Coefficients Of Fractional Parentage Research Articles

A FORTRAN program for the CFPs of a boson system with F spin

A simple FORTRAN program called CFPBF calculates the fractional parentage coefficients (CFPs) of identical boson system with F spin. The program CFPBF is based on an algorithm which uses the generalized Wigner-Eckart theorem of semisimple compact Lie groups. It is not only convenient for the discussion of the role of F spin but also more efficient for the CFPs calculation in the proton-neutron interacting boson model.

Coefficients of fractional parentage of a boson system with F-spin

Wavefunctions of bosons each with angular momentum I and F-spin 1/2 are classified according to the group chain U(2N) contains/implies (U(N) contains/implies O(N) contains/implies O(3))*SU(2). The boson number, seniority, reduced F-spin, total angular momentum, and total F-spin quantum numbers are used to label the wavefunctions. The corresponding coefficients of fractional percentage (CFP) are factorized into F-spin and angular momentum parts. The F-spin parts are calculated by an analytical formula and the angular momentum parts are calculated by recursion technique. The isoscalar factors of U(2N) contains/implies U(N) contains/implies O(N) and the F-spin part of the CFP for the cases F= Fmax and F=Fmax-1 are given explicitly.

A new version of CFPSIB: fractional parentage of an identical boson system

A new version of the program CFPSIB is introduced. Using the same recurrent formula as that used in the program CFPSIB, but a different calculation procedure, the fractional parentage coefficients (CFPs) of an identical boson system with well-defined seniority can be evaluated by this new program. It is shown that this new version (named RCFPSIB) is more efficient than the program CFPSIB, especially for a system including a relatively large number of bosons, each with higher angular momentum l.

A new Fortran program for CFPs of an identical fermion system

A new Fortran program called CFPSIF is introduced. Using the new recurrent formula obtained by us the program CFPSIF calculates the fractional parentage coefficients (CFPs) of an identical fermion system with well-defined seniority. The recurrent process is controlled by the multiplicity of the irreducible representations (IRREPs) of group O(3) in the reduction SP ( N) ⊃ O(3). It is shown that the program CFPSIF is more efficient than the program GENESIS, especially for a system with a relatively large number of fermions.

Second quantization and coefficients of fractional parentage

AbstractThe coefficients of fractional parentage (CFP), treated by tradition as antisymmetrizing multipliers, do not exist as such in the second‐quantization representation, because, therein, by definition, there is nothing to antisymmetrize. This by no way means that the above‐mentioned representation is less relevant in theoretical atomic spectroscopy than is any other one. On the contrary, all the well‐known mathematical apparatus is still successfully applied, alongside with numerous new useful features. Moreover, the analysis presented here yields some results not so obvious from the point of view of the coordinate representation, namely, the consequent application of the second‐quantization representation led us to an expansion of many‐shell wave functions employing analogs of many‐shell CFP, which appeared to be considerably simpler than the traditional CFP expansion. Also, a generalization of Redmond's formula for the case of many‐shell CFP and a straightforward method of deriving summation rules for products of CFP are clear in the second‐quantization representation and are sketched here. © 1994 John Wiley & Sons, Inc.

Hyperspherical functions with arbitrary permutational symmetry.

An algorithm is formulated for the construction of many-particle permutational symmetry adapted functions in hyperspherical coordinates. A recursive procedure is proposed, introducing hyperspherical coefficients of fractional parentage (hscfps). These coefficients are the eigenvectors of the transposition class sum of the symmetric group in an appropriate basis. Only the matrix element of the transposition of the last two particles has to be calculated in each step. This matrix element is obtained by using the hscfps calculated in the preceding step as well as the Raynal-Revai and the T coefficients. The results are applicable to the study of the atomic, molecular, and nuclear few-body problem.

Harmonic Oscillator SU3 States with Arbitrary Permutational Symmetry

Many-particle harmonic oscillator states that belong to a given SU 3 irreducible representation and are at the same time characterized by a Yamanouchi symbol specifying their permutational symmetry, are constructed recursively, using the SU3 coefficients of fractional parentage. These coefficients are the eigenvectors of the two-cycle class operators of the symmetric group in the appropriate basis. In the recursion procedure only the matrix element of the transposition of the last two particles has to be calculated in each step. This matrix element is obtained by using the SU 3 Racah coefficients. The significance of this procedure for nuclear physics as well as for quark calculations is briefly pointed out.

Fractional Parentage Coefficients with Explicit Seniority and Isospin11The project supported by Doctoral Program Foundation of Institution of Higher Education of China and National Natural Science Foundation of China.

A recurrent formula of coupling coefficients for identical fermions with single angular momentum and isospin is given. Using the edicit coupling coefficients obtained previously and the recurrent formula here, the algorithm of CFPs can be improved.

Symmetries of Fractional Parentage Coefficients with Isospin11The project supported by Doctoral Programme Foundation of Institution of Higher Education of China and National Natural Science Foundation of China.

A system of identical fermions with single angular momentum j and isospin is under consideration. The classification of its wave functions, the factorization and the symmetry of fractional parentage coefficients are discussed by using the generalized Wigner-Eckart theorem of semisimple compact Lie groups.

A program to calculate fractional parentage coefficients for jj-coupling states with equivalent particles

The n−1→ n fractional parentage coefficients (FPC's) for the jj-coupling states with equivalent particles in a j shell are calculated by using the branching rules for the unitary group U(2 j+1) and the recurrence formula for the FPC's.

A new FORTRAN program for the CFP's of a system with identical bosons

A simple FORTRAN program called CFPSIB, which determines coefficients of fractional parentage (CFP) for a system with identical bosons, is introduced. By using the new recurrent relation of the CFP directly labelled by seniority and the analytical formulae of the multiplicity of an irreducible representation (irrep) of group O (3) in an irrep of group O ( N), the program is faster and more efficient for a system with a relatively large number of bosons.

Quark-like fractional parentage for the atomic f shell

The states of the atomic f shell are constructed by coupling together four quark-like objects belonging to the eight-dimensional spinor irreducible representation (irrep) (1/2 1/2 1/2) of SO(7). Two parity labels are also required. Transformations among the eight components of each quark lead to the groups U(8) and SO(8), which can be used to augment the labels provided by the groups SO(7) and G2 that Racah (1943,1949) used in his classic analysis. Coefficients of fractional parentage (CFP) have been calculated for the quark configurations q2, q3 and q4. To allow for the automorphisms of SO(8), the groups SO(7)' and SO(7)" of Labarthe (1980) are used as alternative intermediaries to SO(7) between SO(8) and G2. Parallel tables of CFP are provided. As an example of the usefulness of the quark model, a correspondence is established between a three-electron operator t4 used in configuration-interaction studies and a two-quark operator. Some matrix elements of the latter in q4 are calculated by means of the quark CFP, thereby giving the corresponding matrix elements of t4 in the electronic configurations f6 and f7. By using the irreps of SO(7)', an unexpected proportionality between the sets of matrix elements can be accounted for.

Symmetry adapted formulation of the generalized Brillouin theorem

AbstractThe singly excited functions satisfying Brillouin theorem are expressed as linear combinations of configuration‐state functions for any spin and spatial symmetries (atomic or molecular) and for any reference wave function. The generality of the formulation is ensured by the use of the irreducible tensor method that can be adapted to any symmetry point group of interest. The expansion coefficients are simply written as products of fractional parentage coefficients, spin‐ and orbit‐recoupling coefficients, and phase factors. The formalism is illustrated for some atomic (Kh) and molecular (C∞v, C3v, and Td) configurations. Group theoretical techniques are also used to correlate the Brillouin conditions within a chain of groups.

Coefficients of fractional parentage for a double-l boson system

The generalised coefficients of fractional parentage (CFP) for a double-l boson system are defined. The formulae of CFP with seniority are presented when the states of this system are classified according to two group chains. The calculations are done in the scheme of second quantisation. The results show that the computation of CFP in this way is very efficient.

Nonspurious harmonic oscillator states in single particle coordinates

A procedure for the construction of nonspurious harmonic oscillator wave functions with arbitrary permutational symmetry has recently been proposed. The resulting wave functions are expressed in terms of normalized Jacobi coordinates, and involve a new type of harmonic oscillator coefficients of fractional parentage. A simple algorithm to transform these states from the Jacobi coordinates to the single particle coordinates is presented. This is a generalization to an arbitrary number of particles of the harmonic oscillator transformation from center-of-mass and relative coordinates to single particle coordinates.

A new recursion relation of CFP for the system of identical bosons

A new recursion relation of coefficients of fractional parentage (CFP) with seniority provides an efficient algorithm for computation. As a result of the generalised Wigner-Eckart theorem of semisimple Lie groups, the CFP are factorised into isoscalar factors (ISF) of the symmetry group of n bosons. The evaluations are done in the scheme of second quantisation. An analytic formula of the multiplicity of an irreducible representation (IR) of O(3) in an IR of O(N) is presented. A complete label (including seniority) of the states for the system of bosons, each with angular momentum l, is also presented.

Non-spurious harmonic oscillator states with arbitrary symmetry

An algorithm for the construction of non-spurious harmonic oscillator wave functions with arbitrary permutational symmetry is formulated by adapting a procedure recently developed for building general single shell wave functions. The harmonic oscillator functions, expressed in Jacobi coordinates, are calculated recursively using a new type of harmonic oscillator coefficients of fractional parentage. These coefficients are the eigenvectors of the two-cycle class operator of the permutation group in the appropriate basis. The matrix elements of the class operators are evaluated by using a specific version of the harmonic oscillator brackets. The significance of this procedure to atomic, molecular and nuclear physics is pointed out. The presently proposed method is expected to be computationally very efficient.

Shell model calculations: An efficient new algorithm

AbstractWe describe an efficient new algorithm which extends the range of feasible shell model calculations. This algorithm is applicable to single shell and multiple shell configurations, where two or more quantum numbers (e.g., L and S) are required to label the states within each shell. The algorithm proceeds by factoring the shell model Hilbert space into a product of subspaces, one for each angular momentum. N‐particle wave functions are built up recursively from N – 1 particle wave functions. Three kinds of N – 1‐ to N‐particle coefficients are required to carry out the construction of N‐particle electron (or fermion) states from N – 1 particle states. These are (1) coefficients of fractional parentage (CFPs) within a single shell, (2) outerproduct isoscalar factors (OISFs) within a single angular momentum subspace, and (3) innerproduct isoscalar factors (IISFs) which describe how multishell states within the complementary angular momentum subspaces are combined to form totally antisymmetric wave functions. All three types of N – 1‐ to N‐particle coefficients are generated recursively using a single powerful and efficient matrix diagonalization algorithm. Matrix elements of single particle creation and annihilation operators are expressed in terms of single particle CFPs, OISFs, and IISFs. We also describe an efficient algorithm for computing matrix elements of products of creation and anihilation operators by inserting and summing over complete sets of intermediate states. This is the Feynman‐like sum over path overlaps procedure. Timing benchmarks are presented comparing the new Drexel University shell model (DUSM) code with a state of the art shell model code.

The Fractional Parentage Coefficients for a Double-j Fermion System

The coefficients of fractional parentage (CFPs) for a double-j fermion system are defined. The CFPs are given when the wave functions are classified according to the group chains U(N) ⊃ U(N1) × U(N2) ⊃ SP(N1) × SP(N2) ⊃ O(31) × O(32) ⊃ O(3) and U(N) ⊃ SP(N) ⊃ SP(N1) × SP(N2) ⊃ O(31) × O(32) ⊃ O(3). Using these CFPs many repeating calculations can be avoided.

A Recurrent Formula for Fractional Parentage Coefficients of Identical Fermions

A recurrent formula of fractional parentage coefficients (CFPs) with explicit seniorities for identical fermions in the coupling scheme is given. The multiplicity of states with fixed seniority and angular momentum is also given. Using irreducible tensor method, the CFPs are factorized completely along a group chain. It is found that the CFPs are directly related to the group reduction coefficients. Them are useful for constructing an efficient iteration method of calculating the CFPs.