Rotational Spectra

Incident electromagnetic waves can excite the rotational levels of molecules provided they have an electric dipole moment. The electromagnetic field exerts a torque on the molecule. The spectra for rotational transitions of molecules is typically in the microwave region of the electromagnetic spectrum. The rotational energies for rigid molecules can be found with the aid of the Shrodinger equation. The diatomic molecule can serve as an example of how the determined moments of inertia can be used to calculate bond lengths.

The illustration at left shows some perspective about the nature of rotational transitions. The diagram shows a portion of the potential diagram for a stable electronic state of a diatomic molecule. That electronic state will have several vibrational states associated with it, so that vibrational spectra can be observed. Most commonly, rotational transitions which are associated with the ground vibrational state are observed. 左边的示意图展示了旋转跃迁的某些特性。该图显示了双原子分子稳定电子态的一部分势能图。该电子态将与多个振动态相关联，因此可以观察到振动光谱。通常，与基态振动态相关的旋转跃迁会被观察到。

Rotational Energies

The classical energy of a freely rotating molecule can be expressed as rotational kinetic energy

where x, y, and z are the principal axes of rotation and I_x represents the moment of inertia about the x-axis, etc. In terms of the angular momenta about the principal axes, the expression becomes

The formation of the Hamiltonian for a freely rotating molecule is accomplished by simply replacing the angular momenta with the corresponding quantum mechanical operators.

Diatomic Molecules

For a diatomic molecule the rotational energy is obtained from the Schrodinger equation with the Hamiltonian expressed in terms of the angular momentum operator.

More detail 更多内容

where J is the rotational angular momentum quantum number and I is the moment of inertia.

Rotational transitions 旋转跃迁

Determining the rotational constant B 确定旋转常数B enables you to calculate the bond length R. The allowed transitions for the diatomic molecule are regularly spaced at interval 2B. The measurement and identification of one spectral line allows one to calculate the moment of inertia and then the bond length. 这使你能计算键长R。双原子分子的允许跃迁在间隔2B处规律地分布。测量和识别一条光谱线允许你计算转动惯量，从而得到键长。

Rotational Transitions, Diatomic

For a rigid rotor diatomic molecule, the selection rules for rotational transitions are ΔJ = +/-1, ΔM_J = 0 .

	The rotational spectrum of a diatomic molecule consists of a series of equally spaced absorption lines, typically in the microwave region of the electromagnetic spectrum. The energies of the spectral lines are 2(J+1)B for the transitions J -> J+1. 双原子分子的旋转光谱由一系列等间距的吸收线组成，通常位于电磁波谱的微波区域。谱线的能量为2(J+1)B，对应于J到J+1的跃迁。
	For real molecules like the example of HCl , the successive transitions are a bit lower than predicted because centrifugal distortion lengthens the molecule, increasing its moment of inertia. 对于像HCl这样的真实分子，连续的跃迁略低于预测值，因为离心畸变使分子变长，从而增加了其转动惯量。

Moment of Inertia, Diatomic

The moment of inertia about the center of mass is 关于质心的转动惯量是 From the center of mass definition 从质心定义出发 and substituting to eliminate r1 and r2 gives where μ is called the "reduced mass." 并将 r₁ 和 r₂ 代入以消去它们，得到其中 μ 称为‘约化质量’。