Computational quantum chemistry (DFT, intro)

From quantum mechanics to computer calculation

Computational quantum chemistry aims to solve the electronic Schrödinger equation for real molecules. The time-independent Schrödinger equation reads:

Ĥ ψ = E ψ

where Ĥ contains electronic kinetic energy, electron–electron repulsion, and nucleus–electron attraction terms. The problem is exactly solvable for H and H₂⁺; for every other system, systematic approximations are required.

The Born-Oppenheimer approximation is the first: nuclei, being far heavier, are treated as fixed while electrons redistribute. This yields a potential energy surface (PES) as a function of nuclear coordinates.

Hartree-Fock: the single-determinant limit

The Hartree-Fock (HF) method represents the many-electron wavefunction by a single Slater determinant:

ψ_HF = (1/√N!) det[χ₁χ₂…χ_N]

where χ_i are spin-orbitals. Each electron experiences the mean field of all others (self-consistent field, SCF). The HF energy is the best achievable with one determinant.

The energy missing relative to the exact non-relativistic value is the correlation energy:

E_corr = E_exact − E_HF (< 0)

It represents roughly 1 % of total energy yet can account for 100 % of a reaction energy. Post-HF methods (MP2, CCSD, CCSD(T)) recover this correlation at increasing computational cost (O(N⁵) to O(N⁷)).

Density Functional Theory (DFT)

DFT rests on the Hohenberg-Kohn theorems (1964): 1. The ground-state energy is a unique functional of the electron density ρ(r). 2. The variational principle ensures the true density minimises the energy.

The DFT energy is written:

E[ρ] = T[ρ] + V_ee[ρ] + V_ext[ρ]

The problematic term is the exchange-correlation energy E_xc[ρ], whose exact form is unknown. Kohn and Sham (1965) reformulated the problem by introducing a fictitious non-interacting electron system with the same density as the real one, reducing DFT to a tractable orbital problem.

B3LYP is historically the most cited functional in organic chemistry; PBE dominates for solids and physics.

Basis sets: the language of calculation

Kohn-Sham orbitals are expanded in a basis set — a set of known functions. Two main families exist:

Gaussian-type functions (GTF): φ(r) = N · r^l · exp(−αr²). Two-electron integrals are evaluated analytically. Pople bases (6-31G, 6-311+G*), Dunning bases (cc-pVDZ, aug-cc-pVTZ), Ahlrichs bases (def2-SVP, def2-TZVP).

Plane waves: used in solid-state physics with pseudopotentials; not suited to isolated molecules.

The star notation (one or two asterisks) indicates polarisation functions (d on heavy atoms, p on H). Diffuse functions (prefixed with + or aug-) are essential for anions and excited states.

Reading a Gaussian output file

A typical Gaussian calculation (B3LYP/6-31G* on water) produces structured output:

1. Header: charge, multiplicity, basis, functional, version. 2. Initial geometry: coordinates in Ångströms (Z-matrix or Cartesian). 3. SCF cycles: convergence of energy (~10⁻⁸ hartree) and density. 4. HF/DFT energy: in hartree (1 Eh = 2625.5 kJ/mol). 5. Optimised geometry: if opt is requested — bond lengths (Å), angles (°). 6. Harmonic frequencies: if freq — values in cm⁻¹ (negative = transition structure). 7. NBO/Mulliken charges: electron density analysis.

"Essentially, all models are wrong, but some are useful." — George Box. In DFT, B3LYP/6-31G* remains useful to explore trends, even though def2-TZVP/PBE0-D3 is preferable for publication.

The key quantity for comparing molecules is the thermochemical Gibbs energy G = E_elec + ZPE + H_therm − TS, accessible with the freq=temperature keyword.

Limitations and outlook

Standard DFT underestimates van der Waals (London) dispersion. Grimme's empirical corrections (D3, D3BJ) or the non-local VV10 functional remedy this. Multi-reference transition-metal states (Fe, Mn in enzymes) remain difficult for DFT; CASSCF/NEVPT2 is required. DFT-scale calculations for thousands of atoms are possible with DFTB (approximate tight-binding) or QM/MM methods.