Closed shell atom

The data show that the differential cross section is dominated by the direct scattering amplitude / the modulus for the spin-flip amplitude g is in general an order of magnitude smaller. Thus the spin—orbit interaction has only a small influence on the cross section, which is mainly influenced by the Coulomb interaction, exchange, and charge-cloud polarisation. 

As discussed in the last chapter, electron—photon (e, e y) measurements yield much more information on the scattering process than simple inelastic differential cross section measurements. In particular the population of magnetic sublevels can be obtained, which can be visualised by the corresponding charge-cloud probability distribution (fig. 8.1). The set of parameters discussed in the last chapter must be enlarged when polarised 

P4 is again a Stokes parameter for unpolarised electrons. P5 and Pe must be zero if Px = 0. This can be seen by taking a mirror reflection in the scattering plane. Nonzero values of P5 and Pe would violate parity conservation if Px = 0, since the electron beam geometry would remain unchanged unless there is an x-component of Pe. Similarly the tilt (e) in the charge-cloud distribution out of the scattering plane indicated in fig. 9.7 must vanish unless Px 0. 

Due to parity conservation A a)x = 0 for a = 45° and 135°. These parameters can be related to the state multipoles T j)) (8.22) describing the atomic state, which depend on the electron polarisation components as follows (Bartschat et al, 1981) 

Here we have simplified the state multipole notation (8.22) to show only the angular momentum of the excited state. 

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We write for the ground state wave function of the closed shell atom or molecule a Slater determinant for the N electrons... 

As an example of the interest to scrutinise the UHF solution, one may quote the Bea problem [19]. The bond is weak but it takes plaee at short interatomic distance and is definitely not the dispersion well which one might expect from two closed shell atoms (and which occurs in Mga and heavier eompounds). Quantum chemical calculations only reproduce this bond when using large basis sets and extensive Cl calculations [20]. It is amazing to notice that the UHF solution gives a qualitatively correct behaviour, and suggests a physical interpretation of this bond since in... 

The peculiar behavior of H might be relevant to understand the hydrogen bond, which deforms the electronic cloud of the proton. On the other hand, it is surprising to discover an anomalous behavior for a closed-shell atom like He. However, it has been demonstrated in helium-atom-scattering that interactions between He atoms... 

Gordon R. G. and Kim Y. S. (1971). Theory of the forces between closed-shell atoms and molecules. T Chem. Phys., 56 3122-3133. 

Finally the BERTHA technology has been applied to relativistic density functional theory by Quiney and Belanzoni [36]. This showed that the method works well for closed shell atoms as compared with benchmark calculations using finite difference methods, and there have been promising parallelization studies [37] which should in future greatly extend the range of application of the code. 

If a closed-shell atom is placed in an electric field of strength F, it undergoes a change in energy of... 

Closed-shell atoms are considered here, because in open-shell systems the photoionization process is generally determined by more than three matrix elements (five parameters). 

The derivation of the HF equations for atoms other than helium follows along the same lines as presented above, and one gets for closed-shell atoms, in atomic units, [Sla60]... 

Within the dipole approximation (jphoton = 1) and for the example of photoionizing an wp-electron from a closed-shell atom (J = 0), the photoelectron angular momentum is either t = 0 or 2 and, hence,... 

For the general relation valid for closed-shell atoms see, for example, [Sch92a] ... 

From the abundance of relations which have been worked out for statistical tensors, only those needed to derive expressions for photon-induced electron emission from closed-shell atoms will be quoted here. (In this context it should be mentioned that many different conventions are in use which are not always compatible and which lead to different phases and/or numerical values. Here the convention of [Fer65] is employed.) The necessary expressions are ... 

This work done is path-independent since V x R(r) = 0. For systems of certain symmetry such as closed shell atoms or open-shell atoms in the central-field approximation, the jellium and structureless-pseudopotential models of a metal surface considered here, etc., the work Wxc (r) and Wt (r) are separately path-independent since for these cases Vx xc(r) = VxZt (r) = 0. 

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