Question

Explain why in high-spin octahedral complexes, orbital contributions to the magnetic moment are only important for \(d^{1}, d^{2}, d^{6}\) and \(d^{7}\) configurations.

Short answer

High-spin octahedral complexes with \(d^1, d^2, d^6,\) and \(d^7\) configurations have significant orbital contributions due to unpaired electrons in degenerate orbitals facilitating orbital mixing.

Step-by-step solution

Understanding High-Spin Octahedral Complexes

In high-spin octahedral complexes, the crystal field splitting energy (_{ ext{cf}}) is relatively low, causing electrons to occupy higher energy orbitals rather than pair up in lower ones. This results in a maximum number of unpaired electrons, corresponding to the spin state of the complex.

Analyzing Orbital Contributions

The magnetic properties of a complex are determined by both spin and orbital contributions. While spin contributions are straightforward and arise from unpaired electrons, orbital contributions require the presence of specific degenerate orbitals that can undergo mixing. These conditions make orbital contributions significant only for certain electronic configurations.

Identifying Degenerate Orbitals

Degenerate orbitals are those of identical energy levels. For configurations like \(d^1\), \(d^2\), \(d^6\), and \(d^7\), the electronic arrangements allow \(t_{2g}\) and \(e_g\) orbitals to participate in interactions that lead to orbital contributions to the magnetic moment. This arises because the electrons in these configurations can occupy different orbitals of the same energy, enabling movement and orbital mixing.

Explaining the Specific Configurations

For \(d^1\) and \(d^2\), the presence of one or two electrons in degenerate \(t_{2g}\) orbitals allows for maximum orbital movement. In \(d^6\) and \(d^7\) configurations, a similar scenario arises, where backfilling to nearly full \(t_{2g}\) orbitals and having partially filled \(e_g\) orbitals contributes to significant orbital participation in the magnetic properties.

Concluding Why Other Configurations Lack Orbital Contributions

For other electron counts (^{3-5}, ^{8-10}), the orbital contributions are minimized either due to lack of unpaired spinning in the case of completely filled \(t_{2g}\) orbitals or absence of electron movement between degenerate states, as they reach stability. Thus, their magnetic moments are mostly attributable to spin contributions.

Key concepts

Orbital Contributions

In high-spin octahedral complexes, the magnetic moment is influenced not only by the spin of unpaired electrons but also by how these electrons occupy degenerate orbitals, leading to what we call orbital contributions. These orbital contributions arise when unpaired electrons occupy orbitals that can potentially undergo mixing due to their similar energy levels.

- **Degenerate Orbitals:** These are orbitals that have the same energy level. In these complexes, the ability of electrons to freely move and interact within these orbitals enhances magnetic properties via orbital contributions. - **Importance of Different Configurations:** For specific electron configurations like \(d^1\), \(d^2\), \(d^6\), and \(d^7\), orbital contributions are significant because they have available degenerate orbitals that can participate in electron movement. This is due to having less parallel orientation, allowing for electron mixing across orbitals of the same energy.

If there are no suitable degenerate orbitals, as seen in other configurations, the orbital contributions to the magnetic moment become negligible.

Magnetic Moment

The magnetic moment of a high-spin octahedral complex represents the magnetic strength and orientation resulting from both spin and orbital contributions. This value essentially quantifies how strongly the complex is magnetized.

### Spin Contributions Spin contributions to the magnetic moment are relatively straightforward. They originate from unpaired electrons, calculated by the number of these electrons in the complex. More unpaired electrons mean a higher magnetic moment due to spin contributions.

### Orbital Contributions However, orbital contributions add an additional dimension. While spin contribution is derived from just counting unpaired electrons, orbital contributions depend on the electron's capacity to move within degenerate orbitals. This happens because such movements create additional magnetic effects, which enhance the total magnetic moment.

Thus, not only does the number of unpaired electrons matter, but also where and how these electrons are arranged in the orbitals, affecting the total magnetic response of the complex.

Crystal Field Splitting

Crystal field splitting is crucial in understanding high-spin octahedral complexes, as it dictates how an octahedral field affects the electron distribution in the d-orbitals. This concept involves the separation of d-orbitals into two sets of energy levels when ligands approach the metal ion.

### How It Works In an octahedral complex, the incoming ligands cause d-orbitals to split into different energy levels; \(e_g\) and \(t_{2g}\). The degree of splitting, denoted as \(\Delta_{cf}\), determines whether electrons will pair up in lower energy \(t_{2g}\) orbitals or occupy higher energy \(e_g\) orbitals.

### High-Spin vs. Low-Spin In a high-spin complex, \(\Delta_{cf}\) is smaller, which means electrons prefer to remain unpaired in higher orbitals rather than pair up in the lower ones. This results in a maximum number of unpaired electrons, affecting the spin state of the complex and contributing to the overall magnetic properties. Conversely, in low-spin complexes, \(\Delta_{cf}\) is large, so electrons pair up in lower orbitals first, resulting in fewer unpaired electrons.

Electron Configurations

The electron configurations of the metal ions in high-spin octahedral complexes significantly influence their properties, particularly magnetic behavior and the role of orbital contributions. Understanding these configurations helps in predicting both the magnetic and chemical nature of the complex.

### Impact of Electron Configuration - **\(d^1\) and \(d^2\):** In these configurations, electrons reside in the \(t_{2g}\) set, allowing for unpaired electron movement within degenerate levels, enhancing orbital contribution.- **\(d^6\) and \(d^7\):** These configurations allow for a mix between \(t_{2g}\) and \(e_g\) orbitals with the potential for movement, particularly when \(t_{2g}\) is nearly filled or \(e_g\) begins to fill.
### Limited Effect in Other Configurations Configurations like \(d^3-d^5\) or \(d^8-d^{10}\) witness minimal orbital contributions. In these states, electrons fill orbitals in a way that restricts such movement, thereby reducing the effect of orbital contributions. This distinction is vital for predicting the magnetic moment through electron configuration analysis.