Quantum Chemistry — Molecular Orbitals & HF Theory

Reach for molecular orbital theory and Hartree-Fock when Lewis structures or localized bond pictures start feeling too coarse, which usually happens once you care about antibonding orbitals, delocalization, magnetic behavior, or why a computation needs a starting wavefunction. Quantum chemistry models electrons as wavefunctions, not particles in fixed paths, so instead of asking which electron "belongs" to which bond, you ask which allowed electron patterns exist for the whole molecule and how the electrons fill them.

The Procedure For Building A Molecular Orbital Picture

Working an MO problem follows a fixed sequence. The five steps below take you from atomic orbitals to a Hartree-Fock-level description.

1. Start from atomic orbitals. Identify which atomic orbitals are close enough in energy and compatible enough in symmetry to combine. If they are not, the combination is weak or negligible.
2. Build molecular orbitals. Combine the chosen atomic orbitals into bonding and, when allowed, antibonding orbitals. A molecular orbital is approximated as $\psi = c_A \phi_A + c_B \phi_B$, where the coefficients say how much each atomic orbital contributes. Constructive overlap raises electron density between the nuclei and gives a bonding orbital; destructive overlap puts a node between them and gives an antibonding orbital.
3. Fill electrons carefully. Place electrons from lower to higher energy while respecting spin rules.
4. Check bond order. In the simple MO-filling picture, use $\text{bond order} = (N_b - N_a)/2$, where $N_b$ and $N_a$ are electrons in bonding and antibonding orbitals.
5. Add the Hartree-Fock idea. Treat the molecular orbitals as unknowns solved self-consistently in an average electron-electron field, rather than labeled by hand.

For a many-electron molecule the exact electronic Schrodinger equation is usually unsolvable in closed form, so Hartree-Fock (HF) supplies a practical approximation. Each electron is described by a one-electron spin-orbital, but the orbitals are solved together because each electron feels the average effect of all the others. The total wavefunction is a Slater determinant, which enforces the antisymmetry electrons require. 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.

Worked Example: Running The Steps On H2

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

Step 1 and 2: the two $1s$ orbitals combine into a lower-energy bonding orbital, often $\sigma_{1s}$, and a higher-energy antibonding orbital, $\sigma_{1s}^*$. Step 3: both electrons enter the bonding orbital with opposite spins; none go into the antibonding orbital. Step 4: with $N_b = 2$ and $N_a = 0$,

which matches a single bond. Step 5: a real HF calculation would not label the two orbitals by hand. It would solve for the best orbitals within the mean-field approximation, then use them to compute an approximate electronic energy. HF includes an explicit exchange term but does not capture all electron correlation, which is why more accurate methods are often built on top of it.

Try The Sequence Yourself

Run the same five steps on $H_2$, $He_2$, and $H_2^+$. Use the identical bonding-and-antibonding framework and change only the electron count. Predict each bond order before checking: you should find $H_2^+$ at $1/2$ and $He_2$ at $0$. Watching the bond order fall as you add antibonding electrons is the fastest way to see why filling an orbital can weaken a bond.

Where The Procedure Breaks Down

Each step has a characteristic failure mode worth watching:

• Treating an orbital as a literal path. It is part of a wavefunction model that predicts electron distribution and energy, not a trajectory.
• Combining incompatible atomic orbitals (step 1). Orbitals need compatible symmetry and comparable energy to mix effectively.
• Over-trusting the bond-order formula (step 4). $(N_b - N_a)/2$ is a counting rule for the standard introductory picture, not a universal bonding analysis.
• Treating Hartree-Fock as exact (step 5). HF is an approximation; when electron correlation matters strongly it can be qualitatively incomplete. It is also not the same as density functional theory, which uses the electron density as its central variable rather than a Slater determinant of orbitals.

When This Approach Pays Off

Use molecular orbitals and HF to predict whether filling an orbital strengthens or weakens a bond, to describe delocalized electrons across a whole molecule, to estimate approximate molecular energies and orbital shapes, to build a starting point for more accurate methods, and to connect spectroscopy, magnetism, and reactivity to electron structure. In computational chemistry HF is often the first serious approximation, structured enough to be useful and still tractable for many systems, which is exactly why the five-step routine above is worth practicing.