Atomic electron affinities as a `litmus test'

The electron affinities of the first-and second-row atoms have often been used as benchmarks for high-level electronic structure methods (see e.g. the introductions to Refs.[57,58] for reviews). Because electron affinities involve a change in the number of electrons correlated in the system, they are very taxing tests for any electron correlation method; in addition, they involve a pronounced change in the spatial extent of the wave function, making them very demanding in terms of the basis set as well.

Until recently, three of the first-and second-row atomic electron affinities were imprecisely known experimentally (B, Al, and Si): this situation was changed very recently by high-precision measurements for recent experiments for B[59], Al[60,61], and Si[62].

The approach we have chosen here for the SCF and valence correlation components is summarized in our notation as W[n,Q,56,56,Q56] for the first-row atoms, and W[n,Q,Q5,Q5,TQ5] for the second-row atoms. The effect of inner-shell correlations was assessed at the CCSD(T)/MTav5z level, while Darwin and mass-velocity corrections were evaluated at the ACPF/MTav5z level. Finally, spin-orbit splittings were calculated at the CASSCF-CI level with the spdf part of a MTav5z basis set. (For technical reasons, the h functions were omitted in both the scalar relativistic and spin-orbit calculations, as were the g functions in the latter.)

Our best computed results are compared with experiment in Table I, where results from recent calibration studies are also summarized.

Agreement between computed and observed values can be described without reservation as excellent: the mean absolute error amounts to 0.0009 eV. The fact that this accuracy is obtained systematically and across the board strongly suggest that the 'right result was obtained for the right reason'. Upon eliminating the corrections for imperfections in CCSD(T), i.e. restricting ourselves to W[56,56,Q56] for first-row atoms and W[Q5,Q5,TQ5] for second-row atoms, the mean absolute error increases by an order of magnitude to 0.009 eV, i.e. about 0.2 kcal/mol. As we shall see below, this is essentially the type of accuracy we can obtain for molecules without corrections for CCSD(T) imperfections.

The importance of Darwin and mass-velocity corrections increases, as expected, with increasing Z, and its contribution becomes quite nontrivial for atoms like Cl. It is therefore to be expected that, e.g., in polar second-row molecules like ClCN or SO₂ they will contribute substantially to TAE as well.

The importance of inner-shell correlation effects is actually largest for Al, because of the small gap between valence and sub-valence orbitals in the early second-row elements.

Table II compares the convergence behavior of the extrapolated valence correlation contributions as a function of the largest basis set used, both using the Martin three-term and Helgaker two-term formulas. While both formulas appear to give the same answer if the underlying basis sets are large enough, the two-term formula is by far the more stable towards reduction of the basis sets used in the extrapolation. Since the use of W[Q56,Q56,Q56] is hardly an option for molecules, the two-term formula appears to be the formula of choice.

Following the suggestion of a referee, we have considered (Table II) the performance of some other extrapolation formulas for the valence correlation energy. As a point of reference, we have taken an ``experimental valence correlation contribution to EA'', which we derived by subtracting all computed contributions other than the valence correlation energy from the best experimental EAs. While some residual uncertainties may remain in some of the individual contributions, these should be reliable to 0.001 eV on average.

As seen in Table II, performance of the geometric series extrapolation[55] A+B/Cⁿis outright poor: in fact, for extrapolation from AV{D,T,Q}Z results the error is twice as large as that caused by not extrapolating at all. If AV{T,Q,5}Z basis sets are used, mean absolute error drops to 0.015 eV, which is still an order of magnitude larger than for the A+B/l³ extrapolation, and only slightly better than not extrapolating at all. Finally, for AV{Q,5,6}Z basis sets, the error is three times smaller than complete omission of extrapolation, but three times larger than that of using any of the following formulas: A+B/l³ [56], A+B/l^C [13], or A+B/(l+1/2)⁴+C/(l+1/2)⁶ [13]. All three of the latter yield a mean absolute error of about 0.001 eV, on about the same order of accuracy as the reference values. For the smallest basis set series AV{D,T,Q}Z, the mixed exponential-Gaussian extrapolation[63] $A+B/\exp(l-1)+C/\exp((l-1)^2)$ represents a very substantial improvement over A+B/C^l, and actually exhibits the second-best performance (after A+B/l³). For the AV{T,Q,5}Z series which is of greatest interest here, the Halkier et al. A+B/l³formula by far outperforms the other formulas considered.

In short, it appears to be established beyond doubt that the two-term formula of Helgaker and coworkers[56] is the extrapolation method of choice, with the Martin three-term formula A+B/l^C the second-best choice provided basis sets of AV{T,Q,5}Z quality are used, and the mixed exponential-Gaussian formula if only AV{D,T,Q}Z basis sets are used.

Computed spin-orbit contributions to the electron affinities are compared in Table III to values obtained from observed fine structures[64,59,61]. While small deviations appear to persist, these may at least in part be due to higher-order spin-orbit effects which were neglected in the calculation rather than to deficiencies in the electronic structure treatment. At any rate, to the accuracy relevant for our purpose (establishing spin-orbit corrections to molecular binding energies) it appears to be immaterial whether the computed or the experimentally derived values are used.

Finally, the convergence of the SCF component is so rapid that it appears to be essentially irrelevant which extrapolation formula is used -- the amount bridged by the extrapolation is on the order of 0.0001 eV.