# At what point in the Hartree-Fock formalism is the Fock operator introduced?

I seem to be having a conceptual problem with HF theory as to where the Fock operator comes from. In order to explain the context of my conceptual problem, I will include how HF was derived according to the resources I have used:

Starting from the electronic time-independent Schrodinger equation (assume Born-Oppenheimer approximation), the electronic Hamiltonian containing the correlated motion of the set of all electrons $$\boldsymbol{\vec{r}}$$: $$\hat{H}_{elec}(\boldsymbol{\vec{r}})\Psi_{elec}(\boldsymbol{\vec{r}})=E_{elec}\Psi_{elec}(\boldsymbol{\vec{r}})$$ $$\hat{H}_{elec}(\boldsymbol{\vec{r}})=-\sum_{i=1}^{N}\frac{1}{2}\nabla_{i}^{2}-\sum_{i=1}^{N}\sum_{k=1}^{M}\frac{Z_{k}}{\vec{r}_{ik}}+\sum_{i Step 1: Assume (incorrectly) no electron correlation. The many-electron electronic Hamiltonian operator $$\hat{H}_{elec}$$ becomes separable into one-electron Hamiltonian operators $$\hat{h}_{i}$$: $$\hat{H}_{elec}(\boldsymbol{\vec{r}})=\sum_{i=1}^{N}\hat{h}_{i}(\vec{r}_{i})$$ $$\hat{h}_{i}(\vec{r}_{i})=\frac{1}{2}\nabla_{i}^{2}-\sum_{k=1}^{M}\frac{Z_{k}}{\vec{r}_{ik}}$$ As a result, the many-electron wavefunction $$\Psi_{elec}$$ is expressed as the product of one-electron wavefunctions $$\psi_{i}$$, and the total electronic energy $$E_{elec}$$ is the sum of one-electron energies $$\epsilon_{i}$$: $$\Psi_{elec}(\boldsymbol{\vec{r}})=\prod_{i=1}^{N}\psi_{i}(r_{i})$$ $$E_{elec}=\sum_{i=1}^{N}\epsilon_{i}$$ Step 2: The assumption that electrons are not correlated is severe; however, use the product of one-electron wavefunctions as a first order approximation for which is expanded upon in the following steps. $$\Psi_{elec}(\boldsymbol{\vec{r}})\approx\prod_{i=1}^{N}\psi_{i}(r_{i})$$ The one-electron product wavefunction must be modified to include spin and the antisymmetry of electrons: $$\Psi(\vec{\boldsymbol{x}})=\frac{1}{\sqrt{N!}}\begin{vmatrix} \chi_{i}(\vec{x}_{i})&\chi_{j}(\vec{x}_{i})&\cdots&\chi_{N}(\vec{x}_{i})\\ \chi_{i}(\vec{x}_{j})&\chi_{j}(\vec{x}_{j})&\cdots&\chi_{N}(\vec{x}_{j})\\ \vdots&\vdots&\ddots&\vdots\\ \chi_{i}(\vec{x}_{N})&\chi_{j}(\vec{x}_{N})&\cdots&\chi_{N}(\vec{x}_{N}) \end{vmatrix}$$ where $$\chi_{i}(\vec{x}_{i})=\psi_{i}(\vec{r}_{i})\sigma_{i}(\omega_{i})$$

Step 3: Generate an energy expression using the exact many-electron electronic Hamiltonian operator and antisymmetric wavefunction: $$E_{elec}=\left \langle \Psi(\vec{\boldsymbol{x}})|\hat{H}_{elec}|\Psi(\vec{\boldsymbol{x}}) \right \rangle=\sum_{i=1}^{N}h_{i}+\sum_{i where the one-electron energies are: $$h_{i}=\left \langle i|\hat{h}_{i}|i \right \rangle=\int\chi_{i}(\vec{x}_{i})\left [\sum_{i=1}^{N}\left(-\frac{1}{2}\nabla_{i}^{2}-\sum_{k=1}^{M}\frac{Z_{k}}{\vec{r}_{ik}}\right)\right]\chi_{i}(\vec{x}_{i})d\vec{x}_{i}$$ and the two-electron energies are: $$J_{ij}=\left [ ii|jj \right ]=\iint\chi_{i}^{\ast}(\vec{x}_{i})\chi_{i}(\vec{x}_{i})\left [\frac{1}{r_{ij}} \right ]\chi_{j}^{\ast}(\vec{x}_{j})\chi_{j}(\vec{x}_{j})d\vec{x}_{j}d\vec{x}_{i}$$ $$K_{ij}=\left [ ij|ji \right ]=\iint\chi_{i}^{\ast}(\vec{x}_{i})\chi_{j}(\vec{x}_{i})\left [ \frac{1}{r_{ij}} \right ]\chi_{j}^{\ast}(\vec{x}_{j})\chi_{i}(\vec{x}_{j})d\vec{x}_{j}d\vec{x}_{i}$$ Step 4: Minimize energy of antisymmetric wavefunction with respect to all spin orbitals $$\chi_{a}(\vec{x}_{i})$$, with the constraint that all spin orbitals remain orthonormal during energy minimization. Lagrange's method of undetermined multipliers is implemented to find an energy minimum under the orthonormality constraint, where the Lagrange function $$L\left[\left\{\chi_{a}\right\}\right]$$ is expressed as follows: $$L\left[\left\{\chi_{a}\right\}\right]=E_{0}\left[\left\{\chi_{a}\right\}\right]-\sum_{a=1}^{N}\sum_{b=1}^{N}\epsilon_{ab}(\int\chi_{a}^{\ast}\chi_{b}-\delta_{ab})$$ The Lagrange function is then differentiated with respect to the set of spin orbitals and the following expression results: $$\hat{f}_{i}(x_{i})\chi_{a}(x_{i})=\sum_{b=1}^{N}\epsilon_{ba}\chi_{b}(x_{i})$$

$$\hat{f}_{i}(\vec{x}_{i})=\hat{h}_{i}(\vec{x}_{i})+\sum_{b=1}^{N}\int\left [\chi_{b}^{\ast}(\vec{x}_{j})\frac{1}{r_{ij}}\left ( 1-\hat{P}_{ab} \right )\chi_{b}(\vec{x}_{j})d\vec{x}_{j} \right]$$

$$\hat{H}_{elec}(\boldsymbol{\vec{x}})\approx\sum_{i=1}^{N}\hat{f}_{i}(\vec{x}_{i})$$

Confusion: The moment I get confused occurs around around the mathematics surrounding step 4. After generating a minimum energy expression and differentiating the Lagrange function with respect to the set of spin orbitals, somehow the noncanonical form of the Hartree-Fock equations results, containing the approxmate Fock operator.

It seems to me that the exact form of the many-electron electronic Hamiltonian operator is used when the energy expression is generated (step 3), but at the end of energy minimization with the orthonormality restraint (step 4) we end up with the approximate many-electron electronic Hamiltonian.

My Question: At what point is the Fock operator introduced?

A) Does the Fock operator come in because of Step 2— as the natural result of generating a minimum energy expression by assuming the many-electron wavefunction takes the form of an antisymmetrized determinant of one-electron spin orbitals?

B) Does the Fock operator come in during Step 4— as the result of substituting an approximate expression for two-electron integrals in place of the exact expression for two-electron integrals while generating a minimum energy expression?

C) Does the Fock operator comes from somewhere else that I did not think of?

D) Am talking out my shorts, and I have no idea what is actually going on? (very possible)

Nevertheless, I appreciate anyone who is able to take the time to help me clear my confusion.