What are some examples of orbitals

Orbitals are single electron wave functions (mostly abbreviated with φ or ψ (small psi)) in quantum mechanics. The square of the magnitude of a wave function is interpreted as the probability of the electron that it describes. in the Orbital model there are no circular orbits as in Niels Bohr's atomic model and no other, defined orbits (trajectories). Rather, developments in quantum mechanics led to the realization that the exact location of the electrons, due to Werner Heisenberg's uncertainty relation, cannot be described exactly, only their distribution can be described stochastically.

Since the probability of the electrons being located asymptotically tends to zero with the distance from the atomic nucleus and extends to infinity, the orbital is chosen as the room in which the electron in question is with about 90% probability. This gives you spaces that roughly correspond to the size of the atoms. The boundary surfaces are surfaces with the same probability of presence (isosurfaces). The distances between the greatest probabilities of meeting an electron within the orbitals correspond to the orbital distances calculated by Niels Bohr.

The direct interpretation of orbitals as wave functions is only possible with single electron systems. In multi-electron systems, however, orbitals in Slater determinants are used to construct multi-electron wave functions. Such orbitals can be determined by Hartree-Fock, Kohn-Sham calculations (see: density functional theory (quantum physics)) or MCSCF calculations (MCSCF: Multiconfiguration Self Consistent Field), but are usually not clearly defined (different sets of orbits represent the same Multi-particle wave function).


Orbitals are determined using the four quantum numbers n, l, ml and s classified, sometimes by n, l, j and mj, where:

  • n (Principal quantum number, Value range: n = 1, 2, 3, ...) describes the main energy level that an electron possesses. It corresponds to a certain extent to the shell n of Bohr's model of the atom. The main quantum number describes a range in which the probability of an electron being located is very high. The larger n becomes, the further away the electron moves from the atomic nucleus, and its energy also increases. The maximum number of electrons in a shell is 2n2.
  • l (Secondary quantum number, orbital angular momentum quantum number, Value range: l = 0, 1, ..., (n - 1)) describes the orbital angular momentum of the electron
and thus the "shape" of the orbital. The letters s, p, d, f, g are often found in the literature as a designation for the secondary quantum number, derived from the English adjectives for the corresponding spectral lines: sharp, principal, diffuse, fundamental (continued alphabetically). The number of subshells is equal to the main quantum number. For n = 3, three subshells are possible l = 0, 1, 2. The number of orbitals per subshell is limited to 2l + 1. The letter q is often used instead of the letter l for the secondary quantum number.
  • ml (Magnetic quantum number, Value range: ml = -l, - (l - 1), ... 0, ... + (l - 1), + l) describes the spatial orientation that the orbital adopts with respect to an external magnetic field. The resulting orbitals are energetically the same, only when a magnetic field is applied from the outside can they be differentiated. The following applies to the projection of the angular momentum vector onto the direction of the magnetic field:
  • s (Spin (magnet) quantum number, s = +1/2 or s = -1/2) Their existence is interpreted as the viewing angle (spin) (1/2: 720-degree symmetry) of the electrons. An orbital can accommodate two electrons with opposing spin (Pauli principle). The spin quantum number is also denoted by ms designated.
For every angular momentum quantum number there is a magnetic quantum number, so there are the quantum numbers ml (Value range -l, ..., + l) and ms (possible values ​​+1/2 and -1/2).

Often the orbital angular momentum and the spin become the total angular momentum of an electron with the quantum number j added (value range | l - s |, | l - s | + 1, ..., l + s), the associated magnetic quantum number is then mj.

Characteristic shapes

The orbitals for the various l numbers have characteristic (rough) shapes, which are qualitatively retained even with higher n-values. For historical reasons, each l is assigned a specific letter:

Surnamewritten out value of lLook
s orbital sharp l = 0 radially symmetrical
p orbital principal l = 1 dumbbell-shaped in the three spatial axes
d orbital diffuse l = 2 crossed double dumbbells
f orbital fundamental l = 3 rosette-shaped

(The designations s, p, d and f come from spectroscopy and are only used for designation. A g-orbital with l = 4 occurs theoretically for an atom with the ordinal number 121. The designation follows the same as for the subsequent h-orbital ( l = 5) the alphabet.)

Strictly speaking, the orbitals only characterize the possible eigenstates of the electron waves, such as those in one-electron systems, such as hydrogen atom H or helium ions He+, Lithium ions Li+ etc. occur. The wave functions belonging to the orbitals (see also spherical surface functions) result from the stationary Schrödinger equation of a one-electron system. Despite this limitation, however, knowledge of the rough shape of the orbitals, which is also preserved in multi-electron systems, is sufficient to answer many qualitative questions about the structure of substances.

It should be noted that the orbitals shown in the literature are often Not are the eigen-states of the angular momentum operator. For example, the eigen-states of L.z (Angular momentum in z-direction) only the one eigenstate for the eigenvalue m = 0 and represented as pz designated. The ones with px and py designated orbitals Not the corresponding eigenstates for m = -1 and m = 1 but superpositions of these eigenstates. (They are eigen-states of Lx or Ly , but not with Lz commute!) This is not a problem for the conclusions as long as the corresponding wave functions are orthogonal.

Quantum theory

The orbitals result from the non-relativistic quantum theory according to the following calculation: The interaction between electron and atomic nucleus is described in a simplified manner by the Coulomb potential, the atomic nucleus is assumed to be fixed. The Hamilton operator for the one-electron system is

Since the Hamilton operator commutes with the angular momentum operator, form H, L.2 and L.z a complete system of commuting operators. So there are common eigen-states of these three operators. The states are determined by the three associated quantum numbers n, l and m. The Schrödinger equation can be broken down into a radius-dependent and an angle-dependent part. The eigenfunctions are the product of spherical functions (eigenfunctions of the angular momentum operator) and a radial component φnl(r).


Some symmetries of chemical bonds seem to contradict the characteristic shapes of the orbitals. These bond symmetries can only be understood through the formation of hybrid orbitals. These are orbitals that occur in multi-particle wave functions (see above).

See also

Categories: Atomic Physics | Physical chemistry | Theoretical chemistry