where d 3 3)2 ( L x - -- i3x j3x j i i>j Thus the Gegenbauer polynomials play a role in the theory of hyper spherical harmonics which is analogous to the role played by Legendre polynomials in the familiar theory of 3-dimensional spherical harmonics; and when d = 3, the Gegenbauer polynomials reduce to Legendre polynomials. The familiar sum rule, in 'lrlhich a sum of spherical harmonics is expressed as a Legendre polynomial, also has a d-dimensional generalization, in which a sum of hyper spherical harmonics is expressed as a Gegenbauer polynomial (equation (3-27": The hyper spherical harmonics which appear in this sum rule are eigenfunctions of the generalized angular monentum 2 operator A , chosen in such a way as to fulfil the orthonormality relation: VIe are all familiar with the fact that a plane wave can be expanded in terms of spherical Bessel functions and either Legendre polynomials or spherical harmonics in a 3-dimensional space. Similarly, one finds that a d-dimensional plane wave can be expanded in terms of HYPERSPHERICAL HARMONICS xii "hyperspherical Bessel functions" and either Gegenbauer polynomials or else hyperspherical harmonics (equations ( 4 - 27) and ( 4 - 30) ) : 00 ik*x e = (d-4)!!A~oiA(d+2A-2)j~(kr)C~(~k'~) 00 (d-2)!!I(0) 2: iAj~(kr) 2:Y~ (["2k)Y (["2) A A=O ). l). l)J where I(O) is the total solid angle. This expansion of a d-dimensional plane wave is useful when we wish to calculate Fourier transforms in a d-dimensional space.

"The Theory of Atomic Spectra," surrrrnanzlllg all that was then known about the quantum theory of free atoms; and in 1961, J.S. Griffith published "The Theory of Transition Metal Ions," in which he combined the ideas in Condon and Shortley's book with those of Bethe, Schlapp, Penney and Van Vleck. All this work, however, was done by physicists, and the results were reported in a way which was more accessable to physicists than to chemists. In the meantime, Carl J. Ballhausen had been studying quantum theory with W. Moffitt at Harvard; and in 1962 (almost simultaneously with Griffith) he published his extremely important book, "Introduction to Ligand Field Theory." This influential book was written from the standpoint of a chemist, and it became the standard work from which chemists learned the quantum theory of transition metal complexes. While it treated in detail the group theoretical aspects of crystal field theory, Carl J. Ballhausen's book also emphasized the limitations of the theory. As he pointed out, it is often not sufficient to treat the central metal ion as free (apart from the influence of the charges on the surrounding ligands): - In many cases hybridization of metal and ligand orbitals is significant. Thus, in general. a molecular orbital treatment is needed to describe transition metal complexes. However, much of the group theory developed In connection with crystal field theory can also be used in the molecular orbital treatment.

This volume contains all of the invited lectures presented at the NATO ARWon "New Methods in Quantum Theory" held in Halkidiki, Greece from May 14th to May 19th, 1995. This survey of new perspectives, techniques and results in quantum theory contains 26 chapters by leading quantum chemists and physicists from 14 countries. The book covers a wide range of topics, though the emphasis throughout is on new approaches and their interrelationships. Topics covered include dimensional scaling, the hyperspherical method applied both to reactive scattering theory and to bound state problems, chaotic behaviour, large-order perturbation theory, complex eigenvalues and quasistationary states, semiclassical methods, cusps in hyperaccurate wave functions, density functional theory, relativistic quantum theory, and quantum Monte Carlo methods. I hope very much that the various lectures presented at this Advanced Research Workshop will be of as great interest to the reader as they were to the participants. The organization of the ARW would have been impossible without the generous funding by NATO, which is gratefully acknowledged. I sincerely thank Drs. J. -M. Cadiou, Assistant Secretary General for Scientific and Environmental Affairs, and J. A. Rausell-Colom, Programme Director for Priority Area on High Technology, North Atlantic Treaty Organization, who helped me throughout the various stages of the ARW.

n Angular Momentum Theory for Diatomic Molecules, R R method of trees, 3 construct the wave functions of more complicated systems for ex- ple many electron atoms or molecules. However, it was soon realized that unless the continuum is included, a set of hydrogenlike orbitals is not complete. To remedy this defect, Shull and Lowdin [273] - troduced sets of radial functions which could be expressed in terms of Laguerre polynomials multiplied by exponential factors. The sets were constructed in such a way as to be complete, i. e. any radial fu- tion obeying the appropriate boundary conditions could be expanded in terms of the Shull-Lowdin basis sets. Later Rotenberg [256, 257] gave the name "Sturmian" to basis sets of this type in order to emp- size their connection with Sturm-Liouville theory. There is a large and rapidly-growing literature on Sturmian basis functions; and selections from this literature are cited in the bibliography. In 1968, Goscinski [138] completed a study ofthe properties ofSt- rnian basis sets, formulating the problem in such a way as to make generalization of the concept very easy. In the present text, we shall follow Goscinski's easily generalizable definition of Sturmians.

n Angular Momentum Theory for Diatomic Molecules, R R method of trees, 3 construct the wave functions of more complicated systems for ex- ple many electron atoms or molecules. However, it was soon realized that unless the continuum is included, a set of hydrogenlike orbitals is not complete. To remedy this defect, Shull and Lowdin [273] - troduced sets of radial functions which could be expressed in terms of Laguerre polynomials multiplied by exponential factors. The sets were constructed in such a way as to be complete, i. e. any radial fu- tion obeying the appropriate boundary conditions could be expanded in terms of the Shull-Lowdin basis sets. Later Rotenberg [256, 257] gave the name "Sturmian" to basis sets of this type in order to emp- size their connection with Sturm-Liouville theory. There is a large and rapidly-growing literature on Sturmian basis functions; and selections from this literature are cited in the bibliography. In 1968, Goscinski [138] completed a study ofthe properties ofSt- rnian basis sets, formulating the problem in such a way as to make generalization of the concept very easy. In the present text, we shall follow Goscinski's easily generalizable definition of Sturmians.

Dimensional scaling offers a new approach to quantum dynamical correlations. This is the first book dealing with dimensional scaling methods in the quantum theory of atoms and molecules. Appropriately, it is a multiauthor production, derived chiefly from papers presented at a workshop held in June 1991 at the A~rsted Institute in Copenhagen. Although focused on dimensional scaling, the volume includes contributions on other unorthodox methods for treating nonseparable dynamical problems and electronic correlation. In shaping the book, the editors serve three needs: an introductory tutorial for this still fledgling field; a guide to the literature; and an inventory of current research results and prospects. Part I treats basic aspects of dimensional scaling. Addressed to readers entirely unfamiliar with the subject, it provides both a qualitative overview, and a tour of elementary quantum mechanics. Part II surveys the research frontier. The eight chapters exemplify current techniques and outline results. Part III presents other methods, including nonseparable dynamics, and electron correlation in pseudomolecular excited states of atoms. Although procrustean conformity was not imposed, unifying and complementary themes are emphasized throughout the book.

Dimensional scaling offers a new approach to quantum dynamical correlations. This is the first book dealing with dimensional scaling methods in the quantum theory of atoms and molecules. Appropriately, it is a multiauthor production, derived chiefly from papers presented at a workshop held in June 1991 at the A~rsted Institute in Copenhagen. Although focused on dimensional scaling, the volume includes contributions on other unorthodox methods for treating nonseparable dynamical problems and electronic correlation. In shaping the book, the editors serve three needs: an introductory tutorial for this still fledgling field; a guide to the literature; and an inventory of current research results and prospects. Part I treats basic aspects of dimensional scaling. Addressed to readers entirely unfamiliar with the subject, it provides both a qualitative overview, and a tour of elementary quantum mechanics. Part II surveys the research frontier. The eight chapters exemplify current techniques and outline results. Part III presents other methods, including nonseparable dynamics, and electron correlation in pseudomolecular excited states of atoms. Although procrustean conformity was not imposed, unifying and complementary themes are emphasized throughout the book.

where d 3 3)2 ( L x - -- i3x j3x j i i>j Thus the Gegenbauer polynomials play a role in the theory of hyper spherical harmonics which is analogous to the role played by Legendre polynomials in the familiar theory of 3-dimensional spherical harmonics; and when d = 3, the Gegenbauer polynomials reduce to Legendre polynomials. The familiar sum rule, in 'lrlhich a sum of spherical harmonics is expressed as a Legendre polynomial, also has a d-dimensional generalization, in which a sum of hyper spherical harmonics is expressed as a Gegenbauer polynomial (equation (3-27": The hyper spherical harmonics which appear in this sum rule are eigenfunctions of the generalized angular monentum 2 operator A , chosen in such a way as to fulfil the orthonormality relation: VIe are all familiar with the fact that a plane wave can be expanded in terms of spherical Bessel functions and either Legendre polynomials or spherical harmonics in a 3-dimensional space. Similarly, one finds that a d-dimensional plane wave can be expanded in terms of HYPERSPHERICAL HARMONICS xii "hyperspherical Bessel functions" and either Gegenbauer polynomials or else hyperspherical harmonics (equations ( 4 - 27) and ( 4 - 30) ) : 00 ik*x e = (d-4)!!A~oiA(d+2A-2)j~(kr)C~(~k'~) 00 (d-2)!!I(0) 2: iAj~(kr) 2:Y~ (["2k)Y (["2) A A=O ). l). l)J where I(O) is the total solid angle. This expansion of a d-dimensional plane wave is useful when we wish to calculate Fourier transforms in a d-dimensional space.

"The Theory of Atomic Spectra," surrrrnanzlllg all that was then known about the quantum theory of free atoms; and in 1961, J.S. Griffith published "The Theory of Transition Metal Ions," in which he combined the ideas in Condon and Shortley's book with those of Bethe, Schlapp, Penney and Van Vleck. All this work, however, was done by physicists, and the results were reported in a way which was more accessable to physicists than to chemists. In the meantime, Carl J. Ballhausen had been studying quantum theory with W. Moffitt at Harvard; and in 1962 (almost simultaneously with Griffith) he published his extremely important book, "Introduction to Ligand Field Theory." This influential book was written from the standpoint of a chemist, and it became the standard work from which chemists learned the quantum theory of transition metal complexes. While it treated in detail the group theoretical aspects of crystal field theory, Carl J. Ballhausen's book also emphasized the limitations of the theory. As he pointed out, it is often not sufficient to treat the central metal ion as free (apart from the influence of the charges on the surrounding ligands): - In many cases hybridization of metal and ligand orbitals is significant. Thus, in general. a molecular orbital treatment is needed to describe transition metal complexes. However, much of the group theory developed In connection with crystal field theory can also be used in the molecular orbital treatment.

This book contains a collection of essays and articles by John Scales Avery discussing the severe problems and challenges which the world faces during the 21st century. Human civilization and the biosphere are threatened by catastrophic climate change. Unless rapid steps are taken to replace fossil fuels by 100% renewable energy, we risk passing a tipping point beyond which uncontrollable feedback loops could produce a 6th extinction event comparable to those observed in the geological record. Another serious threat to human civilization and the biosphere is the danger of a catastrophic thermonuclear war. Over a long period of time there is an ever-increasing risk that such a war will occur by accident or miscalculation. Thirdly, there is threat of an extremely serious and widespread famine, produced by the climate change, rapidly-growing populations, and the end of the fossil fuel era. We must urgently address all three challenges.