Basis sets linear dependence issues

Of cmcial importance in designing a basis set is the issue of linear dependence, because this affects the numerical stability of the atomic or molecular calculations that use the basis sets. Linear dependence in the primitive set can be controlled by the use of even-tempered or well-tempered basis sets, which minimize the linear dependence by construction. However, such basis sets tend to be larger than energy-optimized basis sets, where linear dependence problems can become significant as the basis set size increases. 

Except for occasional discussions of the basis set dependence of the results, the numerical implementation issues such as grid integration techniques, electron-density fitting, frozen-cores, pseudopotentials, and linear-scaling techniques, are omitted. 

The choice of basis sets is a particularly important technical issue for this type of calculation. Specifically, the small, highly contracted basis sets such as used in Ref. [16] generally are insufficient to calculate the real parts of dielectric functions and, consequently, the energy loss functions. We started from Partridge s 16sllp set [17], contracted the seven tightest s-functions and the four tightest p-functions, removed the most diffuse s and the two most diffuse p-functions to avoid approximate linear dependencies, and added a full set of three d-functions with exponents equal to those of the... 

With the use of increasingly parallel architectiues in modem computers and the growing awareness that basis set truncation errors are frequently the dominant source of error in contemporary electronic structure studies, significant improvements in the accuracy of such studies can be expected in the new few years. The key issue in making this progress is the construction of large basis sets which will afford an accurate representation of atomic and molecular wavefunctions whilst avoiding computational linear dependence. This review was completed in September, 1985. 

The set of all PAOs so generated (the standard PAO set) constitutes a nonorthogonal, linearly dependent, incomplete set of local functions, strictly orthogonal to aU WFs. The problem of the adequacy of the standard set to describe excitations is a delicate issue, because the basis sets usually adopted to solve the HF problem for... 

The main problem to be addressed in the choice between scalar and 2-spinor basis sets is that of linear dependence, which can affect the convergence of the SCF procedure. Several issues arise from the basis set choice that have consequences for linear dependence. The first of these is the choice of RKB or UKB in the representation of the small component of the j = I - 1/2 spinors. UKB has more severe linear dependence problems than RKB, due to the overrepresentation of the small component. However, RKB requires some manipulation of the integrals, because the nl and (n + 2)1 integrals are not generated as part of the same angular momentum shell. This issue affects both scalar and 2-spinor basis sets. 

---