Basis-Set Convergence

Basis-set convergence asks whether the chosen finite space represents the state and observable accurately enough. It is a different limit from converging an iterative eigensolver inside that space. A stable energy at one cutoff can coexist with inaccurate derivatives, contact densities, polarizabilities, or small energy differences.

Helpful background. Rayleigh–Ritz states when nested subspaces give variational upper bounds.

The two limits of an eigenvalue calculation

Label the representation by $N$ and the eigensolver tolerance by $\eta$. The computed energy can be organized as

$$E_{N,\eta}=E_\infty+\delta_{\mathrm{basis}}(N) +\delta_{\mathrm{solve}}(N,\eta).$$

Here $\delta_{\mathrm{basis}}(N)$ is the error left after exactly solving the finite problem, and $\delta_{\mathrm{solve}}(N,\eta)$ is the additional solver error. Tighten $\eta$ until further changes are much smaller than the differences between adjacent $N$. Then study $N$. If the solver tolerance is comparable to those differences, the apparent convergence order is contaminated.

For conforming nested spaces and a fixed semibounded self-adjoint Hamiltonian, a ground-state Ritz energy cannot increase as the space grows. A finite-difference sequence or a sequence of separately optimized, non-nested bases need not follow this pattern. Even a monotone sequence can converge to the wrong limit if the chosen spaces never become rich enough to approximate the target state.

Keep all other approximations fixed while changing one representation parameter. For a plane-wave calculation, the reciprocal cutoff and cell size regulate different effects. For oscillator functions, the number of functions and oscillator length jointly control spatial reach and short-distance resolution. A single cutoff may improve both at unequal rates.

Why observables can lag behind energies

Write a normalized trial state near an isolated exact ground state as

$$|\psi\rangle=\sqrt{1-\epsilon^2}|0\rangle+\epsilon|v\rangle, \qquad \langle0|v\rangle=0,\quad\|v\|=1.$$

Stationarity gives

$$\langle H\rangle-E_0 =\epsilon^2(\langle v|H|v\rangle-E_0).$$

For a general observable $A$, the error instead contains $2\epsilon\,\operatorname{Re}\langle0|A|v\rangle$ at first order. Thus two energies agreeing to many digits can coexist with a visibly unconverged dipole moment. Unbounded observables require still more care: small Hilbert-space norm error does not alone control high moments or derivatives.

Converge the observable actually needed, along with energies, residuals, and relevant sum rules. Excitation energies should be tracked by symmetry and overlaps; reordering near a crossing can make a smooth calculation look discontinuous.

Extrapolation with a stated model

Suppose independent analysis or an established basis family motivates

$$E_N=E_\infty+aN^{-p}+o(N^{-p}).$$

For known $p$ and a fixed ratio $q>1$,

$$E_\infty\approx \frac{q^pE_{qN}-E_N}{q^p-1}.$$

Use at least one additional size to test the asymptotic assumption. Fitting an exponent to three noisy points and reporting an exact-looking intercept is not an uncertainty estimate. Compare fit windows and account for subleading terms. Exponential convergence may occur for smooth functions in a suitable spectral basis, whereas cusps and nonsmooth boundaries can impose algebraic convergence.

For an explicit check, let $E_N=1/2+2/N^2$. At $N=10$ and $20$, the energies are $0.52$ and $0.505$. The $p=2$ extrapolation gives $(4E_{20}-E_{10})/3=0.5$. Using $p=1$ gives $0.49$, below the true energy even though both raw values were upper bounds. Extrapolated values do not inherit a variational guarantee.

Exercises

A sequence has $E_{10}=0.510$, $E_{20}=0.5025$, and $E_{40}=0.500625$. Estimate the leading power and limiting energy.

Solution

The difference ratio is $(0.510-0.5025)/(0.5025-0.500625)=4=2^p$, so $p=2$ for this model. Either adjacent pair yields $E_\infty=0.5$. Additional sizes or physical error analysis are still needed before assuming that the simple model describes a real calculation.

References

• Bai, Z., J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, eds., Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM, 2000, online text — eigenpair error versus approximation-space error.
• Dunning, T. H., Jr., “Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen,” Journal of Chemical Physics 90, 1007–1023 (1989), doi:10.1063/1.456153 — systematic molecular basis families.