First Principles: Ab Initio Quantum Chemistry Quiz

Hartree Fock is a mean field theory. While it accounts for exchange (Pauli principle), it fails to account for the instantaneous "correlation" between electrons, leading to the correlation energy error.

This is due to the Variational Principle. Since the HF wavefunction is an approximation (a single Slater determinant), the energy calculated will always be higher than or equal to the exact solution of the Schrödinger equation.

To satisfy the Pauli Exclusion Principle, the wavefunction must change sign when two electrons are swapped. A determinant naturally possesses this property, making it the standard way to represent many-electron wavefunctions.

These methods build upon the HF starting point by mixing in excited states or using perturbation theory to account for the fact that electrons avoid each other more than the mean field model suggests.

[Image comparing Gaussian and Slater type orbital shapes] While STOs better represent the "cusp" at the nucleus and the long-range decay, GTOs are mathematically easier for computers to integrate. We use "Contracted" Gaussians to mimic the shape of a Slater orbital.

An example is the STO-3G basis set. While computationally very cheap, minimal basis sets lack the flexibility to accurately describe the polarization and contraction of electron clouds during chemical bonding.

By doubling the number of functions for valence orbitals, the model allows the "size" of the orbital to adjust based on the chemical environment. This is often combined with polarization functions (e.g., 6-31G*).

CCSD(T) is highly accurate for small to medium molecules. Its size consistency ensures that the energy of two separated molecules is calculated correctly relative to the single system, which is vital for thermochemistry.

Since HF accounts for everything except instantaneous electron-electron repulsion, the "missing" energy is defined as the correlation energy. Recovering this energy is the primary goal of Post-HF methods.

Because the field an electron feels depends on the orbitals of the other electrons, you must guess the orbitals, calculate the field, find new orbitals, and repeat until the field and orbitals no longer change.

MP2 is the simplest and most common Post-HF method. It provides a significant improvement over HF for systems dominated by dispersion (van der Waals) forces at a relatively low computational cost.

In a molecule, a p-orbital might need to lean or bulge toward a bonding partner. Adding d-functions (higher angular momentum) allows the s and p orbitals to "hybridize" or polarize, yielding better geometries.

HF scales formally as $O(N^4)$, though modern techniques can reduce this for very large systems. In contrast, high-level methods like CCSD(T) scale as $O(N^7)$, explaining why they are limited to smaller molecules.

Size extensivity is a formal requirement for a method to be reliable for large systems. Methods that are not extensive (like some versions of CI) give increasingly poor results as the system size increases.

As you add more functions (moving from DZ to TZ to QZ), the wavefunction has more flexibility to minimize the energy. The CBS limit represents the best possible energy that specific level of theory can provide.