Fundamental quantum QSAR (Q²SAR) equation: Extensions, nonlinear terms and generalizations within extended Hilbert-Sobolev spaces

The quantitative structure activity relationship (QSAR) equation was analyzed from the point of the expectation value structure in extended Sobolev and Hilbert spaces. A new quantum mechanical related formalism, where the typical scalar Schrödinger wave function was extended, was derived. The extended wave function formalism forwarded the idea that the wave function and its gradient could be collected together to form a new mathematical entity.