Politzer formula

Equations (3.20) and (3.21) represent an identity in Hartree-Fock theory. (The Hellmann-Feynman and virial theorems are satisfied by Hartree-Fock wavefunc-tions.) The particular interest offered by (3.21) lies in the fact that 7 = 1 appears to be the characteristic homogeneity of both Thomas-Fermi [62,75,76] and local density functional theory [77], in which case (3.20) gives the Ruedenberg approximation [78], E = v,e,-, while (3.21) gives the Politzer formula [79], E = Vne-... 

The thus computed potential energies, (Eq. (10.14)), were rescaled with the help of the experimental total energy, using the Politzer formula [79]... 

Remembering that, given spherical symmetry, it is dr = A-irr dr, we see that Eq. (3.35) is our counterpart of the Politzer-Parr Thomas-Fermi-like formula (3.2) describing the valence region of atoms. Here we must stress that Eq. 3.35) represents a valence energy in which relaxation ejfects are included. It was... 

The Politzer-Parr partitioning of molecular energies in terms of atomic-bke contributions results in an exact formula for the nonrelativistic ground-state energy of a molecule as a sum of atomic terms that emphasizes the dependence of atomic and molecular energies on the electrostatic potentials at the nuclei. 

Politzer P. Atomic and molecular energy and energy difference formulae based upon electrostatic potentials at nuclei. In March NH, Deb BM, eds. The Single-Particle Density in Physics and Chemistry. London Academic, 1987 59-72. 

P. Politzer, in Single-Particle Density in Physics and Chemistry. N. H. March and B. M. Deb, Eds., Academic Press, New York, 1987, Chap. 3. Atomic and Molecular Energy and Energy Differences Formulas Based upon Electrostatic Potentials at Nuclei. 

The very simplest one is rooted in Politzer s formula = Vi/7 connecting Ek, the energy of atom k, to VklZk , the total potential at its nucleus with charge Zk- The 7 parameters are treated as constants for each type k =H, C. etc and their proper selection is known to give reasonably accurate results. The valence energy = — E " calculated on this basis (approximation A) for atom k is not a direct product of, but is consistent with, our core-valence separation in real space. 

Approximation B lends itself to very convenient tests if we rewrite our basic formula as follows El = f il — ikhD k - If accuracy turns out to be virtually the same as that of approximation A. This can be understood because the latter obeys to practically the same formula, namely E — /7fc k -Still the fact remains that we cannot justify both our approximations, A and B, because their parameters, 7, 7 and 7, cannot be simultaneously constant. This is so because 1/7 is the weighted average of 1/7 and hk (Section 2) and Vk. at le2ist, is certainly not to be treated as a constant. This criticism illustrates the pitfalls of these formulas when used with approximate constant )k parameters, notwithstanding the acceptable numerical results. On the other hand, this definition of the average I/7 pin-points the place of Politzer s formula in the framework of core-valence theory in real space as a simple form of the basic relationship E). = El + EJP". 

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