Electron density

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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 .

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 .

By definition, the integral of the electron density, which extends over the entire area of ​​space , must be equal to the number of electrons:

The typical electron density for conduction electrons is in metallic solids , in the F-layer of the ionosphere only .

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:

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):

In words: one holds any electron in place and adds up the probabilities of all possible arrangements of the other electrons.

After presenting the expected value in the usual form:

the associated operator can be identified as:

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.

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

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