Table Of ContentPublished by
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Library of Congress Cataloging-in-Publication Data
Names: Avery, James Emil, author. | Avery, John, 1933– author.
Title: Hyperspherical harmonics and their physical applications / James Emil Avery (University of
Copenhagen, Denmark), John Scales Avery (University of Copenhagen, Denmark).
Description: New Jersey : World Scientific, 2017. | Includes bibliographical references and index.
Identifiers: LCCN 2017026745 | ISBN 9789813229297 (hardcover : alk. paper)
Subjects: LCSH: Spherical harmonics--Problems, exercises, etc. |
Scattering (Physics)--Problems, exercises, etc.
Classification: LCC QC20.7.S645 A94 2017 | DDC 515/.785--dc23
LC record available at https://lccn.loc.gov/2017026745
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2018 by World Scientific Publishing Co. Pte. Ltd.
Printed in Singapore
October19,2017 9:2 ws-book9x6 HypersphericalHarmonicsandtheirPhysicalApplications 10690-main pagevii
Contents
Preface xiii
1. HARMONIC FUNCTIONS 1
1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The canonical decomposition of a homogeneous polynomial 3
1.3 More general canonical decompositions . . . . . . . . . . . 8
1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2. GENERALIZED ANGULAR MOMENTUM 11
2.1 Eigenfunctions of generalized angular momentum . . . . . 11
2.2 Hyperspherical harmonics . . . . . . . . . . . . . . . . . . 12
2.3 Generalized solid angle . . . . . . . . . . . . . . . . . . . . 13
2.4 Hyperangular integration . . . . . . . . . . . . . . . . . . 14
2.5 A general theorem for hyperangular integration . . . . . . 15
2.6 Chains of subgroups . . . . . . . . . . . . . . . . . . . . . 17
2.7 Degeneracy of hyperspherical harmonics . . . . . . . . . . 18
2.8 Representations of Λ . . . . . . . . . . . . . . . . . . . . 19
s,t
2.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3. GEGENBAUER POLYNOMIALS 25
3.1 The generating function for Gegenbauer polynomials . . . 25
3.2 Properties of Gegenbauer polynomials . . . . . . . . . . . 27
3.3 Determination of the constant K . . . . . . . . . . . . . . 29
λ
3.4 The generalization of Pm(u·u(cid:48)) . . . . . . . . . . . . . . 30
l
3.5 The standard tree. . . . . . . . . . . . . . . . . . . . . . . 33
3.6 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.7 Nonstandard trees . . . . . . . . . . . . . . . . . . . . . . 40
3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4. FOURIER TRANSFORMS IN d DIMENSIONS 45
4.1 Notation and basic properties . . . . . . . . . . . . . . . . 45
4.2 Expansions of a d-dimensional plane wave . . . . . . . . . 46
4.3 The Green’s function of ∆ . . . . . . . . . . . . . . . . . . 48
4.4 Hyperspherical Bessel transforms . . . . . . . . . . . . . . 49
4.5 An alternative expansion of a plane wave. . . . . . . . . . 50
4.6 The Fourier convolution theorem . . . . . . . . . . . . . . 52
4.7 Harmonic analysis for non-Euclidean spaces . . . . . . . . 53
4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5. FOCK’S TREATMENT OF HYDROGENLIKE
ATOMS AND ITS GENERALIZATION 55
5.1 Fock’s original treatment . . . . . . . . . . . . . . . . . . . 55
5.2 Generalization of Fock’s treatment . . . . . . . . . . . . . 60
5.3 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6. D-DIMENSIONAL HYDROGENLIKE ORBITALS IN
DIRECT SPACE 67
6.1 Generalization of the 3-dimensional solution . . . . . . . . 67
6.2 Orthonormality . . . . . . . . . . . . . . . . . . . . . . . . 70
6.3 d-dimensional Coulomb Sturmians . . . . . . . . . . . . . 71
6.4 Potential-weighted orthonormality . . . . . . . . . . . . . 74
6.5 Fourier transforms of Coulomb Sturmians . . . . . . . . . 76
6.6 Use of d-dimensional Coulomb Sturmians as a basis. . . . 76
6.7 Cases where the potential may have angular dependence . 78
6.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7. GENERALIZED STURMIANS 81
7.1 Generalized Sturmians and many-particle bound state
problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.2 Sturmians and the many-particle Schr¨odinger equation . . 83
7.3 Momentum-space orthonormality of generalized Sturmian
basis sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.4 Sturmian expansions of d-dimensional plane waves . . . . 85
7.5 Iteration of the Schr¨odinger equation . . . . . . . . . . . . 86
7.6 Molecular spectra . . . . . . . . . . . . . . . . . . . . . . . 89
7.7 Goscinskian configurations in atomic physics. . . . . . . . 91
7.8 Derivation of the secular equations . . . . . . . . . . . . . 93
7.9 The Large-Z approximation . . . . . . . . . . . . . . . . . 96
7.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8. CHOOSING APPROPRIATE HYPERSPHERICAL
REPRESENTATIONS 103
8.1 The alternative Coulomb Sturmians of Aquilanti and
Coletti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.2 Transformations between matrix representations . . . . . 107
8.3 An example: Alternative Coulomb Sturmians
corresponding to real spherical harmonics . . . . . . . . . 108
8.4 The d-dimensional case . . . . . . . . . . . . . . . . . . . . 115
8.5 The d-dimensional Schr¨odinger equation in momentum
space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.6 Shibuya-Wulfman orbitals . . . . . . . . . . . . . . . . . . 120
8.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
9. MOLECULAR INTEGRALS FROM
HYPERSPHERICAL HARMONICS 127
9.1 Molecular integrals using exponential-type orbitals . . . . 127
9.2 Many center Sturmians. . . . . . . . . . . . . . . . . . . . 128
9.3 Overlap integrals involving Coulomb Sturmians . . . . . . 129
9.4 Shibuya-Wulfman integrals . . . . . . . . . . . . . . . . . 132
9.5 Matrices representing kinetic energy and nuclear attraction 134
9.6 1-center densities in terms of 2k Sturmians . . . . . . . . . 137
9.7 Interelectron repulsion integrals between two 1-center
densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
9.8 2-center densities in terms of 2k Sturmians . . . . . . . . . 143
9.9 Generalized scattering factors . . . . . . . . . . . . . . . . 150
9.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
10. LAGRANGIANS FOR PARTICLES AND FIELDS 153
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 153
10.2 Cyclic coordinates . . . . . . . . . . . . . . . . . . . . . . 156
10.3 Hamilton’s unified formulation . . . . . . . . . . . . . . . 158
10.4 Normal modes. . . . . . . . . . . . . . . . . . . . . . . . . 161
10.5 Molecular vibrations and rotations . . . . . . . . . . . . . 166
10.6 Lagrangian densities for fields . . . . . . . . . . . . . . . . 167
10.7 Electromagnetic potentials . . . . . . . . . . . . . . . . . . 169
10.8 Metric tensors . . . . . . . . . . . . . . . . . . . . . . . . . 173
10.9 The Laplace-Beltrami operator . . . . . . . . . . . . . . . 177
10.10 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
10.11 Huygens’ principle . . . . . . . . . . . . . . . . . . . . . . 183
10.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
11. COORDINATE TRANSFORMATIONS FOR N BODIES 187
11.1 Transforming the kinetic energy operator. . . . . . . . . . 187
11.2 The Laplace-Beltrami operator . . . . . . . . . . . . . . . 191
11.3 Transformation of the kinetic energy operator . . . . . . . 192
11.4 A simple example . . . . . . . . . . . . . . . . . . . . . . . 193
11.5 Jacobi coordinates of a 3-body system with equal masses . 194
11.6 Normal mode transformations . . . . . . . . . . . . . . . . 195
11.7 A simple model for interatomic forces in molecules . . . . 196
11.8 A simple example . . . . . . . . . . . . . . . . . . . . . . . 196
11.9 Separability in the harmonic approximation . . . . . . . . 200
11.10 The Morse potential . . . . . . . . . . . . . . . . . . . . . 201
11.11 Exact solutions for the Morse oscillator. . . . . . . . . . . 202
11.12 Morse oscillator eigenfunctions as a basis . . . . . . . . . . 203
11.13 Rotations and vibrations of diatomic molecules . . . . . . 205
11.14 The rotational and vibrational modes of molecules and
clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
11.15 Normal modes. . . . . . . . . . . . . . . . . . . . . . . . . 208
11.16 A simple example . . . . . . . . . . . . . . . . . . . . . . . 208
11.17 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
12. SOME ILLUSTRATIVE EXAMPLES 211
12.1 Matrix elements of the kinetic energy operator . . . . . . 211
12.2 Matrix elements of the potential energy . . . . . . . . . . 211
12.3 Products of basis functions . . . . . . . . . . . . . . . . . 213
12.4 Examples of basis sets . . . . . . . . . . . . . . . . . . . . 213
12.5 The problem of slow convergence . . . . . . . . . . . . . . 215
12.6 2-electron atoms: the results of Das et al. . . . . . . . . . 216
12.7 Approximate relativistic corrections. . . . . . . . . . . . . 219
12.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Appendix A THE D-DIMENSIONAL HARMONIC OSCILLATOR 221
A.1 Harmonic oscillators in one dimension . . . . . . . . . . . 221
A.2 Creation and annihilation operators for harmonic
oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
A.3 A collection of harmonic oscillators . . . . . . . . . . . . . 225
A.4 d-dimensional isotropic harmonic oscillators . . . . . . . . 226
A.5 Fourier transforms of 3-dimensional harmonic oscillator
wave functions . . . . . . . . . . . . . . . . . . . . . . . . 228
A.6 The hyperspherical Bessel transform of the radial function 229
A.7 Coupling coefficients for harmonic oscillator wave functions 230
Appendix B MOLECULAR INTEGRALS FOR
SLATER-TYPE ORBITALS 233
B.1 Definition of STO’s . . . . . . . . . . . . . . . . . . . . . . 233
B.2 Expansion of an arbitrary function of s = kr in terms of
Coulomb Sturmian radial functions . . . . . . . . . . . . . 233
B.3 Evaluation of molecular integrals . . . . . . . . . . . . . . 236
B.4 STO overlap integrals . . . . . . . . . . . . . . . . . . . . 239
B.5 Interelectron repulsion integrals with STO’s . . . . . . . . 240
B.6 Checks in the atomic case . . . . . . . . . . . . . . . . . . 242
Bibliography 247
Index 275
Preface
Everyone working in theoretical physics and chemistry is familiar with the
beauty and utility of spherical harmonics. Solutions to the Schr¨odinger
equationareveryoftenobtainedbyseparatingtheproblemintoanangular
partandaradialpart. Theangularpartoftheproblemisthensolvedusing
the elegant theorems involving spherical harmonics. For example, in elec-
tronic structure theory, the wavefunction is built up from radial functions
andsphericalharmonics,andinscatteringproblems,oneusesexpansionsof
planewavesintermsofsphericalharmonicsandsphericalBesselfunctions.
Remarkably,allofthetheoremsinvolving3-dimensionalharmonicshave
simple and beautiful d-dimensional generalizations. For example, a d-
dimensional plane wave may be expanded in terms of hyperspherical har-
monics and functions of the hyperradius which we call “hyperspherical
Bessel functions”.
Hyperspherical harmonics have shown themselves to be extremely use-
ful, both in nuclear physics and in reactive scattering theory. However,
their use has been confined to specialists with very strong backgrounds
in mathematics. The aim of this book is to change the theory of hyper-
spherical harmonics from an esoteric field, mastered by specialists, into an
easily-used tool with a place in the working kit of all theoretical physicists,
theoretical chemists and mathematicians. The theory that we present in
this book is accessible without knowledge of Lie-groups and representation
theory,andcanbeunderstoodwithanordinaryknowledgeofcalculus. The
book is accompanied by programs and exercises designed for teaching and
practical use.
Butwhydoweclaimthatthetheoryofhypersphericalharmonicsought
to be part of the tool-kit of all theoretical workers in the physical sciences?
Let us next try to answer this question.
In solving the Schr¨odinger equation for many-electron systems, the
starting point is often the Hartree-Fock approximation, which is an
independent-particlemodel. Insuchamodel,eachelectronisthoughtofas
moving in the attractive field of the nuclei and an averaged repulsive field
due to all the other electrons. Electron correlation effects are then added
using the formalism of configuration interaction.
In nuclear physics, however, the wave function of the nucleus is usually
sohighlycorrelatedthatanindependent-particlemodelsuchastheHartree-
Fockapproximationisnotagoodstartingpoint. Instead,onetriestosolve
themany-nucleonSchr¨odingerequationdirectly, ina(3N−6)-dimensional
space. In order to simplify such solutions, nuclear physicists have long
relied on the theory of hyperspherical harmonics. But even in the study
of electronic structure, one encounters systems that are so strongly corre-
lated that the starting point must be a many-dimensional wave-equation.
Furthermore, hyperspherical harmonics allow difficult molecular integrals
involving exponential-type orbitals to be evaluated both rapidly and accu-
rately.
In Chapter 1 of this book, we approach the theory of hyperspheri-
cal harmonics by starting from the theory of harmonic polynomials in a
d-dimensional space. Harmonic polynomials are just homogeneous poly-
nomialsthatsatisfytheLaplaceequation. Thed-dimensionalgeneralization
is very easy: d-dimensional harmonic polynomials are just homogeneous
polynomials of d Cartesian coordinates that also satisfy the generalized
Laplace equation, and hyperspherical harmonics are harmonic polynomials
in the Cartesian coordinates of the unit hypersphere.
Chapter 2 introduces generalized angular momentum in d-dimensional
spaces. Hyperspherical harmonics are the eigenfunctions of the generalized
angular momentum operator. They arise from irreducible representations
of the rotation group SO(d) and form complete basis sets, so that one can
synthesize any smooth angular function. In this chapter, we also present a
powerful hyperangular integration theorem that makes integration of poly-
nomials on the unit sphere trivial.
In Chapter 3, we discuss the properties of Gegenbauer polynomials,
which are sometimes called ultra-spherical polynomials. The Gegenbauer
polynomials play the same role for hyperspherical harmonics as Legendre
polynomials play in the theory of spherical harmonics. In fact, for d = 3,
GegenbauerpolynomialsarejustLegendrepolynomials. Thereadermaybe
familiar with the expansion of 3-dimensional plane waves in terms of Leg-
endre polynomials multiplied by spherical Bessel functions. Analogously, a
d-dimensionalplanewaveisasumofhypersphericalBesselfunctionstimes
Gegenbauerpolynomials. Gegenbauerpolynomialsalsohelpustogenerate
d-dimensional hyperspherical harmonics.
Chapter 4 introduces a number of tools for working with Fourier trans-
forms in d-dimensional spaces. These tools will be used throughout the
book. Of particular importance is the hyperspherical Bessel transform,
which is shown to be no more difficult than the three dimensional spher-
ical Bessel transform. It is also shown that the Green’s function of the
d-dimensional Laplace operator is the generating function for Gegenbauer
polynomials, just as the Green’s function of the 3-dimensional Laplacian
generates the Legendre polynomials. Finally, we show properties of d-
dimensional plane wave expansions both in terms of hyperspherical har-
monic, and in terms of general complete basis sets.
In an extremely brilliant early paper, the Russian physicist V. Fock
([Fock,1935],[Fock,1958])showedthattheSchr¨odingerequationforhydro-
genlike atoms in reciprocal space can be solved by means of 4-dimensional
hyperspherical harmonics. To obtain these solutions, he projected 3-
dimensional momentum-space onto the surface of a 4-dimensional unit
sphere.
Fock’s momentum-space solutions, when transformed back into direct
space, are identical with the Coulomb Sturmian basis sets which were in-
troduced into theoretical physics and theoretical chemistry by Per-Olov
L¨owdin and others ([L¨owdin, 1955]). Coulomb Sturmian basis sets have
very desirable completeness properties, and for this reason they are widely
used. Fock’smomentum-spacecalculationwassoongeneralizedtoddimen-
sions, and it was shown that solutions to the d-dimensional hydrogenlike
Schr¨odinger equation can be expressed in terms of (d+1)-dimensional hy-
perspherical harmonics. The d-dimensional Coulomb Sturmians also have
desirablecompletenessproperties. Chapters5and6discussthemomentum
space work of Fock and its generalizations to d dimensions.
The name “Sturmians” was due to M. Rotenberg ([Rotenberg,
1962],[Rotenberg, 1970]), who wished to emphasize the connection of these
functions with Sturm-Liouville theory. Later, Osvaldo Goscinski ([Goscin-
ski, 2003]) generalized the concept of Sturmian basis sets. He discussed
many-particleisoenergeticbasissetswhicharesolutionstotheSchr¨odinger
equation with weighted potentials. In such a scheme, the weighting factor
attached to the potential takes on the role of the eigenvalue. In Chapter
7, we demonstrate the usefulness of Goscinskian configurations. A much
more in-depth treatment of Goscinskian basis sets is found in our previous
book ([Avery and Avery, 2006]).
There are many ways to construct hyperspherical harmonic basis sets.
