Hydrogen, atomic Schroedinger equation

The reason a single equation = ( can describe all real or hypothetical mechanical systems is that the Hamiltonian operator H takes a different form for each new system. There is a limitation that accompanies the generality of the Hamiltonian and the Schroedinger equation We cannot find the exact location of any election, even in simple systems like the hydrogen atom. We must be satisfied with a probability distribution for the electron s whereabouts, governed by a function (1/ called the wave function. 

Using atomic units, the Schroedinger equation for ground-state hydrogen is... 

By this time, we have introduced so many approximations and restrictions on our wave function and energy spectrum that is no longer quite legitimate to call it a Schroedinger equation (Schroedinger s initial paper treated the hydrogen atom only.) We now write... 

The Schroedinger equation has not been solved for all of the 102 elements. The mathematics is both difficult and time-consuming, and solution is practical only for very simple systems such as the hydrogen atom or the He+ ion. The characteristics of the electrons in the more complicated atoms are set down by analogy. However, studies of the spectra of almost all elements have been carried out such studies indicate that these analogies were, for the most part, properly taken and that other atoms may be described in the same quantum terms that are used to describe the hydrogen atom. 

The Schroedinger equation for the hydrogen molecule that corresponds to equation 1 for the atom is... 

The Schroedinger equation cannot be solved exactly except for very simple systems like the hydrogen atom. For molecules, we must be satisfied with an approximate solution of H V(r) = is F(r). In recent years, owing to the work of Pople, Gordon, and others, agreement between MO approximations and such experimental results as exist has been brought to a level that makes quantum thermochemistry competitive with experimental thermochemistry in reliability. 

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