Spin-Coupled States

Why doesn t Cu + disproportionate One important reason is that the difference in occupation is not in the 4s orbital, bnt in the much smaller 3d orbital. In fact, the 4s and 5s orbitals do not get occupied as easily as the 6s orbital. This may have to do with the relativistic effects becoming important in the lower part of the periodic table. The valence states Zn+ and Cd+ are, in fact, missing, while Hg+ and T1+ exist. Ga and In are very rare, meaning that it is hard to reduce Ga + or + to Ga+ and 

respectively. Ga +, In + and TF+ are all missing, since a system with a single s orbital occupied is very unstable. 

In the spin-coupled cases with equal valence, the coupling forces are of the same type as in ordinary chemical bonding, say between two nitrogen atoms. The only difference is that the bonding forces are smaller since there is usnally an or in between the metal ions. 

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The theoretical problem is to understand electron pair transfer in the limit when Hubbard U 0 or even becomes negative. We will find that in this limit there is interaction between two many-electron states, the charged state, and the spin-coupled state. We will find that the motion of electron pairs is only possible in this correlated situation. We will also find that strong coupling to the vibrational states (phonons) is directly related to electron pair mobility. 

If AG is too large for thermal transition between the phases, optical transitions may take place between the spin-coupled states and the charged states. As Equation... 

Consider two different sites, M+ and M, in a disproportionated system. We first assume symmetric geometry of the enviromnent, so that M+M has the same energy as M"M+. Since the spin-coupled state MM is unstable, two electrons may be transferred at the same time from M" to M+, forming M+ and M. This is one of the rare situations when a transfer of electron pairs is possible (and, in fact, the only possibility) in a chemical system. 

The ground state is Tj. Since the interaction between aa and bb is vanishing, the first excited state is Ta. Using T j and Ta, it is possible to form a time-dependent wave function that starts from aa at t = 0 and then oscillates between aa and bb, that is, between the sites. lf C2l is much larger than Ci, which is the case when the spin-coupled state is the ground state, this oscillation will be insignificant. 

We may first study the limit case when the charged state is the ground state and no spin-coupled state exists below the crossing point of the M+M" state and the M M state. One such case is the insulator TlBr2. The difference in bond length clearly... 

Cgo" ") might be stable in one end of the series and the spin-coupled state... 

Qo -,Qo -) in the other end. Almost degeneracy somewhere in between leads to an interaction between the charged states and the spin-coupled states. This interaction will now be further investigated. 

The site charges are originally the same, eqnal to Z. After disproportionation, the charges (Z - 1) and (Z -i-1) alternate. The Bom radius R can be taken as the lattice constant. In atomic units, the Bom free energy (Section 6.2.2) difference between the spin-coupled state and the charged state is... 

R is about 20 Bohr in fullerides. AG(Born) is thus equal to -1/20 Hartree or -1.3 eV. This lowers the energy difference between the spin-coupled state and the charged state, thus Hubbard U decreases. 

If ( )2 is occupied by a spin-up electron and ([>3 with a spin-down electron, we obtain (after symmetrization) the spin-coupled state. If ( 2 is occupied by two electrons or ( )3 is occupied by two electrons, we obtain the charged state. With these occupations the bond lengths remain the same, since in 2 and 4>3 two nonzero MOs are never on adjacent atoms. 

Alternatively, we may occupy the upper orbitals in Figures 17.9 and 17.10 by one electron with spin up, while the lower orbitals are occupied by one electron with spin down. Adding the other possible wave function of this type with opposite spins leads to a spin-coupled state, since the spins are different on adjacent sites. Thus, if the intermediate valence state is present at a low energy, spin coupling occurs, as in the undoped cuprates. It should be noted that we are dealing with chemical effects, which are not easily calculated and not easily understood. What is important is that in addition to these two possibilities, an electron pair current state is possible if the charged state and spin-coupled state are almost degenerate. 

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