Electron-pair

Electron pairs are as important in physics as in chemistry. Below a critical temperature, Tc, some metals become superconductors (SC) without resistance. Experiments show that the charge carriers are electron pairs. Calculated heat capacities using the Bose-Einstein statistics agree very well with the experimental results, and this is also consistent with pair formation among some of the electrons. 

However, physicists and chemists tend to have different views on electron pairing. Physicists view electrons as free in metals, and feel obliged to find an attractive force that makes pairing between equally charged particles possible. To chemists, on the other hand, pairing is commonplace (bond pairs, lone pairs, disproportionation). In most cases, electron pairs are strongly bound, and have large activation barriers for pair transfer, but there is no particular reason why this should be the case. 

In this chapter, we will take the chemist s view, and use the same theory that we have used throughout this book. We will show that pair transfer is possible without activation energy under special circumstances, independent of the binding energy. 

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Figure Al.7.4. Schematic illustration of two Si atoms as they would be oriented on the (100) surface, (a) Bulk-tenuiuated structure showing two dangling bonds (lone electrons) per atom, (b) Synnnetric dimer, in which two electrons are shared and each atom has one remaining dangling bond, (c) Asynnnetric dimer in which two electrons pair up on one atom and the otiier has an empty orbital.

Flere two electrons occupy the 1 s orbital (with opposite, a and p spins) while the other electron pair resides in 2s-2p polarized orbitals in a maimer that instantaneously correlates their motions. These polarized orbital... 

Let us consider another example. In describmg the n electron pair of an olefin, it is important to mix in doubly excited configurations of the fomi (n ). The physical importance of such configurations can again be made clear by using the identity... 

In this example, tlie two non-orthogonal polarized orbital pairs involve mixing the k and k orbitals to produce two left-right polarized orbitals as depicted in figure B3.1.7. Here one says that the n electron pair undergoes left-right correlation when the (n configuration is introduced. 

For these reasons, in the MCSCF method the number of CSFs is usually kept to a small to moderate number (e.g. a few to several thousand) chosen to describe essential correlations (i.e. configuration crossings, near degeneracies, proper dissociation, etc, all of which are often tenned non-dynamicaI correlations) and important dynamical correlations (those electron-pair correlations of angular, radial, left-right, etc nature that are important when low-lying virtual orbitals are present). 

The need for such large CSF expansions should not be surprising considering (i) that each electron pair requires at least two CSFs to fomi polarized orbital pairs, (ii) there are of the order of N N - 1 )/2 = X electron pairs for N electrons, hence (iii) the number of temis in the Cl wavefiinction scales as 1. For a molecule containing ten electrons, there could be 2 =3.5x10 temis in the Cl expansion. This may be an overestimate of the number of CSFs needed, but it demonstrates how rapidly the number of CSFs can grow with the number of electrons. 

Stabilizing resonances also occur in other systems. Some well-known ones are the allyl radical and square cyclobutadiene. It has been shown that in these cases, the ground-state wave function is constructed from the out-of-phase combination of the two components [24,30]. In Section HI, it is shown that this is also a necessary result of Pauli s principle and the permutational symmetry of the polyelectronic wave function When the number of electron pairs exchanged in a two-state system is even, the ground state is the out-of-phase combination [28]. Three electrons may be considered as two electron pairs, one of which is half-populated. When both electron pahs are fully populated, an antiaromatic system arises ("Section HI). 

By using the determinant fomi of the electronic wave functions, it is readily shown that a phase-inverting reaction is one in which an even number of election pairs are exchanged, while in a phase-preserving reaction, an odd number of electron pairs are exchanged. This holds for Htickel-type reactions, and is demonstrated in Appendix A. For a definition of Hilckel and Mbbius-type reactions, see Section III. 

The concept of phase change in chemical reactions, was introduced in Section I, where it was shown that it is related to the number of electron pairs exchanged in the course of a reaction. In every chemical reaction, the fundamental law to be observed is the preservation pemiutational symmetry of... 

A more general classification considers the phase of the total electronic wave function [13]. We have treated the case of cyclic polyenes in detail [28,48,49] and showed that for Hiickel systems the ground state may be considered as the combination of two Kekule structures. If the number of electron pairs in the system is odd, the ground state is the in-phase combination, and the system is aromatic. If the number of electron pairs is even (as in cyclobutadiene, pentalene, etc.), the ground state is the out-of-phase combination, and the system is antiaromatic. These ideas are in line with previous work on specific systems [40,50]. 

Even of electron pairs exchanged Anti-aromatic system... 

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. 

Finally, the distinction between Huckel and Mobius systems is considered. The above definitions are valid for Hiickel-type reactions. For aromatic Mobius-type reations, the reverse holds An ATS is formed when an even number of electron pairs is re-paired. 

A phase change takes place when one enantiomer is converted to its optical isomer. As illustrated in Figure 9, when the chiral center is a tetra-substituted carbon atom, the conversion of one enantiomer to the other is equivalent to the exchange of two electron pairs. This transformation is therefore phase inverting. 

Next, we consider one pair of it electrons and one pair of cj elections. The cj electrons may originate from a CH or from a CC bond. Let us consider the loop enclosed by the three anchors formed when the electron pair comes from a C-H bond. There are only three possible pairing options. The hydrogen-atom originally bonded to carbon atom 1, is shifted in one product to carbon atom 2,... 

The next simplest loop would contain at least one reaction in which three electron pairs are re-paired. Inspection of the possible combinations of two four-electron reactions and one six-electron reaction starting with CHDN reveals that they all lead to phase preseiwing i p loops that do not contain a conical intersection. It is therefore necessary to examine loops in which one leg results in a two electron-pair exchange, and the other two legs involve three elechon-pair exchanges fip loops). As will be discussed in Section VI, all reported products (except the helicopter-type elimination of H2) can be understood on the basis of four-electron loops. We therefore proceed to discuss the unique helicopter... 

The transformation of ethylene to the carbene requires the re-pairing of three electron pairs. It is a phase-preserving reaction, so that the loop is an ip one. The sp -hybridized carbon atom formed upon H transfer is a chiral center consequently, there are two equivalent loops, and thus conical intersections, leading to two enantiomers. 

In the reaction, compound A transforms to B the total number of electron pairs is preserved, but at least four electrons are assumed to change spin partners. 

With 4) containing a normalization factor and all permutations over the atomic orbital wave functions i (1 = 1,2,... 2n). Likewise, if all electron pairs were exchanged in a cyclic manner, the product wave function, 4>b, has the fonn ... 

According to Eq. (A.4), if < 0, the ground state will be the in-phase combination, and the out-of-phase one, an excited state. On the other hand, if > 0, the ground state will be the out-of-phase combination, while the in-phase one is an excited state. This conclusion is far reaching, since it means that the electronic wave function of the ground state is nonsymmetric in this case, in contrast with common chemical intuition. We show that when an even number of electron pairs is exchanged, this is indeed the case, so that the transition state is the out-of-phase combination. 

Table 2.6 shows the electron affinities, for the addition of one electron to elements in Periods 2 and 3. Energy is evolved by many atoms when they accept electrons. In the cases in which energy is absorbed it will be noted that the new electron enters either a previously unoccupied orbital or a half-filled orbital thus in beryllium or magnesium the new electron enters the p orbital, and in nitrogen electron-pairing in the p orbitals is necessary. 

Unlike the forces between ions which are electrostatic and without direction, covalent bonds are directed in space. For a simple molecule or covalently bonded ion made up of typical elements the shape is nearly always decided by the number of bonding electron pairs and the number of lone pairs (pairs of electrons not involved in bonding) around the central metal atom, which arrange themselves so as to be as far apart as possible because of electrostatic repulsion between the electron pairs. Table 2.8 shows the essential shape assumed by simple molecules or ions with one central atom X. Carbon is able to form a great many covalently bonded compounds in which there are chains of carbon atoms linked by single covalent bonds. In each case where the carbon atoms are joined to four other atoms the essential orientation around each carbon atom is tetrahedral. 

The shapes indicated in Table 2.8 are only exact in cases in which all the electron pairs are equivalent, i.e. they are all bonding pairs. 

Double and triple covalent bonds can be formed between elements by the sharing of two or three electron pairs respectively. Consider the formation of ethene (ethylene), C2H4 ... 

These apparent anomalies are readily explained. Elements in Group V. for example, have five electrons in their outer quantum level, but with the one exception of nitrogen, they all have unfilled (I orbitals. Thus, with the exception of nitrogen. Group V elements are able to use all their five outer electrons to form five covalent bonds. Similarly elements in Group VI, with the exception of oxygen, are able to form six covalent bonds for example in SF. The outer quantum level, however, is still incomplete, a situation found for all covalent compounds formed by elements after Period 2. and all have the ability to accept electron pairs from other molecules although the stability of the compounds formed may be low. This... 

Phosphorus pentafluoride PF will readily accept an electron pair from a fluoride ion F to form the stable hexafluorophosphate(V) anion PF C. This ion is isoelectronic with SF. and neither SF nor PF show any notable tendency to accept further electron pairs, though there is some evidence for the existence of an SF ion. 

The formation of a fourth covalent bond by the aluminium atom results in spatial rearrangement from the trigonal planar, for three bonding electron pairs, to tetrahedral, for four bonding electron pairs. 

This compound, which contains atoms arranged tetrahedrally around the boron atom, can readily be isolated from a mixture of dimethyl ether and boron trichloride. On occasions a chlorine atom, in spite of its high election affinity, will donate an electron pair, an example being found in the dimerisation of gaseous monomeric aluminium chloride to give the more stable Al2Clg in which each aluminium has a tetrahedral configuration ... 

The strongly electronegative (p. 49) chlorine atom becomes a chloride ion, the proton accepting the electron pair donated by the nitrogen atom. A similar reaction occurs when ammonia is passed into water, but to a much lesser extent as oxygen in water is a poorer donor of the electron pair ... 

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