New definitions of the effective nuclear charge and its application to estimate the matrix element ⟨n,l|rβ|n′,l′⟩

New definitions of the effective nuclear charge are proposed to evaluate the orbital wave function and the dipole matrix element $⟨n,l\left|r\right|n^{\prime },l^{\prime }⟩$ for the non-hydrogenic atoms or ions. It is found that the commonly used effective nuclear charge defined by the orbital energy is insufficiently precise to estimate the orbital wave functions and matrix elements $⟨n,l\left|r\right|n^{\prime },l^{\prime }⟩$. Instead, the effective nuclear charge defined by $⟨r⟩$ or $⟨r^2⟩$ are more advantageous. It is shown that the effective nuclear charge method becomes increasingly precise to predict wave functions and matrix elements as the orbital angular momentum, the principle quantum number and the degree of ionization increase. Good accuracy is achieved when principal quantum numbers are identical $n=n^{\prime }$ or when both orbital quantum numbers $l$ and $l^{\prime }$ are non-zero. When $s$-orbitals are involved ($l$ or $l^{\prime }$ equal to zero) the precision is decreasing.