Knowledge Bank

The Direct Numerical Diagonalization (DND) technique has been utilized for the study of the potential and dipole moment functions of $\mathrm{<mrow\; data-mjx-texclass="ORD">12</mrow>}C\mathrm{<mrow\; data-mjx-texclass="ORD">16</mrow>}O2$. The present implementation is a three dimensional formulation, including three dimensionless normal coordinates, harmonic oscillator basis states, and a Given's diagonalization scheme. The $\mathrm{<mrow\; data-mjx-texclass="ORD">12</mrow>}C\mathrm{<mrow\; data-mjx-texclass="ORD">16</mrow>}O2$ potential has been determined by a nonlinear least squares fitting procedure for matching the DND eigenvalues to experimentally derived vibrational energy levels. Inaccuracies due to lack of basis set completeness (i.e., finiteness of the Hamiltonian matrix) and due to truncation of the potential function expansion are discussed. A comparison of the final eigenstate energies with those determined by the twice-contact transformed Hamiltonian technique is also shown. The eigenvectors resulting from the diagonalization are then used to similarity transform the matrix formed from a Taylor series expansion of the dipole moment operator. The solution of the Hamiltonian (i.e., the isolated, mechanical problem) is thus utilized to transform dipole moment derivatives into transition moments (i.e., the nonisolated, electrical problem). Inversion of this transform makes possible the calculation of dipole moment coefficients. Methods are discussed for dealing with the sign ambiguities arising from determining the transition moments as square roots of band strengths. ``Hot'' and difference band strengths can then be calculated from overtone and combination band strength data. The eigenvectors and calculated band strengths (especially the combination vs difference band strengths) are compared with $Suzuki\prime s[1]$ DND and $Borde´s[2]$ contact transformation work. Also covered is the extension of this technique to future calculations of individual line strengths for general polyatomic molecules.