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Molecular Integrals for Exponential-Type Orbitals Using Hyperspherical Harmonics
Abstract
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 continues to dominate computational quantum chemistry, because of the ease with which difficult molecular integrals may be evaluated when Gaussians are used as a basis.
In the present chapter, we hope to contribute to a new movement in quantum chemistry, in which ETO’s will not only be able to produce more accurate results than could be obtained using Gaussians, but also will compete with Gaussian technology in the speed of integral evaluation. The method presented here makes use of V. Fock’s projection of three-dimensional momentum-space onto a four-dimensional hypersphere. Using this projection, Fock was able to show that the Fourier transforms of Coulomb Sturmian basis functions are very simply related to four-dimensional hyperspherical harmonics.
With the help of Fock’s relationships and the theory of hyperspherical harmonics we are able to evaluate molecular integrals based on Coulomb Sturmians both rapidly and accurately. The method is then extended to Slater-Type Orbitals by using a closed-form expression for expanding STO’s in terms of Coulomb Sturmians. A general theorem is presented for the rapid evaluation of the necessary angular and hyperangular integrals. The general methods are illustrated by a few examples.
... [1] With the help of Fock's relationships and the theory of hyperspherical harmonics, we are able to evaluate molecular integrals based on Coulomb Sturmians both rapidly and accurately. [2,3] The method is then extended to STO's using a closed-form expression for expanding STO's in terms of Coulomb Sturmians. [4] Coulomb Sturmian basis sets were introduced into quantum theory by L€ owdin, Shull and others. ...
... To evaluate the angular and hyperangular integrations needed for the calculations described earlier, we make use of the hyperangular integration theorem presented in Refs. [3] and [6]. Let x 1 ; . . . ...
... A detailed discussion of the evaluation of the coefficients b l 00 l 0 ;l 1 can also be found in Refs. [2] and [3], as can the complete evaluation of the d 3 p integral in Eq. (34). ...
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 evaluate generalized scattering factors for ETOs which allow first-order density matrices to be measured experimentally using high-quality X-ray diffraction data. © 2015 Wiley Periodicals, Inc.
... CS functions proved to be especially easy to work with since their momentum-space representation by hyperspherical harmonics allows efficient calculation of multi-centre integrals, opening the way for efficient molecular calculations. [25][26][27][28] The Coulomb Sturmian construction generalises well and generalised Sturmian basis sets preserving many useful Sturmian properties can be constructed. These allow us, for example, to build N-particle basis functions that include important geometric properties of the physical system under consideration at the level of the basis. ...
... In the future, we plan to add support for further basis function types in gint and molsturm, in particular, molecular and generalised Sturmians. [25][26][27][28] Via gscf, multiple SCF schemes are available, namely, Roothaan's repeated diagonalisation, 83 Pulay's commutator direct inversion of the iterative subspace (DIIS), 86 and the truncated optimal damping algorithm (tODA), 75 an approximation of the optimal damping algorithm, 77 which is more suitable for the contraction-based interface of gscf. During the SCF procedure, molsturm automatically switches between the available schemes, trying to balance the convergence rate and expense of the individual algorithms. ...
We present the design of a flexible quantum-chemical method development framework, which supports employing any type of basis function. This design has been implemented in the light-weight program package molsturm, yielding a basis-function-independent self-consistent field scheme. Versatile interfaces, making use of open standards like python, mediate the integration of molsturm with existing third-party packages. In this way both rapid extension of the present set of methods for electronic structure calculations as well as adding new basis function types can be readily achieved. This makes molsturm well-suitable for testing novel approaches for discretising the electronic wave function and allows comparing them to existing methods using the same software stack. This is illustrated by two examples, an implementation of coupled-cluster doubles as well as a gradient-free geometry optimisation, where in both cases, an arbitrary basis functions could be used. molsturm is open-source and can be obtained from https://molsturm.org.
... There, we are able to transform the difficult interelectron repulsion integrals (ERI) from integrals over 6 dimensions down to integrals over 3 dimensions, and which can be evaluated efficiently using hyperspherical harmonics. [74][75][76][77][78][79] If an analogous result can be derived for fullerene harmonics such that interelectron repulsion reduces from 4dimensional integrals (i.e, two surface variables) down to integrals over only 2 dimensions, surface-restricted electron repulsion integrals would be possible to calculate efficiently using simple numerical quadratures. However, even if it is possible, this may be a difficult task -and the harmonic transformation may turn out to be expensive. ...
Can we solve electronic wave equations absent a coordinate system? The question arises from the wish to treat polyhedral molecules such as fullerenes as two-dimensional closed surfaces. This would allow us to study electronic structure on intrinsic surface manifolds, which can be derived directly from the bond structure. The wave equation restricted to the (non-Euclidean) surface could then be solved without reference to any three-dimensional geometry of the molecule, and hence without the need for quantum chemical geometry optimization. The resulting 2D system can potentially be solved several orders of magnitude faster than the full wave equation. However, because these curved surfaces do not admit any simple coordinate system, we must devise methods that can do without. In this paper, I describe how surface geometries can be derived from fullerene bond graphs as combinatorial objects, and how electronic structure may be studied by solving wave equations directly on these intrinsic surface manifolds, without needing to find three-dimensional geometries. The goal is approximation methods that are rapid enough to systematically analyze entire isomer spaces consisting of millions of molecules, so as to identify structures with desired properties.
... (29) and (48), exactly and rapidly. These programs are based on a general theorem for angular and hyperangular integration, which we have presented and discussed in some of our previous books and papers (See, for example, [4], Chapter 2, or [30]). The radial p-integrals in Eq. (48) can be evaluated in closed form automatically using Mathematica. ...
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 Sturmian basis sets to solve the N-body wave equation.
Employing Fourier‐transform technique, this work derives the molecular multicenter integrals for the modified Slater‐type orbitals (STOs) with arbitrary principal and angular quantum numbers, in which the spherical harmonics parts are the homogeneous solutions of original STOs. The integrands include spherical Bessel functions and associated Legendre functions. One‐electron three‐center nuclear attraction integral and two‐electron four‐center Coulomb repulsion integral are presented as numerical integrations in two dimensions and three dimensions, respectively. A simple Python script is provided to run the formulas. The correctness and computational efficiency of the script have been verified.
We consider an 1-electron model Hamiltonian, whose potential energy corresponds to the Coulomb potential of an infinite wire with charge Z distributed according to a Gaussian function. The time independent Schrödinger equation for this Hamiltonian is solved perturbationally in the asymptotic limit of small amplitude vibration (Gaussian function width close to zero). We propose to use the naturally polarized functions so-obtained, as orbital basis sets for quantum chemical calculations. In particular, they should be well suited to perform electron-nucleus mean field configuration interaction calculations. Since the free-parameters of the model have the remarkable property to factorize the perturbative corrections to the eigenfunctions, these corrective part in factor can be simply added as additional functions to standard basis sets, leaving it to the molecular orbital calculation to optimize the free parameters within molecular orbital coefficients.
A discussion of basis sets consisting of exponentially decaying Coulomb Sturmian functions for modeling electronic structures is presented. The proposed basis-set construction selects Coulomb Sturmian functions using separate upper limits to their principal, angular momentum, and magnetic quantum numbers. Their common Coulomb Sturmian exponent is taken as a fourth parameter. The convergence properties of such basis sets are investigated taking the second- and third-row atoms at the Hartree-Fock level as examples. Thereby, important relations between the values of the basis-set parameters and the physical properties of the electronic structure are recognized. Furthermore, a connection between the optimal, i.e., minimum-energy, Coulomb Sturmian exponent and the average Slater exponents values obtained by Clementi and Raimondi [J. Chem. Phys. 38, 2686 (1963)] is made. These features of Coulomb Sturmian basis sets emphasize their ability to correctly reproduce the physical features of the Hartree-Fock wave functions. As an outlook, the application of Coulomb Sturmian discretizations for molecular calculations and post-Hartree-Fock methods is briefly discussed.
We present the design of a flexible quantum-chemical method development framework, which supports employing any type of basis function. This design has been implemented in the light-weight program package molsturm, yielding a basis-function-independent self-consistent field scheme. Versatile interfaces, making use of open standards like python, mediate the integration of molsturm with existing third-party packages. In this way, both rapid extension of the present set of methods for electronic structure calculations as well as adding new basis function types can be readily achieved. This makes molsturm well-suitable for testing novel approaches for discretising the electronic wave function and allows comparing them to existing methods using the same software stack. This is illustrated by two examples, an implementation of coupled-cluster doubles as well as a gradient-free geometry optimisation, where in both cases, arbitrary basis functions could be used. molsturm is open-sourced and can be obtained from http://molsturm.org.
This chapter discusses different types of basis sets for electronic structure calculations with the main focus on Gaussian type basis sets for molecular calculations. After a brief overview of the basis set approximation in quantum chemistry, it summarizes and discusses the basic requirements of an ideal basis set. The chapter reviews approximations to the ideal basis set. It talks about the most widely used types of functions in basis sets, namely Slater and Gaussian type orbitals (GTOs) and plane waves, giving examples of other types of functions in use. Strategies for contracting, optimizing, and augmenting Gaussian basis sets are discussed with specific examples of widely used basis sets. Basis sets designed for electric and magnetic properties as well as relativistic calculations are discussed along with the concept of basis set limiting results in terms of basis set extrapolations and composite methods based upon them. Basis set incompleteness and basis set superposition errors are also discussed.
Algorithms for the efficient calculation of two-electron integrals in the newly developed mixed ramp-Gaussian basis sets are presented, alongside a Fortran90 implementation of these algorithms, RampItUp. These new basis sets have significant potential to (1) give some speed-up (estimated at up to 20% for large molecules in fully optimised code) to general-purpose Hartree-Fock (HF) and density functional theory quantum chemistry calculations, replacing all-Gaussian basis sets, and (2) give very large speed-ups for calculations of core-dependent properties, such as electron density at the nucleus, NMR parameters, relativistic corrections, and total energies, replacing the current use of Slater basis functions or very large specialised all-Gaussian basis sets for these purposes. This initial implementation already demonstrates roughly 10% speed-ups in HF/R-31G calculations compared to HF/6-31G calculations for large linear molecules, demonstrating the promise of this methodology, particularly for the second application. As well as the reduction in the total primitive number in R-31G compared to 6-31G, this timing advantage can be attributed to the significant reduction in the number of mathematically complex intermediate integrals after modelling each ramp-Gaussian basis-function-pair as a sum of ramps on a single atomic centre.
Recently the author has given two modifications of a nonlinear extrapolation method due to Levin and Sidi, which enable one to accurately and economically compute certain infinite integrals whose integrands have a simple oscillatory behavior at infinity. In this work these modifications are extended to cover the case of very oscillatory infinite integrals whose integrands have a complicated and increasingly rapid oscillatory behavior at infinity. The new method is applied to a number of complicated integrals, among them the solution to a problem in viscoelasticity. Some convergence results for this method are presented.
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-electron molecules is also discussed. Basis sets of this type are shown to have many advantages over other types of ETO's, especially the property of automatic scaling.
A new method is presented for calculating interelectron repulsion integrals for molecular Coulomb Sturmian basis sets. This makes use of an expansion of densities in terms of 2. k-Sturmians, and the interelectron repulsion integrals are then calculated by a method based on the theory of hyperspherical harmonics.A rudimentary software library has been implemented and preliminary benchmarks indicate very good performance: On average 40. ns, or approximately 80 clock cycles, per electron repulsion integral. This makes molecular Coulomb Sturmians competitive with Gaussian type orbitals in terms of speed, and is three to four orders of magnitude faster than methods based on expanding the exponential type orbitals in Gaussians. A full software library will be made available during autumn 2013.
Recently the author has given two modifications of a nonlinear extrapolation method due to Levin and Sidi, which enable one to accurately and economically compute certain infinite integrals whose integrands have a simple oscillatory behavior at infinity. In this work these modifications are extended to cover the case of very oscillatory infinite integrals whose integrands have a complicated and increasingly rapid oscillatory behavior at infinity. The new method is applied to a number of complicated integrals, among them the solution to a problem in viscoelasticity. Some convergence results for this method are presented.
The simple connection between the Slater orbitals, venerable in quantum chemistry, and the
Coulomb Sturmian orbitals, more recently employed in atomic and molecular physics, is pointed out explicitly in view of the renewed interest in both as basis sets in applied quantum mechanics. Research in Slater orbitals mainly concerns multicentre, many-body integrals, whereas that on Sturmians exploits their orthonormality
and completeness with no need of continuum states. An account of recent progress is outlined, also with reference to relationships between the two basis sets, and with the momentum space and hyperspherical harmonics representations.
To exploit hyperspherical harmonics (including orthogonal transformations)
as basis sets to obtain atomic and molecular orbitals, Fock projection into
momentum space for the hydrogen atom is extended to the mathematical ddimensional
case, higher than the physical case d ˆ 3. For a system of N particles
interacting through Coulomb forces, this method allows us to work both in a
d ˆ 3…N ¡ 1† dimensional con®guration space (on eigenfunctions expanded on a
Sturmian basis) and in momentum space (using a …d ‡ 1†-dimensional hyperspherical
harmonics basis set). Numerical examples for three-body problems are
presented. Performances of alternative basis sets corresponding to di erent
coupling schemes for hyperspherical harmonics have also been explicitly obtained
for bielectronic atoms and H‡2 (in the latter case, also in the Born±Oppenheimer
approximation extending the multicentre technique of Shibuya and Wulfman).
Among the various generalizations and applications particularly relevant is the
introduction of alternative expansions for multidimensional plane waves, of use
for the generalization of Fourier transforms to many-electron multicentre problems.
The material presented in this paper provides the starting point for numerical
applications, which include various generalizations and hierarchies of approximation
schemes, here brie¯y reviewed.
In this Review we show several mathematical techniques that can be used in molecular calculations using Slater orbitals, like the transformation of the Hamiltonian, derivatives of spherical harmonics with respect to the angles, and angular transformations. We treat several kinds of integrals in detail: the exchange, Coulomb and hybrid repulsion and exchange correlated integrals, and the three-center nuclear attraction ones. The Molecular Orbital (MO) wave function is compared with the elliptical one. It is discussed how MO wave functions using Slater orbitals can be extended to Configuration Interaction and Hylleraas-Configuration Interaction calculations. Finally, some CI calculations on Hydrogen molecule using these techniques are shown.
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 energy. The advantage of such an isoenergetic basis set is that every basis function has the correct turning point behavior needed for efficient synthesis of the wave function.
The book also discusses methods for automatic generation of symmetry-adapted basis sets. Calculations using the generalized Sturmian method are presented and compared with experimental results from the NIST database. The relationship of Sturmians to harmonic polynomials and hyperspherical harmonics is also described. Methods for treating angular functions and angular integrals by means of harmonic projection are discussed, and these methods are shown to be especially useful for relativistic calculations. A final chapter discusses application of the generalized Sturmian method to the calculation of molecular spectra.
We discuss a generalization of the resolution of the identity by considering one-body resolutions of two-body operators, with particular emphasis on the Coulomb operator. We introduce a set of functions that are orthonormal with respect to 1r(12) and propose that the resulting "resolution of the Coulomb operator," r(12) (-1)=mid R:phi(i)phi(i)mid R:, may be useful for the treatment of large systems due to the separation of two-body interactions. We validate our approach by using it to compute the Coulomb energy of large systems of point charges.
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 approximation, and then to reduce the resulting many-electron wave equation to a single-electron equation by means of the Hartree-Fock approximation. The effects of correlation are then added by means of a configuration interaction calculation. Multiconfigurational SCF calculations still adhere to this basic framework. The problem with this approach is that in order to adequately describe the effects of correlation, a very large number of configurations are needed. Accurate calculations even on small molecules can strain the power of the largest modern computers; and the resulting multiconfigurational wave functions are often so complicated that they are difficult to interpret.
Molecular electronic structure calculations are usually based upon the LCAO-MO approach in which the molecular orbitals (MO’s) are represented by linear combinations of atomic orbitals (AO’s). For the quantum mechanical calculation of energies and properties of molecules one has to evaluate the matrix elements of the Hamiltonian. With the AO’s being centered at the various atomic nuclei in the molecule under consideration, these matrix elements become the infamous multicenter molecular integrals: The use of AO’s which resemble physically meaningful one-electron wave functions for the one-nucleus one-electron Coulomb problem (i.e. the hydrogen atom) like exponentially decaying Slater-type orbitals (STO’s) leads to multicenter integrals which are notoriously difficult to evaluate for poly-atomic molecules.
It is easy to prove that atomic and molecular orbitals must decay exponentially at long-range. They should also possess cusps when an electron approaches another particle (a peak where the ratio of orbital gradient to function gives the particle charge). Therefore, hydrogen-like or Slater-type orbitals are the natural basis functions in quantum molecular calculations. Over the past four decades, the difficult integrals led computational chemists to seek alternatives. Consequently, Slater-type orbitals were replaced by Gaussian expansions in molecular calculations (although they decay more rapidly and have no cusps). From the 1990s on, considerable effort on the Slater integral problem by several groups has led to efficient algorithms which have served as the tools of new computer programs for polyatomic molecules. The key ideas for integration: one-center expansion, Gauss transform, Fourier transform, use of Sturmians and elliptical coordinate methods are presented here, together with their advantages and disadvantages, and the latest developments within the field. Recent advances using symbolic algebra, pre-calculated and stored factors and the state-of-the art with regard to parallel calculations are reported. At times, high accuracy is not required and at others speed is unimportant. A recent approximation separating the variables of the Coulomb operator will be described, as well as its usefulness in molecular calculations. There is a renewed interest in the use of Slater orbitals as basis functions for configuration interaction (CI) and Hylleraas-CI atomic and molecular calculations, and in density functional and density matrix theories. In a few special cases, e.g. three particles, symmetry conditions lead to simple explicit pair-correlated wave-functions. Similarly, advantages of this basis are considerable for both absolute energy and fixed-node error in quantum Monte Carlo (QMC). The model correlated functions may be useful to build up Jastrow factors. These considerations will be dealt with in the context of modern computer hardware and its rapid development.
This chapter focuses on scattering problems that have made significant use of Sturmians. The use of Sturmians to include effects of the continuum has the appeal of a systematic development, but there are other ways to include the continuum. One advantage of pseudo-states over Sturmians is that they are applicable to inelastic scattering problems, but by using linear combinations of Sturmians the same end can be achieved. The subsequent interpretation of the phase shifts in terms of elastic scattering cross sections remains unaltered from the usual hydrogenic treatment. The appeal of the Sturmians, however, lies in their ability to describe continuum admixtures to the closed channels, thus their usefulness can be retained if they are used only for this purpose. Elastic scattering from a two-particle system in its ground state occurs between the lowest two of these points; between the second and third points an inelastic channel is open, another inelastic channel opens between the third and fourth points.
The basic properties of the polynomials p
n
(x) that satisfy the orthogonality relations $$ \int_a^b {{p_n}(x)} {p_m}(x)\rho (x)dx = 0\quad (m \ne n) $$
((2.0.1))
hold also for the polynomials that satisfy the orthogonality relations of a more general form, which can be expressed in terms of Stielties integrals $$ \int_a^b {{p_n}(x)} {p_m}(x)dw(x) = 0\quad (m \ne n), $$
((2.0.2))
where w(x) is a monotonic nondecreasing function (usually called the distribution function). The orthogonality relation (2.0.2) is reduced to (2.0.1) in the case when the function w(x) has a derivative on (a, b) and w′(x) = ϱ(x). For solving many problems orthogonal polynomials are used that satisfy the orthogonality relations (2.0.2) in the case when w(x) is a function of jumps, i.e. the piecewise constant function with jumps ϱ
i
at the points x = x
i
. In this case the orthogonality relation (2.0.2) can be rewritten in the form $$ \sum\limits_i {{p_n}({x_i})pm} ({x_i}){\rho_i} = 0\quad (m \ne n). $$
((2.0.3))
KeywordsDifference EquationOrthogonal PolynomialRecursion RelationDiscrete VariableHermite PolynomialThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
We analyze the properties of Slater-type functions (STFs) and B functions with respect to the Fourier transformation which is one of the most important methods for the evaluation of multicenter integrals. Although B functions have a much more complicated structure than STFs, their Fourier transform is probably the simplest of all expontential-type basis functions. Accordingly, in multicenter integrals, B functions seem to have much more attractive properties than STFs. We demonstrate this by analyzing shifting operators which make it possible to increase quantum numbers of the orbitals. Again we find that the shifting operators of B functions can be applied much more easily and yield much more compact results than the corresponding operators for STFs. We also show that the extremely compact convolution and Coulomb integrals of B functions are direct consequences of the simple Fourier transform of B functions.
A method is suggested for setting up analytic atomic wave functions which form good approximations to Hartree's functions. These functions are of the form Sigmacrne-ar, where the exponent a as well as c and n vary from one term to another. The constants are determined for 1, 2, and 3-quantum electrons by fitting Hartree's values numerically for five selected atoms, and interpolation methods are presented for dealing with the intermediate atoms. A method is suggested for setting up exactly orthogonal functions, with no loss of accuracy. It is shown that the analytic wave functions are the solutions of central field problems in which the field is slightly different for different quantum numbers, on account of the inaccuracy in the function, but a table shows that the discrepancy between this and the correct field is small over the region where the wave function is large. Suggestions are made for future work, on the one hand in extending the tables, on the other in using the wave functions in investigating atomic energies, exchange integrals, etc.
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.
The troublesome problem of developing cusps in ordinary molecular wave functions can be avoided by working with momentum-space wavefunctions for these have no cusps. The need for continuum wavefunctions can be eliminated if one works with a hydrogenic basis set in Fock's projective momentum space. This basis set is the set of R_4 spherical harmonics and as a consequence one may obtain, solely by the ordinary angular momentum calculus, algebraic expressions for all the integrals required in the solution of the momentum space Schrodinger equation. A number of these integrals and a number of R_4 transformation coefficients are tabulated. The method is then applied to several simple united-atom and l.c.a.o. wavefunctions for H^+_2, and ground state energies and corrected wavefunctions are obtained. It is found in this numerical work that the method is most appropriate at internuclear distances some-what less than the equilibrium distance. In Fock's representation both l.c.a.o. and unitedatom approximations become exact as the internuclear distance approaches zero. The united-atom expansion can be viewed as an eigenvalue equation for the root-mean-square momentum, p_0 = surd(-2E). In the molecule, the matrix operator corresponding to p_0 is related to the operator for the united-atom by a sum of unitary transformations, one for each nucleus in the molecule.
Two new classes of nonlinear transformations, the D-transformation to accelerate the convergence of infinite integrals and the d-transformation to accelerate the convergence of infinite series, are presented. In the course of the development of these transformations two interesting asymptotic expansions, one for infinite integrals and the other for infinite series, are derived. The transformations D and d can easily be applied to infinite integrals ∫∞0f(t)dt whose integrands f(t) satisfy linear differential equations of the form f(t)=Σmk=1pk(t)fk(t) and to infinite series Σ∞r=1f(r) w terms f(r) satisfy a linear difference equation of the form f(r)=Σmk=1pk(r)Δkf(r), such that in both cases the pk have asymptotic expansions in inverse powers of their arguments. In order to be able to apply these transformations successfully one need not know explicitly the differential equation that the integrand satisfies or the difference equation that the terms of the series satisfy; mere knowledge of the existence of such a differential or difference equation and its order m is enough. This broadens the areas to which these methods can be applied. The connection between the D- and d-transformations with some known transformations in shown. The use and the remarkable efficiency of the D- and d-transformations are demonstrated through several numerical examples. The computational aspects of these transformations are described in detail.
This communication deals with the general theory of obtaining numerical electronic wave functions for the stationary states of atoms and molecules. It is shown that by taking Gaussian functions, and functions derived from these by differentiation with respect to the parameters, complete systems of functions can be constructed appropriate to any molecular problem, and that all the necessary integrals can be explicitly evaluated. These can be used in connexion with the molecular orbital method, or localized bond method, or the general method of treating linear combinations of many Slater determinants by the variational procedure. This general method of obtaining a sequence of solutions converging to the accurate solution is examined. It is shown that the only obstacle to the evaluation of wave functions of any required degree of accuracy is the labour of computation. A modification of the general method applicable to atoms is discussed and considered to be extremely practicable.
The method of superposition of configurations is examined in its application to the helium atom in two cases: a 21×21 matrix including all configurations up to 〈6s〉2, and a 20×20 matrix with all configurations up to the 4‐quantum level, including angular terms. A new radial limit is established at −2.87900±0.00003, and this is used to discuss the convergence of such expansions in Legendie functions. The variation with scale factor is discussed in detail. The wave functions are analyzed in terms of natural spin orbitals (NSO's), which seem to have many advantages. The first NSO bears a striking resemblance to the Hartree‐Fock function, and the first two together provide a close approximation to the solution of the extended Hartree‐Fock equations with different orbitals for different electrons. An energy of −2.877924 is obtained for the best (u, v) function found. An analysis of the results suggests that inner orbitals may be better represented by pure exponentials than by Hartree‐Fock orbitals whenever additional correlational degrees of freedom are permitted. Expressed in approximate NSO form, the wave function is almost invariant to choice of basis set, provided that the latter is reasonably chosen. In particular, the necessity of including continuum terms along with the discrete hydrogen‐like set is demonstrated.
The internal 0(4) symmetry group of the nonrelativistic hydrogen atom is
discussed and used to relate the various approaches to the bound-state
problems. A more general group 0(1,4) of transformations is shown to
connect the various levels, which appear as basis vectors for a
continuous set of unitary representations of this noncompact group.
A new technique is presented for the solution of Poisson's equation in spherical coordinates. The method employs an expansion of the solution in a new set of functions defined herein for the first time, called ‘spectral forms’. The spectral forms have spherical harmonics as their angular part, but use a new set of radial functions that automatically statisfy the boundary conditions, up to a multiplicative constant, on the Poisson solution. The resultant problem reduces to a set of simultaneous equations for the expansion coefficients . The matrix A is block diagonal in the spherical harmonic indices l,m and is independent of any parameters. The simultaneous equations may be solved by LU decomposition. The LU decomposition only needs to be done once and multiple right hand sides (B-vectors) can be treated by a matrix-vector multiply. For a parallel computing platform, each such B-vector may be dealt with on a separate processor. Thus the algorithm is highly parallel. This technique may be used to calculate Coulomb energy integrals efficiently on a parallel computer.
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 to achieve this ambitious goal for molecular systems of ever-increasing size. This book focuses on non-mainstream methods to solve the molecular electronic Schrödinger equation. Each method is based on a set of core ideas and this volume aims to explain these ideas clearly so that they become more accessible. By bringing together these non-standard methods, the book intends to inspire graduate students, postdoctoral researchers and academics to think of novel approaches. Is there a method out there that we have not thought of yet? Can we design a new method that combines the best of all worlds?
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 many-particle system is usually so difficult that it requires approximate methods, the most common of which is expansion of the eigenfunctions in terms of basis functions that obey the boundary conditions of the problem. The computational effort needed in such problems can be much reduced by making use of symmetry-adapted basis functions. The conventional method for generating symmetry-adapted basis sets is through the application of group theory, but this can be difficult. This book describes an easier method for generating symmetry-adapted basis sets automatically with computer techniques. The method has a wide range of applicability, and can be used to solve difficult eigenvalue problems in a number of fields. The book is of special interest to quantum theorists, computer scientists, computational chemists and applied mathematicians.
Momentum space hydrogenic orbitals can be regarded as orthonormal and complete Sturmian basis sets and explicitly given in terms of (hyper)-spherical harmonics on the 4-D hypersphere S3. Among the alternative coordinate systems that allow separation of variables, the usual ones involving parameterizations of the sphere S3 by circular functions correspond to canonical subgroup reduction chains; we also investigate harmonic “elliptic” sets (as, e.g., obtained by parameterizations in terms of Jacobi elliptic functions). In this article we list the canonical hydrogenic Sturmian sets and the orthogonal transformations connecting them. The latter enjoy useful three-term recurrence relationships that allow their efficient calculations even for large strings. We also consider modifications needed when the conservation of the symmetry of Sturmians with respect to parity. Finally, we discuss some properties of elliptic hydrogenic Sturmians and their relations with canonical Sturmians. Because elliptic Sturmians cannot be expressed in closed form, it is important to find expansions in a suitable basis set and calculate the transformation coefficients. We derive three-term recursion relationships fulfilled by the coefficients of the transformation between elliptic Sturmians and canonical Sturmians. A concluding discussion on the connections between configuration space and momentum space hydrogenic Sturmians completes this article. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
A general ab initio package using Slater-type atomic orbitals is presented. This package, called STOP, uses the one-center two-range expansion method to evaluate the multicenter electronic integrals. Thoroughly optimized numerical techniques, in particular, convergence accelerators and suitable Gauss quadratures, are used in the algorithms which provide accurate numerical values for all these integrals. STOP thus provides wavefunctions for general molecular structures at the self-consistent field level for the first time over a Slater-type orbital basis. © 1996 John Wiley & Sons, Inc.
IntroductionGeneral ConsiderationsOne-electron, Two-center IntegralsThree-center Nuclear Attraction IntegralsElectron Repulsion IntegralsComputational Details and Results
A program for computing all the integrals appearing in molecular calculation with Slater-type orbitals is reported. The program is mainly intended as a reference for testing and comparing other algorithms and techniques. An analysis of the performance of the program is presented, paying special attention to the computational cost and the accuracy of the results. Results are also compared with others obtained with Gaussian basis sets of similar quality. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1284–1293, 1998
Molecular overlap-like quantum similarity measurements imply the evaluation of overlap integrals of two molecular electronic densities related by Dirac delta function. When the electronic densities are expanded over atomic orbitals using the usual LCAO-MO approach (linear combination of atomic orbitals), overlap-like quantum similarity integrals could be expressed in terms of four-center overlap integrals.It is shown that by introducing the Fourier transform of delta Dirac function in the integrals and using the Fourier transform approach combined with the so-called B functions, one can obtain analytic expressions of the integrals under consideration. These analytic expressions involve highly oscillatory semi-infinite spherical Bessel functions, which are the principal source of severe numerical and computational difficulties.In this work, we present a highly efficient algorithm for a fast and accurate numerical evaluation of these multicenter overlap-like quantum similarity integrals over Slater type functions. This algorithm is based on the approach due to Safouhi. Recurrence formulae are used for a better control of the degree of accuracy and for a better stability of the algorithm. The numerical result section shows the efficiency of our algorithm, compared with the alternatives using the one-center two-range expansion method, which led to very complicated analytic expressions, the epsilon algorithm and the nonlinear transformation.
a b s t r a c t We discuss a resolution of the Coulomb operator, r À1 12 ¼ j/ i ih/ i j, into a one-particle basis. We show that the Laguerre polynomials generate a resolution with attractive computational properties and we apply it to the calculation of Coulomb and exchange energies in hydrogenic ions, the H 2 molecule, and the Be atom. Rapid convergence is observed in all cases and a theoretical reason for this is discussed.
This work gives new, highly accurate optimized gaussian series expansions for the B functions used in molecular quantum mechanics. These functions are generally chosen because of their compact Fourier transform, following Shavitt. The inverse Laplace transform in the square root of the variable is used for Gauss quadrature in this work. Two procedures for obtaining accurate gaussian expansions have been compared for the required extended precision arithmetic. The first is based on Gaussian quadratures and the second on direct optimization. Both use the Maple computer algebra system. Numerical results are tabulated and compared with previous work. Special cases are found to agree before pushing the optimization technique further. The optimal gaussian expansions of B functions obtained in this work are available for reference. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009
In this work, Slater-type atomic orbitals (STOs) are expanded over B functions to exploit the compact Fourier transform of these functions. The analytic representations of three-center nuclear attraction integrals and multicenter bielectronic integrals over B functions involve semiinfinite highly oscillatory integrals, which can be transformed into infinite series. These integrals must be evaluated precisely and quickly. This work describes an efficient method for the evaluation of these integrals based on the nonlinear SD transformation, recently developed by Safouhi. The SD method was applied to multicenter integrals over B functions, and it is shown that this method is highly efficient compared with alternatives previously used. The SD approach should lead to a definitive suite of ab initio Slater software. The convergence properties were analyzed and they showed that the approximations obtained using the SD approach converge to the exact values of the semiinfinite integrals without any constraint. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
Die Schrdinger-Gleichung fr das Wasserstoffatom im Impulsraum erweist sich als identisch mit der Integralgleichung fr die Kugelfunktionen der vierdimensionalen Potentialtheorie. Die Transformationsgruppe der Wasserstoffgleichung ist also die vierdimensionale Drehgruppe; dadurch wird die Entartung der Wasserstoffniveaus in bezug auf die Azimutalquantenzahl l erklrt. Die aus der potentialtheoretischen Deutung der Schrdinger-Gleichung folgenden Beziehungen (Additionstheorem usw.) erlauben mannigfache physikalische Anwendungen. Die Methode ermglicht, die unendlichen Summen, die in der Theorie des Compton-Effektes an gebundenen Elektronen und in verwandten Problemen auftreten, fast ohne Rechnung auszuwerten. Unter Zugrundelegung eines vereinfachten Atommodells lassen sich ferner explizite Ausdrcke fr die Dichtematrix im Impulsraum, fr Atomformfaktoren, fr das Abschirmungspotential usw. aufstellen.
Hydrogenoid orbitals, i.e. the solutions to the Schrödinger equation for a central Coulomb field, are considered in mathematical dimensions d = 2 and d > 3 different from the physical case, d = 3. Extending known results for d = 3, Sturmian basis sets in configuration (or direct) space — corresponding to variable separation in parabolic coordinates — are introduced as alternatives to the ordinary ones in spherical coordinates: extensions of Fock stereographic projections allow us to establish the relationships between the corresponding momentum (or reciprocal) space orbitals and the alternative forms of hyperspherical harmonics. Properties of the latter and multi-dimensional Fourier integral transforms are exploited to obtain the matrix elements connecting the alternative basis sets explicitly in terms of Wigner's rotation matrix elements for d = 2 and generalized vector coupling (or Hahn) coefficients for d > 3. The use of these orbitals as complete and orthonormal expansion basis sets for atomic and molecular problems is briefly commented.
The properties of a set of eigenfunctions closely related to Schroedinger wave functions are exploited to deal with certain three-body problems. These functions, termed Sturmian functions, have a distinct advantage over Schroedinger functions when used as an expansion basis since they form a complete set without a continuum, regardless of the potential existing between the particles. This enables one to examine, for example, the mechanism of polarization of the deuteron in elastic N-D scattering. Further use of the Sturmian functions can be made in describing elastic scattering of electrons and positrons from hydrogen.A detailed calculation of elastic scattering of positrons from atomic hydrogen is carried out. It is shown that Sturmian functions converge rapidly when they are used as an expansion basis, and provide phase shifts which are slightly greater than rigorous lower bounds provided by a variational treatment.
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 used previously to study atomic spectra. We show that when the method is applied to diatomic molecules, all of the relevant integrals are pure functions of the parameter s = kR, and therefore they can be evaluated once and for all and stored.
A method for computing the overlap integral of two Slater orbitals of arbitrary l and integer n>=l on arbitrary centers is developed. The expressions permit the elimination in a simple way of the numerical difficulties that arise when nonlinear parameters are close together. The complexity of a calculation using the results is discussed and compared with other approaches. Numerical examples and qualitative estimates are given to indicate the accuracy of the method.
We present analytic refinements and applications of the deformed atomic densities method [Fernández Rico, J.; López, R.; Ramírez, G. J Chem Phys 1999, 110, 4213-4220]. In this method the molecular electron density is partitioned into atomic contributions, using a minimal deformation criterion for every two-center distributions, and the atomic contributions are expanded in spherical harmonics times radial factors. Recurrence relations are introduced for the partition of the two-center distributions, and the final radial factors are expressed in terms of exponential functions multiplied by polynomials. Algorithms for the practical implementation are developed and tested, showing excellent performances. The usefulness of the present approach is illustrated by examining its ability to describe the deformation of atoms in different molecular environments and the relationship between these atomic densities and some chemical properties of molecules.
Harmonic polynomials, hyperspherical harmonics, and Sturmians Fundemental World of Quantum Chemistry, A Tribute Volume to the Memory of Per-Olov Löwdin
- J S Avery
J. S. Avery, Harmonic polynomials, hyperspherical harmonics, and Sturmians, in: E. Brändas, E. Kryachko (Eds.), Fundemental World of Quantum
Chemistry, A Tribute Volume to the Memory of Per-Olov Löwdin, Kluwer,
2003, pp. 261-296.
New computational methods in the quantum theory of nanostructures
- J E Avery
J. E. Avery, New computational methods in the quantum theory of nanostructures, Ph.D. thesis, University of Copenhagen (2011).