In our course on physical chemistry, which involves MOT, we have been taught that in the LCAO approach, the wave function for a molecule … say hydrogen ion ($\ce{H2+}$), can be approximated by a linear combination of two atomic orbitals.
From what I could gather from the very little I know about variational method is that this linear combination is going to be then optimised to give the minimum energy to find the ground state of the molecule in question. Is this correct?
Also, if we are simply using the linear combination as a test/guess function for a variational method, why should we restrict ourselves to only (say $+$ and $-$) two combinations? That is, how does the rule "Number of molecular orbitals = Number of atomic orbitals constituting them" arise?
Also, why is there the restriction in form of the rule that "Only atomic orbitals of close energies and proper symmetries combine to give MOs"?
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