Interval finite element - Revision history

en>Finell: /* 1-dimension example */ Copy edit (needs more)

2014-06-02T02:07:46Z

1-dimension example: Copy edit (needs more)

← Older revision		Revision as of 04:07, 2 June 2014
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	~~Within~~ [~~[computational chemistry]], the '''Slater–Condon rules''' express integrals of one~~- ~~and two-body operators over [[wavefunction~~]~~]s constructed as~~ [~~[Slater determinant]~~]~~s of~~ [~~[orthonormality\|orthonormal]~~] [~~[molecular orbital\|orbitals]] in terms of the individual orbitals~~. In doing so, the original integrals involving ''N''-electron wavefunctions are reduced to sums over integrals involving at most two molecular orbitals, or in other words, the original 3''N'' dimensional integral is expressed in terms of many three- and six-dimensional integrals.		My name is Suzette Highett. I life in Red Bank (United States).<br>xunjie 女優の李冰冰金馬は共同アディダスウィメンズ2010年春夏キャンペーンを開始した。
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	The rules are used in deriving the working equations for all methods of approximately solving the Schrödinger equation that employ wavefunctions constructed from Slater determinants. These include [[Hartree-Fock]] theory, where the wavefunction is a single determinant, and all those methods which use Hartree-Fock theory as a reference such as [[Møller-Plesset perturbation theory]], and [[Coupled cluster]] and [[Configuration interaction]] theories.		my web page - [http://www.hps-heerbrugg.ch/admin/eddie/editor/chrome/ chrome hearts リング]

	In 1929 [[John C. Slater]] derived expressions for diagonal matrix elements of an approximate Hamiltonian while investigating atomic spectra within a perturbative approach.<ref>{{cite journal\|last=Slater\|first=J. C.\|year=1929\|title=The Theory of Complex Spectra\|journal=Phys. Rev.\|volume=34\|issue=10\|pages=1293–1322\|doi=10.1103/PhysRev.34.1293\|pmid=9939750\|bibcode=1929PhRv...34.1293S}}</ref> The following year [[Edward Condon]] extended the rules to non-diagonal matrix elements.<ref>{{cite journal\|last=Condon\|first=E. U.\|year=1930\|title=The Theory of Complex Spectra\|journal=Phys. Rev.\|volume=36\|issue=7\|pages=1121–1133\|doi=10.1103/PhysRev.36.1121 \|bibcode = 1930PhRv...36.1121C }}</ref> In 1955 [[Per-Olov Löwdin]] further generalized these results for wavefunctions constructed from non-orthonormal orbitals, leading to what are known as the '''Löwdin rules'''.<ref>{{cite journal\|last=Löwdin\|first=Per-Olov\|year=1955\|title=Quantum Theory of Many-Particle Systems. I. Physical Interpretations by Means of Density Matrices, Natural Spin-Orbitals, and Convergence Problems in the Method of Configurational Interaction\|journal=Phys. Rev.\|volume=97\|issue=6\|pages=1474–1489\|doi=10.1103/PhysRev.97.1474\|bibcode=1955PhRv...97.1474L}}</ref>

	~~==Mathematical background==~~

	In terms of an [[antisymmetrizer\|antisymmetrization]] operator (<math>\mathcal{A}</math>) acting upon a product of ''N'' orthonormal [[spin-orbital]]s (with '''r''' and ''σ'' denoting spatial and spin variables), a determinantal wavefunction is ''denoted'' as

	:<math>\|\Psi\rangle = \mathcal{A}(\phi_{1}(\mathbf{r}_{1}\sigma_{1})\phi_{2}(\mathbf{r}_{2}\sigma_{2})\cdots\phi_{m}(\mathbf{r}_{m}\sigma_{m})\phi_{n}(\mathbf{r}_{n}\sigma_{n})\cdots\phi_{N}(\mathbf{r}_{N}\sigma_{N})).</math>

	~~A wavefunction differing from this by only a single orbital (the ''m'''th orbital) will be denoted as~~

	:<math>\|\Psi_{m}^{p}\rangle = \mathcal{A}(\phi_{1}(\mathbf{r}_{1}\sigma_{1})\phi_{2}(\mathbf{r}_{2}\sigma_{2})\cdots\phi_{p}(\mathbf{r}_{m}\sigma_{m})\phi_{n}(\mathbf{r}_{n}\sigma_{n})\cdots\phi_{N}(\mathbf{r}_{N}\sigma_{N})),</math>

	~~and a wavefunction differing by two orbitals will be denoted as~~

	:<math>\|\Psi_{mn}^{pq}\rangle = \mathcal{A}(\phi_{1}(\mathbf{r}_{1}\sigma_{1})\phi_{2}(\mathbf{r}_{2}\sigma_{2})\cdots\phi_{p}(\mathbf{r}_{m}\sigma_{m})\phi_{q}(\mathbf{r}_{n}\sigma_{n})\cdots\phi_{N}(\mathbf{r}_{N}\sigma_{N})).</math>

	For any particular one- or two-body operator, ''Ô'', the Slater–Condon rules show how to simplify the following types of integrals:<ref name="piela">{{cite book\|last=Piela\|first= Lucjan\|title=Ideas of Quantum Chemistry\|publisher=Elsevier Science\|location=Amsterdam\|year=2006\|chapter=Appendix M\|isbn=0-444-52227-1}}</ref>

	~~:<math> \langle\Psi\|\hat{O}\|\Psi\rangle, \langle\Psi\|\hat{O}\|\Psi_{m}^{p}\rangle,\ \mathrm{and}\ \langle\Psi\|\hat{O}\|\Psi_{mn}^{pq}\rangle.</math>~~

	~~Matrix elements for two wavefunctions differing by more than two orbitals vanish unless higher order interactions are introduced.~~

	~~==Integrals of one-body operators==~~

	One body operators depend only upon the position or momentum of a single electron at any given instant. Examples are the [[Kinetic energy#Quantum_mechanical kinetic energy of rigid bodies\|kinetic energy]], [[Electric dipole moment\|dipole moment]], and [[Angular momentum coupling\|total angular momentum]] operators.

	~~A one-body operator in an ''N''-particle system is decomposed as~~

	~~:<math>\hat{F} = \sum_{i=1}^{N}\ \hat{f}(i).</math>~~

	~~The Slater–Condon rules for such an operator are:<ref name="piela"/><ref name="szabo">{{cite book\|last=Szabo\|first=Attila\|coauthors=Ostlund, Neil S.\|title=Modern Quantum Chemistry~~ : ~~Introduction to Advanced Electronic Structure Theory\|publisher=Dover Publications\|location=Mineola, New York\|year=1996\|chapter=Ch. 2.3.3\|isbn=0-486-69186-1}}<~~/~~ref>~~

	~~:<math>~~
	~~\begin{align}~~
	~~\langle\Psi\|\hat{F}\|\Psi\rangle &= \sum_{i=1}^{N}\ \langle\phi_{i}\|\hat{f}\|\phi_{i}\rangle, \\~~
	~~\langle\Psi\|\hat{F}\|\Psi_{m}^{p}\rangle &= \langle\phi_{m}\|\hat{f}\|\phi_{p}\rangle, \\~~
	~~\langle\Psi\|\hat{F}\|\Psi_{mn}^{pq}\rangle &= 0.~~
	~~\end{align}~~
	</~~math>~~

	~~==Integrals of two-body operators==~~

	~~Two-body operators couple two particles at any given instant~~. ~~Examples being the electron~~-~~electron repulsion, [[Magnetic dipole-dipole interaction\|magnetic dipolar coupling]], and total angular momentum-squared operators~~.

	~~A two-body operator in an ''N''-particle system is decomposed as~~

	~~:<math>\hat{G} = \frac 12 \sum_{i=1}^{N}\sum_{{j =1}\atop{j\neq i}}^{N}\ \hat{g}(i,j).<~~/~~math>~~

	~~The Slater–Condon rules for such an operator are:<ref name="piela"~~/~~><ref name="szabo"~~/>

	~~:<math>~~
	~~\begin{align}~~
	\langle\Psi\|\hat{G}\|\Psi\rangle &= \frac{1}{2}\sum_{i=1}^{N}\sum_{j=1\atop{j\neq i}}^{N}\ \bigg(\langle\phi_{i}\phi_{j}\|\hat{g}\|\phi_{i}\phi_{j}\rangle - \langle\phi_{i}\phi_{j}\|\hat{g}\|\phi_{j}\phi_{i}\rangle\bigg), \\
	~~\langle\Psi\|\hat{G}\|\Psi_{m}^{p}\rangle &= \sum_{i=1}^{N}\ \bigg(\langle\phi_{m}\phi_{i}\|\hat{g}\|\phi_{p}\phi_{i}\rangle - \langle\phi_{m}\phi_{i}\|\hat{g}\|\phi_{i}\phi_{p}\rangle\bigg), \\~~
	~~\langle\Psi\|\hat{G}\|\Psi_{mn}^{pq}\rangle &= \langle\phi_{m}\phi_{n}\|\hat{g}\|\phi_{p}\phi_{q}\rangle - \langle\phi_{m}\phi_{n}\|\hat{g}\|\phi_{q}\phi_{p}\rangle,~~
	~~\end{align}~~
	</~~math>~~
	~~where~~
	:<math>\langle\phi_{i}\phi_{j}\|\hat{g}\|\phi_{k}\phi_{l}\rangle = \int\mathrm{d}\mathbf{r}\int\mathrm{d}\mathbf{r}'\ \phi_{i}^{}(\mathbf{r})\phi_{j}^{}(\mathbf{r}')g(\mathbf{r},\mathbf{r}')\phi_{k}(\mathbf{r})\phi_{l}(\mathbf{r}').</~~math>~~

	~~==References==~~
	~~{{reflist}}~~

	~~{{DEFAULTSORT:Slater-Condon rules}}~~
	~~[[Category:Computational chemistry]]~~
	~~[[Category:Quantum chemistry]~~]

en>Chris the speller: /* 1D Example */replaced: calcuated → calculated using AWB

2013-07-07T16:58:08Z

1D Example: replaced: calcuated → calculated using AWB

← Older revision		Revision as of 18:58, 7 July 2013
Line 1:		Line 1:
	~~If you are looking for a specific plugin~~, ~~then you can just search for~~ the ~~name~~ of the ~~plugin~~. ~~This one is one of~~ the ~~most beneficial features of Word~~ - ~~Press as this feature allows users~~ to ~~define~~ the ~~user roles. Step-4 Testing: It~~ is ~~the foremost important~~ of ~~your Plugin development process~~. ~~Out~~ of the ~~various designs of photography identified these days~~, ~~sports photography~~ is ~~preferred most~~, ~~probably for the enjoyment~~ and ~~enjoyment associated with it~~. In the ~~most current edition you can customize your retailer layout and display hues and fonts similar~~ to ~~your site or blog~~. <br><br>~~These folders~~ as ~~well as files have to copied and~~ the ~~saved~~. ~~If you wish to sell your services or products via internet using your website~~, ~~you have to put together on the website the facility for trouble~~-~~free payment transfer between customers~~ and the ~~company~~. ~~You are able to set them within your theme options~~ and ~~so they aid the search engine to get~~ a ~~suitable title and description for the pages that get indexed by Google~~. ~~These four plugins will make~~ this ~~effort easier and~~ the ~~sites run effectively~~ as ~~well as make other widgets added to a site easier to configure. W3C compliant HTML~~ and a ~~good open source powered~~ by ~~Word~~ - ~~Press CMS site is regarded as~~ the ~~prime minister.~~ <br><br>~~Photography is an entire activity in itself~~, and ~~a thorough discovery~~ of ~~it is beyond~~ the ~~opportunity~~ of ~~this content~~. ~~But if you~~ are ~~not willing to choose cost to~~ the ~~detriment~~ of ~~quality~~, ~~originality~~ and ~~higher returns, then go for a self~~-~~hosted wordpress blog and increase the presence of your business~~ in ~~this new digital age~~. ~~Setting Up Your Business Online Using Free Wordpress Websites~~. ~~To turn the Word~~ - ~~Press Plugin on~~, ~~click Activate on the far right side~~ of the ~~list. So~~, ~~if you are looking online to hire dedicated Wordpress developers~~, ~~India PHP Expert can give a hand you in each~~ and ~~every best possible way~~. <br><br>~~If you adored this article so you would like to receive more info about [http~~://~~miniURL.fouiner.info/wordpress_backup_plugin_55395 wordpress dropbox backup]~~ i ~~implore you to visit the website. You can add keywords but it is best to leave this alone. Cameras with a pentaprism~~ (~~as in comparison to pentamirror~~) ~~ensure that little mild is lost before it strikes your eye~~, ~~however these often increase the cost of the digital camera considerably. One of the great features of Wordpress is its ability to integrate SEO into your site. Fast Content Update~~ - ~~It's easy to edit or add posts with free Wordpress websites. OSDI~~, ~~a Wordpress Development Company based on ahmedabad~~, ~~India.~~ <br><br>~~Yet~~, overall, less than 1% of websites presently have mobile versions of their websites. Sanjeev Chuadhary is an expert writer who shares his knowledge about web development through their published articles and other resource. Offshore Wordpress development services from a legitimate source caters dedicated and professional services assistance with very simplified yet technically effective development and designing techniques from experienced professional Wordpress developer India. This is because of the customization that works as a keystone for a SEO friendly blogging portal website. Customers within a few seconds after visiting a site form their opinion about the site.		Within [[computational chemistry]], the '''Slater–Condon rules''' express integrals of one- and two-body operators over [[wavefunction]]s constructed as [[Slater determinant]]s of [[orthonormality\|orthonormal]] [[molecular orbital\|orbitals]] in terms of the individual orbitals. In doing so, the original integrals involving ''N''-electron wavefunctions are reduced to sums over integrals involving at most two molecular orbitals, or in other words, the original 3''N'' dimensional integral is expressed in terms of many three- and six-dimensional integrals.

			The rules are used in deriving the working equations for all methods of approximately solving the Schrödinger equation that employ wavefunctions constructed from Slater determinants. These include [[Hartree-Fock]] theory, where the wavefunction is a single determinant, and all those methods which use Hartree-Fock theory as a reference such as [[Møller-Plesset perturbation theory]], and [[Coupled cluster]] and [[Configuration interaction]] theories.

			In 1929 [[John C. Slater]] derived expressions for diagonal matrix elements of an approximate Hamiltonian while investigating atomic spectra within a perturbative approach.<ref>{{cite journal\|last=Slater\|first=J. C.\|year=1929\|title=The Theory of Complex Spectra\|journal=Phys. Rev.\|volume=34\|issue=10\|pages=1293–1322\|doi=10.1103/PhysRev.34.1293\|pmid=9939750\|bibcode=1929PhRv...34.1293S}}</ref> The following year [[Edward Condon]] extended the rules to non-diagonal matrix elements.<ref>{{cite journal\|last=Condon\|first=E. U.\|year=1930\|title=The Theory of Complex Spectra\|journal=Phys. Rev.\|volume=36\|issue=7\|pages=1121–1133\|doi=10.1103/PhysRev.36.1121 \|bibcode = 1930PhRv...36.1121C }}</ref> In 1955 [[Per-Olov Löwdin]] further generalized these results for wavefunctions constructed from non-orthonormal orbitals, leading to what are known as the '''Löwdin rules'''.<ref>{{cite journal\|last=Löwdin\|first=Per-Olov\|year=1955\|title=Quantum Theory of Many-Particle Systems. I. Physical Interpretations by Means of Density Matrices, Natural Spin-Orbitals, and Convergence Problems in the Method of Configurational Interaction\|journal=Phys. Rev.\|volume=97\|issue=6\|pages=1474–1489\|doi=10.1103/PhysRev.97.1474\|bibcode=1955PhRv...97.1474L}}</ref>

			==Mathematical background==

			In terms of an [[antisymmetrizer\|antisymmetrization]] operator (<math>\mathcal{A}</math>) acting upon a product of ''N'' orthonormal [[spin-orbital]]s (with '''r''' and ''σ'' denoting spatial and spin variables), a determinantal wavefunction is ''denoted'' as

			:<math>\|\Psi\rangle = \mathcal{A}(\phi_{1}(\mathbf{r}_{1}\sigma_{1})\phi_{2}(\mathbf{r}_{2}\sigma_{2})\cdots\phi_{m}(\mathbf{r}_{m}\sigma_{m})\phi_{n}(\mathbf{r}_{n}\sigma_{n})\cdots\phi_{N}(\mathbf{r}_{N}\sigma_{N})).</math>

			A wavefunction differing from this by only a single orbital (the ''m'''th orbital) will be denoted as

			:<math>\|\Psi_{m}^{p}\rangle = \mathcal{A}(\phi_{1}(\mathbf{r}_{1}\sigma_{1})\phi_{2}(\mathbf{r}_{2}\sigma_{2})\cdots\phi_{p}(\mathbf{r}_{m}\sigma_{m})\phi_{n}(\mathbf{r}_{n}\sigma_{n})\cdots\phi_{N}(\mathbf{r}_{N}\sigma_{N})),</math>

			and a wavefunction differing by two orbitals will be denoted as

			:<math>\|\Psi_{mn}^{pq}\rangle = \mathcal{A}(\phi_{1}(\mathbf{r}_{1}\sigma_{1})\phi_{2}(\mathbf{r}_{2}\sigma_{2})\cdots\phi_{p}(\mathbf{r}_{m}\sigma_{m})\phi_{q}(\mathbf{r}_{n}\sigma_{n})\cdots\phi_{N}(\mathbf{r}_{N}\sigma_{N})).</math>

			For any particular one- or two-body operator, ''Ô'', the Slater–Condon rules show how to simplify the following types of integrals:<ref name="piela">{{cite book\|last=Piela\|first= Lucjan\|title=Ideas of Quantum Chemistry\|publisher=Elsevier Science\|location=Amsterdam\|year=2006\|chapter=Appendix M\|isbn=0-444-52227-1}}</ref>

			:<math> \langle\Psi\|\hat{O}\|\Psi\rangle, \langle\Psi\|\hat{O}\|\Psi_{m}^{p}\rangle,\ \mathrm{and}\ \langle\Psi\|\hat{O}\|\Psi_{mn}^{pq}\rangle.</math>

			Matrix elements for two wavefunctions differing by more than two orbitals vanish unless higher order interactions are introduced.

			==Integrals of one-body operators==

			One body operators depend only upon the position or momentum of a single electron at any given instant. Examples are the [[Kinetic energy#Quantum_mechanical kinetic energy of rigid bodies\|kinetic energy]], [[Electric dipole moment\|dipole moment]], and [[Angular momentum coupling\|total angular momentum]] operators.

			A one-body operator in an ''N''-particle system is decomposed as

			:<math>\hat{F} = \sum_{i=1}^{N}\ \hat{f}(i).</math>

			The Slater–Condon rules for such an operator are:<ref name="piela"/><ref name="szabo">{{cite book\|last=Szabo\|first=Attila\|coauthors=Ostlund, Neil S.\|title=Modern Quantum Chemistry : Introduction to Advanced Electronic Structure Theory\|publisher=Dover Publications\|location=Mineola, New York\|year=1996\|chapter=Ch. 2.3.3\|isbn=0-486-69186-1}}</ref>

			:<math>
			\begin{align}
			\langle\Psi\|\hat{F}\|\Psi\rangle &= \sum_{i=1}^{N}\ \langle\phi_{i}\|\hat{f}\|\phi_{i}\rangle, \\
			\langle\Psi\|\hat{F}\|\Psi_{m}^{p}\rangle &= \langle\phi_{m}\|\hat{f}\|\phi_{p}\rangle, \\
			\langle\Psi\|\hat{F}\|\Psi_{mn}^{pq}\rangle &= 0.
			\end{align}
			</math>

			==Integrals of two-body operators==

			Two-body operators couple two particles at any given instant. Examples being the electron-electron repulsion, [[Magnetic dipole-dipole interaction\|magnetic dipolar coupling]], and total angular momentum-squared operators.

			A two-body operator in an ''N''-particle system is decomposed as

			:<math>\hat{G} = \frac 12 \sum_{i=1}^{N}\sum_{{j =1}\atop{j\neq i}}^{N}\ \hat{g}(i,j).</math>

			The Slater–Condon rules for such an operator are:<ref name="piela"/><ref name="szabo"/>

			:<math>
			\begin{align}
			\langle\Psi\|\hat{G}\|\Psi\rangle &= \frac{1}{2}\sum_{i=1}^{N}\sum_{j=1\atop{j\neq i}}^{N}\ \bigg(\langle\phi_{i}\phi_{j}\|\hat{g}\|\phi_{i}\phi_{j}\rangle - \langle\phi_{i}\phi_{j}\|\hat{g}\|\phi_{j}\phi_{i}\rangle\bigg), \\
			\langle\Psi\|\hat{G}\|\Psi_{m}^{p}\rangle &= \sum_{i=1}^{N}\ \bigg(\langle\phi_{m}\phi_{i}\|\hat{g}\|\phi_{p}\phi_{i}\rangle - \langle\phi_{m}\phi_{i}\|\hat{g}\|\phi_{i}\phi_{p}\rangle\bigg), \\
			\langle\Psi\|\hat{G}\|\Psi_{mn}^{pq}\rangle &= \langle\phi_{m}\phi_{n}\|\hat{g}\|\phi_{p}\phi_{q}\rangle - \langle\phi_{m}\phi_{n}\|\hat{g}\|\phi_{q}\phi_{p}\rangle,
			\end{align}
			</math>
			where
			:<math>\langle\phi_{i}\phi_{j}\|\hat{g}\|\phi_{k}\phi_{l}\rangle = \int\mathrm{d}\mathbf{r}\int\mathrm{d}\mathbf{r}'\ \phi_{i}^{}(\mathbf{r})\phi_{j}^{}(\mathbf{r}')g(\mathbf{r},\mathbf{r}')\phi_{k}(\mathbf{r})\phi_{l}(\mathbf{r}').</math>

			==References==
			{{reflist}}

			{{DEFAULTSORT:Slater-Condon rules}}
			[[Category:Computational chemistry]]
			[[Category:Quantum chemistry]]

en>Sun Creator: Typo fixing, typos fixed: Mid point → Midpoint (2), , → , using AWB

2012-08-04T00:51:38Z

Typo fixing, typos fixed: Mid point → Midpoint (2), , → , using AWB

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If you are looking for a specific plugin, then you can just search for the name of the plugin. This one is one of the most beneficial features of Word - Press as this feature allows users to define the user roles. Step-4 Testing: It is the foremost important of your Plugin development process. Out of the various designs of photography identified these days, sports photography is preferred most, probably for the enjoyment and enjoyment associated with it. In the most current edition you can customize your retailer layout and display hues and fonts similar to your site or blog. These folders as well as files have to copied and the saved. If you wish to sell your services or products via internet using your website, you have to put together on the website the facility for trouble-free payment transfer between customers and the company. You are able to set them within your theme options and so they aid the search engine to get a suitable title and description for the pages that get indexed by Google. These four plugins will make this effort easier and the sites run effectively as well as make other widgets added to a site easier to configure. W3C compliant HTML and a good open source powered by Word - Press CMS site is regarded as the prime minister. Photography is an entire activity in itself, and a thorough discovery of it is beyond the opportunity of this content. But if you are not willing to choose cost to the detriment of quality, originality and higher returns, then go for a self-hosted wordpress blog and increase the presence of your business in this new digital age. Setting Up Your Business Online Using Free Wordpress Websites. To turn the Word - Press Plugin on, click Activate on the far right side of the list. So, if you are looking online to hire dedicated Wordpress developers, India PHP Expert can give a hand you in each and every best possible way. If you adored this article so you would like to receive more info about [http://miniURL.fouiner.info/wordpress_backup_plugin_55395 wordpress dropbox backup] i implore you to visit the website. You can add keywords but it is best to leave this alone. Cameras with a pentaprism (as in comparison to pentamirror) ensure that little mild is lost before it strikes your eye, however these often increase the cost of the digital camera considerably. One of the great features of Wordpress is its ability to integrate SEO into your site. Fast Content Update - It's easy to edit or add posts with free Wordpress websites. OSDI, a Wordpress Development Company based on ahmedabad, India. Yet, overall, less than 1% of websites presently have mobile versions of their websites. Sanjeev Chuadhary is an expert writer who shares his knowledge about web development through their published articles and other resource. Offshore Wordpress development services from a legitimate source caters dedicated and professional services assistance with very simplified yet technically effective development and designing techniques from experienced professional Wordpress developer India. This is because of the customization that works as a keystone for a SEO friendly blogging portal website. Customers within a few seconds after visiting a site form their opinion about the site.