Four simple three-electron systems Table 10.1. C2v characters. [Pg.126]

Consider the Dirac-Fock equations for a three-electron system Is nlj. Formally they fall into one-electron Dirac equations for the orbitals l5 and nlj with the potential [Pg.290]

Table 5-2. [dentification of Couiombic and Exchange Integrals for the Three-Electron System [Pg.195]

In this chapter we describe four rather different three-electron systems the it system ofthe allyl radical, the HeJ ionic molecule, the valence orbitals ofthe BeHmolecule, and the Li atom. In line with the intent of Chapter 4, these treatments are included to introduce the reader to systems that are more complicated than those of Chapters 2 and 3, but simple enough to give detailed illustrations of the methods of Chapter 5. In each case we will examine MCVB results as an example of localized orbital treatments and SCVB results as an example of delocalized treatments. Of course, for Li this distinction is obscured because there is only a single nucleus, but there are, nevertheless, noteworthy points to be made for that system. The reader should refer back to Chapter 4 for a specific discussion of the three-electron spin problem, but we will nevertheless use the general notation developed in Chapter 5 to describe the results because it is more efficient. [Pg.125]

Lithium has one more electron but the Is orbital is already full. The third electron must go into the next lowest orbital and that is the 2s. In this three-electron system, like that of the two-electron He atom, the three 2p orbitals are higher in energy than the 2s orbital. By the time we come to boron, with five electrons, the 2s is full as well and we must put the last electron into a 2p orbital. It doesn t matter which one they are degenerate. [Pg.94]

F. A. Matsen, J. Phys. Chem. 68, 3282 (1964). Spin-Free Quantum Chemistry. II. Three-Electron Systems. [Pg.23]

We now have a significant difiference from the case of two electrons in a singlet state, namely, we have two spin functions to combine with spatial functions for a solution to the ESE rather than only one. For a doublet three-electron system our general solution must be [Pg.55]

V i hen the Slater determinant is expanded, a total of N1 terms results. This is because there are N different permutations of N eleefrons. For example, for a three-electron system with spin orbitals X2 and xs the determinant is [Pg.59]

In Chapter 5 we give an analysis of VB functions that is general for any number of electrons. In order to motivate some of the considerations we discuss there we first give a detailed example of the requirements when one is to constmct an antis5munetric doublet eigenfunction of the spin for a three-electron system. Pauncz[36] has written a useful workbook on this subject. [Pg.53]

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]

These authors proposed a uniform mean-field HF (UMHF) procedure which involves orbital occupancy constraints and correction of resonance energy by non-empirical factors. This UMHF method yields the dissociation energies of three-electron systems in satisfactory agreement with accurate calculation performed in the same basis set. [Pg.26]

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