Cracking the Quantum Nut: My First Encounter with Hartree-Fock

From the beginning of my third year at university, I became deeply fascinated by quantum chemistry. I wasn’t just eager to learn about it — I wanted to apply my understanding of quantum principles to real scientific problems. My first real encounter with this field came during my fourth year, when my supervisor assigned me a simple DFT (Density Functional Theory) task. At first, I thought, “Wow, this is easy — just a few clicks in Gaussian software!” I even felt a bit disappointed that it didn’t seem to involve much chemistry. But that was my naive perspective. After doing some research, I quickly realized how challenging and profound DFT actually is. To truly grasp it, one must first understand the Hartree–Fock self-consistent field theory (HF). So, what are you waiting for?

When it comes to problems about the electronic structure of atoms and molecules then HF is a very efficient way to do that. So, as a computational chemistry enthusiast you can not completely ignore HF theory.  

From the Schrodinger equation we have:

Therefore, to perform a complete molecular calculation, we must construct the Hamiltonian (H) for a multi-electron system. H is a combination of potential energy and kinetic energy. 

For multielectron we have kinetic energy of electrons, the kinetic energy of nuclei, the Coulomb attraction between electrons and nuclei, repulsion between electrons and repulsion between nuclei, respectively. Thus the total hamiltonian for multielectron is given in the following figure.  

We have not gone through the derivation because we can get it from different textbooks. As you know, the mass of electrons is much much lower than the nucleus itself. Thus kinetic energy of nuclei can be overlooked. This approximation is made by Max Born and called Born Oppenheimer approximation. We can neglect the second and last term. Then we got an equation like the figure below. This approximation made our calculation easier. The first and second term is a one electron problem. But the last term is two electron problems. The solution of this part depends on coordinates of two electrons. Thus it is a very arduous process. 

Here, Hartree comes into the picture. He attempted to solve this problem by assuming that electrons are indistinguishable and introducing the mean-field theory. The idea is, instead of counting each electron repulsion he assumes that take one electron and considers the repulsion of that particular electron with the average potential field created by the rest of the electron. Hence, we can replace last term with: 

Thus, the updated Hamiltonian and the corresponding wave equation can be expressed as follows:

Now, let's jump to the wave function. As we know, the interelectron repulsion term does not permit separation of the Schrodinger equation into one-electron equations. Therefore, the product of wave function ψ using H-like orbitals should not be expected to yield correct energy in the case of many electron atoms. Hartree assumed that the wave function of an n-electron atom is a simple product of one electron spin-orbitals. 

Up until this point it is okay and it is undeniable that Hartree-method is the first most important endeavour to solve multielectronic problems in quantum chemistry. But there are several issues. The first problem is that if we swap two electrons then it leaves the function unchanged. That means two electrons can occupy the same state. Which is forbidden by Pauli’s exclusion principle. In short, electrons wave function must be antisymmetric with respect to the exchange of electrons. Because it ignores the permutational antisymmetry of the wave function it does not count exchange interactions. This exchange interaction is a result of pure quantum mechanical effect. So, Hartree-method has some limitations. To solve these problems, Vladimir Fock extended Hartree’s idea by including the Pauli exclusion principle explicitly. 

Let’s construct an antisymmetric wave function for three electrons. To construct a three‑electron wavefunction, the Pauli exclusion principle requires the total wavefunction to be antisymmetric: it changes sign when the electrons are exchanged. Fock used the determinantal form of the wave function which is also known as Slater determinants. The normalized wave function for three electron atom with the Slater determinants is written as:

The term outside the determinant ensures unit normalization, provided that the occupied spin-orbitals are orthonormal. Here, each row is labelled with an electron and each column with a spin-orbital. The Slater determinant introduces some correlation for the electron with parallel spin but does not account for the electron with anti-parallal spin (dynamic correlation). Thus it is an uncorrelated method. But as we include the exchange term here we obviously need to add an exchange operator (see the following equation) in our hamiltonian. So, we have to update our hamiltonian by the following term: 

So far, we have come across a multielectron Schrodinger equation with Hartree-mean field independent electron spin-orbital approximation and we also introduced the Fock method to ensure antisymmetry (incorporate exchange property).  Thus our final Hartree-Fock equation will be:

As we see in the equation ,this is not a linear equation. The Fock operator ​ itself depends on all the occupied orbitals (through the Coulomb and Exchange operators). But these are the orbitals we are trying to find! We have to solve it iteratively. To solve it:

1. Make an initial guess for the set of molecular orbitals. 

2. Use these guessed orbitals to build the Coulomb and Exchange operators, and thus the Fock operator

3. Solve the eigenvalue equation

4. Compare the new orbitals (or the new energy) with the previous ones. If they are not the same (within a chosen threshold), go back to step 2 using the new orbitals.

5. Repeat this process until the output orbitals are identical to the input orbitals. The system is now "self-consistent," and the potential field is consistent with the electron distribution it produces.

This process is self consistent thus we say “Hartrr-Fock self consistent field” theory.  One last important thing is that the  last version of Hartree-Fock equation is not used by computers to calculate the electronic structure of an atom or molecule. It requires more mathematical formalism for example Roothaan-Hall Equations to make it suitable for calculation. We’ll explore that topic in more detail in another article. Until then, goodbye!

References:

1. Prasad, R. K. (2014). Quantum chemistry (4th rev. ed.). New Age International Publishers. ISBN 978-81-224-2408-9

2. https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Advanced_Theoretical_Chemistry_(Simons)/06%3A_Electronic_Structure/6.03%3A_The_Hartree-Fock_Approximation

3. https://insilicosci.com/hartree-fock-method-a-simple-explanation/