Odd Numbers of Electrons

Molecules and polyatomic ions containing an odd number of electrons 

Molecules and polyatomic ions in which an atom has fewer than an octet of valence 

In the vast majority of molecules and polyatomic ions, the total number of valence electrons is even, and complete pairing of electrons occurs. However, in a few molecules and polyatomic ions, such as CIO2, NO, NO2, and O2, the number of valence electrons is odd. Complete pairing of these electrons is impossible, and an octet around each atom cannot be achieved. For example, NO contains 5 -I- 6 = 11 valence electrons. The two most important Lewis structures for this molecule are 

Which of the Lewis structures for NO is dominant based on analysis of the formal charges  

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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. 

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. 

In this chapter, recent advances in the theory of conical intersections for molecules with an odd number of electrons are reviewed. Section II presents the mathematical basis for these developments, which exploits a degenerate perturbation theory previously used to describe conical intersections in nonrelativistic systems [11,12] and Mead s analysis of the noncrossing rule in molecules with an odd number of electrons [2], Section III presents numerical illustrations of the ideas developed in Section n. Section IV summarizes and discusses directions for future work. 

Industrially nitrogen monoxide is prepared by the catalytic oxidation of ammonia as an intermediate in the manufacture of nitric acid (p. 238). The molecule of nitrogen monoxide contains an odd number of electrons and can be represented as... 

Cobalt has an odd number of electrons, and does not form a simple carbonyl in oxidation state 0. However, carbonyls of formulae Co2(CO)g, Co4(CO)i2 and CoJCO),6 are known reduction of these by an alkali metal dissolved in liquid ammonia (p. 126) gives the ion [Co(CO)4] ". Both Co2(CO)g and [Co(CO)4]" are important as catalysts for organic syntheses. In the so-called oxo reaction, where an alkene reacts with carbon monoxide and hydrogen, under pressure, to give an aldehyde, dicobalt octacarbonyl is used as catalyst ... 

This result applies when the number of up spins equals the number of down spins and so is not applicable to systems with an odd number of electrons. The correlation energy functional was also considered by Vosko, Wdk and Nusarr [Vosko et al. 1980], whose expression is ... 

Free radicals are species that contain unpaired electrons The octet rule notwithstand mg not all compounds have all of their electrons paired Oxygen (O2) is the most famil lar example of a compound with unpaired electrons it has two of them Compounds that have an odd number of electrons such as nitrogen dioxide (NO2) must have at least one unpaired electron... 

We say the molecule AB has been ionized by electron impact The species that results called the molecular ion, is positively charged and has an odd number of electrons—it IS a cation radical The molecular ion has the same mass (less the negligible mass of a single electron) as the molecule from which it is formed... 

To define the state you want to calculate, you must specify the multiplicity. A system with an even number of electrons usually has a closed-shell ground state with a multiplicity of 1 (a singlet). Asystem with an odd number of electrons (free radical) usually has a multiplicity of 2 (a doublet). The first excited state of a system with an even number of electrons usually has a multiplicity of 3 (a triplet). The states of a given multiplicity have a spectrum of states —the lowest state of the given multiplicity, the next lowest state of the given multiplicity, and so on. 

If the number of electrons, N, is even, you can have a closed shell (as shown) where the occupied orbitals each contain two electrons. For an odd number of electrons, at least one orbital must be singly occupied. In the example, three orbitals are occupied by electrons and two orbitals are unoccupied. The highest occupied molecular orbital (HOMO) is /3, and the lowest unoccupied molecular orbital (LUMO) is 11/4. The example above is a singlet, a state of total spin S=0. Exciting one electron from the HOMO to the LUMO orbital would give one of the following excited states ... 

Having gained one electron, the alkyl halide is now negatively chaiged and has an odd number of electrons. It is an anion radical. The extra electron occupies an antibonding orbital. This anion radical fragments to an alkyl radical and a halide anion. 

There are some notable exceptions. Free radicals are molecules that contain an odd number of electrons. Since the number of spin up and spin down electrons in a free radical cannot be equal, the spin up and spin down electron clouds cannot be identical. 

A more general way to treat systems having an odd number of electrons, and certain electronically excited states of other systems, is to let the individual HF orbitals become singly occupied, as in Figure 6.3. In standard HF theory, we constrain the wavefunction so that every HF orbital is doubly occupied. The idea of unrestricted Hartree-Fock (UHF) theory is to allow the a and yS electrons to have different spatial wavefunctions. In the LCAO variant of UHF theory, we seek LCAO coefficients for the a spin and yS spin orbitals separately. These are determined from coupled matrix eigenvalue problems that are very similar to the closed-shell case. 

A radical is highly reactive because it contains an atom with an odd number of electrons (usually seven) in its valence shell, rather than a stable, noble-gas octet. A radical can achieve a valencC Shel octet in several wavs, for example, the radical might abstract an atom and one bonding electron from another reactant, leaving behind a new radical. The net result is a radical substitution reaction ... 

Both Cope and Claisen rearrangements involve reorganization of an odd number of electron pairs (two tt bonds and one a bond), and both react by suprafacial pathways (Figure 30.13). 

A change either from thermal to photochemical or from an even to an odd number of electron pairs changes the outcome from conrotatory/antarafacial to dis-rotatory/suprafacial. A change from both thermal and even to photochemical and odd causes no change because two negatives make a positive. 

Radical (Section 5.2) A species that has an odd number of electrons, such as the chlorine radical, Cl. ... 

The octet rule accounts for the valences of many of the elements and the structures of many compounds. Carbon, nitrogen, oxygen, and fluorine obey the octet rule rigorously, provided there are enough electrons to go around. However, some compounds have an odd number of electrons. In addition, an atom of phosphorus, sulfur, chlorine, or another nonmetal in Period 3 and subsequent periods can accommodate more than eight electrons in its valence shell. The following two sections show how to recognize exceptions to the octet rule. 

Nitrogen dioxide, N02 (oxidation number -t-4), is a choking, poisonous, brown gas that contributes to the color and odor of smog. The molecule has an odd number of electrons, and in the gas phase it exists in equilibrium with its colorless dimer N204. Only the dimer exists in the solid, and so the brown gas condenses to a colorless solid. When it dissolves in water, NOz disproportionates into nitric acid (oxidation number +5) and nitrogen oxide (oxidation number +2) ... 

Chlorine dioxide has an odd number of electrons and is a paramagnetic yellow gas. Despite the environmental damage it creates, it is often used to bleach paper pulp, because it can oxidize the various pigments in the pulp without degrading the wood fibers. 

In carrying out he calculations we use essentially the same procedure as in the case of benzene and naphthalene. As an additional simplification, however, we neglect entirely all the excited states of the molecule, since their contribution to the total energy is comparatively small, and since they would complicate the calculations tremendously if retained. Another slight modification of the procedure is necessitated by the fact that a free radical possesses an odd number of electrons, one of which must remain unpaired. This is taken care of formally by introducing a phantom orbit X with an accompanying phantom electron which is paired with the odd electron.4 In the subsequent... 

The concentration of ozone in the stratosphere is lower than predicted from reactions 1-4. This is due to the presence of trace amounts of some reactive species known as free radicals. These species have an odd number of electrons and they can speed up reaction 4 by means of catalytic chain reactions. Nitrogen oxides, NO and NO2, which are naturally present in the stratosphere at levels of a few parts per billion (ppb), are the most important catalysts in this respect. The reactions, first suggested by Paul Crutzen (2) and by Harold Johnston (3) in the early 1970 s, are as follows ... 

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