The flaws in Rioux' and Rioux & Kroger's (RK) derivation


In order to derive the kinetic energy of the electron in the 1s state of the H atom RK assume an infinitely hard sphere of radius R. This means that for all space r > R the potential energy for the electron becomes infinite and, hence, its wavefunction vanishes at the boundary. For this system the quantum mechanical solutions are well known, e.g. Walter Kauzmann, Quantum Chemistry, Academic Press, NY, 1957, p.187, and p.104 ff. They have often been used by solid state physicists in the early years of quantum mechanics and especially for an approximation which has become known as the "jellium" model. However, they are not at all suitable for the H atom. The shells of the eigenstates follow the sequence 1s<1p<1d<2s<1f<2p<1g,..., whereas the Coulomb potential of a central positive charge induces the well known H-atom shells 1s<2s,2p<3s,3p,3d<4s,...

Moreover, RK's solution does not belong to the results of the spherical cavity. Instead they use a one-dimensional box with infinite walls, posed on a diameter of the spherical cavity with which they start the derivation! By this trick they obtain a kinetic energy which is within 9% of the correct minimal value of Neumark (see in the main page), which they then apply later in the text. So its actually a double error: Wrong model, wrong theory! Science usually does not proceed with "The end justifies the means", especially not for a pedagogical aim.

As far as the kinetic energy of H(1s) is concerned, the size of the Kimball sphere is given by the root mean square radius of an unlimited spherically symmetric probability distribution and is exact. The character of a model comes into play, when he, and Neumark, Kleiss ..., compute the potential energy of electrostatic interaction by assuming a uniform charge cloud of that dimension. The Kimball sphere is not infinitely hard, because overlap is possible. The main essence of Kimball's model is in fact, that the spherical clouds are not hard: They do flexibly adjust to the electrostatic fields of the nuclei (and other clouds) whereby they exert the necessary "pressure" by the finite +/- cost of increasing or decreasing their kinetic energy when pressed together or expanded. This leads to the stationary state of the molecular structure as a mechanical equilibrium between the two opposing forces: Coulomb attraction (repulsion) and increasing (decreasing) electron pressure by adjusting all R(i) of the molecule. In every Kimball calculation the minimization of the energy balances these forces exactly! Look at the animation of building ethanol, to see these forces in action during the minimization process.

Ch.K. Jørgensen, Chimia 31(1977)445 exemplifies this criticism in more detail. The same errors exist in later publications of Frank Rioux where he presents a de Broglie-Bohr model, e.g Chem Educator 2007,12,243.

To end on a friendly note: RK in both articles use Neumark's Ekin = 9/8R2 after their false start! Hereafter, the text of the publications is correct and interesting.-