion phenomena (20– 23, 61), and especially gravitational (62–64) and electrostatics (65) problems. The Schrödinger equation occurs in many quantum mechanics problems, as, for example, one-particle problems (a harmonic oscillator [56, 66], a particle on a sphere [37] and in a spherical box [67], a free particle [7, 68], angular momentum of a single particle [6, 27, 29, 35, 39, 56, 69]); twoparticle problems (hydrogen-like atoms [2–8, 26, 29, 34, 35, 37, 38, 45 ],5 a rigid rotator [4, 28, 70], rotation of a diatomic 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 composite system [27, 74 ], rotation of a polyatomic linear molecule [75], quantum-size effects in colloidal chemistry [76]). Functions in Terms of Surface Spherical Harmonics Surface 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 spherical harmonics (eq 3), four- and higher-dimensional surface spherical harmonics (62, 64, 77 ), tensor and vector surface spherical harmonics (61, 63, 64, 78), spherical harmonics with spin (spinors) (79, 80), generalized spherical harmonics (81 ); in magnetic resonance spectroscopy: Y2m( θ, ϕ) in positional vectors of atomic nuclei (44 ); in quantum chemistry: in atomic orbitals—hydrogen-like orbitals (2–8, 26, 29, 34, 35, 37, 38, 45), Slater’s and Gaussian orbitals (36, 46 ), Xα orbitals (47 ), numerical Hartree–Fock orbitals (48), and others. 336

1

2

2

Concluding Remarks There are many fields in which surface spherical harmonics or their squares occur and are used to define and calculate various quantities, or are simply solutions of some differential equations solved in spherical coordinates. In some fields such as quantum chemistry, there is a great need for exact graphical representation of these functions, especially in spherical coordinates. The first step for undergraduates when introduced to these functions is to understand what they are and to learn necessary mathematical concepts for further considerations. The next task is to become familiar with the rules of graphical representation of the harmonics and their squares, then to try to make accurate hand-drawings, and finally to visualize the exact size, shape, and symmetry of these functions. Acknowledgments Thanks to Márcia M. C. Ferreira and Sanja Tomić for reading this manuscript and giving me constructive comments and suggestions to improve it. Notes 1. Laplace introduced this equation into the theory of potential in 1782. A brief historical sketch of this subject with references can be found in some monographs (19 ). 2. The character ψ is not so commonly used for various spherical harmonics as T, Y, P. 3. Since the general solution of the Laplace’s equation cannot be found, the solution defined by eq 3 or any other most general solution is always less general than the general solution. 4. Boundary condition means a function (solution) that satisfies certain conditions (properties of the function or its derivative) in certain points or regions of its variables, to give a unique solution to a problem. 5. Three simplifications are understood here: relativistic effects are ignored, an ideal ground state is assumed, and the state is in absence of external fields. In a real situation, the surface spherical harmonics that are angular parts of atomic orbitals are significantly changed owing to the relativistic effects (83 ), external fields (84, 85 ) and electronic transitions (85 ). 6. Both the surface and the solid spherical harmonics could be classified according to the number of their variables (two-, three-, four-, or many-variable harmonics; or analogously, three-, four-, five-, and higher-dimensional harmonics), or kind of their variables (scalar, vector, or tensor harmonics). The surface spherical harmonics that are subject of this paper (eqs 5–9) are three-dimensional (or two-variable) and scalar functions.

Literature Cited 1. White, H. E. Phys. Rev. 1931, 37, 1416–1424. Friedman, H. G., Jr.; Choppin, G. R.; Feuerbacher, D. G. J. Chem. Educ. 1964, 41, 354–358. Becker, C. J. Chem. Educ. 1964, 41, 358–360. 2. White, H. E. Introduction to Atomic Spectra; McGraw-Hill: New York, 1934; Chapter IV. 3. Metz, C. R. Physical Chemistry; McGraw-Hill: New York, 1976; Chapter 12. 4. Pilar, F. L. Elementary Quantum Mechanics; McGraw-Hill: New York, 1968; Chapter 7.


Research: Science & Education 5. Anderson, J. M. Introduction to Quantum Chemistry; Benjamin: New York, 1969; Chapter 6. 6. Eisberg, R.; Resnick, R. Quantum Physics, 2nd ed.; Wiley: New York, 1985; Chapter 7. 7. Levich, B. G.; Myamlin, V. A.; Vdovin, Yu. A. Theoretical Physics; North-Holland: Amsterdam, 1973; Vol. 3, Chapter 4. 8. Pauling, L.; Wilson, E. B. Introduction to Quantum Mechanics; McGraw-Hill: New York, 1935; Chapter V. 9. Breneman, G. L. J. Chem. Educ. 1988, 65, 31–33. Breneman, G. L.; Parker, O. J. J. Chem. Educ. 1992, 69, A4–A7. Ferreira, M. M. C.; Porto, M. E. G. Quim. Nova 1993, 16, 589–595. Zimmermann, R. L.; Olness, F. I. Mathematica for Physics; Addison-Wesley: Reading, MA, 1995; cover page and p 364. See also Pines, A. Isr. J. Chem. 1992, 32, 137–144. 10. Elcoro, L.; Perez-Mato, J. M.; Madariaga, G. Acta Crystallogr., Sect. A 1994, A50, 182–193. Elcoro, L.; Perez-Mato, J. M. Acta Crystallogr., Sect. B 1994, B50, 294–306. 11. Cremer, D.; Pople, J. A. J. Am. Chem. Soc. 1975, 97, 1354–1358. Boeyens, J. C. A. J. Cryst. Mol. Struct. 1978, 8, 317–320. Musin, R. N.; Schastnev, P. V. Zh. Strukt. Khim. 1976, 17, 419–425. Minyaev, R. M.; Minkin, V. I.; Kletsky, M. E. Zh. Org. Khim. 1978, 14, 449–455. 12. Margenau, H.; Murphy, G. M. The Mathematics of Physics and Chemistry, 2nd ed.; Van Nostrand: Princeton, NJ, 1964; Chapter 5. 13. Boas, M. L. Mathematical Methods in the Physical Sciences; Wiley: New York, 1966; Chapter 4. 14. Korn, A. G.; Korn, T. M. Mathematical Handbook, 2nd ed.; McGraw-Hill: New York, 1968; Chapter 6. 15. Azároff, V. Elements of X-ray Crystallography; McGraw-Hill: New York, 1968; Chapter 2. Glusker, J. P.; Lewis, M.; Rossi, M. Crystal Structure Analysis for Chemists and Biologists; VCH: New York, 1994; p 56. 16. Sirotin, Yu. I.; Shaskolskaya, M. P. Fundamentals of Crystal Physics; Mir: Moscow, 1982; pp 23–27. 17. Lippincott, K. Astronomy; Dorling Kindersley: London, 1995. Levy, D. H. The Sky: A User’s Guide; Cambridge University Press: Cambridge, 1995; especially pp 9–10. Phillips, K. J. H. Guide to the Sun; Cambridge University Press: Cambridge, 1995; especially pp 307–313, 352–365. 18. Robinson, A. H. Elements of Cartography, 2nd ed.; Wiley: New York, 1963; Chapter 2. 19. Byerly, W. E. Harmonic Functions, 4th ed.; Wiley: New York, 1906. Hobson, E. W. The Theory of Spherical and Ellipsoidal Harmonics; Cambridge University Press: Cambridge, 1955. Byerly, W. E. Spherical, Cylindrical, and Ellipsoidal Harmonics; Dover: New York, 1959. 20. Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover: New York, 1965; Chapter 8. Arsenin, V. Ya. Basic Equations and Special Functions of Mathematical Physics; Iliffe: London, 1968; Chapter 12. Hildebrand, F. B. Advanced Calculus for Applications, 2nd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1976; especially pp 511–513. 21. Sneddon, I. N. Elements of Partial Differential Equations; McGrawHill: New York, 1957. 22. Mathews, J.; Walker, R. L. Mathematical Methods of Physics, 2nd ed.; Benjamin: Menlo Park, CA, 1970; Chapter 8. 23. Pilar, F. L. Elementary Quantum Mechanics; McGraw-Hill: New York, 1968; Chapters 6, 7. 24. Anderson, J. M. Introduction to Quantum Chemistry; Benjamin: New York, 1969; Chapter 4, Appendix 2. 25. Levich, B. G.; Myamlin, V. A.; Vdovin, Yu. A. Theoretical Physics; North-Holland: Amsterdam, 1973; Vol. 3, Chapter 3. 26. Harnwell, G. P.; Stephens, W. E. Atomic Physics; McGraw-Hill: New York, 1955; Chapter 3. 27. Landau, L. D.; Lifshitz, E. M. Quantum Mechanics, 2nd ed.; Pergamon: Oxford, 1965; Chapter IV. 28. Flurry, R. L. Quantum Chemistry; Prentice-Hall: Englewood Cliffs, NJ, 1983; Chapter 3. 29. Lowe, J. P. Quantum Chemistry; Academic: New York, 1978; Chapter 4. 30. Messiah, A. Quantum Mechanics; North-Holland: Amsterdam, 1967; Vol. 1, Appendix B.

31. Butkov, E. Mathematical Physics; Addison-Wesley: Reading, MA, 1968; Chapters 9, 14, especially pp 375, 380, 618. Jackson, J. D. Mathematics for Quantum Mechanics; Benjamin: New York, 1962; Appendix B and references cited therein. 32. Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in Fortran, 2nd ed.; Cambridge University Press: Cambridge, 1992; pp 246–247. 33. Jackson, J. D. Classical Electrodynamics, 2nd ed.; Wiley: New York, 1975; Chapter 3. 34. Troup, G. Understanding Quantum Mechanics; Methuen: London, 1968; Chapter 8. 35. Schiff, L. T. Quantum Mechanics, 3rd ed.; McGraw-Hill Kogakusha: Tokyo, 1968; Chapter 4. 36. McWeeny, R. Methods of Molecular Quantum Mechanics, 2nd ed.; Academic: London, 1978; Appendix 1. 37. Atkins, P. W. Molecular Quantum Mechanics, 2nd ed.; Oxford University Press: Oxford, 1988; Chapter 4. 38. Fermi, E. Notes on Quantum Mechanics; University of Chicago: Chicago, 1960; Sects. 6–8. 39. Fock, V. A. Fundamentals of Quantum Mechanics; Mir: Moscow, 1986; Part II, Chapter IV. 40. International Tables for Crystallography; Shmueli, U., Ed.; Kluwer: Dordrecht, 1993; Vol. B, Section 1.2. 41. Hansen, N. K.; Coppens, P. Acta Crystallogr., Sect. A 1978, A34, 909– 921. Su, Z.; Coppens, P. Acta Crystallogr., Sect. A 1994, A50, 636–643. 42. McMullan, R. K.; Kvick, Å. Acta Crystallogr., Sect. B 1990, B46, 390–399. 43. Finney, J. L. Acta Crystallogr., Sect. B 1995, B51, 447–467. Michael, K. H.; Lamoen, D.; David, W. I. F. Acta Crystallogr., Sect. A 1995, A51, 365–374. 44. Andrew, E. R. Nuclear Magnetic Resonance; Cambridge University Press: Cambridge, 1956; Sects. 5.2., 5.3. 45. Margenau, H.; Murphy, G. M. The Mathematics of Physics and Chemistry, 2nd ed.; Van Nostrand: Princeton, NJ, 1964; Chapter 11. Landau, L. D.; Lifshitz, E. M. Quantum Mechanics, 2nd ed.; Pergamon: Oxford, 1965; Chapter V. Flurry, R. L. Quantum Chemistry; Prentice-Hall: Englewood Cliffs, NJ, 1983; Chapter 5. Messiah, A. Quantum Mechanics; North-Holland: Amsterdam, 1967; Vol. 1, Chapter IX. 46. Clementi, E. Modern Techniques in Computational Chemistry: MOTTEC-91; ESCOM: Leiden, 1991; Chapter 2. 47. Hall, M. B.; Fenske, R. F. Inorg. Chem. 1972, 11, 768–775. Burnsten, B. E.; Fenske, R. F. J. Chem. Phys. 1977, 67, 3138– 3145. Burnsten, B. E.; Jensen, J. R.; Fenske, R. F. J. Chem. Phys. 1978, 68, 3320–3321. 48. Herman, F.; Skillman, S. Atomic Structure Calculations; PrenticeHall: Englewood Cliffs, NJ, 1963; Chapter One. Hartree, D. R. The Calculation of Atomic Structures; Wiley: New York, 1957; especially Chapter 1. Fernández Pacios, L. J. Comp. Chem. 1993, 14, 410– 421; 1995, 16, 133–145.s 49. Cotton, A. F. Chemical Applications of Group Theory, 3rd ed.; Wiley: New York, 1990; Appendix IV. Brackman, C. M. Acta Crystallogr., Sect. A 1987, A43, 270–283. Su, Z.; Coppens, P. Acta Crystallogr., Sect. A 1994, A50, 408–409. 50. Kettle, S. F. A. J. Chem. Educ. 1969, 46, 339–341. Companion, A. L.; Trevor, D. J. J. Chem. Educ. 1975, 52, 710–714. Krishnamurthy, R.; Schaap, W. B. J. Chem. Educ. 1969, 46, 799– 810; 1970, 47, 433–446. 51. Metz, C. R. Physical Chemistry; McGraw-Hill: New York, 1976; Chapter 15. Takazava, H.; Ohba, S.; Saito, Y. Acta Crystallogr., Sect. B 1989, B45, 432–437. 52. Hirshfeld, F. L. Acta Crystallogr., Sect. B 1971, B27, 769–781. Harel, M.; Hirshfeld, F. L. Acta Crystallogr., Sect. B 1975, B31, 162–172. Hirshfeld, F. L. Isr. J. Chem. 1977, 16, 226–229. 53. Anderson, J. M. Introduction to Quantum Chemistry; Benjamin: New York, 1969; Chapter 14. Levich, B. G.; Myamlin, V. A.; Vdovin, Yu. A. Theoretical Physics; North-Holland: Amsterdam, 1973; Vol. 3, Chapter 11. Landau, L. D.; Lifshitz, E. M. Quantum Mechanics, 2nd ed.; Pergamon: Oxford, 1965; Chapter V. Messiah, A. Quantum Mechanics; North-Holland: Amsterdam, 1967; Vol. 1, Chapter IX.


337

Research: Science & Education 54. Stevens, E. D.; Coppens, P. Acta Crystallogr., Sect. B 1980, B36, 1864–1876. Coppens, P. Trans. Am. Crystallogr. Assoc. 1990, 26, 91–103. Craven, B. M.; Stewart, R. F. Trans. Am. Crystallogr. Assoc. 1990, 26, 41–54. 55. International Tables for Crystallography; Shmueli, U., Ed.; Kluwer: Dordrecht, 1993; Vol. B, pp 253–254. Navaza, J. Acta Crystallogr., Sect. A 1987, A43, 645–653. 56. Mayer, M. G.; Jensen, J. H. D. Elementary Theory of Nuclear Shell Structure; Wiley: New York, 1955; Appendix. 57. Jackson, J. D. Classical Electrodynamics, 2nd ed.; Wiley: New York, 1975; Chapters 4, 16. Gelessus, A.; Thiel, W.; Weber, W. J. Chem. Educ. 1995, 72, 505–508. 58. See Problems 9.4 and 16.6 in Jackson, J. D. Classical Electrodynamics, 2nd ed.; Wiley: New York, 1975; pp 461, 777. 59. Manch, W.; Fernelius, W. C. J. Chem. Educ. 1961, 38, 192–201. 60. Spiegel, M. R. Fourier Analysis; McGraw-Hill:New York, 1974; Chapter 7. 61. Sandberg, V. D. J. Math. Phys. 1978, 19, 2441–2446. 62. Hylleraas, E. A. Z. Phys. 1932, 74, 216–224. 63. Regge, T.; Wheeler, J. A. Phys. Rev. 1957, 108, 1063–1069. 64. Lifshitz, E. M.; Khalatnikov, I. M. Adv. Phys. 1963, 12, 185– 249. 65. Jackson, J. D. Classical Electrodynamics, 2nd ed.; Wiley: New York, 1975; especially Chapters 1–4. 66. Anderson, J. M. Introduction to Quantum Chemistry; Benjamin: New York, 1969; Sect. 5-4. Landau, L. D.; Lifshitz, E. M. Quantum Mechanics, 2nd ed.; Pergamon: Oxford, 1965; Chapter V. 67. Varandas, A. J. C.; Martins, L. J. A. J. Chem. Educ. 1986, 63, 485–486. 68. Levich, B. G.; Myamlin, V. A.; Vdovin, Yu. A. Theoretical Physics; North-Holland: Amsterdam, 1973; Vol. 3, Chapter 4. Landau, L. D.; Lifshitz, E. M. Quantum Mechanics, 2nd ed.; Pergamon: Oxford, 1965; Chapter V. Messiah, A. Quantum Mechanics; North-Holland: Amsterdam, 1967; Vol. 1, Chapter IX. 69. Pilar, F. L. Elementary Quantum Mechanics; McGraw-Hill: New York, 1968; Chapter 6. Anderson, J. M. Introduction to Quantum Chemistry; Benjamin: New York, 1969; Chapter 4. Levich, B. G.; Myamlin, V. A.; Vdovin, Yu. A. Theoretical Physics; NorthHolland: Amsterdam, 1973; Vol. 3, Sects. 30, 36. Pauling, L.; Wilson, E. B. Introduction to Quantum Mechanics; McGraw-Hill: New York, 1935; Chapter XV. Margenau, H.; Murphy, G. M. The Mathematics of Physics and Chemistry, 2nd ed.; Van Nostrand: Princeton, NJ, 1964; Chapter 11. Messiah, A. Quantum Mechanics; North-Holland: Amsterdam, 1967; Vol. 1, Chapter IX, Appendix B. Atkins, P. W. Molecular Quantum Mechanics, 2nd ed.; Oxford University Press: Oxford, 1988; Chapter 6. 70. Anderson, J. M. Introduction to Quantum Chemistry; Benjamin: New York, 1969; Chapter 4. Margenau, H.; Murphy, G. M. The Mathematics of Physics and Chemistry, 2nd ed.; Van Nostrand: Princeton, NJ, 1964; Chapter 11. Harnwell, G. P.; Stephens, W.

338

E. Atomic Physics; McGraw-Hill: New York, 1955; Chapter 5. 71. Anderson, J. M. Introduction to Quantum Chemistry; Benjamin: New York, 1969; Chapter 4. Pauling, L.; Wilson, E. B. Introduction to Quantum Mechanics; McGraw-Hill: New York, 1935; Chapter X. Harnwell, G. P.; Stephens, W. E. Atomic Physics; McGraw-Hill: New York, 1955; Chapter 5. Schiff, L. T. Quantum Mechanics, 3rd ed.; McGraw-Hill Kogakusha: Tokyo, 1968; Chapter 12. 72. Eisberg, R.; Resnick, R. Quantum Physics, 2nd ed.; Wiley: New York, 1985; Chapter 15. Mayer, M. G.; Jensen, J. H. D. Elementary Theory of Nuclear Shell Structure; Wiley: New York, 1955; especially Chapter VIII. 73. Nemirovskii, P. E. G. Nuclear Models; E.&F. N. Spon: London, 1963. 74. Pilar, F. L. Elementary Quantum Mechanics; McGraw-Hill: New York, 1968; Chapter 12. McWeeny, R. Methods of Molecular Quantum Mechanics, 2nd ed.; Academic: London, 1978; Appendix 2. 75. Anderson, J. M. Introduction to Quantum Chemistry; Benjamin: New York, 1969; Chapter 4. Atkins, P. W. Molecular Quantum Mechanics, 2nd ed.; Oxford University Press: Oxford, 1988; Chapter 11. 76. Nedeljković, J. M.; Patel, R. C.; Kaufman, P.; Joyce-Pruden, C.; O’Leary, N. J. Chem. Educ. 1993, 70, 342–346. 77. Fock, V. Z. Phys. 1936, 98, 145–154. Bander, M.; Itzykson, C. Rev. Mod. Phys. 1966, 38, 330–345. 78. Jackson, J. D. Classical Electrodynamics, 2nd ed.; Wiley: New York, 1975; Chapter 16. Stone, A. J. Inorg. Chem. 1981, 20, 563–571. 79. Fock, V. A. Fundamentals of Quantum Mechanics; Mir: Moscow, 1986; Part II, 239–245. 80. Clementi, E. Modern Techniques in Computational Chemistry: MOTTEC-91; ESCOM: Leiden, 1991; Chapters 4, 5, 15. 81. Fock, V. A. Fundamentals of Quantum Mechanics; Mir: Moscow, 1986; Part II, pp 322–325. 82. See, for example Anderson, J. M. Introduction to Quantum Chemistry; Benjamin: New York, 1969; Appendix 2. Pauling, L.; Wilson, E. B. Introduction to Quantum Mechanics; McGraw-Hill: New York, 1935; Chapter IX. Jackson, J. D. Classical Electrodynamics, 2nd ed.; Wiley: New York, 1975; especially Chapter 3. Clementi, E. Modern Techniques in Computational Chemistry: MOTTEC-91; ESCOM: Leiden, 1991. Dewar, M. J. S.; Thiel, W. Theoret. Chim. Acta (Berl.) 1977, 46, 89–104. De Bondt, H. L.; Blaton, N. M.; Peeters, O. M.; De Ranter, C. J. Z. Kristallogr. 1993, 208, 167– 174. Satten, R. A. J. Chem. Phys. 1957, 27, 286–299. 83. White, H. E. Introduction to Atomic Spectra; McGraw-Hill: New York, 1934; Chapter IX. Powell, R. E. J. Chem. Educ. 1968, 45, 558–563. Szabo, A. J. Chem. Educ. 1969, 46, 678–680. McKelvey, D. R. J. Chem. Educ. 1983, 60, 112–116. 84. Harnwell, G. P.; Stephens, W. E. Atomic Physics; McGraw-Hill: New York, 1955; Chapter 4. 85. Schiff, L. T. Quantum Mechanics, 3rd ed.; McGraw-Hill Kogakusha: Tokyo, 1968; Chapter 12.


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