Space quantisation

With the introduction of electronic angular momentum, we have to consider how the spin might be coupled to the rotational motion of the molecule. This question becomes even more important when electronic orbital angular momentum is involved. The various coupling schemes give rise to what are known as Hund s coupling cases they are discussed in detail in chapter 6, and many practical examples will be encountered elsewhere in this book. If only electron spin is involved, the important question is whether it is quantised in a space-fixed axis system, or molecule-fixed. In this section we confine ourselves to space quantisation, which corresponds to Hund s case (b). 

We have derived the total Hamiltonian expressed in a space-fixed (i.e. non-rotating) coordinate system in (2.36), (2.37) and (2.75). We can now simplify the electronic Hamiltonian 3Q,i by transforming the electronic coordinates to the molecule-fixed axis system defined by (2.40) because the Coulombic potential term, when expressed as a function of these new coordinates, is independent of 0, ip and x From a physical standpoint it is obviously sensible to transform the electronic coordinates in this way because under the influence of the electrostatic interactions, the electrons rotate in space with the nuclei. We shall take the opportunity to refer the electron spins to the molecule-fixed axis system in this section also, and leave discussion of the alternative scheme of space quantisation to a later section. Since we assume the electron spin wave function to be completely separable from the spatial (i.e. orbital) wave function,... 

Rotational Hamiltonian for space-quantised electron spin [1 67... 

As before, this Hamiltonian can be converted into a Hamiltonian for a diatomic molecule with space-quantised spin by making the replacements,... 

Figure 6.3. Space-quantised orientations for the angular momentum vector til oI a state with 1=1. The length of til is [/(/ + 1 ) 2]1/2.

We can specify the conditions for the above sequences of reactions to be possible in terms of thermodynamic quantities provided we can ascribe these to electronically excited states. This is normally possible for excited states in bulk condensed phases because these become thermalised , i.e. vibrationally and rotationally equilibrated with their enviromnent, extremely rapidly (usually within a few picoseconds), long before they undergo any chemical reaction. It is however not possible to assume thermalisation in space-quantised stractures such as quantum dots, in which relatively long-lived hot carriers are generated by photoexcitation. Indeed, the very slowness of thermalisation in space-quantised stractrrres makes it possible to envisage photoconversion devices in which hot carriers can deliver more work than wotrld be thermodynamically possible with thermalised carriers. Nozik discusses such possibihties in Chapter 3. 

The leitmotifs of these devices include bespoke dye sensitisers, space-quantised nanoscale structures that enable hot carrier or multiple exciton generation, molecular and solid-state junction architectures that lead to efficient exciton dissociation and charge separation, and charge collection by percolation through porous or mesoscale phases. Another common theme underlying the devices discussed in this book is the... 

The multiplicity of the terms cannot be explained on the assumption of a point electron and a central field of force. It was first ascribed to a space quantisation of the orbit of the radiating electron with respect to an axis in the core, and later to a spin of the electron itself (r/. p. 155). 

This implies a somewhat modified type of space quantisation, since by (19) ... 

The values A and 8 of the molecular electronic states arising from a given state L, 8 of the united atom are even easier to find than in (a). All we have to do is to subject L to space quantisation with respect to the internuclear line, giving as possible values of the component... 

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