What gets measured, what gets computed in quantum chemistry

The methods you'll find on this site — Hartree-Fock, DFT, TDDFT, coupled cluster, configuration interaction — all produce numbers. The question is which of those numbers a real chemist or spectroscopist actually uses, and how those numbers correspond to what an instrument in a lab actually measures. This page is the dictionary between the two columns. Read it before deciding which method to learn or which calculation to run.

What an experiment measures

Four broad classes of experimental observable, in roughly increasing equipment cost:

1. Structural data — where the atoms are

X-ray crystallography measures the diffraction pattern of a crystal, from which bond lengths and angles are refined. Gas-phase electron diffraction does something similar for small molecules in the vapor. Microwave spectroscopy gives rotational constants, which in turn yield moments of inertia — and thus bond lengths and angles — to $\sim 0.001$ Å accuracy for small enough molecules. The output is a list of nuclear positions, or equivalently a bond-length / bond-angle table.

2. Energetic data — heats, barriers, stabilities

Calorimetry measures reaction enthalpies $\Delta H_{\mathrm{rxn}}$ directly. Equilibrium constants at multiple temperatures give Gibbs free energies $\Delta G$ via van 't Hoff plots. Bond dissociation energies come from photoacoustic calorimetry, kinetics, or thermochemistry tables. Atomization energies are the headline number for "how stable is this molecule" — energy required to dissociate into separated atoms.

3. Spectroscopic data — peaks at frequencies

Each technique gives a spectrum: intensity versus frequency (or wavelength). The peaks are transitions; the intensities are transition strengths.

• UV-Vis absorption — electronic excitations, transitions to excited states. Peak positions ${\omega }_n$ are excitation energies; integrated peak areas are oscillator strengths $f_n$.
• IR / Raman — vibrational transitions; peaks at normal-mode frequencies, intensities tied to dipole-moment derivatives (IR) or polarizability derivatives (Raman).
• NMR — chemical shifts $\delta$ (in ppm) of individual nuclei, plus J-couplings between them. The shift reports on the local electronic environment via the shielding tensor.
• EPR / ESR (for open-shell systems) — g-factors and hyperfine couplings.
• Photoelectron spectroscopy — ejected-electron kinetic energies give ionization potentials, one per occupied orbital, with sub-eV resolution.
• X-ray absorption / emission — core-level spectra; sensitive to oxidation state, coordination, local geometry.

4. Kinetic data — how fast does a reaction go

Rate constants $k(T)$ measured at multiple temperatures, fit to an Arrhenius or Eyring form, yield activation energies $E_a$ (Arrhenius) or activation Gibbs energies $\Delta G^{‡}$ (Eyring). These are the transition-state-theory observables that a model has to match.

What a calculation produces

Start from a Hamiltonian and a method (HF, DFT, MP2, CCSD, CCSD(T), TDDFT, CASSCF, FCI, …) and you can compute the following. Roughly ordered from cheapest to most expensive:

Total electronic energy E

$E=⟨\Psi |\hat{H}|\Psi ⟩$ for the ground-state wavefunction. By itself a single number; differences between geometries / charge states / spin states are what you actually use. Most things on this list reduce to "compute $E$ twice and subtract."

Equilibrium geometry

Minimize $E$ over nuclear positions $\mathbf{R}$. Output: a stationary point with $\partial E/\partial \mathbf{R}=0$. Bond lengths and angles read directly off the result. Quality depends on the method: HF is usually 0.01-0.02 Å too short on bonds; DFT is method-dependent (B3LYP, PBE, BLYP each have their own systematic errors); CCSD(T) is typically within 0.005 Å of experiment for small molecules.

Vibrational frequencies and intensities

The Hessian $H_{ij}={\partial }^{2}E/\partial R_i\partial R_j$ at the equilibrium geometry. Diagonalize; eigenvalues divided by reduced masses give normal-mode frequencies, eigenvectors give normal-mode displacement patterns. IR intensities come from $|\partial \mathbf{\mu }/\partial Q_k{|}^2$ (dipole derivative along mode $k$); Raman from polarizability derivatives.

Excitation energies and oscillator strengths

The headline output of TDDFT (the Casida equation), CIS, EOM-CC, or CASPT2. Excitation energies ${\omega }_n$ are eigenvalues; oscillator strengths $f_n=(2/3){\omega }_n|⟨0|\hat{\mathbf{r}}|n⟩{|}^2$ come from the eigenvectors as transition dipole moments. Compared directly to UV-Vis peaks.

Electron density and density-derived properties

$\rho (\mathbf{r})={\sum }_i|{\psi }_i(\mathbf{r}){|}^2$ for occupied orbitals. From $\rho$ you get the dipole moment $\mathbf{\mu }=-\int \mathbf{r}\rho d\mathbf{r}$, higher multipoles, the electrostatic potential, and atomic charges (Mulliken, Hirshfeld, NBO, AIM, ChelpG — they all partition $\rho$ differently and there's no unique answer).

Linear-response properties

Polarizability $\alpha (\omega )$, hyperpolarizabilities, NMR shielding tensors $\sigma$, EPR g-tensors, optical rotation. Each is a response of the ground state to a weak perturbation (electric field, magnetic field, electron spin), computed via the same coupled-perturbed machinery that underlies TDDFT.

Ionization potentials and electron affinities

Compute $E(\text{neutral})$ and $E(\text{cation})$ separately; their difference is the adiabatic IP. The vertical IP keeps the geometry fixed. Same for EA with anion. Higher-quality: GW or IP-EOM-CC give the spectrum of IPs (one per orbital) directly without separate cation calculations.

Reaction barriers and transition states

Find a saddle point on the potential energy surface — geometry where $\partial E/\partial \mathbf{R}=0$ and the Hessian has exactly one negative eigenvalue. The energy difference between the saddle point and the reactants is the classical activation energy. Combined with vibrational frequencies, transition-state theory gives a rate constant $k(T)$.

The bridge

Direct correspondences, in the same row-by-row style as the comp-neuro version:

Which method for which quantity

The point of having so many methods is that no one method does everything well. The right tool depends on which quantity you need:

One-sentence summary

A quantum chemistry calculation is almost always answering "what will an experiment measure?" — and the menu of quantities is small enough to fit on one page. Pick the experimental observable you want to model, find the corresponding row of the bridge section, choose a method that handles that row well, and run it. Everything else on this site is the machinery that fills in the rightmost two columns of the table.

Related on this site

• Hartree-Fock — the starting point most other methods build from.
• Functional derivatives — the math machinery underlying DFT and TDDFT response theory.
• Casida equation (TDDFT) — excitation energies and oscillator strengths for medium molecules.
• Liouville-Lanczos — same response theory at solid-state scale via Krylov methods.
• Same idea, comp-neuro version — what gets measured and modeled in brain networks.