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# Diatomic Spectral Database, 1Σ-Ground State Molecules

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### 2. 1Σ-Ground State Molecules

The majority of the diatomic molecules studied by microwave techniques have 1Σ electronic ground states. The rotational levels of these species can be described by the quantum numbers J, F, and v as defined in section 2.b. Selection rules for rotational transitions observed are:

$$\Delta J = + 1 ~ , \quad \Delta F = 0, \pm 1 ~ , \quad {\rm and} \quad \Delta v = 0 \quad .$$

(eq 1)

#### a. Molecular Parameters and Energy Level Formulation

Throughout the present review the Dunham formulation [3] for the energy level expressions is employed wherever possible. In a few cases where observational data is limited, the traditional spectroscopic formulation is followed, e.g., the use of B0, D0, etc. Dunham's treatment shows that the energy levels can be written as:

$$W_{v,J} = \Sigma_{l,j}\left(v + \frac{1}{2}\right)^l J^j \left(J + 1\right)^j \quad ,$$

(eq 2)

where l and j are summation indices, v and J are the vibrational and rotational quantum numbers, respectively, and the Yl,j are coefficients which are equivalent to the traditional spectroscopic constants from electronic band spectra, with the exception of Be and ωe which are defined slightly differently in the two formulations. (See references [3] or [4] for the exact relations.) Since the isotopic dependence of these coefficients is often utilized, the following list of determinable constants summarizes the isotopic relations most often encountered:

$$\begin{eqnarray*} Y_{10} &\cong&\omega_{\rm e} \propto ~ 1/\mu_{\rm r}^{1/2}\\ Y_{20} &\cong& - \omega_{\rm e} x_{\rm e} \propto ~ 1/\mu_{\rm r} \\ Y_{30} &\cong& \omega_{\rm e} y_{\rm e} \propto ~ 1/\mu_{\rm r}^{3/2} \\ Y_{40} &\cong& \omega_{\rm e} z_{\rm e} \propto ~ 1/\mu_{\rm r}^2 \end{eqnarray*}$$

(eq 3)

$$\begin{eqnarray*} Y_{01} &\cong& B_{\rm e} \propto ~ 1/\mu_{\rm r} \\ Y_{11} &\cong& - \alpha_{\rm e} \propto ~ 1/\mu_{\rm r}^{3/2} \\ Y_{21} &\cong& \gamma_{\rm e} \propto ~ 1/\mu_{\rm r}^2 \\ Y_{02} &\cong& -D_{\rm e} \propto ~ 1/\mu_{\rm r}^2 \\ Y_{12} &\cong& - \beta_{\rm e} \propto ~ 1/\mu_{\rm r}^{5/2} \\ Y_{03} &\cong& - H_{\rm e} \propto ~ 1/\mu_{\rm r}^3 \end{eqnarray*}$$

(eq 4)

where the molecular reduced mass,

$$\mu_{\rm r} ~=~ \frac{M_1M_2}{M_1 + M_2}$$

(eq 5)

with M1 and M2 representing the atomic masses of atom 1 and 2, respectively. For a more detailed discussion of the energy level formulations for diatomic molecules, see references [3, 4, 5, and 6].

$$\begin{eqnarray*} \nu_{v,J^\prime\leftarrow J^{\prime\prime}} &=& 2[Y_{01} + Y_{11}(v+{\textstyle\frac{1}{2}} ) + Y_{21}(v+{\textstyle\frac{1}{2}} )^2 + ...] J^\prime \\ &~&+ 4[Y_{02} + Y_{12}(v+{\textstyle\frac{1}{2}} ) + ...] (J^\prime)^3 \\ &~&+ [Y_{03} + Y_{13}(v+{\textstyle\frac{1}{2}} ) + ...] [6(J^\prime)^5 + 2(J^\prime)^3] \end{eqnarray*}$$

(eq 6)

An equivalent expression in terms of the traditional constants is

$$\nu_{v,J^\prime\leftarrow J^{\prime\prime}} = 2B_v J^\prime - 4D_v (J^\prime)^3 + H_v[6 (J^\prime)^5 + 2(J^\prime)^3]$$

(eq 7)

where

$$\begin{eqnarray*} B_v&=&B_{\rm e} - \alpha_{\rm e}(v+{\textstyle\frac{1}{2}} ) + \gamma_{\rm e}(v+{\textstyle\frac{1}{2}} )^2 ~ ... \\ D_v&=&D_{\rm e} - \beta_{\rm e}(v+{\textstyle\frac{1}{2}} ) ~ ... \\ H_v&=&H_{\rm e} ~ ... ~ . \end{eqnarray*}$$

(eq 8)

Hyperfine structure is observable in a majority of the molecules tabulated here. Hyperfine structure stems from nuclear electric quadrupole interaction with the electric field gradient at the nucleus, magnetic interaction of nuclear spin with the field produced by molecular rotation, and interaction between the two nuclear spins. Basically, only the nuclear quadrupole and spin-rotation effects have been observed in microwave rotational spectra, while all of the hyperfine structure interactions of a number of diatomics have been determined from molecular beam electric resonance studies. Since the treatment of these effects can become quite complex and is often handled individually for each case, the reader is referred to the literature cited for particular formulations. Rather detailed general treatments of hyperfine structure in molecular spectra can be found in refs. [4], [5], and [7].

The most common case observed is that for diatomic molecules which contain one nucleus with nuclear spin, I ≤ 1. In this case. the nuclear electric quadrupole and spin-rotation interactions from first order perturbation theory add to the rotational energy via:

$$W_{hfs} = -eQq_v f(I, J, F) + (c/2) [F(F+1) - I(I+1) - J(J+1)]$$

(eq 9)

where f(I, J, F) is Casimir's function which is tabulated in Appendix I of ref. [4]. Here F is the total angular momentum quantum number. where

$$F = J + I, ~ J + I - 1, ~ ... ~ |J - I| ~ .$$

(eq 9a)

#### b. List of Symbols and Conversion Factors

Symbols Definition
I or (Ii) Angular momentum quantum number of nuclear spin for one (or ith) nucleus.
S Resultant angular momentum number of electron spins. Σ is the projection of S on the molecular axis. (Hund's case (a)).
Λ Absolute value of the projection of the resultant orbital angular momentum on the molecular axis. (Hund's cases (a) and (b)).
Ω Absolute value of the projection of the total electronic angular momentum on the molecular axis.
J Resultant total angular momentum quantum number, excluding nuclear spins.
N Rotational angular momentum quantum number, excluding electron and nuclear spins, in the case where electron spin is present.
F1 Resultant angular momentum quantum number including nuclear spin for one nucleus.
F Resultant total angular momentum quantum number.
X Quantum number employed when F1 is not a good quantum number. This value simply numbers the levels from lowest to highest energy for the same F quantum number.
v Vibrational quantum number.
Σ, Π, Δ Electronic state designation for which Λ = 0, 1, 2, respectively. The symbols XA . . . precede the state designation whereby X refers to the ground state.
Yl,j Dunham coefficients.
′ or ″ Prime or double prime is used to distinguish the upper (′) and lower (″) levels in a transition. They occur as superscripts on the quantum numbers.
(. . .) Parentheses in the numerical listings contain measured or estimated uncertainties. For example, the value 1.407(83) should be interpreted as 1.407±0.083. Thus the value in parentheses refers to the last significant digits given. If X appears in parentheses, the uncertainty was indeterminate
$$\nu_{v,J^\prime\leftarrow J^{\prime\prime}}$$ Rotational transition frequency between the J′ and J″ levels of the vth vibrational state (MHz).
ωe, ωeχe, . . . Coefficients in the power series for the vibrational energy where G = ωe(v + 1/2) - ωeχe(v + 1/2)2 + . . . (cm-1).
Be Rotational constant for the equilibrium internuclear distance. Be = h/8π2µrre2 (MHz).
Bv, Dv, Hv . . . Coefficients in the power series for rotational energy.
α, γ Coefficients in the power series representing Bv.
De, β Coefficients in the power series representing Dv.
µr Reduced mass, u.
re Equilibrium internuclear distance (Å).
eQqv[M] Nuclear electric quadrupole coupling constant of nucleus M for the vth vibrational state (MHz).
cM Spin-rotation hyperfine structure constant related to nucleus M (kHz).
c3, c4 Spin-spin hyperfine structure constants. Occasionally denoted as dT and dS, respectively (kHz).

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Created April 24, 2018, Updated November 15, 2019