Self-consistent Hartree-Fock based random phase approximation and the spurious state mixing

We use a fully self-consistent Hartree-Fock (HF) based continuum random phase approximation (CRPA) to calculate strength functions S(E) and transition densities ρ_t(r) for isoscalar giant resonances with multipolarities L=0, 1, and 2 in ⁸⁰Zr nucleus. In particular, we consider the effects of spurious state mixing (SSM) in the isoscalar giant dipole resonance and extend the projection method to determine the mixing amplitude of the spurious state so that properly normalized S(E) and ρ_t(r), having no contribution due to SSM, can be obtained. For the calculation to be highly accurate, we use a very fine radial mesh (0.04 fm) and zero smearing width in HF-CRPA calculations. We use our most accurate results as a basis to (i) establish the credibility of the projection method, employed to eliminate the SSM, and (ii) to assess the consequences of the common violations of self-consistency, often encountered in actual implementation of HF based CRPA and discretized RPA (DRPA) published in the literature, on the values of S(E) and ρ_t(r). This is achieved by varying the radial mesh size, the particle-hole interaction, the smearing parameter, and the particle-hole energy cutoff used in the HF-RPA calculations.