# Predicting magnetic moment of a metal complex computationally

Recently, for entrance exams, I have been dealing with a lot of weird compounds. One of them is $$\ce{K3[Mn(CN)6]}$$.

The objective was to predict the magnetic moment of the above complex.
I predict that the number of unpaired electrons in the metal is 2 which straight away implies that the magnetic moment must be 2.8 but since it is a mere speculation considering the strong-ligand $$\ce{CN^-}$$ I would like to confirm it by modelling the compound. The data for the given complex is not available/accessible for me.

An article available online says to calculate the average of Gross Orbital Population (in Gaussian09). Trying it out gave an output of 1.

Hence, I would like to know how do I find the magnetic moment computationally (maybe using Gaussian, Gamess, Terachem, OCRA, etc.).

As a sidenote, since NMR uses magnetic field, can it be useful for computing magnetic moment?

• +1. Welcome to the site, and thanks for bringing the question here !!! We hope to see much more of you!! Commented Sep 22, 2020 at 3:46
• Your value of 2.8 seems to only be based on the # of unpaired electrons, but according to this it also will have a nuclear component. Commented Sep 23, 2020 at 1:04
• @NikeDattani Yes, I remember someone on researchgate telling that it is molecular magnetic moment but their wording was not clear. Should I clear in my question that I want spin only? I am asking so because I do not think it will be possible to seperate it out and that the spin only magnetic moment will be dominant (octahedral symmetry should help?) ---------This is just a speculation Commented Sep 23, 2020 at 3:10
• @NikeDattani magnetic properties are dominated by the electrons since the Bohr magneton is way bigger than the nuclear magneton by a ratio of m_e/m_p ~ 1826, which is the reason why we can also use the Born-Oppenheimer principle. Commented Sep 23, 2020 at 7:12

The main question here is whether the question makes sense for $${\rm K}_3[{\rm Mn(CN)}_6]$$ as a molecular complex.

Looks like the material has a solid state structure https://materials.springer.com/isp/crystallographic/docs/sd_1100190 which you could with solid state methods. As a complex it's not obvious where the potassiums would go, so you would probably start out by getting rid of the potassiums in the +1 charge state to get $$[{\rm Mn(CN)}_6]^{3-}$$. If you further assumed the $${\rm CN}$$ to be $${\rm CN}^-$$, you would get a +3 oxidation state for $${\rm Mn}$$, and two unpaired electrons, see e.g. https://www.quora.com/What-is-the-hybridisation-of-Mn-CN-6-3; but your question was about a computational value.

You could get the magnetic moment on the metal by running some calculations on the $$[{\rm Mn(CN)}_6]^{3-}$$ complex. However, this is not nearly as it sounds like: transition metal complexes are often challenging due to near-degeneracies, which means that the numbers you get from your calculations may be complete garbage (I'm not sure whether this is the case for this complex): density functional approximations may be unreliable, whereas wave function methods may require a very fine balance between static and dynamic correlation to make the computed ground state match the experimental one. Setting up the necessary model and making sure that the computed number is converged with respect to all parameters is a huge hassle.

This is an interesting question, but I fear it may be quite a lot of work to get a reliable answer from theory!

• The linked website for crystallographics is a bit restrictive for me because I do not have a licence yet. Also, the Quora link is broken (although I know the hybridisation and how to find it). You could get the magnetic moment on the metal by running some calculations .What are the calculations you are referring to, and how to run them? (Note: It will also be helpful if I get to know how to determine mag.moment for molecules like $O_2$ etc.. Commented Sep 23, 2020 at 7:46
• Any standard electronic structure method: Hartree-Fock, DFT, coupled-cluster, MCSCF, etc. The problem is that these are not usually simple for transition metal complexes: the results are wrong since you're missing out important physics of the problem. In contrast, O2 is almost trivial: you get the experimental triplet ground state even at Hartree-Fock level; it is about 2.3 eV below the singlet. Commented Sep 23, 2020 at 9:19
• Okay, so practically using a computer, it looks impossible with current methods(i.e. will take a large amount of time and computation). And what you said about O2...Do I need to output all HF calculations and compute the total of "gross orbital population"? (same as the suggested link in the question's body) Commented Sep 23, 2020 at 9:30
• Yes: it's simple to get some number, but it's a wholly different question what is the right number. For the O2 calculation you would just run the triplet state and look at the spin population. Commented Sep 24, 2020 at 10:16

While this is an old question, let me chip in: These cyanides are essentially a potassium salt of the metal cyanide complex, so you can focus on the $$\ce{[Mn(III)(CN)_6]^{3-}}$$ complex. As for magnetism, coordination complexes can have a spin and orbital contribution to their moment. In octahedral field, simply from the placement of d electrons (low spin state, due to the strong ligand field), you can have an educated guess about the spin state and the expected orbital degeneracy. If there is no orbital degeneracy, spin only momentum is your best guess for magnetic moment, and no fancy calculation will give you any more accurate.

If there is an orbital degeneracy, technically speaking you should calculate that contribution, however here is a catch: in reality, this orbital contribution is often quenched (i.e. very little or zero, due to deformation of the complex). So the educated guesses in these situation a close-to zero orbital contribution (with the exception of Co(II) complexes, which are often close to fully unquenched orbital contribution). If there is a partial quenching, you are in big trouble for several reasons:

1. the actual quenching is most probably sensitive to the environment of the complex ion, what you can or cannot model,
2. if your workhorse calculation method is DFT, the wavefunction can represent only $$L=0$$ states, so it will be hard to get accurate information about orbital multiplicity of the state.

If you have a good understanding of the geometry (eg you have X-ray) or you have a more rigid ligands around the metal, you can build a Ligand Field model to estimate $$L$$, but in most cases it is not really accurate.