We're starting to develop some calculations of the hyperfine parameters (electric field gradient, spin-rotation coupling, spin-spin coupling) with CFOUR, and we reached a limitation of the public version, which is the impossibility to get the parameters for levels other than the ground vibrational state.

We wish to extend this study to v=1-4 of a closed shell "light" diatomic molecules, as well to triatomic molecules (still at relatively low energy).

Which package(s) could you recommend?

Possibly, the rotational constant, vibrational energies and transition intensities will be calculated. Open-source or free packages are a plus.

  • $\begingroup$ In addition to my answer below, the ADF package was used to calculate hyperfine parameters for diatomics in this paper and Eq. 9 of that paper explains how they got the spin-rotation coupling constant. Also spin-spin and spin-rotation constants were calculated (again for diatomics) in this paper but there doesn't seem to be any citation to any software, which suggests that everything was done (including MCSCF calculations!) with home-built software written by this single author (wow!). $\endgroup$ Commented Jun 17, 2021 at 21:37

1 Answer 1



Given a well-behaved electronic potential energy curve for a diatomic molecule, LEVEL will "automatically locate and calculate expectation values for all vibration–rotation levels". This means that for any operator $\hat{M}$, you can calculate the expectation values for each vibrational level $v$ and each associated rotational level $J$:

$$ \langle \psi_{v,J}| \hat{M}|\psi_{v,J}\rangle, \tag{1} $$

which means that you can certainly calculate properties for $v=1-4$ with $J=0$ (as well as also with higher $J$ values). This program was a project that Bob Le Roy worked on since the 1960s, and had contributions from several others over the decades, including myself. The latest version available online was from 2016, so I pushed the most recent version I could find today into Github Here (let me know if you have any trouble compiling it, while I work on adding a Makefile and sample input file for you).


  • Free and open-source
  • Extremely accurate, and extremely fast (orders of magnitude faster and more black-box than other radial Schroedinger solvers I know some others have used)
  • Widely used and with a rich history spanning several decades
  • There's not a lot of other programs available with this type of functionality


  • Only works for diatomics
  • As you are a CFOUR user, you might be used to getting ground state vibrational properties in a much more black-box way. Here you'd have to build the electronic potential energy curve with a program like CFOUR, then enter it into LEVEL and climb a slight learning curve as you need to check convergence and such more manually in LEVEL than you would for single-point energy calculations (within a basis set and level of theory) in CFOUR.
  • You may have to define $\hat{M}$ yourself in the FORTRAN code if the particular $\hat{M}$ that interests you is not available by default (since I've been using this software since about 15 years ago, I can probably help you with this if you want to collaborate).

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