Hamiltonian electronic/nuclear, distinguishing

In Section 1.19 we classified the electronic wave functions of homonuclear diatomic molecules as g or u, according to whether they were even or odd with respect to inversion g and u refer to inversion of the electronic coordinates with respect to the molecule-fixed axes. This is to be distinguished from the inversion of electronic and nuclear coordinates with respect to space-fixed axes, which was discussed in this section. The electronic Hamiltonian for a diatomic molecule is... 

Care must be exercised to distinguish the concept of adiabatic Floquet dynamics introduced here, which refers to an adiabatic time-evolution, or to the slow variations of the Floquet basis with time, from the concept of adiabatic representation defined in the previous section, which refers to the slow variations of the electronic Hamiltonian (Floquet or not) with respect to nuclear motions (i.e., the noncommutativity of the electronic Hamiltonian Hei and the nuclear KE operator Tjv). Where confusion is possible and to be avoided, we shall refer to this concept of adiabaticity related to the BO approximation as the R-adiabaticity, while adiabaticity in actual time evolution will be termed t-adiabaticity. Non-adiabatic effects in time evolution are due to a fast variation of the (Floquet) Hamiltonian with time, causing Floquet states to change rapidly in time, to the extent that in going from one time slice to another, a resonance may be projected onto many new resonances as well as diffusion (continuum) states [40], and the Floquet analysis breaks down completely. We will see in Section 5 how one can take advantage of such effects to image nuclear motions by an ultrafast pump-probe process. 

From now on, we will not consider the vibrational-rotational problem but concentrate solely on the electronic problem of (2.11) to (2.14). We thus drop the subscript elec and only consider electronic Hamiltonians and electronic wave functions. Where it is convenient or necessary, we will distinguish between the electronic energy of (2.13) and the total energy of (2.14), which includes nuclear-nuclear repulsion. 

At present it is not possible to place properly the clamped nuclei electronic Hamiltonian in the context of the full problem, including nuclear motion. However if the nuclei were treated as distinguishable particles, even when formally identical, then some of difficulties that arise from the consideration of nuclear permutations would not occur. But it would still be necessary to be able to justify the choice of subsets of permutations among identical particles when such seem to be required to explain experimental results. A particular difficulty arises here for it is not possible to distinguish between isomers nor is it possible to specify a molecular geometry, unless it is possible to distinguish between formally identical particles. 

It seems fair to say that if one treats the nuclei as distinguishable particles and takes the sum of the electronic energy obtained as an eigenvalue of the clamped nuclei Hamiltonian and the classical nuclear repulsion energy as a potential, the geometrical structure of the minimum in... 

With such a Hamiltonian, it seems reliable to distinguish also the nuclear /(R) and electronic ke (r R) part in the wave function... 

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