Spin-paired

CIDNP involves the observation of diamagnetic products fonned from chemical reactions which have radical intemiediates. We first define the geminate radical pair (RP) as the two molecules which are bom in a radical reaction with a well defined phase relation (singlet or triplet) between their spins. Because the spin physics of the radical pair are a fiindamental part of any description of the origins of CIDNP, it is instmctive to begin with a discussion of the radical-pair spin Hamiltonian. The Hamiltonian can be used in conjunction with an appropriate basis set to obtain the energetics and populations of the RP spin states. A suitable Hamiltonian for a radical pair consisting of radicals 1 and 2 is shown in equation (B1.16.1) below [12]. 

WFth all semi-empirical methods, IlyperChem can also perform psendo-RIfF calculations for open -shell systems. For a doublet stale, all electrons except one are paired. The electron is formally divided into isvo "half electron s" with paired spins. Each halfelec-... 

In a metal of molar volume, V, , these energy levels are filled with paired-spin electrons up to a maximum energy level described by... 

The spins of two electrons are said to be paired if one is T and the other 1 (Fig. 1.43). Paired spins are denoted Tl, and electrons with paired spins have spin magnetic quantum numbers of opposite sign. Because an atomic orbital is designated by three quantum numbers (n, /, and mt) and the two spin states are specified by a fourth quantum number, ms, another way of expressing the Pauli exclusion principle for atoms is... 

When drawing a box diagram, show the electrons in different orbitals of the same subshell with parallel spins electrons sharing an orbital have paired spins. 

The relative size of atomic orbitals, which is found to increase as their energy level rises, is defined by the principal quantum number, n, their shape and spatial orientation (with respect to the nucleus and each other) by the subsidiary quantum numbers, Z and m, respectively. Electrons in orbitals also have a further designation in terms of the spin quantum number, which can have the values +j or — j. One limitation that theory imposes on such orbitals is that each may accommodate not more than two electrons, these electrons being distinguished from each other by having opposed (paired) spins, t This follows from the Pauli exclusion principle, which states that no two electrons in any atom may have exactly the same set of quantum numbers. 

S0, S], S2,. .. represent so-called singlet states in which all the electrons have paired spins, and T1( T2,. .. represent triplet states in which two electrons have unpaired spins. The energy levels of both ground (S0) and activated states (Sj, S2,. ..) are subdivided into vibrational and rotational energy levels. The vibrational energy levels are shown in Figure 11.2. Differences in rotational levels are very small and can be ignored for the present discussion. 

The ground state (0 kJ/mol) for the CL molecule is represented by the term symbol 3v . The first excited state (92 kJ/mol above the ground state) is a 1 singlet (electrons spin paired with both electrons in either the n x or the n y level). The 1 v state with paired spin electrons, one each in the 7i v and n y levels, is the next excited level 155 kJ/mol above the ground state. Reduction of 02 by one electron yields the superoxide ion (02), a radical anion. Reduction by two electrons yields the peroxide ion, (02 ). Bond lengths and bond orders for these are given in Table 4.2. As noted in equation 4.2, the reduction potential for 02 in the presence of protons is thermodynamically favorable. Therefore, reversible binding of O2 to a metal can only be achieved if competition with protons and further reduction to superoxide and peroxide are both controlled.8... 

A ground-state helium atom has two paired electrons in the Is orbital (Is2). The electrons with paired spin occupy the lowest of the quantised orbitals shown below (the Pauli exclusion principle prohibits any two electrons within a given quantised orbital from having the same spin quantum number) ... 

Fig. 3 A conceptual sketch showing approaches to molecular ferrimagnets (paired spins in the open-shell molecules have been left out).

A number of cases have been reported in the literature where the exchange between pairs of ions in a lattice is piedominant, i.e., ions aie exchange coupled in pairs because of their proximity. An example of this is copper acetate 168), where the copper ions occur in pairs which are relatively close, so that the spins of pairs of copper ions are coupled together to form a singlet state (paired spins) and a triplet state (unpaired spins). Resonance measurements 158) permit determination of the magnitude and sign of J. 

Note that the density of states increases with increasing energy—the higher the energy, the more states there are in the interval dE. In metals, the valence electrons fill up the states from the lowest energy up with paired spins. For sodium, for example, each atom contributes one 3s electron and the electrons from all the atoms in the crystal occupy the levels in Figure 4.1 until all the electrons are used up. The highest occupied level is called the Fermi level. 

Show that the oxygen dimer is predicted to be paramagnetic since the highest occupied level is a doubly degenerate n state in which the two electrons will have parallel spin by Hund s rule. (This agrees with experiment unlike valence bond theory s prediction of diamagnetism due to all electrons being paired spin-up-spin-down.)... 

Electrons having the same spin strongly repel each other and tend to occupy different regions of space. This is a result of a fundamental law of nature known as the Pauli exclusion principle. It states that total wave functions (including spin) must change their signs on exchange of any pair of electrons in the system. Briefly, this means that if two electrons have the same spin they must have different spatial wave functions (i.e., different orbitals) and if they occupy the same orbital they must have paired spins. The Pauli principle and the so-called Pauli repulsive forces 1 have lar-... 

Normally two electrons pair up to form each bond. This is a consequence of the Pauli exclusion principle—two electrons must have paired spins if they are both to occupy the same region of space between the nuclei and thereby attract both nuclei. The definition of a bond as a shared pair of electrons, however, is overly resirictive, and wc shall see that the early emphasis on electron pairing in bond formation is unnecessary and even misleading. 

---