Interactions spin wavefunctions

Thus far in this chapter we have considered single-spin systems only. The zero-field interaction that we worked out in considerable detail was understood to describe interaction between unpaired electrons localized all on a single paramagnetic site with spin S and with associated spin wavefunctions defined in terms of its m5-values, that is, (j) = I ms) or a linear combination of these. However, many systems of potential interest are defined by two or more different spins (cf. Figure 5.2). By means of two relatively simple examples we will now illustrate how to deal with these systems in situations where the strength of the interaction between two spins is comparable to the Zeeman interaction of at least one of them S Sh B Sa. 

There are numerous interactions which are ignored by invoking the Born-Oppenheimer approximation, and these interactions can lead to terms that couple different adiabatic electronic states. The full Hamiltonian, H, for the molecule is the sum of the electronic Hamiltonian, the nuclear kinetic energy operator, Tf, the spin-orbit interaction, H, and all the remaining relativistic and hyperfine correction terms. The adiabatic Born-Oppenheimer approximation assumes that the wavefunctions of the system can be written in terms of a product of an electronic wavefunction, (r, R), a vibrational wavefunction, Xni( )> rotational wavefunction, and a spin wavefunction, Xspin- However, such a product wave-function is not an exact eigenfunction of the full Hamiltonian for the... 

The arrows in reactions (15)-(17) are used to illustrate that upon the change in the multiplicity of the excited molecules, the spin wavefunction for the radical ion that interacted with the neutral species also changed [36]. Unlike Borokov s analysis [36], it is not assumed in the model that the spin correlated radical pair completely loses its spin correlation upon reaction. Instead for reactions (15), (16) and (17) the wavefunction belonging to the ion-pair is multiplied by the appropriate wavefunction of the excited species (i.e. the collapsed ir of the ion-pair forming the excited neutral species). For example, in reaction (15) the total wavefunction for the (S+, e )t + is... 

In the case of a closed-shell molecule where all of the electrons in the occupied molecular orbitals are paired, the wavefunction representation is as described in Equation 9-29. Numerous cancellations will occur when integrating over these spin wavefunctions due to the orthonormality of the spin functions a and p. Electrons with like spins will interact and electrons with unlike spins will not interact. The function represented in Equation 9-29 is termed a restricted Hartree-Fock (RHF) wavefunction. An example of a closed shell molecule is carbon monoxide. There are a total of 14 electrons that are paired up in seven molecular orbitals (see Figure 9-5). 

Electrons, protons and neutrons and all other particles that have s = are known as fennions. Other particles are restricted to s = 0 or 1 and are known as bosons. There are thus profound differences in the quantum-mechanical properties of fennions and bosons, which have important implications in fields ranging from statistical mechanics to spectroscopic selection mles. It can be shown that the spin quantum number S associated with an even number of fennions must be integral, while that for an odd number of them must be half-integral. The resulting composite particles behave collectively like bosons and fennions, respectively, so the wavefunction synnnetry properties associated with bosons can be relevant in chemical physics. One prominent example is the treatment of nuclei, which are typically considered as composite particles rather than interacting protons and neutrons. Nuclei with even atomic number tlierefore behave like individual bosons and those with odd atomic number as fennions, a distinction that plays an important role in rotational spectroscopy of polyatomic molecules. 

The essence of this analysis involves being able to write each wavefunction as a combination of determinants each of which involves occupancy of particular spin-orbitals. Because different spin-orbitals interact differently with, for example, a colliding molecule, the various determinants will interact differently. These differences thus give rise to different interaction potential energy surfaces. 

In Pauli s model, the sea of electrons, known as the conduction electrons are taken to be non-interacting and so the total wavefunction is just a product of individual one-electron wavefuncdons. The Pauli model takes account of the exclusion principle each conduction electron has spin and so each available spatial quantum state can accommodate a pair of electrons, one of either spin. 

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