In Wikipedia there is a list of Slater-Koster matrix elements, but not all of them are listed because "matrix elements that are not listed in this table can be constructed by permutation of indices and cosine directions of other matrix elements in the table". I checked the original article and the authors provide the same matrix elements.
However, I do not understand how to apply these permutations. For instance, in the list above, we have the matrix elements $$E_{z,x^2-y^2} = \frac{\sqrt{3}}{2} n(l^2 - m^2) V_{pd\sigma} - n(l^2 - m^2) V_{pd\pi}$$
$$E_{z,3z^2-r^2} = n [n^2 - (l^2 + m^2) / 2] V_{pd\sigma} + \sqrt{3} n (l^2 + m^2) V_{pd\pi},$$ where $l,m,n$ are the cosine directors. Other matrix elements of a $p_z$ orbital with other $d$ orbitals are however not listed. How can we construct, for example, the matrix element $E_{z,zx}$ from the given matrix elements?
