# Is the Self Consistent Field an actual field?

I know this is really stupid. But bear with me.

In everything I have read so far, the Self Consistent Field (SCF) is always used to describe a method or a process, namely, that of iteratively solving the Schrodinger's Equation to generate new wavefunctions, using them for a new potential, and solving until convergence.

If SCF is not an actual field, why is it named as such? And then when we describe the SCF space what are we really talking about? Does the iterative process create a series of objects that spans a field?

For SCF: https://en.wikipedia.org/wiki/Hartree%E2%80%93Fock_method or if you want a textbook for reference, there's always Szabo and Ostlund's Modern Quantum Chemistry.

• It might not hurt to add a link so those who don't recognize SCF or SE might have a chance to learn something also. Possibly en.wikipedia.org/wiki/Hartree%E2%80%93Fock_method or maybe tlchm.bris.ac.uk/webprojects2002/grant/webcomp/scf.html or even www3.nd.edu/~johnson/Class01F/chap3a.pdf but you may have something you like better.
– uhoh
Commented Jul 25, 2017 at 18:30
• Thanks @uhoh I added the wiki, but really liked the other two links you shared. Commented Jul 25, 2017 at 18:35
• @Karl I understand what you were trying to say, but it could have been expressed a lot more precisely without the judgement and overtones. Please be mindful of Be Nice. Commented Jul 25, 2017 at 22:33
• I'm tempted to ask "what is an actual field", but I'm allergic to rabbit holes.
– uhoh
Commented Jul 26, 2017 at 8:34
• I'm sorry, I should have clarified. I meant a mathematical field - the kind where division is defined, like the reals or complex field. Commented Jul 26, 2017 at 15:07

The notion of a self-consistent field comes from the orbital approximation - that a multi-electron wavefunction can be approximated as multiple one-electron wavefunctions.

At that point, the question remains - how does each of those electrons interact with the remaining $N-1$ electrons?

Well, there's an electric field from the electron density, right? Maybe you want to call this an electrostatic potential -- but Hartree considered it a spherically-symmetric electric field.

The problem, of course, is that since you don't know the final charge density, you have to make a guess and iterate until it's "self consistent."

Thus, SCF has been associated first with HF theory, and other related methods, including Kohn-Sham DFT.

• so there's a strictly physical basis for the name and concept, not a mathematical one? Commented Jul 25, 2017 at 18:51
• I'd have to go check Hartree's papers, but AFAIK, the "field" part is a physical one -- referring to the electric field experienced by the electron in an orbital. Commented Jul 25, 2017 at 18:58
• The "field" part can't really be separated from the mathematical construct of a field, but the more physically common field - an assignment of a value to all points in a space - not the algebraic field I think you're thinking of. Here the relevant physical field is the scalar field of charge density or its derivative, the vector field of electric field. Commented Jul 27, 2017 at 0:34
• Ah, yes, @user213305 makes a great distinction -- the term has nothing to do with an algebraic field. Commented Aug 1, 2017 at 17:24