Optimal Radial Decay of Molecular Orbitals

In Problem AlO.l we proposed a trial form for the Is orbital in which the decay with distance from the nucleus is controlled by a parameter 

For f Ibohr , the lCTg+) orbital becomes more spread out this has the effect of reducing the kinetic energy, as the MO has less curvature than when we use the AO decay constant in the basis. The more diffuse orbital also increases the potential energy, as the electron spends more time away from the nuclei. This wins out and the energy increases compared with the f = 1 reference. 

For f 1 bohr , the lag+) orbital becomes more compact and the potential energy goes down. Of course, this also confines the electron more closely and increases the curvature of the wavefunction, and thus the kinetic energy. Near to unity the net effect is a lowering of the total energy, and a minimum is found for t = 1.238 bohr. At higher values of the decay constant the kinetic energy increases more rapidly than the potential falls. 

Note that now the electron kinetic energy is greater than the atomic reference state and stability is provided by the lower potential energy. The shrinking of the AOs around the nuclei increases the electron-nuclear favourable interactions and now outweighs the nuclear-nuclear repulsion. 

An important point to note from this study of a relatively simple system is that the AO functions as derived for isolated atoms are not the ideal basis for constructing MOs. In real molecular systems the orbital shape adapts to the potential it experiences. For quantitative work we require basis sets that can reproduce this by responding to the potential 

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