Dual basis sets

Even if the conditions for persistent, lost, and new eigenvalues are completely clear for the exact eigenvalue problems to the operators T and Tt, it is considerably more difficult to translate them to the approximate eigenvalue problems associated with the application of the bi-variational principle for the operators T and Tt to truncated basis sets. In this connection, the relations (A. 1.40-1.49) may turn out to be useful in formulating the problem. Some of the computational aspects, particularly the choice of the dual basis sets, are further discussed in reference A. 

R. Jurgens-Lutovsky and J. Almidf, Chem. Phys. Lett., 178, 451 (1991). Dual Basis Sets in Calculation of Electron Correlation. 

The previous analysis by the dual substituent parameter equation of substituent effects in the naphthalene series provided support for the scale, especially for sets involving nonconjugating positions (2p). The available data yield six basis sets which presumably give a critical analysis and, in particular, provide distinctions between conjugative (three sets) and nonconjugative positions (three sets). The results (using the earlier symbolism (2p)) are given in Table X. 

Feasible x) and y) give upper and lower bounds on the optimal value of the objective function, which in the 2-RDM problem is the ground-state energy in a finite basis set. The primal and dual solutions, x) and y), sie feasible if they satisfy the primal and dual constraints in Eqs. (107) and (108), respectively. The difference between the feasible primal and the dual objective values, called the duality gap fi, which equals the inner product of the vectors x) and z). 

It should be observed that one can conveniently describe the calculation of approximate eigenfunctions in terms of truncated basis sets by using the concept of outer projections (7) of the operators involved. If F = F(t and d> = < > for k = 1,2,., m are the dual bases chosen, the operator... 

Since one has the general property W = H, it is convenient to introduce a dual basis 4 and having the property = iff. For this purpose, we will now introduce a linearly independent set 0 = l5 2,..., m consisting of m complex functions, which have the additional property that the overlap matrix A = <0 10) is nonsingular, and make the dual choice of bases ... 

In order to reduce the size of the PW basis set pseudo potentials (PP) of the dual-space type [12,13] are used. The latest implementation of the GPW method [34] has been done within the CP2K program and the corresponding module is called Quickstep [32]. In this implementation the linear scaling calculation of the GPW KS matrix elements is combined with an optimizer based on orbital transformations [33]. This optimization algorithm scales linearly in the number of basis functions for a given system size and, in combination with parallel computers, it can be used for systems with several thousands of basis functions [33,34]. 

In addition to the chosen set of orbitals , we introduce the dual basis ... 

Quantum mechanics can also be formulated in terms of an alternate Hilbert space whose elements are operators, with the density operator p and the typical measurables F among them. A variety of complete sets of basis operators in the space may be constructed, for example, as the tensor product of a basis set of the Hilbert space vectors > and its dual, yielding elements of the form... 

Also, apart from the compilation of dual family basis sets for the series potassium to element (118) [30], and the Universal Gaussian Basis of Malli et al. [26], there are few extensive series for the heavier elements which can be explored for systematic investigations. Among the tasks that need to be undertaken is a more complete derivation of higher quality basis sets, including polarization and correlation functions, and a systematic investigation of even-tempered basis sets. 

After a successful conversion of the raw data in (he filial x(k) function, the last step of daia analysis consists of the determination of the struc dual parameters rJ% N - and a,. To do (his, one dies by variation of these parameters according to equation (10.4), to describe (he experimental y(A) function optimally with a minimal basis set. i.e. preferably few baekscatierers. Frequently, the experimental EXaFS function is. however, first dismantled by means of the Fourier filtering ... 

A fundamental characteristic of the FPA is the dual extrapolation to the one-and n-particle electronic-structure limits. The process leading to these limits can be described as follows (a) use families of basis sets, such as the correlation-consistent (aug-)cc-p(wC)VnZ sets [51,52], which systematically approach completeness through an increase in the cardinal number n (b) apply lower levels of theory with extended [53] basis sets (typically direct Hartree-Fock (HF) [54] and second-order Moller-Plesset (MP2) [55] computations) (c) use higher-order valence correlation treatments [CCSD(T), CCSDTQ(P), even FCI] [5,56] with the largest possible basis sets and (d) lay out a two-dimensional extrapolation grid based on the assumed additivity of correlation increments followed by suitable extrapolations. FPA assumes that the higher-order correlation increments show diminishing basis set dependence. Focal-point [2,49,50,57-62] and numerous other theoretical studies have shown that even in systems without particularly heavy elements, account must also be taken for core correlation and relativistic phenomena, as well as for (partial) breakdown of the BO approximation, i.e., inclusion of the DBOC correction [28-33]. 

Before discussing the CIM-CC calculations for the (H20) clusters with basis sets containing diffuse functions, where the earlier, dual-environment CIM scheme encounters problems with accurately reproducing the relative energies of the canonical CC calculations, we examine the performance of the modified, singleenvironment CIM scheme introduced in Section 4.1 (cf., also, W. Li and P. Piecuch, unpublished manuscript) in the calculations for the ten lowest-energy stmctures of the (H20) clusters with n = 10, 12, 14, and 16, as described by the 6-31G(d) basis set. We chose the 6-31G(d) basis set, so that we could perform the canonical CCSD calculations for systems as big as (H20)i6. Due to the prohibitively large computer costs, we were unable to perform the canonical CR-CC(2,3) or CCSD(T) calculations for (H20) with n = 14 and 16 that could provide the relevant reference... 

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