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Register a new userCombined theory of complete orthonormal sets of quasirelativistic and relativistic sets of wave functions, and Slater orbitals of spin -1/2 particles in position, momentum and four dimensional spaces
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Authors
I.I.Guseinov
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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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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.
Using the complete orthonormal basis sets of nonrelativistic and quasirelativistic orbitals introduced by the author in previous papers for particles with arbitrary spin the new analytical relations for the -component relativistic tensor wave functions and tensor Slater orbitals in coordinate, momentum and four-dimensional spaces are derived, where. The relativistic tensor function sets are expressed through the corresponding nonrelativistic and quasirelativistic orbitals. The analytical formulas for overlap integrals over relativistic tensor Slater orbitals with the same screening constants in coordinate space are also derived.
Using the complete orthonormal sets of radial parts of nonrelativitistic exponential type orbitals (2,1, 0, 1, 2, ...) and spinor type tensor spherical harmonics of rank s the new formulae for the 2(2s+1)-component relativistic spinors useful in the quantum mechanical description of the arbitrary half-integral spin particles by the generalized Dirac equation introduced by the author are established in position, momentum and four-dimensional spaces, where 1/ 2, 3 / 2, 5 / 2, ... s = . These spinors are complete without the inclusion of the continuum. The 2(2s+1)component spinors obtained are reduced to the independent sets of two-component spinors defined as a product of complete orthonormal sets of radial parts of orbitals and twocomponent spinor type tensor spherical harmonics. We notice that the new idea presented in this work is the unified treatment of half-integral spin and scalar particles in position, momentum and four-dimensional spaces. Relations presented in this study can be useful in the linear combination of atomic orbitals approximation for the solution of different problems arising in the relativistic quantum mechanics when the orthonormal basis sets of relativistic exponential type spinor wave functions and Slater type spinor orbitals in position, momentum and four -dimensional spaces are employed.
The analytical relations in position, momentum and four-dimensional spaces are established for the expansion and one-range addition theorems of relativistic complete orthonormal sets of exponential type spinor wave functions and Slater spinor orbitals of arbitrary half-integral spin. These theorems are expressed through the corresponding nonrelativistic expansion and one-range addition theorems of the spin-0 particles introduced by the author. The expansion and one-range addition theorems derived are especially useful for the computation of multicenter integrals over exponential type spinor orbitals arising in the generalized relativistic Dirac-Hartree-Fock-Roothaan theory when the position, momentum and four-dimensional spaces are employed.
The 2(2s+1)-component relativistic basis spinors for the arbitrary spin particles are established in position, momentum and four-dimensional spaces, where s=0,1 / 2,1, 3 / 2, 2, ... . These spinors for integral- and half-integral spins are reduced to the independent sets of one- and twocomponent spinors, respectively. Relations presented in this study can be useful in the linear combination of atomic orbitals approximation for the solution of generalized Dirac equation of arbitrary spin particles introduced by the author when the orthogonal basis sets of relativistic exponential type spinor wave functions and Slater type spinor orbitals in position, momentum and four -dimensional spaces are employed.
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