Opposed spins

An s orbital is spherically symmetrical and can contain a maximum of two electrons with opposed spins. A p orbital has a solid figure-of-eight shape there are three equivalent p orbitals for each principal quantum number they correspond to the three axes of rectangular coordinates. 

The d and f orbitals have more complex shapes there are five equivalent d orbitals and seven equivalent f orbitals for each principal quantum number, each orbital containing a maximum of 2 electrons with opposed spins. 

Electrons that have the same value of ms (i.e., both + or both — ) are said to have parallel spins. Electrons that have different ms values (i.e., one + and the other — ) are said to have opposed spins. 

The Pauli exclusion principle has an implication that is not obvious at first glance. It requires that only two electrons can fit into an orbital, since there are only two possible values of m,. Moreover, if two electrons occupy the same orbital, they must have opposed spins. Otherwise they would have the same set of four quantum numbers. 

To show how orbital diagrams are obtained from electron configurations, consider the boron atom (Z = 5). Its electron configuration is ls22s22p1. The pair of electrons in the Is orbital must have opposed spins (+j, or f j). The same is true of the two electrons in the 2s orbital. There are three orbitals in the 2p sublevel. The single 2p electron in boron could be in any one of these orbitals. Its spin could be either up or down. The orbital diagram is ordinarily written... 

With the next element, carbon, a complication arises. In which orbital should the sixth electron go It could go in the same orbital as the other 2p electron, in which case it would have to have the opposite spin, [. It could go into one of the other two orbitals, either with a parallel spin, f, or an opposed spin, Experiment shows that there is an energy difference among these arrangements. The most stable is the one in which the two electrons are in different orbitals with parallel spins. The orbital diagram of the carbon atom is... 

Strategy Start with the electron configuration, obtained as in Section 6.5. Then write the orbital diagram, recalling the number of orbitals per sublevel, putting two electrons of opposed spin in each orbital within a completed sublevel, and applying Hund s rule where sublevels are partially filled. 

According to this model, a covalent bond consists of a pair of electrons of opposed spin within an orbital. For example, a hydrogen atom forms a covalent bond by accepting an electron from another atom to complete its Is orbital. Using orbital diagrams, we could write... 

Let us consider lithium as an example. In the usual treatment of this metal a set of molecular orbitals is formulated, each of which is a Bloch function built from the 2s orbitals of the atoms, or, in the more refined cell treatment, from 2s orbitals that are slightly perturbed to satisfy the boundary conditions for the cells. These molecular orbitals correspond to electron energies that constitute a Brillouin zone, and the normal state of the metal is that in which half of the orbitals, the more stable ones, are occupied by two electrons apiece, with opposed spins. 

The two electrons in this configuration are said to be paired electrons, meaning that they are in the same energy level, with opposing spins. Opposing spins cancel, so paired electrons have zero net spin. 

Nonbonding electrons usually occur in pairs with opposing spins. 

In the ground state of helium, according to this model, the two electrons are in the Is orbital with opposing spins. The ground-state wave function is... 

A nucleus under study by nuclear magnetic resonance techniques is affected by other nuclei in the same molecule. This phenomenon is known as spin-spin coupling. The effect arises (in adjacent nuclei) from the two electrons joining the nuclei in a covalent bond. Suppose the energy of states in which the electrons in the bond have opposing spins is lower than the state in which the electron spins are parallel. Then the AE between the two states (in this case a negative number) is called the coupling constant, J, expressed in frequency units, Hz. Internuclear... 

The chemical shift of this triplet is quoted as the centre point of the triplet (the peak of relative intensity 2), which corresponds to the situation in which the effect of orte of the methylene protons on the magnetic field of the methyl group is cancelled by the other (opposing spins), and so this peak has the same chemical shift as if there were no protons on the carbon adjacent to the methyl group. 

For example, 6 P, (six triplet P one) state of mercury signifies that the total energy of the state corresponds to n = 6 the orbital angular momentum is L— 1 the multiplicity is three hence it is a triplet energy state and the spins of the two valence electrons must be parallel (S = 1) and the particular value of 3 is 1 (/= 1). Since a normal mercury atom, has a pair of electrons with opposed spin in the S orbital, this must be an excited energy state, where a 6S electron is promoted to a 6P state. 

In general two electrons, with opposed spins, may occupy each atomic orbital. There is one s orbital in each electron shell, with a given value of the total quantum number n three p orbitals, corresponding to mi = — 1, 0, and +1, in each shell beginning with the L shell five d orbitals in each shell beginning with the M shell, and so on. The numbers of electrons in completed subshells and shells of an atom are shown in Table 2-3. Note that there are alternative ways of naming the shells. 

Fig. 2-15.—The approximate sequence of energy values for atomic orbitals, the lowest circle representing the most stable orbital (Is). Each circle represents one atomic orbital, which can be ocoupied either by one electron or by two electrons with opposed spins.

The formal results of the quantum-mechanical treatment of valence, developed by Heitler, London, Born, Weyl, Slater, and other investigators, can be given the following simple statement an atom can form an electron-pair bond for each stabh orbital, the bond being of the type described for the hydrogen molecule and owing its stability to the same resonance phenomenon. In other words, for the formation of an eleo-tron-pair bond two electrons with opposed spins and a stable orbital of oach of the two bonded atoms are needed. 

Ib methyl fluoride two electrons with opposed spins are concentrated along the C—F bond. The fluorine atom is, in consequence of correlation, presumably not cylindrically symmetrical about the bond direction, but somewhat hexafoliate. In water and dimethyl ether the two unshared electron pairs of the oxygen atom, despite the effect of correlation, are directed toward two corners of the tetrahedron that has its other two corners determined by the two bonds. 

The theory of resonance has been applied to many problems in chemistry. In addition to its use in the discussion of the normal covalent bond (involving the interchange of two electrons, with opposed spins, between two atoms) and to the structure of molecules for which a single valence-bond structure does not provide a satisfactory description, it has rendered service to chemistry by leading to the discovery of several... 

The exclusion principle permits only two electrons, which must have opposed spins, to occupy either one of the orbitals the third electron must occupy the other orbital.6... 

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