Basis set limit

The basis sets that we have considered thus far are sufficient for most calculations. However, for some high-level calculations a basis set that effectively enables the basis set limit to be achieved is required. The even-tempered basis set is designed to achieve this each function m this basis set is the product of a spherical harmonic and a Gaussian function multiplied... 

Castro Jorge universal Available for H(20.v) through Lr(32.v25/)20r/15/). For actually reaching the inhnite basis set limit to about seven digits of accuracy. 

CBS—n [n = 4, Lq, Q, APNO) Available for FI through Ne. For estimating the inhnite basis set limit. This implies a series of calculations with different basis sets, some of which are large sets. 

The rows of the chart correspond to increasingly larger basis sets. The specific basis sets cited there serve as examples, illustrating the additional types of functions added as you move down any column. The bottom row of the chart represents a completely flexible basis set, and the cells in it correspond to the basis set limit for each specified theoretical method. 

All of the geometries predicted with the 6-31IG basis set are quite accurate. Adding two sets of diffuse functions yields a more accurate structure. However, adding additional polarization functions does not significantly affect the results. 6-311++G(d,p) thus appears to achieve the basis set limit for this model chemistry. ... 

The following table lists the predicted bond energy of hydrogen fluoride computed with various methods using the 6-311-H-G(3df,3pd) basis set. We chose this basis set because it is near the basis set limit for this problem errors that remain can be assumed to arise from the method itself and not from the basis set. 

The CBS models use the known asymptotic convergence of pair natural orbital expansions to extrapolate from calculations using a finite basis set to the estimated complete basis set limit. See Appendix A for more details on this technique. 

The filled and hollow circles indicate the contributions of each successive natural orbital. The filled circles correspond to complete shells. Only these points are useful for extrapolating to the complete basis set limit. 

SCF procedure is begun, and then used in each SCF iteration. Formally, in the large basis set limit the SCF procedure involves a computational effort which increases as the number of basis functions to the fourth power. Below it will be shown that the scaling may be substantially smaller in acmal calculations. 

The main advantage of the ANO and cc basis sets is the ability to generate a sequence of basis sets which converges toward the basis set limit. For example, from a series of... 

To correct for electron correlation beyond QCISD(T) and basis set limitations, an empirical correction is added to the total energy. 

A MP2/6-311- -G(2df,2p) calculation is carried out, which automaticaUy yields the corresponding HF energy. The MP2 result is extrapolated to the basis set limit by the pair natural orbital method. 

The counterpoise corrected complexation energy is given as A eompLexaUan — AEcp- For regular basis sets this typically stabilizes at the basis set limiting value much earlier than the uncorrected value, but this is not necessarily the case if diffuse functions are included in tlie basis set. 

The calculated ioi as a function of basis set and electron correlation (valence electrons only) at the experimental geometry is given in Table 11.8. As the cc-pVXZ basis sets are fairly systematic in how they are extended from one level to the next, there is some justification for extrapolating the results to the infinite basis set limit (Section 5.4.5). The HF energy is expected to have an exponential behaviour, and a functional form of the type A + 5exp(—Cn) with n = 2-6 yields an infinite basis set limit of —76.0676 a.u., in perfect agreement with the estimated HF limit of -76.0676 0.0002 a.u. ... 

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. 

First consider the dipole operator (O = r). The matrix elements on rhs of eq. 17 are thus just the dipole transition moments, and the commutator becomes C = -ip. As the exact solution (complete basis set limit) to the RPA is under consideration, we may use eq. 10 to obtain... 

If the basis set is mathematically complete, then the equation holds precisely. In practice, one has to work with an incomplete finite basis set and hence the equality is only approximate. Results close to the basis set limit (the exact HF solutions) can nowadays be found, but for all practical intents and purposes, one needs to live with a basis set incompleteness error that must be investigated numerically for specific applications. 

One specific problem becomes very acute in wavefunction based methods the basis set problem. The introduction of a finite basis set is not highly problematic in HE theory since the results converge quickly to the basis set limit. This is, unfortunately, not true in post-HE theory where the results converge very slowly with basis set size - which is another reason why the methods become computationally intractable for more than a few heavy atoms (heavy being defined as nonhydrogen in this context). These problems are now understood and appropriate approaches have been defined to overcome the basis set problem but a detailed description is not appropriate here. 

Dickson and Becke, 1996, use a basis set free numerical approach for obtaining their LDA dipole moments, which defines the complete basis set limit. In all other investigations basis sets of at least polarized triple-zeta quality were employed. Some of these basis sets have been designed explicitly for electric field response properties, albeit in the wave function domain. In this category belong the POL basis sets designed by Sadlej and used by many authors as well as basis sets augmented by field-induced polarization (FTP) func-... 

In Table 12-5 we compare the binding energies computed using several hybrid functionals and basis sets, attempting to approach the basis set limit for each functional in a systematic (but not necessarily cost effective) way. At first we note the reasonable performance of all functionals. The converged results, however, indicate a slight tendency to underestimate the experimental value by about 1-2 kcal/mol. This trend is slightly more emphasized for... 

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