10.4.5: The Magnitude of Parameters eσ, eπ and Δ

In the angular overlap model (AOM), the field splitting, \(Δ\), can be shown to result from a combination of the interaction parameters eσ and eπ. For example, in octahedral geometry,

\[Δ = 3eσ - 4eπ \label{eq1} \]

in which the value of eπ is positive for a pi donor but negative for a pi acceptor.

The magnitude of these parameters varies from one complex to another. As we have seen previously in examination of ligand field theory, both the metal ion and the identity of the ligand play a role in determining the magnitude of the field splitting. As outlined in the previous section on ligand field theory, the identity of the metal influences the d orbital splitting. Even for the same element, an increase in charge translates into a significant increase in the splitting parameter. For this reason, aqueous \(\ce{Co(II)}\) ion is high spin, whereas the aqueous \(\ce{Co(III)}\) ion is low spin.

The spectrochemical series tells us that the ligands also play a major role in determining the magnitude of the field splitting, with pi donors producing a smaller splitting than pi acceptors. The splitting produced by an aquo ligand is much smaller than the produced by a cyano ligand, for example.

Spectroscopic data can be used to determine the parameters eσ and eπ. A series of examples for different ligands coordinated to the \(\ce{Cr^{3+}}\) ion are outlines in Table \(\PageIndex{1}\).

Table \(\PageIndex{1}\): Angular Overlap Parameters in Octahedral Cr(III) Complexes¹

Ligand	eσ	eπ	Δ (cm-1)
π acceptors
-CN	7,530	-930	26,310
pyridine	6,150	-330	19,770
σ donors
en	7,260	assumed 0	21,780
NH3	7,180	assumed 0	21,540
π donors
HO-	8,600	2,150	17,200
H2O	7,550	1,850	15,250
F-	8,200	2,000	16,600
Cl-	5,700	980	13,180
Br-	5,380	950	12,430
I-	4,100	670	9,620

It is worth noting that the parameters observed for Cr(III), although mostly consistent with the spectrochemical series, are not identical to those seen in Co(III). In particular, the places of water and hydroxide are switched in the series. Once again, the identity of the metal plays a role in the strength of interaction with the ligand, and certain ligands may be observed to interact more strongly with some metals than with others.

Example \(\PageIndex{1}\)

Calculate Δ for tetrahedral \(\ce{Ni^{2+}}\) when coordinated with the following ligands, given the parameters eσ and eπ.¹

1. \(\ce{PPh3}\) with \(eσ = 5,000 \, \text{cm}^{-1}\) and \(eπ = -1,750 \, \text{cm}^{-1}\)
2. \(\ce{Cl^{-}}\) with \(eσ = 3,900\, \text{cm}^{-1}\) and \(eπ = 1,500 \, \text{cm}^{-1}\)
3. \(\ce{Br^{-}}\) with \(eσ = 3,600\, \text{cm}^{-1}\) and \(eπ = -1,000 \, \text{cm}^{-1}\)

Solution

This question is a direct application of Equation \ref{eq1}:

1. \(\ce{PPh3}\): \[\begin{align*} Δ &= 3eσ - 4eπ \\[4pt] &= 3(5,000) - 4(-1,750) cm^{-1} \\[4pt] &= 22,000\, \text{cm}^{-1}.\end{align*} \nonumber \]
2. \(\ce{Cl^{-}}\): \[\begin{align*}Δ &= 3(3,900) - 4(1,500) \\[4pt] &= 5,700 \, \text{cm}^{-1}.\end{align*} \nonumber \]
3. \(\ce{Br^{-}}\): \[\begin{align*}Δ &= 3(3,600) - 4(1,000) \\[4pt] &= 6,800 \, \text{cm}^{-1}.\end{align*} \nonumber \]

Example \(\PageIndex{2}\)

AOM parameters eσ were calculated from spectroscopic data for a series of bidentate nickel complexes with octahedral geometry, \(\ce{Ni(en')2(NCS)2}\).²

• en' = \(\ce{H2NCH2CH2NH2}\) with eσ = 4,010 cm-1
• en' = \(\ce{(CH3)2NCH2CH2N(CH3)2}\) with eσ = 3,165 cm-1
• en' = \(\ce{(CH3CH2)2NCH2CH2NH2}\) with eσ = 2,485 cm-1 (\(\ce{-NEt2}\)); eσ = 4,650 cm-1 (\(\ce{-NH2}\))

Explain the reasons for the differences in the eσ values.

Solution

Greater steric hindrance appears to decrease the interaction between the sigma donor and the metal. Thus, the magnitude of eσ decreases from the least hindered \(\ce{-NH2}\) group to the most hindered \(\ce{NEt2}\) group.

References

1. Figgis, B. N.; Hitchman, M. A. Ligand Field Theory and Its Applications. Wiley-VCH: New York, 2000, p. 71.
2. Lever, A. B. P.; Walker, I. M.; McCarthy, P. J.; Mertes, K. B.; Jircitano, A.; Sheldon, R. "Crystallographic and Spectroscopic Studies of Low-Symmetry Nickel(II) Complexes Possessing Ling Nickel-Nitrogen Bonds", Inorg. Chem. 1983, 22, 2252-8.