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| In [[quantum mechanics]], specifically [[time-dependent density functional theory]], the '''Runge–Gross theorem''' ('''RG theorem''') shows that for a [[many-body system]] evolving from a given initial [[wavefunction]], there exists a [[one-to-one mapping]] between the potential (or potentials) in which the system evolves and the density (or densities) of the system. The potentials under which the theorem holds are defined up to an additive purely time-dependent function: such functions only change the phase of the wavefunction and leave the density invariant. Most often the RG theorem is applied to molecular systems where the [[electronic density]], ''ρ''('''r''',''t'') changes in response to an external [[scalar potential]], ''v''('''r''',''t''), such as a time-varying electric field.<ref>{{cite book|last=Marques|first=Miguel A. L.|coauthors=Eberhard K. U. Gross|title=Time-Dependent Density Functional Theory, in A Primer in Density Functional Theory|editor=Carlos Fiolhais, Fernando Nogueira, and Miguel Marques|publisher=Springer|year=2003|pages=144–151|isbn=978-3-540-03083-6|url=http://books.google.com/?id=mX793GABep8C&printsec=frontcover}}</ref>
| | If you are looking for a specific plugin, then you can just search for the name of the plugin. Good luck on continue learning how to make a wordpress website. Step-4 Testing: It is the foremost important of your Plugin development process. If you loved this informative article and you would like to receive more info with regards to [http://GET7.pw/wordpress_backup_plugin_661556 backup plugin] i implore you to visit our web-site. Dead links are listed out simply because it will negatively have an influence on the website's search engine rating. The top 4 reasons to use Business Word - Press Themes for a business website are:. <br><br>These folders as well as files have to copied and the saved. The higher your blog ranks on search engines, the more likely people will find your online marketing site. 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. It primarily lays emphasis on improving the search engine results of your website whenever a related query is typed in the search box. W3C compliant HTML and a good open source powered by Word - Press CMS site is regarded as the prime minister. <br><br>Here are a few reasons as to why people prefer Word - Press over other software's. The nominee in each category with the most votes was crowned the 2010 Parents Picks Awards WINNER and has been established as the best product, tip or place in that category. After age 35, 18% of pregnancies will end in miscarriage. You can allow visitors to post comments, or you can even allow your visitors to register and create their own personal blogs. Websites using this content based strategy are always given top scores by Google. <br><br>Word - Press installation is very easy and hassle free. php file in the Word - Press root folder and look for this line (line 73 in our example):. Next you'll go by way of to your simple Word - Press site. If you just want to share some picture and want to use it as a dairy, that you want to share with your friends and family members, then blogger would be an excellent choice. Now all you have to do is log into your Word - Press site making use of the very same username and password that you initially had in your previous site. <br><br>Website security has become a major concern among individuals all over the world. Mahatma Gandhi is known as one of the most prominent personalities and symbols of peace, non-violence and freedom. However, you must also manually approve or reject comments so that your website does not promote parasitic behavior. If this is not possible you still have the choice of the default theme that is Word - Press 3. Definitely when you wake up from the slumber, you can be sure that you will be lagging behind and getting on track would be a tall order. |
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| The Runge–Gross theorem provides the formal foundation of time-dependent density functional theory. It shows that the density can be used as the fundamental variable in describing quantum [[many-body system]]s in place of the wavefunction, and that all properties of the system are [[Functional (mathematics)|functionals]] of the density.
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| The theorem was published by [[Erich Runge]] and [[Eberhard K. U. Gross]] in 1984.<ref name="RG"/> As of January 2011, the original paper has been cited over 1,700 times.<ref>[[ISI Web of Knowledge]] cited reference search, 7 January 2011.</ref>
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| ==Overview==
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| The Runge–Gross theorem was originally derived for electrons moving in a [[scalar field|scalar external field]].<ref name="RG">{{cite journal|last=Runge|first=Erich|coauthors=E. K. U. Gross|year=1984|title=Density-Functional Theory for Time-Dependent Systems|journal=Phys. Rev. Lett.|volume=52|issue=12|pages=997–1000|doi=10.1103/PhysRevLett.52.997|bibcode=1984PhRvL..52..997R}}</ref> Given such a field denoted by ''v'' and the number of electron, ''N'', which together determine a [[Molecular Hamiltonian|Hamiltonian]] ''H<sub>v''</sub>, and an initial condition on the wavefunction Ψ(''t'' = ''t''<sub>0</sub>) = Ψ<sub>0</sub>, the evolution of the wavefunction is determined by the [[Schrödinger equation]] | |
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| :<math>\hat{H}_v(t)|\Psi(t)\rangle=i\frac{\partial}{\partial t}|\Psi(t)\rangle.</math>
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| At any given time, the ''N''-electron wavefunction, which depends upon 3''N'' spatial and ''N'' [[Spin (physics)|spin]] coordinates, determines the [[electronic density]] through integration as
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| :<math>\rho(\mathbf r,t)=N\sum_{s_1} \cdots \sum_{s_N} \int \ \mathrm d\mathbf r_2 \ \cdots \int\ \mathrm d\mathbf r_N \ |\Psi(\mathbf r_1,s_1,\mathbf r_2,s_2,...,\mathbf r_N,s_N,t)|^2.</math>
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| Two external potentials differing only by an additive time-dependent, spatially independent, function, ''c''(''t''), give rise to wavefunctions differing only by a [[phase factor]] exp(-''ic''(''t'')), and therefore the same electronic density. These constructions provide a mapping from an external potential to the electronic density:
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| :<math>v(\mathbf r,t)+c(t)\rightarrow e^{-ic(t)}|\Psi(t)\rangle\rightarrow\rho(\mathbf r,t).</math>
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| The Runge–Gross theorem shows that this mapping is invertible, modulo ''c''(''t''). Equivalently, that the density is a functional of the external potential and of the initial wavefunction on the space of potentials differing by more than the addition of ''c''(''t''):
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| :<math>\rho(\mathbf r,t)=\rho[v,\Psi_0](\mathbf{r},t)\leftrightarrow v(\mathbf r,t)=v[\rho,\Psi_0](\mathbf r,t)</math>
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| ==Proof==
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| Given two scalar potentials denoted as ''v''('''r''',''t'') and ''v''<nowiki>'</nowiki>('''r''',''t''), which differ by more than an additive purely time-dependent term, the proof follows by showing that the density corresponding to each of the two scalar potentials, obtained by solving the Schrödinger equation, differ.
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| The proof relies heavily on the assumption that the external potential can be expanded in a [[Taylor series]] about the initial time. The proof also assumes that the density vanishes at infinity, making it valid only for finite systems.
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| The Runge–Gross proof first shows that there is a one-to-one mapping between external potentials and current densities by invoking the [[Heisenberg picture|Heisenberg equation of motion]] for the current density so as to relate time-derivatives of the current density to spatial derivatives of the external potential. Given this result, the continuity equation is used in a second step to relate time-derivatives of the electronic density to time-derivatives of the external potential.
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| The assumption that the two potentials differ by more than an additive spatially independent term, and are expandable in a Taylor series, means that there exists an integer ''k'' ≥ 0, such that
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| :<math>u_{k}(\mathbf{r})\equiv\left.\frac{\partial^k}{\partial t^k}\big(v(\mathbf{r},t)-v'(\mathbf{r},t)\big)\right|_{t=t_0}</math>
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| is not constant in space. This condition is used throughout the argument. | |
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| ===Step 1===
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| From the [[Heisenberg picture|Heisenberg equation of motion]], the time evolution of the [[Probability current|current density]], '''j'''('''r''',''t''), under the external potential ''v''('''r''',''t'') which determines the Hamiltonian ''H''<sub>''v''</sub>, is
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| :<math>i\frac{\partial\mathbf j(\mathbf r,t)}{\partial t}=\langle\Psi(t)|[\hat{\mathbf{j}}(\mathbf r),\hat{H}_v(t)]|\Psi(t)\rangle.</math>
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| Introducing two potentials ''v'' and ''v''<nowiki>'</nowiki>, differing by more than an additive spatially constant term, and their corresponding current densities '''j''' and '''j'''<nowiki>'</nowiki>, the Heisenberg equation implies
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| :<math>
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| \begin{align}
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| i\left.\frac{\partial}{\partial t}\big(\mathbf j(\mathbf r,t)-\mathbf j'(\mathbf r,t) \big)\right|_{t=t_0} &= \langle\Psi(t_0)|[\hat{\mathbf{j}}(\mathbf r),\hat{H}_{v}(t_0)-\hat{H}_{v'}(t_0)]|\Psi(t_0)\rangle,\\
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| &=\langle\Psi(t_0)|[\hat{\mathbf{j}}(\mathbf r),\hat{V}(t_0)-\hat{V}'(t_0)]|\Psi(t_0)\rangle,\\
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| &= i\rho(\mathbf r,t_0)\nabla\big(v(\mathbf{r},t_0)-v'(\mathbf{r},t_0)\big).
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| \end{align}
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| </math>
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| The final line shows that if the two scalar potentials differ at the initial time by more than a spatially independent function, then the current densities that the potentials generate will differ infinitesimally after ''t''<sub>0</sub>. If the two potentials do not differ at ''t''<sub>0</sub>, but ''u''<sub>''k''</sub>('''r''') ≠ 0 for some value of ''k'', then repeated application of the Heisenberg equation shows that | |
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| :<math>i^{k+1}\left.\frac{\partial^{k+1}}{\partial t^{k+1}}\big(\mathbf j(\mathbf r,t)-\mathbf j'(\mathbf r,t)\big)\right|_{t=t_0}=i\rho(\mathbf r,t)\nabla i^k\left.\frac{\partial^{k}}{\partial t^{k}}\big(v(\mathbf{r},t)-v'(\mathbf{r},t) \big)\right|_{t=t_0},</math>
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| ensuring the current densities will differ from zero infinitesimally after ''t''<sub>0</sub>.
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| ===Step 2===
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| The electronic density and current density are related by a [[continuity equation]] of the form
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| :<math>\frac{\partial\rho(\mathbf r,t)}{\partial t}+\nabla\cdot\mathbf j(\mathbf r,t)=0.</math>
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| Repeated application of the continuity equation to the difference of the densities ''ρ'' and ''ρ''<nowiki>'</nowiki>, and current densities '''j''' and '''j'''<nowiki>'</nowiki>, yields
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| :<math>
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| \begin{align}
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| \left.\frac{\partial^{k+2}}{\partial t^{k+2}}(\rho(\mathbf r,t)-\rho'(\mathbf r,t))\right|_{t=t_0}&=-\nabla\cdot\left.\frac{\partial^{k+1}}{\partial t^{k+1}}\big(\mathbf j(\mathbf r,t)-\mathbf j'(\mathbf r,t)\big)\right|_{t=t_0},\\
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| &=-\nabla\cdot[\rho(\mathbf r,t_0)\nabla\left.\frac{\partial^k}{\partial t^k}\big(v(\mathbf{r},t_0)-v'(\mathbf{r},t_0)\big)\right|_{t=t_0}],\\
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| &=-\nabla\cdot[\rho(\mathbf r,t_0)\nabla u_k(\mathbf r)].
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| \end{align}
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| </math>
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| The two densities will then differ if the right-hand side (RHS) is non-zero for some value of ''k''. The non-vanishing of the RHS follows by a [[reductio ad absurdum]] argument. Assuming, contrary to our desired outcome, that
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| :<math>\nabla\cdot(\rho(\mathbf r,t_0)\nabla u_k(\mathbf r)) = 0,</math>
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| integrate over all space and apply Green's theorem.
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| :<math>
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| \begin{align}
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| 0&=\int\mathrm d\mathbf r\ u_k(\mathbf r)\nabla\cdot(\rho(\mathbf r,t_0)\nabla u_k(\mathbf r)),\\
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| &=-\int\mathrm d\mathbf r\ \rho(\mathbf r,t_0)(\nabla u_k(\mathbf r))^2+\frac{1}{2}\int \mathrm d\mathbf S\cdot\rho(\mathbf r,t_0)(\nabla u_k^2(\mathbf r)).
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| \end{align}
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| </math>
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| The second term is a surface integral over an infinite sphere. Assuming that the density is zero at infinity (in finite systems, the density decays to zero exponentially) and that ∇''u<sub>k''</sub><sup>2</sup>('''r''') increases slower than the density decays,<ref>{{cite journal|last=Dhara|first=Asish K.|coauthors=Swapan K. Ghosh|year=1987|title=Density-functional theory for time-dependent systems|journal=Phys. Rev. A|volume=35|issue=1|pages=442–444|doi=10.1103/PhysRevA.35.442|bibcode = 1987PhRvA..35..442D }}</ref> the surface integral vanishes and, because of the non-negativity of the density,
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| :<math>\rho(\mathbf r,t_0)(\nabla u_k(\mathbf r))^2=0,</math>
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| implying that ''u<sub>k''</sub> is a constant, contradicting the original assumption and completing the proof.
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| ==Extensions==
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| The Runge–Gross proof is valid for pure electronic states in the presence of a scalar field. The first extension of the RG theorem was to time-dependent [[Statistical ensemble (mathematical physics)|ensembles]], which employed the [[Liouville equation#Quantum mechanics|Liouville equation]] to relate the Hamiltonian and [[density matrix]].<ref>{{cite journal|last=Li|first=Tie-cheng|coauthors=Pei-qing Tong|year=1985|title=Hohenberg-Kohn theorem for time-dependent ensembles|journal=Phys. Rev. A|volume=31|issue=3|pages=1950–1951|doi=10.1103/PhysRevA.31.1950|bibcode = 1985PhRvA..31.1950L }}</ref> A proof of the RG theorem for multicomponent systems—where more than one type of particle is treated within the full quantum theory—was introduced in 1986.<ref>{{cite journal|last=Li|first=Tie-Cheng|coauthors=Pei-qing Tong|year=1986|title=Time-dependent density-functional theory for multicomponent systems|journal=Phys. Rev. A|volume=34|issue=1|pages=529–532|doi=10.1103/PhysRevA.34.529|bibcode = 1986PhRvA..34..529L }}</ref> Incorporation of magnetic effects requires the introduction of a [[vector potential]] ('''A'''('''r''')) which together with the scalar potential uniquely determine the current density.<ref>{{cite journal|last=Ghosh|first=Swapan K.|coauthors=Asish K. Dhara|year=1988|title=Density-functional theory of many-electron systems subjected to time-dependent electric and magnetic fields|journal=Phys. Rev. A|volume=38|issue=3|pages=1149–1158|doi=10.1103/PhysRevA.38.1149|bibcode = 1988PhRvA..38.1149G }}</ref><ref>{{cite journal|last=Vignale|first=Giovanni|year=2004|title=Mapping from current densities to vector potentials in time-dependent current density functional theory|journal=Phys. Rev. B|volume=70|issue=20|pages=201102|doi=10.1103/PhysRevB.70.201102|arxiv = cond-mat/0407682 |bibcode = 2004PhRvB..70t1102V }}</ref> Time-dependent density functional theories of [[superconductivity]] were introduced in 1994 and 1995.<ref>{{cite journal|last=Wacker|first=O. -J.|coauthors=R. Kümmel and E. K. U. Gross|year=1994|title=Time-Dependent Density-Functional Theory for Superconductors|journal=Phys. Rev. Lett.|volume=73|issue=21|pages=2915–2918|doi=10.1103/PhysRevLett.73.2915|bibcode=1994PhRvL..73.2915W}}</ref><ref>{{cite journal|last=Rajagopal|first=A. K.|coauthors=F. A. Buot|year=1995|title=Time-dependent functional theory for superconductors|journal=Phys. Rev. B|volume=52|issue=9|pages= 6769–6774|doi=10.1103/PhysRevB.52.6769|bibcode = 1995PhRvB..52.6769R }}</ref> Here, scalar, vector, and [[pairing potential|pairing]] (''D''(''t'')) potentials map between current and [[anomalous density|anomalous]] (Δ<sub>IP</sub>('''r''',''t'')) densities.
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:Runge-Gross theorem}}
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| [[Category:Density functional theory]]
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| [[Category:Theorems in quantum physics]]
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