# 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:

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