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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). Quantum theory is a theory of physics that uses Plancks constant. ...
In physics, the term state is used in several related senses, each of which expresses something about the way a physical system is. ...
Electron atomic and molecular orbitals In atomic physics and quantum chemistry, the electron configuration is the arrangement of electrons in an atom, molecule, or other physical structure (eg, a crystal). ...
Properties In chemistry and physics, an atom (Greek ἄτομος or átomos meaning indivisible) is the smallest particle still characterizing a chemical element. ...
In science, a molecule is a group of atoms in a definite arrangement held together by chemical bonds. ...
Generally, quantization is the state of being constrained to a set of discrete values, rather than varying continuously. ...
In physics, continuous spectrum refers to a range of values which may be graphed to fill a range with closely-spaced or overlapping intervals. ...
As with classical potentials, the potential energy is usually set at zero to infinity, leading to a negative potential energy for bound electron states. In physics, a bound state is a composite of two or more building blocks (particles) that behaves as a single object. ...
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. A quantum state is any possible state in which a quantum mechanical system can be. ...
The energy levels of two or more physical states are said to be degenerate when they have the same value. ...
The following sections of this article gives an overview over the most important factors that determine the energy levels of atoms and molecules.
Atoms
Intrinsic energy levels
Orbital state energy level Assume 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. The energy levels of an electron around a nucleus are given by: In chemistry, an atomic orbital is the region in which an electron may be found around a single atom. ...
 , where  is the Rydberg constant (typically between 1 eV and 103 eV), Q is the charge of the atom's nucleus,  is the principal quantum number, e is the charge of the electron, h is Planck's constant, and c is the speed of light. The Rydberg constant, named after physicist Janne Rydberg, is a physical constant discovered when measuring the spectrum of hydrogen, and building upon results from Anders Jonas Ångström and Johann Balmer. ...
The electronvolt (symbol eV, or, rarely and incorrectly, ev) is a unit of energy. ...
In atomic physics, the principal quantum number symbolized as n is the first quantum number of an atomic orbital. ...
A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ...
A line showing the speed of light on a scale model of Earth and the Moon The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. It is the speed of all electromagnetic radiation...
The Rydberg levels depend only on the principal quantum number .
Fine structure splitting Fine 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. 1. ...
Hyperfine structure Spin-nuclear-spin coupling (see hyperfine structure). Typical magnitude 10 − 4 eV. In atomic physics, hyperfine structure is a small perturbation in the energy levels (or spectra) of atoms or molecules due to the magnetic dipole-dipole interaction, arising from the interaction of the nuclear magnetic dipole with the magnetic field of the electron. ...
Electrostatic interaction of an electron with other electrons If 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 fields
Zeeman effect -
The interaction energy is: U = − μB with μ = qL / 2m The Zeeman effect (IPA ) is the splitting of a spectral line into several components in the presence of a magnetic field. ...
Zeeman effect taking spin into account This takes both the magnetic dipole moment due to the orbital angular momentum and the magnetic momentum arising from the electron spin into account.
Due to relativistic effects (Dirac equation), the magnetic moment arising 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). In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. ...
Stark effect Interaction with an external electric field (see Stark effect). The Stark effect is the splitting of a spectral line into several components in the presence of an electric field. ...
Paschen-Back effect For 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.
Molecules Roughly speaking, a molecular energy state, i.e. an eigenstate of the molecular Hamiltonian, is the sum of an electronic, vibrational, rotational and translational component, such that: In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...
The molecular Hamiltonian is an operator in quantum chemistry and atomic, molecular, and optical physics which describes the motions of electrons and nuclei in a polyatomic molecule. ...
 where Eelectronic is an eigenvalue of the electronic molecular Hamiltonian (the value of the potential energy surface) at the equilibrium geometry of the molecule. In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...
The electronic Hamiltonian for a multi-electron molecule in atomic units is: where is the vector position of electron with vector components in Bohr radii, is the charge of fixed nucleus a in units of the elementary charge, is the vector position of nucleus with vector components in Bohr radii. ...
A potential energy surface is generally used within the adiabatic or Born–Oppenheimer approximation in quantum mechanics and statistical mechanics to model chemical reactions and interactions in simple chemical and physical systems. ...
Geometry of the water molecule Molecular geometry or molecular structure is the three dimensional arrangement of the atoms that constitute a molecule, inferred from the spectroscopic studies of the compound. ...
The molecular energy levels are labelled by the molecular term symbols. In molecular physics, the molecular term symbol is a shorthand expression of the group representation and angular momenta that characterize the state of a molecule, i. ...
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. Molecular physics is the study of the physical properties of molecules and of the chemical bonds between atoms that bind them into molecules. ...
Quantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics and quantum field theory to address issues and problems in chemistry. ...
In physics, a bound state is a composite of two or more building blocks (particles or bodies) that behaves as a single object. ...
A quantum state is any possible state in which a quantum mechanical system can be. ...
Crystalline Materials Crystalline materials are often characterized by a number of important energy levels. The most important ones are the top of the valence band, the bottom of the conduction band, the Fermi energy, the vacuum level, and the energy levels of any defect states in the crystals. In solids, the valence band is the highest range of electron energies where electrons are normally present at zero temperature. ...
In semiconductors and insulators, the conduction band is the range of electron energy, higher than that of the valence band, sufficient to make the electrons free to accelerate under the influence of an applied electric field and thus constitute an electric current. ...
The Fermi energy is a concept in quantum mechanics referring to the energy of the highest occupied quantum state in a system of fermions at zero temperature. ...
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