One-range Addition Theorems Research Articles

Use of Coulomb-Yukawa Like Correlated Interaction Potentials of Integer and Noninteger Indices and One-range Addition Theorems for Ψα-ETO in Evaluation of Potential of Electric Field Produced by Molecule

Using Coulomb-Yukawa like correlated interaction potentials of integer and noninteger indices the series expansion formulae in terms of multicenter overlap integrals of three complete orthonormal sets of <TEX>${\psi}^{\alpha}$</TEX>‒exponential type orbitals and linear combination coefficients of molecular orbitals are established for the potential of electrostatic field produced by the charges of molecule, where <TEX>$\alpha$</TEX> = 1, 0, ‒1, ‒2,<TEX>${\cdots}$</TEX>. The formulae obtained can be useful for the study of interaction between atomic--molecular systems containing any number of closed and open shells when the <TEX>${\psi}^{\alpha}$</TEX>‒exponential type basis functions and Coulomb-Yukawa like correlated interaction potentials are used in the Hartree-Fock-Roothaan and explicitly correlated approximations. The final results are valid for the arbitrary values of parameters of correlated interaction potentials and orbitals. As an example of application, the calculations have been performed for the potential energy of interaction between electron and molecule <TEX>$H_2O$</TEX> using combined Hartree-Fock-Roothaan equations suggested by the author.

Evaluation of Multicenter Multielectron Integrals Using One-range Addition Theorems in Terms of STOs for STOs and Coulomb-Yukawa Like Correlated Interaction Potentials with Integer and Noninteger Indices

Using one-range addition theorems for Slater type orbitals (STOs) and Coulomb-Yukawa like correlated interaction potentials (CIPs) introduced by the author, the series expansion formulae are derived for the multicenter multielectron integrals. The expansion coefficients occurring in these relations are presented through the overlap integrals of two STOs. The convergence of series expansion relations is tested by calculating concrete cases. The accuracy of the results is quite high for quantum number, screening constants and location of orbitals. The final results are especially useful in the calculation of multielectron properties for atoms and molecules when Hartree-Fock-Roothaan (HFR) and explicitly correlated methods are employed.

Application of combined open shell Hartree–Fook–Roothaan theory to molecules using symmetrical one-range addition theorems of Slater type orbitals

We study the efficiency of symmetrical one-range addition theorems for Slater type orbitals which are used in the calculation of multicenter integrals arising in the combined open shell Hartree–Fook–Roothaan equations for molecules. As an example of application, the calculations are performed for the ground state of electronic configuration of molecules NH3 and CH2 that have the closed and open shells, respectively. The results of computer calculations for the orbital and total energies (in a.u.), and linear combination coefficients for symmetrical molecular orbitals are presented.

One-Range Addition Theorems in Terms of Ψα -ETOs for STOs and Coulomb–Yukawa Like Correlated Interaction Potentials of Integer and Noninteger Indices

The expansion formulas in terms of complete orthonormal sets of Ψα-exponential type orbitals are established for the Slater type orbitals and Coulomb–Yukawa-like correlated interaction potentials of integer and noninteger indices. These relations are used in obtaining their unsymmetrical and symmetrical one-range addition theorems. The final results are especially useful in the calculations of multicentre multielectron integrals occurring when Hartree–Fock–Roothaan and explicitly correlated methods are employed.

On the analyticity of Laguerre series

The transformation of a Laguerre series f(z) = ∑∞n=0λ(α)nL(α)n(z) to a power series f(z) = ∑∞n=0γnzn is discussed. Since many nonanalytic functions can be expanded in terms of generalized Laguerre polynomials, success is not guaranteed and such a transformation can easily lead to a mathematically meaningless expansion containing power series coefficients that are infinite in magnitude. Simple sufficient conditions based on the decay rates and sign patterns of the Laguerre series coefficients λ(α)n as n → ∞ can be formulated which guarantee that the resulting power series represents an analytic function. The transformation produces a mathematically meaningful result if the coefficients λ(α)n either decay exponentially or factorially as n → ∞. The situation is much more complicated—but also much more interesting—if the λ(α)n decay only algebraically as n → ∞. If the λ(α)n ultimately have the same sign, the series expansions for the power series coefficients diverge, and the corresponding function is not analytic at the origin. If the λ(α)n ultimately have strictly alternating signs, the series expansions for the power series coefficients still diverge, but are summable to something finite, and the resulting power series represents an analytic function. If algebraically decaying and ultimately alternating Laguerre series coefficients λ(α)n possess sufficiently simple explicit analytical expressions, the summation of the divergent series for the power series coefficients can often be accomplished with the help of analytic continuation formulae for hypergeometric series p+1Fp, but if the λ(α)n have a complicated structure or if only their numerical values are available, numerical summation techniques have to be employed. It is shown that certain nonlinear sequence transformations—in particular the so-called delta transformation (Weniger 1989 Comput. Phys. Rep. 10 189–371 (equation (8.4-4)))—are able to sum the divergent series occurring in this context effectively. As a physical application of the results of this paper, the legitimacy of the rearrangement of certain one-range addition theorems for Slater-type functions (Guseinov 1980 Phy. Rev. A 22 369–71, Guseinov 2001 Int. J. Quantum Chem. 81 126–29, Guseinov 2002 Int. J. Quantum Chem. 90 114–8) is investigated.

UNSYMMETRICAL AND SYMMETRICAL ONE-RANGE ADDITION THEOREMS FOR SLATER TYPE ORBITALS AND COULOMB–YUKAWA-LIKE CORRELATED INTERACTION POTENTIALS OF INTEGER AND NONINTEGER INDICES

Using one-center expansion relations for the Slater type orbitals (STOs) of noninteger principal quantum numbers in terms of integer nSTOs derived in this study with the help of ψa-exponential type orbitals (ψa-ETOs, a = 1, 0, -1, -2,…), the general formulas through the integer nSTOs are established for the unsymmetrical and symmetrical one-range addition theorems for STOs and Coulomb–Yukawa-like correlated interaction potentials (CIPs) with integer and noninteger indices. The final results are especially useful for the computations of arbitrary multicenter multielectron integrals that arise in the Hartree–Fock–Roothaan (HFR) approximation and also in the correlated methods based upon the use of STOs as basis functions.

Use of unsymmetrical one-range addition theorems of Slater type orbitals in molecular electronic structure determination

In this work, the applicability of the unsymmetrical one-range addition theorems obtained from the use of complete orthonormal sets of Ψα-exponential type orbitals (Ψα-ETOs, where α = 1, 0, − 1, − 2, ...) to the study of electronic structure of molecules is demonstrated using minimal basis sets of Slater type orbitals (STOs). As an example of application of unsymmetrical one-range addition expansion method to evaluate the multicenter electronic integrals, the calculation has been performed for the ground state of BH 3 molecule. The results of computer calculations for the orbital and total energies, and linear combination coefficients of symmetrized molecular orbitals are presented.

On the unified treatment of complete orthonormal sets of functions in coordinate, momentum, and four dimensional spaces and their expansion and one-range addition theorems

On the unified treatment of complete orthonormal sets of functions in coordinate, momentum, and four dimensional spaces and their expansion and one-range addition theorems

Unified treatment of complete orthonormal sets of functions in coordinate, momentum and four-dimensional spaces and their expansion and one-range addition theorems

The simpler formulas are derived for the complete orthonormal sets of exponential- type orbitals, momentum space orbitals and hyperspherical harmonics and their expansion and one-range addition theorems. The continuum states are not properly included in these functions. The analytical formulas are also obtained for the overlap integrals over Ψα-ETOs, their extensions to momentum and four-dimensional spaces and STOs with the same screening constants using addition and expansion theorems derived in this paper. The complete orthonormal sets of functions and their expansion and one-range addition theorems obtained can be useful in the study of different quantum mechanical problems when the coordinate, momentum or four-dimensional spaces employed.

New complete orthonormal sets of hyperspherical harmonics and their one-range addition and expansion theorems

In this study, the complete orthonormal sets of phi(alpha)-momentum space orbitals (where alpha = 1, 0, -1, -2, ...) obtained from the psi(alpha)-ETO in coordinate representation (I.I. Guseinov, J. Mol. Model., 9 (2003) 135) are reduced to the complete orthonormal sets of hyperspherical harmonics (HSH) by means of a Fock transformation of the radial momentum to an angular variable. It is shown that the group of transformations is the four-dimensional rotation group O(4) and, therefore, the HSH presented in this work are the complete orthonormal sets of functions. For these functions, the one-range addition and expansion theorems are obtained. The formulae for HSH and their addition and expansion theorems derived in this work can be used to evaluate the multicenter integrals that arise when exponential-type basis functions are used in atomic and molecular calculations.

One-range addition theorems for combined Coulomb and Yukawa like central and noncentral interaction potentials and their derivatives

In this work we present the combined one-range addition theorems for arbitrary Coulomb and Yukawa like central and noncentral interaction potentials and their first and second derivatives occurring in the Hartree–Fock–Roothaan approximation and explicitly correlated methods, and also in the study of electric field and its gradient induced by electrons at the nuclei of a molecule. The relationships obtained for addition theorems are valid for the arbitrary parameters of potentials.

One-range addition theorems for derivatives of integer and noninteger u Coulomb–Yukawa type central and noncentral potentials and their application to multicenter integrals of integer and noninteger n Slater orbitals

Using properties of regular solid spherical harmonics the one-range addition theorems for first and second derivatives of combined Coulomb ( η=0) and Yukawa ( η>0) type potentials f u v s ( η , r ⇀ ) = r u − v − 1 e − η r T v s ( x , y , z ) , where T v s ( x , y , z ) = [ 4 π / ( 2 v + 1 ) ] 1 / 2 r v S v s ( θ , φ ) , with integer and noninteger values of indices u are presented. With the help of these addition theorems and complete orthonormal sets of ψ α -ETOs, where α=1, 0, −1, −2,…, a large number of series expansion formulae for arbitrary one-electron multicenter integrals over integer and noninteger n STOs and their derivatives are established . The final results are expressed in terms of two-center overlap integrals between noninteger n STOs. The series expansion formulae obtained in this work is especially useful for the study of electronic structure and electron–nuclei interaction properties of atoms, molecules, and solids. It should be noted that the results for the noninteger indices of potentials and STOs presented in this paper were not appeared in our past publications.

One-Range Addition Theorems for Yukawa-Like Central and Noncentral Interaction Potentials and Their Derivatives

Abstract Using addition theorems for Slater-type orbitals, obtained by the author with the help of complete orthonormal sets of Ψα-Exponential type orbitals (α = 1, 0, −1, −2, ...), the one-range addition theorems for arbitrary Yukawa-like central and noncentral interaction potentials and their first and second derivatives are established. The addition theorems derived for the interaction potentials and the Slater orbitals are especially useful for machine computations of arbitrary multielectron multicenter integrals that arise in the Hartree–Fock–Roothaan approximation, and also in the Hylleraas-correlated wave-function method, which play a significant role in the theory and applications to quantum mechanics of atoms, molecules, and solids. The obtained relationships are valid for arbitrary nonzero screening constants of the potentials.

One-range addition theorems for derivatives of complete orthonormal sets of ψ α-ETOs

A large number of one-range addition theorems for first and second derivatives of complete orthonormal sets of ψ α -ETOs ( α=1,0,−1,−2,…) through the new basic auxiliary functions is presented. The basic auxiliary functions are constructed from the finite linear combinations of ψ α -ETOs. These theorems are especially useful in the study of electric field and its gradient induced at the nuclei of a molecule by its electrons. The relationships obtained are valid for the arbitrary values of parameters and location of orbitals.

One-range addition theorems for Coulomb interaction potential and its derivatives

Abstract In this study, using the addition theorems for STOs obtained by the author with the help of complete orthonormal sets of Ψ α -ETOs ( α = 1, 0, −1, −2, …) a large number of one-range addition theorems for Coulomb interaction potential (CIP) and its fist and second derivatives is presented. The final results are expressed in terms of a single noncentral potential function. The addition theorems derived are especially useful for the computation of multicenter integrals over STOs occurring in the Hartree–Fock–Roothaan equations, and also in the study of electric field and electric field gradient induced at the nuclei of a molecule by its electrons.

One-range addition theorems for derivatives of Slater-type orbitals.

Using addition theorems for STOs introduced by the author with the help of complete orthonormal sets of psi(alpha)-ETOs (Guseinov II (2003) J Mol Model 9:190-194), where alpha=1, 0, -1, -2, ..., a large number of one-range addition theorems for first and second derivatives of STOs are established. These addition theorems are especially useful for computation of multicenter-multielectron integrals over STOs that arise in the Hartree-Fock-Roothaan approximation and also in the Hylleraas function method, which play a significant role for the study of electronic structure and electron-nuclei interaction properties of atoms, molecules, and solids. The relationships obtained are valid for arbitrary quantum numbers, screening constants and location of STOs.