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A quantum mechanical system can only be in certain states, so that only certain energy levels are possible. The term energy level is most commonly used in reference to the electron configuration in atoms or molecules. In other words, the energy spectrum can be quantized (see continuous spectrum for the more general case). As with classical potentials, the potential energy is usually set to zero at infinity, leading to a negative potential energy for bound electron states. Energy levels are said to be degenerate, if the same energy level is obtained by more than one quantum mechanical state. They are then called degenerate energy levels. The following sections of this article gives an overview over the most important factors that determine the energy levels of atoms and molecules.
AtomsIntrinsic energy levelsOrbital state energy levelAssume an electron in a given atomic orbital. The energy of its state is mainly determined by the electrostatic interaction of the (negative) electron with the (positive) nucleus, calculatable using the principal quantum number n. The energy levels of an electron around a nuclues with charge Z are given by En = − RyZ2 / n2, where Ry is the Rydberg constant. The typical magnitude is 1w/.103 eV. The Rydberg levels depend only on the principal quantum number n. However, there are many interactions that lead to small changes to this energy level (corrections), which can be calculated involving the other orbital electron quantum numbers l, ml, ms. The more accurate description of the electron wavefunction often leads to the splitting of the energy levels and therefore removes energy level degeneracy. Fine structure splittingFine structure arises from relativistic kinetic energy corrections, spin-orbit coupling (magnetic dipole interaction of the electron spin with the magnetic field due to the movement in the electric field of the nucleus) and the Darwin term (contact term interaction of s-shell electrons inside the nucleus). Typical magnitude 10 − 3 eV. Hyperfine structureSpin-nuclear-spin coupling (see hyperfine structure). Typical magnitude 10 − 4 eV. Electrostatic interaction of an electron with other electronsIf there is more than one electron around the atom, electron-electron-interactions raise the energy level. These interactions are often neglected if the spatial overlap of the electron wavefunctions is low. Energy levels due to external fieldsZeeman effectSee main article: Zeeman effect. The orbital angular momentum of the electron corresponds to a magnetic momentum, interacting with the outer magnetic field (electromagnetic interaction). The interaction energy is: U = − μB with μ = qL / 2m Zeeman effect taking spin into accountThis takes both the magnetic dipole moment due to the orbital angular momentum and the magnetic momentum arrising from the electron spin into account. Due to relativistic effects (Dirac equation), the magnetic moment arriving from the electron spin is μ = − μBgs with g the gyro-magnetic factor (about 2). μ = μl + gμs The interaction energy therefore gets UB = − μB = μBB(ml + gms). Stark effectInteraction with an external electric field (see Stark effect). Paschen-Back effectFor strong magnetic fields, the quantum numbers l,s,j,mj are not "good" any more and the Zeeman splitting does not give a correct description of the energy levels. This is known as the Paschen-Back effect. MoleculesRoughly speaking, a molecular energy state, i.e. an eigenstate of the molecular Hamiltonian, is the product of an electronic, vibrational, rotational and translational component, such that: where Eelectronic is an eigenvalue of the electronic molecular Hamiltonian (the value of the potential energy surface) at the equilibrium geometry of the molecule. The molecular energy levels are labelled by the molecular term symbols. The specific energies of these components vary with the specific energy state and the substance. In molecular physics and quantum chemistry, an energy level is a quantized energy of a bound quantum mechanical state. See also? |
- Atomic Orbital
- Electron configuration
- Perturbation theory (quantum mechanics)
- Computational chemistry
- Potential energy surface
- Molecular term symbol
- Spectroscopy