Spin parallel

Taking into account the preference for parallel spins the configurations illustrated in Figure 7.39 are obtained unambiguously. Flowever, this is not the case for d, d, d and d ... 

In Chapter 6, I discussed the open-shell HF-LCAO model. 1 considered the simple case where we had ti doubly occupied orbitals and 2 orbitals all singly occupied by parallel spin electrons. The ground-state wavefunction was a single Slater determinant. I explained that it was possible to derive an expression for the electronic energy... 

As mentioned in the start of Chapter 4, the correlation between electrons of parallel spin is different from that between electrons of opposite spin. The exchange energy is by definition given as a sum of contributions from the a and /3 spin densities, as exchange energy only involves electrons of the same spin. The kinetic energy, the nuclear-electron attraction and Coulomb terms are trivially separable. 

Figure 13.1 (a) Nuclear spins are oriented randomly in the absence of an external magnetic field but (b) have a specific orientation in the presence of an external field, B0. Some of the spins (red) are aligned parallel to the external field while others (blue) are antiparallel. The parallel spin state is slightly lower in energy and therefore favored. 

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. 

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... 

This vanishing of the probability density for rx — r2 and Ci == C2 means that it is unlikely for two electrons having parallel spins to be in the same place (rx = r2). The phenomenon is called the Fermi hole and we note that it is a direct consequence of the Pauli principle for electrons with the same spin. 

Fig. 1. The Fermi hole for electrons with parallel spins.

It is clear that, for electrons with parallel spins, the auxiliary condition (Eq. II.2) gives rise to a correlation effect which very closely resembles the correlation effect coming from the Coulomb repulsion in the Hamiltonian for = 2 the Fermi hole replaces to a certain degree the Coulomb hole. This means that, if... 

FIGURE 1.43 (a) Two electrons are said to be paired if they have opposite spins (one clockwise, the other counterclockwise), (b) Two electrons are classified as having parallel spins if their spins are in the same direction in this case, both T. 

If more than one orbital in a subshell is available, add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals. 

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 atomic numbers (Z), electronic configurations, and numbers of unpaired electrons for five ions are listed in the following table. Assume that all unpaired electrons have parallel spins. Indicate the element symbol, charge, and energy state (that is, ground state or excited state) for each of the five cases. 

If more than one molecular orbital of the same energy is available, the electrons enter them singly and adopt parallel spins (Hund s rule). 

The stability of sexivalent chromium, in the chromate ion and related ions, can also be understood. The chromic complexes, involving tervalent chromium, make use of d2sp3 bond orbitals, the three remaining outer electrons of the chromium atom being in three of the 3d orbitals, with parallel spins. The resonance energy of these three atomic electrons in a quartet state helps to stabilise the chromic compounds. However, if all of the nine outer orbitals of the chromium atom were available for bond formation, stable compounds might also be expected... 

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