Translation:On the theory of the spectra of diatomic molecules

by Lev Landau

612

On the theory of the spectra of diatomic molecules

Von L. Landau in Leningrad

Received on 13 November 1926

 

The model of the diatomic molecule as a rotator with internal momentum is treated by means of the new quantum mechanics. The familiar band theory is obtained for the frequencies. All intensities are calculated. The behaviour of the molecule in electric and magnetic fields (Stark and Zeeman effects in the bands) has been examined.

 

The Hamiltonian for a diatomic molecule is

(1)$H=U+{\frac {1}{2m}}\sum {\boldsymbol {p}}^{2}+{\frac {1}{2M_{1}}}{\boldsymbol {P}}_{1}^{2}+{\frac {1}{2M_{2}}}{\boldsymbol {P}}_{2}^{2},$^[1]

where the small letters refer to electrons and the capital letters to the nuclei; $U$ denotes the potential energy.

In order to separate the translatory motion of the molecule as a whole, we make the following transformation of coordinates:

(2)${\overline {\boldsymbol {r}}}={\boldsymbol {r}}-{\frac {M_{1}{\boldsymbol {R}}_{1}+M_{2}{\boldsymbol {R}}_{2}}{M_{1}+M_{2}}},\quad {\boldsymbol {R}}={\boldsymbol {R}}_{2}-{\boldsymbol {R}}_{1},\quad {\boldsymbol {C}}={\frac {m\sum {\boldsymbol {r}}+M_{1}{\boldsymbol {R}}_{1}+M_{2}{\boldsymbol {R}}_{2}}{\sum m+M_{1}+M_{2}}};$

the new coordinates of the electrons are their vector distances from the centre of mass of the nuclei, ${\boldsymbol {R}}$ is the distance between the nuclei, and ${C}$ is the radius vector of the centre of mass of the molecule. After this transformation the Hamiltonian becomes

(3)$H=U+{\frac {1}{2m}}\sum {\overline {\boldsymbol {p}}}^{2}+{\frac {1}{2(M_{1}+M_{2})}}\left(\sum {\overline {\boldsymbol {p}}}\right)^{2}+{\frac {1}{2}}\left({\frac {1}{M_{1}}}+{\frac {1}{M_{2}}}\right){\boldsymbol {P}}^{2}+{\frac {1}{2(\sum m+M_{1}+M_{2})}}\gamma ^{2}$

where $\gamma$ is the momentum corresponding to the coordinate ${\boldsymbol {C}}$. It is seen that only the last term in equation (3) relates to the translatory motion; it is therefore of no further interest here, and we shall ignore it. We shall also omit the bar from ${\boldsymbol {r}}$ and ${\boldsymbol {p}}$. The angular momentum is then

(4)${\boldsymbol {M}}=\sum {\boldsymbol {r}}\times {\boldsymbol {p}}+{\boldsymbol {R}}\times {\boldsymbol {P}}.$

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1. All coordinates and momenta which occur are treated as matrices.