Knowledge Bank

The vibrational F matrix can be written in the parameterized form $F=Lo-2C+1AC-1Lu-1$ where $Lo$ is determined from the G matrix alone; A is from the vibration frequencies; and C is an orthogonal matrix of N(N-1)/2 parameters, $\varphi ij$. The restoring forces for internal coordinates or symmetry coordinates are calculated as a function of the $\varphi ij$ parameters and the variation principle is used as a tool to suggest that these parametric restoring forces must take their maximum possible values. Conditions on the restoring forces and energy are summarized in the equation $Fsteep=\Sigma iNTiDii-1/2$. The critical points of $Fstp$ or certain “intersection” points (e.g. $T1F22-1/2=T2F11-1/2$ for N = 2) determine the $\varphi ij$ parameters and may specify several possible F matrices; the unique one has the largest restoring forces associated with it. The vibrational assignment must be independently known. In the $Fstp$ equation the $Ti$ are geometrical factors derived using the virial theorem (e.g. $T1=$ R and $T2=2$ $\cos \theta$ for $C2\nu$ triatomic molecules) and the $Dii$ are cofactors of the diagonal force constants. The method is illustrated with two, three and four dimensional examples and $OF2$, $NF3$, NOF, $H2S2$ and other molecules will be discussed. The force constants determined from the vibration frequencies of a simple molecule compare very well with those determined by conventional methods using isotopic substitution, Coriolis coupling constants and centrifugal distortion constants.