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.

Interpretation of bands of the CuH, HCl, CH, and CO types, and statement of the postulates. Recent work on the classification of electron levels in molecules in analogy with those of atoms is discussed; the $\mathrm{NO}\gamma$ bands (third positive nitrogen bands) are classified as ${}_{}{}^2{}_{}{}^{}S\to {}_{}{}^2{}_{}{}^{}P_{1,2}^{}{}_{}{}^{}$, transitions and it is suggested that Jevons' SnCl bands are due to the transition ${}_{}{}^2{}_{}{}^{}S\to {}_{}{}^2{}_{}{}^{}P_{1,2}^{}{}_{}{}^{}$ in the ${\mathrm{Sn}}^{+}$ ion in ${\mathrm{Sn}}^{+}$${\mathrm{Cl}}^{-}$. The quantities $\epsilon$ and $\sigma$ of the Kratzer-Kramers and Pauli formula are interpreted as components of an electronic inner quantum number $j_e$ for the molecule as a whole, or in some cases as individual inner quantum numbers $j_e$ for the two atoms of the molecule; in all cases it is found possible to assume that the $j_e$ values for molecular states are identical with Sommerfeld's $j$ values for corresponding atomic states.—Kratzer's interpretation of the violet CN bands ($j=\mathrm{integral}$, $j_e=\rho \sim \pm \frac{1}{2}$ for both initial and final states) is adopted with slight modification; in addition, these bands are ascribed to a ${}_{}{}^2{}_{}{}^{}S\to {}_{}{}^2{}_{}{}^{}S$ transition, probably of a single outer electron. The same interpretation applies to a number of analogous band spectra, emitted by BO, C${\mathrm{O}}^{+}$, ${\mathrm{N}}_2^{+}$, SiN, and AlO. The CuH, AgH, and AuH bands, and the infrared halogen halide bands, hitherto classed as essentially similar to the violet CN bands (all these bands have $P$- and $R$-form $P$ and $R$ branches only), are now interpreted as due to ${}_{}{}^1{}_{}{}^{}S\to {}_{}{}^1{}_{}{}^{}S$ electronic transitions, with $j=\mathrm{h}\mathrm{a}\mathrm{l}\mathrm{f}-\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}$, and $j_e=0\mathrm{}$ for both initial and final states; the infrared CO bands probably have the same structure. This new interpretation makes understandable the single character of the lines in bands of the CuH type, as compared with the doublets of CN, and at the same time gives expression, in the $j$ and $j_e$ values, to the contrasting facts that CN contains an odd number of electrons, and molecules like CuH and HCl an even number. The importance of Kratzer's analysis of the CH bands ($j=\mathrm{integral}$, ${\rho }^{\prime }=\pm \frac{1}{2}$, ${\sigma }^{\prime }=0\mathrm{}$,—the value of ${\sigma }^{\prime }$ represents a revision of Kratzer's ${\sigma }^{\prime }=\frac{1}{2}$,— ${\rho }^{\prime \prime }=\pm \frac{1}{2}$, ${\sigma }^{\prime \prime }=1\mathrm{}$) is emphasized. Since CH is an odd molecule like CN, the analysis confirms the possibility of $\rho =\pm \frac{1}{2}$ and of $j=\mathrm{integral}$ for odd molecules, which were assumed by Kratzer in the case of CN. The presence of $Q$ branches may be attributed to the presence of ${\sigma }^{\prime \prime }$. The values ${\rho }^{\prime \prime }=\pm \frac{1}{2}$ and ${\sigma }^{\prime \prime }=1\mathrm{}$ are respectively attributed to the H atom in its normal state and to the C atom in a ${}_{}{}^1{}_{}{}^{}P$ excited state; the normal state of H is thereby classified as ${}_{}{}^2{}_{}{}^{}S$, as for the alkali metals, in confirmation of recent work in line spectra.—The observed structure of the CO Angstrom bands, including the presence of $Q$ branches and the characteristic missing lines, is explained by the assumptions $j=\mathrm{half}\mathrm{integral}$ (in accord with the assumptions made for CuH and other even molecules), ${\rho }^{\prime }=0\mathrm{}$, ${\sigma }^{\prime }=0\mathrm{}$, ${\rho }^{\prime \prime }=0\mathrm{}$, ${\sigma }^{\prime \prime }=1\mathrm{}$. The values of $j_e$ here (${j_e}^{\prime }=0\mathrm{}$, ${j_e}^{\prime \prime }=1\mathrm{}$) are in complete agreement with Birge's interpretation of these bands, based on entirely different grounds, as ${}_{}{}^1{}_{}{}^{}S\to {}_{}{}^1{}_{}{}^{}P$.—With but slight generalization, the leading assumptions and conclusions involved in the above interpretations can be stated in the form of three simple postulates, which are stated on p. 503 ($q.v$.) and will not be repeated here. With the help of these postulates, a systematic interpretation of most known band spectra is possible. As a corollary of the postulates it is concluded that the normal state of all diamagnetic gases (${\mathrm{H}}_2$, ${\mathrm{N}}_2$, CO, HCl, etc.) is probably a ${}_{}{}^1{}_{}{}^{}S$ state ($j_e=0\mathrm{}$); the paramagnetism of NO is attributed to its double ${}_{}{}^2{}_{}{}^{}P$ normal state ($j_e=\sigma =\frac{1}{2} \mathrm{and} 1\frac{1}{2}$), while that of ${\mathrm{O}}_2$ is perhaps due to a ${}_{}{}^3{}_{}{}^{}S$ normal state ($j_e=1\mathrm{}$). An incidental result of the postulates is the disappearance of quarter-integral values of $j$ and $j_e$.

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