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I'm a beginner in spectroscopy and I have troubles understanding papers describing values of spectroscopic constants. This paper, for example, describes constants $T_e, R_e, \omega_e, \omega_ex_e, B_e, \alpha_e, D_0$ and $D_e$.

I can see, that $R_e$ is an internuclear distance, but I have no idea, what are the remaining constants.

Could you provide some brief explanation? Preferably with picture of some potential energy curve(s), if possible, as I find it much easier to understand such concepts from schemes.

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You can find plots and further explanations online (as I easily did). This is a homework-type question, yet I do feel that the information request for diatomic molecular spectroscopic constants warrants at least this partial answer. See this writeup for the "Vibronic Absorption Spectrum of Molecular Iodine" for more details and illustrations.

From NIST:

T$_e$ minimum electronic energy (cm$^{-1}$)

R$_e$ internuclear distance (Å)

ω$_e$ vibrational constant – first term (cm$^{-1}$)

ω$_e$x$_e$ vibrational constant – second term (cm$^{-1}$)

B$_e$ rotational constant in equilibrium position (cm$^{-1}$)

α$_e$ rotational constant – first term (cm$^{-1}$)

D$_e$ centrifugal distortion constant (cm$^{-1}$)

β$_e$ rotational constant – first term, centrifugal force (cm$^{-1}$)

From Wikipedia:

D$_0$ dissociation energy (cm$^{-1}$)

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The subscript $e$ means measurements relating to the minimum of the internuclear separation, i.e. at the bottom of the potential energy. Many textbooks do not use the subscript $e$ at all but the meaning is the same.

The $D_0$ by contrast is the dissociation energy measured from the lowest vibrational energy level ($n=0$), some books also quote $D_e$ which is the energy from the lowest point on the potential energy and so this is $D_0$ plus the zero point energy. However, $D_e$ may also be the centrifugal distortion constant and has the same units as $B_e$. The context in which it is used will determine what it means, i.e. if the use is in determining the rotational energy $D_e$ (or just $D$) will be the centrifugal distortion constant.

The term $x_e$ is the anharmonicity constant and is dimensionless and $\alpha_e$ is the vibration-rotation coupling constant, and has the same units as $B_e$. (It describes the fact that as the molecule has a different bond length in vibrational level $n=1$ than in $n=0$, and that in $n=2$ from $n=1$ etc, consequently the rotational constant $B$ has to be modified). The other terms are described in the first answer.

However, there is a wrinkle, the fact that $\omega$ is used and not $v$ for frequency means that angular frequencies are being used. They are related as $\omega =2\pi v$. The frequency $\omega $ has units $\mathrm{radian\,s ^{-1}}$ and $v$ units of $\mathrm{\,s^{-1}}$. Some books use $v$ some $\omega$; it is really quite messy you just have to be careful.

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