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Hydrocarbon Spectral Database, Molecular Parameters and Energy

2. Molecular Parameters and Energy Level Formulation

The discussion which follows deals with the most Common cases which will provide the user with the essential definition of quantum numbers, molecular parameters and basic relations employed in the analysis of rotational spectra. For the reader interested in a more detailed description of polyatomic rotational spectral measurements and analysis, we refer to texts on this subject by Townes and Schawlow [4], Gordy and Cook [5], Wollrab [6] and Kroto [7] which have both detailed and excellent discussions of all facets of rotational spectra. The spectroscopic notation employed follows, as closely as possible, the recommendations of the Joint Commission for Spectroscopy of the International Astronomic Union and the International Union of Pure and Applied Physics [8].

2.a. Linear and Symmetric Top Molecules

The principal axes of a linear rigid rotor are along the molecular bond or a-axis for which the inertial moment is zero, and perpendicular to the bond axis in two orthogonal planes through the center-of-mass of the molecule. These are called the b- and c-axes whereby Ic > Ib > Ia determines the labeling of the principal axes. For linear molecules Ib = Ic, and Ia = 0, and the rotational constant, B, is related to the moment of the inertia as


$$ B = {\displaystyle \frac{h^2}{8\pi^2 I_b}} ~ . $$

(eq 1)

The selection rules for rotational transitions of a linear polyatomic molecule are ΔJ = 0, ±1, and Δ = 0, ±1; where J is the total angular momentum quantum number excluding nuclear spin and is the vibrational angular momentum quantum number which arises in degenerate bending vibrational states.

Since molecules are not rigid, the effects of molecular vibrations and centrifugal distortion must be included in the model in order to accurately fit the observed rotational spectra. The rotational energy levels are represented as:


$$ F(J) = B_v[J(J+1) - \ell^2] - 
   D_v[J(J+1) - \ell^2]^2 + H_v[J(J+1) - \ell^2]^3 ~ , $$

(eq 2)

where Bv, is the rotational constant for the nth vibrational state, and Dv, and Hv, are the centrifugal distortion constants. The rotational constant can be expressed in terms of its equilibrium value, Be, and rotation-vibration interaction constants, αi, as


$$ B_v = B_{v_1,v_2,v_3} = B_e - \sum_{i=1,2,3} \alpha_i 
            \left( vi + \frac{d_i}{2} \right) ~ , $$

(eq 3)

neglecting higher order terms. Within this level of approximation the frequencies of rotational transitions from lower state J′′ to upper state J′ = J′′ + 1 are expressed:


$$ \nu_{J^\prime\leftarrow J^{\prime\prime}} = 2B_v  J^\prime - 
            4 D_v \left[ (J^\prime)^3 - J^\prime \, \ell^2 \right] + 
            6 J^\prime H_v \left[ (J^\prime)^4 + \frac{1}{3}~ J^{\prime\,2} 
            - 2(J^\prime)^2 \, \ell^2  + \ell^4 \right] ~ . $$

(eq 4)

The treatment of rotational transitions in excited vibrational states requires additional terms to account for the rotation-vibrational interactions. The symmetry species of excited vibrational states are designed as Σ, Π, Δ, etc., when  = 0, ±1, ±2, etc., respectively. One of the most common rotation-vibration interactions is -type doubling in Π states. In this case each JJ + 1 transition has two components which are indicated as L (lower) and U (upper) components in the tables which follow. The doublet separation is represented as: qv(v + 1)(J + 1). In addition ΔJ = 0 transitions are observable with the frequency expressed as: v = (qv/2)(v+1)J(J+1). These transitions are also included in the spectral tables. Other rotation-vibration interactions, such as Fermi resonance, often must be included for particular measurements. Since the level of approximation and method of analysis is dependent on the extent and quality of the spectral measurements available, the user should refer to the literature references cited in the tables for details concerning the analysis. For more general treatments of -type doubling and resonance interactions see the texts mentioned earlier [4 to 7] or the review by D.R. Lide [9].

2.b. Symmetric Top Molecules

Like linear molecules, in a symmetric top one of the principal axes must be coincident with the molecular symmetry axis, which must also be the axis with nonzero dipole moment. In a prolate symmetric top (Ia < Ib = Ic), the a-axis lies along the symmetry and in an oblate symmetric top (Ia = Ib < Ic). The rotational energy levels for the ground vibrational state of a symmetric top are represented as


$$E_{J,K} = B\,J(J+1)+(A-B)K^2 -D_J J^2(J+1)^2 -
 D_{J\,K} J(J+1) K^2 -D_K \,K^4 ~ ,$$

(eq 5)

including the first order (P4) centrifugal distortion terms. The selection rules are ΔJ = 0, ±1, ΔK = 0. The frequency for a J + 1 ← J and K ← K rotational transition is


$$\begin{eqnarray*}v = 2B_{\rm o} J^\prime - 4D_J(J^\prime)^3 
-2D_{JK} J^\prime K^2 &+& H_{JJJ}(J^\prime)^3[(J^\prime+1)^3 -(J^\prime+1)^3]\\
&+& 4H_{JJk}(J^\prime)^3\, K^2 + 2H_{KKJ} \, J^\prime \, K^4 ~ .\end{eqnarray*}$$

(eq 6)

which includes the P6 centrifugal distortion terms.

As in the case of linear molecules, vibrationally excited states can exhibit -type doubling which arises from the degenerate bending vibrational modes. Formulas for the rotational levels are given in the references cited in the molecular parameter tables here, e.g., propyne references, as well as in some of the text books referenced here [5] to [7].

2.c. Asymmetric Top Molecules

The majority of polyatomic molecules fall in the asymmetric-top category. When the three principal moments of inertia of a molecule differ, the molecule is classified as an asymmetric top. The energy level formulation for a rigid asymmetric top is considerably more complex than that for symmetric-tops or linear molecules. With the exception of low rotational levels, the rotational energy and transitions cannot be conveniently expressed in simple algebraic terms. Since references [4] to [7] provide excellent discussions of the usual methods employed in solving the basic rigid asymmetric rotor Hamiltonian: H2 = A Pa2 + B Pb2 + C Pc2 as well as the more complex Hamiltonian which includes centrifugal distortion H = H2 + H4 + H6 + ... , we will not delve into any details of the quantum mechanical formulation, but concentrate on describing the quantum number designations employed in the tables to follow, and provide the basic relationships between the different molecular constant notations used by various authors.

The rotational energy levels are characterized by the three quantum numbers JK-1,K+1 in the King-Hainer-Cross notation. Here, since S = 0, J is used rather than N for the rotational angular momentum. When S ≠ 0 we will use NK-1,K+1 to designate the rotational state and J for rotation plus electron spin and orbital angular momenta. The K-1 subscript is the K value in the limiting case of prolate symmetric-top and K+1 corresponds to the limiting case for an oblate symmetric-top. Ray's asymmetry parameter, κ, is often used to characterize the degree of asymmetry:


$$ \kappa = ~\frac{2B-A-C}{A-C} ~ . $$

(eq 7)

When A ≈ B, κ approaches +1 for the oblate case and when B ≈ C, κ approaches -1 for the prolate case.

(1) Selection Rules

In general an asymmetric rotor can exhibit three types of pure rotational transitions if the molecule has nonzero components of the electric dipole moment in the direction of the a, b, and c principal axes. For an asymmetric rotor the selection rules for a-type transitions are:


$$ \Delta J = 0, \pm 1; ~ \Delta K_{-1} = 0, \pm 2, ...; ~ 
             \Delta K_{+1} = \pm 1, \mp 3, ... ~ , $$

(eq 8)

for b-type transitions:


$$ \Delta J = 0, \pm 1; \quad \Delta K_{-1} = \pm 1, \pm 3, ...; \quad  
             \Delta K_{+1} = \mp 1, \mp 3, ... ~ , $$

(eq 9)

for c-type transitions:


$$ \Delta J = 0, \pm 1; \quad \Delta K_{-1} = \pm 1, \pm 3; \quad  
             \Delta K_{+1} = 0, \mp 2 ~ . $$

(eq 10)

When a molecule has a symmetry axis one must also examine the nuclear spin statistics that influence both the selection rules and the populations of the rotational levels.

(2) Rotational and Centrifugal Distortion Constants

Until approximately 1970 the Kivelson and Wilson [10] formulation of the Hamiltonian for a non-rigid asymmetric rotor was widely employed in analyzing rotational spectra. With the parameter notation employed by Kirchhoff [11] the Kivelson-Wilson Hamiltonian is:


$$ {\cal H} = A^\prime P_a^2 + B^\prime P_b^2 + C^\prime P_c^2 + 
           {\textstyle\frac{1}{4}} ~\sum_{\alpha,\beta} ~ \tau_{\alpha\alpha\beta\beta}^\prime 
           P_\alpha^2 P_\beta^2 ~ , $$

(eq 11)

where α,β = a, b, or c. For a planar molecule the following planarity relations reduce the six linear combinations of distortion constants to four and provide the determinable parameters shown in column 1 of Table 2.1:


$$ \tau_{acac} = \tau_{bcbc} = 0  $$

(eq 12a)

$$ \tau_{aacc} = \frac{C^2}{A^2} ~\tau_{aaaa} +  
 \frac{C^2}{B^2} ~\tau_{aabb} ~ , $$

(eq 12b)

$$ \tau_{bbcc} = \frac{C^2}{B^2} ~\tau_{bbbb} +  
 \frac{C^2}{A^2} ~\tau_{aabb} ~ , $$

(eq 12c)

$$ \tau_{cccc} = \frac{C^2}{A^2} ~\tau_{aacc} +  
 \frac{C^2}{B^2} ~\tau_{bbcc} ~ , $$

(eq 12d)

For non-planar molecules Dreizler et al. [12, 13] found that the Kivelson-Wilson distortion constants were indeterminant. Watson [14, 15] introduced a new relationship which allows the Kivelson-Wilson Hamiltonian to be expressed in terms of five independent centrifugal distortion coefficients, or linear combinations of taus, which eliminates the indeterminancy noted by Dreizler et al. Much of the recent analysis of rotational spectra follows Watson's reformulation [16, 17] in the form of a reduced Hamiltonian which simplified the computation of the energy levels.

Since there is not a unique unitary transformation which allows the nine Kivelson-Wilson parameters to be reduced to eight determinable parameters, several variations of the Watson reduced Hamiltonian are commonly employed in practice. The two most often employed result in the determinable parameters listed in columns 2 and 3 of Table 2.1. In reanalyzing the microwave spectra here both Kirchhoff's [11] formulation has been used as well as Watson's A-reduction [17]. See reference [11] for additional details. The second commonly used formulation is described in detail by Gordy and Cook [5]. Yamada and Winnewisser [18] have examined the effects of employing different reductions for the three King, Hainer and Cross axis representations Ir, IIr, and IIIr [19]. They provide a useful set of relations between the spectroscopic constants determined in the various reduction procedures and discuss the implications of the τ defect when employing the planarity conditions. When the spectral data require a higher order Hamiltonian, such as inclusion of P6 terms, generally the first-order perturbation treatment suggested by Watson [17] has been used.

2.d. Molecules with Doublet Electronic Ground States

The spectra and spectral analysis for molecules with one or more unpaired electrons are substantially more complex. The known hydrocarbon species that have doublet, S = 1/2, ground states are CH, CH2, and CH4. In addition to the molecular rotational angular momentum, N, the interactions from electronic spin, S, and nuclear spin, I, must be included in the Hamiltonian. Depending on the magnitude of the various interactions, one of the following three coupling schemes are used in limiting cases:

  1. N + S = J;     J + I = F
  2. S + I = G;     G + N = F
  3. N + I = E;     E + S = F

These interactions and the Hamiltonian for such molecules are discussed by Lin [20], Van Vleck [21], Curl and Kinsey [22] and others. Curl and Kinsey [22] have summarized the spectroscopic constant notation employed in the various formulations and developed an alternate method which can be applied to the hydrocarbon species. Since none of these species have been reanalyzed in the present work, the notation employed in the publications cited is followed in the present tables of spectroscopic constants.


Sensor Science Division

Created April 18, 2018, Updated November 15, 2019