Springs and Bolts: Ekin and V
in preparation

Victor F. Weisskopf [1], first proposed a visually and mathematically effective formulation of steric repulsion as “kinetic energy pressure.” Steric space-filling or “hardness” properties are generally understood to originate in the Pauli exclusion principle (wavefunction antisymmetry for exchange of identical electrons or other fermions), which limits the maximum occupancy of any spatial orbital to two electrons of opposite spin. Equivalently, this principle prevents electron pairs from crowding into the same spatial region, because their orbitals cannot maintain mutual orthogonality without incurring additional oscillatory “ripple patterns” (nodal features) that increase the 2nd-derivative “curvature,” and thus the kinetic energy of the orbital. Attempted compression of filled orbitals must therefore result in increasingly severe ripple-like nodal features in the outer overlap region, analogous to the inner nodal features that maintain orthogonality to core electrons of the same symmetry, with the resulting kinetic energy increase acting as “pressure” to resist further compression.-
quoted from [2]
"Kinetic energy pressure" (a physical misnomer of mixing extensive with intensive quantity!) already starts with singly and doubly occupied clouds, i.e. in the H and He atoms. Hans Hellmann has extensively anticipated and described this 1933 [1a]. Obviously, this has initially nothing to do with Pauli's Principle but is, together with the electrostatic force, the main ingredient of the molecular world. As soon as two electrons with the same spin are involved, Pauli's Principle joins in and we have the stereochemical
effects of the occupied (spin-)pairs, Weisskopf describes: Given nuclei and electrons, zero point electron pressure, electrostatic interaction and Pauli's Principle then make up the troika which creates and maintains chemistry, i.e. the material world as we know it on our planet. AFAIK no elementary current chemistry text develops this clearly. Kinetic energy is rarely even mentioned. It is implicit in the abstract concept of (complex valued, non observable) wavefunction, which is used instead but must not be applied for qualitative arguments. Wavemechanics is the potent language of computational chemistry. It is not understandable with the mathematical background at highschool level in constrast to kinetic energy and electron density which are.
Although the inventors of the statistical theory, Thomas and Fermi, had completely realized this in 1927/28 it took almost three generations and a robust Walter Kohn [5] for its scientific acceptance (it will take another three generations to percolate to highschool chemistry). With Kimballs quantitative model one can learn and teach these essential facts in the simplest way possible (but not simpler! to quote A. Einstein). Thomas and Fermi [3][4] have given a lucid derivation of the connection of electronic kinetic energy with threedimensional electron density. Except for the rigorous proof of Hohenberg & Kohn (1964) [5] they had already realized that the description of matter does not need a phase space of 6N dimensions for N electrons, as wavemechanics requires, but can be developed from the electronic density observable in three-space. All this is masterly presented and explained in Parr & Yang [6], a bible of Density Functional Theory, of which Hartree-Fock Theory, the basis of traditional quantum chemistry, is a special case.

[1] Weisskopf, V.W., Science 1975, 187, 605-612
[1a] H. Hellmann, Z.f.Physik 85(1933)180-190
[2] F. Weinhold et al., official preprint (2012)
[3] L.H.Thomas, Proc. Cambr. Phil. Soc. 25(1927)542 (atoms)
[4] E. Fermi, Z.f.Physik 36(1926)902 (new statistics), 48(1928)73 (atoms, PT)
[5] P.Hohenberg and W. Kohn Phys.Rev 136(1964)B864-B871. This paper gives a beautifully
simple proof that the 3D electronic density of a molecule or extended structure contains the full information of the complete multibody ground state wavefunction; hence, any ground state property of the system can be computed from it. The problem is, that the exact functional to do this, is not known.
[6] R.G.Parr & W.Yang, Density Functional Theory of Atoms and Molecules, Oxford University Press, 1989, ISBN 0-19-504279-4

Cloud overlap, Kinetic energy functional

The comment in G.F Neumark's thesis, loc.cit p.30: The outer clouds touch the central one, but not necessarily each other needs explanation:
1) The central clouds, defining the backbone and/or the center of a molecule, are core clouds touching bonding clouds. Penetration, i.e. cloud overlap, between those does not happen because of huge repulsion forces coming from the kinetic electron pressure and electrostatic interaction. E. Wigner has computed a Pauli repulsion function which is, however, not a step function with an infinite wall. Suppose we have a hydrocarbon chain with -C-C-C- , then C(1s2) cores of radius Rc = ~0.26 au. are touching C-C bonding clouds (in chain) and C-H clouds ("outer" clouds), both with Rcc or Rch of ~1.2 au. Penetration of C-C or C-H into C cores changes both clouds' radii, causing a change of kinetic energy δT/δR which is about (1.2/0.26)3 = 100 times larger
than penetration between bonding, "outer" clouds, would cost. This is the rationale for setting up a molecular Kimball skeleton with touching clouds, neglecting the possibility of overlap. Electrostatic repulsion adds to this effect. In other words: The small core clouds are too hard for penetration.
2) Kimball's one electron cloud has a kinetic energy T = 9/8R2 au, the same within about 1.8% of the Thomas-Fermi (TF) kinetic energy functional. Kimball and his PhD students (except J.D. Herniter [7], who mentions this as a possible problem) have not discussed how a second, spin compensated, electron changes this value. They just double the constant and use T = 9/4R2 au for the two electron cloud. This turns out to be corroborated by the calculations. Strangely, G.Kimball has nowhere cited the well known results of Thomas-Fermi theory, which have predated his model by 20 years, nor are they treated in [8].

[7] PhD thesis, Columbia Univ., N.Y. 1956, CA 51(1957)9285g, pp.35-36
[8] H.Eyring, J.Walter & G.E.Kimball: "Quantum Chemistry", J.Wiley, N.Y., 1944

Derivation of T[TF] functional

In [6] the derivation of the TF functional is relegated to [9]. This reference treats the TFD(irac) electron gas in a metallic phase but does not derive the TF
kinetic energy functional, although this is very similar. H.Hellmann [10] did it with much pedagogical flair. I want to show you, how he arrives at the ρ5/3 dependence in the following facsimile of pages 7-10 from [10].

[9] Donald A. McQuarrie, Indiana Univ.: "Statistical Mechanics", Harper & Row N.Y. 1976, pp.164-166
[10] Hans Hellmann: "Einführung in die Quantenchemie", Franz Deuticke, Leipzig & Wien, 1937, pp.1-9