5. Basis sets, convergence and error budgets
Position in the course: Lesson 5 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
Prerequisites: Variational principle and SCF matrices
1. Representation and physics are different errors
A finite basis restricts orbital shapes; Hartree-Fock restricts the many-electron state to a determinant. Enlarging the basis improves the representation within the same ansatz but cannot restore omitted correlation. Compare calculations only when the Hamiltonian, reference type, geometry and convergence conditions agree. A lower energy from a different approximation is not a clean measure of basis improvement.
Gaussian orbitals use factors such as \(x^ly^mz^n e^{-\alpha r^2}\). Their products are analytically convenient for integrals, but a finite Gaussian expansion does not reproduce the electron-nucleus cusp exactly. Several primitive Gaussians may be contracted into one fixed basis function. Splitting valence flexibility, adding polarization angular functions and adding diffuse long-range functions address different missing features.
2. Derive a cusp diagnostic
For a nucleus of charge \(Z\), consider the local spherical part \(\psi(r)\simeq\psi(0)(1+cr)\). The Laplacian of the linear term is \(2c\psi(0)/r\). In atomic units the singular terms in \([-\nabla^2/2-Z/r]\psi\) cancel only if \(-c-Z=0\). Thus the spherical-average cusp condition is \(\psi'(0)/\psi(0)=-Z\). Every centered Gaussian has zero first derivative at the origin; a finite sum therefore misses that exact cusp. The energy may still converge well because the singular region has small volume, but contact properties can be more demanding.
3. Nested variational spaces and numerical conditioning
For exactly optimized HF in nested basis spaces, the ground-state minimum cannot increase: the old determinant remains available. Common named basis families are not always strictly nested, so monotonicity is not guaranteed between arbitrary basis labels. Near-linear dependence makes the overlap matrix have tiny eigenvalues. Orthogonalization multiplies those directions by \(s_i^{-1/2}\), amplifying roundoff. Report the threshold used to delete directions and inspect whether the property changes when it is varied.
4. Worked interaction-energy error
Define \(\Delta E=E_{AB}-E_A-E_B\) at fixed specified geometries. In a finite dimer basis, fragment \(A\) can borrow functions on \(B\), lowering its energy more than in its isolated basis. A counterpoise construction computes the fragments with the partner's functions retained as ghost functions:
If teaching values are \(E_{AB}^{AB}=-100.020\), \(E_A^A=-60.000\), \(E_B^B=-40.000\), the uncorrected interaction is \(-0.020\) hartree. If ghost-basis values are \(-60.006\) and \(-40.004\), the corrected value is \(-0.010\) hartree. Half the apparent binding was basis borrowing in this illustrative arithmetic. Counterpoise is a finite-basis diagnostic, not a universal replacement for basis convergence; geometry deformation must be accounted for separately.
5. Construct a practical error budget
Converge the target property, not just an unrelated total energy. Separate SCF residual error, integral thresholds, basis incompleteness, correlation truncation, geometry uncertainty and Hamiltonian approximations such as scalar relativity or pseudopotentials. For energy differences, use balanced settings for every term. A cancellation that works for one reaction may fail for another. Report changes under controlled refinements rather than claiming a universal accuracy from a method name.
6. Exercises and solutions
Why add diffuse functions to an anion? Its weakly bound outer density extends farther. Tight core functions cannot efficiently represent that tail. Excessively diffuse functions can also worsen conditioning.
Does a converged SCF energy establish the complete-basis limit? No. It establishes stationarity in a chosen finite representation. Repeat with a systematic basis enlargement and examine the quantity of interest.
Why may a dipole converge differently from an energy? Dipoles weight the spatial distribution and tails. Variational stationarity protects energy errors to a degree that does not automatically protect arbitrary expectation values.
7. References and study connections
- DePrince group academic HF tutorial
- DePrince group academic tutorial collection
- Psi4 official coupled-cluster documentation
- Helgaker, Jørgensen and Olsen: Molecular Electronic-Structure Theory
The derivations and toy arithmetic are original teaching constructions. No molecular simulation is reported here.
8. Related theory and practice
Quantum mechanics · Density functional theory · Quantum Monte Carlo · Gaussian