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International Journal of Quantum Chemistry

Second quantization and coefficients of fractional parentage

Andrius Bernotas, J M Kaniauskas

Published May 5, 1994• 2 citations

Abstract

Abstract The 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.

DOI: 10.1002/qua.560500406

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Abstract The 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.

Abstract A new procedure for large-scale calculations of the coefficients of fractional parentage (CFP) for many-particle systems is presented. The approach is based on a simple enumeration scheme for antisymmetric N particle states, and we suggest an efficient method for constructing the eigenvectors of two-particle transposition operator

P_{N_1 ,N}

in a subspace where N 1 and N 2 = N − N 1 nucleons basis states are already antisymmetrized. The main result of this paper is that according to permutation operators

P_{N_1 ,N}

eigenvalues we can distinguish totally asymmetrical N particle states from the other states with lower degree of asymmetry.

Abstract In the independent particle model, the total wave function is expressed as a linear combination of products of single particle wave functions. Fractional parentage coefficients give the part of the expression depending on the symmetry properties of the wave function. An operator is introduced which picks out the antisymmetric part of any n-particle wave function. The operator is given explicit form, and an explicit formula for the fractional parentage coefficients is deduced. The total fractional parentage coefficients can be split into a part depending on the orbital vectors, and a part depending on the spin (and isotopic spin) vectors. The separate parts can have symmetries other than total antisymmetry. Operators are introduced which pick out states of definite symmetry from any state. The operators are given explicit form, and explicit formulae for orbital, or spin, fractional coefficients are deduced. Detailed results are given for the three, four and five particles cases.

Reordering of operators in the second quantization formalism is described by using state operators as basic operators, where the state operator is either a cluster of creation operators or that of annihilation operators specified by a set of quantum numbers of any representation. The commutation relation between state operators is simply expressed in terms of the coefficient of fractional parentage (cfp), the factor specific to state operator (de) composition. Wick’s theorem, prevalently applied to a string of creation and annihilation operators, is extended to a string of state operators. Similar extension is done for the contraction theorem.

A new procedure for large-scale calculations of the coecients of fractional parentage (CFP) for manyparticle systems is presented. The approach is based on a simple enumeration scheme for antisymmetric N particle states, and we suggest an ecient method for constructing the eigenvectors of two-particle transposition operator PN1;N in a subspace where N1 and N2 = N N1 nucleons basis states are already antisymmetrized. The main result of this paper is that according to permutation operators PN1;N eigenvalues we can distinguish totally asymmetrical N particle states from the other states with lower degree of asymmetry.

The construction of symmetric states of N particles with angular momentum j=1 is presented. Elliott's result (1958), dealing with the angular momentum content of the symmetric subset of states belonging to the set of states of N particles with angular momentum j=1 is obtained using Racah algebra for the angular momentum coupling. The explicit functional dependence, on the total angular momentum, of the coefficients of fractional parentage (CFP) associated with one, two, three and four removed particles is found for arbitrary values of N. This dependence is expressed through the simple algebraic functions of J and N.