New equations for derivatives of vibrational frequencies

A new equation is presented that expresses first partial derivatives of eigenvalues of the secular equation in terms of first derivatives of coefficients of the secular equation and the eigenvalues themselves. The utility of this equation is demonstrated using the example of isotopic shifts in the vibrational frequencies of molecules with second-order submatrices. Simple expressions for second, third, and forth derivatives of the eigenvalues are derived for this case.