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| {{Merge|electron density|date=September 2010|discuss=talk:electron density}}
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| In [[quantum mechanics]], and in particular [[quantum chemistry]], the '''electronic density''' is a measure of the probability of an [[electron]] occupying an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial variables and is typically denoted as either ''ρ''('''r''') or ''n''('''r'''). The density is determined, through definition, by the normalized ''N''-electron [[wavefunction]] which itself depends upon 4''N'' variables (3''N'' spatial and ''N'' [[Spin (physics)|spin]] coordinates). Conversely, the density determines the wave function modulo a phase factor, providing the formal foundation of [[density functional theory]].
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| ==Definition==
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| The electronic density corresponding to a normalized ''N''-electron [[wavefunction]] (with '''r''' and ''s'' denoting spatial and spin variables respectively) is defined as<ref>{{cite book|last1=Parr|first1=Robert G.|last2=Yang | first2= Weitao|title=Density-Functional Theory of Atoms and Molecules|publisher=Oxford University Press|location=New York|year=1989|isbn=0-19-509276-7}}</ref>
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| :<math>
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| \begin{align}
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| \rho(\mathbf{r})&=N\sum_{{s}_{1}} \cdots \sum_{{s}_{N}} \int \ \mathrm{d}\mathbf{r}_2 \ \cdots \int\ \mathrm{d}\mathbf{r}_N \ |\Psi(\mathbf{r},s_{1},\mathbf{r}_{2},s_{2},...,\mathbf{r}_{N},s_{N})|^2, \\
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| &= \langle\Psi|\hat{\rho}(\mathbf{r})|\Psi\rangle,
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| \end{align}
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| </math>
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| where the operator corresponding to the density observable is | |
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| :<math>\hat{\rho}(\mathbf{r}) = \sum_{i=1}^{N}\sum_{s_{i}}\ \delta(\mathbf{r}-\mathbf{r}_{i}).</math>
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| In [[Hartree-Fock]] and [[density functional theory|density functional]] theories the wave function is typically represented as a single [[Slater determinant]] constructed from ''N'' orbitals, ''φ''<sub>''k''</sub>, with corresponding occupations ''n''<sub>''k''</sub>. In these situations the density simplifies to
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| :<math>\rho(\mathbf{r})=\sum_{k=1}^N n_{k}|\varphi_k(\mathbf{r})|^2.</math>
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| ==General Properties==
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| From its definition, the electron density is a non-negative function integrating to the total number of electrons. Further, for a system with kinetic energy ''T'', the density satisfies the inequalities<ref name="lieb83">{{cite journal|last=Lieb|first=Elliott H.|year=1983|journal=International Journal of Quantum Chemistry|volume=24|issue=3|pages=243–277|title=Density functionals for coulomb systems|doi=10.1002/qua.560240302}}</ref>
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| :<math>\frac{1}{2}\int\mathrm{d}\mathbf{r}\ \big(\nabla\sqrt{\rho(\mathbf{r})}\big)^{2} \leq T.</math>
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| :<math>\frac{3}{2}\left(\frac{\pi}{2}\right)^{4/3}\left(\int\mathrm{d}\mathbf{r}\ \rho^{3}(\mathbf{r})\right)^{1/3} \leq T.</math>
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| For finite kinetic energies, the first (stronger) inequality places the square root of the density in the [[Sobolev space]] ''H''<sup>1</sup>('''R'''<sup>3</sup>). Together with the normalization and non-negativity this defines a space containing physically acceptable densities as
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| :<math>
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| \mathcal{J}_{N} =
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| \left\{ \rho \left| \rho(\mathbf{r})\geq 0,\
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| \rho^{1/2}(\mathbf{r})\in H^{1}(\mathbf{R}^{3}),\
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| \int\mathrm{d}\mathbf{r}\ \rho(\mathbf{r}) = N
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| \right.\right\}.
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| </math>
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| The second inequality places the density in the [[Lp space|''L''<sup>3</sup> space]]. Together with the normalization property places acceptable densities within the intersection of ''L''<sup>1</sup> and ''L''<sup>3</sup> – a superset of <math>\mathcal{J}_{N}</math>.
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| ==Topology==
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| The [[ground state]] electronic density of an [[atom]] is conjectured to be a [[Monotonic function|monotonically]] decaying function of the distance from the [[atomic nucleus|nucleus]].<ref>{{cite journal|last1=Ayers|first1=Paul W.|last2=Parr | first2= Robert G.|year=2003|title=Sufficient condition for monotonic electron density decay in many-electron systems|journal=International Journal of Quantum Chemistry|volume=95|issue=6|pages=877–881|doi=10.1002/qua.10622}}</ref>
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| ===Nuclear cusp condition===
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| The electronic density displays cusps at each nucleus in a molecule as a result of the unbounded electron-nucleus Coulomb potential. This behavior is quantified by the Kato cusp condition formulated in terms of the spherically averaged density, <math>\bar{\rho}</math>, about any given nucleus as<ref>{{cite journal|last=Kato|first=Tosio |year=1957|title=On the eigenfunctions of many-particle systems in quantum mechanics|journal=Communications on Pure and Applied Mathematics|volume=10|issue=2|pages=151–177|doi=10.1002/cpa.3160100201}}</ref>
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| :<math>\left.\frac{\partial}{\partial r_{\alpha}}\bar{\rho}(r_{\alpha})\right|_{r_{\alpha}=0} = -2Z_{\alpha}\bar{\rho}(0).</math>
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| That is, the radial derivative of the spherically averaged density, evaluated at any nucleus, is equal to twice the density at that nucleus multiplied by the negative of the [[atomic number]] (''Z'').
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| ===Asymptotic behavior===
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| The nuclear cusp condition provides the near-nuclear (small ''r'') density behavior as
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| :<math>\rho(r) \sim e^{-2Z_{\alpha}r}\,.</math> | |
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| The long-range (large ''r'') behavior of the density is also known, taking the form<ref>{{cite journal|last1=Morrell|first1=Marilyn M.|last2=Parr|first2=Robert. G.|last3=Levy|first3=Mel|year=1975|title=Calculation of ionization potentials from density matrices and natural functions, and the long-range behavior of natural orbitals and electron density|journal=Journal of Chemical Physics|volume=62|issue=2|pages=549–554|doi=10.1063/1.430509|bibcode = 1975JChPh..62..549M }}</ref>
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| :<math>\rho(r) \sim e^{-2\sqrt{2\mathrm{I}}r}\,.</math>
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| where I is the [[ionization energy]] of the system.
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| ==Response Density==
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| Another more-general definition of a density is the "linear-response density".<ref>
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| {{cite journal |doi = 10.1063/1.447489 |title = On the evaluation of analytic energy derivatives for correlated wave functions |year = 1984 |last1 = Handy |first1 = Nicholas C. |last2 = Schaefer |first2 = Henry F. |journal = The Journal of Chemical Physics |volume = 81 |pages = 5031|bibcode = 1984JChPh..81.5031H |issue = 11 }}</ref><ref>{{ cite journal | doi = 10.1021/j100181a030 | title = Analysis of the effect of electron correlation on charge density distributions | year = 1992 | last1 = Wiberg | first1 = Kenneth B. | last2 = Hadad | first2 = Christopher M. | last3 = Lepage | first3 = Teresa J. | last4 = Breneman | first4 = Curt M. | last5 = Frisch | first5 = Michael J. | journal = The Journal of Physical Chemistry | volume = 96 | pages = 671 | issue = 2}}</ref> This is the density that when contracted
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| with any spin-free, one-electron operator yields the associated property defined as the derivative of the energy.
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| For example, a dipole moment is the derivative of the energy with respect to an external magnetic field and
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| is not the expectation value of the operator over the wavefunction. For some theories they are the same when
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| the wavefunction is converged. The occupation numbers are not limited to the range of zero to two, and therefore
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| sometimes even the response density can be negative in certain regions of space.<ref>{{cite journal
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| | last1 = Gordon | first1 = Mark S. | last2 = Schmidt | first2 = Michael W. | last3 = Chaban | first3 = Galina M.
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| | last4 = Glaesemann | first4 = Kurt R. | last5 = Stevens | first5 = Walter J. | last6 = Gonzalez |first6 = Carlos
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| | year = 1999
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| | title = A natural orbital diagnostic for multiconfigurational character in correlated wave functions
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| | journal = J. Chem. Phys.
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| | volume = 110
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| | issue = 9
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| | pages = 4199–4207
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| | doi = 10.1063/1.478301|bibcode = 1999JChPh.110.4199G }}
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| </ref>
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| ==See also==
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| *[[Probability current|Current density]]
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Electronic Density}}
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| [[Category:Atomic physics]]
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| [[Category:Quantum chemistry]]
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| [[Category:Density functional theory]]
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| [[de:Elektronendichte]]
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| [[es:Densidad (mecánica cuántica)]]
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| [[fr:Densité électronique]]
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| [[it:Densità elettronica]]
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| [[pl:Gęstość elektronowa]]
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| [[ru:Электронная плотность]]
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