# Atomorbital - Atomorbital

Representation of different orbitals of the first and second electron shells .
Top row: Representation of the probability densities${\displaystyle |\Psi ({\vec {r}})|^{2}}$the orbitals as point clouds.
Bottom row: Representation of isosurfaces of${\displaystyle |\Psi ({\vec {r}})|^{2}}$. The isosurface is chosen so that the electron is within the volume enclosed by the isosurface with a 90% probability.

In the quantum mechanical models of atoms, an atomic orbital is the spatial wave function of an individual electron in a quantum mechanical state , usually in a stationary state. His formula symbol is mostly${\displaystyle \varphi }$(little Phi ) or${\displaystyle \psi }$(little psi ). The square of the amount${\displaystyle |\psi ({\vec {r}})|^{2}}$describes as a density function the spatial distribution of the probability of being with which the electron is at the location${\displaystyle {\vec {r}}=(x,y,z)}$can be found ( Born's probability interpretation of quantum mechanics). Together with the specification of whether the spin is aligned parallel or anti-parallel to a fixed axis or to the orbital angular momentum of the electron, an orbital describes the state of the electron completely.

In the older atomic models according to Niels Bohr ( Bohr's atomic model , 1913) and Arnold Sommerfeld ( Bohr-Sommerfeld's atomic model , 1916) an orbital describes an exact electron orbit selected by the quantization rules. This idea was abandoned in quantum mechanics in favor of a diffuse distribution of the probability of the electron being located. The quantum mechanical atomic orbital extends for bound electrons from the atomic nucleus in the center outwards to infinity, where the probability of being asymptotically tends to zero. The most probable distance from the atomic nucleus for the innermost orbital is equal to the radius of Bohr's 1st circular orbit.

An orbital is usually represented clearly by the surface of the smallest possible volume, inside of which the electron is located with a high (e.g. 90%) probability (see illustration). This results in bodies that roughly correspond to the size and shape of the atoms, as they are noticeable in chemical molecules , condensed matter and the kinetic theory of gases.

The most common atomic orbitals are those that result for the single electron of the hydrogen atom as solutions to the Schrödinger equation of the hydrogen problem and were first published in 1926. They have different shapes that come with${\displaystyle \psi _{nlm_{l}}({\vec {r}})}$with the lower index from the main quantum number ${\displaystyle n,}$ the orbital angular momentum quantum number ${\displaystyle l}$ and the magnetic quantum number ${\displaystyle m_{l}}$ consists.

In the orbital model for atoms with several electrons it is assumed that the electrons are distributed over the orbitals taking into account the Pauli principle . Such a state is called the electron configuration and is often a useful approximation for the structure of the atomic shell , although this is made even more complicated by additional electron correlations.

To describe electrons in molecules, molecular orbitals are formed as a linear combination of atomic orbitals. Electrons in solids are described by orbitals, which have the form of Bloch wave functions .

This article only looks at bound electrons in atoms. A simplification of the orbital model is the shell model .

## presentation

Representation of the probability density of the 1s orbital using a (very fine) point cloud

As the full graph of a wave function ${\displaystyle \Psi \colon \mathbb {R} ^{3}\to \mathbb {R} }$ required four dimensions (or five dimensions, if ${\displaystyle \Psi \colon \mathbb {R} ^{3}\to \mathbb {C} }$), a complete three-dimensional representation is not directly possible (but, for example, with the aid of color coding as shown in the table below). As is known from the hydrogen atom, the have eigenfunctions${\displaystyle \Psi ({\vec {r}})}$the stationary Schrödinger equation ${\displaystyle H\Psi ({\vec {r}})=E\Psi ({\vec {r}})}$ a radial part ${\displaystyle R(r)}$ and an angle component ${\displaystyle Y_{l}^{m}(\theta ,\phi )}$:

${\displaystyle \Psi ({\vec {r}})=R(r)Y_{l}^{m}(\theta ,\phi )}$

These shares can be subscribed separately. Often, however, images of orbitals show a representation of the probability density${\displaystyle |\Psi ({\vec {r}})|^{2}}$ (and thus indirectly the orbitals ${\displaystyle \Psi ({\vec {r}})}$). The probability density could be visualized particularly clearly as a point cloud : If the probability density is high, many points are drawn; if the probability density is small, few points are drawn. However, since the probability density is unequal to zero at almost all points (with the exception of the node points of the wave function) in space, an orbital cannot be completely represented in this way - since one would have to continue to draw points to infinity. Instead, one goes over to drawing isosurfaces with the same probability density, implicitly by

${\displaystyle {\text{const}}=|\Psi ({\vec {r}})|^{2}=|R(r)|^{2}|Y_{l}^{m}(\theta ,\phi )|^{2}}$

are defined. By scanning different angles${\displaystyle \theta ,\phi }$one learns something about the shape of the isosurface and thus something about the "shape of the orbital". The shape of the orbital is given by a spherical surface function ${\displaystyle Y_{l}^{m}(\theta ,\phi )}$given. Often the constant is chosen so that the probability of finding the electron in the space enclosed by the isosurface is 90%.

It is not uncommon for an isosurface to be represented by ${\displaystyle |\Psi ({\vec {r}})|^{2}}$the area corresponding to the argument of${\displaystyle \Psi ({\vec {r}})}$ colored (as in the picture of the p orbital).

A simple way of schematically representing the occupation of atomic orbitals is the Pauling notation .

## classification

Atomic orbitals can be represented by three quantum numbers ${\displaystyle n,l,m_{l}}$are set and then can accommodate two electrons of opposite spin . Alternatively, atomic orbitals can be represented by four quantum numbers ${\displaystyle n,l,j,m_{j}}$ can be set and then offer space for only one electron at a time.

### Principal quantum number n: shell

The principal quantum number ${\displaystyle n=1,2,3,\dotsc }$refers to the shell (also called K-shell, L-shell, M-shell, ...) to which the orbital belongs. In Bohr's atomic model there is${\displaystyle n}$the energy level , starting with the lowest, the basic state ${\displaystyle n=1.}$

The bigger ${\displaystyle n}$the lower the binding energy of the electron and thus the greater the probability of finding the electron further away from the atomic nucleus. This also applies to atoms with several electrons. In the case of interactions between atoms that come close to each other (such as collisions of gas molecules, space filling in condensed matter, chemical bonds ), the electrons with the largest principal quantum number therefore play the most important role (the electrons of the valence shell ).

The number of ${\displaystyle (nlm_{l})}$-Orbitals in a shell arises too ${\displaystyle n^{2}.}$Taking into account the Pauli principle , the bowl can with a maximum of${\displaystyle 2\cdot n^{2}}$Electrons are occupied, then it is complete. The corresponding atoms belong to the noble gases .

### Minor or orbital angular momentum quantum number l

#### Form

The minor or orbital angular momentum quantum number ${\displaystyle l=0,1,2,\dotsc ,(n-1)}$ inside a shell describes the amount ${\displaystyle |{\vec {l}}|=\hbar \cdot {\sqrt {l(l+1)}}}$the orbital angular momentum of the electron. With the quantum number${\displaystyle m_{l}}$together, the angle-dependent "shape" of the orbital is determined. It is for all principal quantum numbers (note${\displaystyle n>l}$) same.

Instead of the digits 0, 1, 2 etc. , the secondary quantum number is usually designated in the literature by the letters s, p, d, f, g etc., derived from the original designations for the corresponding spectral lines ; this concrete meaning has long since become insignificant:

${\displaystyle p_{z}}$-Orbital ${\displaystyle d}$-Orbital ${\displaystyle 4p}$-Orbital
Simplified form of a p orbital ${\displaystyle (l=1)}$.
The color stands for the sign of the wave function. An isosurface of${\displaystyle |\Psi ({\vec {r}})|^{2}.}$
Simplified shapes of the various d orbitals (respectively ${\displaystyle l=2}$). For the respective orbitals is an isosurface of the probability density${\displaystyle |\Psi ({\vec {r}})|^{2}}$ shown. Shape of a 4p orbital ${\displaystyle (l=1,\,m_{x}=0)}$.
The color stands for the sign of the wave function.
Name former meaning Minor quantum number Form number ${\displaystyle 2l+1}$
s-Orbital sharp ${\displaystyle \,l=0}$ spherically symmetrical 01
p-Orbital principal ${\displaystyle \,l=1}$ dumbbell shaped 3 000 A2
d-Orbital diffuse ${\displaystyle \,l=2}$ crossed double dumbbells 05
f-Orbital fundamental ${\displaystyle \,l=3}$ rosette shaped 07
g-OrbitalA1 (alphabetical continuation) ${\displaystyle \,l=4}$ rosette-shaped 09
h-OrbitalA1 (alphabetical continuation) ${\displaystyle \,l=5}$ rosette-shaped 11

Remarks:

A1Can occur as an excited state . For the ground state , it is theoretically only expected for atoms from atomic number 121 .
A2 According to the three spatial axes.

Strictly speaking, the orbitals only characterize the stationary electron waves in systems with only one electron (e.g. hydrogen atom H, helium ion He + , lithium ion Li 2+ , etc.). Since the shape of the orbitals is roughly retained even in multi-electron systems, their knowledge is sufficient to answer many qualitative questions about chemical bonds and the structure of substances.

It should be noted that the orbitals shown in the literature are sometimes not the eigenstates of the magnetic quantum number${\displaystyle m_{l}}$the z-component of the angular momentum operator ${\displaystyle {\hat {l}}_{z}}$are. For example, of the p orbitals, only one eigenstate becomes the eigenvalue ${\displaystyle m_{l}{\mathord {=}}0}$shown and designated as p z . However , the orbitals labeled p x and p y are not the corresponding eigen-states for${\displaystyle m_{l}=\pm 1,}$but are their superpositions . They are eigenstates of the operators${\displaystyle {\hat {l}}_{x}}$ or. ${\displaystyle {\hat {l}}_{y},}$ each to ${\displaystyle m_{x,y}{\mathord {=}}0,}$ but not with ${\displaystyle {\hat {l}}_{z}}$commute. This is not a problem for the conclusions as long as the corresponding wave functions are orthogonal .

#### Lower shell

The bigger ${\displaystyle l}$the greater is at solid ${\displaystyle n}$ the mean distance of the electron from the atomic nucleus:

• At ${\displaystyle l=0}$ the orbital is spherical and also has at ${\displaystyle r=0,}$ so in essence, a non-vanishing probability of presence.
• The maximum value ${\displaystyle l=n-1}$ corresponds to Bohr's circular path, here the probability of stay is concentrated at the radius calculated in Bohr's model.

Since the inner electrons shield the attractive nuclear charge in atoms with several electrons , the binding energy of the outer electrons is reduced. Since the mean nuclear distances depend on the secondary quantum number, the same result${\displaystyle n}$different energy levels within the same shell depending on the secondary quantum number. These are also called the lower shells of the main shell (to solid${\displaystyle n}$) designated.

The number of subshells per shell is equal to the main quantum number ${\displaystyle n}$:

• For ${\displaystyle n=1}$ there is only the 1s shell.
• For ${\displaystyle n=3}$ are three sub-shells ${\displaystyle l=0,1,2}$ possible, which are designated with 3s, 3p, 3d.

There are per lower shell ${\displaystyle 2l+1}$ Orbitals (each with a different magnetic quantum number ${\displaystyle m_{l}}$, s. following section) what on total${\displaystyle n^{2}}$ Orbitals per shell leads.

### Magnetic quantum number m l : inclination of the angular momentum vector

The magnetic quantum number

${\displaystyle m_{l}=-l,-(l-1),\dotsc ,0,\dotsc ,(l-1),l}$

gives the z component ${\displaystyle m_{l}\hbar }$of the orbital angular momentum vector with respect to a (freely selected) z-axis. This clearly corresponds to an angle of inclination

${\displaystyle \cos \vartheta ={\frac {m_{l}}{\sqrt {l(l+1)}}}.}$
• At ${\displaystyle m_{l}=+l\Leftrightarrow \cos \vartheta ={\text{max}}\Leftrightarrow \vartheta \approx 0^{\circ }}$is the orbital angular momentum (roughly) parallel to the axis,
• at ${\displaystyle m_{l}=-l\Leftrightarrow \cos \vartheta ={\text{min}}\Leftrightarrow \vartheta \approx 180^{\circ }}$(roughly) antiparallel .

That given ${\displaystyle l}$ I agree ${\displaystyle 2l+1}$different values ​​are possible is called directional quantization.

If there is no external field, they have ${\displaystyle 2l+1}$individual orbitals of a subshell have the same energy. On the other hand, in the magnetic field, the energy within the lower shell splits into${\displaystyle 2l+1}$equidistant values ​​( Zeeman effect ), ie each individual orbital then corresponds to a separate energy level.

### Magnetic spin quantum number m s

In the case of the lighter atoms, the electron spin only needs to be taken into account in the form that every orbital${\displaystyle \psi _{nlm_{l}}}$can be occupied by exactly one electron pair whose two electrons have opposite magnetic spin quantum numbers according to the Pauli principle (${\displaystyle m_{s}=\pm {\tfrac {1}{2}}}$).

### Total angular momentum j and magnetic quantum number m j

The spin-orbit interaction becomes stronger towards the heavy atoms . It causes the energy of a sub-shell to be split with certain${\displaystyle n>1,l>0}$ in two sub-shells, depending on the value of the total angular momentum ${\displaystyle j=l\pm {\tfrac {1}{2}}.}$ The magnetic quantum number ${\displaystyle m_{j}=-j,-(j-1),\dotsc ,+j}$ passes through ${\displaystyle 2j+1}$Values. Each of these orbitals can be occupied by an electron, so the total number of places remains the same. The value for${\displaystyle j}$ as a lower index to the symbol for ${\displaystyle nl}$appended, e.g. B.${\displaystyle 2p_{3/2}.}$

## Quantum theory

The orbitals result from the non-relativistic quantum theory as follows: The interaction between electron and atomic nucleus is described by the Coulomb potential , the atomic nucleus is assumed to be fixed. The Hamilton operator for the one-electron system is

${\displaystyle {\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V(r)}$

with the potential

${\displaystyle V(r)={\frac {Ze}{r}}}$.

Since the Hamilton operator commutes with the angular momentum operator, form ${\displaystyle {\hat {H}},}$ ${\displaystyle {\hat {l}}^{2}}$ and ${\displaystyle {\hat {l}}_{z}}$a complete system of commuting observables . There are common eigen-states for these three operators, which are represented by the three associated quantum numbers${\displaystyle n,l,m_{l}}$ are determined.

The Schrödinger equation

${\displaystyle {\hat {H}}\cdot \psi _{n,l,m_{l}}(r,\vartheta ,\phi )=E_{n,l,m_{l}}\cdot \psi _{n,l,m_{l}}(r,\vartheta ,\phi )}$

can be broken down into a radius and an angle-dependent part. The eigenfunctions${\displaystyle \psi _{n,l,m_{l}}}$are the product of a spherical surface function ${\displaystyle Y_{lm_{l}}(\vartheta ,\varphi )}$ (Eigenfunction of the angular momentum operator) and a radial function ${\displaystyle \Phi _{nl}(r)\colon }$

${\displaystyle \psi _{n,l,m_{l}}(r,\vartheta ,\phi )=Y_{lm_{l}}(\vartheta ,\varphi )\cdot \Phi _{nl}(r)}$

These are up ${\displaystyle n{\mathord {=}}3}$shown standardized in the following table. Designate${\displaystyle a_{0}}$the Bohr radius and${\displaystyle Z}$ the atomic number.

The orbitals shown in the following table are all aligned around the z-axis because they are eigenfunctions of the ${\displaystyle {\hat {l}}_{z}}$Operator. For the alignment of an orbital with a given angular momentum${\displaystyle l}$ in any other direction one must linear combinations of the wave functions for the different ones ${\displaystyle m_{l}}$form. The graphic representation shows a volume with the probability density on its surface${\displaystyle |\psi ({\vec {r}})|^{2}}$is constant. The colors encode the complex phase of the wave function.

Complex wave functions in hydrogen-like atoms
Orbital Wave function of the orbital Form des Orbitals ${\displaystyle \psi ({\vec {r}})}$ (not to scale)
${\displaystyle n}$ ${\displaystyle l}$ ${\displaystyle m_{l}}$ ${\displaystyle \psi _{n,l,m_{l}}(r,\theta ,\phi )}$
1s 1 0 00 ${\displaystyle {\frac {1}{\sqrt {\pi }}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}e^{-\textstyle {\frac {Zr}{a_{0}}}}}$
2s 2 0 00 ${\displaystyle {\frac {1}{4{\sqrt {2\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}\left(2-{\frac {Zr}{a_{0}}}\right)e^{-\textstyle {\frac {Zr}{2a_{0}}}}}$
2p0 2 1 00 ${\displaystyle {\frac {1}{4{\sqrt {2\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}{\frac {Zr}{a_{0}}}e^{-\textstyle {\frac {Zr}{2a_{0}}}}\cos \theta }$
2p-1/+1 2 1 ±1 ${\displaystyle {\frac {1}{8{\sqrt {\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}{\frac {Zr}{a_{0}}}e^{-\textstyle {\frac {Zr}{2a_{0}}}}\sin \theta e^{\pm i\phi }}$
3s 3 0 00 ${\displaystyle {\frac {1}{81{\sqrt {3\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}\left(27-18{\frac {Zr}{a_{0}}}+2{\frac {Z^{2}r^{2}}{a_{0}^{2}}}\right)e^{-\textstyle {\frac {Zr}{3a_{0}}}}}$
3p 0 3 1 00 ${\displaystyle {\frac {\sqrt {2}}{81{\sqrt {\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}\left(6-{\frac {Zr}{a_{0}}}\right){\frac {Zr}{a_{0}}}e^{-\textstyle {\frac {Zr}{3a_{0}}}}\cos \theta }$
3p -1 / + 1 3 1 ±1 ${\displaystyle {\frac {1}{81{\sqrt {\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}\left(6-{\frac {Zr}{a_{0}}}\right){\frac {Zr}{a_{0}}}e^{-\textstyle {\frac {Zr}{3a_{0}}}}\sin \theta e^{\pm i\phi }}$
3d 0 3 2 00 ${\displaystyle {\frac {1}{81{\sqrt {6\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}{\frac {Z^{2}r^{2}}{a_{0}^{2}}}e^{-\textstyle {\frac {Zr}{3a_{0}}}}(3\cos ^{2}\theta -1)}$
3d -1 / + 1 3 2 ±1 ${\displaystyle {\frac {1}{81{\sqrt {\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}{\frac {Z^{2}r^{2}}{a_{0}^{2}}}e^{-\textstyle {\frac {Zr}{3a_{0}}}}\sin \theta \cos \theta e^{\pm i\phi }}$
3d -2 / + 2 3 2 ±2 ${\displaystyle {\frac {1}{162{\sqrt {\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}{\frac {Z^{2}r^{2}}{a_{0}^{2}}}e^{-\textstyle {\frac {Zr}{3a_{0}}}}\sin ^{2}\theta e^{\pm 2i\phi }}$

## Natural orbital

A natural orbital is an orbital that does not result as an eigenfunction of a Hamilton operator , but rather as an eigenfunction of a one-electron density operator . This is obtained from a given many-particle state, which can also contain electron correlations, for example, and thus goes beyond the scope of a single-particle model. The electron configuration formed with the natural orbitals gives the best approximation to the initially given many-particle state that is possible with a single-particle model.

## Time dependence

If orbitals are defined as eigenfunctions of an operator that corresponds to an energy, then these orbitals are stationary within the framework of the selected model. Examples of this are the Hartree-Fock orbitals as eigenfunctions of the Fock operator${\displaystyle {\hat {F}}}$and the Kohn-Sham orbitals, which are eigenfunctions of the Kohn-Sham-Hamilton operator. In contrast to this, the so-called natural orbitals, as eigenfunctions of the reduced one-electron density operator , are not stationary.

## Hybridization

Some symmetries of chemical bonds seem to contradict the characteristic shapes of the orbitals. These bonds can be understood through the formation of hybrid orbitals , which can be formed in the presence of electrons with different orbital angular momentum, if they are energetically almost equivalent (see above).

## Multi-electron wave functions

The interpretation of orbitals as wave functions of an electron is only possible with single-electron systems. A wave function for N electrons can then be constructed by inserting N orbitals into a Slater determinant . This guarantees the antisymmetry of the entire wave function, which is necessary for fermions , but can not represent electron correlations beyond that. In order to approximate the electron-electron interaction, the orbitals can be calculated using Hartree-Fock , Kohn-Sham calculations (see: Density functional theory in quantum physics) or MCSCF invoices (MCSCF: Multiconfiguration Self Consistent Field). But it always remains valid that differently chosen orbitals, if they are linearly independent linear combinations of the original ones, mathematically result in the same Slater determinant, so that one cannot unambiguously infer from a given multi-particle wave function what the individual occupied orbitals are.