Quantum Chemistry — Molecular Orbitals & HF Theory

Quantum chemistry explains bonding by modeling electrons as wavefunctions, not as particles sitting in fixed little paths. For many intro questions, the key ideas are molecular orbital theory, which lets orbitals spread across a whole molecule, and Hartree-Fock theory, which gives a practical way to approximate those orbitals.

If you only keep one picture in mind, use this one: instead of asking which electron "belongs" to which bond, ask which allowed electron patterns exist for the whole molecule and how the electrons fill them.

What Molecular Orbitals Are

A molecular orbital, or MO, is an allowed one-electron wavefunction for a molecule within the model being used. In practice, chemists often approximate an MO by combining atomic orbitals:

$$\psi = c_A \phi_A + c_B \phi_B$$

Here, $\phi_A$ and $\phi_B$ are atomic orbitals and the coefficients say how much each orbital contributes.

When two atomic orbitals combine constructively, the electron density between the nuclei increases and a bonding orbital can form. When they combine destructively, a node appears between the nuclei and an antibonding orbital can form.

That is the core MO idea. Bonding depends on the shape and energy of orbitals spread over the molecule, not only on a localized shared pair drawn between two atoms.

How Hartree-Fock Approximates Molecular Orbitals

For a many-electron molecule, the exact electronic Schrodinger equation is usually not solvable in closed form. Hartree-Fock, often abbreviated HF, gives a practical approximation.

In HF, each electron is described by a one-electron spin-orbital, but those orbitals are solved together because each electron feels the average effect of all the others. The total electronic wavefunction is written as a Slater determinant, which enforces the antisymmetry required for electrons.

The method is called self-consistent field because the orbitals used to build the average field must also be the orbitals that come out of solving the equations. So HF is not just "guess an orbital." It is "guess, solve, update, and repeat until the input and output agree closely enough."

HF includes an explicit exchange term, but it does not capture all electron correlation. That missing correlation is one reason more accurate methods are often built on top of HF rather than stopping there.

Worked Example: The $H_2$ Molecular Orbital Picture

$H_2$ is the cleanest starting example because each hydrogen contributes one $1s$ atomic orbital and one electron.

Those two $1s$ orbitals can combine into:

• a lower-energy bonding orbital, often written $\sigma_{1s}$
• a higher-energy antibonding orbital, often written $\sigma_{1s}^*$

Both electrons go into the lower-energy bonding orbital with opposite spins. None go into the antibonding orbital.

In the simple diatomic MO filling picture, bond order is

$$\text{bond order} = \frac{N_b - N_a}{2}$$

where $N_b$ is the number of electrons in bonding orbitals and $N_a$ is the number in antibonding orbitals.

For $H_2$, $N_b = 2$ and $N_a = 0$, so

$$\text{bond order} = \frac{2 - 0}{2} = 1$$

That matches the idea of a single bond. It also shows why MO theory is useful: if electrons are placed in the antibonding orbital, the bond order drops and the bond is expected to weaken.

What does HF add here? In a real calculation, HF would not just label two orbitals by hand. It would solve for the best orbitals of the molecule within the mean-field approximation, then use those orbitals to compute an approximate electronic energy.

The Intuition That Makes This Click

The safest intuition is not "electrons orbit like planets." It is "electrons occupy allowed wave patterns."

If an orbital puts more electron density between nuclei, that usually supports bonding. If an orbital creates a node between nuclei, that usually weakens bonding. HF then asks for the best set of occupied orbitals when every electron is included in an average, self-consistent way.

This is why MO theory is especially useful for delocalization. If electron density is naturally spread over several atoms, a whole-molecule orbital picture is often cleaner than forcing the electrons into strictly local bonds.

When Quantum Chemistry Uses This Model

Molecular orbitals and HF theory are useful when you want to:

1. predict whether filling an orbital strengthens or weakens a bond
2. describe delocalized electrons across a whole molecule
3. estimate approximate molecular energies and orbital shapes
4. build a starting point for more accurate quantum chemistry methods
5. connect spectroscopy, magnetism, and reactivity to electron structure

In practice, HF is often the first serious approximation in computational chemistry because it is structured enough to be useful and still tractable for many systems.

Common Mistakes

Thinking An Orbital Is A Literal Path

An orbital is not a literal trajectory. It is part of a wavefunction model that helps predict electron distribution and energy.

Treating Hartree-Fock As Exact

HF is an approximation. If electron correlation matters strongly, HF can give qualitatively incomplete or numerically poor results.

Using Bond Order As If It Always Applies

The formula

$$\text{bond order} = \frac{N_b - N_a}{2}$$

is a simple MO-counting rule that works in the standard introductory MO picture. It is not a universal replacement for every bonding analysis in every molecule.

Assuming Any Two Atomic Orbitals Mix Well

The orbitals must have compatible symmetry and comparable enough energy to mix effectively. If those conditions fail, the combination is weak or negligible.

Mixing Up Hartree-Fock And Density Functional Theory

Both are major electronic-structure approaches, but they are not the same method. HF is built from a Slater determinant of orbitals in a mean field, while DFT uses the electron density as its central variable.

When To Use Molecular Orbital Theory And HF

Use this topic when Lewis structures or localized bond pictures start feeling too coarse. That usually happens when you care about antibonding orbitals, delocalization, magnetic behavior, or why a computational method needs an initial wavefunction model.

If you want to try your own version, compare $H_2$, $He_2$, and $H_2^+$. Filling the same bonding and antibonding framework with different electron counts is one of the fastest ways to see why molecular orbital theory changes bonding predictions.