Application of Truncated Multipolar Expansion Pair Functions in Atomic Structure Calculations. II. Irreducible Tensor Operator Form of the Strongly Orthogonal Pair Functions. Reduction of Orthogonality Constraints

1971 ◽  
Vol 49 (1) ◽  
pp. 118-132 ◽  
Author(s):  
K. Jankowski
Keyword(s):  
Closed Shell ◽  
Tensor Operator ◽  
Irreducible Tensor ◽  
Method Of Calculation ◽  
Irreducible Tensor Operator ◽  
Orthogonality Constraints ◽  
Operator Form ◽  
Orthogonal Pair ◽  
Multipolar Expansion ◽  
Pair Function

The truncated multipolar expansion pair functions (TMEPF) defined recently by Jankowski have been used to formulate an approach which takes into account correlation effects within a pair of electrons in the presence of an N electron sea in a closed shell configuration. An irreducible tensor operator form of the strongly orthogonal pair function components has been derived. The analysis of the expressions obtained leads to a systematic reduction of the orthogonality constraints to be imposed on this class of trial functions. The results of this analysis for several types of pair functions have been given. Finally, the method of calculation of matrix elements between orthogonally projected pair function components in terms of radial integrals is presented.

Theory of spin-orbit coupling in atoms I. Derivation of the spin-orbit coupling constant

1962 ◽  
Vol 270 (1340) ◽  
pp. 127-143 ◽  
Keyword(s):  
Exact Expression ◽  
Orbit Coupling ◽  
Tensor Operator ◽  
Spin Orbit Coupling ◽  
Spin Orbit ◽  
Matrix Elements ◽  
Coupling Constant ◽  
Hartree Fock ◽  
Operator Form ◽  
Hartree Potential

An exact expression for the spin-orbit coupling constant is derived within the Hartree-Fock description of the atom by considering the two body mutual spin-orbit interaction between electrons. The interaction is rewritten in tensor operator form and the contribution of outer electron-core interactions to the coupling constant is calculated. We find that the usual expression < 3F/r8r > where V is the Hartree potential is only approximate, and that certain exchange type terms, which arise because we are dealing with a two-body interaction and determinantal wave function, must also be included. These exchange terms are not simply related to the ordinary electrostatic exchange. The resulting expression for the spin-orbit coupling constant is given in terms of radial integrals which can be calculated using Hartree or Hartree—Fock wave functions. We also discuss the effective magnetic Hamiltonian to be used for the calculation of matrix elements within an atomic configuration.

387: The Optimal Timing, Cutoff and Method of Calculation of Preoperative PSA Velocity to Predict Relapse after Prostatectomy: A Report from Search

2007 ◽  
Vol 177 (4S) ◽  
pp. 128-129
Author(s):  
Christopher R. King ◽  
Stephen J. Freedland ◽  
Martha K. Terris ◽  
William J. Aronson ◽  
Christopher J. Kane ◽  
...  
Keyword(s):  
Optimal Timing ◽  
Method Of Calculation ◽  
Psa Velocity
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