en>Addbot: Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q1333561

2013-03-15T09:10:05Z

Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q1333561

← Older revision		Revision as of 11:10, 15 March 2013
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	~~The name~~ of the ~~writer~~ is ~~Figures but it~~'s ~~not~~ the ~~most masucline title out there~~. ~~California~~ is ~~where her house~~ is ~~but she requirements to move simply because~~ of ~~her family~~. ~~What I love doing~~ is ~~doing ceramics but I haven~~'~~t produced~~ a ~~dime with it. Managing individuals has been his day job~~ for a ~~while~~.<br><br>~~Here~~ is ~~my blog post~~ - [~~http~~:/~~/musical~~.~~sehan~~.ac.kr/~~?document_srl~~=~~2110508 std home test~~]		In [[physics]] and [[quantum chemistry]], specifically [[density functional theory]], the '''Kohn–Sham equation''' is the [[Schrödinger equation]] of a fictitious system (the "'''Kohn–Sham system'''") of non-interacting particles (typically electrons) that generate the same [[electronic density\|density]] as any given system of interacting particles.<ref>
			{{cite journal
			\| last1 = Kohn \| first1 = Walter
			\| last2 = Sham \| first2 = Lu Jeu
			\| year = 1965
			\| title = Self-Consistent Equations Including Exchange and Correlation Effects
			\| journal = [[Physical Review]]
			\| volume = 140 \| pages = A1133–A1138 \| issue = 4A
			\| doi = 10.1103/PhysRev.140.A1133
			\|bibcode = 1965PhRv..140.1133K }}</ref><ref>
			{{cite book
			\| last1 = Parr \| first1 = Robert G.
			\| last2 = Yang \| first2 = Weitao
			\| year = 1994
			\| title = Density-Functional Theory of Atoms and Molecules
			\| publisher = [[Oxford University Press]]
			\| isbn = 978-0-19-509276-9
			}}</ref> The Kohn–Sham equation is defined by a local effective (fictitious) external potential in which the non-interacting particles move, typically denoted as ''v<sub>s</sub>''('''r''') or ''v''<sub>eff</sub>('''r'''), called the '''Kohn–Sham potential'''. As the particles in the Kohn–Sham system are non-interacting fermions, the Kohn–Sham wavefunction is a single [[Slater determinant]] constructed from a set of [[Atomic orbital\|orbital]]s that are the lowest energy solutions to

			:<math>\left(-\frac{\hbar^2}{2m}\nabla^2+v_{\rm eff}(\mathbf r)\right)\phi_{i}(\mathbf r)=\varepsilon_{i}\phi_{i}(\mathbf r)</math>

			This [[Eigenvalue, eigenvector and eigenspace#Eigenvalue equation\|eigenvalue equation]] is the typical representation of the '''Kohn–Sham equations'''. Here, ''ε<sub>i''</sub> is the orbital energy of the corresponding Kohn–Sham orbital, ''φ<sub>i''</sub>, and the density for an ''N''-particle system is

			:<math>\rho(\mathbf r)=\sum_i^N \|\phi_{i}(\mathbf r)\|^2.</math>

			The Kohn–Sham equations are named after [[Walter Kohn]] and [[Lu Jeu Sham\|Lu Jeu Sham (沈呂九)]], who introduced the concept at the [[University of California, San Diego]] in 1965.

			==Kohn–Sham potential==

			In [[density functional theory]], the total energy of a system is expressed as a [[Functional (mathematics)\|functional]] of the charge density as

			:<math> E[\rho] = T_s[\rho] + \int d\mathbf r\ v_{\rm ext}(\mathbf r)\rho(\mathbf r) + V_{H}[\rho] + E_{\rm xc}[\rho]</math>

			where ''T<sub>s</sub>'' is the '''Kohn–Sham [[kinetic energy]]''' which is expressed in terms of the Kohn–Sham orbitals as

			:<math>T_s[\rho]=\sum_{i=1}^N\int d\mathbf r\ \phi_i^*(\mathbf r)\left(-\frac{\hbar^2}{2m}\nabla^2\right)\phi_i(\mathbf r),</math>

			''v''<sub>ext</sub> is the external [[Potential energy\|potential]] acting on the interacting system (at minimum, for a molecular system, the electron-nuclei interaction), ''V<sub>H</sub>'' is the Hartree (or Coulomb) energy,

			:<math> V_{H}={e^2\over2}\int d\mathbf r\int d\mathbf{r}'\ {\rho(\mathbf r)\rho(\mathbf r')\over\|\mathbf r-\mathbf r'\|}.</math>

			and ''E''<sub>xc</sub> is the exchange-correlation energy. The Kohn–Sham equations are found by varying the total energy expression with respect to a set of orbitals to yield the Kohn–Sham potential as

			:<math>v_{\rm eff}(\mathbf r) = v_{\rm ext}(\mathbf{r}) + e^2\int {\rho(\mathbf{r}')\over\|\mathbf r-\mathbf r'\|}d\mathbf{r}' + {\delta E_{\rm xc}[\rho]\over\delta\rho(\mathbf r)}.</math>

			where the last term

			:<math>v_{\rm xc}(\mathbf r)\equiv{\delta E_{\rm xc}[\rho]\over\delta\rho(\mathbf r)}</math>

			is the exchange-correlation potential. This term, and the corresponding energy expression, are the only unknowns in the Kohn–Sham approach to density functional theory. An approximation that does not vary the orbitals is [[Harris functional]] theory.

			The Kohn–Sham orbital energies ''ε<sub>i</sub>'', in general, have little physical meaning (see [[Koopmans' theorem]]). The sum of the orbital energies is related to the total energy as

			:<math>E = \sum_{i}^N \varepsilon_i - V_{H}[\rho] + E_{\rm xc}[\rho] - \int {\delta E_{\rm xc}[\rho]\over\delta\rho(\mathbf r)} \rho(\mathbf{r}) d\mathbf{r}</math>

			Because the orbital energies are non-unique in the more general restricted open-shell case, this equation only holds true for specific choices of orbital energies (see [[Koopmans' theorem]]).

			== References ==
			{{reflist}}

			{{DEFAULTSORT:Kohn-Sham equations}}
			[[Category:Density functional theory]]

en>Yobot: WP:CHECKWIKI error fixes + general fixes using AWB (8024)

2012-03-19T16:16:20Z

WP:CHECKWIKI error fixes + general fixes using AWB (8024)

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The name of the writer is Figures but it's not the most masucline title out there. California is where her house is but she requirements to move simply because of her family. What I love doing is doing ceramics but I haven't produced a dime with it. Managing individuals has been his day job for a while.<br><br>Here is my blog post - [http://musical.sehan.ac.kr/?document_srl=2110508 std home test]

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en>Addbot: Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q1333561

en>Yobot: WP:CHECKWIKI error fixes + general fixes using AWB (8024)