Atomic orbital
An atomic orbital is the quantum-mechanical models of atoms spatial wave function of a single electron in a quantum mechanical state, usually in a steady state. His symbols is often ( small phi) or (small Psi ). The magnitude-squared describes the spatial distribution of the probability with which the electron can be found at the site, as density function ( Born's probability interpretation of quantum mechanics). Together with the information such as the spin is aligned to a fixed axis or orbital angular momentum of the electron, the electron orbital describes a state completely.
Unlike the older models of Niels Bohr ( 1913) and Arnold Sommerfeld ( 1916) an atomic orbital describes no exact electron path, but a diffuse distribution of the probability of finding the electron. This extends for bound electrons from the nucleus in the center outward to infinity, where she goes asymptotically to zero. The most probable distance from the nucleus is equal to the radius of the first Bohr orbit for the innermost orbital.
Clearly it is an orbital usually through the surface of the smallest possible volume is limited, inside which the electron is large (eg 90% ) probability of staying (see figure). We thus obtain the body that the size and shape of the atoms correspond roughly how they manifest themselves in chemical molecules, condensed matter and the kinetic theory of gases.
For the single electron of the hydrogen atom, the different orbitals arise as solutions of the Schrödinger equation of the hydrogen problem, which was first published in 1926. They have various shapes that are designated, the lower index of the principal quantum number, the orbital angular momentum quantum number and the magnetic quantum number is.
In the orbital model for atoms with several electrons, it is assumed that the electrons are distributed taking into account the Pauli principle on the orbitals. Such a state is called electron configuration and often represents a useful approximation for the in reality much more complicated structure of the atomic shell
For the description of electrons in molecules atomic orbitals are linearly combined to form molecular orbitals.
Electrons in solids are described by orbitals which have the form of Bloch wave functions.
This article deals only with bound electrons in atoms.
Representation of orbitals
Since the complete graph of a wave function four dimensions needed (or five dimensions, if applicable), is a complete representation in three dimensions is not directly possible (only with the help of color coding as in the table below ). As is known from the hydrogen atom, the eigenfunctions of the stationary Schrödinger equation have a radial part and an angular part. These units can be subscribed separately. Often, however, show pictures of orbitals a representation of the probability density (and thus, indirectly, of the orbitals ). Particularly illuminating the probability density could be visualized as a point cloud: Is the probability density large, so many points are plotted; is the probability density small, a few points are drawn. However, since the probability density is at almost all points (except the nodes of the wave function ) in the space non-zero, an orbital can nevertheless not completely shown in this way - since you continue to infinity would have to draw points. Instead, it is about to draw iso-surfaces of equal probability density that are implicitly defined. By sampling different angle we learn something about the shape of the isosurface and thus about the " shape of the orbital ." The shape of the orbital is given by a spherical harmonic. Frequently, the constant is chosen so that the probability of finding an electron in the region enclosed by the iso-surface area is 90 %.
Not infrequently, when displaying an iso-surface of the surface according to the sign of colored (as in the image of the p- orbital ).
Classification
Atomic orbitals are classified according to the three quantum numbers, sometimes by.
Principal quantum number n: shell
The principal quantum number designates the tray to which the part of the orbital. In the Bohr model, the energy level indicates, starting with the lowest, the ground state.
The bigger, the less further away to find the binding energy of the electron, and thus the greater the probability of finding the electron from the nucleus. This also applies to a plurality of atoms with electrons. When interactions between atoms that come close ( such as impacts of gas molecules, space filling in condensed matter, chemical bonding ) therefore play the electrons with the highest principal quantum number the most important role ( the electrons of the valence shell ).
The number of orbitals in a shell is given by. Taking into account the Pauli principle, the shell may be filled with a maximum of electrons, then it is complete. The corresponding atoms belong to the noble gases.
Incidental or orbital angular momentum quantum number l
Form
The addition or orbital angular momentum quantum number within a shell describes the amount of orbital angular momentum of the electron, and thus the "shape" of the orbital, which remains high even at high main quantum numbers.
Instead of the numbers 0, 1, 2, ... can be found in the literature usually the letters s, p, d, f, g as a term for the quantum number, derived from the English adjectives for the corresponding spectral lines; this concrete meaning has long become insignificant:
Notes:
The orbitals characterize strictly speaking, only the stationary electron waves in systems with only one electron ( such as hydrogen atom H, helium ion He ion, lithium ion Li2 , etc.). Since the shape of the orbitals is also in many-electron systems in approximately obtained their knowledge sufficient to answer many qualitative questions on chemical bonding and building materials.
It should be noted that the orbitals shown in the literature are sometimes not the eigenstates of magnetic quantum number of the z- component of the angular momentum operator. For example, is represented by the p orbitals only one eigenstate for eigenvalue and called pz. However, the orbitals denoted by px and py are the corresponding eigenstates for, but are their superpositions. They are eigenstates of the operators or, in each case to, but do not commute with. For the conclusions that is not a problem as long as the corresponding wave functions are orthogonal.
Lower shell
The larger, the larger the average distance of the electron from the atomic nucleus:
- Wherein the orbital is spherical and has at, that is in the core, a non-zero probability.
- The maximum value corresponds to the Bohr orbit, here the probability is concentrated at the calculated in the Bohr model radius.
Since in multi- electron atoms, the inner electron attracting shield the nuclear charge, decreases the binding energy of outer electrons. This different energy levels arise depending on the quantum number or internuclear distance within the same shell, which are also called sub-shells.
The number of subshells each shell is equal to the principal quantum number:
- For it is only the shell 1s.
- Three bottom shells are possible and are denoted by 3s, 3p, 3d.
Per subshell there are orbitals (each with a different magnetic quantum number, see the following section), leading to a total of orbitals per shell.
Magnetic quantum number ml: inclination of the angular momentum vector
The magnetic quantum number
Describes the slope of the angular momentum vector with respect to a ( freely selectable ) z axis:
- When it is (for example ) parallel to the axis,
- At (around ) antiparallel.
The fact that at a given exactly different values are possible, is called space quantization.
If no external field is applied, have the individual orbitals of a subshell same energy. Meanwhile, in the magnetic field splits the energy within the lower shell in equidistant values ( Zeeman effect ), i.e. each corresponds to a separate orbital energy level.
Magnetic spin quantum number ms
For the lighter atoms one needs to take into account the electron spin only in the form that each orbital can be occupied by exactly one pair of electrons, whose two electrons have by the Pauli principle opposite magnetic spin quantum numbers.
Total angular momentum j and magnetic quantum number mj
Among the heavy atoms toward the spin -orbit interaction is stronger. It causes the splitting of the energy of a lower shell to be fixed in two sub-shells, according to the value of the total angular momentum. The magnetic quantum number by running values . Each of these orbitals can be occupied by one electron, so that the total number of places remains the same. In the name of the value as an index to the icon is added, for example.
Quantum theory
For non-relativistic quantum theory, the orbitals arise as follows: the interaction between the electron and the nucleus is facilitated by the Coulomb described, assuming the nucleus as a fix. The Hamiltonian for the one-electron system is:
Since the Hamiltonian commutes with the angular momentum operator, form, and a complete system of commuting observables. So there are common eigenstates, which are determined by the three quantum numbers associated to these three operators.
The Schrödinger equation
Can be decomposed into a radius and an angle-dependent part. The eigenfunctions are the product of a spherical harmonic ( eigenfunction of the angular momentum operator ) and a radial function:
These are shown normalized to the following table. Here denote the Bohr radius and the atomic number.
Orbitals shown are all oriented around the Z- axis because it is inherent functions of the operator. For alignment of orbitals with a given orbital angular momentum in any other direction you have to form linear combinations of the wave functions to the different.
Time dependence
Become orbitals defined as eigenfunctions of operators that correspond to an energy, then these orbitals are stationary. Examples are the Hartree -Fock orbitals as Eigenfuntionen of Fockoperators and the Kohn -Sham orbitals, the eigenfunctions of the Kohn -Sham Hamiltonian are. In contrast, the so-called natural orbitals, that is, the characteristic functions of the one-electron reduced density matrix is not stationary.
Hybridization
Some symmetries of chemical bonds seem to contradict the characteristic shapes of the orbitals. This binding symmetries can be understood only through the formation of hybrid orbitals that occur in many-particle wave functions ( see above).
Many-electron wave functions
The direct interpretation of orbitals as wave functions is only possible when single-electron systems. For many-electron systems, but these orbitals are used in Slater determinants to construct many-electron wave functions. Such orbitals (see: Density functional theory ( quantum physics ) ) by Hartree -Fock, Kohn -Sham calculations or MCSCF calculations ( MCSCF: Multi Configuration Self Consistent Field) to be determined. It should be noted that one can not clearly infer the individual occupied orbitals of a given many- wave function, since different sets of linear combinations of the orbitals may mathematically provide the same multiparticulate wave function.