Knowledge Bank

Vibrational analyses are performed with ab initio potential-energy and electric dipole-moment surfaces to obtain off-diagonal matrix elements. For molecules possessing N modes of non-degenerate vibrations and two states characterized by the quantum numbers v = $(v1,v2,\dots ,vN)$ and $w=(w1,w2,\dots ,wN)$, it is shown that $$ \langle v|\mu|w\rangle=(\sum^{N}{{i}=1}\Gamma{v_{i}w_{i}})(G_\sim+\sum^{N}{i=1}{A}{i}V_{i}+\sum^{N}{{i}\leq 1}B{ij}V_{i}V_{j}+\ldots), $$ where \begin{eqnarray*} \Gamma_{v_{i}w_{i}}={\begin{array}{c}(w_{i}!/v_{i}!)^{1/2}, w_{i}>v_{i},\ 1,w_{i}>v_{i},\ (/v_{i}!)/w_{i}!^{1/2}, {w}{i}<v{i},\end{array} \end{eqnarray*} $Vi=1/2(vi+wi+1)$, and $G\sim ,Ai$, and $Bij$ are constant vectors independent of V. Components of these vectors are given in terms of dimension less normal-coordinate expansion coefficients of the potential through cubic, and of the dipole moment through quartic, terms. From the vibrational transition moments, the corresponding band intensities in the infrared absorption spectrum of $H2O$ and HDO at room temperature are computed for all of the transitions with $$ \sum^{3}{i=1}v{i}\leq 3,\sum^{3}{{i}=1}w{i}\leq 3, \mbox{ and }\sum^{3}{{i}=1}|w{i}-v_{i}\leq 3. $$