Hydrogen atom expectation values

Recurrence relation for hydrogen-atom expectation values... 

The bonding molecular orbital of a hydrogen molecule in its lowest energy (ground) state contains both electrons from the individual hydrogen atoms. The value of if/ (and therefore also i/r ) is large between the nuclei, precisely as expected since the electrons are shared by both nuclei to form the covalent bond. 

Because particles have wavelike properties, we cannot expect them to behave like pointlike objects moving along precise trajectories. Schrodinger s approach was to replace the precise trajectory of a particle by a wavefunction, i]i (the Greek letter psi), a mathematical function with values that vary with position. Some wavefunctions are very simple shortly we shall meet one that is simply sin x when we get to the hydrogen atom, we shall meet one that is like e x. 

A search for intermolecular bonds resulted in one possible hydrogen bond between hydroxyl 013 and lactone carbonyl Ol. The distance between 01 and 013 is 2.85 A, a value well within the range expected for OH-O hydrogen bonds (25). The hydrogen atom position for hydroxyl 013 was chosen to be along the 013-01 vector. The hydrogen position was not evident in the difference electron density map, presumably due to problems modeling the 013 position. 

This mechanism accounts for the positive value of p, namely, electron-withdrawing substituents increase the acidity of the hydrogen atom. If the reaction occurred by abstraction of the hydride ion, one would expect a negative value of p. 

The expectation values of powers and inverse powers of r for any arbitrary state of the hydrogen-like atom are defined by... 

Show explicitly for a hydrogen atom in the Is state that the total energy is equal to one-half the expectation value of the potential energy of interaction between the electron and the nucleus. This result is an example of the quantum-mechanical virial theorem. 

Calculate the expectation value for 1/r for the electron in the Is state of a hydrogen atom. 

For the cobalt-based system the molecularity of the transition state indicated by the reaction order is H3C0C4O4 and the reactants are H2 and HCoCCO). Thus, two hydrogen atoms start with values of v 3200 cm 1 and one with v 1830 cm"1. If in the transition state the strong H-H bond is not yet completely broken, then we should expect to find the H atom originally attached to cobalt bound to carbon or oxygen (v 2900-3400 cm"1) in the transition state. 

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