# SCF procedure performed for some quantum-chemical methods, but not others. Why?

The SCF procedure is used to find solutions to the Schrödinger equation, and obtain energy of the molecular system. It uses a "guess-and-check" approach, where an initial guess is needed to solve the equation, giving the charge density, which can be used again and again until the results become self-consistent.

So, why don't some quantum-chemical methods perform SCF, while others like HF do?

Most computation methods write the wavefunction($\Psi$) in terms of Slater determinants ($\Phi$)

$\Psi = \sum_i^L c_i\Phi_i$

which in turn are made from MOs ($\phi$), which are linear combinations of basis functions ($\chi$)

$\phi_i = \sum_\mu^K C_{\mu i}\chi_\mu$

The molecular energy depends on the MO coefficients ($C_{\mu i}$) in a non-linear fashion, so if they are variationally optimized then you need to iterate (i.e. SCF). This is the case for HF (where $L$ = 1 and $c_1$ = 1) and MCSCF (where both $\{c_i \}$ and $\{C_{\mu i} \}$ are optimized).

For methods like CI, perturbation theory, and coupled cluster, the MO coefficients are not changed, so you don't need an SCF.

It can also be said more generally that there are two main methods for solving the Schroedinger equation: variation and perturbation. When you use variation, you alter some coefficients of basis "things" (basis functions in SCF, Slater determinants in CI or both in MC-SCF) to make the initial guess of WAVE FUNCTION for the solution more and more consistent with the Hamiltonian of the system. When you do perturbation it is the HAMILTONIAN itself which is gradually improved to arrive at more precise wave function (which should be already made consistent with the initial Hamiltonian, though). Perturbation method is very general and hence versatile, used not only for energy calculations; historically, it was the first expansion beyond the HF - DFT came in much later and does not compute the actual wave function, only the electron density (and also the energy therefore).

Answering @Umkay comment on whether DFT uses SCF:

DFT (more precisely, Kohn-Sham version of it) DOES perform SCF. The crucial point is that DFT uses a different approach to the description of the system than the usual SCF/perturbation methods. These methods try to find the most reasonable approximation to the wave function of the system; DFT instead takes notice of many properties being a direct function of election density instead of the wave function itself; and, as the density has only three arguments (x, y and z) instead of 3*N for N-electron wave function, it tries to go round and compute the density not from the "best-approximated" wavefunction. It appears, though, that quite efficient way for this is to construct a fictional system which is defined to have the same density as the "true" one but has a wave function which is less undertaking to compute. Namely, this system have elections without direct interaction between them (i.e., without corresponding terms in the Hamiltonian) but, like in HF, having some self-consistent term describing their interactions with less computational burden. The difference with the HF is that KS-DFT uses this term to describe interactions between electrons which arguably can be described by such a term: the correlated motion of [distinct pairs of] electrons. In HF it is only the electrostatic field created by all the electrons which is described in this way (KS-DFT uses also that but not just that). The probable impossibility to have such CORRELATIONAL term is supported by the still ongoing search for the "best" density functional. Hence also the debate on whether DFT counts as an ab initio method. More importantly, it means that the orbital-related properties computed with DFT are quite nominally corresponding to those of the real system - DFT was simply not designed to get them, after all. On the other hand, one may say that to get density "good enough", you probably also need the "fictional system" close enough to the real one. For example, it was once shown by Stowasser and Hoffman in 1999 that for ionization properties KS-DFT orbitals look very similar to the "actual" orbitals for this case - the Dyson orbitals - computed with some high-level method.

But KS-DFT still makes use of SCF for computing the wave function of its "fictional" system. And there are the double-hybrid functionals which also employ perturbation approach. But these are not quite consistent with the initial formulation of DFT, because, as A. Becke noted in 2014 review, they use unoccupied orbitals which are merely by-products of employing SCF for DFT as they have no direct connection with the electron density at all.

• Welcome to Chem.SE, Igors! Nice to see you made the hop over from CCL. – hBy2Py Mar 16 '18 at 13:41
• Since DFT does not compute the actual wave function, it doesn't perform SCF, correct? Because I read somewhere on this site that SCF covers both HF and DFT, as the Kohn-Sham equations are solved using the SCF procedure identical to HF. – user60347 Mar 16 '18 at 22:31
• @hBy2Py: I just got to read the CC Daily mail and saw that I am capable of answering :-) – Igors Mihailovs Mar 17 '18 at 10:12
• @Umkay: I have edited the answer itself, I hope it does rely to your comment. – Igors Mihailovs Mar 17 '18 at 10:59