Bayesian Error Estimation in Density-Functional Theory

We present a practical scheme for performing error estimates for density-functional theory calculations. The approach, which is based on ideas from Bayesian statistics, involves creating an ensemble of exchange-correlation functionals by comparing with an experimental database of binding energies for molecules and solids. Fluctuations within the ensemble can then be used to estimate errors relative to experiment on calculated quantities such as binding energies, bond lengths, and vibrational frequencies. It is demonstrated that the error bars on energy differences may vary by orders of magnitude for different systems in good agreement with existing experience.

Figure 1

Exchange enhancement factors as a function of dimensionless electron density gradient. The gray lines show enhancement factors drawn from the ensemble $\exp (-C(\theta )/T)$. The dashed, dotted, and full lines show enhancement factors for PBE, RPBE, and the best fit, respectively.

Figure 2

Histograms of actual error relative to the BEE $(O_{\mathrm{bf}}-O_{\exp })/{\sigma }_{\mathrm{BEE}}(O)$ for different quantities. Top: atomization or cohesive energies. Middle: bond lengths and lattice constants. Bottom: vibrational frequencies and bulk moduli. The dashed lines show the expected normal distribution. Experimental numbers are taken from Refs. 22, 27, 28, 34, 35.

Figure 3

Upper panel: Calculated ensemble for cohesive energy ($x$ axis) and bcc-fcc energy difference ($y$ axis) for a copper crystal. The BEE’s are indicated by error bars. The inset uses rescaled axes. Lower panel: Calculated ensemble for binding energy ($x$ axis) and bridge-top energy difference for CO on a Cu(100) surface. Values for the experimentally preferred states [23, 36] (fcc and top) are indicated by vertical dotted lines. Units are in eV.