Variational Wave-Function and Energies

The stationary Schrodinger equation may be integrated to its eigen-values by means of the variational principle respecting the minimizing of the total (eigen) energy of the system  

With this principle, actually all, natural systems may be quantum mechanically treated with the aid of trial-and-optimized wave-function to determine the correspondent eigen-energies. In next some of the most representative systems are to be accordingly treated from nuclear, to atomic, to molecular, and to solid-state level or matter s organization. 

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The effort in the first stage of the calculations has been focused on generating very accurate variational wave functions and energies for the rotationless vibrational states of the HD+ ion. As mentioned, this system has been studied by many researchers and very accurate, virtually exact nonrelativistic energies have been published in the literature [112]. This includes the energy for the highest vibrational v = 22 state, which is only about 0.4309 cm below the D - - H+ dissociation limit. 

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