Multiplet states

Filatov, M., Shaik, S., 1999, Application of Spin-Restricted Open-Shell Kohn-Sham Method to Atomic and Molecular Multiplet States , J. Chem. Phys., 110, 116. 

Core electron ejection normally yields only one primary final state (aside from shake-up and shake-off states). However, if there are unpaired valence electrons, more than one final state can be formed because exchange interaction affects the spin-up and spin-down electrons differently. If a core s electron is ejected, two final states are formed. If a core electron of higher angular momentum, such as a 2p electron, is ejected, a large number of multiplet states can result. In this case it is difficult to resolve the separate states, and the usual effect of unpaired valence electrons is... 

Similarly, application of the properties of the spin-shifting operators, S , allows one to obtain the relations connecting the 1-TRDMs corresponding to different multiplet states. Thus, by considering the action of the spin-shifting operator S+ on pure spin states,... 

This spectrum has been compared with the BIS measurement on Nd metaP i.e. of the homologous lanthanide. Trivalent Nd has a localized 4f initial state configuration. For U, a 5 f or a 5 f initial state configuration are usually assumed, with a tetravalent or a trivalent core respectively. While in Nd the 4f and 4f multiplet states, as evaluated in an atomic-like Russell-Saunders scheme, can be well recognized in the XPS/BIS combined results, and are well separated from a (weak) d-emission at the Fermi edge, in U the occupied states and the empty states spectra join in a continuous band at Ep. Therefore, only the symmetry of 5 f states, given by the position of the main peaks in the joint spectrum, can be recognized with certainty. 

We see from the above discussion that staying within the DFT it is not possible to describe the multiplet structure of the d-shell (incidentally all the success stories reported so far are limited to the p-shells [82,113] whereas in the d-shells only average energies of several multiplet states would be reproduced [114]). In this context it seems to be necessary to analyze the attempts to achieve it which are available in the literature (see [87,115]). These attempts, however, have a nature absolutely different from the DFT itself so they will be described in an appropriate place below. 

When it comes to the analysis of similar approaches stemming from the DFT the numerous attempts to cope with the multiplet states must be mentioned [113,154]. In these papers attempts are made to construct symmetry dependent functionals capable to distinguish different multiplet states in a general direction proposed by [99,100]. It turns out, however, that the result [113] is demonstrated for the lower multiplets of the C atom which are all Roothaan terms. It is not clear whether this methodology is going to work when applied to the d-shell multiplets which may be either non-Roothaan ones or even nontrivially correlated multiple terms. 

Since it is the exchange interaction that determines to a large extent the energy separation of the multiplet states to first order, the physical basis for the triplet preference thus vanishes. In that case, other electron correlation effects tend to stabilize the singlet selectively. 

The development of a pseudo-angular momentum theory of (t2g)n states, which is rooted in the point group under consideration can thus proceed in the following way one first converts the T2 basis into a T1 basis. The 7 functions can then be coupled to multiplet states using the coupling theory of the full... 

Inelastic neutron scattering, on the other hand, usually employs a monochromatic neutron beam and records the intensity of the scattered neutron beam as a function of neutron kinetic energy. Such inelastic collision spectra are monitored as a function of the applied field and the (usually low) temperature. The observed peaks then represent the energy differences of thermally populated and excited unpopulated multiplet states. Inelastic neutron scattering experiments can be conducted using triple-axis, backscattering, or time-of-flight spectrometers. 

The eigenstates of a many-electron system (multiplet states for open shell systems) can be obtained by solving the many-electron Dirac equation,... 

In all of the transition metals the d shell. scries to the left in the Solid State Table—the itinerant picture is appropriate and the effects of the corresponding r/ bands require further attention. It should be mentioned at the outset that though the itinerant picture is appropriate for all of the transition metals, there is a complication analogous to the multiplet states that will be discussed in Section... 

A ie(d" s-d s ) of the 3d series are compared with experiment in Fig. 5. The experimental results correspond to the lowest-lying term for each configuration, averaged over multiplet states with different J. On the other hand, the central-field calculations often correspond to an average over states of different L arising from the open-shell configurations, so that the comparison is not entirely clean. Ideally, of course, one would like to compare state-by-state but it is not clear how to do this within a theory based st tly on orbitals (it is not obvious how to work with eigenfunctions of. and if one... 

That is, values of the off-diagonal matrix elements of the BJ S, S-uncoupling operator, within a 2S+1An multiplet state, must be much smaller than the energy separation between multiplet components that result from differences between diagonal matrix elements of Hso. Since matrix elements of the S-uncoupling operator are proportional to J, at sufficiently high-J case (a) ceases to be as good an approximation as case (b). 

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