I was reviewing many electronic computational methods recently and got a bit confused about relation between different approaches. In particular, what is the essential difference between Hartree–Fock (HF) and tight binding (TB)? I am not that deeply familiar with theory. And just for note I am physicist more than chemist.

I know that HF is more general method not essentially relying on LCAO approximation. But say we mainly use LCAO for HF anyway in most cases. In TB the Hamiltonian matrix elements $H_{ij}$ are mainly used from some tabulated values or fitting with experimental data. This is good for speedy calculations.

But can we just use LCAO orbitals and calculate matrix elements analytically using general many-body Hamiltonian? Will it not become HF method essentially?