Explanation of the spectrochemical series of transition metal ions

[HAYAO]

TL;DR Summary

Is there a crystal-field/ligand-field theory explanation to the spectrochemical series of transition metal ions within a period?

The spectrochemical series of metals, under the circumstances that same ligands are used and that it is in an octahedral coordination, is given by:
Mn²⁺ < Ni²⁺ < Co²⁺ < Fe²⁺ < V²⁺ < Fe³⁺ < V³⁺ < Co³⁺ < Mn⁴⁺ < Mo³⁺ < Rh³⁺ < Ru³⁺ < Pd⁴⁺ < Ir³⁺ < Pt⁴⁺

When I was skimming through a textbook to teach my lab members spectrochemical series, I was stuck about the ordering of the metals. Textbooks usually state that crystal-field increases in strength with increasing oxidation number, and that it also increases down a group in the periodic table. I can understand this since the former decreases the ionic radii, and the latter increases the distribution of the d-orbital, allowing ligands to interact stronger (=stronger crystal-field).

Now, I know that crystal-field theory or ligand-field theory doesn't fully explain coordination chemistry, but there are something I definitely have problem understanding.

For example, how would the metals within the same period with same oxidation be explained? I can understand that 4th period divalent metal ions will often have high-spin electron configuration. In that sense, Mn²⁺ and beyond should have electrons in the anti-bonding e_g orbitals, which will decrease the interaction between the metal and the ligand, lowering the crystal-field. However, I can't explain Ni²⁺ < Co²⁺ < Fe²⁺. Unless the t_2g orbitals are increasing in orbital energy with increasing atomic number, it should be the other way around because ionic radius should decrease with increasing atomic number.

Some of the elements also seem to not follow the trends of oxidation, such as Mn⁴⁺ and Pd⁴⁺. What would be the factor? Is it something that can be explained by crystal-field/ligand-field theory or not? If not, then that's fine. I'm just wondering for educational purposes.

 

[TeethWhitener]

I’m not sure there’s a simple explanation within crystal field theory of the trend. As you note, ionic radius decreases with increasing atomic number, but electron-electron repulsion within the d-orbitals increases with increasing atomic number, and that is a main driver of the ligand field splitting parameter—the Racah B parameter in particular. It’s the balance of those two effects (ionic radius and electron-electron repulsion) that will ultimately give you the splitting energy which actually determines the order of the spectrochemical series. And I’m not sure that there’s a simple way to intuit that balance without actually calculating the Coulomb and exchange integrals.

 

[HAYAO]

I’m not sure there’s a simple explanation within crystal field theory of the trend. As you note, ionic radius decreases with increasing atomic number, but electron-electron repulsion within the d-orbitals increases with increasing atomic number, and that is a main driver of the ligand field splitting parameter—the Racah B parameter in particular. It’s the balance of those two effects (ionic radius and electron-electron repulsion) that will ultimately give you the splitting energy which actually determines the order of the spectrochemical series. And I’m not sure that there’s a simple way to intuit that balance without actually calculating the Coulomb and exchange integrals.

Yes, the nephelauxetic effect was what I speculated for Ni²⁺ < Co²⁺ < Fe²⁺, but I wasn't confident enough to say that. But I also thought that I would need the benefit of hindsight to be able to explain the series, because it would be a balance with the ionic radius. Even things like ionic radius requires hindsight.

I never really liked the crystal-field theory because sometimes the explanations contains chicken or the egg problem.

Thank you for your insight.

 

[TeethWhitener]

When I replied the first time, I wasn't in my office, so I didn't have access to all my books to give you a really full answer. There's a fantastic book called "Concepts and Models of Inorganic Chemistry" by Douglas, McDaniel, and Alexander that goes over a lot of this stuff in minute detail. Having skimmed a few of the relevant sections, I think I can probably give a little more information.

I'm not quite sure it's so much the nephelauxetic effect as it is the spin-pairing energy. The nephelauxetic effect is really an observation of how the interelectron repulsion changes in going from the free metal ion to a complex. As a result, the magnitude of the nephelauxetic effect is very much dependent on how much the formation of covalent bonds delocalizes the electrons around the metal--therefore, it is very ligand-dependent. The spin-pairing energy, on the other hand, directly impacts Dq (the field splitting energy) of the metal, and is dictated by the exchange integral. Fe²⁺ is d⁶, Co²⁺ is d⁷, and Ni²⁺ is d⁸, and intuitively, the exchange integral is going to increase in the order d⁸ < d⁷ < d⁶ (since the number of term symbols/degeneracy of states increases in this order). So the spin pairing energy is going to be larger for Fe²⁺ than Co²⁺, and the trend will continue for Ni²⁺. The increase in spin pairing energy will push the metal farther up the spectrochemical series. It also (at least heuristically) explains the observation that Fe(NH₃)₆²⁺ is high-spin and Co(NH₃)₆²⁺ is low-spin.

 

[HAYAO]

Thanks for the advice. I'll look into it, when I get to the library of my college.

I got nephelauxetic effect with spin-pairing confused. Yeah, you are right. Nephelauxetic effect comes AFTER coordination with ligands and it's ligand dependent. Sorry about that.

But I'm still a bit confused. I thought that the field-splitting energy and spin-pairing energy were two separate effects. My understanding is that the balance between the spin-pairing energy and crystal-field splitting energy is what causes low- and high- spin complexes. Also, the spin-pairing energy is the sum of Coulomb repulsion (which raises energy) and exchange (which lowers energy), with the former being a lot bigger effect than the latter.
For example, Fe(OH)₆ complex (Fe²⁺ is d⁶) has a crystal field splitting Dq of 9350 cm^-1, Coulomb repulsion energy of 19600 cm^-1, and exchange energy of -2000 cm^-1.
At low spin, LFSE is -22400 cm^-1, Coulomb repulsion is 3x19600 cm^-1, exchange is 6x(-2000) cm^-1, giving a total of 24360 cm^-1.
At high spin, LFSE is -3740 cm^-1, Coulomb repulsion is 1x19600 cm^-1, exchange is 4x(-2000) cm^-1, giving a total of 7860 cm^-1.
This is why Fe(OH)₆ complex is high spin. Or at least, this was the line of argument of how low- and high-spin complexes are determined, back when I studied inorganic chemistry.

The reason I wasn't confident with the electron repulsion being the reason why the spectrochemical series Ni²⁺ < Co²⁺ < Fe²⁺ was because it was in conflict with the argument above that crystal-field splitting and spin-pairing energy is a two different thing.

 

[TeethWhitener]

But I'm still a bit confused. I thought that the field-splitting energy and spin-pairing energy were two separate effects.

You're right. I'm not sure exactly what I was thinking when I wrote that, but you can probably ignore it.

The reason I wasn't confident with the electron repulsion being the reason why the spectrochemical series Ni2+ < Co2+ < Fe2+ was because it was in conflict with the argument above that crystal-field splitting and spin-pairing energy is a two different thing.

I don't know if they're in conflict. The spectrochemical series is an empirically observed trend, whereas crystal field theory is just a model. But yes, I think the best explanation is probably that electron repulsion dominates the trend.

 

[HAYAO]

I see, thank you very much for your insight. I'm a bit more confident with the explanation now.