Knowledge Bank

When the secular equation in the form $|H-E\lambda |=0$, where H is the matrix product GF or FG, is expanded into a polynomial of degree n in $\lambda$, a set of n algebraic equations in the n unknowns $Fij$ is obtained. The degrees of these equations are $1,2,3\dots ,n$. respectively, and the number of solutions (sets of force constants) is $n!=1-2.3\dots n$. Each set of force constants corresponds to one of the possible sets of frequency assignments. When the F matrix is diagonal and the frequencies are all different, the force constants are all real, positive, and different. As an illustration, the six sets of force constants for the valence force treatment of a nonlinear, unsymmetrical, triatomic molecular model are computed, and the corresponding frequency assignments are determined.