Chapter IV. New Computational Methods of Analytical Derivation of Exchange Kernels

Abstract

The exchange kernels due to the antisymmetrization are derived in an analytical way with the symbolical use of computer memory by taking the α + 16O system as a typical example. Starting with the harmonic oscillator shell model wave functions around the centres as internal states of nuclei, we present new algorisms of deriving analytically the generator coordinate kernel (GCM kernel). We have developed two different algorisms for this purpose; one is named the UFPN and the other the “Rendez-Vous”. Contents of such algorisms are explained in detail. Transformation from the GCM kernel to the resonating group one (RGM one) is also analytically carried out by the aid of computer memory through differential formulas of the GCM kernel. Here, we make use of explicit form Bell's polynomials for performing complex differentiation. A new GCM kernel free from the spurious c.m. component is presented for cases with different oscillator parameters. An algorism of procedure to eliminate the c.m. component is the same as that of deriving the RGM kernel from the GCM one. Considerations are given to a simple method of calculating the direct potential, decomposition method of the kernel and a method of expanding the GCM kernel. The methods mentioned in this report will extend applicability of microscopic study of nucleus-nucleus interactions.