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Natural Sciences     Social Sciences     Humanities     Arts     Applied Arts and Sciences     Recreation      Click on first button above for full list of workgroups An encyclopedia project—and more! Welcome to Citizendium, an endeavor to achieve the highest standards of writing, reliability, and comprehensiveness through a unique collaboration between Authors and Editors. We welcome people from all walks of life who want to share their knowledge and expertise freely by writing well-researched and authoritative articles. Please read through our easy registration procedures, then join us. We have 16,602 articles at different stages of development, of which 0 are expert-approved. Eduzendium students in higher education write here as part of their courses. We welcome everyone who has knowledge, broad or narrow, about any subject. Our community is collegial and congenial; everyone writes under his or her real name. Even this, our home page, can be edited by any Citizen. Sign up—we need both Authors and Editors. Get started: How to collaborate | CZ:Quick Start We have a Charter to provide a framework for governing Citizendium. See our own Citizendium article for our history Why Citizendium? | Myths and Facts | Why we contribute to CZ Follow us on Friendfeed, Twitter (new drafts, general news), Facebook or Google+ Join us Join up here and start a new article or improve existing ones. See Recent Changes—an overview of articles we are writing now. Wikipedians are very welcome! Comment on articles nominated for approval. You can discuss any article on its Talk page. From HERE, you can get started, get technical help, see our policies, and explore our articles and organization. Our help system (still under development) We Need Support From Donations! Please help! We have a continuing need for funds to pay for hosting our servers. See our CZ:Financial report for complete details. Please make your donations here. See also our sister project, WatchKnow, a free, non-profit, K-12 educational video website.

Some of our finest Approved Articles (0) Developed Articles (1,130) (16,602 total articles) Who dares to teach must never cease to learn. — John Cotton Dana (1856–1929), American librarian and museum director.        —add a quotation about knowledge or writing •Today in History | •Did You Know? Featured Article about In computational molecular physics and solid state physics, the Born-Oppenheimer approximation is used to separate the quantum mechanical motion of the electrons from the motion of the nuclei. The method relies on the large mass ratio of electrons and nuclei. For instance the lightest nucleus, the hydrogen nucleus, is already 1836 times heavier than an electron. The method is named after Max Born and Robert Oppenheimer, who proposed it in 1927. Rationale The computation of the energy and wave function of an average-size molecule is a formidable task that is alleviated by the Born-Oppenheimer (BO) approximation.The BO approximation makes it possible to compute the wave function in two less formidable, consecutive, steps. This approximation was proposed in the early days of quantum mechanics by Born and Oppenheimer (1927) and is indispensable in quantum chemistry and ubiquitous in large parts of computational physics. In the first step of the BO approximation the electronic Schrödinger equation is solved, yielding a wave function depending on electrons only. For benzene this wave function depends on 126 electronic coordinates. During this solution the nuclei are fixed in a certain configuration, very often the equilibrium configuration. If the effects of the quantum mechanical nuclear motion are to be studied, for instance because a vibrational spectrum is required, this electronic computation must be repeated for many different nuclear configurations. The set of electronic energies thus computed becomes a function of the nuclear coordinates. In the second step of the BO approximation this function serves as a potential in a Schrödinger equation containing only the nuclei—for benzene an equation in 36 variables. The success of the BO approximation is due to the high ratio between nuclear and electronic masses. The approximation is an important tool of quantum chemistry, without it only the lightest molecule, H2, could be handled; all computations of molecular wave functions for larger molecules make use of it. Even in the cases where the BO approximation breaks down, it is used as a point of departure for the computations. Historical note The Born-Oppenheimer approximation is named after M. Born and R. Oppenheimer who wrote a paper [Annalen der Physik, vol. 84, pp. 457-484 (1927)] entitled: Zur Quantentheorie der Molekeln (On the Quantum Theory of Molecules). This paper describes the separation of electronic motion, nuclear vibrations, and molecular rotation. A reader of this paper who expects to find clearly delineated the BO approximation—as it is explained above and in most modern textbooks—will be disappointed. The presentation of the BO approximation is well hidden in Taylor expansions (in terms of internal and external nuclear coordinates) of (i) electronic wave functions, (ii) potential energy surfaces and (iii) nuclear kinetic energy terms. Internal coordinates are the relative positions of the nuclei in the molecular equilibrium and their displacements (vibrations) from equilibrium. External coordinates are the position of the center of mass and the orientation of the molecule. The Taylor expansions complicate the theory tremendously and make the derivations very hard to follow. Moreover, knowing that the proper separation of vibrations and rotations was not achieved in this work, but only eight years later [by C. Eckart, Physical Review, vol. 46, pp. 383-387 (1935)] (see Eckart conditions), chemists and molecular physicists are not very much motivated to invest much effort into understanding the work by Born and Oppenheimer, however famous it may be. Although the article still collects many citations each year, it is safe to say that it is not read anymore, except maybe by historians of science. Footnotes