The Assignment of Quantum Numbers for Electrons in Molecules. I

Abstract

Quantum numbers, notation, closed shells, molecular states.---The problem of making a complete assignment of quantum numbers for the electrons in a (non-rotating) diatomic molecule is considered. A tentative assignment of such quantum numbers is made in this paper (cf. Table III) for most of the known electronic states of diatomic molecules composed of atoms of the first short period of the periodic system. The assignments are based mainly on band spectrum, and to a lesser extent on ionization potential and positive ray, data. The methods used involve the application and extension of Hund's theoretical work on the electronic states of molecules. Although the actual state of the electrons in a molecule, as contrasted with an atom, cannot ordinarily be expected to be described accurately by quantum numbers corresponding to simple mechanical quantities, such quantum numbers can nevertheless be assigned formally, with the understanding that their mechanical interpretation in the real molecule (obtainable by an adiabatic correlation) may differ markedly from that corresponding to a literal interpretation. With this understanding, a suitable choice of quantum numbers for a diatomic molecule appears to be one corresponding to an atom in a strong electric field, namely, quantum numbers ${n}_{\ensuremath{\tau}}$, ${l}_{\ensuremath{\tau}}$, ${\ensuremath{\sigma}}_{{l}_{\ensuremath{\tau}}}$, and ${s}_{\ensuremath{\tau}}({s}_{\ensuremath{\tau}}=\frac{1}{2} \mathrm{always})$ for the $\ensuremath{\tau}'\mathrm{th}$ electron, and quantum numbers $s$, ${\ensuremath{\sigma}}_{l}$, and ${\ensuremath{\sigma}}_{s}$ for the molecule as a whole (${\ensuremath{\sigma}}_{{l}_{\ensuremath{\tau}}}$ and ${\ensuremath{\sigma}}_{s}$ represent quantized components of ${l}_{\ensuremath{\tau}}$ and $s$, respectively, with reference to the line joining the nuclei). These quantum numbers may be thought of as those associated with the imagined "united atom" formed by bringing the nuclei of the molecule together. A notation is then proposed whereby the state of each electron and of the molecule as a whole can be designated, e.g. ${(1{s}^{s})}^{2}{(2{s}^{p})}^{2}{(2{s}^{s})}^{2}(2{p}^{p})$, $^{2}P$ for a seven-electron molecule with ${\ensuremath{\sigma}}_{l}=1$, $s=\frac{1}{2}$; in a symbol such as $2{s}^{p}$ the superscript denotes ${l}_{\ensuremath{\tau}}$, the main letter, ${\ensuremath{\sigma}}_{{l}_{\ensuremath{\tau}}}$, thus $2{s}^{p}$ means that the electron in question has ${n}_{\ensuremath{\tau}}=2$, ${l}_{\ensuremath{\tau}}=1$, ${\ensuremath{\sigma}}_{l\ensuremath{\tau}}=0$. Electrons with ${\ensuremath{\sigma}}_{{l}_{\ensuremath{\tau}}}=0, 1, 2, \ensuremath{\cdots}$, are referred to as $s, p, d, \ensuremath{\cdots}$, electrons. It is shown that in a molecule it is usually natural to define a group of equivalent electrons giving a resultant ${\ensuremath{\sigma}}_{l}=0$, $s=0$ as a closed shell; in this sense, two $s$ electrons, or four $p, or d, f, \ensuremath{\cdots}$, electrons form a closed shell. The possible molecular states corresponding to various electron configurations are deduced by means of the Pauli principle (cf. Table I, and Appendix).Promoted electrons, binding energy, bonding power, and relation of molecular to atomic electron states.---As Hund has shown, some of the electrons must undergo an increase in their $n$ values (principal quantum numbers) when atoms unite to form a molecule. Such electrons are here called promoted electrons. The electrons in a molecule may be classified according to their bonding power, positive, zero, or negative. Electrons whose presence tends to hold a molecule together, as judged by the fact that their removal from a stable molecule causes a decrease in the energy of dissociation $D$ or an increase in the equilibrium internuclear separation ${r}_{0}$ may be said to have positive bonding power, and are identified with, or defined as, bonding electrons. The definitions of bonding power in terms of changes of $D$, and of changes of ${r}_{0}$, are unfortunately not in general equivalent, and we must accordingly distinguish "energy-bonding-power" and "distance-binding-power". On the whole, promoted electrons should tend to show negative energy-bonding-power, unpromoted electrons positive energy-bonding-power, but much should depend on "orbit dimensions."Certain rules governing the relations of the electronic states of a molecule to those of its dissociation products are discussed; in addition to theoretical rules established by Hund in regard to ${\ensuremath{\sigma}}_{l}$ and $s$ values, another, presumably less strict, rule is here proposed, namely that the ${\ensuremath{\sigma}}_{{l}_{\ensuremath{\tau}}}$ values of all the atomic electrons before union should be preserved in the molecule (${\ensuremath{\sigma}}_{{l}_{\ensuremath{\tau}}}$ conservation rule). Selection rules for electronic transitions are also discussed; in addition to rules given by Hund, the following are proposed: $\ensuremath{\Delta}{l}_{\ensuremath{\tau}}=\ifmmode\pm\else\textpm\fi{}1$ for intense transitions; $\ensuremath{\Delta}{\ensuremath{\sigma}}_{{l}_{\ensuremath{\tau}}}=0, \ifmmode\pm\else\textpm\fi{}1$.Results.---The key to the assignment of quantum numbers made here is found in the fact that the molecules BO, C${\mathrm{O}}^{+}$, and CN show an inverted $^{2}P$ state instead of the normal $^{2}P$ which should occur if this state were analogous to the ordinary $^{2}P$ states of the Na atom. The existence of such a low-lying inverted $^{2}P$ indicates that in these molecules there exists a closed shell of $p$ electrons from which one is easily excited. It is concluded that this is a ${(2{p}^{p})}^{4}$ shell. The identification of two other closed shells, of $s$ electrons, very likely ${(3{s}^{p})}^{2}$ and ${(3{s}^{s})}^{2}$, follows; the electrons in these and the ${(2{p}^{p})}^{4}$ shell are roughly equal in energy of binding. According to this interpretation, the electron jumps involved in the band spectra of BO, CN, C${\mathrm{O}}^{+}$, and ${\mathrm{N}}_{2}^{+}$ are more analogous to X-ray than to optical electron transitions. From this beginning, proceeding to CO, ${\mathrm{N}}_{2}$, NO, ${\mathrm{O}}_{2}$, ${\mathrm{O}}_{2}^{+}$, ${\mathrm{F}}_{2}$, ${\mathrm{C}}_{2}$, etc., a self-consistent assignment of quantum numbers is built up for most of the known states of the various molecules treated in this paper. The spectroscopic analogies of CN, ${\mathrm{N}}_{2}$, NO, etc., to Na, Mg, Al are justified and the partial failures of these analogies, such as the chemical resemblance of CN to a halogen, are explained. Nearly all the hitherto observed ionization potentials of the molecules discussed can be accounted for by the removal of a single electron from one or another of the various closed shells supposed to be present. The ${\mathrm{N}}_{2}^{+}$ band fluorescence produced by short wave length ultraviolet light (Oldenberg) is accounted for as the expected result of photo-ionization of a $3{s}^{p}$ electron. The steadily decreasing heat of dissociation in the series ${\mathrm{N}}_{2}$-NO-${\mathrm{O}}_{2}$-${\mathrm{F}}_{2}$ is accounted for by the successive addition of promoted $3{p}^{p}$ electrons with strong negative bonding power. Starting from ${\mathrm{N}}_{2}$, whose normal state corresponds to a $^{1}S$ configuration of closed shells, we add one $3{p}^{p}$ electron to give the $^{2}P$ normal state of NO and ${\mathrm{O}}_{2}^{+}$, two to give the $^{3}S$ normal state of ${\mathrm{O}}_{2}$, four to give a closed shell, ${(3{p}^{p})}^{4}$, which accounts for the $^{1}S$ normal state of ${\mathrm{F}}_{2}$.In ${\mathrm{N}}_{2}$ (probably also in ${\mathrm{O}}_{2}$ and the other homopolar molecules, but data are too few), band systems for which $\ensuremath{\Delta}{l}_{\ensuremath{\tau}}\ensuremath{\ne}1$ are notably lacking, thus giving support to Hund's predicted selection rule for homopolar molecules; in the analogous heteropolar molecule CO, many systems occur with $\ensuremath{\Delta}{l}_{\ensuremath{\tau}}=0$, although they are probably weaker, as expected, than those for which $\ensuremath{\Delta}{l}_{\ensuremath{\tau}}=\ifmmode\pm\else\textpm\fi{}1$. On account of this strict selection rule in ${\mathrm{N}}_{2}$, certain levels should be metastable, in particular the final level of the afterglow ($\ensuremath{\alpha}$) bands of active nitrogen. There is evidence for the existence of a strict selection rule $\ensuremath{\Delta}s=1$ in homopolar molecules.