Hartree-Fock equation equivalent forms

Equivalently, time-dependent canonical Hartree-Fock equations are assumed to take the same form as the time-dependent Schrodinger equation. 

This fact is most forcefully illustrated in the second of the three equivalent forms of the Hartree-Fock equation ... 

Note that performing a unitary transformation will not affect the average of the electronic Hamiltonian but only the HF wave function by a phase factor of unity modulus. Under these circumstances, the famous Hartree-Fock equation results from the successive equivalent forms ... 

The Kohn-Sham equations would indeed be equivalent to the Schrodinger equation if the universal exchange-correlation functional were known. Since it has not yet been discovered, practical forms of Kohn-Sham theory are (like Hartree-Fock theory) well-defined approximations to Schrodinger theory. 

---