Tutorial 6a Kimball Tutorial 7 Tutorial 8
Content: Chemical structure, 3rd part: From molecules composed of atoms to molecules “composed” of nuclei and electrons
Kimball’s model deals with the latter notion, taking the Rutherford-Bohr model rigorously into quantum chemistry.
Where do we go from here?
ES 28 December/07 Sept 2017//2002//1982
Clash of Paradigms
Let us look at two structural concepts in chemistry and pick some more issues expressed in R.G.Woolleys paper. If you don’t like that, just skip this chapter! However, you would miss a hot (hidden) controversy!
was the final proof of a molecular “constitution”. Read a classic for this: Paul Karrer, “Textbook of Organic Chemistry”, 14th (last) edition, 1963, Thieme Stuttgart (of which I have read and tried to remember every word, and reproduced many successfully during two oral examinations by Karrer!).
2) Then came the electron-nuclear atom model of Rutherford-Bohr, G.N. Lewis, W. Pauli, E. Schrödinger - the birth of theoretical chemistry. This branch has had ups and downs but is now established beyond any doubt and is consulted, cited and used by the same famous names and some new ones, equally famous, like R.S. Mulliken, J.A.Pople and W. Kohn, or, on the experimental side D.Hodgkin (building on what has been started by the Bragg's), R.Ernst and K.Wüthrich. The theoreticians did'nt tell and the chemists did not ask, that they use completely different ideas of a molecular structure: A molecule is a unique system, not built from atoms like a wall with bricks, but characterized by a 3D electron density (with electronic kinetic zeropoint energy) shaped by the electric field of the embedded nuclei which are kept at their locations in the molecule by an equilibrium of attracting and repelling electrostatic forces. It has stability if its local energy minimum can at least carry the zeropoint kinetic energy of the nuclei. And this notion is now proven, not by chemical decomposition studies, but by non destructive methodology, mainly by a large array of refined spectroscopic instrumentation: microwave, infrared, Raman, NMR, UV-Vis, X-ray, and several others. Some of these measure precisely the coordinates of the nuclei within the structure. Perhaps, mass spectrometry is still used similarly to one of the older chemical tools - carefully disintegrate a molecule and identify the fragments remaining to "build" a plausible, qualitative structure (meaning "shape" but no coordinates).
state of CH4 and its tetrahedral shape is completely derivable from electrostatics and Pauli's principle, applied to the C+6 and 4 H+ nuclei and 10 electrons with zeropoint energy! Nothing else is necessary. It is much more difficult to derive CH4 from a neutral C and 4 neutral H atoms, a procedure not understandable by highschool mathematics, nor provable in the laboratory! The same is true for bonding angles ∠HNH or ∠HOH in NH3, H2O, respectively. How theoreticians come to quantitative answers in those cases is their scientific process. They may be using parts of atomic wavefunctions - orbitals defined in spherical symmetry ! - to approximate the unknown molecular wavefunction in a variational treatment, or they may be doing it with sine and cosine functions, gaussian bell shape functions, any other asymptotically complete (orthogonal) function set, or without it (like James-Coolidge or W. Kołos and L. Wolniewicz for H2), the ingredients are electrons and nuclei, not atoms.
Where do we go from here?
We continue carefully to explore the limits of the simple, well understandable model of Kimball, and simultaneously, notions we may falsify anytime. No jargon!
We’re still not farther than the ground states of very simple molecules. But a large part of modern, quantitative, knowledge in chemistry has been contributed by all varieties of spectroscopy. They probe energystates of molecules accessible by electromagnetic radiation, absorption and emission of “light” from radiowaves to X-rays. They are just a tiny bit, e.g. NMR-spectroscopy, or a big chunk, e.g. X-rays, above the ground state.
The applicability of Kimball’s model to describe excited electronic states has not been widely examined, yet.
It is manifest that the main features of the excited states of the H atom can be easily computed, e.g. the famous Balmer series, by introducing the principle quantum number n, in a very short summary:
The finer details, the "finestructure" (or even "hyperfinestructure"), the spectroscopists have so masterly revealed in the pioneer episode of atom physics and quantum mechanics, are beyond this simple model, however (see G. Herzberg, Atomic Spectra and Atomic Structure, Prentice Hall, Inc, 1937, or the famous bible by E.U.Condon & G.H.Shortley, The Theory of Atomic Spectra, Cambridge 1935,1951,1989).
Let's see, whether we can do something with the He spectrum. The main light absorption from the ground state at a wavelength of 58.43339 nm leads to the He[1s2p] state. This is stable but may soon emit a light quantum to "fall back" into the ground state or, very rarely, into a state unreachable from the ground state, He[1s2s], emitting light with a wavelength of 2058.6904 nm.
We now model the (1P0) state for He(0) to P(+13), determine the excitation energy, and compare it with experiment. This is an exercise to learn how we can represent an atomic p-function with Kimball’s model. The result is very good without any parametrization, except for using Slater’s value for the screening constant of He σ=0.3.
Here is the He[1s2s] (1S) excited state. This is not reachable by electromagnetic radiation from the ground state, because the “transition moment” = 0 between s-states and hence the excitation ns ← ms is forbidden (no “antenna” from central sphere1 to central sphere2! There is a “selection rule” Δℓ = ±1 for allowed transitions, with ℓ the angular momentum quantum number. For s states ℓ=0, for p states ℓ=1). As mentioned above, the excited state He[1s2p] and all other [1s,np], n>1, states are allowed to relax into the [1s,ns] states with emission of a photon. Many of these transitions are observed and thus states with [1s,ns] known.
And here is the excited state. This is, again, not reachable by electromagnetic radiation from the ground state, because the “transition moment” = 0 between s- and d-states and hence the excitation nd ← ms is forbidden. For d states ℓ=2, so Δℓ = 2, not ±1. States with ℓ=1,3 can combine with d-states, however, from where the energy of [1s3d] is known up tp F(+7).
Part of the success of these Kimball models of excited states of He-like atoms may be due to the fact, that we deal with singly occupied electron clouds and singlet states, only.
However, we may also try simple molecules and atoms with more than 2 electrons with the same scheme:
Look at the B-1Σu+, C-1Πu, and E-1Σg+ states of the hydrogen molecule H2, and on the atomic
We conclude this tutorial by summing up:
■ modern chemical theory uses the electron-nuclear model for quantitatively describing matter - molecules and solids of all kinds. Except for methods probing matter deeply down into the 1s electrons of heavy atoms, like some results of NMR, or X-ray spectroscopies, atoms do not manifest themselves within molecules, but nuclei do. Chemical reactions are the stage of the topmost, least stable, electrons. They behave uniquely in the way of reflecting the molecular structure and identity, not of some conglomerate of constituting atoms.
■ Kimball’s model can successfully model excited states of atoms and molecules.