Hydrogen near equilibrium distance

The optimum value of c is determined by the variational principle. If c = 1, the UHF wave function is identical to RHF. This will normally be the case near the equilibrium distance. As the bond is stretched, the UHF wave function allows each of the electrons to localize on a nucleus c goes towards 0. The point where the RHF and UHF descriptions start to differ is often referred to as the RHF/UHF instability point. This is an example of symmetry breaking, as discussed in Section 3.8.3. The UHF wave function correctly dissociates into two hydrogen atoms, however, the symmetry breaking of the MOs has two other, closely connected, consequences introduction of electron correlation and spin contamination. To illustrate these concepts, we need to look at the 4 o UHF determinant, and the six RHF determinants in eqs. (4.15) and (4.16) in more detail. We will again ignore all normalization constants. 

Figure 3. Evolution of normalized NILS hydrogen concentration C, IC vs. normalized distance R lb ahead of the crack tip for crack size r/ = 1.9 mm (a) near crack tip solution, (b) solution over the entire uncracked ligament. The parameter b denotes the crack tip opening displacement which varies with time as the hydrogen pressure increases toward its final value of 15 MPa over 1 sec. The parameter C =2.659x10 H atoms/m ( = 3.142x10 H atoms per solvent atoms) denotes the hydrogen concentration on the inner wall-surface and crack faces in equilibrium with the hydrogen gas.

In hydrogen fluoride the situation is different. For this molecule the ionic curve and the covalent curve are nearly coincident in the neighborhood of the equilibrium internuclear distance. In conse-... 

A dissociation equilibrium where the acidic hydrogen is two atoms removed from the aromatic ring has a Hammett p value near to unity due to the similar distances between the substituent and reaction centre in the reference dissociation. 

Dipole moment function of the hydrogen fluoride molecule as a function of the separation distance between H and F in angstroms. This curve is based on spectroscopic data. (From Sileo, R.N. and Cool, T.A., /. Chem. Phys., 65,117,1976 Gough, T.E. et al., Faraday Discuss. Chem. Soc., 71,77,1981.) The equilibrium separation distance is at 0.92 A, and in the vicinity of the equilibrium, the dipole moment curve is very nearly linear. 

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