Quantum mechanics of many-electron systems

By first-principles calculation we mean solving the Schrodinger equation 

Consider the helium atom with two electrons. The coordinate of electron 1 is ri while the coordinate of electron 2 is ra. The Schrodinger equation is then 

From now on, we will call the single-particle functions y/a and y/b orbitals . A solution in the form of Equation (4) is called the Hartree approximation. It would be exact if the electrons did not interact. The problem, however, is that not only do electrons interact, they must be indistinguishable from each other. Consequently, 

It turns out that, for particles with half-integral spin (such as electrons), only the antisymmetric wavefunctions (Eqn. 6b) are allowed. This is a rather abstract statement of the Pauli Exclusion Principle. The antisymmetric wavefunction has an interesting property if the two single-particle orbitals are the same, then the two electrons cannot 

SlatdtiioA Once we have more than two electrons in the atom or 

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Dirac, P. A. M. 1929 Quantum mechanics of many-electron systems. Proc. R. Soc. Land A 123, 714-733. 

The conclusion that it may be possible to formulate the quantum mechanics of many-electron systems solely in terms of the single-particle density was put on a firm foundation by the two Hohenberg-Kohn theorems (1964), which are stated below, without proof. 

P. A. M. Dirac, "Quantum Mechanics of Many-Electron Systems," Proc.RSL A123 (1929) 714733. See Kragh, Dirac, 76. 

Quantum Mechanics of Many-Electron Systems." Proc.RSL A123 (1929) 714733. 

Dirac, the discoverer of the relativistic one-electron equation, thought that relativity would be unimportant in chemistry (P. A. M. Dirac, Quantum Mechanics of Many-Electron Systems , Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 1929, 123(192), 714). Why was he mistaken ... 

P.A. M. Dirac, Quantum mechanics of many electron systems," Proceedings of the Royal Society of London A123 (1929) 714-733, on 714. 

Proceedings of the Royal Society of London Quantum Mechanics of Many-Electron Systems (p. 714) Volume 123, Number 792, 6 April 1929... 

Thirty years ago, Hohenberg and Kohn [1] proved a theorem that soon provided an alternative way to solve the Schrodinger equation [2]. Since its introduction in 1926, solving the Schrodinger equation has been the focus of a great deal of attention. Two years later, in his paper on the quantum mechanics of many-electron systems, Dirac stated [3] ... 

Perspective on "Quantum mechanics of many-electron systems" 182... 

Perspective on Quantum mechanics of many-electron systems... 

There is, however, one paper of Dirac [5] that keeps being cited, namely the one at which we want to have a look now. Most people who cite this paper hardly know that its title is Quantum mechanics of many-electron systems and are unaware of its scientific context. It deals mainly with the relation between permutation symmetry and spin and contains a formula which relates the expectation value of the operator of electron exchange to the total spin of the state. 

Nascimento M, Barbosa A (2003) Quantum mechanics of many-electrons systems and the theories of chemical bond. In Brandas EJ, Kryachko E (eds) Fundamental world of quantum chemistry, vol 1. Kluwer, Dordrecht, p 371... 

Dirac PAM (1929) Quantum Mechanics of many-electron systems. Proc R Soc Lond A123 714... 

We underlined the word explanation to emphasize that, besides computations, there is a part of modeling that necessitates interpretation of computations. In 1929, when Dirac wrote his paper on Quantum Mechanics of Many-Electron Systems (from which the above quotations were taken), he could not have anticipated the emergence of fast computers that would made it possible to get involved in heavy computations. But heavy computations do not eliminate the need for an explanation of the main features of complex systems. True, what was complex in 1929 need not be equally complex a hundred years later, but a hundred years later other complex systems will emerge, to which the above Dirac quotation will apply as it did in 1929. Development of fast computers tends to give an impression or illusion that there are no limits to computational capabilities. In this age of euphoria of computerization, some may have overlooked the fact that, as computers may have no limitations, there is also no limit to the complexity of molecular systems, particularly as we drift into molecular biology and cell... 

II. ELEMENTS OF THE QUANTUM MECHANICS OF MANY-ELECTRON SYSTEMS... 

In this section we recapitulate a few selected elements of the quantum mechanics of many-electron systems that are useful for understanding and using DFT. 

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