Electron density

The electron density or , in physics, is a charge carrier density that indicates the location-dependent number of electrons per volume ( density function ). From a mathematical point of view, it is a scalar field of three-dimensional spatial space . ${\displaystyle n({\vec {r}})}$ ${\displaystyle n_{e}({\vec {r}})}$

It is a measured variable ( unit ) that is often used in the description of molecules and solids ( density functional theory ) in order to avoid complicated, high-dimensional wave functions or quantum mechanical state vectors . It is also used in plasma physics , in X-ray structure analysis (as a Fourier transform of the structure factor ) and in semiconductor physics . ${\displaystyle m^{-3}}$

By definition, the integral of the electron density, which extends over the entire area of ​​space , must be equal to the number of electrons: ${\displaystyle V}$ ${\displaystyle N}$

${\displaystyle N_{e}=\int _{V}n_{e}({\vec {r}})dV.}$

The typical electron density for conduction electrons is in metallic solids , in the F-layer of the ionosphere only . ${\displaystyle n_{\mathrm {e} }}$ ${\displaystyle 10^{28}\,\mathrm {m} ^{-3}}$ ${\displaystyle 10^{12}\,\mathrm {m} ^{-3}}$

Expected value of the electron density operator

In general, in quantum mechanics, measured quantities are identified with Hermitian operators whose eigenvectors represent the states in which the system assumes a sharp measured value with regard to the measured quantity, and whose eigenvalues ​​correspond to the associated measured values ​​themselves.

The electron density is identified as the expected value of the electron density operator:

${\displaystyle n({\vec {r}}):=\langle \Psi |{\hat {n}}({\vec {r}})|\Psi \rangle .}$

This operator must meet the following properties:

• Integrability of the expected value (more strictly: the integral over the entire volume must correspond to the number of particles)
• Positive semi-finiteness : expectation value must be greater than or equal to 0 everywhere

By identifying the electron density as the marginal distribution of the probability density ( absolute square of the wave function):

${\displaystyle n({\vec {r}})=N\sum _{s_{1}}\dots \sum _{s_{N}}\int d{\vec {r_{2}}}\dots \int d{\vec {r_{N}}}|\Psi (\mathbf {r} ,s_{1},\mathbf {r} _{2},s_{2},\dots ,\mathbf {r} _{N},s_{N})|^{2}}$

In words: one holds any electron in place and adds up the probabilities of all possible arrangements of the other electrons. ${\displaystyle {\vec {r}}}$

After presenting the expected value in the usual form:

${\displaystyle {\begin{aligned}n({\vec {r}})&=\langle \psi |{\hat {n}}({\vec {r}})|\psi \rangle \\&=\sum _{s_{1}}\dots \sum _{s_{N}}\int d{\vec {r_{1}}}\dots \int d{\vec {r_{N}}}\Psi (\mathbf {r} _{1},s_{1},\mathbf {r} _{2},s_{2},\dots ,\mathbf {r} _{N},s_{N})^{*}\left(\sum _{i=1}^{N}\delta ({\vec {r_{i}}}-{\vec {r}})\right)\Psi (\mathbf {r} _{1},s_{1},\mathbf {r} _{2},s_{2},\dots ,\mathbf {r} _{N},s_{N})\end{aligned}}}$

the associated operator can be identified as: ${\displaystyle {\hat {n}}({\vec {r}}_{1},\dots ,{\vec {r}}_{N},{\vec {r}})=\sum _{i=1}^{N}\delta ({\vec {r_{i}}}-{\vec {r}})}$

and one recognizes that it is not an operator in the strict sense, since it does not convert a square-integrable function into a square-integrable function and therefore does not satisfy the definition of an operator in the space of square-integrable functions .

There is thus no particle density operator , but there is a linear functional ( distribution ) whose integral kernel is commonly referred to as the particle density operator .

In the sense of the topology induced by the 2-norm, it is a non-continuous linear functional on the locally absolutely Lebesgue integrable functions.

Here in particular the absolute Lebesgue integrable functions of the form are valid for and with the extension of a distribution theory known Delta distributions using Delta consequences to represent. ${\displaystyle \phi ={\overline {\psi _{1}}}\psi _{2}}$ ${\displaystyle \psi _{1},\psi _{2}\in L^{2}(\mathbb {R} ^{n},\mathbb {C} )}$ ${\displaystyle \delta ({\vec {r}}_{i}-{\vec {r}})}$ ${\displaystyle L_{loc}^{1}(\mathbb {R} ^{n},\mathbb {C} )}$

Within the Hartree-Fock approximation , the electron density is obtained from the sum of the orbital densities:

${\displaystyle {\hat {n}}({\vec {r}})=\sum _{i=1}^{N}\phi _{i}({\vec {r}}){\overline {\phi }}_{i}({\vec {r}}).}$

Web links

• 3D illustration of the electron density in atoms