The

G.N.Lewis Model

## The protagonists H, H |

Details for the calculations are given in the Tutorial; original Kimball version, experimental numbers in parenthesis.## H-Atom: Variable R, Parameter ZEkin = 9/8R ^{2}; Vne = -3Z/2R;
Vee = 0; Vnn = 0Etot = Ekin + Vne; Min[Etot(R)] = -Z ^{2}/2 Eh; R = 3/2Z a_{o} (exact match of experiment)Extension with principle quantum number n produces the H-atom spectrum. ## H |

## HeH
similar to H |

1) Electron-electron repulsion of pair in same cloud, spins different:
2) High bond energy, low distance in HThis produces a screening of the nuclear charge for each electron. Assuming equal density and distribution of both electrons over the cloud gives a screening constant of Slater found σ = 0.3. Applying this to:
^{+}(1s^{1})^{2}/2 Eh; σ = 0.3 gives good energies for all Z(1s^{2}) He homologues as shown in table as K' model.^{-} gives Etot = -0.49 (-0.5275) Eh for σ = 0.3. With σ = 0.4 it is -0.36 Eh. Both cases yield an unstable hydride ion in the gas phase because E(H1s) = -0.5 Eh, i.e. more stable than the hydride. The experiment has an EA = 0.0275 Eh.
But even the Hartree-Fock limit produces no better result.Hence, the error originates from neglecting correlation energy: The electrons of the pair do not independently move through the whole cloud (space). They prefer being as close to the nuclei as possible but simultaneously avoid each other as much as possible. There is no simple scheme to correct this gross error in Kimball's model nor in Hartree-Fock theory (see any text of QC)! In order to rationalize Slater's value of σ = 0.3 we try to model it:a) e1 is mainly in an inner sphere with r1 while e2 is in an equal volume outer spherical shell R-r1 -> σ = 0.37 b) the average distance of e1 and e2 is <r> = R -> σ = 0.3333 (best try and better than 0.3 for higher shells) c) each electron is predominantly near the center of mass of the other halfsphere -> σ = 0.4444; even worse! d) apply E. Wigner's correlation hole function: would destroy the simplicity of Kimball's model Conclusion: we accept Slater's screening constant for 1s^{2} and determine others by parametrization with a good (correlation corrected) QC result. This applies to all electron pairs in Kimball calculations and is shown in the examples of the main page.
_{2}^{+}, H_{2}, H_{3}^{+}:This is an effect of the inability of Kimballs ansatz to model the density cusps at the nuclei, partially relieved by FSGO, floating spherical gaussian orbitals, which destroy the simplicity of the K-model. _{2}^{+}: Etot = -0.7276 (-0.6026) Eh; Bond energy 0.2276 (0.1026) Eh; Bond length 0.829 (1.058) Å.
Energy minimum reached, Virial theorem correct, Hellman-Feynman force equlibrium true! (the Variation Principle does not apply!). These three pillars of the model cannot guarantee quality of the result. In fact H_{2}^{+} is the worst case of anything computed hitherto with the K-model, a failure.- What can be done: Two variables: R,r, one parameter Z(=1). Easy: Change the "effective" nuclear charge - often used in QC functions - not applicable here: There is no screening constant to adjust! We have no criterion for how much Z=1 should be lowered and thus destabilize the molecular ion. Next: Tweak the kinetic energy, i.e. R, of the bonding electron: With a kinetic energy parameter of k=1.21 we get Etot = -0.6013 (-0.6026) Eh; Bond energy 0.1013 (0.1026) Eh; Bond length 1.0032 (1.058) Å. Not bad! Another approach: compose two half clouds: Gives what we hope for, and is even pleasing as a more elliptical molecule, but the sins immediate penalty manifests itself by a bad Virial ratio and non vanishing Hellmann-Feynman forces! Conclusion: We can ± save the best case of a didactical introduction of chemical bonding with an adjustment of the kinetic energy of the K-model from 9/8R ^{2} to 1.21*9/8R^{2}. Remember, that Heisenberg's uncertainty relation requires 9/8R^{2} as a minimum.
_{2}: Etot = -1.21 (-1.1745) Eh; Bond energy 0.21 (0.1745) Eh; Bond length 0.7216 (0.7414) Å. Bond energy +20%, length -2.7% error. Same trend as in H_{2}^{+}, although much smaller. Remedy: 2-electron problem, hence apply He-adjusted screening constant σ = 0.3 ? Wrong! Both errors larger, because missing cusp modelling increased by higher effective charge. Tweak σ to the other side? Yes: σ = 0.4 -> σ = 0.4165 brings both errors down to 0.3%. Rationale: An increase in the screening constant goes in the direction of modelling a cusp near each proton, and that is, how Kimball should be corrected. |