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Content: Kimball’s model of LiH; Hellmann-Feynman theorem

ES 28 Feb 2017//18 Oct//17 Aug//14 Jul 2016//2 Mar 2015//2002//1982

If this tutorial is a step too steep from the last, I shall insert a halftstep, soon

General LiH

assuming two doubly occupied electron clouds with radii R1,R2 are “touching” at a distance xr2=R1+R2 on the x-axis, which does not impair generality. A and a are in arbitrary positions nearby. Let’s find the energy minimum. Initial and final positions, and cloud sizes are shown on the last pages.

(ES 2 January/28 February 2015)

Summary of quantitative results:

Visualization and discussion

At start of this computation, the two nuclei are at arbitrary positions Li0, H0, while two doubly occupied electron clouds (of arbitrary size) hover in space, “touching” each other. We know, that this is an artificial situation, becoming proper physics/chemistry when the final state of an equilibrium has been reached. Since we have located the two clouds on the x-axis, the y1,z1 and y2,z2 components of the initial nuclear positions have vanished at equilibrium. The ratio of the clouds radii should be about 3, as the ratio of nuclear charges. It is 3.54 because we have been “urged” (by the calibration) to increase the kinetic energy of the larger H- cloud by a factor of 1.23.. = k2, as explained in Tutorial2 with .

This is the (local) energy minimum the FindMinimum routine has determined. Both nuclei have found “their” cloud to drop as near to the center as possible under the influence of all interactions of clouds-nuclei, nuclei-nuclei, and cloud-cloud. The dimension of the diatomic molecule and the different components of energy found, are given above.

The cyan spheres mark the eccentricities of the Li+3 nucleus (5 times magnified) and of the proton in the outer cloud (normal scale).

The charges in these volumes attract the two nuclei to their respective cloud centers. The counter force is the nuclear repulsion and the other interactions: the Li nucleus is right to its cloud center because the outer cloud tears stronger at it than the proton pushes it away. The proton is also right because the repulsion by the Li+3 nucleus predominates the attraction by the small ) cloud.

This simple example shows the delicate charge balances that finally decide about the structure being formed as an equilibrium of forces. That is a special aspect (the "electrostatic theorem") of the generally valid Hellmann-Feynman theorem, which predicates (in mathematical form rather than the following wording):* "In a stationary molecular or extended structure each nucleus is kept at its position by the attraction forces, summed over the complete electron charge of the system (with its zeropoint kinetic energy), compensated by the nuclear repulsion of all other nuclei".* It is hard to check this theorem in quantum chemical computations, since it strictly holds for the true (many body) wavefunction (and its exact density distribution), only. The location of the nuclei is fairly well known, hence the repulsion force - size and direction - is easily calculated with a good structure at hand. For computing the attraction forces between nuclei and electrons, however, the distribution must be known much better than usually determined; especially the tails of the superposed approximate wavefunctions are rarely well defined.

In Kimball’s model that is much easier, since we do not integrate over functions vanishing at infinity, but can simply add the contribution of every spatially restricted electron cloud which represents an average of the infinite distribution. Hence, a good (Kimball) computation satisfies the Virial- and the Hellmann-Feynman theorems (it may lack in other aspects, being such a simple model!). It determines a stationary system in thermodynamic equilibrium (minimum energy) and in a state of mechanical equilibrium (sum of forces and sum of momenta, in polynuclear systems, vanish). Look at the complete Hellmann-Feynman electrostatic analysis of LiH.

Many chemists do not know, that the first, epochal, computation of the H2 molecule by W. Heitler & F. London (1927) does not satisfy either theorem by a large margin and, hence, is neither a stationary nor a mechanically stable solution of the bonding problem (that and other failings made Wolfgang Pauli criticize Walter Heitler (originally in German): "Your solution for H2 is wrong at small distances of the protons, it is wrong at large distances, why do you believe that it's correct inbetween?", at a talk of Heitler in the "Naturforschende Gesellschaft, Zuerich", ca. 1950! Pauli had derived, in 1922(!) (see D. Wallace's history of the 'Hellmann-Feynman Theorem'), a *generalized Hellmann-Feynman theorem* and thus contributed to its understanding. Hence, he knew that Heitler-London's H2 theory does not fulfill this fundamental principle. See: Frank Jensen, Introduction to Computational Chemistry, 2nd ed., 2007, J. Wiley & Sons, N.Y., pp. 322,339,572). If you really want to understand the Virial- and the Hellmann-Feynman theorems and their importance for the theory of chemical bonding, read the excellent account in I.N.Levine, Quantum Chemistry, 2nd ed. 1975, pp. 367-380 or Wallaces mathematical text, cited.

Many chemistry textbooks (among them even Quantum Chemistry texts) do not mention the *Hellmann-Feynman* (electrostatic) theorem (or its physical content) for the ground state equilibrium of molecules and thus fail to explain chemical bonding in scientific terms! (Chemists might challenge this statement, physicists, except Heitlers pupils, will agree!).

Ionic or Covalent ?

For a long time it was customary in the teaching of chemistry to make an almost exclusive distinction between “ionic” and “covalent” bonds. The ionic bond was explained with Coulombs law describing the attraction/repulsion between +/- charged, *spherical* ions with fixed radii. The radii were not explained, just taken from “experiment”, although no measurement of radii has ever been made, just cleverly compiled tables, e.g. by the famous Victor Moritz Goldschmidt or Linus Pauling (among others, assuming some connections of additivity and space requirements of spherical +/- ions. These are, of course, early ideas of Kimball spheres but not endowed with zeropoint kinetic energy and hence not understandable nor computable - just "a fact of nature" as one of my teachers (Nobel laureate Paul Karrer) used to say when a question was posed, he could not answer)! The covalent bond was unexplained and referred to G.N.Lewis (who could not explain it either, at his time), just a bonding electronpair.- Why should electrons so much like to pair up, repelling each other? Why do they like to live as four pairs in an octet? Lewis’ genius abstracted these strange notions from an overwhelming body of experimental data as soon as he heard about the Rutherford-Bohr model of the atom.

LiH was considered an “ion pair”, a “non existent” part of the well understood, stable {LiH} "ionic lattice" (no chemical stockroom sells LiH gas!). The problem has a sociological component:

Elementary chemistry has not digested almost 100 years of spectroscopic research (mostly done by physicists (sic)!) which is collected e.g. in the bible: K.P. Huber & G. Herzberg, “Molecular Spectra and Molecular Structure”, “Constants of diatomic molecules”, Van Nostrand, 1979 (first appeared in 1950). The information collected in this book was the experimental basis for the development of theoretical chemistry! It contains observed data about more than 600 stable diatomic molecules of which less than 20 are known to elementary chemistry. That’s fine, if only the criteria for their selection had been communicated.- Here, the criteria are simply didactical. The stable diatomic species H_{2}^{+}, H_{2}, HeH^{+}, He_{2}^{+}, LiH, Li_{2}^{+}, and Li_{2}, in this order, are just the simplest molecules to start understanding chemical bonding. Include the fact that HeH, He_{2} are not stable in the neutral groundstate (D^{0}_{0} ~ 0). Only H_{2} belongs to the selected species of basic chemistry (and is usually incorrectly explained).- But wait! We talk about CH4 already in our next tutorial!

Here is the summary of observations for the LiH molecule as scanned on p.382 of Huber-Herzberg (ask your mentor to explain the tables).

Let us compare LiH as described above with an ion pair Li(+)H(-): Red: Ion pair, Blue: LiH from above

If conditions for a LiH molecule are switched on the ion pair in red relaxes immediately to the blue spheres, the proton moves away and negative charge flows towards the Li-core. We now have a shared electron pair but still an electric dipole. All this happens outside the {LiH} crystal. The change for the Li He-like core radius is much smaller. We have it exaggerated a bit to avoid the coincidence of the two colors; the gray Li+3 nucleus moves so slightly that we have not drawn it twice. Note, that the outer cloud stabilizes with a smaller radius and an energy drop. Hence the onset of covalency happens spontaneously. This now is a polar molecule, or an ion pair with a covalent part! We don’t mince words ... Ionic and covalent bonding mix continously and both are caused by electrostatic forces.

A very good QC computation [CCSD(T)/cc-pVTZ, optimized] gives the "Atomic Mulliken charges" of the polar LiH molecule. Whereas the ion pair would have (+1)(-1) we see that the bond is formed with a large change of charge distribution in the molecule, leaving just a bit more than ±1/4 electronic charges on each nucleus.

Do not fall into the obvious trap to take the boundaries drawn in the graphics above as rigid! Remember, Kimball's free clouds are just one of the simplest computational tools to approximate the very complicated spatial electron density distribution. The boundaries drawn are the locus of an average measure of that distribution. A more sophisticated computation gives a better approximation of the density between the two cusps: The density contours are going through a neck or a "mountain pass" near the position where the two Kimball clouds touch, see comparison. In LiH it is impossible to say exactly where the Li "atom" ends and the H "atom" begins. Such a border would have a complicated shape and go somewhere through the bonding cloud. Instead, we have a molecule with its characteristic structure and electron distribution, not an agglomerate of two identifiable atoms. These do only manifest themselves if you destroy the molecule into Li and H atoms by administering the "dissociation energy" to the LiH molecule and finally form {Li} and H2 elements. The Li core and the bonding electron pair are separated by the condition of "orthogonality" (of the underlying wavefunction), if you want to hear the quantum chemical parlance. In Kimball's model that is expressed by two separated but touching spherical clouds, each occupied by two electrons with compensated spins ↑↓.

Observant readers may wonder why this LiH model with a single bond looks so much different from their image of a H2 molecule. Could one not use a large doubly occupied ) cloud, into which one puts a proton and a small )(+) cloud, a “bit” larger than a H(+) nucleus? One can do this as shown below and in Li2case

This second LiH model is so similar to H2, that it mimicks even details: The embedded small Li(+) sphere is seen from the proton like another singly charged “proton”. Hence, the distances of either nucleus to the center are identical and half the radius of the outer sphere, as in H2. In molecular beams the scattering of particles - electrons, neutrons, lightquanta a.o. - at LiH molecules can be measured and e.g. a scattering diameter determined. The two models of LiH let us expect quite different results. Hence, experiment could decide which model is nearer the truth.

We know that H2 and LiH are chemically very different molecules. H2 gas is nonpolar and very reluctant to condense to a liquid, boiling near 20 Kelvin/1 bar. LiH is difficult to observe as a free molecule. It needs high temperatures and molecular beam/high vacuum conditions. Otherwise, it will condense to a white crystalline solid salt {LiH} well known to many chemists who do synthetic chemistry. It has a dipolmoment of 0.88 Debye. The electronic structure, even in the simple Kimball image, should bear upon these chemical differences. This is much better realized with the first model above.

This is the H2 like LiH: