Physics:Quantum degenerate orbitals
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Degenerate orbitals are atomic or molecular orbitals that have the same energy within a specified Hamiltonian or approximation. In quantum mechanics, degeneracy means that two or more distinct quantum states correspond to the same energy eigenvalue.[1] For orbitals, this usually means that several wave functions differ in shape, orientation, or quantum numbers while remaining equal in energy.
Orbital degeneracy is a central idea in atomic structure, molecular orbital theory, spectroscopy, and ligand-field theory. It explains why the three p orbitals of an isolated atom have equal energy, why the five d orbitals are degenerate in a spherical field, and why this equality may be broken by molecular geometry, external fields, electron-electron interactions, or relativistic effects.[2]
Definition
In the language of quantum mechanics, an orbital is degenerate with another orbital when both are eigenfunctions of the same Hamiltonian and have the same energy eigenvalue. If the Hamiltonian is written as , two orbitals and are degenerate when
with and representing different states. The number of independent states with the same energy is called the degree of degeneracy.[3]
Degeneracy is always defined relative to a model. Orbitals that are degenerate in an idealized central potential may split when additional physical effects are included. For this reason, the phrase "degenerate orbitals" normally implies a stated or understood approximation, such as the nonrelativistic hydrogen atom, an isolated free atom, or a molecule with a specified symmetry.
Atomic orbitals
In a one-electron atom described by a Coulomb potential, the nonrelativistic energy depends only on the principal quantum number . Orbitals with different angular momentum quantum numbers can therefore have the same energy. This large degeneracy is a special feature of the ideal hydrogen-like atom.[4]
In many-electron atoms, electron-electron repulsion and shielding remove much of this degeneracy. Orbitals with the same principal quantum number but different angular momentum usually have different energies. For example, the 2s and 2p orbitals are degenerate in the simplest hydrogenic model, but not in most many-electron atoms.
Within a free atom that has spherical symmetry, orbitals belonging to the same subshell remain degenerate with respect to their magnetic quantum number. The three 2p orbitals, often labeled 2px, 2py, and 2pz, have equal energy in an isolated atom. The five 3d orbitals are likewise degenerate before external fields or chemical environments are considered.[5]
Molecular orbitals
In molecular orbital theory, degeneracy is closely tied to molecular symmetry. Orbitals that transform together as a multidimensional irreducible representation of the molecular point group have the same energy, provided the symmetry is not broken.[6]
For example, in many linear molecules, two pi bonding orbitals can be degenerate because they are oriented perpendicular to the molecular axis in equivalent directions. Similar degeneracies occur for pi antibonding orbitals. If the molecule bends, distorts, or enters a lower-symmetry environment, the formerly degenerate orbitals can split into orbitals with different energies.
Degenerate molecular orbitals are important in electron filling. According to the Pauli principle, each orbital can hold two electrons with opposite spin. Hund's rule states that electrons occupy degenerate orbitals singly with parallel spins before pairing, when this arrangement is allowed by the electronic structure model.[7]
Splitting of degeneracy
Degenerate orbitals may cease to be degenerate when the symmetry or the Hamiltonian changes. Common mechanisms include:
- spin-orbit coupling, which contributes to fine structure in atomic spectra;
- electric fields, which can split levels through the Stark effect;
- magnetic fields, which split magnetic sublevels through the Zeeman effect;
- molecular distortions and crystal fields, which remove equivalence between orbital directions;
- electron correlation and exchange effects in many-electron systems.
The removal of degeneracy is often called lifting the degeneracy. In perturbation theory, a perturbing Hamiltonian must be diagonalized within the degenerate subspace before ordinary energy corrections can be assigned.[8]
Examples
p orbitals in atoms
The three p orbitals of an isolated atom are degenerate when the atom is described by a spherically symmetric Hamiltonian. They differ in orientation but are equivalent under rotations. A directional perturbation, such as an external electric field or a bonding environment, can make one direction energetically different from another.
d orbitals in transition-metal complexes
In an isolated transition-metal ion, the five d orbitals are degenerate in a spherical field. In an octahedral ligand field, they split into two sets: the lower-energy set and the higher-energy set. In a tetrahedral field, the ordering is reversed. This splitting is the basis of much of ligand-field theory and helps explain the colors and magnetic properties of many coordination compounds.[2]
pi orbitals in linear molecules
Linear molecules often have pairs of degenerate pi molecular orbitals. The degeneracy follows from rotational symmetry around the molecular axis. If the molecule is bent or otherwise lowered in symmetry, the two pi orbitals may no longer remain equal in energy.
Relation to symmetry
Degeneracy usually reflects symmetry, but the connection is not absolute. Symmetry can require degeneracy when the relevant states form a multidimensional representation. Conversely, two orbitals can have the same energy accidentally, without a symmetry requiring it. Such cases are called accidental degeneracies and may disappear when the Hamiltonian is changed slightly.[9]
In practical chemistry and spectroscopy, recognizing whether degeneracy is symmetry-required or accidental is important. Symmetry-required degeneracy is robust as long as the symmetry is preserved. Accidental degeneracy is generally more sensitive to perturbations.
See also
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References
- ↑ Atkins, Peter; Friedman, Ronald (2011). Molecular Quantum Mechanics (5th ed.). Oxford University Press. pp. 55-59. ISBN 978-0199541423.
- ↑ 2.0 2.1 Miessler, Gary L.; Fischer, Paul J.; Tarr, Donald A. (2014). Inorganic Chemistry (5th ed.). Pearson. pp. 337-351. ISBN 978-0321811059.
- ↑ Griffiths, David J. (2005). Introduction to Quantum Mechanics (2nd ed.). Pearson Prentice Hall. pp. 153-157. ISBN 978-0131118928.
- ↑ Bethe, Hans A.; Salpeter, Edwin E. (1957). Quantum Mechanics of One- and Two-Electron Atoms. Springer. pp. 17-25. ISBN 978-3540048022.
- ↑ Housecroft, Catherine E.; Sharpe, Alan G. (2018). Inorganic Chemistry (5th ed.). Pearson. pp. 664-675. ISBN 978-1292134147.
- ↑ Cotton, F. Albert (1990). Chemical Applications of Group Theory (3rd ed.). Wiley. pp. 102-110. ISBN 978-0471510949.
- ↑ Atkins, Peter; Jones, Loretta; Laverman, Leroy (2016). Chemical Principles: The Quest for Insight (7th ed.). W. H. Freeman. pp. 332-337. ISBN 978-1464183959.
- ↑ Sakurai, J. J.; Napolitano, Jim (2011). Modern Quantum Mechanics (2nd ed.). Addison-Wesley. pp. 311-319. ISBN 978-0805382914.
- ↑ Tinkham, Michael (2003). Group Theory and Quantum Mechanics. Dover Publications. pp. 19-25. ISBN 978-0486432472.
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