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 $n_{\tau }$, $l_{\tau }$, ${\sigma }_{l_{\tau }}$, and $s_{\tau }(s_{\tau }=\frac{1}{2} \mathrm{always})$ for the $\tau \text{'}\mathrm{th}$ electron, and quantum numbers $s$, ${\sigma }_l$, and ${\sigma }_s$ for the molecule as a whole (${\sigma }_{l_{\tau }}$ and ${\sigma }_s$ represent quantized components of $l_{\tau }$ 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. ${\left(1\mathrm{}{s}^s\right)}^2{\left(2\mathrm{}{s}^p\right)}^2{\left(2\mathrm{}{s}^s\right)}^2\left(2\mathrm{}{p}^p\right)$, ${}_{}{}^2{}_{}{}^{}P$ for a seven-electron molecule with ${\sigma }_l=1\mathrm{}$, $s=\frac{1}{2}$; in a symbol such as $2s^p$ the superscript denotes $l_{\tau }$, the main letter, ${\sigma }_{l_{\tau }}$, thus $2s^p$ means that the electron in question has $n_{\tau }=2\mathrm{}$, $l_{\tau }=1\mathrm{}$, ${\sigma }_{l\tau }=0\mathrm{}$. Electrons with ${\sigma }_{l_{\tau }}=0, 1, 2, \cdots$, are referred to as $s, p, d, \cdots$, electrons. It is shown that in a molecule it is usually natural to define a group of equivalent electrons giving a resultant ${\sigma }_l=0\mathrm{}$, $s=0\mathrm{}$ as a closed shell; in this sense, two $s$ electrons, or four $p, or d, f, \cdots$, 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).