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We use a fully self-consistent Hartree$-$Fock (HF) based continuum randomphase approximation (CRPA) to calculate strength functions $S(E)$ andtransition densities $\rho_t(r)$ for isoscalar giant resonances withmultipolarities $L = 0$, 1 and 2 in $^{80}Zr$ nucleus. In particular, weconsider the effects of spurious state mixing (SSM) in the isoscalar giantdipole resonance (ISGDR) and extend the projection method to determine themixing amplitude of spurious state so that properly normalized $S(E)$ and$\rho_t(r)$ having no contribution due to SSM can be obtained. For thecalculation to be highly accurate we use a very fine radial mesh (0.04 fm) andzero smearing width in HF$-$CRPA calculations. We first use our most accurateresults as a basis to establish the credibility of the projection method,employed to eliminate the SSM, and then to investigate the consequences of thecommon violation of self-consistency, in actual implementation of HF based CRPAand discretized RPA (DRPA), as often encountered in the published literature.The HF$-$DRPA calculations are carried out using a typical box size of 12 fmand a very large box of 72 fm, for different values of particle-hole energycutoff ranging from 50 to 600 MeV.Comment: Revtex file(39 pages) and figures(20 PS files

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