the energy (Hamiltonian) operator. For the rigid rotor in a field-free space, the energy operator corresponds to the kinetic energy of the system: where is reduced Planck constant and is the Laplacian. The Laplacian is given above in terms of spherical polar coordinates. The energy operator written in terms of these coordinates is: This operator appears also in the Schrödinger equation of the hydrogen atom after the radial part is separated off. The eigenvalue equation becomes The symbol represents a set of functions known as the spherical harmonics. Note that the energy does not depend on . The energy is -fold degenerate: the functions with fixed and have the same energy. Introducing the rotational constant , we write, In the units of reciprocal length the rotational constant is, with c the speed of light. If cgs units are used for , , and , is expressed in cm−1, or wave numbers, which is a unit that is often used for rotational-vibrational spectroscopy. The rotational constant depends on the distance . Often one writes where is the equilibrium value of (the value for which the interaction energy of the atoms in the rotor has a minimum). A typical rotational absorption spectrum consists of a series of peaks that correspond to transitions between levels with different values of the angular momentum quantum number () such that , due to the selection rules (see below). Consequently, rotational peaks appear at energies with differences corresponding to an integer multiple of . Selection rules Rotational transitions of a molecule occur when the molecule absorbs a photon [a particle of a quantized electromagnetic (em) field]. Depending on the energy of the photon (i.e., the wavelength of the em field) this transition may be seen as a sideband of a vibrational and/or electronic transition. Pure rotational transitions, in which the vibronic (= vibrational plus electronic) wave function does not change, occur in the microwave region of the electromagnetic spectrum. Typically, rotational transitions can only be observed when the angular momentum quantum number changes by . This selection rule arises from a first-order perturbation theory approximation of the time-dependent Schrödinger equation. According to this treatment, rotational transitions can only be observed when one or more components of the dipole operator have a non-vanishing transition moment. If is the direction of the electric field component of the incoming electromagnetic wave, the transition moment is, A transition occurs if this