  We are interested in computing a simple eigenpair $(\lambda,{\bf x})$ of a
large non-Hermitian matrix $A$, by a general harmonic Rayleigh-Ritz projection
method. Given a search subspace $\mathcal{K}$ and a target point $\tau$, we
focus on the convergence of the harmonic Ritz vector $\widetilde{\bf x}$ and
harmonic Ritz value $\widetilde{\lambda}$. In [{Z. Jia}, {\em The convergence
of harmonic Ritz values, harmonic Ritz vectors, and refined harmonic Ritz
vectors}, Math. Comput., 74 (2004), pp. 1441--1456.], Jia showed that for the
convergence of harmonic Ritz vector and harmonic Ritz value, it is essential to
assume certain Rayleigh quotient matrix being {\it uniformly nonsingular} as
$\angle({\bf x},\mathcal{K})\rightarrow 0$. However, this assumption can not be
guaranteed theoretically for a general matrix $A$, and the Rayleigh quotient
matrix can be singular or near singular even if $\tau$ is not close to
$\lambda$. In this paper, we abolish this constraint and derive new bounds for
the convergence of harmonic Rayleigh-Ritz projection methods. We show that as
the distance between ${\bf x}$ and $\mathcal{K}$ tends to zero and $\tau$ is
satisfied with the so-called {\it uniform separation condition}, the harmonic
Ritz value converges, and the harmonic Ritz vector converges as
$\frac{1}{\lambda-\tau}$ is well separated from other Ritz values of $(A-\tau
I)^{-1}$ in the orthogonal complement of $(A-\tau I)\widetilde{\bf x}$ with
respect to $(A-\tau I)\mathcal{K}$.
