Hydrogen-type orbitals in terms of Gaussian functions (original) (raw)
Related papers
Atomic orbitals revisited: generalized hydrogen-like basis sets for 2nd-row elements
Theoretical Chemistry Accounts, 2018
In the present work, we revisit the problem of atomic orbitals from the positions mostly dictated by semiempirical approaches in quantum chemistry. To construct basis set, having proper nodal structure and simple functional form of orbitals and representing atomic properties with reasonable accuracy, authors propose an Ansatz based on gradual improvement of hydrogen atomic orbitals. According to it, several basis sets with different numbers of variable parameters are considered and forms of orbitals are obtained for the 2nd-row elements either by minimization of their ground state energy (direct problem) or by extracting from atomic spectra (inverse problem). It is shown that so-derived three-and four-parametric basis sets provide accurate description of atomic properties, being, however, substantially provident for computational requirements and, what is more important, simple to handle in analytic models of quantum chemistry. Since the discussed Ansatz allows a generalization for heavier atoms, our results may be considered not only as a solution for light elements, but also as a proof of concept with possible further extension to a wider range of elements.
Wiley Interdisciplinary Reviews: Computational Molecular Science, 2012
Electronic structure methods for molecular systems rely heavily on using basis sets composed of Gaussian functions for representing the molecular orbitals. A number of hierarchical basis sets have been proposed over the last two decades, and they have enabled systematic approaches to assessing and controlling the errors due to incomplete basis sets. We outline some of the principles for constructing basis sets, and compare the compositions of eight families of basis sets that are available in several different qualities and for a reasonable number of elements in the periodic table.
Distributed Gaussian orbitals for molecular calculations: application to simple systems
Molecular Physics
In this article, the possible use of sets of basis functions alternative with respect to the usual atom-centred orbitals sets is considered. The orbitals describing the inner part of the wavefunction (i.e. the region close to each nucleus) are still atomic Gaussian functions: tight Gaussian orbitals having different angular momenta and large exponential coefficients, centred on each nucleus. On the other hand, the outer part of the wavefunction is described through a set of s-type distributed Gaussian orbitals: s-type Gaussians having a unique fixed exponent, and whose fixed centres are placed on a uniform mesh of points evenly distributed in the region surrounding all the atoms of the molecule. The resulting basis sets are applied to various oneelectron systems in order to assess the capability to describe different types of oneelectron wavefunctions. Moreover, the hydrogen atom and the dihydrogen cation, for which accurate solutions exist, are also considered for comparison, to assess the effectiveness of the proposed approach. Preliminary results concerning the treatment of electron correlation, necessary for a quantitatively correct description of manyelectron atoms and molecules, are also presented.
Fourier transform of hydrogen-type atomic orbitals
Canadian Journal of Physics, 2018
Hydrogen type atomic orbitals (HTOs) are important part of other exponential type orbitals (ETOs). These orbitals have some mathematical properties and they are used usually in the theoretical atomic and molecular investigations as special functions to figure out analytical expressions. Fourier transform method (FTM) is a great way to denote basis functions into the momentum space. Because, their Fourier transforms are easier to use in mathematical calculations. In this paper, we obtain new and useful mathematical representations for the Fourier transform of HTOs related with Gegenbauer polynomials and hypergeometric functions, by using recurrence relations of Laguerre polynomials, Rayleigh expansion and some properties of normalized HTOs.
International Journal of Quantum Chemistry, 2019
As part of previous studies, we introduced a new type of basis function named Simplified Box Orbitals (SBOs) that belong to a class of spatially restricted functions which allow the zero differential overlap (ZDO) approximation to be applied with complete accuracy. The original SBOs and their Gaussian expansions SBO-3G form a minimal basis set, which was compared to the standard Slater-type orbital basis set (STO-3G). In the present paper, we have developed the SBO basis functions at double-zeta (DZ) level, and we have assessed the option of expanding the SBO-DZ as a combination of Gaussian functions. Finally, we have determined the quality of the new basis set by comparing the molecular properties calculated with SBO-nG with those achieved with some standard basis sets. K E Y W O R D S ab-initio calculations, box orbitals, confined systems, Gaussian expansion, spatially restricted basis functions 1 | INTRODUCTION: SIMPLIFIED BOX ORBITAL AND SIMPLIFIED BOX ORBITAL-nG FUNCTIONS Spatially restricted functions are distinguished as having a value of zero from a certain distance to the center to which they are referred: r > r o) χ(r,θ,ϕ) = 0. This characteristic makes the integrals S pq , H pq , and (pq|rs) zero when the distance between the centers of the two basis functions χ p and χ q is higher than the sum of the radii associated with those two functions. Therefore, it leads to simplifications analogous to those of the zero differential overlap (ZDO) approximation of the semiempirical methods, but in a completely ab-initio context. This advantage has been proven to be useful for improving the computing calculations as shown by the results obtained, for example, with Ramp functions. [1-3] In previous studies based on the pioneering work of Lepetit et al. [4] and Fdez Rico et al., [5-8] we developed a valid spatially restricted basis set for performing calculations on molecules with atoms from H to Kr, but at "single-zeta level." [9-11] In this paper we have developed an Simplified Box Orbital (SBO) basis set at "double-zeta level," which allows calculations to be performed with similar or higher precision than the well-known standard basis set 6-311G(d) of Pople et al. [12] An SBO is a spatially restricted function achieved through the linear combination of terms (r − r o) 3n for the interval 0 < r < r o and with the value of zero outside this interval. SBO r, θ, ϕ ð Þ= R SBO r ð ÞÁΥ m ℓ θ,ϕ ð Þ ð1Þ with the radial part defined as a piecewise function:
The expansion of hydrogen states in Gaussian orbitals
2004
The convergence properties of Gaussian orbitals are studied by considering a very simple system, the hydrogen atom. We have variationally optimized even-tempered basis sets containing up to 60 s functions for the ground state and the first excited S state of the hydrogen atom, to an accuracy of 10 À15 E h . In addition, we have freely optimized the exponents in basis sets containing up to 12 Gaussians. We have studied the convergence of the total energy, the kinetic energy, the extent of the atom as measured by r 2 , and the Fermicontact interaction at the nucleus in these basis sets as well as in basis sets augmented with additional diffuse or steep functions.
The relationship between alternative separable solutions of the Coulomb problem both in configuration and in momentum space is exploited in order to obtain Sturmian orbitals of use as expansion basis sets in atomic and molecular problems. The usual spherical basis is obtained by separation in polar coordinates. The mathematical properties are explored for a type of basis set for quantum mechanical problems with axial symmetry, examples being diatomic molecules, or atoms under the influence of a uniform electric field. Because of its appropriateness for treatment of the Stark effect in atomic physics, this alternative basis set is called Stark basis. The Stark basis corresponds to separation in parabolic coordinates in configuration space and in cylindrical coordinates in momentum space. Fock's projection onto the surface of a sphere in the four dimensional hyperspace allows us to establish the connections of the momentum space wave functions with hyperspherical harmonics. As an ...