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Register a new userExpansion formulae for one- and two-center charge densities over complete orthonormal sets of exponential type orbitals and their use in evaluation of multicenter-multielectron integrals
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I.I.Guseinov
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The series expansion formulae are established for the one- and two-center charge densities over complete orthonormal sets of exponential type orbitals introduced by the author. Three-center overlap integrals of appearing in these relations are expressed through the two-center overlap integrals between -orbitals. The general formulae obtained for the charge densities are utilized for the evaluation of arbitrary multicenter-multielectron integrals occurring when the complete orthonormal sets of exponential type orbitals are used as basis functions in the Hartree-Fock-Roothaan and explicitly correlated methods. The relationships for charge densities and multicenter-multielectron integrals obtained are valid for the arbitrary quantum numbers, screening constants and location of orbitals.
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In standard convention, the new complete orthonrmal sets of exponential type orbitals (ETOs) are introduced as functions of the complex or real spherical harmonics and modified and -generalized Laguerre polynomials (MPLs and GLPs), where, and is the noninteger or integer (for) frictional quantum number. It is shown that the origin of the ETOs, MLPs and GLPs is the self-frictional quantum forces which are analog of radiation damping or self-frictional forces introduced by Lorentz in classical electrodynamics. The relations for the quantum self-frictional potentials in terms of ETOs, MLPs and GLPs, respectively, are established. We note that, in the case of disappearing frictional forces, the ETOs are reduces to the oringers wave functions for the hydrogen-like atoms in standard convention and, therefore, become the noncomplete.
Three-center nuclear attraction integrals with Slater type orbitals (STOs) appearing in the Hartree-Fock-Roothaan (HFR) equations for molecules are evaluated using one-range addition theorems of STOs obtained from the use of complete orthonormal sets of -exponential type orbitals (-ETOs), where . These integrals are investigated for the determination of the best with respect to the convergence and accuracy of series expansion relations. It is shown that the best values are obtained for . The convergence of three-center nuclear attraction integrals with respect to the indices for is presented. The final results are expressed through the overlap integrals of STOs containing . The hermitian properties of three-center nuclear attraction integrals are also investigated. The algorithm described in this work is valid for the arbitrary values of, and quantum numbers, screening constants and location of orbitals. The convergence and accuracy of series are tested by calculating concrete cases. It should be noted that the theory of three-center nuclear attraction integrals presented in this work is the extension of method described in our previous paper for to the case of (I.I. Guseinov, N. Seckin Gorgun and N. Zaim, Chin. Phys. B 19 (2010) 043101-1-043101-5).
By the use of complete orthonormal sets of nonrelativistic scalar orbitals introduced by the author in previous papers the new complete orthonormal basis sets for two- and four-component spinor wave functions, and Slater spinor orbitals useful in the quantum-mechanical description of the spin- 1/2 particles by the quasirelativistic and Diracs relativistic equations are established in position, momentum and four-dimensional spaces. These function sets are expressed through the corresponding nonrelativistic orbitals. The analytical formulas for overlap integrals over four-component relativistic Slater spinor orbitals with the same screening constants in position space are also derived. The relations obtained in this study can be useful in the study of different problems arising in the quasirelativistic and relativistic quantum mechanics when the position, momentum and four dimensional spaces are employed.
The new combined formulas have been established for the complex and real rotation-angular functions arising in the evaluation of two-center overlap integrals over arbitrary atomic orbitals in molecular coordinate system. These formulas can be useful in the study of different quantum mechanical problems in both the theory and practice of calculations dealing with atoms, molecules, nuclei and solids when the integer and noninteger n complex and real atomic orbitals basis sets are emploed. This work presented the development of our previous paper (I.I. Guseinov, Phys. Rev. A, 32 (1985) 1864).
By the use of complete orthonormal sets of nonrelativistic scalar orbitals introduced by the author in previous papers the new complete orthonormal basis sets for two-and four-component spinor wave functions, and Slater spinor orbitals useful in the quantum-mechanical description of the spin - 1/2 particles by the quasirelativistic and relativistic equations are established in position, momentum and four-dimensional spaces. These function sets are expressed through the corresponding nonrelativistic orbitals. The analytical formulas for overlap integrals over four component relativistic Slater spinor orbitals with the same screening constants in position space are also derived. The relations obtained in this study can be useful in the study of different problems arising in the quasirelativistic and relativistic quantum mechanics when the position, momentum and four dimensional spaces are employed.
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