Basis-set error

From a basis set study at the CCSD level for the static hyperpolarizability we concluded in Ref. [45] that the d-aug-cc-pVQZ results for 7o is converged within 1 - 2% to the CCSD basis set limit. The small variations for the A, B and B coefficients between the two triple zeta basis sets and the d-aug-cc-pVQZ basis, listed in Table 4, indicate that also for the first dispersion coefficients the remaining basis set error in d-aug-cc-pVQZ basis is only of the order of 1 - 2%. This corroborates that the results for the frequency-dependent hyperpolarizabilities obtained in Ref. [45] by a combination of the static d-aug-cc-pVQZ hyperpolarizability with dispersion curves calculated using the smaller t-aug-cc-pVTZ basis set are close to the CCSD basis set limit. 

Let us now consider the errors in the CC3 S( — 2) Cauchy moment (the static polarizability). From the monotonic convergence of the CCSD doubly augmented basis-set calculations of S( — 2) in Table 1 it appears that the difference between the 5Z and 6Z results should give a good estimate of the CCSD basis-set error at the 6Z level. CC3/d-aug-pV5Z 5 ( —2) Cauchy moment is 2.670 a.u. Using the... 

Figure 3. Basis set errors for the HF energy and CISD correlation energy with the final cc-pVnZ-PP basis sets for the ( , ) 5s 4d, (O, ) 5s4(f, and (A, A) 4

Figure 6. Basis set errors for the HF energy and CISD correlation energy with the final cc-pVnZ-PP basis sets for the ( , ) and (O, ) P states of...

Some argue that Equations (2) and (3) lead to an overcorrection of the BSSE, and, indeed, this is still under considerable discussion. Sufficiently large basis sets, usually with multiple sets of polarization functions, seem to overcome most of the BSSE. Therefore, we computed the magnitude of the BSSE for the dissociation energies separately to evaluate our basis set error margins approximately. 

Because of the studies of the basis set dependence of DFT rotational strengths, the errors of many basis sets are well defined. As discussed above, TZ2P and larger basis sets (e.g. cc-pVTZ) are very good approximations to the complete basis set limit. For these basis sets, errors are negligible. Of course, for much smaller basis sets, such as 6-31G, the opposite is true. 

Tabic 4 shows results from [36] for the static and the second harmonic generation hyperpolarizabilities of CO at 694.3 nm. The electronic contributions were obtained from CC3/d-aug-cc-pVTZ calculations carried out at R q = 2.132bohr. These were approximately corrected for remaining basis set errors by adding the difference between CCSD/d-aug-cc-pVQZ and CCSD/d-aug-cc-pVTZ results for the same frequency and internuclear distance. For CO the triples correction for /3 (0) is 1.72 a.u. or =6%. At a wavelength of 694.3 nm the triples correction is already 2.35 a.u. or s7%. Thus, there is in tliis case a notable triples effect on the frequency dispersion. Since there is no information available about correlation contributions beyond CC3, it is difficult to assess the accuracy of these results. 

Given a density functional, DFT calculations also require the choice of an atomic orbital basis set. The larger the basis set the closer calculations approach the complete basis set limit and the smaller the basis set error. At the same time, computational demands increase rapidly with increasing basis set size. The optimum choice of basis set is that which provides the optimum compromise of accuracy and computational effort. 

In Section III.E, EOM ionization potentials and electron affinities are compared with accurate configuration interaction (Cl) results for a number of atomic and molecular systems. The same one-electron basis sets are utilized in the EOM and Cl calculations, allowing for the separation of basis set errors from errors caused by approximations made in the solution of the EOM equation. EOM results are reported for various approximations including those for the extensive EOM theory developed in Section II. Section III.F presents results of excitation energy calculations for helium and beryllium to address a number of remaining difficult questions concerning the EOM method. 

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