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In the context of Hartree-Fock theory, using Roothaan formalism, we write:

FC=SCE

where F is the matrix of the Fock operator, C are the coefficients to be used in the construction of the wave function from the basis set, S is the overlap matrix and E are the energies.

The problem is that the coefficients in C are actually included in F, if we make it explicit. Using Roothaan approach, we just need to choose a basis set to be able to compute all the integrals in F and in S . . . but we need a "density matrix" (a part of F) that takes into account what molecular orbitals are occupied, in order to (partially) account for electron-electron Coulomb interaction.

So the approach is:

  1. Choose basis set and molecular geometry
  2. Compute all the integrals, as stated above
  3. Guess initial density matrix
  4. Solve the matrix equation to find the energies E

We are guided by the variational principle that ensures we cannot "overshoot" in the search for the minimum energy . . . the lower we find, the better (=closer to true ground state energy).

If up to this point i got it right, then my question is the following; after one cycle, as described above, i'd be tempted to say that the correct procedure is this:

I) Use some kind of minimization algorithm to make a new guess for the density matrix, which should be equivalent to saying 'make a new guess for the coefficients in C' II) Repeat the cycle above III) Evaluate if the new energy minimum is below the previous one (or check for another, more convenient, convergence criterion) IV) Repeat until convergence

What instead seems to be the correct way is that, after one cycle, you use the energies to compute a new set of coefficients, which you then use to construct a new density matrix and so on . . . you are now in a cycle (SCF) and you can stop iterating once you achieve convergence according to the criteria you set.

What seems strange to me is that if you guess a density matrix, then solve the Roothaan matrix equation, i would expect the energies to be "consistent" with the guess we've made . . . meaning that we will NOT get new coefficients from the procedure. Could you give me an intuition why this is not the case? Is everything else i've stated above correct?

Thank you.

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  • $\begingroup$ Can you explain why you "expect the energies to be "consistent" with the guess we've made" - I can't see any reason for this to be the case. Maths would be useful. Please edit the question to explain this. $\endgroup$
    – Ian Bush
    Nov 18, 2021 at 9:12
  • $\begingroup$ BTW the minimisation procedure, or at least something conceptually related, is not uncommon in codes that employ a plane wave basis set. $\endgroup$
    – Ian Bush
    Nov 18, 2021 at 9:12
  • $\begingroup$ @IanBush it might be that actually doing maths is required to understand what's going on, but i'm trying to understand the method from a bird's-eye view perspective. what i see is an equation: FC=SCe. If i guess C (and hence part of F) i can solve the equation and find 'e'. Then, how can 'e' correspond to a matrix C that's different from the one i used to find 'e'? It seems that this way the equation is no longer satisfied . . . $\endgroup$ Nov 18, 2021 at 9:28
  • $\begingroup$ Can't see how you can do anything without maths. I really can't see why you think a matrix which is a complex non-linear mapping of a set of vectors Q will have eigenvectors C such that C=Q (or strictly C=QU, where U is a unitary matrix). This is effectively what you are asking. $\endgroup$
    – Ian Bush
    Nov 18, 2021 at 9:47
  • $\begingroup$ @IanBush I'm not sure i follow you in your terminology. Do you mean that the fact I don't "get back" the coefficients I "put into" the matrix F, once i solve the eigenvalue equation, is due to the fact that those coefficients don't appear in F as a linear combination, but are multiplied together? btw I don't think you have to use mathematical notation, especially when you are trying to build an intuition for something, even though of course maths underlies the whole thing. $\endgroup$ Nov 20, 2021 at 16:02

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I think that the problem you have is that once that you introduce a guess density matrix (step 3) you go straight away to solve the Rothan-Hall equations (step 4) which will give you a new density matrix, a new set of molecular molecular orbitals and a new energy.

You could obtain the energy associated to a particular density matrix using the expression for the energy associated to that density matrix enter image description here

where $P_{\mu\nu}$ represent the elements of the density matrix, $H^{core}_{\mu\nu}$ are the elements of the core hamiltonian matrixm and the intregals correspond to the value of the bielectronic integrals.

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    $\begingroup$ That's an interesting perspective, which puts emphasis on how the energy you get by guessing the density matrix is NOT the energy you get if you then solve the eigenvalue equation: FC=SCe. In fact, the eigenvalue equation comes out if we now put the derivative of the energy with respect to all molecular orbital coefficients to 0, and then try to solve the set of N equations that arise from this. So, yeah, I'd say from this angle it's clearer why the coefficients would change (are no longer equal to my first guess) once i solve the eigenvalue equation. I still need to think a bit about this. $\endgroup$ Nov 20, 2021 at 16:09

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