At the minimum, the occupied orbitals are eigensolutions to the Fock operator via a unitary transformation between themselves.
12.
By the fundamental theorem, we may replace the new set by the old set subject to a unitary transformation.
13.
This is because time-dependent unitary transformations relate operators in one picture to the analogous operators in the others.
14.
A common first step beyond this is to remove the remaining time dependence in the Hamiltonian via another unitary transformation.
15.
Localized orbitals may then be found as linear combinations of the delocalized orbitals, given by an appropriate unitary transformation.
16.
If there are more than one such unitary transformations, then we can say that the set is not yet complete.
17.
The time dependence of the quantum state and the operators can be placed according to unitary transformations on the operators and states.
18.
For a complete set of commuting operators, we can find a unique unitary transformation which will simultaneously diagonalize all of them.
19.
The fact that unitary matrices will commute with the Hamiltonian means that the physical quantities calculated do not change even under unitary transformation.
20.
In particular, the time evolution symplectomorphism generated by the classical Hamiltonian deforms to a unitary transformation generated by the corresponding quantum Hamiltonian.
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