Rotational Spectra and Structure of a Linear Molecule

The frequencies of microwave absorption transitions can be used to calculate molecular structure parameters. The structure of nitrous oxide is one which can be readily calculated from the measurement of molecular rotational spectra. The structure of this linear molecule involves only two parameters R_N and R_O as illustrated.

微波吸收跃迁的频率可用于计算分子结构参数。一氧化二氮的结构可以通过测量分子旋转光谱来方便地计算得出。这种线性分子的结构仅涉及两个参数R_N和R_O，如图所示。

分子的旋转能由以下公式给出

The J=0 to J=1 transition of NNO can be measured to determine the moment of inertia I of the molecule.

NO分子的J=0到J=1跃迁可以被测量以确定该分子的转动惯量I。

But if only one moment of inertia is measured, there is not sufficient data to calculate the structure since it involves two unknown parameters. If the isotope ¹⁵N is substituted at position m₁, then another moment of inertia is determined and the two structural parameters can be determined. This strategy of isotopic substitution to gain structural information is a common one, depending upon the fact that the structure of the molecule is determined by the electric properties of the atoms and unaffected by substitution of a different isotope.

但如果仅测量一个转动惯量，由于涉及两个未知参数，无法确定结构。如果在位置 m₁ 处将原子替换为¹⁵N 同位素，则可以确定另一个转动惯量，从而确定两个结构参数。这种通过同位素替换获取结构信息的策略是常见的，其依据是分子的结构由原子的电性质决定，而不会因替换为不同同位素而改变。

以分子键长R_N和R_O为依据的转动惯量是

where M is the total mass of the molecule. The moments of inertia of the two isotopic species can be obtained from the following spectral measurements.

其中M是分子的总质量。两种同位素物种的转动惯量可通过以下光谱测量获得。

将质量值代入转动惯量方程

When moment of inertia values are inserted and the pair of equations factored to eliminate the R_NR_O term, it turns out that the R_N² is negligible. The resulting equation gives a value for R_O.

当转动惯量值被代入，并将两组方程因式分解以消去R N R O项时，发现R N 2可以忽略不计。所得方程给出了R O 的值。

Substituting this back into the moment of inertia equation for the ¹⁴N¹⁴NO species gives a quadratic equation for the bond length R_N.

将此结果代入14N 14O物种的转动惯量方程，得到一个关于键长R_N的二次方程。

This algebraic solution for the structural parameters of a molecule shows the utility of rotational spectra for determining molecular structure. In general, you will need as many equations as there are unknown parameters in the structure. Additional equations can be obtained by measuring the spectra of the same molecule with different isotopes substituted.

这种分子结构参数的代数解表明旋转光谱在确定分子结构方面的效用。通常来说，你需要的方程数量应与结构中未知参数的数量相等。可以通过测量同一分子的不同同位素取代的光谱来获得额外的方程。