Electronic States and Band Spectrum Structure in Diatomic Molecules. I. Statement of the Postulates. Interpretation of CuH, CH, and Co Band-Types

The problem of the interpretation of band-structure from combination relations; ambiguities and criteria for overcoming them. In the analysis of band spectra, it may often be assumed that each observed frequency is due to a combination of terms whose rotational energy parts are of the form $F\left(j\right)=B{m}^2+\cdots =B{(\sqrt{j^2-{\sigma }^2}-\rho )}^2+\cdots$ (Kratzer-Kramers and Pauli term); here $j$ is the quantum number corresponding to the total angular momentum, $\frac{\mathrm{mh}}{2\pi }$ is the nuclear angular momentum, $\frac{\epsilon h}{2\pi }$ and $\frac{\sigma h}{2\pi }$ are electronic angular momentum components parallel and perpendicular to $m$, $\sigma$ being also parallel to the internuclear axis. In the empirical analysis of band-structure in accordance with the combination principle, only differences such as ${\Delta }_{1}F\left(j\right)=F(j+1\mathrm{})-F\left(j\right)\mathrm{}$ and ${\Delta }_{2}F\left(j\right)=F(j+1\mathrm{})-F(j-1\mathrm{})\mathrm{}$ can be obtained. If derived from terms of the Kratzer-Kramers and Pauli type, these are (after expanding to remove the radical) of the forms ${\Delta }_{1}F\left(j\right)=2B(T+\frac{1}{2})-\frac{B\rho {\sigma }^2}{j(j+1\mathrm{})\mathrm{}}+\cdots$ and ${\Delta }_{2}F\left(j\right)=4BT-\frac{2B\rho {\sigma }^2}{(j^2-1)}+\cdots$, where $T=j-\rho$ is defined as the apparent or effective rotational quantum number. Similarly, the most general expression (after expansion to remove the radical) for any band-branch is of the form $\nu =A+B^{\prime }{\left(\Delta T\right)}^2+2\mathrm{}{B}^{\prime }{T}^{\prime \prime }\Delta T+(B^{\prime }-B^{\prime \prime })T^{\prime \prime 2}+f\left(j\right)+\cdots$; primes here refer to the more excited, double primes to the less excited state; $\Delta T=T^{\prime }-T^{\prime \prime }$; $f\left(j\right)=(\frac{{\rho }^{\prime }{\sigma }^{\prime 2}}{j^{\prime 2}}-\frac{{\rho }^{\prime \prime }{\sigma }^{\prime \prime 2}}{j^{\prime \prime 2}})$. P-form, Q-form, and R-form branches are defined as branches for which $f\left(j\right)\mathrm{}$ is zero and $\Delta T$ has the respective values -1, 0 and +1; these are of the forms assumed by $P$, $Q$, and $R$ branches ($\Delta j=-1, 0, +1\mathrm{}$) for the (usual) special case $\Delta T=\Delta j$. In case $\rho$ and $\sigma$ are simultaneously present and large enough so that the small $\rho {\sigma }^2$ term is experimentally detectable, the values of $j$, $\rho$, and $\sigma$ can be determined directly if sufficiently accurate measurements are feasible. But in the usual case that one (or both) of the quantities $\rho$ and $\sigma$ are zero or nearly so, it is evidently impossible to determine $j$, $\rho$, and $\sigma$ without the aid of additional information or assumptions, since the $\Delta F\text{'}\mathrm{s}$ and the forms of the branches are now functions of $T^{\prime }$ and $T^{\prime \prime }$ alone; in such cases it has ordinarily been tacitly assumed in the past, very often erroneously according to the present work, that $\sigma =0\mathrm{}$. Criteria are discussed for overcoming these ambiguities of interpretation. These are (a) the presence or absence of $Q$ branches (b) especially valuable, the absence of particular lines ("missing lines") near the band-origin, and finally (c) the postulates stated below.