Basis set incompleteness

To summarize, the RPPA is a method that can accurately describe relativistic effects, even though the relativistic perturbation operator used in the pseudopotential procedure is acting on the valence space and not the region dose to the nudeus, as this is the case for the correct all-electron relativistic perturbation operator. That is, relativistic effects are completely transferred into the valence space. These effects are also completely transferable from the atomic to the molecular case as the results for Au2 show. If relativistic pseudopotentials are carefully adjusted, they can produce results with errors much smaller than the errors originating from basis set incompleteness, basis set superposition or from the electron correlation procedure applied. 

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. 

The results for the non-BO diagonal polarizability are shown in Table XIII. Our best—and, as it seems, well-converged—value of a, 29.57 a.u., calculated with a 244-term wave function, is slightly larger than the previously obtained corrected electronic values, 28.93 and 28.26 a.u. [88,91]. It is believed that the non-BO correction to the polarizability will be positive and on the order of less than 1 a.u. [92], but it is not possible to say if the difference between the value obtained in this work and the previous values for polarizability are due to this effect or to other effects, such as the basis set incompleteness in the BO calculations. An effective way of testing this would be to perform BO calculations of the electronic and vibrational components of polarizability using an extended, well-optimized set of explicitly correlated Gaussian functions. This type of calculation is outside of our current research interests and is quite expensive. It may become a possibility in the future. As such, we would like the polarizability value of 29.57 a.u. obtained in this work to serve as a standard for non-BO polarizability of LiH. 

BSSE arises from the intrinsic problem that finite basis sets do not describe the monomer and complex forms equally well. For instance, the energy of two monomers calculated in the full dimer basis is not the same as for the dimer. A simple evaluation of the interaction energy (AE) of the two fragments [Equation (1)] is incorrect. This problem is especially serious with small basis sets. Hence, the magnitude of the BSSE can be used as a measure of the basis set incompleteness. 

In Section 6.2.6, we considered approaches to the HF limit derived under the assumption that various aspects of basis-set incompleteness (radial, angular, etc.) could be accounted for in some additive fashion (see Eq. (6.5)). In essence, multilevel methods carry tliis approach... 

Koopmans theorem also implies that the eigenvalue associated with the HF LUMO may be equated with the EA. However, in the case of EAs errors associated with basis set incompleteness and differential correlation energies do not cancel, but instead they reinforce one another, and as a result EAs computed by this approach are usually entirely untrustworthy. 

At a higher level of complexity, correlation energies are computed assuming tliat effects associated with basis-set incompleteness and, say, truncated levels of perturbation theory,... 

A somewhat more common approach is one that does not try explicitly to extrapolate to the HF limit but uses similar concepts to try to correct for some basis-set incompleteness. The assumption is made that the effects of orthogonal increases in basis set size can be considered to be additive (a substantial amount of work suggests that this assumption is typically not too bad, at least for molecular energies), and thus the individual effects can be summed together to estimate the full-basis-set result. This is best illustrated by example. Consider... 

This section summarizes the TDDFT linear response approach to compute optical rotation and circular dichroism. For reasons of brevity, assume a closed shell system, real orbitals, and a complete basis set (see Sect. 2.4 for comments regarding basis set incompleteness issues). From solving the canonical ground state Kohn-Sham (KS) equations,... 

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