Atomic orbitals describe the three-dimensional regions around an atom’s nucleus where an electron is most likely to be found. Each orbital represents a distinct quantum state for an electron, possessing a specific size, shape, and orientation in space. The energy of an electron is linked to the orbital it occupies. Atomic structure includes groups of orbitals that share the exact same energy, a phenomenon known as orbital degeneracy.
Defining Orbital Energy and Degeneracy
A set of degenerate orbitals consists of two or more distinct atomic orbitals that possess precisely the same energy level. In the theoretical case of a single-electron atom, such as hydrogen, the electron’s energy is determined solely by the principal quantum number, designated as \(n\). The principal quantum number relates to the size of the orbital and the electron’s average distance from the nucleus. For example, the \(2s\) orbital and the three \(2p\) orbitals all have the same energy because they share the principal quantum number \(n=2\). This high degree of degeneracy is a consequence of the simple electrostatic attraction between the single electron and the nucleus, with no other electron interactions to consider.
The quantum mechanical description of an orbital relies on three numbers: the principal quantum number (\(n\)), the angular momentum quantum number (\(l\)), and the magnetic quantum number (\(m_l\)). The angular momentum quantum number (\(l\)) defines the shape of the orbital, such as \(s\), \(p\), \(d\), or \(f\). The magnetic quantum number (\(m_l\)) then specifies the orientation of the orbital in three-dimensional space. Degeneracy arises when different combinations of the quantum numbers, specifically different \(m_l\) values, correspond to the same energy value. The total number of orbitals in a set that share the same energy is referred to as the degree of degeneracy.
Identifying Degenerate Sets in Atoms
The concept of degeneracy changes significantly when moving from the simple hydrogen atom to multi-electron atoms. In multi-electron atoms, the energy is no longer dependent only on \(n\); it is also dependent on the angular momentum quantum number \(l\) due to electron-electron repulsion and shielding effects. This effect lifts the degeneracy between subshells, meaning the \(2s\) and \(2p\) orbitals are no longer equal in energy, with \(2s\) being lower than \(2p\).
However, within a specific subshell, the degeneracy remains, provided the atom is isolated and experiences no external field. The \(p\) subshell, for example, is composed of three orbitals (\(p_x, p_y, p_z\)) that are identical in shape but are oriented along the three axes of a coordinate system. Since an isolated atom possesses spherical symmetry, all directions in space are equivalent, meaning the three \(p\) orbitals must have the same energy. The \(d\) subshell similarly consists of five orbitals, denoted as \(d_{xy}, d_{xz}, d_{yz}, d_{x^2-y^2},\) and \(d_{z^2}\), and they form a five-fold degenerate set. The \(f\) subshell contains seven orbitals that are all degenerate.
Mechanisms for Removing Orbital Degeneracy
The degeneracy observed within a subshell is often a theoretical state that disappears when the atom interacts with its environment, a process known as energy splitting. External forces can break the spatial symmetry of the atom, causing the \(m_l\) states to acquire different energies. The Zeeman effect, for instance, involves applying an external magnetic field, which interacts with the electron’s magnetic moment and causes the degenerate set to split into distinct energy levels based on their \(m_l\) values. Similarly, the Stark effect involves the application of an external electric field, also leading to the splitting of the degenerate orbitals.
The most chemically significant mechanism for removing degeneracy occurs when an atom, particularly a transition metal ion, enters a molecular environment. In coordination complexes, the surrounding molecules or ions, known as ligands, generate a non-spherical electric field around the central metal ion. This interaction is described by Crystal Field Theory, which treats the ligands as point charges. In a common geometry, such as an octahedral complex, the five \(d\) orbitals split into two distinct energy groups. The \(d\) orbitals that point directly toward the ligands (\(d_{x^2-y^2}\) and \(d_{z^2}\)) experience greater electrostatic repulsion, raising their energy to form a higher-energy set, while the remaining three \(d\) orbitals, which point between the ligands, form a lower-energy set. The resulting energy difference is termed the crystal field splitting energy, and its magnitude determines the chemical properties and color of the complex.