Electronic States and Band-Spectrum Structure in Diatomic Molecules II. Spectra Involving Terms Essentially of the Form B(j 2 -σ 2 )

Properties of rotational terms of the form $B({j}^{2}\ensuremath{-}{\ensuremath{\sigma}}^{2})$.---Evidence is presented to show that the AlH, ${\mathrm{He}}_{2}$, and certain other bands involve rotational terms essentially of the form $B({j}^{2}\ensuremath{-}{\ensuremath{\sigma}}^{2})$, and are in agreement with the postulates of a previous ${\mathrm{paper}.}^{1}$ Since terms of this form have not hitherto been recognized in practice, their chief empirical properties, as shown by the bands discussed below, are now summarized. (1) There are in general two values of the rotational energy term $F(j)$ for each $j$ value. This type of rotational doubling has previously been confused with that which occurs, due to $\ifmmode\pm\else\textpm\fi{}\ensuremath{\epsilon} (\mathrm{where} \ensuremath{\epsilon}=|\ensuremath{\rho}|)$, when $F(j)$ is essentially of the form $B{(j\ensuremath{-}\ensuremath{\rho})}^{2}$. In the special case where $\ensuremath{\rho}$ and $\ensuremath{\sigma}$ are both zero ($^{1}S$ states), all rotational doubling is absent. (2) The two sets of $F(j)$ values, in doubling of the $\ensuremath{\sigma}$ type, in general differ in respect to the values of $B$ and of a small "secondary" $\ensuremath{\rho}$ whose presence must, at least formally, be admitted; thus ${F}_{i}(j)={B}_{i}[{(j\ensuremath{-}{\ensuremath{\rho}}_{i})}^{2}+\ensuremath{\cdots}(i=A or B)$, with $\ensuremath{\epsilon}l\frac{1}{2}$, or usually \ensuremath{\ll}\textonehalf{}, in practice; a slight difference in the electronic term values may also occur (as e.g., in the ${\mathrm{He}}_{2}$ bands). (3) In general, there are six branches in combinations of two $\ensuremath{\sigma}$-type terms (cf. Eqs. (2) of text); in the $P$ and $R$ branches, combinations occur only between like rotational terms (${{F}^{\ensuremath{'}}}_{A}\ensuremath{\rightarrow}{{F}^{\ensuremath{'}\ensuremath{'}}}_{A}$, and ${{F}^{\ensuremath{'}}}_{B}\ensuremath{\rightarrow}{{F}^{\ensuremath{'}\ensuremath{'}}}_{B}$), while in the $Q$ branches, "crossing over" always occurs (${{F}^{\ensuremath{'}}}_{A}\ensuremath{\rightarrow}{{F}^{\ensuremath{'}\ensuremath{'}}}_{B}$ or ${{F}^{\ensuremath{'}}}_{B}\ensuremath{\rightarrow}{{F}^{\ensuremath{'}\ensuremath{'}}}_{A}$). (4) In the special case of $^{1}P\ensuremath{-}^{1}S$ and $^{1}S\ensuremath{-}^{1}P$ combinations (cf. Eqs. (1) and Fig. 1 of text), there are only three branches, due to the existence for the $^{1}S$ electronic state of only one rotational state for each value of $j$ (these states have the properties of ${F}_{B}$ states); apparent "combination defects" in $P\ensuremath{-}Q\ensuremath{-}R$ combinations result here from the $Q$ crossing over phenomenon. (5) In $^{2}S\ensuremath{-}^{2}P$ and $^{2}P\ensuremath{-}^{2}S$ transitions, the intercombination of the $\ensuremath{\rho}$-type $^{2}S$ term with the $\ensuremath{\sigma}$-type $^{2}P_{1}$ or $^{2}P_{2}$ term gives rise to a special band structure (with six branches) in which for a frequent limiting case there is a practically complete coalescence of one of the $P$ with one of the $Q$ branches, and of one of the $R$ branches with the other $Q$ branch (cf. Eqs. (8), (8A), and (8B) of text). The existence of these band-types is strong evidence for the reality of the $\ensuremath{\rho}$ values +\textonehalf{} and -\textonehalf{} for $^{2}S$ states, and so affords additional confirmation of Kratzer's interpretation of the violet CN bands ($^{2}S\ensuremath{-}^{2}S$ transition) as adopted in ref. 1.The CO, AlH, and ${\mathrm{He}}_{2}$ bands.---Assuming $\ensuremath{\rho}=0$, $\ensuremath{\sigma}=1$ for $^{1}P$ states, it was shown in ref. 1 that the structure and missing lines of the CO \AA{}ngstrom bands are in agreement with the classification $^{1}P\ensuremath{-}^{1}S$ given by Birge. The AlH bands are now shown to fall under the classification $^{1}S\ensuremath{-}^{1}P$, and it is predicted that the fourth positive CO bands, designated as $^{1}S\ensuremath{-}^{1}P$ by Birge, will have the corresponding structure and missing lines.---By adopting the assumption (due to Mecke) that alternate lines are missing in each branch, it is shown that the three-branch "series" bands of helium are of the $^{1}S\ensuremath{-}^{1}P$ type and the three-branch bands $\ensuremath{\lambda}6400$ and 4546 of the $^{1}P\ensuremath{-}^{1}S$ type, while the six-branch band $\ensuremath{\lambda}5733$ appears as the (only known) representative of the $^{1}P\ensuremath{-}^{1}D$ type. Revised values of various constants of the ${\mathrm{He}}_{2}$ molecule in its different electronic states are given in Table I. To account for the alternate missing lines, a system of alternate suppressed rotational levels (cf. Fig. 2) is assumed. The correctness of this system is supported by several lines of evidence in addition to the satisfactory interpretation of the band structures; e.g., it correctly predicts the absence of $^{1}S\ensuremath{-}^{1}S$ and $^{1}P\ensuremath{-}^{1}P$ transitions in ${\mathrm{He}}_{2}$. The relation of the excited states of ${\mathrm{He}}_{2}$ to those of He is discussed briefly.Resonance series and the band spectra of the alkalies and the halogens. The fluorescent radiation emitted by a molecule after excitation by absorption of monochromatic light consists under suitable conditions of a series ("resonance series" of Wood) of roughly equidistant groups of band lines, corresponding to a series of final vibrational states. According to the present analysis, various transitions in even molecules should yield resonance series of the following types: $^{1}S\ensuremath{-}^{1}S$, $P\ensuremath{-}R$ doublets; $^{1}P\ensuremath{-}^{1}P$ (or $^{1}D\ensuremath{-}^{1}D$, etc.), $P\ensuremath{-}Q\ensuremath{-}R$ triplets but with the $Q$ member usually too weak to be noticed; $^{1}P\ensuremath{-}^{1}D$, $^{1}D\ensuremath{-}^{1}P$, $^{1}P\ensuremath{-}^{1}S$, etc., $P\ensuremath{-}Q\ensuremath{-}R$ triplets; $^{1}S\ensuremath{-}^{1}P$, doublet series ($P, R$) for some exciting wave-lengths and singlet series ($Q$ only) for others. The visible ${\mathrm{I}}_{2}$ absorption bands give rise to $P\ensuremath{-}R$ doublets (interpreted as such by Lenz), so that they (and by analogy the corresponding ${\mathrm{Br}}_{2}$ and ${\mathrm{Cl}}_{2}$ bands) are probably $^{1}S\ensuremath{-}^{1}S$ (or perhaps $^{1}P\ensuremath{-}^{1}P$ or $^{1}D\ensuremath{-}^{1}D$). The ${\mathrm{Na}}_{2}$ absorption bands in the green, giving both singlet and doublet resonance series, are probably $^{1}S\ensuremath{-}^{1}P$, and the same is probably true by analogy of most of the familiar alkali metal bands. In further support of this, there is no evidence, in the band structure, of electronic doublets such as appear in the atomic transitions $^{2}S\ensuremath{-}^{2}P_{1,2}$. The relation of the electronic states of ${\mathrm{Na}}_{2}$ to those of Na is briefly discussed.Other band spectra.---Although no detailed analyses are available, the evidence indicates that the red CN, $\mathrm{BO}\ensuremath{\alpha}$, and C${\mathrm{O}}^{+}$ comet tail bands (also the doublet alkaline earth halide bands?) have a structure (cf.(5) under "Properties....," above) essentially the same as that of the ZnH, CdH, and HgH bands, and characteristic of the transition $^{2}S\ensuremath{-}^{2}P_{1,2}$. The "combination" bands of BO and C${\mathrm{O}}^{+}$ and the NO third positive bands (presumably also Jevons' SnCl bands) probably have a similar structure characteristic of $^{2}P_{1,2}\ensuremath{-}^{2}S$ transitions. Predictions are given for the structure of $^{2}P\ensuremath{-}^{2}D$ bands; these should differ from all bands now on record in having ${T}^{\ensuremath{'}}$ and ${T}^{\ensuremath{'}\ensuremath{'}}$ both integral.---The OH, CH and MgH bands are briefly discussed following Table II, which gives a condensed summary of $T$, $\ensuremath{\rho}$ and $\ensuremath{\sigma}$ values, and electronic transitions, for the band types discussed here and in ref. 1. Table II shows that in all known cases, ${j}_{e}$ appears essentially as $\ensuremath{\sigma}$ in $P$ and $D$ states, but as $\ifmmode\pm\else\textpm\fi{}\ensuremath{\epsilon}$ in $S$ states (for $^{1}S$ states ${j}_{e}=0$). A corollary is that $m$ is half-integral in known $S$ states, but not in $P$ or $D$ states.