John Avery

University of Copenhagen · Department of Computer Science

Jacobi coordinates have often been used to treat few-particle quantum systems. In this paper, we propose an alternative coordinate system which is very easily generalized as the number of particles becomes larger. We make use of forms of the Laplace–Beltrami operator that are invariant under general coordinate transformations, and use Coulomb Sturm...

We present a method for evaluating 4-center electron repulsion integrals (ERI) for Slater-type orbitals by way of expansions in terms of Coulomb Sturmians. The ERIs can then be evaluated using our previously published methods for rapid evaluation of Coulomb Sturmians through hyperspherical harmonics. Numerical investigations are made of the efficie...

In this article, we discuss a way in which the theory of hyperspherical harmonics may be used for rapid evaluation of difficult molecular integrals when exponential-type orbitals (ETOs) are used as a basis. One of us (J.E.A.) has implemented the method, and programs are available for general use. As a byproduct of this work, we are also able to eva...

The use of Slater type orbitals in molecular calculations is hindered by the slowness of integral evaluation. In the present paper, we introduce a method for overcoming this problem by expanding STO’s in terms of Coulomb Sturmians, for which the problem of evaluating molecular integrals rapidly has been satisfactorily solved using methods based on...

Exponential-type orbitals are better suited to calculations of molecular electronic structure than are Gaussians, since ETO’s can accurately represent the behavior of molecular orbitals near to atomic nuclei, as well as their long-distance exponential decay. Orbitals based on Gaussians fail in both these respects. Nevertheless, Gaussian technology...

Almost all modern quantum chemistry programs use Gaussian basis sets even though Gaussians cannot accurately represent the cusp at atomic nuclei, nor can they represent the slow decay of the wave function at large distances. The reason that Gaussians dominate quantum chemistry today is the great mathematical difficulty of evaluating interelectron r...

The theory of Sturmians and generalized Sturmians is reviewed. It is shown that when generalized Sturmians are used as basis functions, calculations on the spectra and physical properties of few-electron atoms can be performed with great ease and good accuracy. The use of many-center Coulomb Sturmians as basis functions in calculations on N-electro...

In theoretical physics, theoretical chemistry and engineering, one often wishes to solve partial differential equations subject to a set of boundary conditions. This gives rise to eigenvalue problems of which some solutions may be very difficult to find. For example, the problem of finding eigenfunctions and eigenvalues for the Hamiltonian of a man...

The Schrödinger equation is the master equation of quantum chemistry. The founders of quantum mechanics realised how this equation underpins essentially the whole of chemistry. However, they recognised that its exact application was much too complicated to be solvable at the time. More than two generations of researchers were left to work out how t...

The properties of monomials, homogeneous polynomials and harmonic polynomials in d-dimensional spaces are discussed. The properties are shown to lead to formulas for the canonical decomposition of homogeneous polynomials and formulas for harmonic projection. Many important properties of spherical harmonics, Gegenbauer polynomials and hyperspherical...

A method is proposed for using isoenergetic configurations formed from many-center Coulomb Sturmians as a basis for calculations on N-electron molecules. Such configurations are solutions to an approximate N-electron Schrodinger equation with a weighted potential, and they are thus closely analogous to the Goscinskian configurations that we have us...

In the generalized Sturmian method, solutions to the many-particle Schrödinger equation are built up from isoenergetic sets of solutions to an approximate Schrödinger equation with a weighted potential bn V0(x){\beta_\nu \mathsf{V}_0({\bf x})}. The weighting factors β ν are chosen in such a way as to make all of the members of the basis set corres...

When the Schrödinger equation for a system of N particles interacting through Coulomb forces is expressed in terms of hyperspherical coordinates, it closely resembles the d-dimensional generalization of the hydrogen atom wave equation (where d = 3N). The d-dimensional hydrogenlike wave functions can be found exactly, and a set of these functions, a...

In the usual ab initio method of calculating molecular orbitals, the number of integrals to be evaluated increases as M⁴, where M is the number of basis functions. In this paper, an alternative method is discussed, where the computation time increases much less violently with the number of basis functions. Matrix elements of the deformation potenti...

Coupling coefficients for symmetrized sinusoidal functions in crystals are discussed. These coupling coefficients, which are analogous to Condon-Shortley coefficients, can be used to calculate generalized scattering factors.

Measured Fourier coefficients of the electronic charge density in a crystal can be used to construct the Coulomb potential and the Slater potential. For large values of K, where measurements cannot be made, the Fourier coefficients reduce to those which would result from the assembled neutral atoms, neglecting charge flow due to bonding. Thus the F...

Two formulas derived in previous articles are used to evaluate the matrix elements of a general many-particle Hamiltonian. The first formula is an expansion of the Coulomb potential of the system in terms of hyperspherical harmonics, while the second is a general formula for the evaluation of angular integrals in many-dimensional spaces. These two...

This book describes the generalized Sturmian method, which offers a fresh approach to the calculation of atomic spectra. Generalized Sturmians are isoenergetic solutions to an approximate many-electron Schrödinger equation with a weighted potential. The weighting factors are chosen in such a way as to make all of the solutions correspond to a given...

The generalized Sturmian method is applied to autoionizing states of atoms and ions. If the Goscinskian basis sets allow for a sufficient amount of angular correletion, the calculated energies of doubly-excited (autoionizing) states are found to agree well with the few available experimental energies. A large-Z approximation is discussed, and simpl...

The generalized Sturmian method for atomic and molecular electronic structure calculations is a direct configuration interaction method in which the configurations are chosen to be isoenergetic solutions of an approximate N-electron Schrödinger equation with a weighted potential, βνV0. The weighting factors βν are especially chosen so that all the...

The term “crystal harmonic” is introduced to denote a symmetrized plane wave in the special case where the wave vector is a reciprocal lattice vector. Crystal harmonics, thus defined, have the translational symmetry of the lattice, and they also have the transformation properties of the irreducible representations of the crystal's point group. An e...

When momentum space is projected onto the surface of a unit 4-D hypersphere by means of Fock's mapping, Coulomb Sturmian basis functions can be simply represented in terms of hyperspherical harmonics. The properties of these harmonics can be used to evaluate Shibuya–Wulfman integrals and other integrals that arise when the Sturmian basis functions...

The π-electron distributions, spin densities, and energies of the first triplets of the nucleotide bases, uracil, thymine, cytosine, adenine, and guanine, were investigated in various semiempirical approximations. Results are presented for calculations using the semiempirical form of the closed-shell SCF configuration interaction method, of the dif...

The generalized Sturmian method for obtaining solutions to the many-particle Schrodinger equation is reviewed. The method makes use of basis functions that are solutions of an approximate Schrodinger equation with a weighted zeroth-order potential. The weighting factors are especially chosen so that all the basis functions are isoenergetic with the...

The properties of generalized Sturmian basis sets are reviewed, and functions of this type are used to perform direct configuration interaction calculations on the spectra of atoms and ions. Singlet excited states calculated in this way show good agreement with experimentally measured spectra. When the generalized Sturmian method is applied to atom...

The generalized Sturmian method is a direct configuration interaction method for solving the Schrödinger equation of a many-electron system. The configurations in the basis set are solutions to an approximate Schrödinger equation with a weighted potential βν V0(x), the weighting factors βν being chosen in such a way as to make the set of solutions...

The properties of generalized Sturmian basis sets are reviewed, and functions of this type are used to perform direct configuration interaction calculations on the spectra of atoms and ions. Singlet excited states calculated in this way show good agreement with experimentally measured spectra. When the generalized Sturmian method is applied to atom...

The Perrin-Förster theory of sensitized fluorescence is extended by replacing the Coulomb interaction with its relativistic counterpart, the Breit interaction. The transition matrix element is evaluated in a multipole expansion. The matrix element is found to be modulated by the retardation factor eikR and to contain terms which fall off only as 1/...

Analytical expressions are obtained for the Born-Oppenheimer non-adiabatic coupling terms formed by a given distribution of conical intersections. The gauge-dependent formulae contain explicitly the boundary conditions that have to be obtained by ab initio calculations performed along a circle in the close vicinity of each conical intersection.

Methods are introduced for constructing sets of antisymmetrized many-electron Sturmian basis functions using the nuclear attraction potential of an atom or ion as the basis potential. When such basis sets are used, the kinetic energy term disappears from the secular equation, the Slater exponents are automatically optimized, convergence is rapid, a...

Generalized Sturmian basis sets make it possible to solve the many-particle Schrödinger equation directly, without the use of the SCF approximation. The functions in such a basis set are solutions to the many-particle Schrödinger equation with a weighted “basis potential”, βvV0(x), the weighting factor βv being chosen in such a way as to make all t...

Sturmian methods for solving the Schrödinger equation for an electron moving in the field of a number of nuclei are reviewed. The problem is approached in direct space, although momentum-space techniques, including the use of hyperspherical harmonics, are used to evaluate the necessary integrals. In Part 2, these many-center one-electron solutions,...

The generalized Sturmian approach to quantum mechanical many-body problems is described. The method allows correlated solutions to the many-particle Schrödinger equation to be obtained directly, without the use of the self-consistent-field approximation. As an illustrative example, spectra and polarizabilities are calculated for atoms and ions in t...

The generalized Sturmian method for solving the many-electron Schrödinger equation is reviewed. The method is illustrated with calculations of the core ionization energies of a series of atoms and ions. It is shown that when the “basis potential” is chosen to be the actual attractive potential of the nuclei in the system being studied, convergence...

The generalized Sturmian method for solving the many-electron Schrdinger equation is reviewed. This method yields rapidly convergent solutions directly, without the use of the SCF approximation. As a simple illustrative example, differential cross sections are calculated for inelastic scattering of fast electrons by atoms and ions in the 2-electron...

Methods are introduced for generating many-electron Sturmian basis sets using the actual external potential experienced by an N-electron system, i.e. the attractive potential of the nuclei. When such basis sets are employed, very few basis functions are needed for an accurate representation of the system; the kinetic energy term disappears from the...

The properties of relativistic one-electron Sturmian basis sets are discussed using Goscinski's definition of Sturmians rather than Rotenberg's. The potential-weighted orthonormality relations obeyed by the members of such a set are discussed. Weighted orthonormality relations in momentum space are also derived and used to construct a Sturmian expa...

A formula is derived which allows angular or hyperangular integration to be performed on any function of the coordinates of a D-dimensional space, provided that it is possible to expand the function as a polynomial in the coordinates x1,x2,...,xd. The expansion need not be carried out for the formula to be applied.

Goscinski's definition of Sturmian basis sets is generalized and applied to many–electron systems. It is shown that when such basis sets are used, the kinetic energy term disappears from the secular equation, the nuclear attraction potential is already in diagonal form in the secular equation, the orbital exponents are automatically optimized, and...

The weighted orthonormality relations for many-particle Sturmian basis functions are derived both in momentum space and in position space. It is shown that when these functions are used as a basis, the kinetic energy term disappears from the Schrödinger equation. A general method is developed for constructing many-electron Sturmian basis sets from...

A formalism is described for choosing computationally manageable Sturmian basis sets which are automatically adapted to the requirements of particular physical problems. The method is a generalization and improvement of the potential harmonic technique of Fabre de la Ripelle, which is extensively used in nuclear and atomic physics. The present gene...

Because of the high degeneracy of hyperspherical harmonics, a method is needed for selecting the most important ones for inclusion in hyperangular basis sets. Such a method was developed by M. Fabre de la Ripelle, who showed that the most important harmonics are λ-projections of the product of the potential and a zeroth-order wave function; and he...

The derivation of the reciprocal-space Schrödinger equation is reviewed, as well as Fock's method for solving it for hydrogenlike atoms. It is shown that Fock's solutions (which represent Fourier transformed hydrogenlike orbitals in terms of 4-dimensional hyperspherical harmonics) can be used as basis sets for solving other problems in quantum chem...

The wave functions of Coulomb systems have cusps at points corresponding to two particle coelescences. In this paper, we derive series representing the cusps in terms of hyperspherical harmonics multiplied by functions of the hyperradius. When the hyperspherical method is applied to Coulomb systems, the harmonics which appear in these series should...

The properties of Sturmian basis sets in d-dimensional direct space and d-dimensional momentum space are reviewed, as well as the relationship between hydrogenlike Sturmians and hyperspherical harmonics. The kernel of the reciprocal-space Schrödinger equation is expanded in terms of Strumian basis sets. This expansion allows Shibuya and Wulfman's t...

In Fock's reciprocal-space treatment of the hydrogen atom,k-space is mapped onto the surface of a 4-dimensional hypersphere, and the solutions (apart from an invariant factor) are 4-dimensional hyperspherical harmonics. Fock's method can be generalized to provide solutions for the Schrdinger equation of a charged particle moving in a many-center Co...

In quantum chemistry, the concepts of the Born-Oppenheimer approximation, the Hartree-Fock approximation and configuration interaction have long been dominant. The most common approach in solving the many-particle Schrödinger equation has been to separate the motions of the nuclei from those of the electrons by means of the Born-Oppenheimer approxi...

Angular momentum and angular integrations are discussed from the standpoint of the theory of harmonic polynomials. General formulae are developed which provide alternatives to the usual group theoretical approach. These formulae are illustrated by applications to the calculation of molecular electrostatic potentials, Fourier transforms of charge de...

Hyperspherical harmonics are the d-dimensional generalization of ordinary 3-dimensional spherical harmonics. In this paper, we review some properties of hyperspherical harmonics. We also discuss the generalization of Fock's reciprocal-space treatment of hydrogen-like atoms and show how it leads naturally to Sturmian basis sets. The hyperspherical S...

Some topics in relativistic quantum chemistry are reviewed with special emphasis on 4-currents and 4-potentials. It is shown that, both in molecular quantum theory and in solid-state physics, calculations can include relativistic and magnetic effects by means of 4-currents without an excessive increase in complication, provided that 4-component Dir...

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 Ørsted Institute in Copenhagen. Althoug...

The 4-current formalism is reviewed and applied to the treatment of magnetic effects in quantum chemistry. The electromagnetic interaction operator is expressed in terms of 4-currents and in terms of electron creation and annihilation operators; and a relativistic analogue of Roothaan's equations is derived. Finally, the 4-current formalism is used...

Methods are discussed for iterating the many-particle Schrödinger equation starting from Hartree–Fock wave functions. It is shown that when the Hartree–Fock wave function is expressed as a linear combination of the appropriate basis functions in hyperspherical coordinates the hyperangular integrals needed for iteration can be evaluated analytically...

A Sturmian basis set is a set of solutions to the Schrödinger equation, with the potential scaled in such a way that all the members of the set correspond to the same value of the energy. We discuss, in particular, the set of Sturmian basis functions corresponding to solutions of the d-dimensional hydrogenlike wave equation. These hydrogenlike Stur...

A growing repertoire of electronic structure methods employ the spatial dimensionD as an interpolation or scaling parameter. It is advantageous to transform the Schrdinger equation to remove all dependence onD from the Jacobian volume element and the Laplacian operator; this introduces a centrifugal term, quadratic inD, that augments the effective...

For large values of d = 3N, the radial distribution function of an N-particle system is sharply peaked near the hyperradius rm = (d − 2)/2k0, where k0≡(2/E/)1/2. This fact allows an approximate separation of the hyperradius, leading to many-dimensional hydrogenlike radial solutions. Kindred applications to dimensional scaling are also discussed, wh...

In equations (3–45) and (3–47), we saw that for d = 4, the number of hyperspherical harmonics corresponding to a given value of λ is (λ+1)2. If we let λ +l = n, where n is the principle quantum number of the hydrogen atom wave functions, we can see that this degree of degeneracy is the same as that of hydrogen. (Notice that the meaning of n here di...

In the previous section, we discussed Fock’s method for finding hydrogenlike wave functions in reciprocal space and its generalization to the d-dimensional hydrogenlike problem: $$ \left( { - \frac{1}{2}\Delta - \frac{Z}{r}} \right)\psi = E\psi $$ (6-1) where r is the hyperradius and where ∆ is the generalized Laplacian operator: $$ \Delta = \sum\l...

In Chapter 5, we reviewed V. Fock’s reciprocal-space treatment of hydrogenlike atoms. It has been pointed out by Shibuya and Wulfman (1965), Monkhorst and Jeziorski (1979), Judd (1975) and Koga (1985; 1988) that Fock’s approach can be generalized in such a way as to yield solutions to the reciprocal-space Schrodinger equation for the many-center Co...

A generalization of the Fourier convolution theorem is used to iterate the many-particle Schrödinger equation in momentum space. The method is applied using hyperspherical coordinates, with many-dimensional hydrogenlike wave functions as the starting point for iteration. The problem of angular integration is converted into a problem of differentiat...

It is shown that a generalization of the Fourier convolution theorem can be used to iterate solutions of the many-particle Schrödinger equation in momentum space. The method is developed both with ordinary coordinates and with hyperspherical coordinates, and as an illustration it is applied to electron correlation in the ground state of helium.

High-quality crystallographic measurements are becomming available for a number of materials, and these measurements, when corrected for thermal vibrations, anomalous scattering and multiple scattering, can give a good picture of the density distribution. By supplementing crystallographic measurements with data from atomic Hartree-Fock calculations...

Formulas are derived which allow the direct construction of total orbital angular momentum eigenfunctions for many-particle systems without the use of Clebsch–Gordan coefficients. One of the equations is closely analogous to Dirac' identity for the total spin operator. This equation describes the action of L2 on a function of the particle coordinat...

Solutions to the many-dimensional analogue of the Schrödinger equation for the hydrogen atom are used as a starting point for solving the many-particle Schrödinger equation for systems with Coulomb interactions. It is shown that zeroth-order solutions can be improved by means of perturbation theory, which is simplified by means of a sum rule.

A method is described for using many-dimensional hydrogen-like wave functions as a starting point for constructing solutions to the Schrödinger equation of an N-particle system. The solutions are built up from symmetry-adapted hyperspherical harmonics, multiplied by functions of the hyperradius, r. Approximate asymptotic solutions for large values...

This volume is concerned with an approximate scheme of a very general type; the scheme reduces the many-electron Schroedinger equation, and it assumes that the effective potential in this equation is the sum of the potential from the nuclei, the classical Coulomb potential from the electronic charge-cloud, and a so-called exchange-correlation poten...

A general formula is given for the canonical decomposition of a homogeneous polynomial of order λ in m variables into a sum of harmonic polynomials. This formula, which involves successive applications of the generalized Laplace operator, is proved in the Appendix. It is shown that the group‐theoretical method for constructing irreducible Cartesian...

The phenomenon of biological specificity is described, and a history of discoveries related to the phenomenon is presented. Aspects of biological specificity described include the mechanism of the immune system, chemotherapy, enzyme-substrate specificity, neurotransmitters, autoassembly of viruses, autoassembly of subcellular organelles, differenti...

The arrangement of membrane-bound pigments, proteins, and lipids in the thylakoids of higher plants is described, and the role of the membrane in preventing the back-reaction is discussed. The photosynthetic bacterium halobacterium halobium is also described. A simplified quantum-mechanical picture of the primary process in photosynthesis is presen...

Hyperspherical coordinates are used to construct a matrix representation of a general N-particle Hamiltonian in the case where the interaction is electrostatic. The Yukawa interaction can be treated similarly, as is shown in an appendix. The basis functions used to construct the matrix representation of H are mononomials inthe 3N coordinates of the...

Techniques for applying hyperspherical coordinates to the quantum-mechanical many-body problem are reviewed. An improved method is presented for evaluating matrix elements of the Hamiltonian of a system of particles. This method involves a rotation in the many-dimensional coordinate space of the system, and it can be applied not only to Coulomb pot...

The S-matrix formalism is used to treat the phenomenon of resonance energy transfer (sensitized fluorescence). It is shown that for dipole-allowed transitions and short sensitizer–acceptor separations, the relativistic treatment yields the same result as the nonrelativistic Perrin–Förster theory. For large sensitizer-acceptor separations, long-rang...

General methods are outlined for constructing Fourier coefficients of the Coulomb and Slater potentials in crystals from X-ray diffraction measurements of the charge density supplemented with atomic Hartree-Fock data. Corrections for temperature effects are discussed. Asymptotic expressions for the Fourier coefficients are also given, as well as an...

The· simplest picture of an atom, a molecule or a solid is the picture of its distribution of charge. It is obtained by specifying the positions of the atomic nuclei and by showing how the density, p(E), of the electronic charge-cloud varies from place to place. A much more detailed picture is provided by the many-electron wavefunction. This quanti...

A general formula for angular integrations in many-dimensional spaces (derived in a previous paper) is applied to several problems connected with solution of the Schrödinger equation for many-particle systems. Matrix elements of the Hamiltonian are derived for cases where the potential can be expressed in terms of functions of the generalized radiu...

A general method for performing angular integrations is presented. The method depends on the fact that the integral must be invariant under rotations of the coordinate system, and it also makes use of combinatorial analysis. In most cases the method presented is computationally much faster than alternative methods of angular integration using Condo...

A new method for calculating crystal orbitals in the Hartree-Fock-Slater approximation is proposed. The method makes use of x-ray crystallographic measurements of the deformation density, and uses transferable integrals to treat the neutral–atom potentials. Methods for evaluating matrix elements of neutral-atom potentials are discussed in detail, a...

A method is proposed for evaluating generalized X-ray scattering factors (Fourier transforms of products of atomic orbitals) in the two-center case with Slater-type orbitals. This method is especially appropriate if one of the Slater exponents is considerably larger than the other.

A method for evaluating in closed form the Fourier transform of a product of two atomic orbitals, i.e. the so-called generalized scattering factor, is described for the case where both atomic orbitals are centred on the same atom.

Fourier transform methods introduced by Harris are applied to the evaluation of Frenkel exciton lattice sums. The slowly-convergent direct lattice sum is converted into a rapidly-convergent reciprocal lattice sum which includes all orders in the multipole expansion. A simple example is discussed, and the calculated exciton energy as a function of w...

Fourier transform methods initiated by Geller and Harris are applied to the calculation of optical properties of molecules. Tables of one-electron two-center integrals needed for the accurate computation of molecular absorption and optical activity are calculated by the Fourier transform method. A general theorem is derived which allows the angular...

A theoretical expression for the ellipticity of an exciton band in an oriented rigid polymer is derived by use of the Frenkel exciton model in conjunction with a fully retarded expression for the partial ellipticity of an electronic transition.

Theoretical expressions are derived for the ellipticity and for the rotatory strength of an electronic transition in an oriented molecule using the fully retarded vector potential for the electromagnetic field. The resulting expression for the retarded rotatory strength is shown to reproduce the Rosenfeld-Condon expression in the electric dipole-ma...

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