The General Solution of the Secular Equation of Second Degree, with Application to the Class A1 Vibrations of the Symmetrical Triatomic Molecule

Abstract

Various solutions of the secular equation of second degree, giving the force constants, normal coordinates, and potential energy distributions in terms of the fundamental vibration frequencies and molecular constants, have been found and their significance discussed. General formulas are obtained for certain critical solutions and illustrative numerical results given for a series of eight symmetrical triatomic molecules, with curves covering all possible solutions in three typical cases. The solution in which the ratios of the contributions to the potential energy from the two square terms in the valence-force potential energy expression are mutually reciprocal in the two normal modes is shown to be universally applicable; that in which the normal coordinate for the vibration of higher frequency is identical with the corresponding valence-force symmetry coordinate is also quite satisfactory. Both always give real results. Alternative solutions are suggested for use when the difference in frequency of the two modes is small.