Bare Coulomb Field

Comparison with Bare Coulomb Field.—The origin of the behaviour of the total energy as Z7/3 is easy to understand. Consider a pure Coulomb field with an energy level spectrum 

Now suppose there are closed shells. Then, for a neutral atom we must have 

In the spirit of the TF statistical theory, not only Z but must be taken to be large i.e.Jf P 1, when equation (34) yields, in units of e2a0 

It is clear that equation (36) will have too large a binding of outer shells the self-consistent field reduces equation (36) to the form of equation (32). 

It is not simple to demonstrate that 7/3 is the correct power of Z for real atoms when Z 100. In fact Foldy showed that up to this value of Z, Hartree and Hartree-Fock data are slightly better fitted with Z12/6. Nevertheless, for a non-relativistic theory there can be no doubt that for really large values of Z the total binding energy must vary as Z7/3. We shall see below, when we deal with 

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Proposed Scaling in Self-consistent Field Theory.—March and Parr have proposed an extension of the 1 fZ expansion for atoms, analogous to rearrangement (48), for the case of diatomic molecules. Introducing a scaled length X=RZ, they note first that from the above discussion of the bare Coulomb field one has... 

Bare Coulomb Field.—As a final example, we consider here the result for the chemical potential for a bare Coulomb potential the example considered in another connection in Section 4. If we generalize that discussion to a nucleus with charge Z, using atomic units, and a number of electrons N, then for Jf closed shells we can write... 

The bare Coulombic interaction (p = 1) and interactions of charges with rotating dipoles (p = 4) do not fall into this class, and it has been argued for a long time [30] that in this case one expects analytical ( classical ) behavior. This implies that the system can be described by a mean-field Hamiltonian, in which the interaction of a particle is ascribed to the mean field of all other particles, thus ignoring local fluctuations [10]. In real ionic fluids the... 

The kernel is a well known quantity that appears in several branches of theoretical physics. For example, evaluated for the electron gas, /xc is, up to a factor, the local field correction . To emphasize the correspondence to the effective interaction of Landau s Fermi-liquid theory, to which it reduces in the appropriate limit, /xc plus the bare Coulomb interaction is sometimes called the effective interaction , while in the theory of classical liquids the same quantity is referred to as the Ornstein-Zernicke function. 

Note, in general case the effective Hamiltonian contains also the Coulomb interaction between doped holes and the interaction of quasiparticles via the phonon field. We dropped these terms here, because they do not contribute directly to the spin susceptibility. The hopping matrix element is Uj = tfj e rE /hcot [ + < SiSj >] where r . is the bare hopping inte-... 

To follow up on this idea that the electrostatic field generated by the ion perturbs the transfer potential, one can consider the quantitative effect that an ion at any of the positions in Figure 9 would have upon a bare proton, moving along the indicated axis. The potential energy of this Coulombic interaction would be proportional to the product of the individual charges, divided by their distance from one another. It was found that this sort of potential fit nearly precisely to the perturbation produced on the native proton transfer potential of the isolated (H20-H+-0H2) system. The best fit... 

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