For many-electron systems such as atoms and molecules, it is obviously important that approximate wavefiinctions obey the same boundary conditions and symmetry properties as the exact solutions. Therefore, they should be antisynnnetric with respect to interchange of each pair of electrons. Such states can always be constmcted as linear combinations of products such as [Pg.31]

Figire 7. Betiding potential curves for the A A electronic system of NH2 [25], Full [Pg.499]

Figure 1. Jacobi vectors for a three-nuclei, four-electron system. The nuclei are Pi, P2, P3, and the electrons are ei, 02, 63, 64, |

Figure 6. Bending potential curves for the X Ai, A B electronic system of BH2 [33,34], Full hotizontal lines K —Q vibronic levels dashed lines /f — I levels dash-dotted lines K — 2 levels dotted lines K — 3 levels. Vibronic levels of the lower electronic state are assigned in benf notation, those of the upper state in linear notation (see text). Zero on the energy scale corresponds to the energy of the lowest vibronic level. |

The H4 system is the prototype for many four-elecbon reactions [34]. The basic tetrahedral sfructure of the conical intersection is preserved in all four-electron systems. It arises from the fact that the four electrons are contributed by four different atoms. Obviously, the tefrahedron is in general not a perfect one. This result was found computationally for many systems (see, e.g., [37]). Robb and co-workers [38] showed that the structure shown (a tetraradicaloid conical intersection) was found for many different photochemical transformations. Having the form of a tetrahedron, the conical intersection can exist in two enantiomeric structures. However, this feature is important only when chiral reactions are discussed. [Pg.340]

The so-called Flohenberg-Kolm [ ] theorem states that the ground-state electron density p(r) describing an A-electron system uniquely detemiines tlie potential V(r) in the Flamiltonian [Pg.2179]

The pseudopotential is derived from an all-electron SIC-LDA atomic potential. The relaxation correction takes into account the relaxation of the electronic system upon the excitation of an electron [44]- The authors speculate that ... the ability of the SIRC potential to produce considerably better band structures than DFT-LDA may reflect an extra nonlocality in the SIRC pseudopotential, related to the nonlocality or orbital dependence in the SIC all-electron potential. In addition, it may mimic some of the energy and the non-local space dependence of the self-energy operator occurring in the GW approximation of the electronic many body problem [45]. [Pg.2209]

Perdew J P and Zunger A 1981 Self-interaction correction to density-functional approximations for many-electron systems Phys. Rev. B 23 5048 [Pg.2230]

The war itself also drove the development of improved and miniaturised electronic components for creating oscillators and amplifiers and, ultimately, semiconductors, which made practical the electronic systems needed in portable eddy current test instruments. The refinement of those systems continues to the present day and advances continue to be triggered by performance improvements of components and systems. In the same way that today s pocket calculator outperforms the large, hot room full of intercormected thermionic valves that I first saw in the 50 s, so it is with eddy current instrumentation. Today s handheld eddy current inspection instrument is a powerful tool which has the capability needed in a crack detector, corrosion detector, metal sorter, conductivity meter, coating thickness meter and so on. [Pg.273]

More advanced teclmiques take into account quasiparticle corrections to the DFT-LDA eigenvalues. Quasiparticles are a way of conceptualizing the elementary excitations in electronic systems. They can be detennined in band stmcture calculations that properly include the effects of exchange and correlation. In the [Pg.2208]

We term the in-phase combination an aromatic transition state (ATS) and the out-of-phase combination an antiaromatic transition state (AATS). An ATS is obtained when an odd number of electron pairs are re-paired in the reaction, and an AATS, when an even number is re-paired. In the context of reactions, a system in which an odd number of electrons (3, 5,...) are exchanged is treated in the same way—one of the electron pairs may contain a single electron. Thus, a three-electron system reacts as a four-electron one, a five-electron system as a six-electron one, and so on. [Pg.346]

However, this procedure depends on the existence of the matrix G(R) (or of any pure gauge) that predicates the expansion in Eq. (90) for a full electronic set. Operationally, this means the preselection of a full electionic set in Eq. (129). When the preselection is only to a partial, truncated electronic set, then the relaxation to the truncated nuclear set in Eq. (128) will not be complete. Instead, the now tmncated set in Eq. (128) will be subject to a YM force F. It is not our concern to fully describe the dynamics of the truncated set under a YM field, except to say (as we have already done above) that it is the expression of the residual interaction of the electronic system on the nuclear motion. [Pg.157]

The most important classes of functionalized [60]fullerene derivatives, e.g. methanofullerenes [341, pyrrolidinofullerenes [35], Diels-Alder adducts [34i] and aziridinofullerene [36], all give rise to a cancellation of the fivefold degeneration of their HOMO and tlireefold degeneration of their LUMO levels (figure Cl.2.5). This stems in a first order approximation from a perturbation of the fullerene s 7i-electron system in combination with a partial loss of the delocalization. [Pg.2413]

See also in sourсe #XX -- [ Pg.172 , Pg.173 , Pg.174 , Pg.175 , Pg.176 , Pg.177 , Pg.178 , Pg.179 ]

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