  In this paper, an important discovery has been found for nonconforming
immersed finite element (IFE) methods using the integral values on edges as
degrees of freedom for solving elliptic interface problems. We show that those
IFE methods without penalties are not guaranteed to converge optimally if the
tangential derivative of the exact solution and the jump of the coefficient are
not zero on the interface. A nontrivial counter example is also provided to
support our theoretical analysis. To recover the optimal convergence rates, we
develop a new nonconforming IFE method with additional terms locally on
interface edges. The new method is parameter-free which removes the limitation
of the conventional partially penalized IFE method. We show the IFE basis
functions are unisolvent on arbitrary triangles which is not considered in the
literature. Furthermore, different from multipoint Taylor expansions, we derive
the optimal approximation capabilities of both the Crouzeix-Raviart and the
rotated-$Q_1$ IFE spaces via a unified approach which can handle the case of
variable coefficients easily. Finally, optimal error estimates in both $H^1$-
and $L^2$- norms are proved and confirmed with numerical experiments.
