Pendular-limit representation of energy levels and spectra of symmetric- and asymmetric-top molecules

We report theoretical investigations pertaining to spectroscopy of pendular-state molecules, which are subjected to the large electrostatic interaction between the molecular dipole and a strong external field. After an appropriate coordinate transformation and a power-series expansion with an order parameter $\lambda$ that represents the degree of pendular condition, the zeroth-order Hamiltonian for the pendular limit has been derived. Motions of asymmetric tops in a high field are well described as two-dimensional anisotropic harmonic oscillations, and pendular-state quantum numbers have been introduced to label the energy levels. By using perturbative treatments, energies up to the ${\lambda }^0$ order are represented analytically with the pendular-state quantum numbers. Symmetry considerations are also accomplished by using the group theory appropriate to dipolar rigid bodies of symmetric and asymmetric tops in a uniform electric field. Energy levels and wave functions are classified into irreducible representations of the groups and selection rules for optical transitions are described. Energy-level correlations between the field-free and pendular conditions are also discussed on the basis of the group theoretical considerations. For symmetric and asymmetric tops, pendular-limit selection rules on the quantum numbers are derived by expanding the transition-dipole operators on $\lambda$, and transition strengths are analytically evaluated for all the excitation configurations with each of the transition types. Utilities of the present formulation have been verified by the comparison with exact model calculations based on the matrix diagonalization with a free-rotation basis set.

Figure 1

Variable definitions.

Figure 2

Energy levels of an asymmetric top reproduced by (a) the analytical representation, (b) the analytical representation with the second-order energy correction, and (c) the matrix diagonalization method. The energy levels of different $\mid m\mid$ are separately depicted. The model parameters are described in the text. Levels with the same $v_x,v_y$ are denoted as $(v_x,v_y)$ and connected by thin lines in (a) and (b). All energies are measured from the potential minimum of $-2∕{\lambda }^2=-800.0$.

Figure 3

Calculated spectra of an asymmetric top for a ${\mu }_{Zx}$ type transition from (a) the analytical representation, (b) the analytical representation with the second-order energy correction, and (c) the matrix diagonalization. (d) An expanded view of the $\Delta v_x=+1$, $\Delta v_y=0$ region in (c). The model parameters are described in the text.

Figure 4

Calculated spectra of an asymmetric top for ${\mu }_{Zg}$ type transitions $\left(g=x,y,z\right)$ from (a) the analytical representation, (b) the analytical representation with the second-order energy correction, and (c) the matrix diagonalization. The model parameters are the same as those used in Fig. 3.

Figure 5

Calculated spectra of an asymmetric top for ${\mu }_{Xg}$ type transitions $\left(g=x,y,z\right)$ from (a) the analytical representation, (b) the analytical representation with the second-order energy correction, and (c) the matrix diagonalization.