Noninteger n-Slater Type Orbitals Research Articles

Unified treatment of one-range addition theorems for integer and non-integer n-STO, -GTO and -generalized exponential type orbitals with hyperbolic cosine in position, momentum and four-dimensional spaces

Simpler formulas are derived for one-range addition theorems for the integer and noninteger n generalized exponential type orbitals, momentum space orbitals, and hyperspherical harmonics with hyperbolic cosine (GETO HC, GMSO HC, and GHSH HC) in position, momentum and four-dimensional spaces, respectively. The final results are expressed in terms of one-range addition theorems of complete orthonormal sets of ψα-exponential type orbitals, φα-momentum space orbitals and zα-hyperspherical harmonics. We notice that the one-range addition theorems for integer and noninteger n-Slater type orbitals and Gaussian type orbitals in position, momentum and four dimensional spaces are special cases of GETO HC, GMSO HC, and GHSH HC. The theorems presented can be useful in the accurate study of the electronic structure of atomic and molecular systems.

Use of combined Hartree–Fock–Roothaan theory in evaluation of lowest states of K[Ar]4s 03d 1 and Cr + [Ar]4s 03d 5 isoelectronic series over noninteger n-Slater type orbitals

By the use of integer and noninteger n-Slater Type Orbitals in combined Hartree-Fock-Roothaan method, self consistent field calculations of orbital and lowest states energies have been performed for the isoelectronic series of open shell systems K [Ar]4s^0 3d^1 2(D) (Z=19-30) and Cr+ [Ar] 4s^0 3d^5 6(S) (Z=24-30). The results of calculations for the orbital and total energies obtained from the use of minimal basis sets of integer- and noninteger n-Slater Type Orbitals are given in tables. The results are compared with the extended-basis Hartree-Fock computations. The orbital and total energies are in good agreement with those presented in the literature. The results are accurately and considerably can be useful in the application of non-relativistic and relativistic combined Hartree-Fock-Roothaan approach for heavy atomic systems.

Use of auxiliary functions $${Q_{ns}^q}$$ and $${G_{-ns}^q}$$ in evaluation of multicenter integrals over integer and noninteger n-Slater type orbitals arising in Hartree–Fock–Roothaan equations for molecules

With the help of expansion relations for the two-center Slater type orbitals (STOs) charge densities established by the author from the use of complete orthonormal sets of Ψα-exponential type orbitals (Ψα-ETOs), where α = 1, 0, − 1, − 2, . . . , a large number of series expansion formulas for the multicenter integrals of integer and noninteger n-STOs (ISTOs and NISTOs) occurring in Hartree–Fock–Roothaan (HFR) equations for molecules is derived through the auxiliary functions \({Q_{ns}^q}\) and \({G_{-ns}^q}\) , and one- and two-center basic integrals of ISTOs. The analytical relations for basic integrals are presented. As an example of application, the calculations have been performed for the ground state of electronic configuration of \({{\it CH}_4((1a_1)^{2}(2a_1)^{2}(1t_{2x})^{2} (1t_{2y})^{2} (1t_{2z})^{2},{}^1A_1)}\) using combined HFR theory suggested by the author.

Accurate evaluation of overlap integrals of Slater type orbitals with noninteger principal quantum numbers using complete orthonormal sets of $$\Psi^{\alpha}$$ - exponential type orbitals

In this paper, by the use of complete orthonormal sets of $$\Psi^{\alpha}$$ exponential-type orbitals ( $$\Psi^{\alpha}$$ -ETOs, $$\alpha = 1, 0, -1, -2,\ldots$$ ), the series expansion formulas in line-up coordinate systems are established for the overlap integrals with noninteger n * Slater-type orbitals (NISTOs) in terms of overlap integrals over integer n Slater-type orbitals (ISTOs). The suggested approach guarantees a highly accurate calculation of the noninteger n * overlap integrals with arbitrary values of parameters. The results of computer calculations presented are in a complete agreement with those obtained in the literature using the alternative procedure.

On Evaluation of Overlap Integrals with Noninteger Principal Quantum Numbers

By use of complete orthonormal sets of Ψα exponential-type orbitals the series expansion formulas for the noninteger n* Slater-type orbitals (NISTOs) in terms of integer n Slater-type orbitals (ISTOs) are derived. These formulas enable us to express the overlap integrals with NISTOs through the overlap integrals over ISTOs with the same and different screening constants. By calculating concrete cases the convergence of the series for arbitrary values of noninteger principal quantum numbers and screening constants of NISTOs and internuclear distances is tested. The accuracy of the results is quite high for quantum numbers, screening constants and location of STOs.

Unified treatment of integer and noninteger n multicenter multielectron molecular integrals using complete orthonormal sets of ψ α-ETOs

Using complete orthonormal sets of ψ α exponential-type orbitals (ψ α -ETOs) where α=1,0,−1,… the general formulas are derived for the expansion of noninteger n ∗ Slater-type orbitals (NISTOs) in terms of integer n Slater-type orbitals (ISTO's) at a displaced center. For the expansion of one- and two-center electron charge density with NISTOs the analytical expressions in terms of ISTOs at a new origin are also obtained. The expansion coefficients in these formulas are presented through the two-center noninteger n ∗ overlap integrals. The general formulas obtained for the electron charge density are utilized for the evaluation of multicenter multielectron molecular integrals over NISTOs. The final results are expressed in terms of both the one-center multielectron basic integrals over ISTOs, and the expansion coefficients of the electron charge density with NISTOs. The relationships obtained are valid for the integer and noninteger principal quantum numbers, screening constants and arbitrary location of ISTOs and NISTOs.

Computation of molecular integrals over Slater-type orbitals. IX. Calculation of multicenter multielectron molecular integrals with integer and noninteger n Slater orbitals using complete orthonormal sets of exponential functions

By the use of complete orthonormal sets of Lambda and Coulomb Sturmian exponential-type functions the two kinds of formulas for the expansion of noninteger n ∗ Slater-type orbitals (NISTOs) in terms of integer n Slater-type orbitals (ISTOs) at a displaced center are derived. The analytical expressions in terms of ISTOs are also obtained for the expansion of one- and two-center electron charge density with NISTOs. The expansion coefficients in these formulas are presented through the two-center noninteger- n ∗ overlap integrals. The general formulas obtained for the electron charge density are utilized for the evaluation of multicenter multielectron molecular integrals over NISTOs appearing in the determination of various multielectron properties for molecules when the Hartree–Fock–Roothaan (HFR) approximation is employed. The final results are expressed in terms of both the expansion coefficients of the electron charge density with NISTOs, and the one-center multielectron basic integrals over ISTOs. The relationships obtained are valid for the arbitrary location, integer and noninteger quantum numbers and screening constants of STOs.

Computation of molecular integrals over Slater-type orbitals. X. Calculation of overlap integrals with integer and noninteger n Slater orbitals using complete orthonormal sets of exponential functions

Using two kinds of complete orthonormal sets of exponential-type orbitals, the series expansion formulas are derived for the overlap integrals with noninteger n ∗ Slater-type orbitals (NISTOs) in terms of overlap integrals over integer n Slater-type orbitals with the same and different screening constants. The convergence of the series is tested by calculating concrete cases for arbitrary values of noninteger principal quantum numbers and screening constants of NISTOs and internuclear distances.

Series Expansion for Two-Center Noninteger-n Overlap Integrals

A series in the internuclear distance R is derived for the two-center overlap integral between noninteger-n Slater-type orbitals. There are in general two types of terms: R2N+λ and Rn1+n2+1+N, (N=0,1,2,···; λ = |l1—l2|, |l1—l2| + 1, ···, l1+l2). When n1+n2 is an integer while n1 and n2 are not integers, logarithmic terms arise. The series is for general values of n1, n2, l1, l2, m1, m2, ζ1, and ζ2, and it converges absolutely for R<∞.