Derivation of the Hartree-Fock equation

I have to do some tutoring about basic quantum chemistry and the main purpose is to give a deep insight into Hartree-Fock theory. My plan is to start from expressing the many-electron wavefunction as a Slater determinant and why this is even necessary, etc. From this starting point, one intermediate objective is to give an elaborate and crystal clear derivation of the Hartree-Fock equation and I used my old lecture notes and some textbooks to work out a detailed - and most importantly - comprehensive approach.


But I have to consider some constraints. **Firstly**, the majority of the students are likely to struggle with more sophisticated aspects like the formalism of second quantization (like used in the textbook by Hegaker, Jorgensen and Olsen). Things like functional variation or Lagrange mulitpliers shouldn't be a problem because they were covered in prequisited math lecture series. **Secondly** I realised that every textbook I have used for preparation is leaving out some so called "trivial" parts of the derivation that I need for my own understanding.


So far I have worked out the following key steps to arrive at the general form of the Hartree-Fock equation:


1. Establishing the first approach to the many-electron wave function (the Hartree product)
2. Considering the indistinguishibility of electrons and thus the antisymmetry of the wavefunction with the regard of the exchange of two particles
3. Setting up the Slater determinant to come up for the antisymmetry
4. Deriving the expression for the energy of a Slater determinant (keeping in mind the mean-field approximation)
5. Minimization of the energy with the constraint that the spinorbitals remain orthonormal


I am struggling to formulate the last part as detailed and clear as possible. The best approach I found so far is in *Introduction to Computational Chemistry* by Jensen, but even in this textbook, there are some mathematical "leaps" I don't get.


So the **essence of my question** is: Do you know any resource (textbook or lecture notes, etc.) in which this last step (minimization with the constraint of orthonormality yielding the general form of the Hartree-Fock equation) is explained very detailed and clear ?


Any help is heavily appreciated.


