NIST: Diatomic Molecules - 3-Sigma-Ground State Molecules

3. ³-Ground State Molecules

The O₂ and SO molecules are the only diatomic molecules in this compilation which possess a ³ Sigma

electronic ground state. Since the energy level calculations differ quite markedly from those in section 2, a detailed description of the calculations will be given here. Although a number of authors have treated this problem in slightly different manners than that discussed below, for uniformity we have chosen the formulation which corresponds closest to that employed in the previous section.

In order to describe the rotational spectra of this class, Hund's coupling case (b) was chosen as the starting point. The rotational levels are characterized by the rotational angular momentum quantum number, N, and the resultant angular momentum quantum number, J, which includes the total electron spin angular momentum. If the molecule has nuclei with non-zero nuclear spin, I, these are coupled to J to form the total angular momentum quantum number F, whereby coupling case (b_J) is assumed here. For pure case (b_J) the electric dipole transitions occur with the selection rules: Delta N = ą 1, Delta F = 0, ą 1, and Delta J = 0, ą 1, in the absence of external fields. Since an intermediate coupling case is actually observed, transitions are allowed for Delta N = ą 3. The magnetic dipole transitions occur with the selection rules: Delta N = 0, ą 2 and Delta J = 0, ą 1.

a. Molecular Parameters and Energy Level Formulation

The rotational energy levels may be described with the Hamiltonian [8]:

= 2/3

(3S_z² - S²) + gamma

(N · S) + BN² whereby a molecule fixed cartesian coordinate system is employed with the z-axis along the molecular axis. The first term describes the spin-spin interaction, the second term refers to the spin-rotation interaction and the last term describes the rotational kinetic energy. Since the coefficients lambda

and B are functions of the internuclear distance, r, centrifugal distortion and vibration-rotation interactions arise. If we define the coefficients as follows:

eq10 (eq10)

where $\zeta = \frac{r-r_{\rm e}}{r_{\rm e}}$ , the vibrational state dependence of the molecular parameters is given by:

eq11 (eq11)

where the Dunham coefficients, Y_lj, are defined in section 2 and

(eq12)

(eq13)

The centrifugal distortion terms are defined as:

(eq14)

(eq15)

and

(eq16)

With these definitions, the rotational energy levels are given in the form [9]:

(eq17)

eq18 (eq18)

The sextic terms, H, of the rotational energy are neglected because they cannot be determined from the data presently available for the spectral observations on ³ Sigma electronic ground state molecules. The energy equations are utilized with the selection rules stated above to allow the determination of the molecular constants B, lambda , gamma , D, rho , and delta , for vibrational state italic v . Combining the data available for various vibrational states allows the derivation of potential coefficients, a_i, and the expansion parameters of lambda and gamma .

Magnetic hyperfine structure has been described by Frosch and Foley [10] in terms of the determinable parameters, b and c. The nuclear electric quadrupole hyperfine structure is described by Amano, et al. [11] and results in determination of the constant, eQq, as defined in the discussion of ¹ Sigma ground electronic state molecules.