(a) Sketch a diagram that shows the definition of the crystal-field splitting energy \((\Delta)\) for an octahedral crystal field. (b) What is the relationship between the magnitude of \(\Delta\) and the energy of the d- \(d\) transition for a \(d^{1}\) complex? (c) Calculate \(\Delta\) in \(\mathrm{kJ} / \mathrm{mol}\) if a \(d^{1}\) complex has an absorption maximum at \(545 \mathrm{nm}\).

When delving into the concept of crystal-field theory, an essential model used to understand how metal ions interact with surrounding ligands, we come across the octahedral crystal field. This is a common coordination environment for transition metal complexes where six ligands symmetrically surround a central metal ion, much like the vertices of an octahedron.

In an octahedral field, the energy levels of the five d-orbitals in a free ion split into two distinct groups due to electrostatic interactions with the approaching ligands. Imagine lining up magnets in a specific pattern and observing how a central magnet responds to their placement–similar principles are at work here on a subatomic scale.

The three d-orbitals that become the \(t_{2g}\) set are lower in energy because they are directed between the ligands, resulting in less electrostatic repulsion. Meanwhile, the two d-orbitals that form the \(e_{g}\) set have higher energy as they point directly at the ligands. The energy gap between these two sets of orbitals is termed the crystal-field splitting energy (\(\Delta\)). It's this energy difference that determines many properties of the complex, including its color and magnetic behaviors.

Understanding d-d transitions is crucial as they are responsible for the vivid colors often associated with transition metal complexes. A d-d transition involves an electron jumping from a lower-energy \(t_{2g}\) orbital to a higher-energy \(e_{g}\) orbital within the same d-shell of a transition metal ion.

Think of this process as a tiny energetic leap within an atom. It only occurs when the complex absorbs just the right amount of energy, which typically falls in the visible spectrum for many complexes. The precise amount of energy absorbed corresponds to the difference in energy between \(t_{2g}\) and \(e_{g}\), which is the crystal-field splitting energy, \(\Delta\).

For example, in a \(d^1\) complex, where the metal has a single d-electron, this electron resides in the lower energy \(t_{2g}\) set. When the complex absorbs light, this electron absorbs energy and transitions to the vacant \(e_{g}\) set. So, distinctly, the relationship between the energy of the incoming light and the \(\Delta\) is direct–if you know the energy required for the transition, you can deduce the energy gap \(\Delta\), and vice versa.

Planck's equation, a cornerstone of quantum mechanics, provides the crucial link between energy and wavelength for electromagnetic radiation. It is articulated through the relationship \(E = \frac{hc}{\lambda}\), where \(E\) is the energy of a photon, \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength.

This equation allows us to calculate the energy associated with a photon of a specific wavelength. It's this fundamental understanding that permits scientists to derivate characteristics of atoms and molecules just from knowing the spectral lines they emit or absorb. In the context of a \(d^1\) complex where the wavelength of absorbed light is known (say, 545 nm from our exercise), we can use Planck’s equation to calculate the energy of the d-d transition. That energy, converted to appropriate units such as kilojoules per mole, is directly related to the crystal-field splitting energy \(\Delta\). Thus Planck's equation serves as a bridge that relates the observed color of a complex with the microscopic interactions within the atom.